1. Rockets don't got a "range" like a car running out of gasoline. Instead they have "maximum velocity" called "Delta V".
2. A rocket's propellant tanks are like a "wallet" full of delta-V "money"
3. Each rocket maneuver has a delta-V "cost" which has to be paid with delta-V money. Don't have enough in your wallet? Then you can't afford to do that manuever.
4. A space mission is composed of several manuevers, each with a delta-V cost. The total is the delta-V mission cost.
5. Refilling the propellant tank is like putting more money in the wallet. This is why rocket engineers are trying to figure out how to put gasoline stations in space. Because it sucks trying to do a long trip in your car on one tank.
6. Most rockets are poor, with very little money in their wallet. So they can only afford the cheapest maneuvers
7. One of the cheapest maneuvers is called a "Hohmann transfer." Using one to go from Terra to Mars takes about 5,700 meters per second of delta-V money and 8.6 months of travel time. And about the same to return home.
8. Also you can't just take off on a Mars Hohmann any time you want. Terra and Mars have to be in the proper spots, which happens every 26 months. This is called a "Hohmann Launch Window." And once you reach Mars you have to wait 15.3 months before the return launch window happens.

All those cute starship spec sheets you see with moronic entries like "range" or "maximum distance" betray a dire lack of spaceflight knowledge. Spacecraft ain't automobiles, if they run out of gas they don't drift to a halt. DeltaV is the key.

Mercury's mass is 3.302e23 kg and radius is 2.439e6 m. sqrt[ (6.673e-11 * 3.302e23) / 2.439e6] = 3006 m/s liftoff deltaV.

Mercury's mass is 3.302e23 kg and radius is 2.439e6 m. sqrt[ (1.3346e-10 * 3.302e23) / 2.439e6] = 4251 m/s escape velocity deltaV.

Mercury's surface gravity is 3.70 m/s. Say that the duration of liftoff burn is 30 seconds. Then the gravitational drag would be 3.70 * 30 = 110 m/s Gravitational Drag.

Mercury's basic deltaV cost for liftoff is 3006 m/s. If the spacecraft has an acceleration of 10 g, or 98.1 m/s, then 3006 / 98.1 = 30 seconds Liftoff Burn Duration.

Say the spacecraft can accelerate at 10 g (Terra gravities), or 98.1 m/s. Mercury's surface gravity is 3.70 m/s. So the spacecraft can accelerate at 98.1 / 3.70 = 26.5 Mercury Gravities. Since the deltaV for escape velocity is 4251 m/s, then 4251 / 26.5 = 160 m/s Gravitational Drag (which is close enough for government work to 110 m/s).

Well, for the very unreasonable case where you have an infinitely throttleable rocket, and a 4 g upper limit, you would do something like what’s in the attached. Basically, you keep your accel as high as you can, while staying under a maximum dynamic pressure limit (0.5 * air density * velocity²). Once you’re through the thick part of the atmosphere, density drops off, and you can punch it again. I chose 800 psf for the attached graph, because it is a reasonable upper limit on maximum dynamic pressure (or "Max Q").

In 1925 Walter Hohmann made your life incredibly easier when he discovered Hohmann transfer orbits. Be grateful.

For a Terra-Mars Hohmann with central body being Sol, Terra is starting planet, Mars is destination planet.

• Mass of Sol = 1.9885×10³⁰ kg = M_primary
• Terra's orbital radius = 1.000 AU = 1.496×10¹¹ meters = OrbitRadius_s
• Mars orbital radius = 1.524 AU = 2.280×10¹¹ meters = OrbitRadius_d
• Mass of Terra = 5.9720×10²⁴ kg = M_s
• Mass of Mars = 6.4171×10²³ kg = M_d
• Mean Radius of Terra = 6.3710×10⁶ m = PlanetRadius_s
• Mean Radius of Mars = 3.3895×10⁶ m = PlanetRadius_d
• Terra Parking Orbit Altitude = 300 km = 300,000 m = ParkingOrbitAltitude_s
• Mars Parking Orbit Altitude = 300 km = 300,000 m = ParkingOrbitAltitude_d

Doing the math:

• SemimajorAxis = (OrbitRadius_s + OrbitRadius_d) / 2
• SemimajorAxis = (1.496×10¹¹ + 2.280×10¹¹) / 2
• SemimajorAxis = 1.888×10¹¹ meters
  
  
• μ_primary = 6.674×10^-11 * M_primary
• μ_primary = 6.674×10^-11 * 1.9885×10³⁰ kg
• μ_primary = 1.32715×10²⁰ m³/s²
  
  
• OrbitVelocity_s = Sqrt[ μ_primary / OrbitRadius_s]
• OrbitVelocity_s = Sqrt[1.32715×10²⁰ / 1.496×10¹¹]
• OrbitVelocity_s = 29,785 meters/sec
  
  
• Velocity_s = sqrt[ μ_primary * ((2 / OrbitRadius_s) - (1 / SemimajorAxis))]
• Velocity_s = sqrt[1.32715×10²⁰ * ((2 / 1.496×10¹¹) - (1 / 1.888×10¹¹))]
• Velocity_s = 32,731 meters/sec
  
  
• Velocity_∞s = abs[ Velocity_s - OrbitVelocity_s ]
• Velocity_∞s = abs[ 32,731 - 29,785 ]
• Velocity_∞s = 2,946 meters/sec
  
  
• μ_s = 6.674×10^-11 * M_s
• μ_s = 6.674×10^-11 * 5.9720×10²⁴
• μ_s = 3.9857×10¹⁴
  
  
• ParkingOrbitRadius_s = PlanetRadius_s + ParkingOrbitAltitude_s
• ParkingOrbitRadius_s = 6.3710×10⁶ + 300,000
• ParkingOrbitRadius_s = 6.671×10⁶ m
  
  
• ParkingOrbitCircularVel_s = sqrt[ μ_s / ParkingOrbitRadius_s ]
• ParkingOrbitCircularVel_s = sqrt[ 3.9857×10¹⁴ / 6.671×10⁶ ]
• ParkingOrbitCircularVel_s = 7,730 m/s
  
  
• Velocity_escS = sqrt[ (2 * μ_s) / ParkingOrbitRadius_s ]
• Velocity_escS = sqrt[ (2 * 3.9857×10¹⁴) / 6.671×10⁶ ]
• Velocity_escS =10,931 m/s
  
  
• Velocity_hyperS = sqrt[ Velocity_s∞² + Velocity_sesc² ]
• Velocity_hyperS = sqrt[ 2,946² + 10,931² ]
• Velocity_hyperS = 11,321 m/s
  
  
• DeltaV_s = Velocity_hyperS - ParkingOrbitCircularVel_s
• DeltaV_s = 11,321 - 7,730
• DeltaV_s = 3,592 m/s
  
  
  
  
• OrbitVelocity_d = sqrt[ μ_primary / OrbitRadius_d]
• OrbitVelocity_d = sqrt[1.32715×10²⁰ / 2.280×10¹¹]
• OrbitVelocity_d = 24,126 meters/sec
  
  
• Velocity_d = sqrt[ μ_primary * ((2 / OrbitRadius_d) - (1 / SemimajorAxis))]
• Velocity_d = sqrt[1.32715×10²⁰ * ((2 / 2.280×10¹¹) - (1 / 1.888×10¹¹))]
• Velocity_d = 21,476 meters/sec
  
  
• Velocity_∞d = abs[ Velocity_d - OrbitVelocity_d ]
• Velocity_∞d = abs[ 21,476 - 24,126 ]
• Velocity_∞d = 2,650 m/s
  
  
• μ_d = 6.674×10^-11 * M_d
• μ_d = 6.674×10^-11 * 6.4171×10²³
• μ_d = 4.2828×10¹³
  
  
• ParkingOrbitRadius_d = PlanetRadius_d + ParkingOrbitAltitude_d
• ParkingOrbitRadius_d = 3.3895×10⁶ + 300,000
• ParkingOrbitRadius_d = 3.6895×10⁶ m
  
  
• ParkingOrbitCircularVel_d = sqrt[ μ_d / ParkingOrbitRadius_d ]
• ParkingOrbitCircularVel_d = sqrt[ 4.2828×10¹³ / 3.6895×10⁶ ]
• ParkingOrbitCircularVel_d = 3,407 m/s
  
  
• Velocity_escD = sqrt[(2 * μ_d) / ParkingOrbitRadius_d]
• Velocity_escD = sqrt[(2 * 4.2828×10¹³) / 3.6895×10⁶]
• Velocity_escD = 4,818 m/s
  
  
• Velocity_hyperD = sqrt[Velocity_∞d² + Velocity_escD²]
• Velocity_hyperD = sqrt[2,650² + 4,818²]
• Velocity_hyperD = 5,499 m/s
  
  
• DeltaV_d = Velocity_hyperD - ParkingOrbitCircularVel_d
• DeltaV_d = 5,499 - 3,407
• DeltaV_d = 2,092
  
  
• DeltaV = abs[DeltaV_s] + abs[DeltaV_d]
• DeltaV = abs[2,946] + abs[2,092]
• DeltaV = 3,592 + 2,650
• DeltaV = 5,684 meters/sec

So the Polaris has to be capable of 5,684 m/s of delta V in order to do the Terra-Mars Hohmann transfer from Low Earth Orbit to Low Mars Orbit.

For travel time of a Terra-Mars Hohmann with central body being Sol:

• μ_primary = 1.32715×10²⁰
• SemimajorAxis = 1.888×10¹¹ meters

Doing the math:

• T_h = 0.5 * sqrt[(4 * π² * SemimajorAxis³) / μ_primary]
• T_h = 0.5 * sqrt[(4 * 3.14159² * (1.888×10¹¹)³) / 1.32715×10²⁰]
• T_h = 0.5 * sqrt[2.65684×10³⁵ / 1.32715×10²⁰]
• T_h = 0.5 * sqrt[2,001,915,283,599,362]
• T_h = 0.5 * 44,742,768
• T_h = 22,371,384 seconds = 8.6 months

Delay between Hohmann launch windows for Terra-Mars Hohmann, central body is Sol, Terra is the inferior planet.

• μ_primary = 1.32715×10²⁰
• OrbitRadius_i = 1.496×10¹¹ meters
• OrbitRadius_s = 2.280×10¹¹ meters

Doing the math:

• OrbitPeriod_i = 2 * π * sqrt[OrbitRadius_i³ / μ_primary]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[(1.496×10¹¹)³ / 1.32715×10²⁰]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[3.348071936×10³³ / 1.32715×10²⁰]
• OrbitPeriod_i = 2 * 3.14159 * sqrt[25,227,532,200,580]
• OrbitPeriod_i = 2 * 3.14159 * 5,022,702
• OrbitPeriod_i = 31,558,565 seconds
  
  
• OrbitPeriod_s = 2 * π * sqrt[OrbitRadius_s³ / μ_primary]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[(2.280×10¹¹)³ / 1.32715×10²⁰]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[1.1852352×10³⁴ / 1.32715×10²⁰]
• OrbitPeriod_s = 2 * 3.14159 * sqrt[89,306,800,286,328]
• OrbitPeriod_s = 2 * 3.14159 * 9,450,228
• OrbitPeriod_s = 59,377,531 seconds
  
  
• SynodicPeriod = 1 / ( (1 / OrbitPeriod_i) - (1 / OrbitPeriod_s))
• SynodicPeriod = 1 / ( (1 / 31,558,565) - (1 / 59,377,531))
• SynodicPeriod = 67,359,430 seconds = 26 months = 2.14 years

• 

Angle between planets for Terra-Mars Hohmann.

• r1 = 1.496×10¹¹ meters
• r2 = 2.280×10¹¹ meters

Doing the math:

• α = π * (1 - ( (1/(2*sqrt[2])) * sqrt[(r1/r2 + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[(1.496×10¹¹/2.280×10¹¹ + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[(0.65617 + 1)³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[1.65617³]))
• α = 3.14159 * (1 - (0.35355 * sqrt[4.54271]))
• α = 3.14159 * (1 - (0.35355 * 2.13136))
• α = 3.14159 * (1 - 0.75355)
• α = 3.14159 * 0.24645
• α = +0.77424 radians = +44.36°

Since Phase Angle α is positive, Mars is ahead of Terra.

For the Mars-Terra Hohmann, Phase Angle α = -1.31229 radians, or -75.19° behind Mars.

In a typical round trip, you start at a planet (say, Earth), then execute a Hohmann transfer to another planet (say, Mars). However, you cannot return immediately, since Earth and Mars are then not in the right places. You must wait a certain amount of time before taking another Hohmann transfer back.

How long will that be?

Working out the problem turns out to be relatively straightforward. (For maintaining intuition, I'll continue with the example of visiting Mars, but note that the analysis remains generally applicable.)

First, we define four times:

• t₀: the time of departure from Earth.
• t₁: the time of arrival at Mars.
• t₂: the time of departure from Mars.
• t₃: the time of arrival at Earth.

Next, we define the (heliocentric) angular position of these planets at given times. As time progresses over the planet's orbital period, these angles sweep out 2π radians of passage over a single planetary year (a period of time P_E and P_M for Earth and Mars, respectively). We are only interested in their values at the four points above, though. E.g. θ_M,2 is the angle of Mars at time t₂.

We need to define how these angles relate to each other. Let's call t_H := t₁-t₀ the duration of the Hohmann transfer; notice that it is the same going out as coming back. For the time period t₀ to t₁, the spacecraft is on a Hohmann transfer from Earth to Mars. The angles relate as:

In English, this just says that the (angular) positions of both Earth and Mars advance over the time of the outgoing transfer.

At Mars, we'll wait for some unknown time t_W (waiting time). Again, the planets' angular positions advance:

Finally, coming back, we do another Hohmann:

We still need some more information to solve this problem, though. The first two pieces are that a Hohmann transfer always takes you an angle π around the center:

The first equation says that the spacecraft departing Earth must arrive at Mars after going halfway around the orbit, while the second says that the spacecraft leaving Mars must arrive back at Earth, again after a half-orbit. We need to be more-careful, though. Although the angles for the Earth and Mars can be taken to increase monotonically, one of them might do a full orbit while the other has not. We therefore need to be able to add/subtract multiples of 2π of the angles to get them to agree:

We need just one more equation. The entire problem can be shifted by a constant amount around the Sun. To constrain it, without loss of generality we can simply set:

(I'll also rename the remaining k₁ to just k.)

We can rewrite this huge mess of equations as a matrix equation to impose some semblance of sanity on the complexity:

Solving this is not too difficult (I did it by hand first before checking my result with sympy). The only part of it we care about is the value for t_W, which works out to be:

Because k is just some integer, we can remove that -1. Generalizing the notation a little, so that the home planet has period P₀ and the visited planet has period P₁, we get:

For k, any value can be chosen so long as the resulting t_W is nonnegative.

---

This was the result I gave here (the Google+ question which nerd-sniped me into doing this whole thing). I'm fairly convinced it's correct, but, it's still a little unsatisfactory. Choosing k from the integers is obnoxious. It would be nice to choose it from the natural numbers (i.e., ℕ := {1,2,3,…}), and thereby get an increasing sequence of possible departure dates, starting from a value that is the earliest.

Requiring that t_W ≥ 0, some algebra shows that, if P₀ > P₁, then we must have k ≥ (-2t_H)/(P₀). Similarly, if P₀ < P₁, then k ≤ (-2t_H)/(P₀). And, of course, if P₀ = P₁, then the planets are co-orbital and you cannot travel between them with Hohmann transfers (so we shall assume this is not the case forthwith).

We now want to generate a sequence of k values from natural-valued n values that produces the correct result either way:

Some ad-hoc finagling with the intersection of these lines in the middle produces:

By substituting back into our answer, we can get an equation that tells you every possible waiting time, starting from the first, indexed by n ∈ ℕ = {1,2,3,…}:

I'm less-confident of this result than the previous but, as we'll see below, it seems to work—at-least for the P₀ < P₁ case I tested.

---

Let's check and demonstrate our work by returning to the Earth/Mars example we started with.

The actual values are (note sidereal value for P_i whereas Gregorian calendar used to convert from months):

P₀ = 365.256363004
P₁ = 686.980
t_H =“8.5 months” ≈ 259 days
t_W =“14.9 months” ≈ 454 days

According to our first formula, we get:

k	tW
-3	≈ 1235 days
-2	≈ 455 days
-1	≈ -325 days
0	≈ -1105 days
1	≈ -1885 days
2	≈ -2665 days
3	≈ -3445 days

Therefore, k must be -2 or less. Notice also the separation between the possible waiting times. They should come in multiples of the transfer window times (because if you miss your departure date for the aligned planets, you'll have to wait until the next transfer window). For Earth/Mars, this is “26 months” ≈ 791 days. Indeed, we see that successive launch times differ by roughly this many days. The agreement is to within 1.5% or so, which is pretty good given the imprecision of t_H and the expected t_W, as well as the astronomical fact that the orbits are imperfect.

For our second formula, we have:

n	tW
1	≈ 455 days
2	≈ 1235 days
3	≈ 2015 days

This confirms that the formula works when P₀ < P₁, but the case for P₀ > P₁ remains untested.

      In routing a ship to a planet the two chief considerations are invariably: How much energy will be required and how long will it take? There are literally millions of paths that will lead a ship to Mars. Let us see how these two factors aid in the selection of a route, for some are much easier to follow than others.
     SINCE Mars is exterior to the Earth, the projectile or rocket will have to force its way outward from the Sun—climb uphill so to speak—in order to get there. This means that at the take-off it must be moving faster than the Earth, otherwise it will never be able to make the grade. Now if you were making an urgent business trip by plane from San Francisco to Chicago, for example, you would hardly continue on to Cleveland or Detroit and then double back on yourself. Inst so in aiming for Mars you try not to overshoot the mark but give yourself precisely the right impetus at the start to reach your destination and no more. Calculation shows that the minimum velocity required with respect to the Sun is 19.9 miles per second. (This will vary slightly depending upon what part of the orbit you attempt to reach.)
     The Earth maintains a nearly constant pace in its orbit of 18.5 miles per second. In seeking to reach Mars with as little expenditure of energy as possible we would be foolish not to make use of the Earth’s orbital motion which is already ours for nothing; in fact, we can hardly avoid it. That is, by launching the projectile in the same direction the Earth is headed we need only give it a speed of 1.4 miles per second in order to secure the total of 19.9 required. Also, by starting from the equator at midnight we can pick up an additional 0.3 m.p.s. from the rotation of the Earth. Thus the shell or spaceship will depart this world at the comparatively moderate rate of a trifle over a mile per second—which was very nearly the muzzle velocity of the Big Bertha that shelled Paris.
     (We omit from discussion obstacles that arise through atmospheric resistance, force of surface attraction, et cetera, since such topics would seem to come more properly under the head of “Piloting” rather than “Astragation.” The figures quoted here would have to be greatly modified if purely local planetary problems were included.)

• 

     Without going into the technical details in their entirety, Fig. 12 shows the type of orbit a spaceship would follow in order to reach Mars by the easiest or so-called 180° route. The name comes from the fact that departure takes place when the Earth is on the side of the Sun opposite or 180° from the point where contact is planned with Mars. Only by shooting off along a tangent in this way can the ship acquire all of the Earth’s orbital motion.

---

     TO TAKE the case out of the abstract, suppose that we wished to arrive at Mars when it was passing the perihelion point of its orbit on September 17, 1939 at twelve o’clock noon, Central Standard Time. In order to leave when the Earth is 180° from this point the passengers must be all aboard on February 24, 1939.
     All right. It is now February 24, 1939. Here we go!
     At the start Mars is some one hundred twenty-nine million miles ahead but the ship rapidly cuts the distance down. By March 23rd it is reduced to one hundred two million, by April 24th to seventy-four million, and on ]uly 10th they are separated by a mere thirty-nine million miles. And on September 17th as the passengers are preparing to disembark the steward regretfully announces that they have—missed! By thirty-nine million miles! The ship made the perihelion point all right but Mars was forty-two million miles farther east at that moment.
     This blunder was done purposely to emphasize the obvious fact which most popular writers for some reason more or less ignore, that although there is no trouble in calculating exactly when to leave in order to reach any point on the orbit of Mars at a set time, this implies no obligation whatever on the part of the planet Mars to oblige by being there at that time. Before a ship can be given the green light, the dispatcher must make sure its orbit properly coincides with the positions of the Earth and Mars or else it will fail to make connection at the other end of the line. Thus in the example just cited, although the ship missed badly by starting on February 24th, investigation shows that by trying successively later dates it could have gotten onto a true collision course if departure had been delayed until May 8th. The ship would then have gotten to Mars by January 14, 1940, two hundred fifty-one days later. Astronomical data are sulficiently precise so that the time of transit could be determined to within about an hour if necessary, but it is doubtful if schedules will be tabulated closer than about twelve hours since the last few hundred thousand miles will have to be done by piloting anyhow.
     The minimum energy 180° route is the only one generally considered in most popular articles. But if space travel is going to he limited to a few favorable cases the arrival of a ship from Mars or Venus will be an occasion for a public demonstration, as rare as the pack boat from San Francisco to Pago Pago. Somehow this does not fit in with our picture of transportation in the year 5943 A.D.
     Recently commentators have begun to speak of the “new important great circle” airplanes are opening up across the pole from New York to Chunkiang, Tokyo, and Munnansk. Similarly, there are other possible paths between worlds besides those that require the least effort to follow.
     The bigger the angle between the direction the Earth is moving and the direction the ship takes off for Mars, the more energy—or what amounts to the same thing—the more money it is going to take to make the trip. As already explained, this is because we are using less of the Earth’s motion which is free and expending more of our own which is apt to be very costly. But does anyone doubt that the day will come when the value of an enterprise is reckoned in terms of human necessity rather than such meaningless symbols as ten million dollars or one hundred million dollars or one billion dollars? Let us therefore feel no hesitation about running up a bill on future generations, payable promptly after the first one thousand years.
     It can be safely predicted, however, that in terms of whatever passes for money in 5943, it is going to cost plenty for every day that is pared off the 180° route to Mars. Suppose now that instead of giving Mars a handicap of 180°, we cut the lead down to 90°, and successively smaller angles. How much energy will be needed and how many days will be saved?

• 

     Fig. 13 shows the kind of orbit the ship would follow by the 90° route. It is slightly more elongated than the other and instead of being entirely outside the orbit of the Earth about forty per cent falls inside. The time of transit is cut down from two hundred sixty days to one hundred fifty-six days, a saving of forty per cent. The ship must leave with a speed of 6.2 m.p.s. relative to the surface and head out an angle of 94° with the direction the Earth moves. If we assume that the amount of energy required depends roughly upon the square of the initial velocity, then Mars via 90° is thirty-two hundred per cent more expensive than by way of 180°. The reason is perfectly plain. Before we worked with the Earth in its motion; here we work nearly at right angles to it. In fact, we have to set a course which is actually 4° opposite to the Earth’s orbital velocity, or fire 4° backward, as it were.
     Note the parallel between a plane taking off a carrier into the wind and a spaceship leaving the Earth for another planet. The plane sets a course such that its own speed together with that of the wind will combine to produce a resultant motion toward the objective. Similarly, the spaceship takes off at such an angle that its own speed combined with that of the Earth puts it into the desired orbit. From this point of view the motion of the Earth may be regarded as a steady wind blowing at the rate of 18.5 m.p.s. from the west.

---

• 

     NOW let us put the starting point closer and closer to the position of Mars in its orbit. Let us give Mars a handicap of 80°, 70°, 60°—with respect to the Earth. The orbit the ship must follow is altered drastically as the angle decreases. From a casaba-shaped oval at 90° it collapses through various configurations resembling watermelons, cucumbers, torpedoes, et cetera, until at 10° we obtain a narrow cigar-like figure beyond which there would seem to be little point in pressing matters further. The journey to Mars by the 10° route takes but eighty-eight days. Reducing the angle farther does not appreciably reduce the time beyond a few hours. In fact, as the orbit approaches the limiting figure of a parabola there is an indication that it even increases appreciably.
     The velocity of the ship with respect to the Sun or the velocity of the ship in its orbit in the 10° case is not so great as in the 180° and 90° cases; 15.3 m.p.s. as compared with 19.9 and 19.2. But the velocity of the ship with respect to the Earth is vastly greater: 21.5 m.p.s. as compared with 1.1 and 6.2. The ship gets practically no help from the Earth at all, for it must set a course at an angle of 136° to the Earth’s velocity, or 46° in a direction opposite to the motion of the Earth.
     To reach Mars by the 10° route would be for multimillionaires only, for it would be twelve times more expensive than by 90° and three hundred eighty-two times more than by 180°. If space travel is to be made available to people of moderate means as we understand this term now, parity will have to be fixed at around 0.0001 cent per mile. The longest journey by way of 180° covers three hundred thirty-eight million miles. At this rate a round-trip ticket would cost six hundred seventy-six dollars. But by the 10° route, although the distance is reduced to fifty million miles, the greater energy needed would boost the price up to thirty-eight thousand dollars, or two hundred fifteen dollars per day for mileage alone.
     These highly eccentric routes could be extremely hazardous in addition to being highly expensive. For suppose the driving mechanism failed to work when the time came to land on Mars. If contact could not be eflected or the passengers and crew transferred to another ship by rescue squads, they are doomed to certain destruction. For one of the necessary consequences of choosing a greatly elongated orbit is that it forces you into the Sun at perihelion. In the 10° orbit the ship would whip around the Sun at a distance of two million miles and be speedily converted from a luxurious vehicle for interplanetary travel into a small comet with a strange spectrum composed of strong metallic lines together with a few faint bands of certain well-known carbon compounds.
     It is fun to play with orbits sometimes. Force them to go in certain directions or make drastic alterations in the elements. Many of the orbits of newly discovered asteroids and comets are gradually brought under control by what astronomers have come to call a “cooking” process; that is, little changes are made here and there until the best fit possible with the observations is obtained.

---

• 

     JUST for the devil of it suppose that we take this 10° orbit to Mars and turn it inside out. Not merely turn it end-for-end but force the perihelion point to become the aphelion point, and vice versa. The result is an orbit of exactly the same shape as before but instead of reaching only as far as Mars now extends out to nearly three times the distance of Pluto. The period of an object revolving in this orbit would be three hundred ninety-one years. A path such as a giant interstellar comet might follow—Fig. 15.
     To travel inward from the Sun—to go from Mars to Earth or Earth to Venus—means that the ship must fall toward the Sun or travel more slowly than the planet it leaves behind. To lose energy might seem comparatively easy in contrast to the effort of gaining it, but such is not the case, as anyone who has ever fallen off a train could testify. To reach Venus by the 180° route the ship must move about 1.6 m.p.s. slower than the Earth. The ship, therefore, takes off in the opposite direction the Earth is moving with a speed of 1.6 miles per second. Thus to reach Venus takes practically the same amount of energy required for Mars. The journey is considerably shorter, however, only one hundred forty-six days in all.
     When taking off from a planet the first consideration must always be its orbital velocity of revolution. But for a ship cruising far from any large mass it is questionable whether procedure from point to point should invariably be done by orbit with motors inactive. In many cases it would seem more practicable to take simply the most nearly direct route possible—the straight line.
     Suppose a ship near the orbit of Mars receives orders to meet a convoy at a distance of one million five hundred thousand miles within twenty hours. Some energy woudl have to be spent in getting the ship turned around and headed in the right direction at the necessary speed. But once under way the motors could be cut oif and the ship would continue on in a straight line toward the rendezvous position. The only sensible force acting upon it would be that of the Sun. At the end of twenty hours the ship would have fallen Sunward by five thousand miles and be off its course by 0.2°, scarcely enough to be of consequence. On a long voyage in the vicinity of Venus, however, the effect of the solar attraction might be more serious. In which case an occasional blast to Sunward should be suflicient to maintain a straight-line course.

• 

     When a ship begins to enter the outer satellite system, or what might be called the suburbs of a planet, it will be necessary to abandon strictly orbital motion and proceed by piloting. The ship will of course be aware of all satellites in that sector; nevertheless it will be advisable to exercise the greatest caution at all times. The diagram shows the tangled orbits of the six outer known satellites of Jupiter (known in 1957. In 2021 the number of known satellites is 79). Once these are safely penetrated the four large Galilean satellites and the speedy little fifth moon remain as distinct hazards. Fortunately they revolve in the plane of the planet’s equator so that practically all risk would be eliminated by landing in a high altitude. The greatest danger would be to local traffic moving from one hemisphere to another —Fig. 16.
     But as will be shown in this discussion, other considerations make it very doubtful whether Jupiter and Saturn can ever be successfully colonized.

