Equations - Atomic Rockets

Introduction

Most of these are pretty obvious, but are included for completeness. Be sure to examine the DeltaV nomogram and the Acceleration nomogram. Some of these equations are available in the Atomic Rocket Tiddly Wiki. And don't miss the list of on-line calculators.

Units and Functions

Element	Value
e	2.71828...
kg	kilograms
m	meters
N	newtons
π	3.14159...
s	seconds
w	watts
ln[x]	natural logarithm of x
sqrt[x]	square root of x

Information about the mass and radius of various planets can be found here: http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

Adapted from an image courtesy of NASA

Index

A: Acceleration of spacecraft (m/s²)

A = F / M_c
A = (mDot * V_e) / M_c
A = (mDot * g₀ * I_sp) / M_c

A_pg: Acceleration of spacecraft in terms of planetary gravities (g_p)

A_pg = A / g_p

Δ_v: Spacecraft's total change in velocity capability (deltaV) (m/s)

ΔV = V_e * ln[R]
ΔV = g₀ * I_sp * ln[R]

Δ_vc: Spacecraft's current deltaV capability (m/s)

Δ_vc = V_e * ln[M_c / M_e]
Δ_vc = g₀ * I_sp * ln[M_c / M_e]

Δ_vt: Spacecraft's burn-time deltaV (m/s)

Δ_vt = -V_e * ln[M_c - (mDot * t_b)]

Δ_vd: Planet's gravitational drag (m/s)

DeltaV required to counteract gravitational drag during liftoff or landing.
Δ_vd = g_p * T_l
Δ_vd = D_vo / A_pg

Δ_vo: Planet Surface to Orbit Delta V (m/s)

DeltaV to lift off from planet surface into orbit (or to land from orbit) without taking into account planet's gravitational drag or atmospheric drag.
Δ_vo = sqrt[ (G * P_m) / P_r ]

e: Base of natural logarithms

e = 2.71828...

F: Propulsion system's thrust (N or kg m/s²)

F = M_c * A
F = mDot * V_e
F = mDot * g₀ * I_sp
F = (M_pb * V_e) / T_b
F = (2 * F_p) / V_e

F_p: Propulsion system's thrust power (w)

F_p = (mDot * (V_e²)) / 2
F_p = (F * V_e ) / 2
F_p = (M_pb * (V_e²)) / (2 * T_b)

F/F_p: Propulsion system's thrust efficency (impulse per unit energy) (N/w)

F/F_p = 2 / V_e

F_se: Spacecraft's Specific Exhaust (w/kg)

Watts per kilogram of accelerated mass
F_se = (A * V_e ) / 2

F_sp: Spacecraft's Specific Power (w/kg)

F_sp = F_p / M_e

g₀: Acceleration due to gravity at Terra's surface (m/s²)

g₀ = 9.81

g_p: Acceleration due to gravity at planet's surface (m/s²)

This must be looked up for the planet in question. It is equal to g₀ for Terra.

G: Newton's gravitational constant (N m² kg^-2)

G = 0.00000000006673 = 6.673e-11

GLOW: Spacecraft's maximum gross liftoff weight (kg)

Note that A_pg must be greater than one, preferably much greater to reduce planet's gravitational drag D_vd.
GLOW = F / (A_pg * g_p)
GLOW = (mDot * V_e) / (A_pg * g_p)
GLOW = (mDot * g₀ * I_sp) / (A_pg * g_p)

I_sp: Propulsion system's specific impulse (s)

I_sp = V_e / g₀
I_sp = F / (g₀ * mDot)

mDot or ṁ: Propulsion system's propellant mass flow (kg/s)

Sometimes written as ṁ
mDot = M_pb / T_b
mDot = F / (g₀ * I_sp)
mDot = F / V_e

M: Spacecraft's total (wet) mass (kg)

This is the spacecraft's mass when the propellant tanks are full.
M = M_pt + M_pl + M_ps + M_st
M = M_e * R

M_c: ship's current mass (kg)

This is the spacecraft's mass at this moment in time.
M_c = F / A
M_c = (mDot * V_e) / A
M_c = (mDot * g₀ * I_sp) / A

M_e: Spacecraft's empty (dry) mass (kg)

This is the spacecraft's mass with no propellant, including payload.
M_e = M - M_pt
M_e = M_pl + M_ps + M_st

