# Equations

## Appendix 1

## 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

## Index

### A: Acceleration of spacecraft *(m/s*^{2})

^{2})

### A_{pg}: Acceleration of spacecraft in terms of planetary gravities *(g*_{p})

_{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)*

### e: Base of natural logarithms

### F: Propulsion system's thrust *(N or kg m/s*^{2})

^{2})

### F_{p}: Propulsion system's thrust power *(w)*

### F/F_{p}: Propulsion system's thrust efficency (impulse per unit energy) *(N/w)*

### F_{se}: Spacecraft's Specific Exhaust *(w/kg)*

### F_{sp}: Spacecraft's Specific Power *(w/kg)*

### g_{0}: Acceleration due to gravity at Earth's surface *(m/s*^{2})

^{2})

### g_{p}: Acceleration due to gravity at planet's surface *(m/s*^{2})

^{2})

### G: Newton's gravitational constant *(N m*^{2} kg^{-2})

^{2}kg

^{-2})

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

### I_{sp}: Propulsion system's specific impulse *(s)*

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

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

### M_{c}: ship's current mass *(kg)*

### M_{e}: Spacecraft's empty (dry) mass *(kg)*

### M_{i}: Spacecraft's inert 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_{e}: Power plant energy *(Wh)*

### P_{p}: Power plant power output *(W)*

### P_{al}: Power plant alpha *(kg/W)*

### P_{ed}: Power plant energy density *(Wh/m*_{3})

_{3})

### P_{se}: Power plant specific energy *(Wh/kg)*

### P_{sp}: Power plant specific power *(W/kg)*

### P_{f}: Propellant fraction, percent of spacecraft mass that is propellant

### P_{m}: Planet's mass *(kg)*

### P_{r}: Planet's radius *(m)*

### R: Spacecraft's mass ratio *(dimensionless number)*

### s_{circle}: Surface Area of Circle *(m*^{2})

^{2})

### s_{cone}: Surface Area of Cone *(m*^{2})

^{2})

### s_{cube}: Surface Area of Cube *(m*^{2})

^{2})

### s_{cylinder}: Surface Area of Cylinder *(m*^{2})

^{2})

### s_{ellipse}: Surface Area of Ellipse *(m*^{2})

^{2})

### s_{parallelogram}: Surface Area of Parallelogram *(m*^{2}) (Rocket Fin)

^{2}) (Rocket Fin)

### s_{rectangle}: Surface Area of Rectangle *(m*^{2})

^{2})

### s_{rectangular_prism}: Surface Area of Rectangular Prism *(m*^{2})

^{2})

### s_{sphere}: Surface Area of Sphere *(m*^{2})

^{2})

### s_{square}: Surface Area of Square *(m*^{2})

^{2})

### s_{trapezoid}: Surface Area of Trapezoid *(m*^{2})

^{2})

### s_{triangle}: Surface Area of Triangle *(m*^{2}) (Rocket Fin)

^{2}) (Rocket Fin)

### t_{b}: Duration of current burn *(s)*

### t_{l}: Duration of Liftoff Burn *(s)*

### v_{cone}: Volume of Cone *(m*^{3})

^{3})

### v_{cube}: Volume of Cube *(m*^{3})

^{3})

### v_{cylinder}: Volume of Cylinder *(m*^{3})

^{3})

### v_{rectangular_prism}: Volume of Rectangular Prism *(m*^{3})

^{3})

### v_{sphere}: Volume of Sphere *(m*^{3})

^{3})

### v_{sphere}: Volume of Torus *(m*^{3})

^{3})