---

     ONE of the favorite devices for introducing the solar system to the uninitiated is by means of a broad plain on which divers fruit and vegetables are placed at the proper intervals to represent the Sun and planets.
     On the scale generally adopted, the Sun is a large pumpkin or squash. Mercury thirty-six feet away is by tradition a small pea. Venus and the Earth are larger peas. The Moon nine inches from the Earth is a radish seed, although some authors favor mustard seed for the Moon. ]upiter a quarter of a mile away is an orange. Saturn a smaller orange, and Uranus and Neptune are plums at distances of a mile and a mile and one-half. Pluto at two miles from the central pumpkin is still an uncertain quantity, but probably in the pea class with the Earth and Venus.
     The writer first became aware of this model at about the age of twelve in one of Sir Robert Ball’s numerous monographs on astronomy. Since then it has been tuming up regularly in the popular star books about once or twice a year until now a pronounced allergy has been developed to these fruit-and-vegetable solar systems. There is something irritating about the smug assurance with which each author goes around depositing oranges and radish seeds over that two-thousand-acre field. (A ritual that would certainly cause anyone to be regarded with suspicion of lurking insanity if observed in the act.) You wish somehow there wasn’t such a finality about the whole performance. That just as the author was laying down the final pea for Pluto you could grab his arm and cry, “Your neat little solar system is all wrong! Uranus is closer to the Earth than Mercury and Pluto is not the farthest planet. Distance is more than merely a matter of miles!”
     Anyone making such a ridiculous statement would undoubtedly be considered as of unsound mind himself, an intelligence unhinged possibly by the reading of too much science-fiction. Either that or else a visitor from the future to whom the remark that Pluto is the third nearest planet from the Earth would sound like the most natural thing in the world.
     When we say that the town of A is twenty miles away and B is five miles, therefore B is closer than A, we may be telling the biggest kind of a falsehood. The trouble is we have told only a part of the truth—the geometrical part. If the State has built a beautiful high-gear road to A while the taxpayers on the way to B have been neglected in this respect, then to all intents and purposes A is closer than B. Or perhaps the five miles to B goes through heavy traflic while A is relatively shunned by motorists. There is a convenient term which we might borrow from optics which is applicable here. This is the notion of “effective distance.” If one beam of light goes through a dense flint prism and another through an equal length of air, the former is said to have the longer effective path. Once we begin to take account of obstacles to be overcome or the energy needed to get from place to place our whole scale of measurement calls for immediate revision.
     Space engineers wrestling with the manifold problems of routing vessels throughout the solar system are going to be greatly concerned with energy changes or effective distances and comparatively little with linear changes or geometrical distances. In going from Planet P to Planet Q there are two fundamental factors to be considered: (1) the distance made good toward or away from the Sun; and (2), the relative mass and radius of the planets involved. These factors may combine in all sorts of ways that often lead to energy jumps as surprising as those of the Bohr atom in its heyday. But whereas the Bohr atom led a carefree existence, emitting and absorbing energy at will, a spaceship must reckon continually upon the course of future events. In order to see clearly how the ship’s passage is aifected by the energy conditions encountered, it is first necessary to get a picture of the solar system that is utterly different from any you have ever seen before. But remember that regardless of how weird it may look, it is just as true a representation in its way as the old pumpkin-radish-seed model. Fig. 17.

---

• 

     IMAGINE yourself to be an ant crawling over the outside surface of a vast trumpet-shaped structure one thousand feet tall. It stands upon its small end in an upright position—a highly unstable state of equilibrium. There is no danger of it toppling over or collapsing, however. It is only one foot wide at the base and extends upward Without widening perceptibly almost to the very top. Then from nine hundred eighty-seven feet to one thousand feet it suddenly flares out all around like the stem of a champagne glass to a distance of forty-three hundred feet.
     This trumpet-shaped structure is an Energy-Distance model of the solar system. It is not quite so easy to visualize as the one of flat concentric rings, but then there is no really valid reason why we cannot represent our solar system by an old-fashioned phonograph horn as well as a machine-gun sight. We must imagine also that we are constrained to move over the outside surface of this structure. This means that we are keeping within the plane of the solar system; we cannot drop above or below the plane in which the planets circulate.
     The Sun is supposed to be located at the narrow foot of the model. Its width of one foot represents the diameter of the Sun on the scale we have chosen. Now consider the case of an ant near the Sun who wishes to move away from it; to move outward in space keeping Within the plane of the solar system. Since he is restricted to the surface his only means of doing this is by climbing up the long, steep neck of the figure. Thus to go a very short distance horizontally the ant must do a tremendous lot of hard Work climbing vertically. Eventually at the nine-hundred-eighty-seven-foot level he comes to line drawn around the surface which bears an inscription. Slowly he spells it out letter by letter—ORBIT OF THE PLANET MERCURY. That is, the distance from the Sun to its first member when measured in terms of the energy needed to make the pull constitutes ninety-eight point seven per cent of the whole system.
     Feeling somewhat encouraged, the ant crawls up another six feet and finds a second line marked ORBIT OF THE PLANET VENUS. Two feet more brings it to the orbit of the Earth. The going is much easier now, for the surface is spreading rapidly outward so that to go from one orbit to another requires hardly any work at all. From Jupiter clear on out to the rim which marks the orbit of Pluto is a climb of but eleven inches.
     Millions of ants for countless millions of years might crawl around over such a surface, notice vaguely that it was a lot harder to move over some portions than others, but feel no compulsion to investigate the matter farther. Until one day a certain ant would analyze the situation very minutely and as a result would announce that the intensity of the force varied inversely as the square of the distance from the central vertical axis. This explained immediately why the force was so strong at the lower end where the figure was the slimmest and why it was scarcely perceptible at the upper flared end. Later another ant developed a theory in which the force was ascribed to the curvature of the space itself rather than an inherent property of the matter at the bottom of it.

---

     ON THIS MODEL the orbits of the planets would be nearly circular rings around the extreme top portion, although the orbit of Mercury would dip slightly at one end. That is because most of the orbits are almost perfect circles and experience little change in energy from perihelion to aphelion, except for Mercury which is much more eccentric. The effect would be greatly exaggerated for a comet. At aphelion the orbit would be nearly circular like a planet’s. As the comet nears the Sun its path would begin to drop sharply until the lowest point would be reached at perihelion. Then the comet would zoom up the other side of the column tracing out a path identical with the one going downward in reverse. A comet with an orbit approaching the parabolic like Halley’s would go into a nose dive straight down the long stem and seem on the verge of shooting off the bottom. Then it would suddenly perk back and fly upward at a slowly decreasing pace, leisurely swing around the top rim and almost—but not quite—make connection with its previous path.
     The mass of a comet is so small that it may be disregarded entirely, reducing it to the same social level as the geometrical point, or mere locus in space. A planet, however, has an appreciable mass compared with the Sun, and their case is not so easily set aside. It is hard to represent a planet on our model because they are in the nature of discontinuities in the smooth uniformity of the force field. The only reason why you must take a planet into account at all is because you can get so infernally close to them; right onto their surfaces, in fact. Now the force of surface attraction depends directly upon the mass of a planet—that is why Jupiter pulls so hard—and inversely upon its radius—which is why the tiny white dwarf stars have such incredible strength. Thus even a small planet at its surface can attract much more powerfully than the Sun at a distance of a few million miles. A timely analogy would be that of the lonely soldier pondering how best to spend his week-end leave. He is stirred by thought of the potent attractions of the big city far away, but decides in favor of the small town within easy thumbing distance of camp.
     Although the Sun maintains the space around it in a state of tension that ranges from a steep gradient within the orbit of Mercury out to where it begins to level off beyond Jupiter, there are pockets within it—the planets—where local conditions are sharply reversed. On the Energy-Distance model a planet would appear as a sharp projection or knob depending upon its mass. Jupiter would be a long icicle hanging down almost to the orbit of Mercury. Saturn, Neptune and Uranus would be shorter icicles or stalactites. The Earth, Mars and Venus would be little more than pin points.
     The captain of a spaceship approaching Jupiter would not begin to experience his attraction until within about a million miles or so of the surface. If for some reason he were unaware of the planet’s presence, he would be amazed to find his instruments recording an abrupt reversal in gravitational intensity calculated for that region. He would undergo all the sensations of a man confidently strolling up the side of a hill who was unceremoniously precipitated into a hole in the ground.
     The work required to leave the surface of Jupiter is sufficient to take a ship from the orbit of Mercury to the orbit of Mars. Conversely, a ship that lands on Jupiter would have an equal quantity of work done upon it. (Here we again omit all discussion of practical landing operations.) If space travel can be done on the principle of the storage batery, so that when going downhill or in the direction of increasing gravitational attraction energy can be accumulated, a ship arriving upon jupiter will be fairly bulging with power. But it does not represent any real gain because it will have to be used up again when the time homes to leave. It is like entering a country with a favorable rate of exchange. You are way ahead so long as you stay there, but your wallet flattens out as soon as you cross the border.

---

     THE HUGE MASS of major planets makes it very doubtful whether they can ever be successfully colonized by beings like ourselves. Unless a cheap source of energy becomes available beyond any we can imagine at present—which may easily be the case—these mammoth hulks seem destined to be shunned forever owing to their inordinate tenaciousness. Woe to the skipper who allows his craft to drift within the hold of Jupiter! To approach and disembark is theoretically quite effortless; all done at Jove’s personal expense, in fact. But the traveler soon finds to his dismay that he is fast within a gravitational prison from which escape is possible only by paying an exorbitant ransom. It is one of those easy-to-get-into, hard-to-get-out-of propositions, like promising to make a speech or meet a payment months in advance.
     This must not be taken to signify that space travel is going to be limited by the orbit of Mars. Jupiter, Saturn, and Neptune all have satellites as large as the Moon or Mercury within moderate energy distances. For the inverse square part of Newton's law works both ways; it makes the force-field build up rapidly near a body and also peter out rapidly a few diameters away. Ideal landing fields will undoubtedly be found on Jup III and Jup IV, which according to the latest estimates are a trifle larger than Mercury and so far from Jupiter that his attraction would be a minor consideration.
     Saturn, Neptune, and Uranus are curious examples of massive bodies with feeble surface attractions. They are emasculated, so to speak, because they are unable to make effective use of the matter with which they are endowed. A planet behaves much as if it were a ballbearing surrounded by a film of soap bubble. It attracts at the surface as if its mass were all concentrated at the center. Saturn has eighty-three per cent of Jupiter’s girth but only thirty-three percent of his mass. Result is that Saturn attracts on the surface scarcely more than Earth. But shrink Saturn down by twelve thousand miles—get twelve thousand miles closer to him—and his surface gravity will promptly equal that of Jupiter’s.
     It produces a queer feeling to think that we could walk around over Saturn with little more exertion than on the Earth. Yet Saturn is one of the most powerful disturbing objects in the solar system affecting the motions of Neptune and Jupiter while the Earth can barely produce a tremor as close as Mars or Venus. Which is one for you to figure out.
     Our planet is rather exceptional in that its surface gravity is fairly large, perhaps unduly so compared to the muscular development of its inhabitants. The dinosaurs, for example, were forced out of the race entirely because their size and strength were so out of proportion to their weight. Which might cause one with a bent for ecology to toy with the idea that maybe we are not natives of the Earth at all but creatures originally spawned from some other world of lesser gravitational power. In short, that we need not continue wondering what the Martians are like because we are the descendants of pre-historic Martian invaders!

• 

     Leaving such highly speculative material for other writers in science fiction, it may well be questioned whether the energy difference in a transition from planet to planet is the factor of main importance. There can be no argument that going uphill from an inner orbit to an outer orbit energy will have to be expended in the climb. But when work is being done on a ship as in the drop-down to an inner orbit or surface of a planet, energy will still have to be used in order to cushion the fall. Otherwise a ship would arrive on Jupiter at the rate of around one hundred forty thousand miles an hour.
     An engineer planning a trip, therefore, would probably make a more accurate estimate if he takes into account the total energy involved regardless of which way it is acting. This is plain common sense and agrees with our everyday experience. Thus any astronomer at Mount Wilson can testify that the strain on the leg muscles is the same whether they are used in pulling yourself up the twenty miles to the Observatory, or in bracing yourself on the way down.
     On the basis of the total needed to reach a planet’s surface from the Earth—energy from orbit-to-orbit and from surface-to-surface—the solar system presents such a scrambled appearance that the familiar old astronomer with his fruit and vegetables would never recognize it in a lifetime. Here are the distances to the various members in terms of the distance to Venus, which maintains its position as our nearest neighbor:

EFFECTIVE OR ENERGY DISTANCES TO THE PLANETS

1. EARTH TO VENUS	1.00
2. EARTH TO MARS	1.02
3. EARTH TO PLUTO	2.50
4. EARTH TO URANUS	2.91
5. EARTH TO NEPTUNE	3.91
6. EARTH TO SATURN	4.00
7. EARTH TO MERCURY	4.19
8. EARTH TO JUPITER	7.11
9. EARTH TO SUN	543

     Thus by taking what the mathematician would call the absolute sum of the energy distances to the planets, Pluto becomes a comparatively close object while Mercury is removed to the border of the system. Notice also that the planets fall into rather distinct groups: (1), Venus and Mars; (2), Pluto, Uranus, and Neptune; (3), Saturn and Mercury; (4), Jupiter and his satellite system.
     Other solar systems might be devised even more outlandish than the Energy-Distance Model, yet be just as true a representation as one can readily visualize. Come to think of it, an astronomer’s life is devoted chiefly to sifting illusion from reality. Trying to find where things really belong in this universe. Mars may be forty million miles away but springtime on the Syrtis Major isn’t very much longer than ours.

Terra Space Station and the school ship Randolph are in a circular orbit 22,300 miles above the surface of the Earth, where they circle the Earth in exactly twenty-four hours, the natural period of a body at that distance.

Since the Earth's rotation exactly matches their period, they face always one side of the Earth — the ninetieth western meridian, to be exact. Their orbit lies in the ecliptic, the plane of the Earth's orbit around the Sun, rather than in the plane of the Earth's equator. This results in them swinging north and south each day as seen from the earth. When it is noon in the Middle West, Terra Station and the Randolph lie over the Gulf of Mexico; at midnight they lie over the South Pacific.

The state of Colorado moves eastward about 830 miles per hour. Terra Station and the Randolph also move eastward nearly 7000 miles per hour — 1.93 miles per second, to be finicky. The pilot of the Bolivar had to arrive at the Randolph precisely matched in course and speed. To do this he must break his ship away from our heavy planet, throw her into an elliptical orbit just tangent to the circular orbit of the Randolph and with that tangency so exactly placed that, when he matched speeds, the two ships would lie relatively motionless although plunging ahead at two miles per second. This last maneuver was no easy matter like jockeying a copter over a landing platform, as the two speeds, unadjusted, would differ by 3000 miles an hour.

Getting the Bolivar from Colorado to the Randolph, and all other problems of journeying between the planets, are subject to precise and elegant mathematical solution under four laws formulated by the saintly, absent-minded Sir Isaac Newton nearly four centuries earlier than this flight of the Bolivar — the three Laws of Motion and the Law of Gravitation. These laws are simple; their application in space to get from where you are to where you want to be, at the correct time with the correct course and speed, is a nightmare of complicated, fussy computation.

• 

CONJUNCTION CLASS

For high-thrust rockets, the most fuel-efficient way to get to Mars is called a Hohmann transfer. It is an ellipse that just grazes the orbits of both Earth and Mars, thereby making the most use of the planets’ own orbital motion. The spacecraft blasts off when Mars is ahead of Earth by an angle of about 45 degrees (which happens every 26 months). It glides outward and catches up with Mars on exactly the opposite side of the sun from Earth’s original position. Such a planetary configuration is known to astronomers as a conjunction. To return, the astronauts wait until Mars is about 75 degrees ahead of Earth, launch onto an inward arc and let Earth catch up with them.

Each leg requires two bursts of acceleration. From Earth’s surface, a velocity boost of about 11.5 kilometers per second breaks free of the planet’s pull and enters the transfer orbit. Alternatively, starting from low Earth orbit, where the ship is already moving rapidly, the engines must impart about 3.5 kilometers per second. (From lunar orbit the impulse would be even smaller, which is one reason that the moon featured in earlier mission plans. But most current proposals skip it as an unnecessary and costly detour.) At Mars, retrorockets or aerobraking must slow the ship by about 2 kilometers per second to enter orbit or 5.5 kilometers per second to land. The return leg reverses the sequence. The whole trip typically takes just over two and a half years: 260 days for each leg (increasing astronaut exposure to galactic cosmic radiation) and 460 days on Mars. In practice, because the planetary orbits are elliptical and inclined, the optimal trajectory can be somewhat shorter or longer. Leading plans, such as Mars Direct and NASA’s reference mission, favor conjunction-class missions but quicken the journey by burning modest amounts of extra fuel. Careful planning can also ensure that the ship will circle back to Earth naturally if the engines fail (a strategy similar to that used by Apollo 13).

---

• 

OPPOSITION CLASS

To keep the trip short (reducing astronaut exposure to galactic cosmic radiation), NASA planners traditionally considered opposition-class trajectories, so called because Earth makes its closest approach to Mars—a configuration known to astronomers as an opposition—at some point in the mission choreography. These trajectories involve an extra burst of acceleration, administered en route. A typical trip takes one and a half years: 220 days getting there, 30 days on Mars and 290 days coming back. The return swoops toward the sun, perhaps swinging by Venus, and approaches Earth from behind. The sequence can be flipped so that the outbound leg is the longer one. Although such trajectories have fallen into disfavor—it seems a long trip for such a short stay—they could be adapted for ultrapowerful nuclear rockets or “cycler” schemes in which the ship shuttles back and forth between the planets without stopping.

---

• 

LOW THRUST

Low-thrust rockets such as ion drive save fuel but are too weak to pull free of Earth’s gravity in one go (high specfic impulse but very low thrust). They must slowly expand their orbits, spiraling outward like a car switchbacking up a mountain. Reaching escape velocity could take up to a year, which is a long time to expose the crew to the Van Allen radiation belts that surround Earth. One idea is to use low-thrust rockets only for hauling freight. Another is to move a vacant ship to the point of escape, ferry astronauts up on a “space taxi” akin to the shuttle and then fire another rocket for the final push to Mars. The second rocket could either be high or low thrust. In one analysis of the latter possibility, a pulsed inductive thruster fires for 40 days, coasts for 85 days and fires for another 20 days or so on arrival at the Red Planet.

A VASIMR engine opens up other options. Staying in low gear (moderate thrust but low efficiency), it can spiral low efficiency), it can spiral out of Earth orbit in 30 days. Spare propellant shields the astronauts from radiation. The interplanetary cruise takes another 85 days. For the first half, the rocket upshifts; at the midpoint it begins to brake by downshifting. On arrival at Mars, part of the ship detaches and lands while the rest—including the module for the return flight—flies past the planet, continues braking and enters orbit 131 days later.

Update 11-24-19:  revised delta-vees for Phobos trip,  appended below

The planning of interplanetary flights, such as from Earth to Mars, uses basic orbital mechanics. As long as the spacecraft speed does not reach solar escape speed, the form of the orbital trajectory about the sun will be an ellipse. There are “min energy” trajectories, and there are faster trajectories, but all these trajectories will be ellipses.

The basics of elliptical orbits are given in Figure 1, including just about all the relevant analysis equations, plus a little more. An ellipse is a symmetrical closed curve containing two foci. The central body occupies one focus, the other is unoccupied. These foci are located farther from the center in the more eccentric ellipse. A circular orbit is a sort of “degenerate” ellipse with zero eccentricity, so that the foci come together at the center. I wrote this for a diverse audience of both technical and non-technical people.

Bear in mind that these analytical solutions apply only to a 2-body problem (one object orbiting a central body). The 3-body problem (object and two bodies) requires integration of the equations of motion for an exact solution.

• 

For interplanetary trips, we are talking about trajectories where the central body is the sun. However, this same analysis applies to orbits about the Earth or any other body.  The key features of elliptical orbits are the speeds, which vary around the path,  and the min and max distances from the central body.

“Perihelion” is the end apex of the ellipse closest to the sun, where the speeds are highest. “Aphelion” is the end apex of the ellipse farthest from the sun, where speeds are lowest. The corresponding terms for an orbit about the Earth are “perigee” and “apogee”. For the moon, they are “pericynthion” and “apocynthion”. Thus the abbreviation “per” applies to the closest approach, and “apo” the farthest approach, regardless of the central body.

When looking up tables of characteristics of astronomical bodies, the description of their orbits is usually cast as min and max distances from the central body. Those would be r_per and r_apo, respectively. Their average is the semi-major axis length “a”. The eccentricity “e” and the period “P” are easily computed from these distances, all as given in Figure 1. Further, the distance of the foci from the center of the ellipse “c”, and the length of the semi-minor axis “b”, are also easily computed, and given in the figure, as is the equation for x-y coordinates all along the ellipse, where these x-y coordinate are centered at the center of the ellipse.

The equation giving velocity at any radius “r” from the central body is actually quite simple, as shown in the figure above. Note that “r” is bounded: r_per < r < r_apo. There is no simple equation giving time at any particular point around the elliptical path. Centuries ago, it was said “equal areas are swept out by the radius vector from the central body in equal amounts of time”.

Today, we would say that the time along a segment is proportional to the area swept out by the radius vector from the central body to the object, moving along that segment. You can obtain that time by integrating the area under the segment from one point to another, and adding or subtracting the appropriate triangle area. All of this should be evident upon inspection of Figure 1 above, particularly noting the shaded area.

The “extras” in Figure 1 are the equations for calculating the escape velocity and low circular orbit velocity of any celestial body, plus a form of mechanical energy conservation as you approach (or depart from) a body in an unpropelled state.

Hohmann Min-Energy Transfer

These trajectories are the ones with the minimum velocity requirements for travel. The perihelion of the transfer ellipse is located at the orbit of the Earth on one side of the sun. The apohelion of the transfer ellipse is located at the orbit of Mars on the other side of the sun. See Figure 2. Because the orbit of Earth is slightly elliptical, and Mars more so, these distances vary somewhat. Using the average values gets you into the ballpark, but you should really use the worst case to size your spacecraft’s propulsion capability!

Note that because the Hohmann transfer ellipse is tangent to a very-nearly-circular planetary orbit at each end, then the planetary velocity and transfer orbit velocity vectors are essentially parallel at each end. What is ordinarily a vector subtraction devolves to a simple scalar subtraction, for determining the velocities with respect to the planet, at each end. That is only true when the perihelion or the apohelion of the orbit are located at the appropriate planetary orbit distance from the sun.

Specifically, the velocity vector of the spacecraft with respect to the planet is the velocity vector of the spacecraft with respect to the sun, minus the velocity vector of the planet with respect to the sun:

               spacecraft V_{wrt planet} = V_{wrt sun} – V_{planet wrt sun} where V is a vector velocity

• 

An Approximation to the 3-Body Problem

The trouble the above evaluations right at perihelion and apohelion is that these are 2-body (spacecraft and sun) analyses, and the close-vicinity dynamics of departure and arrival are fundamentally a 3-body problem (spacecraft-sun-planet). As already stated, it takes computerized 3-body analysis to get velocity requirements and detailed localized trajectories exactly right. Yet, you can get very, very close to the velocity requirements with the following approximation technique.

As in the above discussion, you do the vector velocity subtraction to find the velocity of the spacecraft with respect to the planet, at the perihelion and apohelion conditions. For only Hohmann min energy transfer, this calculation devolves to a simple scalar subtraction, because the vectors are parallel. Either way, there is a velocity magnitude involved.

The approximation is to treat this relative velocity with respect to the planet as a velocity “very far from the planet”, and to approximate the pull of the planet’s gravity during the close encounter as an unpowered gravitational acceleration toward the planet, from “very far” to “very close”. This is done with conservation of mechanical energy, based on the “far from planet” velocity magnitude (with respect to the planet), which makes the vector direction more-or-less irrelevant, except to the trajectory details. Conservation of mechanical energy says:

               0.5 m V_far² = 0.5 m V_near² - change-in-PE far-to-near, where m = spacecraft mass

This approximation then makes use of the very convenient fact that the change in potential energy from very far to very near is, in point of fact, numerically equal to the spacecraft kinetic energy associated with the escape velocity of the planet:

               0.5 m V_esc² = change-in-PE far-to-near

Thus, after dividing off the “0.5 m” factors common to all three terms, we have a very simple way to estimate the spacecraft velocity magnitude with respect to the planet, once it is “very close”. This velocity would apply for either entry into planetary orbit, or for the initial direct entry into the local atmosphere for aerobraking (of any type). That simple equation is:

               V_far² = V_near² – V_esc² or V_near = (V_far² + V_esc²)^0.5

Doing the full 3-body problem on the computer refines the actual trajectory to be flown, but does not refine the spacecraft propulsion velocity requirements very much at all, beyond these simple estimates. These estimates are really quite good, and apply to departure as well as arrival.

This is illustrated in Figure 3, for both arrival and departure at both Earth and Mars. For arrival, V_near is denoted as V_int for “interface velocity”, and V_near = V_bo for the “burnout velocity” at departure.

• 

Please note that Mars arrival and departure trajectories are directed retrograde,  with respect to Mars, because the planet’s orbital velocity about the sun exceeds the transfer ellipse apohelion velocity with respect to the sun.  In effect,  the planet literally runs over the spacecraft from behind upon arrival. You want to time your arrival at Mars orbital distance very slightly ahead of Mars’s arrival, so that you don’t miss closing gravitationally with the planet,  or get left behind for not “leading the target” enough. Similarly,  you must accelerate in the retrograde direction escaping from Mars,  so that you end up at the apohelion of your return ellipse,  with the appropriate slower velocity about the sun.

The situation at Earth is different,  because the transfer ellipse perihelion velocity with respect to the sun exceeds the orbital velocity of Earth about the sun.  Thus departures and arrivals are in the posigrade direction with respect to Earth.  Upon arrival,  the spacecraft is literally running into the Earth from behind. It literally runs away from the Earth in a posigrade direction upon departure.

Actual Departure and Arrival “Close-In” Operations

Arrival at,  and departure from, Mars is depicted in Figure 4. These are similar at Earth,  only the numbers are different. You are “close-in” at speed V_near,  which is V_int for arrival.  If Mars (or Earth) were airless, this is the theoretical delta vee you have to “kill” in order to land direct. They are not airless, so your choices are entry into orbit, or some sort of direct aerobraking entry.

• 

If you are entering Mars orbit,  you want to enter it on the side of the planet and specific location where the orbit velocity vector is fairly parallel with your own spacecraft velocity vector.  That gives the smallest delta-vee requirement:

               dV = V_int – V_orbit  on the “correct” side

If you enter orbit on the wrong side,  the delta vee is much larger:

               dV = V_int + V_orbit  on the “wrong” side

Orbital entry dV values need no factoring,  because these are brief impulsive burns in space.  It is not feasible to use low-thrust long burn electric propulsion for this,  at least not for manned craft.  The spiral-in times are months long.

If instead you are aerobraking (whether one pass or multi-pass),  your initial entry interface speed is essentially V_near = V_int.  From there deceleration is by drag,  not propulsion,  until touchdown.  On Mars,  retropropulsive touchdown is required at one level or another,  since terminal parachute speeds are high subsonic at best.  Whatever the terminal velocity is,  that is the theoretical delta-vee you need to “kill” for touchdown.  I recommend that theoretical value be increased by factor 1.5 to cover maneuver and hover allowances.

Departure from Mars is the exact reverse,  but in the same direction as arrival.  You want to end up at V_near = V_bo still close to the planet,  so that your V_far is the aphelion velocity of the transfer ellipse back to Earth.  If departing from orbit,  you burn on the side where the orbital motion is locally retrograde,  so that the delta-vee required is lower:

               dV = V_bo – V_orbit

Departing Mars direct from the surface,  your theoretical dV is V_bo,  and it must be directed retrograde.  This dV needs to be factored-up for small drag and gravity losses.  Recommendations are given in the figure.

While not shown in the figure,  departing Earth is the same process and analysis,  just with everything oriented in the posigrade direction.  If you leave orbit,  you do it on the side where orbital motion is posigrade,  and end up at V_bo in a posigrade direction.  That delta vee is:

               dV = V_bo – V_orbit

If you depart directly from the surface,  your theoretical delta vee is V_bo,  which must be factored-up for gravity and drag losses.  Recommendations are in the figure.

Arriving at Earth has exactly the same values.  If you enter into orbit,  you do it on the side where orbit motion is posigrade,  so that the delta vee is:

               dV = V_int – V_orbit

If you enter the atmosphere for aerobraking,  your entry interface speed is V_int.  Depending upon the design of your spacecraft,  there may or may not be a touchdown burn.  If there is,  it is some terminal speed to “kill”.  I recommend factoring that up by 1.5 for hover and maneuver allowances.

Faster Trajectories

There is no such thing as a “direct flight to Mars”.  All fast trajectories are still elliptical about the sun,  unless your spacecraft propulsion is capable of far greater than solar escape speed.  None are,  at this time in history.

Faster trajectories reduce the travel time at the expense of higher required delta-vees.  There is no way around that bit of physics.  What you want to do is employ the higher-speed end of your transfer ellipse as your trajectory,  and arrive at your destination before you reach the lower speed portion of your ellipse.  Thus for Earth-Mars,  your transfer ellipse perihelion will still be at Earth’s orbital distance,  while your transfer ellipse apohelion will be well beyond the orbit of Mars.

There is no point to putting the transfer ellipse perihelion inward of Earth’s orbit,  because that moves your trip segment towards the slower end of your transfer ellipse.  You would thus average lower speeds over about the same path length.

For any such faster transfer ellipse,  the situation is as depicted in Figure 5.  Your perihelion velocity is higher,  so your departure delta vee is higher.  But the same departure V_bo and V_far calculations apply,  as for the Hohmann ellipse.

You will “get off” the transfer ellipse when your distance from the sun is at Mars’s distance.  You need to time your arrival to be just as Mars gets there.  It will be a real vector subtraction to determine your velocity with respect to Mars at arrival.  Its magnitude is your V_far.  The geometry of closure with the planet is more complicated because of the nonparallel angle between the vectors to be subtracted.  Even so,  just use the “kinetic energy thing” on V_far and V_esc to get V_int.  From there,  it’s the same basic choices for orbital entry or direct landing,  even though the detailed geometries are changed.

• 

Departing Mars is the reverse.  Whether from orbit or direct from the surface,  you will end up at V_bo near the planet.  The “kinetic energy thing” gets you V_far.  That speed and an appropriate direction must add vectorially with the planet’s orbital velocity vector,  to obtain the velocity vector you want for the return trajectory (correct magnitude,  and direction tangent to the transfer ellipse path).  You have to time this such that Earth will be at your perihelion point when you get there.

