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.

Tuesday, April 5, 2011

SpaceX Sanity Check

Elon Musk recently made some amazing claims about the capabilities of SpaceX's Falcon Heavy rocket. He's saying that it will reduce launch costs down to 2200 dollars/kg (1000 dollars/lbm*). If he's right, that'd be truly revolutionary.

*Ugh, pound masses.

Clicking the link at the top of SpaceX's page brings up the Falcon Heavy page. Apparently, the Falcon Heavy replaces Falcon 9 Heavy.

The Falcon Heavy (no 9) promises a payload of 53,000 kg to a shuttle like low earth orbit (LEO)* for launch costs from between 80 million and 125 million dollars. This puts launch costs into the 1510 dollars/kg to 2360 dollars per kg range. That is truly phenomenal.

*200 km radius, 28.5 degrees inclination

The Falcon 9 Heavy, a very similar vehicle, is listed as being able to lift 32000 kg to the same orbit with a launch cost of 95 million dollars, or 2970 dollars/kg. This is still great.

Confusingly, both vehicles are referred to as "Falcon Heavy" on their respective webpages. They're clearly different vehicles, with Falcon Heavy being listed as taller, more massive and having somewhat greater thrust on liftoff. I presume it's a design evolution of the Falcon 9 Heavy with uprated Merlin engines and an improved second stage.

I hate to say it, but this performance seems too good to be true. So I'd like to get a feel for how close the Falcon Heavy comes to violating the fundamental physical constraints on what a rocket can do. So I'll do a sanity check.

SpaceX lists the structural fraction of the side boosters as 1/30*. Space Launch Report estimates the empty mass of a Falcon 9 1st stage as about 16600 kg. That gives a loaded mass of 498000 kg for each of the side boosters with 481400 kg of propellant in each. Presumably, the core booster is very similar to a stock Falcon 9 in structure and tankage. Space Launch Report estimates 315500 kg of propellant in it.

*This is very impressive. Space Launch Report puts the structural fraction for the first stage of the Falcon 9 Block 1 that's flown at about 1/22. So, SpaceX has some improving to do to reach 1/30.

Each of the Falcon Heavy's 27 Merlin engines should produce 630 kN of thrust. At a specific impulse of 275 s (exhaust velocity of 2698 m/s), that gives a propellant consumption of 233.53 kg/s per engine. This gives a side booster burnout (when all of their fuel has been used*) time of 152.7 seconds after launch -- Which is very reasonable.

*The Falcon Heavy is said to have a cross feed system that allows the propellant in the side boosters to be drained completely before using the propellant in the core booster.

The 9 remaining engines of the core stage can continue burning for another 150.1 seconds. How serendipitous that the burnout times are so similar.

Now we can use the rocket equation to calculate the maximum change in velocity (in the absence of atmospheric and gravity drag) for each stage. The rocket equation* is

[; \Delta v = v_{e}\ln{\frac{m_{0}}{m_{1}} ;]

where [; \Delta v ;] is the chage in velocty, [; v_{e} ;] is the exhaust velocity, [; m_{0} ;] is the final mass (at burnout) and [; m_{1} ;] is the loaded mass. Because we're calculating maximum delta-v, we're going to use the higher vacuum specific impulse (and exhaust velocity) for the Merlin engine's [; v_{e} ;]. These are 304 s and 2983 m/s respectively.

*I'm using a Firefox script called TeX The World to render the equations. If you know of a better way to handle equations, please let me know.

At launch, the mass of the Falcon Heavy is 1400000 kg. That's [; m_{0} ;]. At side booster burnout, 962800 kg of propellant have been burnt, leaving [; m_{1} ;] as 437200 kg. This gives a delta-v of 3471.7 m/s.

Now we shed the empty boosters ( 16600 kg each ), leaving a mass of 404000 kg for [; m_{0} ;]. After burning  the 315500 kg of propellant in the core stage, we have just 88500 kg of spacecraft left. The rocket equation gives us a delta-v of 4529.4 m/s for a total velocity change of 8001.1 m/s.

The velocity required to enter a circular orbit is

[; v = \sqrt{\mu/r } ;]

where r is the radius of the orbit, v is the orbital velocity and [; \mu ;] is the standard gravitational parameter, about 398600.4 km^3/s^2 for the earth. For a 200 km orbit (with a radius about the center of the earth of about 6571 km), we'll need a velocity of about 7790 m/s.

Note that, neglecting atmospheric and gravitational drag, the core stage is already travelling faster than this at burnout. And we haven't even accounted for the upper stage or the velocity bonus the rocket gets by launching eastward due to the Earth's rotation.

Optimistically estimating atmospheric and gravity losses at 1200 m/s (apparently, the Shuttle loses about 1330 m/s due to these), we'd still have a velocity of 6801 m/s before adding in the Earth's rotation. From Cape Canaveral into a 28.5 degree orbit, that boost should be about 358 m/s. So, before we've turned on the upper stage, the Falcon Heavy is travelling at about 7159 m/s: pretty close to orbital velocity.

Now things get a bit trickier. For one, the upper stage uses a different nozzle on its Merlin, improving specific impulse to 342 seconds ( 3355 m/s exhaust velocity.) Worse, we don't know the structural fraction of the upper stage, so we don't know how much fuel it has. However, we do know the payload mass, the intial mass the needed delta-v and the exhaust velocity, so we can calculate the structural fraction and see if it's a reasonable number. We just need to rearrange the rocket equation to find [; m_{1} ;]

[; m_{1}=\frac{m_{0}}{{e}^{{\Delta v}/{v_{e}}}} ;]

The remaining mass after the core stage separates from the upper stage is 71900. The necessary delta-v is 7790 m/s - 7159 m/s = 631.0 m/s, giving us a remaining mass of 59573 kg. Subtract away the 53000 kg payload and that leaves 6573* kg for the upper stage structure. 6573 kg/71900kg gives a structural fraction of about 1/11 for the upper stage, which as we saw earlier, is not unreasonable.

*Originally, I had a remaining mass of 3573 kg. Subtraction, my only weakness

So the Falcon Heavy passes my simple sanity check on its performance claims, assuming it can achieve the 1/30 structural fraction for its side mount boosters and keep the aero and gravity losses down. If it can, maybe it really will fulfil Musk's promises and revolutionize spaceflight and make humanity a spacefaring race.