This is the work-room of the spacecraft navigator. The commonly used term is "astrogator", coined by Robert Heinlein.
What's in the astrogation room?
Everything needed for interplanetary navigation. Instrument to determine the ship's current trajectory and calculating devices to plot new trajectories.
There are many navigational instruments. A periscope sextant to take navigational readings, with its azimuth ring. (In THE REVOLT ON VENUS, this is what Roger Manning was looking through when he noticed the atomic bomb attached to the Polaris' tail) There also might be a goniometer, which is used to measure angles. A good-sized telescope, either in a dome or with a coleostat. (The periscope, the telescope, or both will be equipped with a filar micrometer.) Star trackers, star scanners, solar trackers, sun sensors, and planetary limb sensors and trackers. Inertial tracking repeaters (note that the inertial tracker platform will have to be manually realigned every twelve hours because it tends to drift. The star tracker is used for reference.). There might even be a pulsar positioning system.
In addition to navigational instruments there will be other necessary gear. There will be an incredibly precise chronometer. An integrating accelerograph (displaying elapsed time, velocity, and distance in dead-reckoning). An indicator of the spacecraft's current mass ratio. An integral audio recorder and a log book for radio messages and navigational fixes.
Not to mention lots of paper, pencils, slide rules, and ballistic calculators. Or instead a laptop computer with an AstrogateMeTM app.
Secondly, readouts for the ship's navigational and tactical sensors. The big radar scope. Doppler radar and radar altimeter. This might be a separate deck, if you think it is insane to have a single crewperson responsible for navigation, detection, and communication like in Tom Corbett Space Cadet.
Thirdly communication gear, perhaps even with something like a Morse code key for use when radio interference becomes a problem (If this was a Metalunan ship, this is where you'd find the interociter).
There might be a separate communications deck, which is generally called a "radio shack", crewed by a communication officer whose nickname is "Sparks." If this is a military spacecraft this might be the place for the safe containing the code book. Hit the red "incinerate" button to keep the one-time pad and Captain Midnight secret decoder ring from falling into enemy hands. On some ships this safe might be in the captain's cabin.
In a science-fiction universe with Discontinuous ("teleport-like" or "jump") faster-than-light drives in their starships, traditionally the astrogator's job after each instantaneous jump was to establish the location the starship materializes at, with micrometer precision. This has to be done before the astrogation calculation for the next jump can be performed (for each jump you have to know where you are starting from). This is usually done by using spectroscopy to identify three or more stars, to locate the starship's position by triangulation.
Many of the navigational instruments might be mounted inside an "astrodome", which is a blister dome of some strong but transparent material used with a manual sextant as a back-up to the periscope. (Note that astrodomes cause optical distortion that need a mathematical correction.)
If there is an astrodome, the room will have alternative lighting that is all red, like a darkroom. This is to preserve night vision. It should also have a retractable shield. This is to preserve day vision in case the rotation of the ship moves the eye-destroying fury of the Sun into view. The shield is not only useful to keep sunlight out, but to keep the atmosphere in, in case the astrodome is breached or shattered. Not to mention protecting the astrodome from melting if the ship does some aerobraking in a planetary atmosphere.
If the ship spins on its axis for artificial gravity, it might be a good idea to locate the astrodome in the nose of the ship, i.e., at the center of the axis of rotation. A tiny room with the astrodome in it could be counter-spun. So while the ship was spinning, the room would be stationary, freeing the astrogator from the difficulty of making observations of a sky that is madly spinning about. The problem is that if this is a nuclear powered ship, the docking port has to be on the nose. It is possible to rig in a coleostat a shutter that is synchronized with the spin of the ship. This will provide a stroboscopic but steady image if you cannot counter-spin the astrodome.
If the ship is advanced enough to have an actual centrifuge, instead of spinning the entire ship, things will be easier. Just make sure the astrodome is on the stationary part of the ship.
- Orbit Determination: Knowing the spacecraft's current position and velocity, and predicting future position and velocity.
- Flight Path Control: Calculating maneuvers for the pilot to alter the spacecraft's trajectory in the desired direction.
The astrogator is responsible for offering the Captain a range of solutions for the mission the captain orders, plotting the course for the chosen solution, giving the pilot the specifications for the required maneuvers needed to implement the course, and to monitor the progress of the spacecraft along the course while calculating mid course corrections for the pilot in order to keep the ship in the groove.