The easiest way to get the angles for the vector additions is to just plot the transfer ellipse,  and a circle at the Mars distance.  Draw the tangents where they cross,  and measure the angle “a” between them with a protractor.  Otherwise,  when evaluating the points on the trajectory,  compute the slope at the encounter point numerically.  The tan^-1(slope) is numerically its angle “a1” below reference,  where reference is a line parallel to the semi-major axis (a negative angle on the perihelion half of the ellipse,  and positive on the apohelion side).

The encounter coordinates give you the angle a3 of the radius vector at encounter as the value tan^-1(y/(c-x)).  The circle approximation for Mars’s orbit through the encounter point has a tangent normal to that radius vector.  Its angle below reference “a2” is 90^o-radius vector angle a3.  The difference in the angles is the angle “a” between the velocity vectors.  See Figure 6

• 

The easiest way to get the time from perihelion to the Mars encounter point is (again) to plot the transfer ellipse and a circle at the Mars orbit distance.  Bound this with the semi-major axis,  and with the radius vector from the sun to the Mars encounter point.  Then use a planimeter to measure the swept area.

The area of the entire ellipse corresponds to the period of the whole transfer orbit.  Or,  if desired,  half the area of the ellipse corresponds to the travel time from perihelion to apohelion.  The ratio of your planimeter area swept for the trip,  to the ellipse area,  is the same as the ratio of 1-way trip time to orbital period.  Or if ratioed to half the ellipse area,  the ratio of travel time to one-way trip time.If you don’t have a planimeter with which to measure areas on your plot,  then integrate numerically the area under the ellipse curve (relative to semi-major axis) from the perihelion point to the encounter point.  Add or subtract as appropriate the area of the right triangle formed by the encounter vector from the sun to Mars as its hypotenuse.  A spreadsheet would work for this.  (If you have values for a and b,  then you have the equation for the ellipse,  by which to generate coordinate values for x and y.)

Trajectories to Venus or Mercury

Trips to Venus or Mercury work almost the same way as trips outbound to Mars or further.  The difference is that the transfer perihelion is at the destination,  not at Earth.  This is shown in Figure 7.  If Hohmann min energy,  then the apohelion is at Earth’s orbit.  If a faster trajectory,  the apohelion is outward from Earth’s orbit.

• 

Quite frankly,  as fast as these inward-from-Earth trips are with Hohmann min energy ellipses,  there seems little point to the added complications of a faster trajectory.  Venus is only 143 to 149 days away,  and Mercury is only 95 to 117 days,  using min energy Hohmann trajectories.  Compare that with Mars:  235 to 283 days away,  and Ceres-as-typical-of-the-asteroid-belt at 428 to 517 days away.   Times outward of Earth are longer simply because the distances are larger,  and the velocities are lower.

Reference Data for Solar System Bodies

The universal gravitation constant for Newtonian gravity is G = 6.6732E-11 N-m²/kg².  The masses and radius data for some selected bodies are as follows:

Body	mass, kg	eq.R, km	avg.R, km
Sun	1.991E30	695950	695950
Earth	5.979E24	6378.5	6371.3
Mars	6.418E23	3386	3380
Moon	7.354E22	1738.7	1738.3
Phobos	2.72E16	11.3	10.4

Basic orbital data for selected bodies are as follows:

Body	rper, km	rapo, km
Earth	1.4707E8	1.5207E8
Mars	2.0656E8	2.4912E8
Moon	363,299	405,506
Phobos	a = 9408 km

Specific Case Study Numbers for Trips from Earth to Mars

For purposes of typical results,  I presume that both Earth and Mars are at their average distances from the sun,  meaning the radii to their orbits are their “a” values.  I also set the transfer orbit perihelion distance at the Earth orbit distance (in this case its value of “a”) for this study.  What changes is the transfer orbit apohelion distance.

I ran 4 values of transfer orbit apohelion distance:  (1) Hohmann transfer at Mars “a” = 2.28E8 km for comparison,  (2) 3.21E8 km to get a 2-year orbit period so that a free return to Earth is possible,  (3) 4.00E8 km near the inner edge of the asteroid belt,  and (4) 4.671E8 km to get a 3-year orbital period so that a free return to Earth is possible from an orbit whose apohelion is well within the asteroid belt

For all cases,  departure is from low Earth orbit (LEO).  Arrival at Mars could be any of 3 cases:  (1) direct aerobraking entry leading to a landing,  (2) entry into low Mars orbit (LMO),  or (3) rendezvous with and touchdown upon Phobos,  Mars’s inner moon.  Once V_far is determined for each of the transfer orbit cases,  then each arrival sub-case must be analyzed separately.

Finding the encounter point requires solving the equations of the ellipse and the circle models simultaneously,  unless one does this graphically.  The ellipse is centered at the origin of the x-y coordinates,  the circle is not (being centered at the positive-x focus where the central body is located,  in all the figures above depicting this).

Now,  for the transfer ellipse,  the semi-major (a) and semi-minor (b) axis distances,  and the distance to the foci (c),  are all known.  Its center is the origin (0,0).  Its equation is:

               x²/a² + y²/b² = 1

The circle is offset from the origin,  centered at (c,0) to the right of that origin,  with a radius equal to the Mars orbit distance (call it R).  Its equation is therefore:

               (x – c)² + y² = R²

Solving the circle equation for the y² term gets us something we can substitute into the ellipse equation,  getting us one equation in one variable (x) that we can solve:

               y² = R² – (x – c)²                               circle solved for y²
               x²/a² + [R² – (x – c)²]/b² = 1               substitution into ellipse to eliminate y²

We have to expand the squared binomial in the second term,  and then distribute the 1/b² coefficient,  followed by collection of like terms in x² and x:

               x²/a² + [R² – (x² - 2xc + c²)]/b² = 1
               x²/a² + [R² – x² + 2xc – c²]/b² = 1
               x²/a² + R²/b² – x²/b² + 2xc/b² – c²/b² = 1
               (1/a² – 1/b²)x² + (2c/b²)x + (R²/b² – c²/b² -1) = 0

That result is a quadratic equation in standard form Ax² + Bx + C = 0,  where:

               A = 1/a² – 1/b²
               B = 2c/b²
               C = R²/b² – c²/b² -1

for which the most convenient solution is by means of the quadratic formula:

              x = -B/2A +/- (D^0.5)/2A,  where D is the discriminant D = B² – 4AC

For there to be one and only one x solution,  the discriminant must be zero,  so that x = -B/2A.  If the discriminant is positive,  there are two real solutions for x per the formula.  If the discriminant is negative,  there are no real-number solutions at all.

Once we have values for solution x,  the corresponding y coordinates can be determined from either the ellipse equation or the circle equation (we are interested in the positive-y roots for the arrival encounter;  the negative-y roots correspond to the departure point):

               y = +/- [R² – (x – c)²]^0.5                    from the circle equation
               y = +/- [b²(1 - x²/a²)]^0.5                   from the ellipse equation

Once the x-location (along the semi-major axis) is known,  we can integrate numerically under the ellipse curve (relative to the semi-major axis) from the solution x to the perihelion x value.  The area of half the ellipse (to one side of the semi-major axis) is 0.5*pi*a*b.  There is a triangle formed by the radius vector to encounter:  its height is the y coordinate at encounter.  Its base is the focus length c minus the x coordinate.  Thus the triangle area is 0.5*y*(c-x).  To create the area swept by the radius vector,  the area of this triangle subtracts from the integral area,  as long as the encounter x is less than c.  The swept area factor SAF is the swept area divided by the ellipse half area.  This area ratio applies to half the orbit period,  for the one-way trip time.

I used a spreadsheet to do this analysis,  supplemented by hand plots of the orbits to ensure the calculations were getting the right answers.  This process required multiple iterations before I got it “right”.  Figure 8 shows the basic transfer orbit-related data that are independent of the exact nature of Mars arrival.  The basic orbital parameters a,  b,  c are given (blue-highlighted values expressed as Mkm (millions of km) have the most significant figures).  Average,  perihelion,  and apohelion velocities (with respect to the sun) are given in km/s.  The period and half-period values are shown,  with the half-period shown in seconds,  days,  and months.

• 

Also included in the figure are the basics of Earth departure velocities and Mars arrival velocities.  All the Earth departure velocities are tangent to both the transfer orbit and Earth’s orbit,  since the transfer perihelion is always at Earth’s orbit.  The scalar difference between perihelion velocity and Earth’s average orbital velocity is the V_inf value,  typical of “far from Earth” velocity needs,  and measured with respect to Earth.   Adjusted for the effects of Earth’s gravity to a “near Earth” value,  this produces the V_bo values with respect to Earth. 

Mars arrival is a little more complicated,  since the velocities must add vectorially for all but the Hohmann min energy transfer case.  In the middle group in the figure,  the angle “a” is that between the velocity of the spacecraft “V” in its transfer orbit at Mars encounter,  and the velocity vector of Mars that is tangent to its orbit.  Only for the Hohmann case is this angle zero.

The velocity “far from Mars” with respect to Mars is the velocity vector V at angle a relative to Mars’s vector,  minus Mars’s velocity vector.  This is the Mars arrival V_inf in the figure.  I did not include all the spreadsheet details of computing that angle,  but my hand plots showed that I was indeed computing the correct values.  Adjusted for Mars’s gravitational attraction,  this corresponds to a higher V_int “near Mars”.

The bottom group in the figure relates to the swept-area estimated 1-way trip time.  Again,  I didn’t include all the details of the numerical integration,  but my area estimates were indeed confirmed by the hand-plotted orbits.  The blue highlighted data are the transfer orbit parameters expressed as Mkm,  for the most significant figures.

The principal results are plotted versus transfer orbit apohelion distance in Figure 9.  These include the half-period of the orbit,  the “near Earth” departure requirement V_bo with respect to Earth,  the “near Mars” arrival speed V_int with respect to Mars,  and the 1-way trip time.

• 

Note that the 1-way trip time of 8.62 months and the half-period of the transfer orbit (8.62 months) are identical for the Hohmann transfer case,  where the apohelion distance is the average orbital distance of Mars.  This requires 11.57 km/s achieved burnout speed to depart,  and arrives close to Mars at some 5.69 km/s,  more-or less lined up tangent to Mars’s orbit.

The next faster case investigated has a half-period of 12.00 months,  for a full round trip of exactly 2 years.  If the Mars encounter were to fail,  the spacecraft would arrive back at Earth’s orbit just as Earth got there.  This offers the possibility of a free-return abort,  if 2 years in space is tolerable.  Otherwise,  the 1-way trip to Mars is much faster at 4.26 months.  It costs more:  the Earth departure burnout requirement is 12.26 km/s,  and the near-Mars encounter velocity is some 7.40 km/s,  skewed off-tangent at about 34-35 degrees.

The third case used an even 400 million km apohelion distance.  Its total transfer orbit period is 30.3 months,  which is nonresonant with Earth’s period about the sun.  There is no free-return abort using this orbit!  The 1-way trip time is really fast at 3.67 months,  but this costs quite a bit.  The required departure burnout speed is 12.77 km/s,  and the near-Mars encounter velocity is some 7.36 km/s,  skewed about 38 degrees off tangential.  This apohelion is actually into the inner edge of the main asteroid belt.

The final case has a half-period of 18.0 months,  or a full round-trip period of 3 years,  resonant with Earth.  This offers the possibility of a free-return abort,  if 3 years in space is tolerable.  The 1-way trip time is the shortest of the cases investigated,  at 3.40 months.  The cost is high:  13.14 km/s at departure burnout,  and a near-Mars encounter speed of 6.53 km/s to deal with.  That last is skewed about 33-34 degrees off tangential.

Most Mars mission designs will be leaving from orbit about the Earth.  The Spacex Starship is one of those.  Low circular Earth orbit (LEO) has a speed about the Earth of about 7.9 km/s,  departure starts from there and must reach “V_bo”.  At Mars,  the choices are (1) direct entry and retropropulsive touchdown,  (2) entry into low Mars orbit (LMO) with possibly a separate vehicle to deorbit and enter for a retropropulsive landing,  and (3) entry into Phobos’s orbit and touching down propulsively on Phobos.  For that last,  I simply applied a 1.5 factor for maneuver and hover to the escape speed from Phobos,  to estimate a mass ratio-effective delta-vee requirement.

A few of my estimation items are simply assumptions.  I assumed that any of the transits (both outbound and inbound) have a course correction allowance of dV = 0.5 km/s.  I also assumed that any vehicle returning to LMO must rendezvous with another vehicle,  necessitating a rendezvous allowance.  I assumed that allowance to be dV = 0.1 km/s.  Anything deorbiting from LMO to land upon Mars requires an allowance for a deorbit burn.  For this I used dV = 0.05 km/s.  Finally,  I assumed the terminal speed after Mars entry to be about a Mach number,  leading to an estimate of the speed to be “killed” (and about 0.5 Mach on Earth).

A summary of the detailed transit and terminus estimates is given in Figure 10.  Note that the transfer ellipse influences departures and arrivals,  but nothing else.  Delta-vee information is highlighted blue,  and entry speeds for aerobraking are highlighted green.

• 

This same information is rearranged into groups representing each kind of overall mission,  with the data parametric upon the transfer trajectory used.  The presumption is that the return to Earth uses the same transfer ellipse trajectory as the outbound trip to Mars.

Figure 11 shows the velocity requirements (blue) and entry interface speeds (green) for a mission that departs Earth orbit,  makes a direct entry and landing upon Mars,  then makes a direct escape from Mars,  leading to a direct entry and landing at Earth.  Touchdowns are assumed retropropulsive.  There are many mission designs which might use this architecture.  The most notable recent example is Spacex’s proposed “Starship”.

Figure 12 shows the velocity requirements (blue) and entry speeds (green) for a typical orbit-to-orbit mission to Mars.  This is broken down into a transit velocity requirement,  and a separate landing velocity requirement,  since it is likely the lander is a separate vehicle.  Similarly,  the takeoff is separate from the transit home.  Earth return is to LEO,  not a landing.  That is presumed to be a different vehicle.

• 

• 

The Phobos (only) visit is shown in Figure 13.  Because the landing allowance is so small,  it is presumed the transit vehicle also makes the landing.  Transit is from LEO to Phobos orbit,  and from Phobos orbit to LEO for the return.  There is no aerobraking in this scenario.  Upon return to LEO,  it is presumed that some other vehicle is used for the final return-to-Earth landing.

• 

As mentioned above,  I utilized by-hand plots (pencil and paper) to verify correct calculation of many items.  I did not need such a plot for the Hohmann min-energy transit,  but I did for the 3 faster trajectories.  The following two figures are photographs made of those working hand plots,  two plots per page.  Note that I mis-plotted one of them and had to try again.

• 

• 

---

Update 11-24-19:  Revised Delta-Vee Estimates for Phobos Mission

I have corrected an error in estimating Vinf for the Phobos misson.  I used the surface escape speed 5 km/s for the conservation of mechanical energy estimate,  when I should have used the escape velocity out at Phobos's orbital distance,  some 3 km/s.  This reduces the Vnear values substantially,  thus reducing the estimates for delta-vee required.  This changes Figure 13 entirely,  and the Phobos Departure/Arrival details in Figure 10.  The revised data are given in Figure 16 below.

• 

Abstract

     You have a rocket in a high circular orbit around a massive central body (a planet, or the Sun) and wish to escape with the fastest possible speed at infinity for a given amount of fuel. In 1929 Hermann Oberth showed that firing two separate impulses (one retrograde, one prograde) could be more effective than a direct transfer that expends all the fuel at once. This is due to the Oberth effect, whereby a small impulse applied at periapsis can produce a large change in the rocket’s orbital mechanical energy, without violating energy conservation. In 1959 Theodore Edelbaum showed that this effect could be exploited further by using up to three separate impulses: prograde, retrograde, then prograde. The use of more than one impulse to escape can produce a final speed even faster than that of a fictional spacecraft that is unaffected by gravity. We compare the three escape strategies in terms of their final speeds attainable, and the time required to reach a given distance from the central body. To do so, in an Appendix we use conservation laws to derive a “radial Kepler equation” for hyperbolic trajectories, which provides a direct relationship between travel time and distance from the central body. The 3-impulse Edelbaum maneuver can be applied to interplanetary transfers, exploration of the outer solar system and beyond, and (in time reverse) efficient arrival and orbital capture. The physics principles employed are appropriate for an undergraduate mechanics course.

I. INTRODUCTION

     Newton’s laws of motion and universal gravitation allow us to relate the geometrical properties of an orbit to conserved quantities such as angular momentum and energy. Changing the velocity of an orbiting object changes these quantities, and therefore the orbital path. One of the earliest descriptions of such orbital maneuvers was Newton’s cannon, a thought experiment that demonstrates the effects of changing a projectile’s launch speed on its resulting orbit — increasing this speed increases the size and period of the orbit. For launch speeds greater than the local escape speed, the projectile does not return, but follows a hyperbolic escape path and approaches an asymptotic final speed v_∞ far from the central body.

     In the 20th century, pioneers of astrodynamics began to use these principles to chart the courses of hypothetical spacecraft that propel themselves by the directed expulsion of matter carried as fuel. The resulting impulses change their orbital paths. Even for a simple system consisting of a rocket and a massive central body, a variety of maneuvers are possible, such as Hohmann and bi-elliptic transfers between orbits, with differing numbers of impulses employed to use the minimal amount of fuel.

     In this paper we discuss three strategies for a spacecraft in a circular orbit to achieve a high-speed escape with v_∞ > 0. We compare their fuel requirements to achieve a given v_∞, and their travel times to reach a given distance from the central body.

     In Sec. II we start with conservation laws to develop equations that relate orbital speed and distance. In Sec. III we use these to compare three escape strategies in terms of fuel usage. In Sec. IV, with the aid of an Appendix, we derive expressions for the travel time from the original orbit out to a given distance. In Sec. V we conclude by discussing applications and providing suggestions for student investigations.

II. BASIC EQUATIONS FOR ORBITAL MANEUVERS

     We assume that the mass of the spacecraft is tiny relative to the central body, such that any change in the latter’s motion can be ignored. The central body can then be used to define an inertial frame in which we analyze the spacecraft’s motions.

     To simplify our analysis, we consider only impulses of duration much less than the orbital period around the central body. Equivalently, for such idealized impulsive maneuvers the rocket can change its velocity without changing its position, so that the pre- and post-impulse Keplerian orbits intersect. For such impulses to have the greatest effect on a spacecraft’s orbital energy and angular momentum, they must be applied either prograde (in the direction of motion, by expelling exhaust backwards) or retrograde relative to the spacecraft velocity vector. This results in all paths being coplanar, reducing our analysis to two dimensions. (We discuss orbital plane changes briefly in Sec V.)

1. Speed as a function of distance

     Between impulses, an object in freefall in the gravitational field of an isolated spherical body of mass M will have a constant specific energy (where we use “specific” to mean “per unit mass”),

     Here, v is the magnitude of its velocity vector, r is its distance from the central body’s center, and G is the gravitational constant. (This energy properly belongs to the system of the spacecraft and central object, but since the massive central body is assumed stationary, we assign it to the spacecraft for brevity.)

     From Eq. (1), in order to escape with an asymptotic final speed v_∞ at infinity, an unpowered spacecraft at a distance r_dep must have a departure speed v_dep there given by

     For the maneuvers described in this paper, we also need to relate the orbital speed of a spacecraft to its position in a bound elliptical orbit. Let its speed be v_A at apoapsis (farthest from the central body, at distance r_A) and v_P at periapsis (the closest point, distance r_P). Due to the central nature of the gravitational field, the specific angular momentum is conserved and given by

where v_θ is the azimuthal component of the spacecraft’s velocity. Therefore, at the extremes of the ellipse where there is no radial motion,

Combining Eqs. (1) and (4) we can solve for apoapsis and periapsis speeds in terms of the corresponding distances

The first square-root factor in each equation is the circular

2. Relating impulse to fuel usage and energy input

     Typically, a rocket engine is modeled as one that expels its fuel continuously, even for short-duration impulses that represent the idealized limit of minimizing fuel usage and transfer times. Conservation of momentum leads to the rocket equation, attributed to Konstantin Tsiolkovsky, which relates the speed change Δv to the initial and remaining mass of the rocket,

where m_i and m_f are the masses of the rocket (including fuel) before and after the impulse, respectively, and v_ex is the effective exhaust speed. (The gravitational attraction between the rocket and exhaust mass can be ignored.) Equation (6) does not depend on the rate of fuel consumption, and applies to multiple intermittent firings of the rocket engine. For this reason, we shall follow the standard convention of using total Δv as a proxy for fuel cost when comparing strategies to achieve an orbital transfer.

     When a mass of fuel (m_i - m_f) is expelled to obtain a given Δv, the chemical energy converted to kinetic energy is ΔE_chem = ½ (m_i - m_f)v_ex². A fraction of this input energy changes the mechanical energy of the rocket, while the rest is carried away in the exhaust, as we shall discuss in Sec. III.5.

III. ESCAPE FROM A CIRCULAR ORBIT

     Initially, we have a spacecraft in a circular orbit of radius r₀ with orbital speed v₀ = √GM/r0 (This relation can be proven by centripetal arguments, or by setting r_A = r_P = r₀ in Eq. 5.) We desire to achieve a target final speed v_∞ for the smallest total Δv, so that the smallest amount of fuel is required.

1. Single impulse for direct escape

     The simplest escape strategy is for the rocket to fire a single, prograde impulse from the circular orbit, to increase its speed past the local escape speed √2v₀, as shown in Fig. 1(a). To find the specific impulse Δv_D for such a direct escape to produce a final speed v_∞, use Eq. (2) with v_dep = v₀ + Δv_D, and r_dep = r₀, i.e.

This function is plotted as a green dotted line in Fig. 2.

2. Two-impulse escape - the “Oberth maneuver”

     In 1929, Hermann Oberth described a more fuel-efficient strategy to obtain the same v_∞ from a high circular orbit (of radius many times that of the central body). First use a retrograde impulse to drop closer to the body, then apply a prograde impulse at the periapsis of that elliptical orbit, as shown in Fig. 1(b). Author Robert Heinlein (who had worked with Oberth on the pioneering science fiction film Destination Moon) described this maneuver in chapter 7 of his novel The Rolling Stones:

“A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward the Earth, then performing her principal acceleration while as close to the Earth as possible.”

     Fig. 1(a), 1(b), 1(c). Three escape strategies from a counter-clockwise circular orbit, all with a total Δv = 1.1 v₀. Dots show the spacecraft position at intervals of 1/40th of the original circular orbital period. The rocket images mark the position and direction of each impulsive thrust.
     (a) Direct (single-impulse) escape.
     (b) Oberth (2-impulse) escape with r_in = 0.15 r₀.
     (c) Edelbaum (3-impulse) escape with r_out = 1.50 r₀ and r_in = 0.15 r₀.
     (These values were chosen for clarity of plotting, and are not the same as those adopted for Figs. 2 and 3.)

     Reference 7 explains the physics of this Oberth effect in detail. If a rocket body of mass m moves at speed v, a forward specific impulse Δv increases its kinetic energy by ½m(v+Δv)² - ½mv² which increases linearly with v. Since an orbiting rocket travels faster as it moves deeper into the potential well, firing an impulse as close as possible to the central body allows it to maximize the energy gained, in principle without limit as r_P→0. (In practice r_P is limited by the size of the central body).

     The increase in the mechanical energy of the system of rocket+exhaust comes from the chemical energy converted in the impulse ΔE_chem, which is constant regardless of where the impulse was fired. Therefore, any additional energy gained by a fast-moving rocket must be obtained at the expense of the mechanical energy carried by the expelled fuel, as we discuss in Sec. III.5.

     The two-impulse Oberth maneuver is defined by the periapsis distance r_in of the elliptical orbit produced by the initial, retrograde impulse at radius r₀. From Eq. (5) with r_A = r₀ and r_P = r_in, the required velocity change for this first impulse Δv_Ob1 is

where the leading negative sign denotes the retrograde direction. Once the spacecraft reaches periapsis, it will be moving at speed v_P given by Eq. (5) with r_A = r₀ and r_P = r_in. To achieve a final speed v_∞, use Eqs. (2) and (5) with r_dep = r_in to find the second prograde velocity boost Δv_Ob2 from v_P to v_dep,

The total specific impulse for the Oberth escape maneuver is the sum of the absolute values of Eqs. (8) and (9), i.e. Δv_Ob = |Δv_Ob1|+ Δv_Ob2, which simplifies to

Equation (10) is plotted in Fig. 2 as a dashed purple curve for r_in = 0.05r₀. This curve crosses that described by Eq. (7) at Δv_Ob = Δv_D = v₀, independent of r_in, for which the resulting v_∞ = √2v₀, coincidentally equal to the local escape speed from the original orbit. For desired values of v_∞ larger than this, Δv_Ob < Δv_D, so the two-impulse Oberth maneuver will require less fuel than the single-impulse direct escape (cf. Fig. 2).

     As r_in is decreased, the “Oberth advantage” (the difference Δv_D - Δv_Ob) gets larger for v_∞ > √2v₀, since the fuel expelled at periapsis is placed into a lower energy orbit. In the theoretical limit as r_in → 0, the dashed purple curve in Fig. 2 becomes flat, and Eq. (8) shows that Δv_Ob1 → -v₀ for any v_∞. This speed change stops the spacecraft in its original orbit and causes it to fall in radially towards the center of attraction. Then, the tiniest boost Δv_Ob2 at r_in = 0 could produce any value of v_∞ desired.

3. Three-impulse escape - the “Edelbaum maneuver”

     In 1959 Theodore Edelbaum described a coplanar escape maneuver that employs three impulses: a prograde boost to raise apoapsis, followed by a retrograde impulse to cause the spacecraft to fall in to a low periapsis, and thence a final boost to escape. An example is shown in Fig. 1(c). By falling in from a larger radius than the original circular orbit, the resulting faster periapsis speed enhances the Oberth effect, by transferring more of the expelled fuel’s chemical and mechanical energy to the spacecraft.

     The transfer from circular orbit to the intermediate ellipse is defined by the apoapsis radius of that ellipse, r_A = r_out, with periapsis at the original orbit radius, r_P = r₀. Equation (5) gives the required post-impulse velocity v_P, from which we find the necessary boost Δv_Ed1,

The resulting apoapsis speed is also found from Eq. (5). Once the spacecraft reaches this position, it fires a second, retrograde impulse to lower its periapsis distance to a new r_P = r_in < r₀. The required velocity change Δv_Ed2 is then

The speed v_P of the rocket after it falls from apoapsis at rout to periapsis at r_in is given by Eq. (5), and the required velocity for escape from there to a specified v_∞ is found from Eq. (2) with r_dep = r_in. The difference is the final velocity boost Δv_Ed3, i.e.

The total speed change for an Edelbaum escape is the sum of the absolute values of Eqs. (11), (12), and (13), Δv_Ed = Δv_Ed1 + |Δv_Ed2| + Δv_Ed3, and can be simplified to

Equation (14) is plotted in Fig. 2 as a blue dot-dashed curve for r_in = 0.05r₀ and r_out = 2.5r₀. It is also valid for computing Δv_Ob if r_out = r₀, and for Δv_D if r_out = r_in = r₀.

     Compared to the two-impulse Oberth escape maneuver, the Edelbaum escape always has a lower overall Δv for a given v_∞ by an amount

independent of v_∞, with a corresponding savings in fuel expenditure. (Readers can show that Eq. 15 is positive for any r_in < r₀ and r_out > r₀.) This difference can be seen as the constant vertical distance between the dashed purple (Oberth) and dot-dashed blue (Edelbaum) curves in Fig. 2.

     In Fig. 2, the dot-dashed blue curve for Δv_Ed crosses the dotted green curve for Δv_D when

     For larger desired values of v_∞, the Edelbaum maneuver requires a lower total Δv than the direct or Oberth escapes, and so is fuel-optimal for such high-speed escapes.
     In the dual limit r_out → ∞ and r_in → 0, Δv_Ed → (√2 - 1)v₀. This corresponds to a transfer which first sends the spacecraft very far from the central body, then (much later) applies a very small retrograde impulse to nullify its angular momentum, causing it to fall radially inwards to r_in = 0, where a tiny impulse can produce any value of v∞ desired. While this bi-parabolic trajectory has the smallest possible Δv (≈ 0.41v₀) to achieve any desired v_∞, it requires an infinite travel time.

     We can invert and simplify Eq. (14) to give an expression for the mission’s final v_∞ for an available overall Δv provided by the fuel,

This relation also applies to the 2-impulse Oberth escape if one sets r_out = r₀, and to the direct escape if one sets r_out = r_in = r₀.

4. Four or more impulses?

     A valid question is whether additional fuel savings can be achieved using more than three impulses. It has been shown that any fuel-optimal coplanar transfer will consist of no more than three impulses. This also holds true for coplanar bound orbit-to-orbit transfers, which are either Hohmann (two impulses) or bi-elliptic (three). Coplanar escape strategies that produce fuel-optimal transfers with more than three impulses can always be reduced to an equivalent with three or fewer.

     As an example, consider the following 4-impulse strategy: (1) first slow down from circular orbit to an intermediate inner radius r_in, (2) use the Oberth effect there to boost to r_out, (3) slow down again to lower periapsis to r_in, prior to (4) boosting to escape. We have shown above that more efficient escapes are realized as r_out → ∞ and r_in → 0 (to increase periapsis speed and hence the Oberth advantage of the final impulse). Therefore, any intermediate ellipse can be made more fuel-efficient by increasing r_out and decreasing r_in to their mission-constrained extremes. After doing so, the optimal 4-impulse escape degenerates to the same path as the 3-impulse escape and is no more efficient.

     For idealized impulsive maneuvers, a rocket in a closed orbit will always return to the position at which the impulse was applied (unless another impulse is made). A prograde or retrograde impulse at one apsis of an orbit raises or lowers the other apsis. It further follows from Eq. (6) that multiple impulses fired at the same position in orbit are fuel-equivalent to a single, combined impulse.

