How to Calculate Mass Ratio for Flight from LEO to Lunar Surface

When you're designing a moon mission, you'll need look at how you're going to get your stuff to the moon. Let's assume you've figured out what you want to have on the lunar surface and how much that weighs. Given that starting point, here's how to calculate the amount of fuel you'll need to get it there from low earth orbit.

Once you know how much mass you need in LEO, you can review the list of launchers and figure out how you're going to get your payload up to orbit in the first place.

The tables below do this calculation for a simple mission example, flying one rocket from LEO all the way to the moon, to deliver a 100-lb payload the lunar surface, for three different rocket engines. See the notes and definitions for more details about how to use this information.

Rocket Equation Calculations

Rocket Motor Burns

Downloading the Spreadsheets

To make it easy for you to import these numbers into your own spreadsheet program, I've included two versions of the file on line: leo-luna-example.csv and leo-luna-formulas.csv. The first file has the numbers from the tables above; the second one has the formulas.

You can open these files in your web browser and save them as text files. (If you click on the links, the files show show up in a new window in your browser.) Then start up your spreadsheet program and open them as a CSV (comma-delimited) files. The numbers and labels should show up in nice, neat columns. You're on your own for formatting.

The Rocket Equation

The real key to this calculation is the formula for the inverse rocket equation. It shows up in cells that look like this:

=D3*EXP(D4/(D5*D2))

If you look at which cells it's pointing at, you'll see that the formula is

Mo = Mf * EXP( delta_v / ( g * Isp ))

Definitions

Isp	Specific impulse, in seconds. Isp = 460 seconds represents a hydrogen-oxygen rocket such as the Space Shuttle Main Engine. Isp = 444.4 seconds is a typical value for the RL-10 rocket engine. Isp = 260 is a typical value for a hypergolic rocket motor using UDMH (unsymmetrical dimethyl hydrazine) and N2O4 (nitrogen tetroxide) for propellants.
Mf	Final mass. This is the mass landed on the moon. I used pounds (lbs) in this example, but the final mass ratio is dimensionless so it doesn't really matter what units you use as long as you're consisent. Whatever units you use for mass and velocity have to match up with the units you use for the acceleration of gravity.
delta_V	Change in velocity. The second table shows the delta_V values for several rocket motor firings used in this scenario.
g	Standard constant equal to the acceleration of gravity at the earth's surface. In the rocket equation, this term serves to convert units of mass to units of force, so make sure you use a value that matches the units you used for delta_V, and the units you expect to get for mass. I used g = 32.174 ft/sec/sec in this example.
Mo	Initial mass. This is the mass that you land on the moon, plus the mass of the fuel you burned to get there. Note that I did not include dropping tanks, engines, or other parts of the spacecraft; this scenario assumes you're going to fly the same hardware all the way from low Earth orbit to the surface of the moon, and the only mass you lose is fuel. If you want to drop things, set up your spreadsheet to calculate the final mass and initial mass at each staging point. Typical fuel tanks weigh about 3% of the mass of the fuel inside. If you intend to drop them, the tank mass roughly doubles because of the parasitic structure and separation mechanisms. The additional mechanisms and mission complexity will increase the cost of your propulsion package by a factor of 10 or so, so be careful when making broad assumptions about the benefits of staging. If you use staging, calculate each mission phase separately, starting with your end point and working backwards to your starting point.
Delta_M	Change in mass. This is the amount of propellant burned by the rocket. It includes both the fuel and the oxidizer.
Mass Ratio	Initial mass divided by the final mass. Once you've got a number for mass ratio for a given mission, you can use this same ratio to figure out how much initial mass you'll have for any size of payload. For instance, this example lists a 100-pound payload landed on the moon; but if you want to use a hypergolic rocket (Isp=260) to land a 100,000-pound payload on the moon, you can use the same mass ratio to see that you'll need 1,092,000 lbs in low Earth orbit.
TLI	Translunar Injection. This is the delta V you'll need to get your rocket out of low earth orbit and headed on toward the moon. It looks like a suspiciously round number, but 10,000 ft/sec is good to 3 decimal places.
LOI	Lunar Orbit Insertion. This puts your rocket into an orbit around the moon.
DOI	Descent Orbit Initiation. This little burn gets you headed down toward the moon.
Landing	This is the total of all the rocket burns you'll need to get our spacecraft safely onto the lunar surface, arriving as close as you can to zero velocity vertically, down-range, and sideways. It includes some margin, based on Apollo experience, for hovering maneuvers. It also accounts for the fact that during most of this mission phase you will be supporting the mass of your vehicle with your rocket engine.