Nowadays there will be no astrogator. The captain will type the desired mission into their cell phone's astrogator/pilot app and let it do all the math and ship piloting. From an author's point of view this is a disaster due to Burnside's Zeroth Law. One possible solution is to make the personnel on the spacecraft not be "crew" so much as system managers. Rick Robinson points out that you'll need a human astrogator if something drastically unexpected happens. For instance, if the unexpected arrival of a Klingon invasion fleet unexpectedly overlaps the optimal trajectory delivered by the astrogation computer.
Actually calculating interplanetary trajectories is true rocket science, and beyond the scope of this website (translation: I don't know how to do it). If you simply must know how, a good starting text is Fundamentals of Astrodynamics by Roger Bate, ISBN: 0486600610. The book assumes you are already well versed in calculus.
The captain of the spacecraft will ask the astrogator for a mission plan to travel from point A to point B in time T. The astrogator will determine a family of mission plans, with the current ship's delta-V capacity as the upper limit (or the ship will not be capable of performing that mission) and with the captain's specfied mission time as the lower limit (or the captain will be unhappy). You see, a Hohmann trajectory generally uses the least delta-V, but also has the longest possible mission time, and the mission can only start on specific dates ("launch windows") as well. By increasing the delta-V used the mission time can be reduced.
What the astrogator will do is have the navigation computer draw a pork-chop plot, which is a graph with departure times on one axis, arrival times on the other axis, and delta-V requirements drawn as contour lines in the graph. Cross out the areas of delta-V that are too high for the spacecraft, cross out the part of the graph with a mission duration that is too long, what remains are the possible missions.
If it turns out there is no possible mission inside the stated parameters, the astrogator will have to confer with the captain over what is possible.
Once the specific mission is chosen, with delta-V and duration time, the astrogator does the hard part calculating the trajectory, burn vectors, and check-points. If the SF author wants to go full Heinlein and do that, I refer them to Fundamentals of Astrodynamics or equivalent.
The given mission composed of a series of trajectories. At each point where the spacecraft makes a transition from one trajectory to another is a "maneuver". A maneuver is where the spacecraft uses a burn of its rockets to alter its vector to the new trajectory.
For each maneuver, the astrogator will calculate three maneuver parameters for the pilot:
- The Axis of Acceleration (where the ship's nose should be pointing during the burn)
- The required amount of Delta V (pilot will figure the proper engine thrust setting and burn duration for this)
- The starting time of the maneuver (this should happen at the mid-point of the burn duration, pilot will calculate this. Figure burn duration required for delta V, divide by 2, and subtract from astrogator-supplied maneuver time)
These will be passed to the pilot. If the pilot finds a problem (such as the spacecraft not possessing enough propellant reserves to create the required delta V) they will yell at the astrogator, who will have to frantically recalculate to fix the problem.
- Apoapsis In an orbit, the point of the orbit farthest from the astronomical body currently being orbited.
- Periapsis In an orbit, the point of the orbit closest to the astronomical body currently being orbited. Some like to replace the "-apsis" part with the name of the body being orbited, but that gets out of hand real quick. For example "perigee", "perihelion", "pericynthion", and zillions of other unwieldy terms.
- Prograde In the direction of the spacecraft's trajectory, i.e., "forwards". Fun fact: since the trajectory is curved, prograde is actually at a tangent to the trajectory.
- Retrograde In the opposite direction of the spacecraft's trajectory, i.e., "backwards". 180 degrees from Prograde.
- Normal At 90 degrees (perpendicular) to the spacecraft's orbital plane, in the orbital "North" direction (using the "right-hand rule").
- Anti-normal At 90 degrees to the spacecraft's orbital plane, in the orbital "South" direction. 180 degrees from Normal.
- Radial in In the direction of the astronomical body currently being orbited.
- Radial out In the opposite direction of the astronomical body currently being orbited. 180 degrees from Radial in.
When planning maneuvers, astrogators will keep in mind the general rules of orbital mechanics.
Burning with the axis of acceleration pointed in the prograde direction ("burning prograde") will expand the size of the orbit. Burning retrograde will contract the size of the orbit. In both cases, the point on the orbit the spacecraft is currently occupying stays put, that is the center point of the orbit expansion/contraction. Usually the burns are done when the spacecraft is at either the periapsis point (closest to the planet) or the apoapsis point (farthest from the planet).