     For example, if the rocket engine cannot apply the retrograde impulse for the Oberth or Edelbaum maneuver all at once, it can lower its periapsis successively to r_in by multiple firings each time it passes through apoapsis, without any loss of overall fuel efficiency. Students can use the equations in Sec. II to design and evaluate their own multi-impulse maneuvers.

5. Energetics of impulsive escape maneuvers

     How can the same fuel expenditure (and hence energy converted ΔE_chem) produce different final rocket speeds v_∞ for the three escape strategies? The initial mechanical energy of the rocket+fuel system of mass m_i in a circular orbit is E_sys,i = -½m_iv₀² from Eq. (1). After all the fuel is expelled, the mechanical energy added to the system is ΔE_chem from Sec. II.2, which provides the final rocket mass m_f with a speed change Δv given by Eq. (6). The final mechanical energy E_sys,f of the system of rocket+exhaust is then

which is constant for a given Δv and fuel mass expended, and independent of the number of impulses. Therefore, if one wants the rocket to end up with the highest possible mechanical energy m_fε = ½m_fv_∞², one should choose an impulse strategy that leaves the combined exhaust masses with the lowest.

     As an exercise, by expressing v_ex as a multiple of v₀, students can calculate ΔE_chem and the change in kinetic energy of the rocket (due to changes in its speed and mass) after each impulse. The difference gives the mechanical energy change (positive, negative, or zero) of the fuel mass expelled. This will reveal for each strategy how the same final system energy E_sys,f is distributed between the escaping rocket and the total mass of expelled fuel.

6. Comparison with a “no gravity” rocket

     A remarkable result is that both Oberth and Edelbaum escapes can attain a higher v_∞ than a fictional rocket that is unaffected by the central body’s gravity, which boosts tangentially from orbit at r₀ to a constant speed v_∞ = v₀ + Δv, shown as a dotted gray line in Fig. 2. For example, use Eq. (17) to calculate v_∞ for each escape strategy using a mission total Δv = 1.25v₀. With r_in = 0.05r₀, the Oberth escape gives v_∞ = 2.30v₀; additionally setting r_out = 2.5r₀ gives v_∞ = 2.92v₀ for the Edelbaum escape — both faster than v_∞ = 2.25v₀ for the “no gravity” rocket.

     The Oberth and Edelbaum trajectories can end up with larger values of v_∞ because the fuel on board the “no gravity” rocket carries no gravitational potential energy relative to the central body. With no potential energy to “steal” from the expelled fuel, that spacecraft’s kinetic energy gain is limited to a fraction of the fuel’s chemical and kinetic energy only.

IV. TRAVEL TIME COMPARISON

     Compared to a single-impulse direct escape, both the Oberth and Edelbaum escapes can produce faster final speeds for a given total Δv (and thus fuel). However, these maneuvers require additional elliptical orbit segments where the spacecraft moves slowly. For very distant destinations (r >> r₀), the extra time on these segments may not be important, but for intermediate distances there will be a trade-off in total time to destination. In this section we calculate the travel times from the initial circular orbit to a specified distance r from the central body, to determine which of the three strategies will reach that distance in the shortest time.

1. Hyperbolic segment

     All three escape strategies culminate in a hyperbolic escape from a prograde impulse applied at periapsis. For the direct escape, the departure radius r_dep = r₀; for the Oberth and Edelbaum escapes, r_dep = r_in.

     In the Appendix, we derive a general relationship for a hyperbolic orbit between the time since periapsis and the distance r from the central body. Equation (A6) gives the time of flight from a periapsis distance r_P out to a destination distance r . We can rewrite that equation in terms of the original circular speed v₀ and period T₀ = 2πr₀/v₀,

2. Elliptical segments (2- and 3-impulse escapes)

     For the direct impulse, there are no intermediate orbits so the total travel time is just the hyperbolic segment time t_hyp(r₀,r). For the Oberth and Edelbaum transfers, Kepler’s third law provides the time spent on the elliptical segments. The transfer time t_ell(r_A, r_P) between apoapsis and periapsis is half the period of an orbit of semi-major axis (r_A + r_P)/2, as seen in Fig. 1,

     To obtain the total travel time for the two-impulse Oberth maneuver, we must include the time for the infalling segment t_ell(r₀, r_in). For the three-impulse Edelbaum maneuver, we must include two elliptical segment transfer times t_ell(r₀, r_out) + t_ell(r_out, r_in).

3. Shortest travel time for Edelbaum escape

     Expressions for the total travel times for the three escape strategies are summarized in Table I. These are plotted in Fig. 3 as a function of the Edelbaum maneuver “swing out” radius r_out, using the same parameters as for Fig. 2 and a destination distance r = 200r₀. Clearly for this case there is an optimal swing-out radius r_out ≈ 2.5r₀ for the Edelbaum transfer to minimize the travel time. For larger values of r_out, the additional travel time in the first elliptical segment of Fig. 1(c) offsets the Oberth effect’s advantage of gaining a larger v_∞ for the same overall Δv.

     There is no tractable closed-form solution for the time-minimizing r_out as a function of destination distance r and mission constraints Δv_Ed and r_in. Instead, students can use Eq. (17), followed by Eqs. (19) and (20), to evaluate travel times for a range of values of r_out given the values of the other parameters. As the destination distance r increases, the time-optimal value of rout also increases, since the elliptical segments take up a smaller fraction of the overall trip duration, while the larger v_∞ reduces the remaining time spent on the hyperbolic segment.

4. Travel times compared

     Despite their slow starts, spacecraft executing Oberth and Edelbaum escapes can overtake a spacecraft on a direct escape and arrive at a distant destination sooner, as shown in Fig. 3. As discussed in Sec. III.6, a spacecraft on either of the 2- or 3-impulse trajectories can even overtake a fictional rocket that experiences no gravitational attraction to the central body. The travel time t_nog for this putative “no gravity” rocket is simply the length of the linear segment from r₀ to r divided by its constant v_∞ = v₀ + Δv,

and is shown as the dotted gray line in Fig. 3.

V. DISCUSSION

     For a spacecraft that can barely escape the central body (v_∞ ≈ 0), Fig. 2 shows that a single-impulse direct transfer is most efficient in terms of Δv. This is also the case for a spacecraft in a low orbit that cannot approach the central body any closer than its original orbital radius r₀. For a high-speed escape from a high circular orbit, the Oberth and Edelbaum maneuvers can produce a larger v_∞ for a given Δv, as shown in Fig. 2. However, to obtain the benefit of reduced travel times, the destination distance must be much greater than the original orbital radius, since the Oberth and Edelbaum transfers require slow elliptical segments prior to the hyperbolic escape. A mission that calls for a high-speed escape from a lunar-like orbit around Earth (r₀ ≈ 60 Earth radii and v₀ ≈ 1 km/s), or from a high orbit around any planet or asteroid, can benefit from employing the Oberth or Edelbaum maneuver.

     Most mission concepts are constrained by the total Δv available from the fuel on board. If employing either the Oberth or Edelbaum transfer, the inner radius r_in of the intermediate orbit is limited by the radius of the central body (including its atmosphere), and in the case of the Sun, by heating and radiation considerations. (The Parker Solar Probe makes perihelion passes as close as 10 solar radii ≈ 0.05 AU, but an escaping spacecraft would only have to do so once.)

     To date, no mission has used an Oberth or Edelbaum transfer to send a spacecraft to the outer solar system. This is mainly because the “Oberth advantage” over direct escape works best for values of total Δv ≈ v₀ (≈ 30 km/s for a heliocentric orbit at r₀ = 1 AU), which is not currently attainable by chemical rockets. Some mission concepts have proposed hybrid gravity-assist/Oberth trajectories that use a carefully timed fly-by of Jupiter (r_out = 5.2 AU) to provide some of the retrograde Δv_Ed2 necessary to cause the spacecraft to fall in close to the Sun.

     Thus far we have restricted our analysis to coplanar orbits and impulses. However, if one desires to change the plane of the escape hyperbola, the Δv (and fuel cost) to do so is greatly reduced by applying the plane-change impulse simultaneously with the second, retrograde impulse of the Edelbaum transfer at r_out, where the spacecraft is moving slowest.

     An important application of the 3-impulse Edelbaum maneuver is in time reverse, for efficient arrival and capture into a chosen circular orbit when a spacecraft approaches a planetary body at high relative speed. For approaches with v_∞ > v₀, the optimal strategy is shown in Fig 1(c) by reversing the direction of the arrows, and consists of a braking impulse at periapsis (which could be partially achieved by repeated aerobraking passes) into an eccentric orbit, followed by a periapsis-raising prograde impulse, and a third, “circularization burn”.

     For student discussions, Table II summarizes pros and cons of the Edelbaum transfer compared to direct escape and gravity-assist (“slingshot”) trajectories. Students can explore how the functions in Figs. 2 and 3 change as they vary r_in, r_out, and total Δv to plan their own escape and approach maneuvers. They can also extend the analysis presented here to incorporate noncircular initial orbits, which can alter the choice of fuel-optimal transfer strategy. Trajectories can be simulated using software such as Systems Tool Kit (STK), NASA’s General Mission Analysis Tool, Orbiter, or Kerbal Space Program.

• 

      Space manifolds form the boundaries of dynamic channels to provide fast transport to the innermost and outermost reaches of the solar system. Such features are an important element in spacecraft navigation and mission design, providing a window to the apparently erratic nature of comets and their trajectories. In a new report now published on Science Advances, Nataša Todorović and a team of researchers in Serbia and the U.S. revealed a notable and unexpected ornamental structure of manifolds in the solar system. This architecture was connected in a series of arches spreading from the asteroid belt to Uranus and beyond. The strongest manifolds were found linked to Jupiter with profound control on small bodies across a wide and previously unknown range of three-body energies. The orbits of these manifolds encountered Jupiter on rapid time-scales to transform into collisional or escaping trajectories to reach Neptune's distance merely within a decade. In this way, much like a celestial highway, all planets generate similar manifolds across the solar system for fast transport throughout.

Navigating chaos in the solar system

     In this work, Todorović et al. used fast Lyapunov indicator (FLI); a dynamic quantity used to detect chaos, to detect the presence and global structure of space manifolds. They captured the instabilities acting on orbital time scales with the sensitive and well-established numerical tool to define regions of fast transport in the solar system. Chaos in the solar system is inextricably linked to the stability or instability of manifolds forming intricate structures whose mutual interaction can enable chaotic transport. The general properties can be described relative to the planar, circular and restricted three-body problem (PCR3BP) approximating the motion of natural and artificial celestial bodies. While this concept is far from being fully understood, modern geometric insights have revolutionized spacecraft design trajectories and helped build new space-based astronomical observatories to transform our understanding of the cosmos.

The dynamics of space manifolds that allow the grand tour of the solar system via an interplanetary transport network has also contributed to the transit mechanisms of the Jupiter-family comets (JFCs). The JFCs are the evolutionary products of trans-Neptunian objects that continue to evolve through the giant planet region as Centaurs and into the inner solar system. Cometary and asteroidal bodies occupying orbits in the region between Jupiter and Neptune and Centaurs are dynamic and unstable with lifetimes of only a few million years. Astrophysicists usually use vastly diverse time scales to model detailed dynamic pathways that connect different time zones of the outer solar system.

• 

Greeks and Trojans—the global structure of space manifolds

     Todorović et al. considered the short-term (100-year) evolution of massless test particles (TPs) located on orbitals between the main asteroid belt and Uranus. They presented the data in dynamic maps based on two widely used orbit integration packages ORBIT9 and REBOUND while developing a force model containing seven major planets from Venus to Neptune as perturbers alongside the Sun/Jupiter/test particle three-body system. Co-orbital asteroids known as "Greeks" and "Trojans" followed the same orbit as Jupiter but led or trailed the planet by an angular distance.

     The team computed the FLI (fast Lyapunov indicator) across 100 years for a large grid, where lighter regions represented orbits located on stable manifolds and darker regions represented those away from them. The researchers noted an emergence of a large "V-shaped" chaotic structure connected to a series of arches at increasing heliocentric distances and nearly following the Perihelion line of Jupiter. The stable manifolds led to chaotic motion due to complex interactions with the corresponding unstable manifolds. These manifolds were analytically highly complex. Furthermore, as expected, Jupiter was the dominant perturber of the system and responsible for the majority of the rich chaotic architecture—tracked all the way beyond Neptune.

• 

Rapid scattering and collisions, followed by the Centaur: Jupiter-family comets orbital gateway

     To understand the dynamics of manifold- and close-encounter physics in the system, Todorović et al. used software packages to accurately track evolutions through close approaches with Jupiter. Using Jovian-minimum-distance maps for the Greek and Trojan orbital configurations, the team showed how all orbits along the chaotic structures entered Jupiter's Hill sphere during the course of their evolution. To understand close encounter dynamics, the team investigated Lagrange equilibrium points (L₁ and L₂), which define positions in space where the gravitational pull of two large masses precisely equalled the centripetal force required for a small object to move with them. All close-encounter trajectories visited the neighborhood of either L₁ or L₂ Lagrange points, casting light on the poorly understood Greek-Trojan dichotomy of escaped Jupiter Trojan asteroids.

• 

     Among the test particles (TPs) approaching Jupiter, a few dozen directly collided and their Jovicentric distances became less than Jupiter's radius. Nearly 2000 TPs transitioned from bound elliptical orbits to unbound hyperbolic escape orbits as a result of manifold-induced encounters. The transitioning orbits then reached Uranus and Neptune within 38 and 46 years; the fastest test particles arrived in the Neptunian region under a decade. Scattering or collision with Jupiter was at least several orders of magnitude shorter than those previously reported. Todorović et al. next observed the path of comet 39P/Oterma based on previous work conducted more than two decades ago, where the comet closely followed the invariant manifold structures associated with L₁ and L₂. The work showed how the invariant manifolds were the true orbital gateway that seemed to influence the low-inclination orbits closer to the Lagrange points of outer planets.

• 

Outlook on chaotic transport

In this way, Nataša Todorović and colleagues reported manifolds that act across orbital timescales of several decades in this work, in contrast to the tens to thousands of millions orbital revolutions that are traditionally considered. Additional information through quantitative studies will provide deeper insights into the transport between the two belts of minor bodies and the terrestrial planet region. The team expect to combine these observations with theory and simulations to improve the existing understanding of celestial transport. The observed effect of Jupiter-induced, large-scale transport on a decadal time scale is no surprise, since space missions have been historically designed for Jupiter-assisted transport, including flybys of Voyager 1 and Voyager 2.

More information: Nataša Todorović et al. The arches of chaos in the Solar System, Science Advances (2020). DOI: 10.1126/sciadv.abd1313

• 

• 

A free-return trajectory is a trajectory of a spacecraft traveling away from a primary body (for example, the Earth) where gravity due to a secondary body (for example, the Moon) causes the spacecraft to return to the primary body without propulsion (hence the term free).

(ed note: Translation, if an Apollo lunar mission had an engine malfunction in the first half of the journey, the spacecraft would go sailing off into an eccentric Terran orbit and all the astronauts would die. But if it was using a free-return trajectory, the Lunar gravity would automatically sling the spacecraft back towards Terra with no engines needed and the astronauts could land on Terra Firma.)

Earth–Moon

The first spacecraft traveled using free-return trajectory on October 4, 1959 was Russian «Луна-3» (Moon-3). Then, free-return trajectories were introduced by Arthur Schwaniger of NASA in 1963 with reference to the Earth–Moon system. Limiting the discussion to the case of the Earth and the Moon, if the trajectory at some point crosses the line going through the centre of the Earth and the centre of the moon, then we can distinguish between:

• A circumlunar free-return trajectory around the Moon. The spacecraft passes behind the Moon. It moves there in a direction opposite to that of the Moon. If the craft's orbit begins in a normal (west to east) direction near Earth, then it makes a figure 8 around the Earth and Moon.
• A cislunar free-return trajectory. The spacecraft goes beyond the orbit of the Moon, returns to inside the Moon's orbit, moves in front of the Moon while being diverted by the Moon's gravity to a path away from the Earth to beyond the orbit of the Moon again, and is drawn back to Earth by Earth's gravity. (There is no real distinction between these trajectories and similar ones that never go beyond the Moon's orbit, but the latter may not get very close to the Moon, so are not considered as relevant.)

For trajectories in the plane of the Moon's orbit with small periselenum radius (close approach of the Moon), the flight time for a cislunar free-return trajectory is longer than for the circumlunar free-return trajectory with the same periselenum radius. Flight time for a cislunar free-return trajectory decreases with increasing periselenum radius, while flight time for a circumlunar free-return trajectory increases with periselenum radius.

Using the simplified model where the orbit of the Moon around the Earth is circular, Schwaniger found that there exists a free-return trajectory in the plane of the orbit of the Moon which is periodic: after returning to low altitude above the Earth (the perigee radius is a parameter, typically 6555 km) the spacecraft would return to the Moon, etc. This periodic trajectory is counter-rotational (it goes from east to west when near the Earth). It has a period of about 650 hours (compare with a sidereal month, which is 655.7 hours, or 27.3 days). Considering the trajectory in an inertial (non-rotating) frame of reference, the perigee occurs directly under the Moon when the Moon is on one side of the Earth. Speed at perigee is about 10.91 km/s. After 3 days it reaches the Moon's orbit, but now more or less on the opposite side of the Earth from the Moon. After a few more days, the craft reaches its (first) apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, and there is a gravitational interaction. The craft passes on the near side of the Moon at a radius of 2150 km (410 km above the surface) and is thrown back outwards, where it reaches a second apogee. It then falls back toward the Earth, goes around to the other side, and goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved almost half an orbit and is again directly over the craft at perigee.

There will of course be similar trajectories with periods of about two sidereal months, three sidereal months, and so on. In each case, the two apogees will be further and further away from Earth. These were not considered by Schwaniger.

This kind of trajectory can occur of course for similar three-body problems; this problem is an example of a circular restricted three-body problem.

While in a true free-return trajectory no propulsion is applied, in practice there may be small mid-course corrections or other maneuvers.

A free-return trajectory may be the initial trajectory to allow a safe return in the event of a systems failure; this was applied in the Apollo 8, Apollo 10, and Apollo 11 lunar missions. In such a case a free return to a suitable reentry situation is more useful than returning to near the Earth, but then needing propulsion anyway to prevent moving away from it again. Since all went well, these Apollo missions did not have to take advantage of the free return and inserted into orbit upon arrival at the Moon.

Due to the landing-site restrictions that resulted from constraining the launch to a free return that flew by the Moon, subsequent Apollo missions, starting with Apollo 12 and including the ill-fated Apollo 13, used a hybrid trajectory that launched to a highly elliptical Earth orbit that fell short of the Moon with effectively a free return to the atmospheric entry corridor. They then performed a mid-course maneuver to change to a trans-Lunar trajectory that was not a free return. This retained the safety characteristics of being on a free return upon launch and only departed from free return once the systems were checked out and the lunar module was docked with the command module, providing back-up maneuver capabilities. In fact, within hours after the accident, Apollo 13 used the lunar module to maneuver from its planned trajectory to a free-return trajectory. Apollo 13 was the only Apollo mission to actually turn around the Moon in a free-return trajectory (however, two hours after perilune, propulsion was applied to speed the return to Earth by 10 hours and move the landing spot from the Indian Ocean to the Pacific Ocean).

Earth–Mars

A free-return transfer orbit to Mars is also possible. As with the Moon, this option is mostly considered for manned missions. Robert Zubrin, in his book The Case for Mars, discusses various trajectories to Mars for his mission design Mars Direct. The Hohmann transfer orbit can be made free-return. It takes 250 days in the transit to Mars, and in the case of a free-return style abort without the use of propulsion at Mars, 1.5 years to get back to Earth, at a total delta-v requirement of 3.34 km/s. Zubrin advocates a slightly faster transfer, that takes only 180 days to Mars, but 2 years back to Earth in case of an abort. This route comes also at the cost of a higher delta-v of 5.08 km/s. Zubrin claims that even faster routes have a significantly higher delta-v cost and free-return duration (e.g. transfer to Mars in 130 days takes 7.93 km/s delta-v and 4 years on the free return), and thus advocates for the 180-day transfer even if more efficient propulsion systems, that are claimed to enable faster transfers, should materialize. A free return is also the part of various other mission designs, such as Mars Semi-Direct and Inspiration Mars.

However it should be noted that, travel duration (to Mars or back to Earth) and delta-v requirement depend on the departure year (eg. 2020 or 2022 or so on). 2-year-free-return means from Earth to Mars (aborted there) and then back to earth all combine total is 2 years (0.5 yrs + 1.5 yrs). If entry corridor to Mars is limited (eg. +/- 0.5 deg entry with <9km/s speed as in the reference), 2-year-return is not possible for some years and for some years, delta-v kick of 0.6km/s to 2.7km/s at Mars may be needed to reach back to Earth.

NASA published the Design Reference Architecture 5.0 for Mars in 2009, advocating a 174-day transfer to Mars, which is close to Zubrin's proposed trajectory. It cites a delta-v requirement of approximately 4 km/s for the trans-Mars injection, but does not mention to the duration of a free return to Earth.

Abstract

     The use of electric propulsion for Mars has been explored since the 1970’s when men looked to travel beyond the moon. The use of electric propulsion has been recommended in several studies as a low-risk, lower cost approach to the robotic Mars sample return missions. Electric propulsion has been evaluated for delivery of Mars cargo using power systems order of magnitude beyond state-of-the-art. Electric propulsion has also been considered for fast transits to Mars supporting manned exploration activities. Results of generalized electric propulsion transits form Earth to Mars are presented. Trades are presented as a generalized assessment based on spacecraft mass-to-power ratio, trip time, and propulsion system performance including variable and constant specific impulse, and efficiency.

• 

II. Fast Transits

     For the purpose of these trades, a fast transit is any transit faster than the equivalent impulsive low energy transit. NASA has flown several missions to Mars using an opposition trajectory; roughly that of a heliocentric Hohmann transfer to Mars. These low energy transits take approximately 250 days and 180 degrees transit around the sun. Therefore all trades are for only Type I trajectories; less than 180o of heliocentric transit. Example plots of Type I impulsive trajectories are shown in figure 1. In practice, there is a very small penalty, for Mars direct entry missions, to reduce the transit time closer to 200 days. The transfers for Mars Odyssey, Spirit and Opportunity trajectories were 200, 208, and 202 days respectively. The penalty to arrive at Mars much faster than 200 days, however; quickly becomes impractical using impulsive propulsion options.

• 

     The optimal ΔV for a given time of flight fluctuates over the 15 year Earth-Mars cycle due to the eccentricity and inclination of Mars’ orbit. Previous studies have evaluated the ΔV sensitivities and various concepts for alternative architectures with reduced transfer times using impulsive trajectories. While these are beyond the scope of the study, non-electric solutions are not considered viable below 120 days.

     NASA completed several propulsion trades for human Mars missions in the development of the Design Reference Architecture (DRA). The NASA baseline departure is from a 407 km circular orbit using a two-burn Earth escape, to reduce gravity losses, and has transfer times from 175 to 225 days dependent on the launch opportunity. At Mars, the vehicles are inserted in a 1-sol orbit (250 km x 22,793 km) with propulsive insertion for crewed vehicles and both aerocapture and propulsive capture for the cargo vehicles. The crewed Mars Transfer Vehicle (MTV) has an initial mass to low Earth orbit of 356.4 t with a “payload” of 62.8 t and total dry mass fraction of 43%. A high elliptical parking orbit for departure is a higher performance alternative for a reduced departure ΔV. Because chemical propulsion transits are not the focus of these analyses, only reference, impulsive departures were assumed for optimistic results and simplified analysis. The architecture team baselined a nuclear thermal rocket (NTR) with an I_sp capability of 875 - 950s. An optimistic assumption of 1,000 s specific impulse was used for the impulsive comparison analyses. Figure 2 provides the dry mass fraction for an Earth-to-Mars transit using the baseline LEO departure, a high elliptical departure, and an aero Mars Orbit Insertion (MOI) entry from high elliptical departure as a function of transit time.

• 

The total delivered dray mass fraction is simply calculated by Tsiolkovsky’s “rocket” equation, equation (1). Even with optimistic assumptions, an NTR does not have practical delivered mass fractions for transfer times shorter than 2 - 3 months. The high fidelity DRA mission analysis results limit the NTR capability to a minimum potential transfer time of approximately 140 days for the optimal single launch opportunity of 2035.

A. Variable vs. Constant Specific Impulse

     Extensive Copernicus calculations were conducted to quantify the effect of the variable specific impulse for the Earth-Mars transits. Parametric trades included varying the transfer times from Earth to Mars from 30 – 250 days and jet power values between 50 kW to 500 MW. For each case, the initial mass and departure date were optimized to produce the maximum delivered mass. The trades were constrained to a specific impulse range from 2,000 s to 10,000 s. To simplify the parametric trades, the spacecraft was assumed to depart the Earth and arrive at Mars with a V_∞ of 0 km/s.

     The first results, shown in figure 3, are the dry mass fractions delivered as a function of transfer time. For the variable specific impulse, the delivered dry mass fraction is only dependent on transfer time, and is valid for all power levels. The constant specific impulse cases tended to optimized to the lower I_SP boundary of 2,000 s, especially for the very short transfer times. The variable I_SP will also operate at the minimum constrained I_SP near both Earth and Mars, but can operate at a higher I_SP when the spacecraft is transitioning from accelerating away from the Earth and performing the rendezvous with Mars. The ability to leverage both high thrust when necessary and higher specific impulse when there are minimal maneuver inefficiencies results in propellant savings for the variable specific impulse option. In all cases, the variable I_SP outperforms the constant I_SP. Also, the advantage of variable specific impulse increases for the most stressing cases; the very fast transfer times. For example, a 40 day transit with constant specific impulse can only deliver a 10% dry mass fraction while a variable specific impulse transit can deliver a 22% dry mass fraction. This is compared to a 250 day transit that can deliver a 66% versus a 68% dry mass fractions for constant and variable specific impulse cases respectively. Figure 3 illustrates that the transfer time increase required to match performance is significant for the very short transfers; a 70 day variable I_SP transit is comparable to a 100 day constant I_SP transit with respect to mass fraction.

• 

     Increasing power only has a negligible effect on the relative performance because, as will be shown later, the ΔV is primarily dependent on the transfer time with a secondary effort due to the thrust-to-power ratio of the propulsion system. The trades are optimized to produce the maximum final delivered mass for a given transfer time; adding additional power only linearly increases the absolute delivered mass capability. Therefore, the critical comparison is the advantage that can be observed in the delivered mass-to-power ratio, system alpha, for the transits. The delivered mass-topower ratio is defined as the total dry mass that can be delivered to the final orbit for a given power level. Figure 4 provides a comparison of delivered mass-topower ratios for the variable and constant specific impulse. Again, the variable specific impulse always outperforms the constant I_SP solution, but the advantage is relatively small until the transit times become very short. As the transit times fall below 40 days, the comparison begins to diverge; increasing the benefit of a variable specific impulse system. For a 30 day transit, the variable I_SP delivered mass-to-power ratio is more than double the constant I_SP ratio.

• 

B. System Trades

     The majority of the trades are from the baseline LEO departure, using the electric propulsion to escape, performing the transit to Mars, arriving with a V_∞ of 0 km/s, and then spiraling down to Phobos. Data is presented with respect to total system mass-to-power ratios from these four mission milestones: Initial Mass at LEO (IMLEO), Earth escape, Mars Arrival, and Phobos arrival. Note that for the accelerations and mass fractions associated with the fast transits, there is negligible performance difference between a low Mars circular orbit and a Phobos rendezvous at altitude of 5,981 km.

• 

• 

• 

• 

• 

• 

Figure 5. Trajectories to from Earth to Mars for 200 through 25 days.

---

Rendezvous

     The rendezvous mission from LEO to Phobos for various transfer times is shown in figure 5. The transfer time refers to the time between the Earth escape and the arrival at Mars with a V_∞ of 0 km/s. There is additional time associated with the spiral phase from LEO to escape and from Mars arrival to Phobos. The trades are conducted without a duty cycle constraint. From figure 5, the thruster is operating for the majority of the transfer time until the mission durations get very short. For the lower specific impulse and lowest transfer times, the thruster operating time is at its lowest. The thruster operating time as a function of specific impulse for several mission times is shown in figure 6. Figures 5 and 6 show for longer duration missions, the thruster is operating nearly the entire mission. For the shorter duration missions, the thruster accelerates the spacecraft and then has a significant coast period prior to thrusting to rendezvous with Mars.

• 

     It is also important to note that the ΔV changes significantly with trip time, but also to a lesser extent with thrust-to-power ratio. Figure 7 provides the ΔV versus specific impulse and transfer time. The specific impulse dependency is based on a constant efficiency resulting in a lower thrust for the same power. The ΔV and I_SP can be used to calculate the mass fraction through Tsiolkovsky’s equation, but is calculated directly from the trajectory performance output; shown in figure 8.

• 

• 

• 

• 

     The two columns from figure 8 are results assuming either staging of the spacecraft at escape or departure from LEO without staging. Because of the large mass fraction required to raise the vehicle from a low-Earth-orbit to escape, one may prefer to stage the spacecraft. For a crewed mission, it is likely that the vehicle would be raised near escape, potentially into a high elliptical orbit and wait for a crew module to dock with the spacecraft prior to escape. For this scenario, the dry mass fraction delivered from escape is improved by the ΔV savings over the LEO departure. This is shown in figure 9 for all specific impulse and transfer times evaluated. At 2,000 s specific impulse, just over 40% of the departure mass is delivered to Phobos, but 57% of the mass from escape is delivered to Phobos on a 200 days transit.

