Appendix 1: Equations

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_i

A = (mDot * V_e) / M_i

A = (mDot * g₀ * I_sp) / M_i

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)

Δ_v = V_e * ln[R]

Δ_v = g₀ * I_sp * ln[R]

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

Δ_vc = V_e * ln[M_i / M_e]

Δ_vc = g₀ * I_sp * ln[M_i / M_e]

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_i * 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

g₀: Acceleration due to gravity at Earth'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 Earth.

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

mDot = M_pb / T_b

mDot = F / (g₀ * I_sp)

mDot = F / V_e

M: Spacecraft's total mass (kg)

This is the spacecraft's mass when the propellant tanks are full.

M = M_pt + M_pl + M_ps + M_st

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

This is the spacecraft's mass when the propellant tanks are empty.

M_e = M - M_pt

M_e = M_pl + M_ps + M_st

M_i: ship's instantaneous mass (the current mass) (kg)

M_i = F / A

M_i = (mDot * V_e) / A

M_i = (mDot * g₀ * I_sp) / A

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)

Given

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)

Given

M_ts: Spacecraft's thruster system mass(kg)

Given

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)

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

v_cone: Volume of Cone (m³)

v_cone = 0.3333 * π * r² * h

where:

r = base radius (m)
h = height (m)

v_cube: Volume of Cube (m³)

v_cube = s³

where:

s = side (m)

v_cylinder: Volume of Cylinder (m³)

v_cylinder = π * r² * h

where:

r = base radius (m)
h = height (m)

v_{rectangular_prism}: Volume of Rectangular Prism (m³)

v_{rectangular_prism} = l * w * h

where:

l = length (m)
w = width (m)
h = height (m)

v_sphere: Volume of Sphere (m³)

v_sphere = 1.3333 * π * r³

where:

r = radius (m)

v_sphere: Volume of Torus (m³)

v_torus = 19.739 * R * r²

where:

R = radius from the center of the hole to the center of the torus tube (m)
r = radius of the torus tube(m)

V_e: Propulsion system's exhaust velocity (m/s)

V_e = g₀ * I_sp

V_e = F / mDot

Appendix 1: Equations

On This Page

Introduction

Units and Functions

Index

A: Acceleration of spacecraft (m/s2)

Apg: Acceleration of spacecraft in terms of planetary gravities (gp)

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

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

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

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

e: Base of natural logarithms

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

Fp: Propulsion system's thrust power (w)

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

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

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

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

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

Isp: Propulsion system's specific impulse (s)

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

M: Spacecraft's total mass (kg)

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

Mi: ship's instantaneous mass (the current mass) (kg)

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

Mpl: Spacecraft's payload mass(kg)

Mpp: Spacecraft's power plant mass(kg)

Mps: Spacecraft's propulsion system mass(kg)

Mpt: Spacecraft's total propellant mass(kg)

Mst: Spacecraft's structural mass(kg)

Mts: Spacecraft's thruster system mass(kg)

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

Pm: Planet's mass (kg)

Pr: Planet's radius (m)

R: Spacecraft's mass ratio (dimensionless number)

scircle: Surface Area of Circle (m2)

scone: Surface Area of Cone (m2)

scube: Surface Area of Cube (m2)

scylinder: Surface Area of Cylinder (m2)

sellipse: Surface Area of Ellipse (m2)

sparallelogram: Surface Area of Parallelogram (m2) (Rocket Fin)

srectangle: Surface Area of Rectangle (m2)

srectangular_prism: Surface Area of Rectangular Prism (m2)

ssphere: Surface Area of Sphere (m2)

ssquare: Surface Area of Square (m2)

strapezoid: Surface Area of Trapezoid (m2)

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

tb: Duration of current burn (s)

tl: Duration of Liftoff Burn (s)

vcone: Volume of Cone (m3)

vcube: Volume of Cube (m3)

vcylinder: Volume of Cylinder (m3)

vrectangular_prism: Volume of Rectangular Prism (m3)

vsphere: Volume of Sphere (m3)

vsphere: Volume of Torus (m3)

Ve: Propulsion system's exhaust velocity (m/s)

A: Acceleration of spacecraft (m/s²)

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

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

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

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

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

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

F_p: Propulsion system's thrust power (w)

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

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

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

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

I_sp: Propulsion system's specific impulse (s)

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

M_i: ship's instantaneous mass (the current mass) (kg)

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

M_pl: Spacecraft's payload mass(kg)

M_pp: Spacecraft's power plant mass(kg)

M_ps: Spacecraft's propulsion system mass(kg)

M_pt: Spacecraft's total propellant mass(kg)

M_st: Spacecraft's structural mass(kg)

M_ts: Spacecraft's thruster system mass(kg)

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

P_m: Planet's mass (kg)

P_r: Planet's radius (m)

s_circle: Surface Area of Circle (m²)

s_cone: Surface Area of Cone (m²)

s_cube: Surface Area of Cube (m²)

s_cylinder: Surface Area of Cylinder (m²)

s_ellipse: Surface Area of Ellipse (m²)

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

s_rectangle: Surface Area of Rectangle (m²)

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

s_sphere: Surface Area of Sphere (m²)

s_square: Surface Area of Square (m²)

s_trapezoid: Surface Area of Trapezoid (m²)

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

t_b: Duration of current burn (s)

t_l: Duration of Liftoff Burn (s)

v_cone: Volume of Cone (m³)

v_cube: Volume of Cube (m³)

v_cylinder: Volume of Cylinder (m³)

v_{rectangular_prism}: Volume of Rectangular Prism (m³)

v_sphere: Volume of Sphere (m³)

v_sphere: Volume of Torus (m³)

V_e: Propulsion system's exhaust velocity (m/s)