In the diagrams below please remember that Rockets Are Not Arrows. The spacecraft does not have to travel in the direction the nose is pointing, like aircraft do. For instance, in the Burning "Retrograde at Periapsis" diagram below, the spacecraft is traveling counter-clockwise even though its nose is pointed clock-wise and flames are shooting out its rear. The thrust is slowing the ship down, not forcing it to move clock-wise.
Burning prograde at periapsis will raise your apoapsis (move it farther away from planet). Burning retrograde at periapsis will lower your apoapsis (move it closer to the planet).
Burning prograde at apoapsis will raise your periapsis. Burning retrograde at apoapsis will lower your periapsis.
This works if you burn at other points in the orbit besides apoapsis and periapsis, but you'll probably only be using apo and peri. Unless you work at NASA or play a lot of Kerbal Space Program. The general rule is that if you burn prograde at a given point in the orbit, the point in the orbit on the exact opposite side of the planet will increase its orbital radius by the maximum amount, and the points in between will increase their radii by lesser amounts, shading down to an increase of zero at the ship's location.
You "circularize" an orbit by making the periapsis and apoapsis the same distance from the planet, i.e., you make the orbital eccentricity close to zero, thus making the orbit a circle instead of some kind of egg shape.
What Is This Used For?
The main use is Changing Orbits. And the most important example of changing orbits is using a Hohmann transfer to another planet.
If the destination orbit is farther from the primary than the starting orbit (technical term is "superior", Mars' orbit is superior to Terra's):
- BURN 1: Move your current apoapsis outward until it touches the destination orbit by burning prograde at periapsis. Your orbit has been altered into a Transfer Orbit. You have departed from your starting planet and are en route to your destination.
- COAST PHASE: Coast until you reach new apoapsis
- BURN 2: Circularize your orbit by moving your current periapsis outward until it also touches the destination orbit by burning prograde at apoapsis. You have matched the Solar orbit of the destination planet. You will have to burn just a little bit more to start orbiting the planet instead of Sol.
If the destination orbit is closer to the primary than the starting orbit ("inferior orbit"), you do the inverse:
- BURN 1: Move current periapsis inward until it touches the destination orbit with a retrograde apoapsis burn. Your orbit has been altered into a Transfer Orbit
- COAST PHASE: Coast until you reach the new periapsis
- BURN 2: Circularize your orbit by moving current apoapsis inward until it also touches the destination orbit by burning retrograde at periapsis
When trying to rendezvous with a spacecraft or space station, you have to match the position, vector, and velocity. Matching the velocity is a little counter-intuitive. The technical term is "orbital phasing".
Every object in a given circular orbit is moving at the same speed. So if the Polaris is in a 400 km circular orbit 1,000 km behind a Blortch warship in the same orbit, it will never catch up and will never be left behind. It will eternally be 1,000 km behind.
The Polaris will never catch up if they both are in an elliptical orbit either. They will move at different speeds at different parts of the orbit, but always at the same speed at a given point in said orbit so it all evens out. For instance when the Polaris passes through apoapsis it will be moving at the same speed as when the Blortch moves through apoapsis.
The point being: for the Polaris to move faster than the Blortch warship it has to contract the radius of its orbit (lower the altitude). To move slower than the Blortch, the Polaris has to expand the radius of its orbit (raise the altitude).
Here's the confusing part: to lower the altitude of the orbit (thus increasing your orbital velocity) you have to burn retrograde. Burning retrograde means you are slowing down, since you are thrusting contrary to your orbital vector. See the confusion? To speed up, you slow down. Actually, burning retrograde means you are contracting your orbital radius but using naive reasoning it sure looks like you are putting on the brakes.
So here's the deal: the Polaris burns retrograde into a lower orbit than the Blortch, circularizing the orbit at the desired orbital speed.. It moves faster in this orbit, thus catching up with the Blorch. When the Blorch is almost "overhead" (i.e., on a line connecting the planet's center, the Polaris, and the Blorch) the Polaris burns prograde into a higher orbit then circularizes, matching the orbital radius of the Blortch orbit.
Burning radial in (towards the primary) or radial out (away from the primary) will spin the entire orbit in place. This only has a noticable effect if the orbit is egg shaped. The orbit can only be spun a maximum of 90 degrees clockwise or counterclockwise. These burns are not used very much, since it is almost always more efficient to use prograde / retrograde burns to do the same thing.
