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

ElementValue
e2.71828...
kgkilograms
mmeters
Nnewtons
π3.14159...
sseconds
wwatts
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

Index

A: Acceleration of spacecraft (m/s2)

A = F / Mc

A = (mDot * Ve) / Mc

A = (mDot * g0 * Isp) / Mc

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

Apg = A / gp

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

Δv = Ve * ln[R]

Δv = g0 * Isp * ln[R]

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

Δvc = Ve * ln[Mc / Me]

Δvc = g0 * Isp * ln[Mc / Me]

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

DeltaV required to counteract gravitational drag during liftoff or landing.

Δvd = gp * Tl

Δvd = Dvo / Apg

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 ]

e: Base of natural logarithms

e = 2.71828...

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

F = Mc * A

F = mDot * Ve

F = mDot * g0 * Isp

F = (Mpb * Ve) / Tb

F = (2 * Fp) / Ve

Fp: Propulsion system's thrust power (w)

Fp = (mDot * (Ve2)) / 2

Fp = (F * Ve ) / 2

Fp = (Mpb * (Ve2)) / (2 * Tb)

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

F/Fp = 2 / Ve

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

Watts per kilogram of accelerated mass

Fse = (A * Ve ) / 2

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

Fsp = Fp / Me

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

g0 = 9.81

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.

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 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)

Isp: Propulsion system's specific impulse (s)

Isp = Ve / g0

Isp = F / (g0 * mDot)

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

mDot = Mpb / Tb

mDot = F / (g0 * Isp)

mDot = F / Ve

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

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

M = Mpt + Mpl + Mps + Mst

Mc: ship's current mass (kg)

This is the spacecraft's mass at this moment in time.

Mc = F / A

Mc = (mDot * Ve) / A

Mc = (mDot * g0 * Isp) / A

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

This is the spacecraft's mass with no propellant, including payload.

Me = M - Mpt

Me = Mpl + Mps + Mst

Mi: Spacecraft's inert mass (kg)

This is the spacecraft's mass with no propellant and no payload.

Mi = Mps + Mst

Mi = M - (Mpt + Mpl)

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)

Mass of removable items that are being transported. Cargo, crew, consumbables, etc.

Mpl = M - (Mpt + Mps + Mst)

Mpp: Spacecraft's power plant mass(kg)

Given

Mps: Spacecraft's propulsion system mass(kg)

Mps = Mpp + Mts

Mpt: Spacecraft's total propellant mass(kg)

Given

Mst: Spacecraft's structural mass(kg)

Mass of spacecraft less propellant, payload, and propulsion system

Mst = M - (Mpt + Mps + Mpl)

Mts: Spacecraft's thruster system mass(kg)

Given

Pe: Power plant energy (Wh)

Given

Pp: Power plant power output (W)

Given

Pal: Power plant alpha (kg/W)

Pal = Mpp / Pp

Ped: Power plant energy density (Wh/m3)

Ped = Pe / Vpp

Pse: Power plant specific energy (Wh/kg)

Psp = Pe / Mpp

Psp: Power plant specific power (W/kg)

Psp = Pp / Mpp

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

Pf = 1 - (1/R)

Pm: Planet's mass (kg)

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

Pr: 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 / Me

R = (Mpt / Me) + 1

R = ev/Ve)

scircle: Surface Area of Circle (m2)

scircle = π * r2

where:

  • r = radius (m)

scone: Surface Area of Cone (m2)

scone = π * r * ( r + sqrt(r2 + h2))

where:

  • r = base radius (m)/li>
  • h = height (m)

scube: Surface Area of Cube (m2)

scube = 6 * s2

where:

  • s = side (m)

scylinder: Surface Area of Cylinder (m2)

scylinder = (2 * π * r2) + (2 * π * r * h)

where:

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

sellipse: Surface Area of Ellipse (m2)

sellipse = π * r1 * r2

where:

  • r1 = max radius (m)
  • r2 = min radius (m)

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

sparallelogram = b * h

where:

  • b = base (m)
  • h = height (m)

srectangle: Surface Area of Rectangle (m2)

srectangle = l * h

where:

  • l = length (m)
  • h = height (m)

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)

ssphere: Surface Area of Sphere (m2)

ssphere = 4 * π * r2

where:

  • r = radius (m)

ssquare: Surface Area of Square (m2)

ssquare = s2

where:

  • s = side (m)

strapezoid: Surface Area of Trapezoid (m2)

strapezoid = (h / 2) * (b1 + b2)

where:

  • b1 = base 1 (m)
  • b2 = base 2 (m)
  • h = height (m)

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

striangle = 0.5 * b * h

where:

  • b = base (m)
  • h = height (m)

tb: Duration of current burn (s)

Given

tl: Duration of Liftoff Burn (s)

tl = Dvo / A

vcone: Volume of Cone (m3)

vcone = 0.3333 * π * r2 * h

where:

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

vcube: Volume of Cube (m3)

vcube = s3

where:

  • s = side (m)

vcylinder: Volume of Cylinder (m3)

vcylinder = π * r2 * h

where:

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

vrectangular_prism: Volume of Rectangular Prism (m3)

vrectangular_prism = l * w * h

where:

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

vsphere: Volume of Sphere (m3)

vsphere = 1.3333 * π * r3

where:

  • r = radius (m)

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)

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

Ve = g0 * Isp

Ve = F / mDot