• 

• 

     The key parameters for mission design for chemical missions are typically the ΔV and specific impulse to characterize the spacecraft in terms of delivered mass performance. For electric propulsion missions, the key parameters are often the thrust-topower ratio of the propulsion system and the spacecraft power-to-mass ratio. The thrust-to-power ratio has a secondary influence on the mission ΔV and the power-to-mass ratio, i.e. P/m ≈ F/m = a, has a strong influence on the achievable mission trip time. Because the trades are conducted with fixed trip times and propulsion system thrust-to-power ratios, the key performance parameter that can be derived from the analyses is the delivered mass-to-power ratio, alpha. Again, because we are fixing trip time and thruster performance, the resulting parameter is the delivered mass-to-power ratio. The delivered alpha is illustrated in figure 10. Figure 10 is based on a 2,000 s I_SP propulsion system and the power is jet power.

• 

     Figure 10 shows that for every kW of spacecraft power available, a 2,000 s propulsion system can deliver 111 kg to Phobos given a 200 days transfer, 54 kg for a 150 day transfer, 18 kg for a 100 day transfer, etc. Figure 10 also provide the start mass alpha required to deliver those masses. Therefore a 1 kW, 2000 s propulsion system would require a start mass of 273 kg in LEO, it would have a mass of 197 kg at escape, and 120 kg when it arrives at Mars to eventually deliver 111 kg to Phobos. Figure 10 highlights the challenge of very fast transits to Mars; as the trip time decreases, the mass that can be delivered for a given power also decreases exponentially. This is expected because of the ΔV increase as was shown in figure 7. Figure 11 provides the delivered alpha capability for various specific impulse capabilities to Phobos. As will be shown later, figure 9 and 11 can be used for performance trades and feasibility assessments for any payload and trip time to Mars.

     Up until now, all data is based on jet power; assuming each kW of power is available for either thrust or specific impulse following equation (2) assuming 100% efficiency. This is convenient for analysis, but it does not lend itself for evaluating the power and propulsion system requirements based on realistic efficiencies.

• 

• 

     When propulsion system efficiency is addressed, the delivered mass capability for a given power will decrease; linearly if thrust is held constant. The efficiency is the total percentage of electric power generated that is converted into jet power. Losses include everything from power conditioning, ionization losses, beam divergence, etc. Figures 12 - 15 provides the deliver mass performance for a 2,000 s, 3,000 s, 4,000 s, and 5,000 s propulsion system at various efficiencies.

• 

• 

• 

• 

• 

• 

• 

• 

     The data clearly shows two competing constraints, mass fraction and the delivered mass-to-power ratio. If we want to have any appreciable payload at very short transfer times, we desire lower specific impulse option. If we want to have practical dry mass fractions, we desire a higher specific impulse option.

Direct or Constrained Entry Velocities

     One method to greatly reduce the mission ΔV is to allow for a direct entry into the Mars atmosphere. This would allow the propulsion system to accelerate the vehicle to a very high velocity without an arrival ΔV penalty. However, the spacecraft must have some method to reduce velocity at arrival. Crewed missions during the Apollo program experienced entry velocities on the order of 11 km/s at Earth; unmanned entry vehicles for science missions have been designed for up to 15 km/s. Analysis for crewed missions to Mars recommended constraining the entry velocity to 10 km/s; corresponding to a V_∞ of 8.65 km/s at a 300 km entry altitude. Unfortunately, as shown in figure 16, the direct entry velocities greatly exceed the human rated limit for transfer times less than 100 days. Therefore, the fast transits to Mars must leverage the propulsion system to slow down before entering the atmosphere. Nevertheless, the potential for mass savings should not be ignored entirely. For a 40 day transfer, the rendezvous transit ΔV is approximately 50 km/s, but approaches only 26 km/s if direct entry was tolerable. Figure 17 provides the delivered alpha using a direct entry and a comparison to a rendezvous. If a direct entry were tolerable, the performance for transfer times under 50 days appears far more practical than the rendezvous solutions.

• 

• 

• 

     Methods used by scientific spacecraft include entering into a very high elliptical orbit and use aerobraking to reduce the energy of the orbit without onboard propulsion. Unfortunately, the time of the aerobraking sequence would exceed the transfer time to Mars and eliminate any safety benefit achieved by the rapid transit. An aerocapture system is also impractical for such high entry velocities. The human body’s tolerance to vibration is even lower than a directed g-force. Staying within the atmosphere long enough to bleed off 20 - 30 km/s is not something that can be achieved within the tolerances of the human body. It is possible to use the electric propulsion system to constrain the arrival velocity and save approximately 8.5 km/s. The payoff for a constrained entry velocity must be traded with the mass increase imposed on the entry capsule and also the risk increase from a direct entry to Mars without an opportunity to assess the landing site and conditions before entry.

C. Solar vs. Constant Power

     The analysis shown previously is based on having a constant power available to the thrusters. This could be done using a nuclear power source, or by sizing a solar array such that the power fall-off never drops below the propulsion system input power. For missions to Mars, the power drop-off is relatively reasonable. Mars aphelion is only 1.67 AU, and therefore the power drop-off is estimated at approximately 1/2.78. As was shown in figure 5, the majority of the mission ΔV is during the initial departure from the Earth, when the spacecraft is closest to the sun. There is a penalty for the power degradation, but a solar solution may still provide a higher near-term performance than a nuclear alternative. Figure 18 illustrates a trajectory using SEP for a 50 day transit to Mars. The thruster does operate longer during the arrival arc because less power is available. Because the spiral time scales with power, it was initially expected that the spiral to low Mars would be a significant detriment for SEP on crewed missions, however, if the lower delivered alpha is tolerable, the spiral time is not significantly longer than the nuclear option because the arrival mass is lower. The delivered alpha is approximately 75% of the constant power solution; so depending on the alpha of the solar power system, it may not be practical to oversize the solar array. Either the power system will be oversized at 1 AU or the propulsion system with be undersized at 1 AU. The power and propulsion system alphas can be traded with the delivered alpha to assess the benefit of over sizing the solar array or tolerating the performance decrement.

• 

IV. Results

     Mass fraction and system limitations have been traded to illustrate near-term and far-term viable solutions for fast transits to Mars. Analyses are “spreadsheet” level using generalized system assumptions and not full spacecraft design. Several feasibility cases have been run, and those results are provided in the appendix. A few cases are shown using tables 1 and 2. The values in the tables represent the remaining mass-to-power ratio available for payload; calculated by equation (3). The results are based on various system allocations that are dependent on either the power level or the mass fraction of the launched mass. For example, the power, propulsion, and potentially thermal subsystems can be scaled with power, while structures and propellant tanks are subsystems that may scale as a fraction of the launched mass.

Example Case:

     Assume a 60% efficient propulsion system that operates at 5,000 s specific impulse and has an alpha of 0.5 kg/kW and a power generation system alpha of 0.5 kg/kw, a tankage fraction of 10%, and structure, communications, thermal control, etc. are another 10% of the dry mass. The totals for non-payload subsystems are 1 kg/kW and 20% of the dry mass allocation. If we want to evaluate a 100 day transfer staged at escape to Phobos, it has a delivered alpha of 6.5 kg/kW and a delivered dry mass fraction of 62%. The payload capability is:

Then for a payload mass requirement, MPayload, the spacecraft allocation can be simply calculated as:

• 

     Therefore, if we wanted to send a payload to Mars of 50,000 kg with this spacecraft, we would need a power system capable of providing 14.66 MW to a spacecraft with a launch mass of 153,350 kg including 58,020 kg for propellant, and 45,330 kg for the remaining dry mass: 14,660 kg for the power and propulsion subsystems, and 30,670 kg for propellant tanks and remaining subsystems.

     If we try to send the same 50,000 kg payload using a variable Isp system; α_Delivered = 11.16 kg/kW and MFDelivered = 41%, and follow equation 3 we calculate a value for α_Payload of 4.72 kg/kW. This translates to a power and launch mass requirement of 10.6 MW and 288,470 kg respectively. This highlights an important note for using variable specific impulse results based on this methodology; because the trades are based on maximizing the delivered alpha, the variable specific impulse case will always use less power to deliver the same payload mass for a fixed transfer time, but that may come at a higher launch mass. One could employ a methodology to minimize IMLEO, but the purpose of the chosen methodology was to determine the maximum delivered mass-to-power ratio to understand the propulsion and power limiting case. In any methodology the variable specific impulse outperforms the constant specific impulse.

     Table 2 provides the 60% efficient propulsion system solution with no dry mass dependence on either power or launched mass. Table two provides the limiting cases for a 60% efficient system. The values in red signify that the delivered mass fraction is less than 30%. Figure two illustrates that there is an exponential decrease in performance as trip time is decrease, that the highest delivered mass-to-power ratio is achieved at low specific impulse, but that performance comes at the cost of a high propellant mass fraction.

• 

V. Technology Status and Predictions

     While beyond the scope of these analyses, a brief summary on technology status to meet the performance requirments to enable fast transits to Mars is provided. The summary below is not intended to be a comprehensive assessment of the available and required technologies, rather provide a reference point for scaling a few of the key technologies that will be required to enable transfer times to Mars in less than 100 days.

A. Propulsion

     There are several options for electric propulsion systems as the available power scales from 10s of kW to 10s of MW. In many cases the same propulsion system will not be ideal over the entire range of input powers. For example, Hall thrusters may be suitable up to a few hundred kilowatts, but the system may become cumbersome at several megawatts. Also, a magnetoplasmadynamic (MPD) thruster may not make sense for a 100 kW system, but an MPD, ELF, or VASIMR may offers system benefits at > 100 KW, with VASIMR having an added advantage of variable I_SP.

     Though it is beyond the scope of this analysis, a survey of propulsion technologies is relevant to near-term feasibility. Therefore, a few propulsion options are presented in table 3 from an Air Force Research Laboratory status of high power electric propulsion activities.7 For relative comparison, a 200 kW thruster system is shown based on available scaling data. The tables has been updated as new data has become available. These specific mass estimates do not include auxiliary hardware, thermal management, etc. that would increase the system specific mass. Aside from thermal management, there is an expectation that a propulsion system scaled at the megawatt level could be less than 1 kg/kW. Efficiencies in table 1 are thruster efficiencies and does not account for power conditioning and any propulsion related subsystems. Optimistically, total system efficiencies of 40% - 80% may be possible for high power EP systems.

     While the advantage of variable to constant specific impulse may not be a significant driver, there is a distinct advantage in technology development for an evolutionary path of power and propulsion system implementation. Near-term missions are likely to be in the gravity well of the Earth with power levels below 100kW. These systems typically optimize to lower specific impulse. As the power system increases; so will the optimal specific impulse for the mission. While an individual mission may not see a benefit from the variable specific impulse, working out the technology challenges of a system that can vary specific impulse with significant design changes does provide a larger return on investment for intermediate steps. For example, at 50 kW, investments may focus on Hall technology ideal for 2,000 - 3,000 s specific impulses. An intermediate 300 kW system may employ gridded-ion engine technology for 4,000 - 6,000 s operation. When multi-MW power systems become available, the thruster technology may shift again to MPDs, PITs, ELFs, VASIMR, etc. Certainly system level integration and scaling challenges will advance the state-of-the-art, but the thruster challenges will likely require parallel investment of independent technologies. Solving a challenge such as thruster lifetime on a Hall thruster may not be directly applicable to the future systems.

     Also, if one does consider the possibility of multi-hundred megawatt power systems with near zero mass, the divergence of performance for trip times is likely to hold for missions beyond Mars. As the mission performance pushes the boundary of a practical ΔV that can ever be achieved with any system, a variable specific impulse system may offer double the performance capability.

• 

B. Nuclear Power

     As power systems increase to hundreds of kilowatts and multi-megawatts, nuclear power must be considered. The highest fidelity near-term predictions based on NASA studies from the Prometheus project predicted nuclear power system alpha 30 – 40 kg/kW. The baseline power system produced 200 kW with a core mass of 1,569 kg for a core alpha of 7.8 kg/kw before you include the shielding, control system, and power conversion system. The complete reactor module had a system alpha of approximately 16.5 kg/kW. This reactor module is for the power production, but does not include the mass of the heat rejection system; another 12.8 kg/kW. This brings the JIMO total power system alpha to approximately 30 kg/kw. This system alpha is before margin and is still more than an order of magnitude higher than required for transfer times <90 days.

     It is not expected that turbo-Brayton reactor systems will fall below 10 kg/kW, even for larger power output systems. However, alterative lower-TRL options exist including a Magnetohydrodynamic (MHD) system with a projected alpha approaching 1 kg/kW. Even lower system alpha systems have been proposed using Fissioning Plasma Core Reactor (FPCR) with MHD power generation. The high operating temperatures of an MHD generator (efficiencies near 30%) can be combined with heat recovery steam cycles for overall system efficiencies approaching 60%. These studies highlight the potential for significant improvement in system alphas, but provide little insight into the often over looked power management and distribution (PMAD) system and thermal management for the waste heat generated by the power and propulsion systems.

     A broad NASA study predicted the 200 kW class nuclear power specific masses would approach 30 kg/kW, consistent with the higher fidelity Prometheus results, and also predicts that 100 MW class systems could approach 8 kg/kW including the often over looked power PMAD, radiators, and power conversion redundancy for a 10 year mission. Using more aggressive assumptions and shorter lifetime, the study predicts 6 kg/kW may be achievable. The study predicted the power conditioning and control system to have an alpha of 2 kg/kW and the distribution lines with another 0.5 kg/kW. Data from several studies starting with human Mars mission studies in 1996 through the present from throughout NASA has been compiled and is shown in figure 19. The data is consistent with expectations that Brayton technology will be limited near 10 kg/kW while farther term technology may approach power system alphas near 1 kg/kW for 10s – 100s of MW power.

• 

C. Solar Power

     The potential does exist to use solar power for exploration within the inner solar system. Solar array technology continues to improve in cell efficiencies and in system alpha. The expanded use of electric prolusion, and the synergistic benefit of electric propulsion and increasing power availability is driving commercial and government towards higher power systems. The SOA solar arrays are currently 15 - 20 kg/kW. The DARPA Fast Access Spacecraft Testbed (FAST) program has progressed the SOA has with near-term projected goals of 8 kg/kW at 1 AU. NASA studies have far-term predictions approaching 3.3 kg/kW including the array, based on Stretched Lens Array Square Rigger (SLASR) technology with advanced cells, gimbals, booms, power cabling, etc. A point design for a 232kW system had a current best estimate (CBE) system mass of 781 kg. Using an advanced cell, a 100 kW End-of-Life, after 10 years at GEO, stretched lens array design has a predicted mass performance of 1.85 kg/kW.

     Figure 20 illustrates the trend of spacecraft power available over time. The International Space Station (ISS) represents the largest space power asset at over 100 kW. The trends of available power illustrate a doubling of power approximately every four years. Base on this trend, a 1 MW, 10 MW and 100 MW solar power system would be available in 13, 27, and 41 years respectively. This does not preclude the possibility of revolutionary improvements in technology, but the evolutionary advancements will not have 100 MW class solar power systems available until after 2050.

• 

D. Heat Rejection

     Another challenge of very high power generation and propulsion is the waste heat generated by the systems. For use on the interplanetary transfer, assuming a view to deep space, the radiators have an area requirement of approximately 2.2 m²/kW of heat dissipation. The radiators for the International Space Station (ISS) have an area density of 14.64 kg/m². The ISS radiators include mass for deployment support structure. The fast transit electric propulsion vehicle would likely also require deployable radiators. During the NASA Capabilities, Requirements, Analysis, and Integration (CRAI) study, NASA estimated NASA radiators could obtain 10 kg/m² in the near-term with investment. The CRAI team also provided a long-term goal of approaching 2.5 kg/m² using advanced radiator materials and doubling the SOA heat transport capabilities. Therefore the near-term performance would be approximately 22 kg/kW and in the long-term reach 5.5 kg/kW. The majority of the heat waste heat is likely to come from the power generation. If we use the most optimistic assumptions, a 60% efficient power generator and an 80% efficient propulsion system, our total efficiency from thermal heat generated to jet power is only 48%. Therefore, for each kW of jet power, the spacecraft must dissipate 1.08 kW of thermal waste heat. In other words, the long range heat rejection system alpha goal almost 6 kg/kW of electric propulsion jet power. Relative to the performance goals of fast transit, even if the power and propulsion system were without mass, the mass of the heat rejection system alone would prevent the possibility of a transfer to Mars in less than two months. Note this challenge is exacerbated for the nuclear power option. A solar powered option may only need to dissipate 1 – 0.25 kW of waste heat for every 1 kW of jet power produced for 50% - 80% efficient propulsion systems. This would translate into an approximate heat rejection alpha as low as 5.5 – 1.4 kg/kW in the near-term and far-term respectively.

VI. Conclusion

     In 2002, it was reported that 39 transits to Mars may be possible with 200MW NEP system with variable specific impulse using a power system alpha below 1 kg/kW. Results indicate that this may be possible if the power, propulsion, PMAD, thermal control, etc. can approach a combined system alpha approaching 0.5 kg/kW. Variable I_SP may double the performance over constant ISP for trip times less than 40 days; alleviating the requirement closer to 1 kg/kW. In all cases, variable specific impulse can increase both delivered mass fraction and delivered alpha compared to fixed I_SP. State-of-the-art and near-term technology predictions indicate options may exist for a path to achieve very low alpha propulsion systems. However, the power system alpha is far from that necessary to achieve transfers less than 50 days, and would take a revolutionary technology advancement by an order of magnitude; not a evolutionary advancement in Brayton technology. Also, the PMAD and thermal control challenges associated with the tremendous power transmission and conditioning, in addition to waste heat dissipation orders of magnitude beyond those demonstrated, are not likely to be addressed in the near-term. The waste heat rejection system has a total alpha beyond those required for transfers less than 60 days, and there is no plan to achieve the alpha reduction required. Solar array technology may offer the lowest alpha achievable without significant investment in space based nuclear reactors. With advanced solar array technology, combined with high power propulsion system development, crewed missions to Mars may become practical for transfer times of a few months. Therefore, electric propulsion does the have the potential to perform transits to Mars faster than practical with the baselined NTR system. Crewed transits to Mars in a few weeks are not possible using any foreseeable technology available.

---

Appendix

• 

• 

---

• 

• 

---

• 

• 

• 

Question

      NICK T: On a porkchop plot for a given departure date (X axis), why are there two local energy minimums along the arrival date (Y-axis) for a transfer between Earth and Mars? The Hohmann transfer orbit seems to have one optimal solution, but the plot's two minimums seem fairly distinct in time but similar in energy; 15.0 to 15.5 and 15.5 to 16.0 in terms of C3L (something I only vaguely understand; KSP mostly deals in delta-V) (C3L is the delta-V of the Trajectory Injection)
     A NASA page about sending spacecraft to Mars refers to "Type 1" and "Type 2" transfer orbits, which clashes with a perhaps simple view that there should be a singular optimal Hohmann transfer orbit; the smallest ellipse tangent to two orbits. Granted, the planets' orbits are not perfectly circular, but the different transit times (red contours) for local minimums differ by almost a factor of two! (about 200 days and about 400 days) I'm not sure if I'm reading the plot correctly though, as there is a disconnect between the given Earth-Mars plot, and the prose on the page as well, which talks about 7 month (~200 day) or 10 month (~300 day) transfers.

---

Answer

What are porkchop plots?

     When the goal is to send a spacecraft from the Earth to another planet, it's not enough to reach the target planet's orbit. The vehicle has to meet the target planet itself. The amount of energy needed to accomplish this varies widely depending on departure and arrival dates. A porkchop plot is a graphical interplanetary mission planning tool that depicts as a contour plot the required energy as a function of departure date and arrival date. The energy needed by the launch vehicle is key in determining the feasibility of such a mission. A mission plan that requires more energy than a launch vehicle could possibly provide is not feasible. In addition to feasibility, a porkchop plot aids in planning key mission operations and in planning the optimal trajectory between the two planets. The plot shown in the question and replicated below show launch energy. It does not show the change in energy needed at arrival. Other porkchop plots do show this as a second set of contour lines.
     The original post asks three key questions:

• Why is there a gap in porkchop plots?
• Why don’t they use an optimal Hohmann transfer?
• Why are there two local energy minima?

     Before answering the above, it will help to explain how a porkchop plot is constructed.

How are porkchop plots constructed?

     As a mission planning tool, a porkchop plot makes certain simplifying assumptions with regard to reaching the target planet. Later on, more detailed analyses address those simplifying assumptions. The key simplifying assumptions used in making a porkchop plot are the patched conic approximation and impulsive maneuvers.
     These assumptions reduce the problem to one of finding Keplerian orbits about the Sun that take the spacecraft from the vicinity of the Earth to the vicinity of the target planet in the requisite amount of time. Finding such transfer orbits is the subject of Lambert's problem. A number of such transfer orbits might exist. I’ll denote the angle subtended by the departure point, the Sun, and arrival point as θ, with . The principal value of this angle will be between 0° and 180°, inclusive. For now I’ll ignore cases where θ is 0° or 180°. This means that the transfer plane is well-defined and that the number of solutions is finite.
     Lambert's problem does not have closed form solutions; a number of iterative techniques have been developed to find solutions. One solution, “the short way”, or “Type 1” transfers, has the change in true anomaly equal to θ as described above. Another solution, the “long way”, or “Type 2” transfers, has the change in true anomaly equal to 360°-θ. Other solutions may exist as well. For example, one way to transfer from Earth to Mars in 2.5 years is to make more than a complete orbit during the transfer. Porkchop plots typically only show the Type 1 and Type 2 solutions, and typically only show at most one of these two solutions for a given pair of departure and arrival dates. If one of the two solutions is close to optimal, the other solution will inevitably follow a retrograde path and thus will involve huge expenditures of energy. There’s no reason to show these highly sub-optimal solutions.

Why is there a gap in porkchop plots?

     The plot can be cleaned up further by removing cases where the better of the two solutions still involves huge energy expenditures. Huge energy expenditures are obviously going to result when the transfer time is very short or very long. A not so obvious place where this happens is when the angle subtended between the line from the Earth and Sun at departure and the target planet and Sun at arrival is nearly 180°. That the Earth and target planet have slightly different orbital planes means that the transfer plane will be nearly orthogonal to the planetary orbital planes when the transfer angle is close to but not equal to 180°. This makes the approach of having a maneuver at departure and a maneuver at arrival extremely expensive for those transfers that are close to 180°. Removing those very expensive transfers from view is what creates the gap in the porkchop plot.
     This excessive cost for near 180° transfers is to some extent an artifact of the approach used to create a porkchop plot. Adding a third maneuver enables the use of much smaller in-plane maneuvers at the start and end, with a small plane change somewhere along route. There’s a problem, however. This mid-course plane change would necessarily mean thrust from the spacecraft itself. This is undesirable. The spacecraft itself provides very little of the energy with the two burn approach. The energy for the Earth departure comes from the launch vehicle, and in the case of Mars, most of the energy for Mars arrival comes from aerobraking. It’s better to fold that plane change into the maneuvers at departure and arrival so as to keep the thrust needed by the spacecraft down to a minimum.

Why don’t they use an optimal Hohmann transfer?

     An optimal Hohmann transfer doesn’t exist. Hohmann transfers are in-plane maneuvers that transfer from one circular orbit to another that share a common orbital plane. Planetary orbits are slightly inclined with respect to one another and are elliptical rather than circular. Generalizing the concept of a Hohmann transfer to that of a 180° transfer, in most cases that 180° transfer doesn’t exist. When it does, that the orbits are not coplanar and that the orbits are not circular means that this 180° transfer is no longer optimal.

Why are there two local energy minima?

     There aren’t just two local energy minima. There are a countable infinite number of local minima. A porkchop plot only shows the first two. The other solutions take even longer than Type 1 and Type 2 transfers and are more sensitive to errors.

      Changing direction causes ΔV (change in velocity), often more than a change in speed. Compare the velocity vectors below. When going the same direction, the difference is 1 km/s. When at right angles the difference is 5 km/s. We know this from driving in traffic. Two cars going almost the same speed hit each other. If they’re in the same lane going the same direction, it’s a mild bump. If one car runs a red light and T-bones a car in cross traffic, the impact is serious:


     This is the strength of a Hohmann transfer orbit. Velocity vectors are pointing the same direction at departure as well as destination. No direction change is needed, only a speed change:


     Note the Hohmann transfer path moves 180 degrees about the sun:


     A Hohmann transfer assumes the departure and destination orbits are co-planar. But what if the destination orbit is inclined?

Orbit Planes and Spherical Trigonometry

     A plane passing through a sphere’s center cuts the sphere along a great circle. A group of planes all sharing a common point can be represented as great circles on a sphere. Since every orbit about the sun is a conic section having the sun as a focus, each orbital plane shares the sun as a common point. Representing the orbital planes as great circles is convenient. There are already a lot of theorems in spherical trigonometry which gives us a suite of tools for looking at angles between orbital planes.
     The shortest path (or geodesic) along a spherical surface between two points is an arc of a great circle. If we set the sphere’s radius to be 1, the arc length is also the angular separation in radians.
     A familar group of great circles are the longitude lines on a globe. The equator is the only great circle among the latitude lines. All the longitude lines are great circles passing through the poles.
     Let’s use the equatorial great circle to represent the departure plane. Recall the Hohmann transfer moves 180 degrees about the center. In this illustration, latitude and longitude for departure and destination is (0°, 0°) and (7°, 180°). The only great circle connecting these points is a polar orbit nearly 90° from the departure and destination planes! Big plane changes at departure and destination destroys the virtue of a Hohmann orbit.


     I’ve also tried to demonstrate this in this video:

• 

     The big delta V needed for large plane changes makes the ridge (gap between the two blobs) in a porkchop plot:


     Porkchop plots are drawn by doing iterations of various Lambert Space Triangles. Lambert iterations give polar transfer orbits when departure and destination longitudes differ by 180°.
     Does this mean Hohmann transfers are no good if the destination orbit’s inclined? No, the big plane changes can be avoided with a mid course plane change. Here is a broken plane transfer where a plane change burn is done at the ascending node:


     The line where the destination and departure planes intersect form the ascending and descending nodes. Starting in the departure plane and doing a plane change at the node avoids the two major plane changes. The departure and destination planes differ by an angle called i, for inclination.
     Changing a vector by an angle i takes dv of v * 2 * sin(i/2).


     The Vis Viva Equation tells us v = sqrt(μ(2/r - 1/a)). So v ranges from sqrt (μ((1-e)/(a(1+e)))) at aphelion to sqrt (μ((1+e)/(a(1-e)))) at perihelion. Let's look at a Ceres transfer orbit. An ellipse with a 1.88 a.u. semi major axis and eccentricity 0.47 will have speeds ranging from 36 km/s (at perihelion) to 13 km/s (at apohelion). Inclination's about 10.6 degrees. So plane change ranges from 36 km/s   * 2 * sin(10°/2) to 13 km/s   * 2 * sin(10°/2) or from 6.7 to 2.4 km/s. Is the a 2.4 km/s plane change at aphelion the best we can do?  No, it's possible to have less plane change expense.
     Launch is at the perihelion of an outbound Hohmann orbit. If the launch coincides with a node, the entire plane change can be done during earth departure or at arrival. Then the delta V entails a speed change as well as a direction change. Doing a single plane change/speed change burn saves delta V as shown by this diagram:


     Law of cosines tells us for a triangle a, b, c, a² + b² - 2ab cos(i) = c². In this case, i is the angle between a and b and c is the delta V needed from the combined plane change and speed change.
     At aphelion, a combined speed change/plane change only costs 0.76 km/s more than the speed change alone.
     When launching deep in earth’s gravity well, we enjoy an Oberth benefit. Ceres' gravity well lends a little Oberth benefit at the destination. If the line of nodes coincides with transfer orbit's line of apsides, plane change can cost as little as 0.52 km/s extra.
     This indicates as much plane change as possible should be made at departure and arrival. What sort of plane changes should we make to minimize the angle of the midcourse plane change?
     The fattest part of an orange slice is right in the middle:


     The angular separation at launch has to be some part of the orange slice. To minimize the angle between transfer plane and destination plane, the angular separation at launch should be in the middle. Having the transfer plane intersect the destination plane 90° from launch minimizes plane change angle.


     An object on an elliptical path moves slower as it moves further from the sun, so doing plane changes further out are cheaper. The 90º from launch is a minimum. There will be a larger plane change angle 100 degrees from launch, but velocity will be slower. Also plane change lessens as flight path angle increases. I hope to talk about this more when I have time.
     But for now I believe this shows that the Lambert iterations greatly exaggerates plane change expense for a Hohmann path where departure and destination points are 180° degrees apart. Most of that plane change expense can be eliminated by choosing a good place to do a midcourse plane change

     A PDF on Broken Plane Maneuvers Fernando Abilleria of NASA Jet Propulsion Laboratory

• 

The Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the Solar System that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through space are redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them, despite lacking an object to orbit. While they use little energy, the transport can take a very long time. Shane Ross has said "Due to the long time needed to achieve the low energy transfers between planets, the Interplanetary Superhighway is impractical for transfers such as from Earth to Mars at present

History

Interplanetary transfer orbits are solutions to the gravitational "restricted three-body problem", which, for the general case, does not have exact solutions, and is addressed by numerical analysis approximations. However, a small number of exact solutions exist, most notably the five orbits referred to as "Lagrange points", which are orbital solutions for circular orbits in the case when one body is significantly more massive.