If your orbital plane is tilted at a different angle with respect to the desired new orbital plane, you will have to match orbital inclination. This is the first step to making a rendezvous with a planet or docking with another spacecraft. This is also notoriously the most expensive maneuver in terms of delta V.
Where the two orbital planes cross each other are two "nodes", the ascending node and the descending node. At either of the nodes, you burn normal or anti-normal (depending upon the angle of the new orbital plane with respect to the old one, at that node). "Normal" means "at 90° to the orbital plane, in direction of right hand rule." After burning an exorbitant amount of propellant, you will have changed to the destination plane.
The procedure is much the same whether one is trying to leave an interplanetary trajectory to enter orbit around a planet or trying to dock to another spacecraft in orbit around the same planet as you are. In the first case the "target" is the orbit around the planet, in the second case the target is the ship one is docking to.
The goal is for your spacecraft to match both the target's position and vector. That is, you want to be at the target spacecraft or planet's location, moving at the same velocity and in the same direction.
First match orbital inclination with the target.
Secondly change orbit shape so that your orbit is at a tangent to the target orbit, preferably at your apoaspsis or periapsis.
Thirdly, rotate the apoapsis and periapsis such that the tangent point will be at the target's location when you arrive.
When you arrive at the tangent point and the target, change orbit shape to match the target's orbit. When making a planetary rendezvous, your spacecraft will commonly have lots of velocity that has to be gotten rid off. Often aerocapture is used to avoid having to burn lots of expensive propellant.
However, if the maneuver is a spacecraft docking, you will fail the rendezvous if you do not take into account the complicated interplay of tidal, Coriolis, and centrifugal forces acting upon the docking spacecraft. The linked paper ("The Delicate Dance of Orbital Rendezvous") goes into these in excruciating detail, including lots of ferocious equations with nasty pointed teeth. If you do read the paper, don't miss section III: The Stranded Astronaut.
Altering the spacecraft's trajectory so that the periapsis is inside the atmosphere of the planet being orbited. The spacecraft will slow down due to atmospheric drag. The general rule is that aerobraking can kill a velocity approximately equal to the escape velocity of the planet where the aerobraking is performed (10 km/s for Venus, 11 km/s for Terra, 5 km/s for Mars, 60 km/s for Jupiter).
Can be a prelude to landing, can also be used to slow the spacecraft into a capture orbit ("aerocapture") without having to expend any expensive propellant.
Warning: if the drag and/or heat from friction becomes too strong, bits of the spacecraft will be torn off or melted away. If the drag becomes monstrously strong the entire spacecraft will be shredded or melted away. If you have an astrodome, be sure to protect it by closing the retractable shield. The plasma sheath of ionized atmosphere will cut off radio communcation.
To get an idea of what the bare minimum is, we will unashamedly be taking a good look at the solution in the computer game Kerbal Space Program. Since that is a game, the designers were forced to distill the controls to the very essentials (because the players will quickly get fed up and leave if they think the game is too complicated). As a matter of fact, that game is so wonderfully educational yet fun, you might be better off if you skipped this section of the website and instead started playing the Kerbal game.
The science fictional astrogation user interface an author invents for their novel does not have to look anything like this. But it does have to offer the same options and functionality.
In Kerbal Space Program there is a solar system map display. This displays the planets in their orbits and the ship in its trajectory (including altitude, position and time of apoapsis and periapsis). To create a maneuver, the player/astrogator uses something called the "maneuver node tool."
In broad over view: player will click on the ship's trajectory to create a new maneuver node. The node has six "controls" on it. By tugging on the controls, the ship's trajectory will be bent in various directions. The player manipulates the the six controls until the desired new trajectory is created. The three components of the the maneuver will be automatically calculated (acceleration axis, delta V, and manuever start time) and displayed on the pilot's Nav Ball.
In more detail:
The position of the maneuver node determines the maneuver starting time. Basically, when the spacecraft crawling along the trajectory reaches the position of the manuever node, it is time to start the manuever.
On the maneuver node, there is one control for each of the six burn directions: prograde, retrograde, normal, anti-normal, radial in, and radial out. Selecting and dragging a given control will set the desired velocity change in that direction. Pulling the control away from the center of the node increases the velocity, pulling it closer decreases it (the equivalent of pulling the control on the opposite side of the node). One can burn in several directions at once, the control will calculate the appropriate axis of thrust and delta V so that it is the equivalent of the vector sum of all desired burns.