M_i: Spacecraft's inert mass (kg)

This is the spacecraft's mass with no propellant and no payload.
M_i = M_ps + M_st
M_i = M - (M_pt + M_pl)

M_pb: Mass of propellant burnt in current burn (kg)

M_pb = mDot * T_b
M_pb = (F * T_b) / (g₀ * I_sp)
M_pb = (F * T_b) / V_e

M_pl: Spacecraft's payload mass(kg)

Mass of removable items that are being transported. Cargo, crew, consumbables, etc.
M_pl = M - (M_pt + M_ps + M_st)

M_pp: Spacecraft's power plant mass(kg)

Given

M_ps: Spacecraft's propulsion system mass(kg)

M_ps = M_pp + M_ts

M_pt: Spacecraft's total propellant mass(kg)

Given

M_st: Spacecraft's structural mass(kg)

Mass of spacecraft less propellant, payload, and propulsion system
M_st = M - (M_pt + M_ps + M_pl)

M_ts: Spacecraft's thruster system mass(kg)

Given or
M_ts = (F / T/W / 9.81

P_e: Power plant energy (Wh)

Given

P_p: Power plant power output (W)

Given

P_al: Power plant alpha (kg/W)

P_al = M_pp / P_p

P_ed: Power plant energy density (Wh/m₃)

P_ed = P_e / V_pp

P_se: Power plant specific energy (Wh/kg)

P_sp = P_e / M_pp

P_sp: Power plant specific power (W/kg)

P_sp = P_p / M_pp

P_f: Propellant fraction, percent of spacecraft mass that is propellant

P_f = 1 - (1/R)

P_m: Planet's mass (kg)

This must be looked up for the planet in question. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

P_r: Planet's radius (m)

This must be looked up for the planet in question. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

R: Spacecraft's mass ratio (dimensionless number)

R = M / M_e
R = (M_pt / M_e) + 1
R = e^(Δ_v/V_e)
R = e^{(Δ_v/ sqrt((2*Fp) / mDot))}
R = 1 / (1-P_f)

s_circle: Surface Area of Circle (m²)

s_circle = π * r²
where:
r = radius (m)

s_cone: Surface Area of Cone (m²)

s_cone = π * r * ( r + sqrt(r² + h²))
where:
r = base radius (m)/li>
h = height (m)

s_cube: Surface Area of Cube (m²)

s_cube = 6 * s²
where:
s = side (m)

s_cylinder: Surface Area of Cylinder (m²)

s_cylinder = (2 * π * r²) + (2 * π * r * h)
where:
r = base radius (m)
h = height (m)

s_ellipse: Surface Area of Ellipse (m²)

s_ellipse = π * r₁ * r₂
where:
r₁ = max radius (m)
r₂ = min radius (m)

s_{parallelogram}: Surface Area of Parallelogram (m²) (Rocket Fin)

s_{parallelogram} = b * h
where:
b = base (m)
h = height (m)

s_rectangle: Surface Area of Rectangle (m²)

s_rectangle = l * h
where:
l = length (m)
h = height (m)

s_{rectangular_prism}: Surface Area of Rectangular Prism (m²)

s_{rectangular_prism} = (2 * l * h) + (2 * l * w) + (2 * h * w)
where:
l = length (m)
w = width (m)
h = height (m)

s_sphere: Surface Area of Sphere (m²)

s_sphere = 4 * π * r²
where:
r = radius (m)

s_square: Surface Area of Square (m²)

s_square = s²
where:
s = side (m)

s_trapezoid: Surface Area of Trapezoid (m²)

s_trapezoid = (h / 2) * (b₁ + b₂)
where:
b₁ = base 1 (m)
b₂ = base 2 (m)
h = height (m)

s_triangle: Surface Area of Triangle (m²) (Rocket Fin)

s_triangle = 0.5 * b * h
where:
b = base (m)
h = height (m)

t_b: Duration of current burn (s)

Given

t_l: Duration of Liftoff Burn (s)

t_l = D_vo / A

T/W: Thrust-to-weight ratio (dimensionless number)

T/W = F / (GLOW * g_p)
where:
F = thrust (N)
GLOW = gross liftoff weight (kg)
g_p = acceleration due to gravity at planet's surface (m/s²) = g₀ or 9.81 for Terra