The key to discovering the Interplanetary Transport Network was the investigation of the nature of the winding paths near the Earth-Sun and Earth-Moon Lagrange points. They were first investigated by Jules-Henri Poincaré in the 1890s. He noticed that the paths leading to and from any of those points would almost always settle, for a time, on an orbit about that point. There are in fact an infinite number of paths taking one to the point and away from it, and all of which require nearly zero change in energy to reach. When plotted, they form a tube with the orbit about the Lagrange point at one end.

The derivation of these paths traces back to mathematicians Charles C. Conley and Richard P. McGehee in 1968. Hiten, Japan's first lunar probe, was moved into lunar orbit using similar insight into the nature of paths between the Earth and the Moon. Beginning in 1997, Martin Lo, Shane D. Ross, and others wrote a series of papers identifying the mathematical basis that applied the technique to the Genesis solar wind sample return, and to Lunar and Jovian missions. They referred to it as an Interplanetary Superhighway (IPS).

Paths

As it turns out, it is very easy to transit from a path leading to the point to one leading back out. This makes sense, since the orbit is unstable, which implies one will eventually end up on one of the outbound paths after spending no energy at all. Edward Belbruno coined the term "weak stability boundary" or "fuzzy boundary" for this effect.

With careful calculation, one can pick which outbound path one wants. This turned out to be useful, as many of these paths lead to some interesting points in space, such as the Earth's Moon or between the Galilean moons of Jupiter. As a result, for the cost of reaching the Earth–Sun L₂ point, which is rather low energy value, one can travel to a number of very interesting points for a little or no additional fuel cost. But the trip from Earth to Mars or other distant location would likely take thousands of years.

The transfers are so low-energy that they make travel to almost any point in the Solar System possible. On the downside, these transfers are very slow. For trips from Earth to other planets, they are not useful for manned or unmanned probes, as the trip would take many generations. Nevertheless, they have already been used to transfer spacecraft to the Earth–Sun L₁ point, a useful point for studying the Sun that was employed in a number of recent missions, including the Genesis mission, the first to return solar wind samples to Earth. The network is also relevant to understanding Solar System dynamics; Comet Shoemaker–Levy 9 followed such a trajectory on its collision path with Jupiter.

Further explanation>

The ITN is based around a series of orbital paths predicted by chaos theory and the restricted three-body problem leading to and from the unstable orbits around the Lagrange points – points in space where the gravity between various bodies balances with the centrifugal force of an object there. For any two bodies in which one body orbits around the other, such as a star/planet or planet/moon system, there are three such points, denoted L₁ through L₃. For instance, the Earth–Moon L₁ point lies on a line between the two, where gravitational forces between them exactly balance with the centrifugal force of an object placed in orbit there. For two bodies whose ratio of masses exceeds 24.96, there are two additional stable points denoted as L₄ and L₅. These five points have particularly low delta-v requirements, and appear to be the lowest-energy transfers possible, even lower than the common Hohmann transfer orbit that has dominated orbital navigation in the past.

Although the forces balance at these points, the first three points (the ones on the line between a certain large mass, e.g. a star, and a smaller, orbiting mass, e.g. a planet) are not stable equilibrium points. If a spacecraft placed at the Earth–Moon L₁ point is given even a slight nudge towards the Moon, for instance, the Moon's gravity will now be greater and the spacecraft will be pulled away from the L₁ point. The entire system is in motion, so the spacecraft will not actually hit the Moon, but will travel in a winding path, off into space. There is, however, a semi-stable orbit around each of these points, called a halo orbit. The orbits for two of the points, L₄ and L₅, are stable, but the halo orbits for L₁ through L₃ are stable only on the order of months.

In addition to orbits around Lagrange points, the rich dynamics that arise from the gravitational pull of more than one mass yield interesting trajectories, also known as low energy transfers. For example, the gravity environment of the Sun–Earth–Moon system allows spacecraft to travel great distances on very little fuel, albeit on an often circuitous route.

Missions

Launched in 1978, the ISEE-3 spacecraft was sent on a mission to orbit around one of the Lagrange points. The spacecraft was able to maneuver around the Earth's neighborhood using little fuel by taking advantage of the unique gravity environment. After the primary mission was completed, ISEE-3 went on to accomplish other goals, including a flight through the geomagnetic tail and a comet flyby. The mission was subsequently renamed the International Cometary Explorer (ICE).

The first low energy transfer using what would later be called the ITN was the rescue of Japan's Hiten lunar mission in 1991. Another example of the use of the ITN was NASA's 2001–2003 Genesis mission, which orbited the Sun–Earth L₁ point for over two years collecting material, before being redirected to the L₂ Lagrange point, and finally redirected from there back to Earth. The 2003–2006 SMART-1 of the European Space Agency used another low energy transfer from the ITN. In a more recent example, the Chinese spacecraft Chang'e 2 used the ITN to travel from lunar orbit to the Earth-Sun L₂ point, then on to fly by the asteroid 4179 Toutatis.

      Wikipedia describes the Interplanetary Transport Network as "… pathways through the Solar System that require very little energy for an object to follow." See this Wikipedia article. They also say "While they use very little energy, the transport can take a very long time."

     Low energy paths that take a very long time? I often hear this parroted in space exploration forums and it always leaves me scratching my head.

     The lowest energy path I know of to bodies in the inner solar system is the Hohmann orbit. Or if the destination is noticeably elliptical, a transfer orbit that is tangent to both the departure and destination orbit. Although I think of bitangential transfer orbits as a more general version of the Hohmann orbit.

• 

     In the case of Mars, a bitangential orbit is 8.5 months give or take a month or two. Is there a path that takes a lot longer and uses almost no energy? I know of no such path.

L1 and L2

     The interplanetary Superhighway supposedly relies on weak stability or weak instability boundaries between L1 and/or L2 regions. Here is an online text on 3 body Mechanics and their use in space mission design. The authors are Koon, Lo, Marsden and Ross. Shane Ross is one of the more prominent evangelists spreading the gospel of the Interplanetary Super Highway.

     The focus of this online textbook is the L1 and L2 regions. From page 10:

• 

     L1 and L2 are necks between realms. In the above illustration the central body is the sun, and orbiting body Jupiter. L1 and L2 are necks or gateways between three realms: the Sun realm, the Jupiter Realm and the exterior realm.

     Travel between these realms can be accomplished by weak stability or weak instability boundaries that emanate from L1 or L2. From page 11 of the same textbook:

• 

My terms for various Lagrange necks

     First letter is the central body, the second letter is the orbiting body.

Earth Moon L1: EML1
Earth Moon L2: EML2

Sun Earth L1: SEL1
Sun Earth L2: SEL2

Sun Mars L1: SML1
Sun Mars L2: SML2

     Since I'm a lazy typist that is what I'll use for the rest of this post.

EML1 and 2

     I am very excited about the earth-moon Lagrange necks. They've been prominent in many of my blog posts. Here's a post entirely devoted to EML2.

     EML1 and 2 are about 5/6 and 7/6 of a lunar distance from earth:

• 

     Both necks move at the same angular velocity as the moon. So EML1 moves substantially slower than an ordinary earth orbit would at that altitude. EML2 moves substantially faster.

     It takes only a tiny nudge and send objects in these regions rolling about the slopes of the effective potential hills. Outside of the moon's influence they tend to fall into ordinary two body ellipses (for a short time).

     Here's the ellipse an object moving at EML1 velocity and altitude would follow if the moon weren't there:

• 

     In practice an EML1 object nudged earthward will near the moon on the fifth apogee. If coming from behind, the moon's gravitational tug can slow the object which lowers perigee.

     Here is an orbital sim where the moon's influence lowered perigee four times:

• 

     I've run sims where repeated lunar tugs have lowered perigees to atmosphere grazing perigees. Once perigee passes through the upper atmosphere, we can use aerobraking to circularize the orbit.

     Orbits are time reversible. Could we use the lunar gravity assists to get from LEO to higher orbits? Unfortunately, aerobraking isn't time reversible. The atmosphere can't increase orbital speed to achieve a higher apogee. And low earth orbit has a substantially different Jacobi constant than those orbits dwelling closer to the borders of a Hill Sphere.

     So to get to the lunar realm, we're stuck with the 3.1 km/s LEO burn needed to raise apogee. But once apogee is raised, many doors open.

     There are low energy paths that lead from EML1 to EML2. EML2 is an exciting location.

• 

     In the above illustration I have an apogee beyond SEL2. But by timing the release from EML2, we could aim for other regions of the Hill Sphere, including SEL1.

     Here is a sim where slightly different nudges send payloads from EML2:

• 

     See how the sun bends the path as apogee nears the Hill Sphere? From EML2 there are a multitude of wildly different paths we can choose. In this illustration I like pellet #3 (orange). It has a very low perigee that is moving about 10.8 km/s. And it got to this perigee with just a tiny nudge from EML2. Pellet # 4 is on it's way to a retrograde earth orbit. Most of the other pellets are saying good bye to earth's Hill Sphere.

     I am enthusiastic about using EML1 and EML2 as hubs for travel about the earth-moon neighborhood. But a little less excited about travel about the solar system.

We've left Earth's Hill Sphere. Now what?

     Recall that EML1 and 2 are ~5/6 and 7/6 of a lunar distance from the earth. SEL1 and 2 are much less dramatic: 99% and 101% of an A.U. from the sun. Objects released from these locations don't vary much from earth's orbit:

• 

     Running orbital sims gets pretty much the same result pictured above.

     Mars is even worse:

• 

     Are there weak instability boundary transfers leading from SEL2 to SML1? I don't think this particular highway exists.

     To get a 1.52 A.U. aphelion, we need a departure Vinfinity of 3 km/s. To be sure EML2 can help us out in achieving this Vinfinity. In other words we could use lunar assists to depart on a Hohmann orbit. But a Hohmann orbit is different from the tube of weak instability boundaries we're led to imagine.

     And once we arrive at a 1.52 aphelion. we have an arrival 2.7 km/s Vinfinity we need to get rid of.

     Pass through SML1 at 2.7 km/s and you'll be waving Mars goodbye. The Lagrange necks work their mojo on near parabolic orbits. And an earth to Mars Hohmann is decidedly hyperbolic with regard to Mars.

     What about Phobos and Deimos? The Martian moons are too small to lend a helpful gravity assist. We need to get rid of the 2.7 km/s Vinf and neither SML1 nor the moons are going to do it for us.

Mars ballistic capture by Belbruno & Toppotu

     Edward Belbruno is another well known evangelist for the Interplanetary Superhighway (though he likes to call them ballistic captures). Belbruno cowrote this pdf on ballistic Mars capture.

     Here is a screen capture from the pdf:

• 

     The path from Earth@Departure to Xc is pretty much a Hohmann transfer. In fact they assume the usual departure for Mars burn. Arrival is a little different. They do a 2 km/s heliocentric circularization burn at Xc (which is above Mars' perihelion). This particular path takes an extra year or so to reach Mars.

     So they accomplish Mars capture with a 2 km/s arrival burn. At first glance this seems like a 0.7 km/s improvement over the 2.7 km/s arrival Vinf.

     Or it seems like an advantage to those unaware of the Oberth benefit. If making the burn deep in Mars' gravity well, capture can be achieved for as little as 0.7 km/s.

     Comparing capture burns it's 2 km/s vs 0.7 km/s. So what do we get for an extra year of travel time? 1.3 km/s flushed down the toilet!

What About Ion Engines?

     "What about ion engines?" a Belbruno defender might object. "They don't have the thrust to enjoy an Oberth benefit. So Belbruno's 0.7 km/s benefit is legit if your space craft is low thrust."

     Belbruno & friends are looking at a trip from a nearly zero earth C3 to a nearly zero Mars C3. In other words from the edge of one Hill Sphere to another.

     So to compare apples to apples I'll look at a Hohmann from SEL2 to SML1. I want to point out I'm not using Lagrange necks as key holes down some mysterious tube. They're simply the closest parts of neighboring Hill Spheres.

     "Wait a minute..." says Belbruno's defender, "We're talking Hall thrusters. So no Hohmann ellipse, but a spiral."

     Low earth orbit moves about 4° per minute. So a low-thrust burn lasting days does indeed result in a spiral. But Earth's heliocentric orbit moves about a degree per day while Mars' heliocentric orbit moves about half a degree per day. At this more leisurely pace, a 4 or 5 day burn looks more an impulsive burn. The transfer between Hill Spheres is more Hohmann-like than the spiral out of earth's gravity well.

     Instead of a 1 x 1.524 AU orbit, the new Hohmann is a 1.01 x 1.517 AU ellipse. The new Hohmann's perihelion is a little slower, the new aphelion a little faster.

     Moreover, SEL2 moves at the same angular velocity as earth. So it's speed is about 101% earth's speed. Likewise SML1 moves at about 99.6% Mars' speed.

     With this revised scenario, aphelion rendezvous delta V is now more like 2.4 km/s. Still, Belbruno's 2 km/s capture burn saves 0.4 km/s.

     0.4 km/s is better than chopped liver, right? Well, recall ion engines with very good I_SP. I'll look at an exhaust velocity of of 30 km/s.

e^2.4/30 - 1 = 0.083
e^2/30 - 1 = 0.069

     So given a 100 tonne payload, rendezvous xenon is 8.3 tonnes for Hohmann vs 6.9 tonnes for Belbruno's ballistic capture.

108.3/106.9 = 1.0134

     We're adding a year to our trip time for a one percent mass improvement? Sorry, I don't see this a great trade-off.

Summary

     The virtually zero energy looooong trips between planets are an urban legend.

     I'll be pleasantly surprised if I'm wrong. To convince me otherwise, show me the beef. Show me the zero energy trajectory from an earth Lagrange neck to a Mars Lagrange neck.

     Until then I'll think of this post as a dose of Snopes for space cadets.

     I'd like to thank Mike Loucks and John P. Corrico Jr. I've held these opinions for awhile but didn't have the confidence to voice them. Who am I but an amateur with no formal training? But talking with these guys I was pleasantly surprised to find some of my heretical views were shared by pros. Without their input I would not have had the guts to publish this post.

• 

In the meantime he had another worry; strung out behind him were several more ships, all headed for Mars. For the next several days there would be frequent departures from the Moon, all ships taking advantage of the one favorable period in every twenty-six months when the passage to Mars was relatively 'cheap', i.e., when the minimum-fuel ellipse tangent to both planet's orbits would actually make rendezvous with Mars rather than arrive foolishly at some totally untenanted part of Mars' orbit. Except for military vessels and super expensive passenger-ships, all traffic for Mars left at this one time.

During the four-day period bracketing the ideal instant of departure ships leaving Leyport paid a fancy premium for the privilege over and above the standard service fee. Only a large ship could afford such a fee; the saving in cost of single-H reactive mass had to be greater than the fee. The Rolling Stone had departed just before the premium charge went into effect; consequently she had trailing her like beads on a string a round dozen of ships, all headed down to Earth, to tack around her toward Mars.

---

Hazel looked them over. 'Mr d'Avril, don't you have something a bit larger?'

'Well, yes, ma'am, I do — but I hate to rent larger ones to such a small family with the tourist season just opening up: I'll bring in a cot for the youngster.'

(ed note: The "tourist season" on Mars starts with the earliest arrival time of a spacecraft on a Terra-Mars Hohmann trajectory)

'See here, I don't want to buy this du — this place. I just want to use it for a while.'

Mr d'Avril looked hurt. 'You needn't do either one, ma'am. With ships arriving every day now I'll have my pick of tenants. My prices are considered very reasonable. The Property Owner's Association has tried to get me to up 'em — and that's a fact'

Hazel dug into her memory to recall how to compare a hotel price with a monthly rental — add a zero to the daily rate; that was it Why, the man must be telling the truth! — if the hotel rates she had gotten were any guide. She shook her head. 'I'm just a country girl, Mr d'Avril. How much did this place cost to build?'

Again he looked hurt 'You're not looking at it properly, ma'am. Every so often we have a big load of tourists dumped on us. They stay awhile, then they go away and we have no rent coming in at all. And you'd be surprised how these cold nights nibble away at a house. We can't build the way the Martians could.'

Hazel gave up. 'Is that season discount you mentioned good from now to Venus departure?'

'Sorry, ma'am. It has to be the whole season.' The next favorable time to shape an orbit for Venus was ninety-six Earth-standard days away — ninety-four Mars days — whereas the 'whole season' ran for the next fifteen months, more than half a Martian year before Earth and Mars would again be in a position to permit a minimum-fuel orbit.

It was true. Even in the bars that catered to inner planet types, the mix was rarely better than one Earther or Martian in ten (Belters). Squinting out at the crowd, Miller saw that the short, stocky men and women were nearer a third.

"Ship come in?" he asked.

"Yeah."

"EMCN?" he asked. The Earth-Mars Coalition Navy often passed through Ceres on its way to Saturn, Jupiter, and the stations of the Belt, but Miller hadn't been paying enough attention to the relative position of the planets to know where the orbits all stood.

• 

      This essay was written in 1952, long before the Mariner space probes gave us our close-up glimpses of the tantalizing red planet. Nevertheless, most of the concepts presented here are still quite valid, though we now know that Mars is even more rugged than anticipated. In particular, the atmospheric pressure is so low (about one-hundredth of Earth’s) (actually more like 0.00628 of Earth) that simple breathing masks will not give sufficient protection; we will have to wear spacesuits.

     So you’re going to Mars? That’s still quite an adventure —though I suppose that in another ten years no one will think twice about it. Sometimes it’s hard to remember that the first ships reached Mars scarcely more than half a century ago and that our colony on the planet is less than thirty years old. (By the way, don’t use that word when you get there. Base, settlement, or whatever you like—but not colony, unless you want to hear the ice tinkling all around you.)

     I suppose you’ve read all the forms and tourist literature they gave you at the Department of Extraterrestrial Affairs. But there’s a lot you won’t learn just by reading, so here are some pointers and background information that may make your trip more enjoyable. I won’t say it’s right up to date—things change so rapidly, and it’s a year since I got back from Mars myself—but on the whole you’ll find it pretty reliable.

     Presumably you’re going just for curiosity and excitement—because you want to see what life is like out on the new frontier. It’s only fair, therefore, to point out that most of your fellow passengers will be engineers, scientists, or administrators traveling to Mars—some of them not for the first time—because they’ve got a job of work to do. So whatever your achievements here on Earth, it’s advisable not to talk too much about them, as you’ll be among people who’ve had to tackle much tougher propositions. I won’t say that you’ll find them boastful: it’s simply that they’ve got a lot to be proud of, and they don’t mind who knows it.

     If you haven’t booked your passage yet, remember that the cost of the ticket varies considerably according to the relative positions of Mars and Earth. That’s a complication we don’t have to worry about when we’re traveling from country to country on our own globe, but Mars can be six times farther away at one time than at another. Oddly enough, the shortest trips are the most expensive, since they involve the greatest changes of speed as you hop from one orbit to the other. And in space, speed, not distance, is what costs money.

     Incidentally, I’d like to know how you've managed it. I believe the cheapest round trip comes to about $30,000, and unless the firm is backing you or you’ve got a very elastic expense account—Oh, all right, if you don’t want to talk about it…

     I take it you’re O.K. on the medical side. That examination isn’t for fun, nor is it intended to scare anyone off. The physical strain involved in space flight is negligible— but you’ll be spending at least two months on the trip, and it would be a pity if your teeth or your appendix started to misbehave. See what I mean?

     You’re probably wondering how you can possibly manage on the weight allowance you’ve got. Well, it can be done. The first thing to remember is that you don’t need to take any suits. There’s no weather inside a spaceship; the temperature never varies more than a couple of degrees over the whole trip, and it’s held at a fairly high value so that all you’ll want is an ultra-lightweight tropical kit. When you get to Mars you’ll buy What you need there and dump it when you return. The great thing to remember is only carry the stuff you actually need on the trip. I strongly advise you to buy one of the complete travel kits—a store like Abercrombie & Finch can supply the approved outfits. They’re expensive, but will save you money on excess baggage charges.

     Take a camera by all means—there’s a chance of some unforgettable shots as you leave Earth and when you approach Mars. But there’s nothing to photograph on the voyage itself, and I’d advise you to take all your pictures on the outward trip. You can sell a good camera on Mars for five times its price here—and save yourself the cost of freighting it home. They don’t mention that in the official handouts.

     Now that we’ve brought up the subject of money, I’d better remind you that the Martian economy is quite different from anything you’ll meet on Earth. Down here, it doesn’t cost you anything to breathe, even though you’ve got to pay to eat. But on Mars the very air has to be synthesized—they break down the oxides in the ground to do this—so every time you fill your lungs someone has to foot the bill. Food production is planned in the same way —each of the cities, remember, is a carefully balanced ecological system, like a well-organized aquarium. No parasites can be allowed, so everyone has to pay a basic tax which entitles him to air, food, and the shelter of the domes. The tax varies from city to city, but averages about $10 a day. Since everyone earns at least ten times as much as this, they can all afford to go on breathing.

     You’ll have to pay this tax, of course, and you’ll find it rather hard to spend much more money than this. Once the basic needs for life are taken care of, there aren’t many luxuries on Mars. When they’ve got used to the idea of having tourists around, no doubt they’ll get organized, but as things are now you’ll find that most reasonable requests won’t cost you anything. However, I should make arrangements to transfer a substantial credit balance to the Bank of Mars—if you’ve still got anything left. You can do that by radio, of course, before you leave Earth.

     So much for the preliminaries; now some points about the trip itself. The ferry rocket will probably leave from the New Guinea field, which is about two miles above sea level on the top of the Orange Range. People sometimes wonder why they chose such an out-of-the-way spot. That’s simple: it’s on the equator, so a ship gets the full thousand-mile-an-hour boost of the Earth’s spin as it takes off—and there’s the whole width of the Pacific for jettisoned fuel tanks to fall into. And if you’ve ever heard a spaceship taking off, you’ll understand why the launching sites have to be a few hundred miles from civilization. (gee, I guess it is just too bad for all those uncivilized natives living there who are rapidly going deaf)

     Don’t be alarmed by anything you’ve been told about the strain of blast-off. There’s really nothing to it if you’re in good health—and you won’t be allowed inside a spaceship unless you are. You just lie down on the acceleration couch, put in your earplugs, and relax. It takes over a minute for the full thrust to build up, and by that time you’re quite accustomed to it. You’ll have some difficulty in breathing, perhaps—it’s never bothered me—but if you don’t attempt to move you’ll hardly feel the increase of weight. What you will notice is the noise, which is slightly unbelievable. Still, it lasts only five minutes, and by the end of that time you’ll be up in the orbit and the motors will cut out. Don’t worry about your hearing; it will get back to normal in a couple of hours.

     You won’t see a great deal until you get aboard the space station, because there are no viewing ports on the ferry rockets and passengers aren’t encouraged to wander around. It usually takes about thirty minutes to make the necessary steering corrections and to match speed with the station; you’ll know when that's happened from the rather alarming “clang” as the air locks make contact. Then you can undo your safety belt, and of course you’ll want to see what it’s like being weightless.

     Now, take your time, and do exactly what you’re told. Hang on to the guide rope through the air lock and don’t try to go flying around like a bird. There’ll be plenty of time for that later: there’s not enough room in the ferry, and if you attempt any of the usual tricks you’ll not only injure yourself but may damage the equipment as well.

     Space Station One, which is where the ferries and the liners meet to transfer their cargoes, takes just two hours to make one circuit of the Earth. You’ll spend all your time in the observation lounge: everyone does, no matter how many times they’ve been out into space. I won’t attempt to describe that incredible view; I’ll merely remind you that in the hundred and twenty minutes it takes the station to complete its orbit you’ll see the Earth wax from a thin crescent to a gigantic, multicolored disk, and then shrink again to a black shield eclipsing the stars. As you pass over the night side you’ll see the lights of cities down there in the darkness, like patches of phosphorescence. And the stars! You’ll realize that you’ve never really seen them before in your life.

     But enough of these purple passages; let’s stick to business. You’ll probably remain on Space Station One for about twelve hours, which will give you plenty of opportunity to see how you like weightlessness. It doesn’t take long to learn how to move around; the main secret is to avoid all violent motions—otherwise you may crack your head on the ceiling. Except, of course, that there isn’t a ceiling since there’s no up or down any more. At first you’ll find that confusing: you’ll have to stop and decide which direction you want to move in, and then adjust your personal reference system to fit. After a few days in space it will be second nature to you.

     Don’t forget that the station is your last link with Earth. If you want to make any final purchases, or leave something to be sent home—do it then. You won’t have another chance for a good many million miles. But beware of buying items that the station shop assures you are “just the thing on Mars.”

     You’ll go aboard the liner when you’ve had your final medical check, and the steward will show you to the little cabin that will be your home for the next few months. Don’t be upset because you can touch all the walls without moving from one spot. You’ll only have to sleep there, after all, and you’ve got the rest of the ship to stretch your legs in.

     If you’re on one of the larger liners, there’ll be about a hundred other passengers and a crew of perhaps twenty. You’ll get to know them all by the end of the voyage. There’s nothing on Earth quite like the atmosphere in a spaceship. You’re a little, self-contained community floating in vacuum millions of miles from anywhere, kept alive in a bubble of plastic and metal. If you’re a good mixer, you’1l find the experience very stimulating. But it has its disadvantages. The one great danger of space flight is that some prize bore may get on the passenger list—and short of pushing him out of the air lock there’s nothing anyone can do about it.

     It won’t take you long to find your way around the ship and to get used to its gadgets. Handling liquids is the main skill you’ll have to acquire: your first attempts at drinking are apt to be messy. Oddly enough, taking a shower is quite simple. You do it in sort of a plastic cocoon, and a circulating air current carries the water out at the bottom.

     At first the absence of gravity may make sleeping difficult—you’ll miss your accustomed weight. That’s why the sheets over the bunks have spring tensioning. They’ll keep you from drifting out while you sleep, and their pressure will give you a spurious sensation of weight.

     But learning to live under zero gravity is something one can’t be taught in advance: you have to find out by experience and practical demonstration. I believe you’ll enjoy it, and when the novelty’s worn off you’ll take it completely for granted. Then the problem will be getting used to gravity again when you reach Mars!

     Unlike the take-off of the ferry rocket from Earth, the breakaway of the liner from its satellite orbit is so gentle and protracted that it lacks all drama. When the loading and instrument checks have been completed, the ship will uncouple from the Space Station and drift a few miles away. You’ll hardly notice it when the atomic drive goes on; there will be the faintest of vibrations and a feeble sensation of weight. The ship’s acceleration is so small, in fact, that you’l1 weigh only a few ounces, which will scarcely interfere with your freedom of movement at all. Its only effect will be to make things drift slowly to one end of the cabin if they’re left lying around.

     Although the liner’s acceleration is so small that it will take hours to break away from Earth and head out into space, after a week of continuous drive the ship will have built up a colossal speed. Then the motors will be cut out and you’ll carry on under your own momentum until you reach the orbit of Mars and have to start thinking about slowing down.

     Whether your weeks in space are boring or not depends very much on you and your fellow passengers. Quite a number of entertainments get organized on the voyage, and a good deal of money is liable to change hands before the end of the trip. (It’s a curious fact, but the crew usually seems to come out on top.) You’ll have plenty of time for reading, and the ship will have a good library of micro-books. There will be radio and TV contact with Earth and Mars for the whole voyage, so you’ll be able to keep in touch with things—if you want to.

     On my first trip, I spent a lot of my time learning my way around the stars and looking at clusters and nebulae through a small telescope I borrowed from the navigation officer. Even if you’ve never felt the slightest interest in astronomy before, you’ll probably be a keen observer before the end of the voyage. Having the stars all around you—not merely overhead—is an experience you’ll never forget.

     As far as outside events are concerned, you realize, of course, that absolutely nothing can happen during the voyage. Once the drive has cut out, you’ll seem to be hanging motionless in space: you’ll be no more conscious of your speed than you are of Earth’s seventy thousand miles an hour around the Sun right now. The only evidence of your velocity will be the slow movement of the nearer planets against the background of the stars—and you’ll have to watch carefully for a good many hours before you can detect even this.

     By the way, I hope you aren’t one of those foolish people who are still frightened about meteors. They see that enormous chunk of nickel-steel in New York’s American Museum of Natural History and imagine that’s the sort of thing you’ll run smack into as soon as you leave the atmosphere—forgetting that there’s rather a lot of room in space and that even the biggest ship is a mighty small target. You’d have to sit out there and wait a good many centuries before a meteor big enough to puncture the hull came along. It hasn’t happened to a spaceship yet.

     One of the big moments of the trip will come when you realize that Mars has begun to show a visible disk. The first feature you’ll be able to see with the naked eye will be one of the polar caps, glittering like a tiny star on the edge of the planet. A few days later the dark areas—the so-called seas—will begin to appear, and presently you’ll glimpse the prominent triangle of the Syrtis Major. In the week before landing, as the planet swims nearer and nearer, you’ll get to know its geography pretty thoroughly.

• 

     The braking period doesn’t last very long, as the ship has lost a good deal of its speed in the climb outward from the Sun. When it’s. over you’ll be dropping down onto Phobos, the inner moon of Mars, which acts as a natural space station about four thousand miles above the surface of the planet. Though Phobos is only a jagged lump of rock not much bigger than some terrestrial mountains, it’s reassuring to be in contact with something solid again after so many weeks in space.

     When the ship has settled down into the landing cradle, the air lock will be coupled up and you’ll go through a connecting tube into the port. Since Phobos is much too small to have an appreciable gravity, you’ll still be effectively weightless. While the ship’s being unloaded the immigration officials will check your papers. I don’t know the point of this; I’ve never heard of anyone being sent all the way back to Earth after having got this far!

     There are two things you mustn’t miss at Port Phobos. The restaurant there is quite good, even though the food is largely synthetic; it’s very small, and only goes into action When a liner docks, but it does its best to give you a fine welcome to Mars. And after a couple of months you’ll have got rather tired of the shipboard menu.