In other words: the astrogator create a maneuver node, play around with the node's six interactive controls to bend the trajectory, until the new bent trajectory looks like what the astrogator wants.
So between the position of the manuever node and the values for the six burn directions the acceleration axis, delta V and maneuver start time can be calculated and relayed to the pilot's nav ball. This sure beats using slide-rules, drawing curves on paper charts, filling out a FORM 235 PILOT MANEUVER ORDER in triplicate, and climbing the ladder to the control deck to give one copy to the pilot.
Remember that prograde / retrograde burns are used to change orbit shape, and normal / anti-normal burns are used to change orbital inclination. Radial out / radial in burns are used to rotate the orbit, but that isn't used very much. Don't forget that normal / anti-normal burns are very expensive in terms of delta V.
During the mission the astrogator will periodically check the spacecraft's current position, vector (speed and direction it is traveling in), and point in time to ensure that the ship is on course. Astrogators know that pilots are only human, and no maneuver is 100% perfect. And they know that astrogators are only human as well, unavoidable perturbations can creep in.
If the spacecraft is leaving the required trajectory, mid-course corrections (Trajectory Correction Maneuver or TCM) will be needed, which the astrogator will calculate. This is a vector that will correct the spacecraft into the desired trajectory.
Say Roger want's to fix the position of the Polaris. From the ephemeris he knows where Terra is, and thus the Sol-Terra line. The ephemeris also tells him where Venus is, and thus the Sol-Venus line. Roger uses the periscopic sextant to measure angle A and angle B. With simple geometry the Polaris' current position is fixed. Of course this is an approximation based on assuming that everything is in the plane of the ecliptic. If the course gets more three dimensional a third angle will be required.
The spacecraft's vector isn't quite so simple. You will have to wait a while, make a second position fix, and calculate what the vector had to be. If you are inside a solar system you can use the observed positions of the planets against the background of stars. The positions can be precalculated at a checkpoint. When that checkpoint is reached, the planet's position is measured with a telescope. If the planet is not at the calculated position, you are off-course. Currently such observations have an accuracy on the order of 5 μ-radians, or about 750 kilometers at one astronomical unit.
Currently I have no idea how to calculate what sort of delta-V requirements TCMs will need. In Proceeding of the Symposium on Manned Planetary Missions 1963/1964 they suggested that with then-current navigation gear the total delta V required for TCM on the Terra-Mars trajectory was typically about 105 m/s and the Mars-Terra trajectory would 92 m/s.
If you are close to a planet, the distance to it can be determined by radar. Further away, the filar micrometer in the periscope can be used to determine the angular size of the planet. Since the planet's diameter is known, simple trigonometry will yield the distance. A filar micrometer is an instrument mounted in a telescope. It displays two cross hairs that can be positioned with dials (one dial rotates the micrometer, the other adjusts the distance between the two cross hairs). Once set, the angular separation between the two cross hairs can be read from the scale.
Astronomers and space engineers are currently working on a way to navigate a spacecraft by using pulsars, see below.
For NASA space probes, and future spacecraft operating in the civilized sections of a solar system, things are easier due to ground support. A ground installation can see the position of your spacecraft relative to that planet. The ground installation optically sees your spacecraft's right ascension and declination. The ship and the installation trade radio pulses with time stamps on them, lightspeed lag yields the distance. Two angles and a distance gives your spacecraft position in spherical coordinates, relative to the planet. The planets position is known, correct for that an you have your spacecraft's position. Doppler radar will even give you the component of your velocity normal to the planet. All this can be had if you've paid your fees to ground installation.
In a dense asteroid drift a variable-baseline stereoscopic radar could come in handy. Look through the double eyepiece and you'll see the surrounding asteroids in 3-D. Use the sweep control to pan the view fore, aft, port, or starboard. The pilot might have one of these as well. Keep in mind that there does not appear to be any "dense astroid drifts" in our solar system, outside of Saturn's rings.
The next section in the Star Trek nav text is how to cope when your subspace radio is non-functional. The astrogator can use naturally occuring pulsars for navigation (Navigator Chekov sniffs "how primitive!"). The pulsars are taking the place of GPS satellites. Observing the pulsar frequency gives its distance from the spacecraft.
You might want to use this handy table of the 14 pulsars used in the map. Hey, if it is good enough for NASA, it's good enough for you. Table 3 has each pulsar's RA (right ascension), DEC (declination) and distance in parsecs (multiply by 3.26 to convert to light years). Use this with the technique I give here to plot your very own three D star map of navigational pulsars.
