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Introduction to the Rocket Equation

To calculate fuel masses, here's the magic equation you need:

Delta_V = Isp * g * ln(Mo/Mf)


Delta_V change in velocity, in units compatible with the value you use for g
Isp specific impulse, in seconds
g acceleration of gravity at the earth's surface
32.174 ft/sec/sec (9.805 m/sec/sec)
ln() natural logarithm function
Mo Mass before the burn
Mf Mass after the burn

Note that the choice of system of units is driven only by the value you use for g.

Now you need some constants to plug in. You can look up the numbers for Delta V in the Reference Mission Timeline in section 4.1.1 of the Artemis Data Book. No matter where you're going on the moon, those Delta V numbers will be within the error band of your assumptions.

Isp depends on the type of fuel and oxydizer:

Cryogenic 440 to 460 seconds
(For an RL-10, 444.4)
Hypergolics 260 to 290 seconds
Nuclear 10^2 to 10^3 seconds, maybe
Antimatter 10^7 seconds

Since nuclear rockets and matter-antimatter engines aren't exactly off-the-shelf technology (yet), just look at the chemical rockets. You'll find more information about the Isp's for sundry rocket engines in section 4.3.9 of the Artemis Data Book.

To calculate the fuel burns, invert the rocket equation:

Delta_V/(Isp * g) = ln(Mo/Mf)

Mo/Mf = exp( Delta_V/(Isp * g) )

Mo = Mf * exp(Delta_V/(Isp * g))

You should be able to plug that line above directly into your spreadsheet program, and then feed it whatever numbers you want to work with. Delta-M for a rocket engine burn is Mo - Mf; that's the total of the fuel plus oxydizer used during the burn.

The RL-10 runs a bit fuel rich. The ratio of oxygen to hydrogen is 5:1 by weight. You can run a perfect stochastic ratio (2H+O --> H20), and use a 7:1 mixture, but it'll cost you Isp because of the heavier exhaust gas.

To figure out how much fuel you need for several burns, start at the end of the mission. You know how much mass you want to deliver to your final destination. Then work backwards.

Assume fuel tanks weigh about 3% of the mass of the fuel they carry. Don't forget to add the fuel tank weight in, because it can add up as you go from stage to stage. Usually you'll want to add in a bit more weight for supporting structure, engine controllers, communication gear, and range safety pyrotechnics. You'll quickly see why we like to throw away fuel tanks; hence, staged rockets.

You can tune it up further by including some numbers for boiling off cryogenic H2 and O2, but it turns out that for the reference mission boil-off doesn't affect any significant digits.

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