     The other item is the centrifuge; I believe that’s compulsory now. You go inside and it will spin you up to half a gravity, or rather more than the weight Mars will give you when you land. It’s simply a little cabin on a rotating arm, and there's room to walk around inside so that you can practice using your legs again. You probably won't like the feeling; life in a spaceship can make you lazy.

     The ferry rockets that will take you down to Mars will be waiting when the ship docks. If you’re unlucky you’ll hang around at the port for some hours, because they can’t carry more than twenty passengers and there are only two ferries in service. The actual descent to the planet takes about three hours, and it’s the only time on the whole trip when you’ll get any impression of speed. Those ferries enter the atmosphere at over five thousand miles an hour and go halfway around Mars before they lose enough speed through air resistance to land like ordinary aircraft.

     You’ll land, of course, at Port Lowell: besides being the largest settlement on Mars it’s still the only place that has the facilities for handling spaceships. From the air the plastic pressure domes look like a cluster of bubbles—a very pretty sight when the Sun catches them. Don’t be alarmed if one of them is deflated. That doesn’t mean that there’s been an accident. The domes are let down at fairly frequent intervals so that the envelopes can be checked for leaks. If you’re lucky you may see one being pumped up—it’s quite impressive.

     After two months in a spaceship, even Port Lowell will seem a mighty metropolis. (Actually, I believe its population is now well over twenty thousand.) You’ll find the people energetic, inquisitive, forthright—and very friendly, unless they think you’re trying to be superior.

     It’s a good working rule never to criticize anything you see on Mars. As I said before, they’re very proud of their achievements—and after all you are a guest, even if a paying one.

     Port Lowell has practically everything you’ll find in a city on Earth, though of course on a smaller scale. You’ll come across many reminders of “home.” For example, the main street in the city is Fifth Avenue—but surprisingly enough you’ll find Piccadilly Circus where it crosses Broadway.

     The port, like all the major settlements, lies in the dark belt of vegetation that roughly follows the Equator and occupies about half the southern hemisphere. The northern hemisphere is almost all desert—the red oxides that give the planet its ruddy color. Some of these desert regions are very beautiful; they’re far older than anything on the surface of our Barth, because there’s been little weathering on Mars to wear down the rocks—at least since the seas dried up, more than 500 million years ago.

     You shouldn’t attempt to leave the city until you’ve become quite accustomed to living in an oxygen-rich, low-pressure atmosphere. You'll have grown fairly well acclimated on the trip, because the air in the spaceship will have been slowly adjusted to conditions on Mars. Outside the domes, the pressure of the natural Martian atmosphere is about equal to that on the top of Mount Everest—and it contains practically no oxygen. So when you go out you’ll have to wear a helmet, or travel in one of those pressurized jeeps they call “sand fleas.”

     Wearing a helmet, by the way, is nothing like the nuisance you’d expect it to be. The equipment is very light and compact and, as long as you don’t do anything silly, is quite foolproof. As it’s very unlikely that you’ll ever go out without an experienced guide, you’ll have no need to worry. Thanks to the low gravity, enough oxygen for twelve hours’ normal working can be carried quite easily —and you’ll never be away from shelter as long as that.

     Don’t attempt to imitate any of the locals you may see walking around without oxygen gear. They’re second-generation colonists and are used to the low pressure. They can’t breathe the Martian atmosphere any more than you can, but like the old-time native pearl divers they can make one lungful last for several minutes when necessary. Even so, it’s a silly sort of trick and they’re not supposed to do it.

     As you know, the other great obstacle to life on Mars is the low temperature. The highest thermometer reading ever recorded is somewhere in the eighties, but that’s quite exceptional. In the long winters, and during the night in summer or winter, it never rises above freezing. And I believe the record low is minus one hundred and ninety!

     Well, you won’t be outdoors at night, and for the sort of excursions you’ll be doing, all that’s needed is a simple thermosuit. It’s very light, and traps the body heat so effectively that no other source of warmth is needed.

     No doubt you’ll want to see as much of Mars as you can during your stay. There are only two methods of transport outside the cities—sand fleas for short ranges and aircraft for longer distances. Don’t misunderstand me when I say “short ranges”—a sand flea with a full charge of power cells is good for a couple of thousand miles, and it can do eighty miles an hour over good ground. Mars could never have been explored without them. You can survey a planet from space, but in the end someone with a pick and shovel has to do the dirty work filling in the map.

     One thing that few visitors realize is just how big Mars is. Although it seems small beside the Earth, its land area is almost as great because so much of our planet is covered with oceans. So it’s hardly surprising that there are vast regions that have never been properly explored, particularly around the poles. Those stubborn people who still believe that there was once an indigenous Martian civilization pin their hopes on these great blanks. Every so often you hear rumors of some wonderful archaeological discovery in the wastelands, but nothing ever comes of it.

     Personally, I don’t believe there ever were any Martians —but the planet is interesting enough for its own sake. You’ll be fascinated by the plant life and the queer animals that manage to live without oxygen, migrating each year from hemisphere to hemisphere, across the ancient sea beds, to avoid the ferocious winter. The fight for survival on Mars has been fierce, and evolution has produced some pretty odd results. Don’t go investigating any Martian life forms unless you have a guide, or you may get some unpleasant surprises. Some plants are so hungry for heat that they may try to wrap themselves around you.

     Well, that’s all I’ve got to say, except to wish you a pleasant trip. Oh, there is one other thing. My boy collects stamps, and I rather let him down when I was on Mars. If you could drop me a few letters while you’re there— there’s no need to put anything in them if you’re too busy —I’d be much obliged. He’s trying to collect a set of spacemail covers portmarked from each of the principal Martian cities, and if you could help—thanks a lot!

      For any given vehicle design,  what one assumes for mission delta-vees,  vehicle weight statements,  course corrections,  and landing burn requirements greatly affects the payload that can be carried.  The effect is exponential:  variation in required mass ratio with changes in delta-vee and exhaust velocity.
     This analysis looks at trips from low Earth orbit to direct entry at Mars,  and for the return,  a direct launch from Mars to a direct entry at Earth.  The scope is min-energy Hohmann transfer plus 3 faster trajectories (see ref. 1). 
     The vehicle under analysis is the 2019 version of the Spacex “Starship” design,  as described in ref. 2.  The most significant items about that vehicle model are the inert mass and the maximum propellant load.  For this study,  the vehicle is presumed fully loaded with propellant at Earth departure,  and at Mars departure.  See also Figure 1.  Evaporative losses are ignored.

• 

     Since a prototype has yet to fly,  the design target inert mass of 120 metric tons is presumed as baseline.  Uncertainty demands that inert mass growth be investigated.  To that end,  the average of that design target and the 200 metric ton inert mass of the so-called “Mark 1 prototype” (that average is some 160 metric tons) is used to explore that effect.
     As currently proposed,  the vehicle has six engines.  Three are the sea level version of the “Raptor” engine design,  and the other three are vacuum versions of the same engine design (basically just a larger expansion bell).  I have already reverse-engineered fairly-realistic performance for these in ref. 3.  Because of the smaller bells,  the sea level engines gimbal significantly,  while the vacuum engines cannot.  Thus it is the sea level engines that must be used to land on Mars as well as Earth:  gimballing is required for vehicle attitude control.

Analysis Process

     As shown in Figure 2,  the analysis process is not a simple single-operation calculation.  The vehicle model provides a weight statement and engine performance.  The mission has delta-vee requirements for departure,  course correction,  and landing,  which must be appropriated factored (in order to get mass ratio-effective values).   There are two sets of analysis:  the outbound leg from Earth to Mars,  and the return leg from Mars to Earth. 
     Each leg analyzes 3 burns.  Earth departure,  and course correction are done with the vacuum “Raptor” engines,  while the landing on Mars is done with the sea level “Raptors” to obtain the necessary gimballing.  Mars departure and course correction are done with the vacuum “Raptor” engines (Mars atmospheric pressure is essentially vacuum).  The Earth landing is done with the sea level “Raptors” to get the gimballing and to get the atmospheric backpressure capability.  

• 

     This analysis is best done in a spreadsheet,  which then responds instantly to changes in one of the constants (like an inert mass or a delta-vee).  That is what I did here. 
     Referring again to Figure 2,  for each burn,  there is an appropriate vehicle ignition mass.  At departure,  it is the ignition mass from the weight statement.  For each subsequent burn,  it is the previous burn’s burnout mass.  Each burn’s burnout mass is its ignition mass divided by the required mass ratio for that burn,  in turn figured from that burn’s delta vee and the appropriate exhaust velocity.
     For each burn,  the change in vehicle mass from ignition to burnout is the propellant mass used for that burn.  For the first burn,  the propellant remaining (after the burn) is the initial propellant load minus the propellant mass used for that burn.  For the subsequent burns,  propellant remaining is the previous value of propellant remaining,  minus the propellant used for that burn. 
     After the final burn,  the propellant remaining cannot be a negative number!  If it is,  one reduces the payload number originally input,  and does all the calculations again.  If this done in a spreadsheet,  this update is automatic!  Ideally,  the propellant remaining should be exactly zero,  but for estimating purposes here,  a small positive fraction of a ton (out of 1200 tons) is “close enough”.
     Thus it is payload that is determined in this analysis.   This particular input (payload mass) is revised iteratively until the final burn’s remaining-propellant estimate is essentially zero.  That is the maximum payload value feasible for the mission case.

Orbits and the Associated Delta Vees

     As indicated in ref. 2,  I have looked at a Hohmann min energy transfer orbit,  and 3 faster transfers with shorter flight times.  All of these are transfer ellipses with their perihelions located at Earth’s orbit.  For Hohmann transfer,  the apohelion is at Mars’s orbit.  For the faster transfers,  apohelion is increasingly far beyond Mars’s orbit.  Why this is so is explained in the reference.  See Figures 3 and 4.
     Note that the overall period of the transfer orbit is important for abort purposes.  If the period is an exact integer multiple of one Earth year,  then Earth will be at the orbit perihelion point simultaneously with anything traveling along that entire transfer orbit.  This offers the possibility of aborting the direct entry and descent at Mars,  if conditions happen to be bad when the encounter happens.  Otherwise,  the spacecraft is committed to entry and descent,  no matter what.  

• 

• 

     The cases examined in ref. 1 were all computed for Earth and Mars at their average distances from the sun.  The larger transfer ellipse with the longer period occurs when both Earth and Mars are at their farthest distances from the sun.  This leads to larger delta vees to reach transfer perihelion velocity for the trip to Mars,  and larger velocity on the transfer orbit for the trip back to Earth.
     Ref 1 has the required velocities and delta-vees,  but the most pertinent data are repeated here:

Transfer	E.depdV, km/s	trip time, days	M. Vint, km/s
Hohmann	3.659	259	5.69
2-yr abort	4.347	128	7.40
No abort	4.859	110	7.36
3-yr abort	5.223	102	6.53
Transfer	M.depdV, km/s	trip time, days	E.Vint, km/s
Hohmann	5.800	259	11.57
2-yr abort	7.548	128	12.26
No abort	7.509	110	12.77
3-yr abort	6.653	102	13.14

     I did not examine the worst cases for all the transfer orbits in ref. 1,  but I do have the  increase in perihelion velocity for the worst case Earth departure on a Hohmann transfer for Mars:  0.20 km/s higher than average.  I also have the increase in apohelion velocity for the worst case Mars departure on a Hohmann transfer for Earth:  0.16 km/s higher than average.
     I cheated here:  I used those worst-case Hohmann increases for all the faster trajectories as well.  That’s not “right”,  but it should be close enough to see the relative size of the effect of worst case over average conditions.  I also used the same additive changes on the entry velocities.
     Because of the precision trajectory requirements for direct entry while moving above planetary escape speed,  some sort of course correction burn or burns will simply be required.  With this kind of analysis,  I have no way to evaluate that need.  So I just guessed:  0.5 km/s delta-vee capability in terms of propellant reserves. 
     Because this is just a guess,  I did not run any sensitivity analysis on it.  However,  the delta-vee budget proposed here is factor 2.5 larger than the difference average-to-worst-case for the trip to Mars,  which suggests it is “plenty”.   It is about factor 3 larger than the difference average-to-worst-case for the return trip to Earth.  You can get a qualitative sense of this effect from examining that average-vs-worst case effect.

Propellant Budgets for Direct Landings

     With this vehicle (or just about any other vehicle),  entry must be made at a shallow angle relative to local horizontal.  Down lift is required to avoid bouncing off the atmosphere,  since entry interface speed V_int exceeds planetary escape speed.  This is true at both Mars and at Earth.  Once speed has dropped to about orbit speed,  the vehicle must roll to up lift,  to keep the trajectory from too-quickly steepening downward. 
     The hypersonics end at roughly local Mach 3 speeds,  which is around 0.7-1 km/s velocity,  near 5 km altitude on Mars,  and near 45 km altitude on Earth  which has about the same air pressure.  Up to that point,  entry at Mars and Earth look very much alike,  excepting the altitude.  After that point they diverge sharply,  as illustrated in Figure 5.

• 

     The descent and landing at Earth require the ship to decelerate to transonic speed,  then pull up to a 90-degree angle of attack (AOA,  measured relative to the wind vector).  Thus,  as the trajectory rapidly steepens to vertical,  the ship executes a broadside “belly-flop” rather like a skydiver. 
     At low altitude where the air is much denser,  the terminal speed in the “belly-flop” will be well subsonic.  I assumed 0.5 Mach,  but that might be a little conservative.  This is the point where AOA increases to 180 degrees (tail-first),  and the landing engines get ignited.  From there,  touchdown is retropropulsive.
     The landing on Mars is quite different.  The ship comes out of hypersonics very close to the surface,  still at high AOA and still very supersonic.  From there,  the ship must pitch to higher AOA and pull up,  actually ascending back toward 5 km altitude.  This ascent is energy management:  speed drops rapidly as altitude increases.  It’s not quite a “tail slide” maneuver,  but it is similar to one. 
     At the local peak altitude,  the ship is moving at about local Mach 1,  and pitches to tail first attitude,  igniting the landing engines.  From there,  touchdown is retropropulsive.  The Martian “air” at the surface is very thin indeed,  as the figure indicates.  It may be that thrust is required to assist lift toward bending the trajectory upward:  the engines would have to be ignited earlier,  and at higher speed,  as indicated in the figure.  Whether this is necessary is just not yet known.
     The low point preceding the local pull-up is at some supersonic speed;  I just assumed about local Mach 1.5,  as indicated in the figure.  That would correspond to a factor 1.5 larger landing delta-vee requirement,  implying a larger landing propellant budget. 
     In either case,  I also use an “eyeball” factor of 1.5 upon the kinematic landing delta-vee,  to cover gravity loss effects,  maneuver requirements,  and any hover or near-hover to divert laterally to avoid obstacles. 
     So,  for purposes of this sensitivity analysis,  the Earth landing is not of much interest,  but the Mars landing is.  The sensitivity analysis looks at the effects of Mach 1.5-sized vs Mach 1-sized touchdown delta-vee.

Analysis Results

     The scope of the sensitivity analysis is illustrated in Figure 6.  As indicated earlier,  the orbital delta-vee increases worst-case-vs-average,  for Hohmann transfer,  were applied additively to the departure delta-vees for the faster trajectories.  No attempt was made to vary the course correction budgets.   Growth in vehicle design inert mass was examined.  An increase in the Mars touchdown delta-vee was examined.  Nothing else was considered.

• 

     The results start with the worst vs average orbital delta-vee sensitivity.  These results are given in Figure 7.  These are the plots from the spreadsheet,  copied and pasted into the figure.  There are 4 such plots in the figure:  the top two are for the outbound journey Earth to Mars.  The bottom two are for the return journey Mars to Earth.  Results for all 4 transfer orbit cases are shown simultaneously by using trip time as the abscissa. 
     Each has 4 data points:  these are for the Hohmann transfer at 259 days flight time,  the 2-year abort orbit at 128 days,  the non-abort orbit at 110 days,  and the 3-year abort at 102 days.  Be aware that the curves are probably not really straight between the Hohmann orbit and the 2-year abort orbit.  I did not run enough fast transfer cases in ref. 1 to get a smooth curve here.
     The most significant thing in the left hand figure for the outbound trip is the about-40 ton loss of max payload between average and worst case for the Hohmann transfer.  This is a lot less than the about-130 ton payload loss using the 2-year abort orbit instead of Hohmann transfer,  or the about-210 ton payload loss for using the 3-year abort orbit. 
     The average-vs-worst-case deficits are somewhat similar on the faster orbits.  The Mars entry interface velocity trend in the right-hand figure is obviously very nonlinear.  Yet,  all the calculated values fall below the entry velocity from low Earth orbit (LEO).  Any heat shield capable of serving for return from LEO will serve this Mars entry purpose,  which would be the governing case if the trip were one-way only.  There’s only a small change in entry speeds for average-vs-worst orbit case in this estimated analysis. 
     The return voyage has trends shaped quite differently.  For Hohmann transfer,  the worst-vs-average payload loss is about 20 tons.  The deficits on the faster orbits should be similar.  The deficit for using the 2-year abort orbit instead of Hohmann is far larger at about 110 tons,  and that’s from a small return payload to begin with. 
     In the right hand Earth entry interface speed plot,  the blue and orange curves in the entry interface plot fall only slightly apart.  Note that all the entry velocities are much higher than the just-below-escape speed seen with Apollo returning from the moon.  The faster transfer orbits,  and even the Hohmann transfer,  are substantially more demanding than a lunar return entry.  It is clearly the direct-entry Earth return that will size the heat shield design!  

• 

     Results for the effects of inert mass growth sensitivity are given in Figure 8.  This is the same 4-plot format as Figure 7.  For the outbound trip to Mars,  the Hohmann mass penalty for inert mass growth is about the same 40 ton deficit as for worst-case orbit distances.  It is similar for the faster trajectories.  It is the return trip that most suffers from vehicle inert mass growth.  We lose about 40 tons from an already small return payload on the Hohmann transfer.  However the 2-year abort trajectory and the no-abort trajectory are entirely infeasible,  with their max payloads calculated as negative.  Note that both the Mars and Earth entry interface velocities are unaffected by this sensitivity.  The orange and blue curves fall right on top of each other.

• 

     The sensitivities to the need for a thrusted pull-up on Mars are given in Figure 9.  This follows the same format as Figures 7 and 8.  Bear in mind that the nominal design lights the engines for touchdown at about Mach 1 speed.  For this analysis,  the engines are ignited earlier,  at about Mach 1.5 flight speed,  to assist lift in pulling up to the Mach 1 “flip”,  to tail-first attitude.  That makes the landing delta vee about 1.5 times larger.  (Note that each case is also factored up by 1.5 further,  to cover any maneuver / hover needs for the touchdown.)
     What the figure shows is about the same 40-ton payload loss on the voyage to Mars to cover the increased landing propellant requirement for the Hohmann transfer.  Effects on the faster transfers are similar.  This trend is comparable to the worst-case orbit losses.  The return payload is entirely unaffected,  as the landing occurs prior to refueling and loading for the trip home.
     Both the Earth and Mars entry interface velocities are unaffected by this Mars thrusted pull-up scenario.  The orange and blue curves fall right on top of each other. 

• 

Final Remarks

     #1.  These results are only approximate!  Real 3-body orbital analysis,  and real entry-trajectory lifting flight dynamics models,  must be used to get better answers.  Nevertheless,  the trends are quite clear from this approximate analysis.
     #2.  Flying on faster transfer orbits will cost a lot of payload capability,  on both the outbound voyage,  and the return voyage.  This effect is much worse on the return voyage,  where the allowable payload is just inherently smaller.
     #3.  The effects of worst-case orbital positions-relative-to-average,  of Mars and Earth,  have a significant effect on payload,  but it is only half or less the effect of choosing faster transfer orbits.
     #4.  The effect of vehicle inert mass growth from the design target of 120 metric tons to an arbitrary but realistic 160 metric tons is comparable to the effect of worst-case vs average orbits on the outbound voyage.  However it has catastrophic effects on the return voyage!  This is enough to prevent faster-than-Hohmann transfers on the voyage home,  for this vehicle model.
     #5.  The effects of needing a thrusted pull-up for the Mars landing is comparable to the effects of worst-case orbit distances on the outbound voyage.  This has no effects upon the return voyage.
     #6.  It is the direct Earth entry velocity that will design the vehicle heat shield for any vehicle capable of making the return.  This is substantially more challenging than was the return from the moon.  For deliberately-designed one-way vehicles to Mars,  the heat shield design requirements are comparable to entry from low Earth orbit.
     #7.  My personal opinions are that thrusted pull-up will be needed,  along with the need to fly when Earth and Mars orbital distances are worst-case,  plus there will be a little inert mass growth (say by 20 metric tons to 140 metric tons vehicle inert mass).  That kind of thing is the proper design point for this vehicle,  not the most rosy projections!  Estimated performance data for this design case (at 140 metric ton inert mass) are in Figure 10 (same basic format as Figures 7,  8,  and 9).   Note that two of the faster transfers home are precluded.  The feasible one has a very small max payload value compared to Hohmann transfer. 

• 

     #8.  Bear in mind that the rather high max allowable payload figures feasible to Mars for Hohmann transfer are incompatible with what can be aboard “Starship” for launch to low Earth orbit.  The payloads for the faster transfers to Mars look more like what can be ferried up to LEO.  That suggests that a faster transfer to Mars is most compatible with the projected “Starship” / “Super Heavy” system design characteristics,  as these were evaluated in references 2 and 3.
     #9.  Bear also in mind that a faster transfer orbit to Mars ought to include abort capability,  in case conditions at arrival prove too bad to attempt the landing.  There is simply not the propellant available to enter orbit and wait for better conditions.  Thus life support supplies must be carried to last the entire period of the transfer orbit,  and a full-capability heat shield for direct Earth entry must be used.
     #10.  The fast transfer home need not be limited by abort capability.  It can be a different transfer orbit than the outbound trip.  Surprisingly,  the shapes of the plotted curves suggest that something faster than the “3-year abort” orbit could be used for the return home.
     #11.  Given a way to combine two payloads to LEO into one “Starship” by cargo transfer operations on orbit,  then (and only then) the very large payloads to Mars indicated for Hohmann transfer become feasible.  Like on-orbit cryogenic refueling,  this on-orbit cargo transfer capability does not yet exist,  not even as a concept (on-orbit refueling at least exists as a concept).

References:

     #1. G. W. Johnson,  “Interplanetary Trajectories and Requirements”,  posted 21 November 2019, 
     #2. G. W. Johnson,  “Reverse-Engineering the 2019 Version of the Spacex “Starship”/ ”Super Heavy” Design”,  posted 22 October 2019, 
     #3. G. W. Johnson,  “Reverse-Engineered “Raptor” Engine Performance”,  posted 26 September 2019, 

INTRODUCTION

     The human exploration of Mars is a daunting undertaking. Safely transporting astronauts to and from Mars will require advances in many areas to develop spacecraft that are up to the challenge. Propulsion systems are one such area. Advanced nuclear propulsion systems (alone or in combination with chemical propulsion systems) have the potential to substantially reduce trip time compared to fully non-nuclear approaches. Shorter trip time reduces risks associated with space radiation, zero gravity, launch and orbital assembly requirements, and many other aspects of long-duration space missions.

Based on the relative orbits of Mars and Earth, the distance between Earth and Mars ranges from 55 to 400 million km over a synodic period of approximately 26 months. Launch (or Earth departure) requirements vary significantly over this cycle.

TABLE 1.1 Mission Scenarios for Crewed Mars Missions

Surface Time	Short stay time on Mars (30 to 90 days) (Opposition class) Long stay time on Mars (~ 500 days) (Conjunction class)
Vehicles	All-up (no separate cargo missions) Cargo missions precede crewed missions
Options for Mars Orbits	Low Mars orbit (e.g., altitude of 200-400 km with an orbital period of 1-2 h) Elliptical Mars orbit with a period of one Martian day Areosynchronous orbit (i.e., spacecraft tracks over the same geographic position on the Mars surface) Base of operations on Phobos
Options for In-Space Propulsion Systems	Nuclear thermal propulsion (NTP) Nuclear electric propulsion (NEP) NEP with chemical augmentation NEP-NTP bimodal Solar electric propulsion (SEP) with chemical augmentation SEP-NTP Chemical Chemical with aeroassist NTP with aeroassist

     Each 26-month cycle is not the same. Propulsion system performance requirements, in terms of the total velocity increment (ΔV) of a round trip Mars mission, vary from one launch opportunity to the next. The ΔV for a particular mission also depends on other mission constraints, particularly the stay time at Mars and the desired trip time.

     There are two classes of crewed missions to Mars: conjunction class and opposition class. Conjunction-class missions have the lowest ΔV requirements. For crewed conjunction-class missions, trip times are typically 180 to 210 days each way, stay times on Mars are typically 500 days or more, and total mission time is around 900 days.4 These are the “long stay” missions in Table 1.1.

     In contrast, one leg of opposition-class missions occurs when the orbital alignment of Earth and Mars is less favorable, but they allow for short stays on the surface of Mars (“short stay” missions in Table 1.1). These missions have higher ΔV requirements and require more propellant, which increases the mass of the Mars vehicle and the number of launch vehicles necessary to lift the required mass to its assembly orbit. Opposition-class missions are characterized by much shorter stay time on Mars (30 to 90 days) and a shorter total mission time (400 to 750 days). An additional complexity of opposition-class missions is that the long leg of the mission typically passes inside Earth’s orbit, generally as close to the Sun as the orbit of Venus, to mitigate the adverse planetary alignment of that leg of the mission. This results in both thermal and radiation challenges for a crewed Mars mission. Representative trajectories for each of the crewed mission scenarios are shown in Figure 1.1.

• 

BASELINE MISSION TO MARS: CREWED OPPOSITION CLASS MISSIONS

     The baseline mission specified by NASA for this report is an opposition-class crewed mission to Mars launched in 2039. This mission would be preceded by cargo missions beginning in 2033 to pre-place surface infrastructure and consumables for the crew. The propulsion system needed for this mission would also be sufficient for conjunction-class missions. The baseline mission has the following parameters:

• Crew mission launch in 2039 opportunity;
• Total crew trip time ≤750 days;
• Split mission with separate crew and cargo vehicles,
  • Same propulsion systems used on all vehicles,
  • Cargo vehicles arrive at Mars prior to first crew departure from Earth;
• Stay time on the Mars surface of 30 days;
• Crew of four, two of whom land on Mars; and
• Vehicle systems, cargo, and propellant launched by multiple launch vehicles to an assembly orbit, which would be either in low Earth orbit or cislunar space.

     In order to meet the requirement for total trip time, with an NEP system Earth departure and Mars capture and departure would be augmented by an additional in-space liquid methane and liquid oxygen (LOX) chemical propulsion system. The NEP system provides acceleration and deceleration in interplanetary space. In contrast, the NTP system provides propulsion for all transit maneuvers. The mission segments and the propulsion system used for each phase of flight are described in Table 1.2.

     As Earth and Mars revolve about the Sun, the most efficient trajectories vary, resulting in varying levels of propulsive requirements (ΔV) over a 15- to 17-year period (see Figure 1.2).

TABLE 1.2 Nuclear Propulsion Architectures for the Baseline Crewed Mars Mission

Propulsion System	TMI	Departure DSM	Mars Capture	TEI	Return DSM	Earth Return
NTP	NTP	NTP	NTP	NTP	NTP	Capsule EDL
NEP/Chemical	NEP/ Chemical	NEP a	NEP/ Chemical	NEP/ Chemical	NEP a	Capsule EDL

^a For some launch opportunities, the total velocity increment (ΔV) requirements for deep space maneuvers will be so great that an NEP system will also need to use its chemical propulsion system to meet the desired trip time.

NOTE: DSM, deep space maneuver EDL, entry, descent, and landing; NEP, nuclear electric propulsion; NTP, nuclear thermal propulsion; TEI, trans-Earth injection; TMI, trans-Mars injection.

• 

     A factor in mission assessment for repeated trips to Mars is the ability of propulsion systems to meet mission ΔV requirements over a series of consecutive launch opportunities without large variability in overall mission parameters, such as propellant mass, which could drive very different launch requirements for different opportunities. This variability is reduced by propulsion systems with high specific impulse (I_sp). Previous studies have shown the impact of NTP for an opposition-class mission in different launch opportunities, although not for the current years of interest. An example of the change in vehicle (propellant) mass with launch date is shown in Figure 1.3 for an advanced chemical system with an I_sp of 480 s and an NTP system with an I_sp of 825 s. The mass variation with launch opportunity for the higher I_sp system is about one half of the variation of the chemical system. Similar benefits would likely be achieved with an NEP system with an I_sp of 2,000 s paired a conventional chemical system. This is particularly important because some launch opportunities are not feasible using purely a chemical system. Flexibility to launch date is a major architectural advantage of the use of nuclear propulsion.

PROPULSION SYSTEM REQUIREMENTS

     Although NEP (Nuclear Electric Propulsion, ion drives and the like) and NTP (Nuclear Thermal Propulsion, NERVA and the like) systems both use nuclear power, they convert this power into thrust in different ways based on different technologies (as will be discussed in Chapters 2 and 3). The performance of rocket propulsion systems is defined by multiple parameters that define how much propellant they use and how much acceleration they can generate. In the case of chemical rockets or NTP systems, the two primary parameters are the I_sp and thrust. For NEP systems, I_sp is important to determine propellant requirements, but thrust and acceleration are defined by multiple parameters: power, thrust efficiency, and specific mass. Thrust efficiency defines how much electric power is converted into thrust power, and the specific mass is defined as the mass of the entire NEP system divided by the electrical power available for the thrusters. NEP systems have a higher I_sp than NTP systems, but they have very low thrust. The megawatt electric (MWe)-class NEP systems proposed to execute the baseline mission therefore require chemical rockets (which have an I_sp that is much lower than either an NTP system or an NEP system) to meet the desired trip time.