Bertolomé Coll at the Observatoire de Paris in France and Albert Tarantola have proposed a system using pulsars as a GPS for the solar system (not for insterstellar space). They suggest using pulsars PSR J0751+1807, PSR J2322+2057, 0711-6830 and 1518+0205B. These form a rough tetrahedron centered on the Solar System. The UK’s National Physical Laboratory and the University of Leicester are working with the European Space Agency to investigate pulsar methods for spacecraft in the solar system. The Royal Astronomical Society is looking further afield at interstellar navigation.
A given pulsar's signal can only be seen from certain locations, so the interstellar navigator needs a large list of pulsars to ensure that at least three on the list are visible from the ship's current location. This is because the beam from the pulsar's magenetic north pole and the beam from the magnetic south pole sweep out along the surface of a a cone centered on either the north or south rotational axis, respectively. If the ship is not on the surface of the cone, the pulsar is invisible. Keep in mind that the surface is rather thick. If you can see a pulsar from the orbit of Mercury, you will still be able to see it from the orbit of Pluto. This means astrogators who stay within the solar system can make do with a list of four pulsars.
There is an easier way than solving simultaneous equations. Use Trilateration. The way I understand it, first you have to rotate the coordinate system so that all three sphere centers have a Z-coord of 0, sphere one's center is at the origin, and sphere two's center is on the X-axis. You then perform trilateration, and rotate the result back to the original coordinate system. I'd go into more detail, but I'm still trying to wrap my brain around the problem. A Google search on "calculation intersection three spheres" will yield all sorts of algorithms and Matlab scripts.
By "easier" I mean "easier than randomly selecting x, y, and z values until you stumble over the solution."
The section below applies mainly to interstellar travel, or situations where the start and ending locations are stationary relative to each other, and the course between is a straight line. This is not true in interplanetary travel, where the start and destination planets are moving in their orbits, and the sun's gravity bends the course into a curved line. If you want to calculate that, read the aforementioned Fundamentals of Astrodynamics.
The astrogator will be performing plenty of math in the astrogation room, with assistance from whatever level of technology they are allowed or have access to. This can range from pencil-and-paper to slide rules to analog computers to ballistic integrators to full blown electronic digital computers.
These tools will be used to calculate the maneuvers for the mission: start time, delta V, and axis of acceleration (perhaps in the form of guide star settings for a coelostats if the pilot has no gyro horizon instrument). If this is a pre-transistor ship, all the books, slide rules and whatnot should be magnetized to stick to the desk, be on tethers, under elastic straps, or otherwise restrained so they don't float around the room. (Or turn into deadly missiles if the spacecraft has to abruptly accelerate. Spacers have a fastidious horror of unsecured objects.) For Tom Corbett fans, the ephemeris is the functional equivalent of Roger's space charts.
Doing astrogation the old school way is a nightmare. They will have a current ephemeris, a book of nine-place logarithms, rulers, dividers, protractors, pads of light green Keuffel & Esser graph paper, realms of scratch paper and lots of pencils. And a pencil sharpener designed to capture every last shaving. You don't want electrically conductive bits of graphite floating into the circuitry.
Logarithms were the mathematical marvel of the age back in the 1600s. Pierre-Simon Laplace called logarithms "...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations."
How do logarithms help so much?
Well, if you are old enough that you were actually taught how to add, subtract, multiply, and divide using pencil and paper you probably noticed that addition was so much easier and quicker than multiplication. Here's the trick: if you take two numbers, convert each number to its corresponding logarithm, add the logarithms, then convert the result from a log back to a number (the antilogarithm), the result is the two numbers multiplied. And if you subtracted logarithms, the result was division.
John Napier popularized this method in his book Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). After that, various people published books full of logarithmic tables. I still have a small pocket book of logarithms I used in junior high in the early 1970s.
Slide rules are even quicker to multiply numbers, because you do not have to do the conversion to logarithms and conversion to antilogarithms. The rule handles that automatically.
Besides multiplication and division, fancy slide rules can also handle trigonometry.
Even better, if in your line of work (say, if you were an astrogator), it is possible to make a specialized slide rule that solves a specific complicated mathematical equation. As an example, here are the instructions to make your very own Nuclear bomb effects circular slide rule.
