RSE #49 RSE #49

Go Figure - Growth Factors

Rocket Science for Earthlings number 49

A continuing series for the gravitationally impaired

The Devil is in the details - No, the Devil is in the Mass Ratio at least for rocket scientists. You get your ISP number when you choose the propellants and you don't have a lot to say about it, but the mass ratio depends on how much money you have to spend. There is always going to be a minimum cost point in rocket design, too heavy is expensive and too light is expensive. Of course upper stages will have better mass ratios than lower stages, but how do you get an idea of what weights will be for each pound of payload?

First find out the velocity needed for the mission and divide it by the number of stages to get an average Delta Vee for each stage. Plug this and the expected ISP into the rocket equation (see somewhere above) and you will come up with a mass ratio number greater than (>) one. There are two ways of expressing mass ratio, > one and < one, and we need a mass ration number less than one, so, (MR - 1)/MR=(MR < one). Or the other way 1/(1-MR)=(MR > one).

Now we can figure the growth factor. Each stage will have two mass ratios, the stage mass ratio, equal to the propellant divided by the propellant plus the stage structural mass (MR=PR/(PR+ST), and the working mass ratio, which includes the mass of the payload (PL) which the stage is pushing, (MR=PR/(PR+ST+PL). The working mass ratio is always lower because of the added mass of the payload. These two mass ratios determine how big the stage should be, the growth factor. The Growth Factor = the working mass ratio / (the stage mass ratio - the working mass ratio), or GF=WMR/(SMR-WMR). Example, for a payload of 1, with a working mass ratio of 0.5, and a stage mass ratio of 0.66, the stage would have a mass of 3.125 the growth factor, 2.0625 of propellant, 1.03125 of structure, and 1 payload.

For the next stage, the payload would be 3.125+1=4.125, times the growth factor of 3.125, its mass would be 12.890625. Things just keep adding up from there on down. At some point it's a good idea to stop staging and start refueling, Low Earth Orbit is a good place.