     NTP and NEP system performance requirements to execute the baseline mission are a topic of ongoing study by NASA. Table 1.3 summarizes the committee’s estimate of those requirements for NTP and NEP systems based on information from multiple sources.

• 

CARGO MISSIONS

     Conjunction-class missions have the lowest possible ΔV requirements because they use minimum energy, or Hohmann-like, trajectories. These trajectories are traditionally cited for cargo missions in which mass efficiency rather than trip time is a priority. Cargo missions also benefit from the higher I_sp of NEP and NTP systems. To ensure delivery of the requisite payloads to Mars before launch of crew, multiple cargo flights are planned as an integral aspect of this enterprise. As discussed in Chapters 2 and 3, using the crew vehicle propulsion system on one or more of the precursor cargo vehicles provides significant risk reduction and valuable flight information about propulsion system reliability, safety, and performance.

SUMMARY

     NASA is presently considering multiple forms of propulsion, including NTP and NEP, in its mission architecture analyses. Opposition-class missions, while reducing crew duration on Mars and total mission time, markedly increase mission ΔV requirements. This mission class introduces a higher sensitivity in propulsion system requirements from one launch opportunity to another, which could be achieved by either an NTP or NEP system. Successful development of an NTP or NEP/chemical system at relevant scale and performance would allow NASA to develop a robust architecture with flexibility across multiple mission opportunities. This report provides a technology assessment of the NTP and NEP development challenges that must be overcome to execute the baseline Mars mission. It is not intended to provide—nor did the committee’s statement of task allow—a comprehensive assessment of all aspects or trade studies associated with how a human Mars exploration mission should be organized, funded, or executed.

My text for this sermon is the set of delta v maps, especially the second of them, at the still ever-growing Atomic Rockets site. These maps show the combined speed changes, delta v in the biz, that you need to carry out common missions in Earth and Mars orbital space, such as going from low Earth orbit to lunar orbit and back.

Here is a table showing some of the missions from the delta v maps, plus a few others that I have guesstimated myself:

Patrol Missions

Mission	Delta V
Low earth orbit (LEO) to geosynch and return	5700 m/s powered (plus 2500 m/s aerobraking)
LEO to lunar surface (one way)	5500 m/s (all powered)
LEO to lunar L4/L5 and return (estimated)	4800 m/s powered (plus 3200 m/s aerobraking)
LEO to low lunar orbit and return	4600 m/s powered (plus 3200 m/s aerobraking)
Geosynch to low lunar orbit and return (estimated)	4200 m/s (all powered)
Lunar orbit to lunar surface and return	3200 m/s (all powered)
LEO inclination change by 40 deg (estimated)	5400 m/s (all powered)
LEO to circle the Moon and return retrograde (estimated)	3200 m/s powered (plus 3200 m/s aerobraking)
Mars surface to Deimos (one way)	6000 m/s (all powered)
LEO to low Mars orbit (LMO) and return	6100 m/s powered (plus 5500 m/s aerobraking)

Entries marked "(estimated)" are not in source table; delta v estimates are mine. ("Plus x m/s aerobraking" means ordinarily the engine would be responsible for that delta V as well, but it can be obtained for free via aerobraking. E.g., LEO to geosynch and return costs 8,200 m/s with no aerobraking)

Two things stand out in this list. One is how helpful aerobraking can be if you are inbound toward Earth, or any world with a substantial atmosphere. Many craft in orbital space will be true aerospace vehicles, built to burn off excess speed by streaking through the upper atmosphere at Mach 25 up to Mach 35.

But what really stands out is how easily within the reach of chemical fuels these missions are. Chemfuel has a poor reputation among space geeks because it barely manages the most important mission of all, from Earth to low orbit. Once in orbit, however, chemfuel has acceptable fuel economy for speeds of a few kilometers per second, and rocket engines put out enormous thrust for their weight.

(ed note: with 4,400 m/s exhaust velocity oxygen-hydrogen chemical rockets:

3100 m/s ΔV requires a very reasonable mass ratio of 2 {50% of wet mass is fuel}

6100 m/s ΔV requires a mass ratio of 4 {75% fuel} which is right at the upper limit of economical mass ratios )

In fact, transport class rocket ships working routes in orbital space can have mass proportions not far different from transport aircraft flying the longest nonstop global routes.

A jetliner taking off on a maximum-range flight may carry 40 percent of its total weight in fuel, with 45 percent for the plane itself and 15 percent in payload. A moonship, the one that gets you to lunar orbit, might be 60 percent propellant on departure from low Earth orbit, with 25 percent for the spacecraft and the same 15 percent payload. The lander that takes you to the lunar surface and back gets away with 55 percent propellant, 25 percent for the spacecraft, and 20 percent payload.

(These figures are for hydrogen and oxygen as propellants, currently somewhat out of favor because liquid hydrogen is bulky, hard to work with, and boils away so readily. But H2-O2 is the best performer, and may be available on the Moon if lunar ice appears in concentrations that can be shoveled into a hopper. Increase propellant load by about half for kerosene and oxygen, or 'storable' propellants.)

(ed note: so the point is that chemical rockets are perfectly adequate for missions to Mars or cis-Lunar space provided there is a network of orbital propellant depots suppled by in-situ resource allocation. An orbital propellant depot in LEO supplied by Lunar ice would do the trick. An orbital depot in Low Mars Orbit supplied by Deimos ice would also be very useful.)

      One of our favorite SF themes is the "Belter Civilization," which usually seeks—and gets—independence from the colonial masters on Earth. Belters make their livings as asteroid miners, and they flit from asteroid to asteroid, slicing up planetoids for the rich veins of metal we'll presumably find in them.

     In the usual story, the miners go off on long prospecting tours, leaving their families on a "settled" rock. The Belt Capital is usually located on Ceres or some other central place which may or may not have been extensively transformed; and when Belters get together, it's always in an asteroid city.

     The Belters don't ever come to Earth or any other planet. Indeed, they regard planets as "holes," deep gravity wells which can trap them and use up their precious fuels. The assumption here is that it's far less costly to flit from asteroid to asteroid than it is to land on a planet or get into close orbit around one.

     Another assumption, generally not stated in the stories, is that fuels are expensive and scarce, and the Belters have to conserve reaction mass; this is why, in the usual Belter story, you conserve both time and energy by never going outside the Belt. Scarce as fuel is, though, I suppose the Belters have a source of it locally or they couldn't contemplate independence. They must have fuel for their ships and energy for their artificial environments. Without those, there'd be no Belter Civilization. Even if we discover something of fabulous value in the Belt we can't operate without energy and fuel.

     Those are not, by the way, the same thing. Nuclear fission reactors and large solar panels could provide enough power for a permanent Belt station, and if there were something valuable enough out there we could put a reactor onto an asteroid now. Rocket fuels are something else again. To make a rocket work, you must have reaction mass: something to get moving fast backwards and dump overboard. Unfortunately, asteroids are rock, and rocks don't make very good rocket fuel. We'll come back to what the Belters might do about that later.

     For the moment, let's see how difficult travel to and in the Belt is. We’ll use the same measure as last time, the total change in ship velocity required to perform the mission. This is called delta v, and you should recall that a ship with a given fuel efficiency and ratio of fuel to non-fuel weight will have a unique calculatable delta v. It doesn't matter whether the pilot uses that delta v in little increments or in one big burn: the sum of velocity changes remains the same.

     Similarly, various mission delta-v requirements can be calculated from the laws of orbital mechanics independent of the ship used. Figure 14 gives the delta-v requirements for getting around the Earth-Jupiter portion of the solar system. We're assuming that getting to Earth orbit is free, whether with the laser launching system I described previously, or with shuttles, or whatever, so all missions to or from Earth begin and end in orbit.

     The first thing we see is that landing on an asteroid isn't much easier than going to Mars; in fact, Ceres is harder to get to than Mars. This is because not only are the asteroids a long way out, but they don't help you catch up to them; they've so little mass that you have to chase them down. Thus, once among the asteroids, you may well want to stay there and not use up all that delta v coming back to Earth.

     Then, too, travel between Ceres and a theoretical asteroid 2 AU out is a lot cheaper than getting to Earth from either of them. (One AU, or astronomical unit, is the distance from Earth to the Sun and is 93,000,000 miles, or 149,500,000 = 1.5 × 10⁸ kilometers.) It takes 8 km/sec to get to the 2 AU rock, but only 3.2 more to get from there to Ceres.

     So far so good, and we're well on the way to developing a Belt Civilization. There's already a small nit to pick, though: although travel to Mars itself is costly, it's as easy to get to Mars orbit as it is to go from asteroid to asteroid. Thus, if a laser-launch system could be built on Mars, making travel to and from Mars orbit cheap, Mars might well become the Belt Capital.

     Politics being what they are, though, perhaps the Martians (well, Mars colonists?) won't like having all those crude asteroid miners on their planet, and the Belters will have to build their own capital at some convenient place such as Ceres or the 2 AU rock, saving both their feelings and some energy. However, we've so far said nothing about how long it takes to get from one place to another. The delta v's in Figure 14 are for minimum energy trips, Hohmann transfer orbits, and to use a Hohmann orbit you must start and finish with origin and destination precisely opposite the Sun. You can't just boom out when you feel like it; you must wait for the precise geometry, otherwise the delta-v requirements go up to ridiculous values.

     You get a launch window once each synodic period. A synodic period is the time it takes two planets, or planetoids, to go around the sun and come back to precisely the same positions relative to each other: from, say, being on opposite sides of the Sun until they’re in opposition again, which is what we need for a Hohmann journey.

     The synodic periods and travel times are given in Figure 15, and our Belt Civilization is in trouble again. Not only does it take 1.57 years to get from Ceres to 2 AU (or vice versa), but you can only do it once each 7 years! Travel to and from Mars isn't a lot better, either. The Belters aren't going to visit their Capital very often, and one wonders if a civilization can be built among colonies that can only visit each other every seven to nine years, spending years in travel times to do it.

     By contrast, you can get from Earth to the flying rocks every year and a half, spending another year-and-a-half in transit. That's no short time either, but it beats the nine years of the Ceres-asteroid visitations.

     Perhaps, though, we haven't been quite fair to the Belters. Asteroids aren't as widely separated as Ceres and our 2 AU rock Most textbooks claim the asteroids are concentrated between 2.1 and 3.3 AU out from the Sun. We'll assume they're all in the same plane (they aren't), so the Belt area works out to 4.6 × 10²⁷ square centimeters. The books say there are about 100,000 asteroids visible with the Palomar Eye, but we want to be fair (and make things simple) so we'll assume there are 460,000 asteroids interesting enough to want to visit, or one every 10²² cm² within the Belt. That means the asteroids lie on an average of 10¹¹ cm apart, which happens to be 10⁶ km or one million kilometers, about three times the distance from Earth to the Moon.

It is incredibly rare when the laws of the universe let you get something for nothing. Give your heartfelt gratitude to Hermann Oberth for uncovering one for you. It will be a lifesaver.

What is the escape velocity 300 kilometers above the surface of Mars?

300 km = 300,000 meters. Mars has a radius of about 3,396,000 meters. So r = 3,396,000 + 300,000 = 3,696,000 meters. Mars also has a mass of 6.4185e23 kg, you can find this in NASA's incredibly useful Planetary Fact Sheets.

• V_esc = sqrt((2 * G * M) / r)
• V_esc = sqrt((2 * 6.67428e-11 * 6.4185e23) / 3,696,000)
• V_esc = sqrt(85,678,000,000,000 / 3,696,000)
• V_esc = sqrt(23,181,000)
• V_esc = 4814 m/s = 4.81 km/s

What is periapsis around the Sun that will give an escape velocity of 200 km/sec?

200 km/sec = 200,000 m/sec. The mass of the Sun is about 1.9891e30 kg.

• r = (2 * G * M) / (V_esc²)
• r = (2 * 6.67428e-11 * 1.9891e30) / (200,000²)
• r = 265,436,115,600,000,000,000 / 40,000,000,000
• r = 6,635,902,890 meters = 6,636,000 kilometers

Given that you are going to travel a parabolic orbit around the Sun that has an escape velocity of 200 km/s at periapsis, you have an initial velocity of 3.2 km/s, and you wish to exit the Oberth Maneuver with a final velocity of 50 km/s, calculate the required Δ_v burn at periapsis.

V_esc = 200 km/s = 200,000 m/s. V_h = 3.2 km/s = 3200 m/s.V_f = 50 km/s = 50,000 m/s.

• Δ_v = sqrt(V_f² + V_esc²) - sqrt(V_h² + V_esc²)
• Δ_v = sqrt(50,000² + 200,000²) - sqrt(3200² + 200,000²)
• Δ_v = sqrt(2,500,000,000 + 40,000,000,000) - sqrt(10,240,000 + 40,000,000,000)
• Δ_v = sqrt(42,500,000,000) - sqrt(40,010,240,000)
• Δ_v = 206,000 - 200,000
• Δ_v = 6,000 m/s = 6 km/s

So by burning 6 km/s of Δ_v, you get an actual Δ_v increase of 46.8 km/s. That's 40.8 km/s for free. Sweet!

      If onboard fuel is available to produce a velocity change, another type of swingby can do even better. This involves a close approach to the Sun, rather than to one of the planets. The trick is to swoop in close to the solar surface and apply all available thrust near perihelion, the point of closest approach.

     Suppose that your ship has a small velocity far from the Sun. Allow it to drop toward the Sun, so that it comes close enough almost to graze the solar surface. When it is at its closest, use your onboard fuel to give a 10 kms/second kick in speed; then your ship will move away and leave the solar system completely, with a terminal velocity far from the Sun of 110 kms/second.

     The question that inevitably arises with such a boost at perihelion is, where did that "extra" energy come from? If the velocity boost had been given without swooping in close to the Sun, the ship would have left the solar system at 10 kms/second. Simply by arranging that the same boost be given near the Sun, the ship leaves at 110 kms/second. And yet the Sun seems to have done no work. The solar energy has not decreased at all. It sounds impossible, something for nothing.

     The answer to this puzzle is a simple one, but it leaves many people worried. It is based on the fact that kinetic energy changes as the square of velocity, and the argument runs as follows: The Sun increases the speed of the spacecraft during its run towards the solar surface, so that our ship, at rest far from Sol, will be moving at 600 kms/second as it sweeps past the solar photosphere. The kinetic energy of a body with velocity V is V²/2 per unit mass, so for an object moving at 600 kms/second, a 10 kms/second velocity boost increases the kinetic energy per unit mass by (610²-600²)/2 = 6,050 units. If the same velocity boost had been used to change the speed from 0 to 10 kms/second, the change in kinetic energy per unit mass would have been only 50 units. Thus by applying our speed boost at the right moment, when the velocity is already high, we increase the energy change by a factor of 6,050/50 = 121, which is equivalent to a factor of 11 (the square root of 121) in final speed. Our 10 kms/second boost has been transformed to a 110 kms/second boost.

     All that the Sun has done to the spaceship is to change the speed relative to the Sun at which the velocity boost is applied. The fact that kinetic energy goes as the square of velocity does the rest.

     If this still seems to be getting something for nothing, in a way it is. Certainly, no penalty is paid for the increased velocity—except for the possible danger of sweeping in so close to the Sun's surface. And the closer that one can come to the center of gravitational attraction when applying a velocity boost, the more gratifying the result.

     Let us push the limits. One cannot go close to the Sun's center without hitting the solar surface, but an approach to within 20 kilometers of the center of a neutron star of solar mass would convert a 10 kms/second velocity boost provided at the right moment to a final departure speed from the neutron star of over 1,500 kms/second. An impressive gain, though the tidal forces derived from a gravitational field of over 10,000,000 gees might leave the ship's passengers a little the worse for wear.

     Suppose one were to perform the swingby with a speed much greater than that obtained by falling from rest? Would the gain in velocity be greater? Unfortunately, it works the other way round. The gain in speed is maximum if you fall in with zero velocity from a long way away. In the case of Sol, the biggest boost you can obtain from your 10 kms/second velocity kick is an extra 100 kms/second. That's not fast enough to take us to Alpha Centauri in a hurry. A speed of 110 kms/second implies a travel time of 11,800 years.

One way to look at the Oberth effect is in terms of gravitational potential energy. In the reference frame of the planet, the sum of kinetic energy and potential energy is conserved.

So, consider that when you do a rocket thrust, your rocket thruster pumps some kinetic energy into the system and then the result is your rocket ship going off in one path and the exhaust going off in another path. The total energy will be equal to your initial energy plus the energy provided by the rocket thruster.

But that total energy is split between the rocket ship and the exhaust. The Oberth effect is an observation that your rocket ship ends up with more energy if the exhaust ends up with less energy. By "dumping" the exhaust when you're lower in the gravity well, it ends up in a lower orbit with less energy. Therefore, your rocket ship ends up with more energy.

Furthermore, it's quite easy to calculate from first principles the benefit of a general Oberth maneuver, and helps to make it understandable.

Let's say we're in circular orbit at a distance r around a planet of mass M, such that our orbital speed around the planet is v_cir. Let's say we want to execute a burn that will give us a hyperbolic excess of v_inf — that is, we want to burn now such that we ultimately end up with a speed at infinity of v_inf. (If this were to commit to a Hohmann transfer orbit, then v_inf would be the Hohmann orbit transfer insertion deltavee.)

So we need to make some burn deltav that will give us a total initial speed of v_ini = v_cir + deltav. deltav is what we want to solve for. Well, after our burn leaves us with a speed of v_ini, we make no other burns, and so we're strictly under the influence of gravity. That means that the total energy immediately after our burn is complete E will be equal to our total energy after we've escaped the planet entirely and have ended up with our proper hypberbolic speed, E':

E = E'

Since total energy is the sum of the kinetic and potential energies, then

K + U = K' + U'

The kinetic energies should be obvious; they're just (1/2) m v² for the circular and hyperbolic excess speeds, respectively. For potential energy, this is also relatively straightforward. The potential energies are similarly easy to find since U(r) = -G m M/r. Initially we're at distance r; finally we're at distance r → ∞. So:

(1/2) m v_ini² - G m M/r = (1/2) m v_inf² + 0

We can simplify this by noting that G m M/r is also just the escape speed from the planet at our initial distance, which we'll call v_esc. Substituting and canceling the (1/2) m terms:

v_ini² - v_esc² = v_inf²

Now just substitute the expanded value for v_ini and solve for deltavee:

deltavee = √(v_inf² + v_esc²) - v_cir

• 

A gravity-well maneuver involves what appears to be a contradiction in the law of conservation of energy. A ship leaving the Moon or a space station for some distant planet can go faster on less fuel by dropping first toward Earth, then performing her principal acceleration while as close to Earth as possible. To be sure, a ship gains kinetic energy (speed) in falling towards Earth, but one would expect that she would lose exactly the same amount of kinetic energy as she coasted away from Earth.

The trick lies in the fact that the reactive mass or 'fuel' is itself mass and as such has potential energy of position when the ship leaves the Moon. The reactive mass used in accelerating near Earth (that is to say, at the bottom of the gravity well) has lost its energy of position by falling down the gravity well. That energy has to go somewhere, and so it does — into the ship, as kinetic energy. The ship ends up going faster for the same force and duration of thrust than she possibly could by departing directly from the Moon or from a space station. The mathematics of this is somewhat baffling — but it works.

8.12 Gravity swingbys.

There is one form of velocity increase that needs neither onboard rockets nor an external propulsion source. In fact, it can hardly be called a propulsion system in the usual sense of the word. If a spacecraft flies close to a planet it can, under the right circumstances, obtain a velocity boost from the planet's gravitational field. This technique is used routinely in interplanetary missions. It was used to get the Galileo spacecraft to Jupiter, and to permit Pioneer 10 and 11 and Voyager 1 and 2 to escape the solar system. Jupiter, with a mass 318 times that of Earth, can give a velocity kick of up to 30 kms/second to a passing spacecraft. So far as the spaceship is concerned, there will be no feeling of onboard acceleration as the speed increases. An observer on the ship experiences free fall, even while accelerating relative to the Sun.

• 

In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist maneuver, or swing-by is the use of the relative movement (e.g. orbit around the Sun) and gravity of a planet or other astronomical object to alter the path and speed of a spacecraft, typically to save propellant and reduce expense.

Gravity assistance can be used to accelerate a spacecraft, that is, to increase or decrease its speed or redirect its path. The "assist" is provided by the motion of the gravitating body as it pulls on the spacecraft. The gravity assist maneuver was first used in 1959 when the Soviet probe Luna 3 photographed the far side of Earth's Moon and it was used by interplanetary probes from Mariner 10 onwards, including the two Voyager probes' notable flybys of Jupiter and Saturn.

Explanation

• 

• 

A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational sphere of influence of a planet. The spacecraft's speed increases as it approaches the planet and decreases while escaping its gravitational pull (which is approximately the same), but because the planet orbits the Sun the spacecraft is affected by this motion during the maneuver. To increase speed, the spacecraft flies with the movement of the planet (taking a small amount of the planet's orbital energy); to decrease speed, the spacecraft flies against the movement of the planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun.

A close terrestrial analogy is provided by a tennis ball bouncing off the front of a moving train. Imagine standing on a train platform, and throwing a ball at 30 km/h toward a train approaching at 50 km/h. The driver of the train sees the ball approaching at 80 km/h and then departing at 80 km/h after the ball bounces elastically off the front of the train. Because of the train's motion, however, that departure is at 130 km/h relative to the train platform; the ball has added twice the train's velocity to its own.

Translating this analogy into space: in the planet reference frame, the spaceship has a vertical velocity of v relative to the planet. After the slingshot occurs the spaceship is leaving on a course 90 degrees to that which it arrived on. It will still have a velocity of v, but in the horizontal direction. In the Sun reference frame, the planet has a horizontal velocity of v, and by using the Pythagorean Theorem, the spaceship initially has a total velocity of √2v. After the spaceship leaves the planet, it will have a velocity of v + v = 2v, gaining around 0.6v.

This oversimplified example is impossible to refine without additional details regarding the orbit, but if the spaceship travels in a path which forms a hyperbola, it can leave the planet in the opposite direction without firing its engine. This example is also one of many trajectories and gains of speed the spaceship can have.

This explanation might seem to violate the conservation of energy and momentum, apparently adding velocity to the spacecraft out of nothing, but the spacecraft's effects on the planet must also be taken into consideration to provide a complete picture of the mechanics involved. The linear momentum gained by the spaceship is equal in magnitude to that lost by the planet, so the spacecraft gains velocity and the planet loses velocity. However, the planet's enormous mass compared to the spacecraft makes the resulting change in its speed negligibly small. These effects on the planet are so slight (because planets are so much more massive than spacecraft) that they can be ignored in the calculation.

Realistic portrayals of encounters in space require the consideration of three dimensions. The same principles apply, only adding the planet's velocity to that of the spacecraft requires vector addition, as shown below.

• 

Due to the reversibility of orbits, gravitational slingshots can also be used to reduce the speed of a spacecraft. Both Mariner 10 and MESSENGER performed this maneuver to reach Mercury.

If even more speed is needed than available from gravity assist alone, the most economical way to utilize a rocket burn is to do it near the periapsis (closest approach). A given rocket burn always provides the same change in velocity (Δv), but the change in kinetic energy is proportional to the vehicle's velocity at the time of the burn. So to get the most kinetic energy from the burn, the burn must occur at the vehicle's maximum velocity, at periapsis. Oberth effect describes this technique in more detail.

Purpose

• 

A spacecraft traveling from Earth to an inner planet will increase its relative speed because it is falling toward the Sun, and a spacecraft traveling from Earth to an outer planet will decrease its speed because it is leaving the vicinity of the Sun.

Although the orbital speed of an inner planet is greater than that of the Earth, a spacecraft traveling to an inner planet, even at the minimum speed needed to reach it, is still accelerated by the Sun's gravity to a speed notably greater than the orbital speed of that destination planet. If the spacecraft's purpose is only to fly by the inner planet, then there is typically no need to slow the spacecraft. However, if the spacecraft is to be inserted into orbit about that inner planet, then there must be some way to slow it down.

Similarly, while the orbital speed of an outer planet is less than that of the Earth, a spacecraft leaving the Earth at the minimum speed needed to travel to some outer planet is slowed by the Sun's gravity to a speed far less than the orbital speed of that outer planet. Thus, there must be some way to accelerate the spacecraft when it reaches that outer planet if it is to enter orbit about it. However, if the spacecraft is accelerated to more than the minimum required, less total propellant will be needed to enter orbit about the target planet. In addition, accelerating the spacecraft early in the flight reduces the travel time.

Rocket engines can certainly be used to increase and decrease the speed of the spacecraft. However, rocket thrust takes propellant, propellant has mass, and even a small change in velocity (known as Δv, or "delta-v", the delta symbol being used to represent a change and "v" signifying velocity) translates to a far larger requirement for propellant needed to escape Earth's gravity well. This is because not only must the primary-stage engines lift the extra propellant, they must also lift the extra propellant beyond that, which is needed to lift that additional propellant. Thus the liftoff mass requirement increases exponentially with an increase in the required delta-v of the spacecraft.

Because additional fuel is needed to lift fuel into space, space missions are designed with a tight propellant "budget", known as the "delta-v budget". The delta-v budget is in effect the total propellant that will be available after leaving the earth, for speeding up, slowing down, stabilization against external buffeting (by particles or other external effects), or direction changes, if it cannot acquire more propellant. The entire mission must be planned within that capability. Therefore, methods of speed and direction change that do not require fuel to be burned are advantageous, because they allow extra maneuvering capability and course enhancement, without spending fuel from the limited amount which has been carried into space. Gravity assist maneuvers can greatly change the speed of a spacecraft without expending propellant, and can save significant amounts of propellant, so they are a very common technique to save fuel.

Limits

• 

The main practical limit to the use of a gravity assist maneuver is that planets and other large masses are seldom in the right places to enable a voyage to a particular destination. For example, the Voyager missions which started in the late 1970s were made possible by the "Grand Tour" alignment of Jupiter, Saturn, Uranus and Neptune. A similar alignment will not occur again until the middle of the 22nd century. That is an extreme case, but even for less ambitious missions there are years when the planets are scattered in unsuitable parts of their orbits.

Another limitation is the atmosphere, if any, of the available planet. The closer the spacecraft can approach, the faster its periapsis speed as gravity accelerates the spacecraft, allowing for more kinetic energy to be gained from a rocket burn. However, if a spacecraft gets too deep into the atmosphere, the energy lost to drag can exceed that gained from the planet's gravity. On the other hand, the atmosphere can be used to accomplish aerobraking. There have also been theoretical proposals to use aerodynamic lift as the spacecraft flies through the atmosphere. This maneuver, called an aerogravity assist, could bend the trajectory through a larger angle than gravity alone, and hence increase the gain in energy.

Even in the case of an airless body, there is a limit to how close a spacecraft may approach. The magnitude of the achievable change in velocity depends on the spacecraft's approach velocity and the planet's escape velocity at the point of closest approach (limited by either the surface or the atmosphere.)

Interplanetary slingshots using the Sun itself are not possible because the Sun is at rest relative to the Solar System as a whole. However, thrusting when near the Sun has the same effect as the powered slingshot described as the Oberth effect. This has the potential to magnify a spacecraft's thrusting power enormously, but is limited by the spacecraft's ability to resist the heat.

An interstellar slingshot using the Sun is conceivable, involving for example an object coming from elsewhere in our galaxy and swinging past the Sun to boost its galactic travel. The energy and angular momentum would then come from the Sun's orbit around the Milky Way. This concept features prominently in Arthur C. Clarke's 1972 award-winning novel Rendezvous With Rama; his story concerns an interstellar spacecraft that uses the Sun to perform this sort of maneuver, and in the process alarms many nervous humans.

A rotating black hole might provide additional assistance, if its spin axis is aligned the right way. General relativity predicts that a large spinning mass produces frame-dragging—close to the object, space itself is dragged around in the direction of the spin. Any ordinary rotating object produces this effect. Although attempts to measure frame dragging about the Sun have produced no clear evidence, experiments performed by Gravity Probe B have detected frame-dragging effects caused by Earth. General relativity predicts that a spinning black hole is surrounded by a region of space, called the ergosphere, within which standing still (with respect to the black hole's spin) is impossible, because space itself is dragged at the speed of light in the same direction as the black hole's spin. The Penrose process may offer a way to gain energy from the ergosphere, although it would require the spaceship to dump some "ballast" into the black hole, and the spaceship would have had to expend energy to carry the "ballast" to the black hole.

The Tisserand parameter and gravity assists

The use of gravity assists is constrained by a conserved quantity called the Tisserand parameter (or invariant). This is an approximation to the Jacobi constant of the restricted three-body problem. Considering the case a comet orbiting the Sun and the effects a Jupiter encounter would have, Tisserand showed that

will remain constant (where a is the comet's semi-major axis, e its eccentricity, i its inclination, and a_J is the semi-major axis of Jupiter). This applies when the comet is sufficiently far from Jupiter to have well-defined orbital elements, and to the extent that Jupiter is much less massive than the Sun and on a circular orbit.

This quantity is conserved for any system of three objects, one of which has negligible mass, and another of which is of intermediate mass and on a circular orbit. For example, the Sun, Earth and a spacecraft, or Saturn, Titan and the Cassini spacecraft (using the semi-major axis of the perturbing body instead of a_J.) This imposes a constraint on how a gravity assist may be used to alter a spacecraft's orbit.

The Tisserand parameter will change if the spacecraft makes a propulsive maneuver or a gravity assist of some fourth object. This is one reason why many spacecraft frequently combine Earth and Venus (or Mars) gravity assists or also perform large deep space maneuvers.