Appendix 1
Equations
Most of these are pretty obvious, but are included for completeness.
Be sure to examine the DeltaV nomogram
and the Acceleration nomogram.
Eric Rozier has implemented
on-line calculators for a few of these equations.
Luke Campbell has an on-line calculator
for the pulse parameters of a laser weapon.
Some of these equations
are available in the Atomic Rocket Tiddly Wiki.
Units and Functions
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
A: Acceleration of spacecraft (m/s2)
A = F / Mi
A = (mDot * Ve) / Mi
A = (mDot * g0 * Isp) / Mi
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Apg: Acceleration of spacecraft in terms of planetary gravities (gp)
Apg = A / gp
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Δv: Spacecraft's total change in velocity capability (deltaV) (m/s)
Δv = Ve * ln[R]
Δv = g0 * Isp * ln[R]
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Δvc: Spacecraft's current deltaV capability (m/s)
Δvc = Ve * ln[Mi / Me]
Δvc = g0 * Isp * ln[Mi / Me]
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Δvd: Planet's gravitational drag (m/s)
DeltaV required to counteract gravitational drag during liftoff or landing.
Δvd = gp * Tl
Δvd = Dvo / Apg
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Δvo: 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.
Δvo = sqrt[ (G * Pm) / Pr ]
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e: Base of natural logarithms
e = 2.71828...
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F: Propulsion system's thrust (N or kg m/s2)
F = Mi * A
F = mDot * Ve
F = mDot * g0 * Isp
F = (Mpb * Ve) / Tb
F = (2 * Fp) / Ve
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Fp: Propulsion system's thrust power (w)
Fp = (mDot * (Ve2)) / 2
Fp = (F * Ve ) / 2
Fp = (Mpb * (Ve2)) / (2 * Tb)
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F/Fp: Propulsion system's thrust efficency (impulse per unit energy) (N/w)
F/Fp = 2 / Ve
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g0: Acceleration due to gravity at Earth's surface (m/s2)
g0 = 9.81
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gp: 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.
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G: Newton's gravitational constant (N m2 kg-2)
G = 0.00000000006673 = 6.673e-11
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GLOW: Spacecraft's maximum gross liftoff weight (kg)
Note that Apg must be greater than one, preferably much greater to reduce
planet's gravitational drag Dvd.
GLOW = F / (Apg * gp)
GLOW = (mDot * Ve) / (Apg * gp)
GLOW = (mDot * g0 * Isp) / (Apg * gp)
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Isp: Propulsion system's specific impulse (s)
Isp = Ve / g0
Isp = F / (g0 * mDot)
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mDot: Propulsion system's propellant mass flow (kg/s)
mDot = Mpb / Tb
mDot = F / (g0 * Isp)
mDot = F / Ve
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M: Spacecraft's total mass (kg)
This is the spacecraft's mass when the propellant tanks are full.
M = Mpt + Mpl + Mps + Mst
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Me: Spacecraft's empty (dry) mass (kg)
This is the spacecraft's mass when the propellant tanks are empty.
Me = M - Mpt
Me = Mpl + Mps + Mst
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Mi: ship's instantaneous mass (the current mass) (kg)
Mi = F / A
Mi = (mDot * Ve) / A
Mi = (mDot * g0 * Isp) / A
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Mpb: Mass of propellant burnt in current burn (kg)
Mpb = mDot * Tb
Mpb = (F * Tb) / (g0 * Isp)
Mpb = (F * Tb) / Ve
Mpl: Spacecraft's payload mass(kg)
Given
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Mpp: Spacecraft's power plant mass(kg)
Given
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Mps: Spacecraft's propulsion system mass(kg)
Mps = Mpp + Mts
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Mpt: Spacecraft's total propellant mass(kg)
Given
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Mst: Spacecraft's structural mass(kg)
Given
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Mts: Spacecraft's thruster system mass(kg)
Given
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Pf: Propellant fraction, percent of spacecraft mass that is propellant
Pf = 1 - (1/R)
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Pm: Planet's mass (kg)
This must be looked up for the planet in question. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html
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Pr: Planet's radius (m)
This must be looked up for the planet in question. http://nssdc.gsfc.nasa.gov/planetary/planetfact.html
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R: Spacecraft's mass ratio (dimensionless number)
R = M / Me
R = (Mpt / Me) + 1
R = e(Δv/Ve)
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scircle: Surface Area of Circle (m2)
scircle = π * r2
where:
r = radius (m)
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scone: Surface Area of Cone (m2)
scone = π * r * ( r + sqrt(r2 + h2))
where:
r = base radius (m)
h = height (m)
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scube: Surface Area of Cube (m2)
scube = 6 * s2
where:
s = side (m)
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scylinder: Surface Area of Cylinder (m2)
scylinder = (2 * π * r2)
+ (2 * π * r * h)
where:
r = base radius (m)
h = height (m)
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sellipse: Surface Area of Ellipse (m2)
sellipse = π * r1 * r2
where:
r1 = max radius (m)
r2 = min radius (m)
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sparallelogram: Surface Area of Parallelogram (m2) (Rocket Fin)
sparallelogram = b * h
where:
b = base (m)
h = height (m)
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srectangle: Surface Area of Rectangle (m2)
srectangle = l * h
where:
l = length (m)
h = height (m)
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srectangular_prism: Surface Area of Rectangular Prism (m2)
srectangular_prism = (2 * l * h) + (2 * l * w) + (2 * h * w)
where:
l = length (m)
w = width (m)
h = height (m)
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ssphere: Surface Area of Sphere (m2)
ssphere = 4 * π * r2
where:
r = radius (m)
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ssquare: Surface Area of Square (m2)
ssquare = s2
where:
s = side (m)
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strapezoid: Surface Area of Trapezoid (m2)
strapezoid = (h / 2) * (b1 + b2)
where:
b1 = base 1 (m)
b2 = base 2 (m)
h = height (m)
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striangle: Surface Area of Triangle (m2) (Rocket Fin)
striangle = 0.5 * b * h
where:
b = base (m)
h = height (m)
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tb: Duration of current burn (s)
Given
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tl: Duration of Liftoff Burn (s)
tl = Dvo / A
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vcone: Volume of Cone (m3)
vcone = 0.3333 * π * r2 * h
where:
r = base radius (m)
h = height (m)
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vcube: Volume of Cube (m3)
vcube = s3
where:
s = side (m)
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vcylinder: Volume of Cylinder (m3)
vcylinder = π * r2 * h
where:
r = base radius (m)
h = height (m)
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vrectangular_prism = l * w * h
where:
l = length (m)
w = width (m)
h = height (m)
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vsphere: Volume of Sphere (m3)
vsphere = 1.3333 * π * r3
where:
r = radius (m)
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vsphere: Volume of Torus (m3)
vtorus = 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)
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Ve: Propulsion system's exhaust velocity (m/s)
Ve = g0 * Isp
Ve = F / mDot
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