Thursday, April 7, 2011

More on the Falcon Heavy

When I did my sanity check, I made some optimistic assumptions about the potential Falcon Heavy performance. For one, I used vacuum exhaust velocity (specific impulse) for the 1st stage prior to side booster separation, which will over estimate the delta-v for that stage. We should be able to keep using the vacuum exhaust velocity for the core stage post separation and for the upper stage, since they'll be operating in near vacuum.

To get a worst case estimate of the needed second stage mass fraction, we can use the deliberately pessimistic sea level exhaust velocity for the entire pre separation first stage.

This reduces the delta-v for the first phase to about 2964 m/s. Add in a less charitable (pessimistic, even) estimate of gravitational and aerodynamic losses, say -1400 m/s, and the velocity at upper stage separation is only 6450 m/s. That leaves 1338.6 m/s for the second stage to make up. Remember, we can use this to solve for the post burn out mass of the second stage. We get 48244 kg -- less than the payload mass.

A more realistic lower bound for performance would be provided by using the average of the sea level and vacuum exhaust velocities for the pre separation 1st stage rocket equation. That average is 2840.5 m/s, which yields a booster burnout velocity of 3471.7 m/s. Plugging this back into our previous calculations (including losses and boost due to the rotation of the Earth) and we end up with an upper stage velocity of 6959.1 m/s. The upper stage now only has to add 830.9 m/s to reach the orbital velocity of 7790 m/s.

Remember, we find the burnout mass from the initial mass, exhaust velocity and delta-v. It's 56127 kg, leaving 3127 kg of structure after the 53000 kg payload is accounted for. That gives a structural fraction of about 1/23 (0.0435) for the upper stage, which is similar to the Falcon 9 core stage structural fraction. Upper stage structural fractions tend to be higher (worse) than core or lower stages because vacuum optimized bell nozzles are longer and thus heavier than nozzles optimized to for liftoff. The payload interface may create additional structural mass as well. Still, if SpaceX can achieve a structural fraction of 1/30 for the side boosters, they may well be able to get 1/23 for the upper stage.

We can tweak the variables in a few other ways to see how tight the constraints on the Falcon Heavy really are. What if they boosters aren't quite as lightly built? What if the core stage is less massive than we've estimated? Launch vehicles are very sensitive to structural mass.

Still, from the information available, Falcon Heavy looks likely to be a workable vehicle. The unknown we've estimated, upper stage mass fraction, is likely to be closer to 1/23 than 1/11.

No comments:

Post a Comment