Airplane pilots still use E6-B flight Computers, which are part circular slide rule and part analog computers. Just in case of instrument failure.
In 1895, a Japanese firm, Hemmi, started to make slide rules from bamboo, which had the advantages of being dimensionally stable (doesn't swell or shrink with the humidity), strong and naturally self-lubricating. It was only later they were made of celluloid, plastic, or painted aluminium.
Of course I personally would be thrilled to have some sort of hand-waved FTL drive that has the side effect of forcing the use of slide rules. It would be so deliciously retro. I keep trying to come up with one, but so far none my inventions has been free of unwanted side effects. It's hard to think of something that will kill a computer but not the crew.
Late breaking news, Karl Gallagher thought of a pretty good reason to use slide rules in his novel Torchship. You can read all about it in the novel.
The Nomogram (Nomograph) or "Alignment Chart" was invented by the French mathematicians Massau and M. P. Ocagne in 1889. It is a set of scales printed on a piece of paper that will solve a specific equation. Given the all but one of the values for the equation, it will solve for the unknown value. A ruler or straight edge is laid across the scales at the points corresponding to the known values, and the unknown value can be read off directly.
These were very popular with engineers up to about the 1950's. They were quicker than using a slide rule, since they were pre-set for a specific equation. Engineers had entire books filled with nomograms.
They also allowed engineers to off-load some of their donkey-work to assistants and apprentices. The tedious bulk calculations were farmed out by giving each assistant a list of values, some blank paper, and a photocopy of the relevevant equation. The assistant might have shakey math skills but it doesn't take much brain power to lay a straight-edge on a diagram.
I have a tutorial on how to make your own nomographs here.
As an example, you can play with my handy-dandy DeltaV nomogram. Download it, print it out, and grab a ruler or straightedge. You can also purchase an 11" x 17" poster of this nomogram at . Standard disclaimer: I constructed this nomogram but I am not a rocket scientist. There may be errors. Use at your own risk.
Say we needed a deltaV of 36,584 m/s for the Polaris, that's in between the 30 km/s and the 40 km/s tick marks on the DeltaV scale, just a bit above the mark for 35 km/s. The 1st gen Gas Core drive has an exhaust velocity of 35,000 m/s, this is at the 35 km/s tick mark on the Exhaust Velocity scale (thoughtfully labeled "NTR-GAS-Open (H2)"). Now, lay the straightedge between the NTR-GAS-Open tick mark on the Exhaust Velocity scale and the "2" tick mark on the Mass Ratio scale. Note that it crosses the DeltaV scale at about 24 km/s, which is way below the target deltaV of 36,584 m/s.
But if you lay the straightedge between the NTR-GAS-Open tick mark and the "3" tick mark, you see it crosses the DeltaV scale above the target deltaV, so you know that a mass ratio of 3 will suffice.
The scale is a bit crude, so you cannot really read it with more accuracy than the closest 5 km/s. You'll have to do the math to get the exact figure. But the power of the nomogram is that it allows one to play with various parameters just by moving the straightedge. Once you find the parameters you like, then you actually do the math once. Without the nomogram you have to do the math every single time you make a guess.
As with all nomograms of this type, given any two known parameters, it will tell you the value of the unknown parameter (for example, if you had the mass ratio and the deltaV, it would tell you the required exhaust velocity).
Note that the Exhaust Velocity scale is ruled in meters per second on one side and in Specific Impulse on the other, because they are two ways of measuring the same thing. In the same way, the Mass Ratio scale is ruled in mass ratio on one side, and in "percentage of ship mass which is propellant" on the other.
Nomograms have an advantage over a raw mathematical equation when it comes to visualizing the range the solution resides in. The value that cannot change becomes the fixed "pivot point", and the straight edge is pivoted to see the various trade-offs. For example:
Download and print out my Transit Time Nomogram.
Let's say that our spacecraft is 1.5 ktons (1.5 kilo-tons or 1500 metric tons). It has a single Gas-Core Nuclear Thermal Rocket engine (NTR-GAS MAX) and has a (totally ridiculous) mass ratio of 20. The equation for figuring a spacecraft's total DeltaV is Δv = Ve * ln[R]. On your pocket calculator, 98,000 * ln = 98,000 * 2.9957 = 300,000 m/s = 300 km/s. Ideally this should be on the transit nomogram, but the blasted thing was getting crowded enough as it is. This calculation is on a separate nomogram found here.
The mission is to travel a distance of 0.4 AU (about the distance between the Sun and the planet Mercury). Using a constant boost brachistochrone trajectory, how long will the ship take to travel that distance?
Examine the nomogram. On the Ship Mass scale, locate the 1.5 kton tick mark. On the Engine Type scale, locate the NTR-GAS MAX tick mark. Lay a straight-edge on the 1.5 kton and NTR-GAS MAX tick marks and examine where the edge crosses the Acceleration scale. Congratulations, you've just calculated the ship's maximum acceleration:2 meters per second per second (m/s2).
For your convenience, the acceleration scale is also labeled with the minimum lift off values for various planets.
So we know our ship has a maximum acceleration of 2 m/s2 and a maximum DeltaV of 300 km/s. As long as we stay under both of those limits we will be fine.
On the Acceleration scale, locate the 2 m/s2 tick mark. On the Destination Distance scale, locate the 0.4 AU tick mark. Lay a straight-edge on the two tick marks and examine where it intersects the Transit time scale. It says that the trip will take just a bit under four days.
But wait! Check where the edge crosses the Total DeltaV scale. Uh oh, it says almost 750 km/s, and our ship can only do 300 km/s before its propellant tanks run dry. Our ship cannot do this trajectory.
The key is to remember that 2 m/s2 is the ship's maximum acceleration, nothing is preventing us from throttling down the engines a bit to lower the DeltaV cost.
This is where a nomogram is superior to a calculator, in that you can visualize a range of solutions. This is the "pivot point" technique I was talking about earlier.
Pivot the straight-edge on the 0.4 AU tick mark (meaning, imagine there is a pin stuck in the nomogram at 0.4 AU that the straight-edge rotates around). Pivot it until it crosses the 300 km/s tick on the Total DeltaV scale. Now you can read the other mission values: 0.4 m/s2 acceleration and a trip time of a bit over a week. Since this mission has parameters that are under both the DeltaV and Acceleration limits of our ship, the ship can perform this mission (we will assume that the ship has enough life-support to keep the crew alive for a week or so).
Of course, if you want to have some spare DeltaV left in your propellant tanks at the mission destination, you don't have to use it all just getting there. For instance, you can pivot around the 250 km/s DeltaV tick mark to find a good mission. You will arrive at the destination with 300 - 250 = 50 km/s still in your tanks.
You can think of analog computers as "steampunk computers." Probably no actual steam but they will have zillions of gears and cams. It uses using tiny electric motors to drive mechanical shafts and gears. These position shafts to represent some mathematical value, and drive cams shaped to represent mathematical functions or statements. It is used to solve navigational equations.
An example is a ballistics integrator.
If you want the precise details about how to make a computer out of cams, differentials, and gears, read Basic fire Control Mechanisms, OP 1140, (1944). It is available as a free download here. Below are just some of the components.
Lynn Albritton wanted a simple project to help learn how to use her CAD software. Her idea of "simple" is a bit more ambitious than mine. She designed an analog computer that calculates sine and cosine. Called an "Ideal Harmonic Transformer", she has made the blueprints available on Thingiverse so those with access to a 3D printer can make one for their very own.
In many of the Heinlein novels, computers capable of doing interplanetary navigation were not portable. Large computers would pre-compute the courses. And do emergency re-computations when they got a panicked radio message from a ship in trouble.
Remember that early computers are going to give their results by spitting out Hollerith punch cards, punched tape/ticker tape, or printed fanfold sheets. Standard CRT monitors displaying text come later, and monitors with cute graphic user interfaces (such as a maneuver node tool) come later still.
Actually there will still probably be manual equipment, in case the computer gets fried by a solar storm or the EMP from a nearby nuclear weapon detonation. A slide rule will be in a box on the hull, with a sign that says "In case of EMP, break glass."
Computers, whether analog or digital, should be of the 'I-tell-you-three-times' variety. It is actually three computers, each of which does the calculation. If operating perfectly, all three answers will be the same. If a malfunction occurs, two answers will agree and one won't. Use the answer the two agree on, which will allow you to get though the burn. Then fix the bad computer, pronto! If all three disagree, it's time to break out the slide rule.
Other critical instruments might be in triplicate as well. If you have one clock, you know the time. If you have two clocks, you are never quite sure, since they probably won't agree with each other. But if you have three clocks, you take a reading from the two clocks with values closest to each other, and assume that the actual time is somewhere in between.