Explaining the Infeasability of Second Stage Reuse

In an interview in MIT in October 2014 [1], Musk said:

“I don’t expect SpaceX’s Falcon line to have a reusable upper stage. With a kerosene based system, the specific impulse isn’t really high enough to do that, and a lot of the missions we do for commercial satellite deployment are geostationary missions. So we’re really going very far out – these are high delta-velocity missions, so to try and get something back from that is really difficult.”

Let’s see if we can explain why geostationary transfer orbit (GTO) missions are the limiting factor in developing second stage reusablity, as Musk says. We start with SpaceX’s description of their capabilities, from their website [2] [3]:

The thing about second stage reuse is that you have to propel the whole of the second stage to orbit, so any extra mass you add to the second stage has to come directly out of the payload. Knowing this, you can immediately see that F9 GTO missions will be by far the most affected – adding two metric tons to the second stage would be a 40% payload loss for this kind of mission, while only being an 16% loss for F9 LEO, a 10% loss for FH GTO, and a 4% loss for FH LEO.

Even without knowing, even to within an order of magnitude, how much mass reusability would add, we can see why Musk says what he says. A third of Falcon 9’s flights so far have been to GTO, and in the past few years, about half of the Atlas V’s, and almost all the Ariane 5’s launches have been to that (or higher) orbit. And already some of F9’s missions have pushed the launch system’s GTO capability to its limit: AsiaSAT 6, AsiaSAT 8, TurkmenAlem52E/MonacoSAT were all Falcon 9 GTO missions which went within a few hundred kilograms of their upper limit. SES-9 (next in manifest at time of writing) is 5300 kg to GTO, which is actually larger than SpaceX’s claimed capability.

For these, even a small increase in the mass of the second stage would have meant the Falcon 9 could no longer launch them. Second stage reuse would thus cost SpaceX significant numbers of GTO launches, until the Falcon Heavy can fly regularly enough to take over that market.

But the question remains – how much mass would a reusability system on the second stage really add? Would it be 100kg, one ton, or ten tons? Obviously as outsiders, we will probably never know an exact number, but as with many things it’s still possible to estimate an order of magnitude for it.

To do this we first need to consider how SpaceX would have implemented stage two reusability if they were pursuing it. Obviously, this is a hypothetical, but SpaceX did make plans to reuse the upper stage some years ago. In a flight animation from 2011 [4], SpaceX show the stage with a heat-shield on the (blunt) nose. Since this is the only configuration with any known SpaceX endorsement, it seems sensible to base our estimates on it.

So what would be main contributions to the mass in this situation? You would need:

a heat shield for the nose
less powerful heat protection for the body
landing legs
an RCS control system
grid fins
extra fuel for the landing
extremely good avionics / deeper throttle
probably some other things.

Not all of these are possible to estimate, but some are.

Heat shielding (nose + body):

Assuming they use the same material, PICA, for stage 2 (S2) as they do for Dragon, we can estimate approximately what the heat shield would weigh. According to NASA, describing a concept for the Orion heat shield [5]:

“A 5.5-m, 28-deg sidewall concept with a total mass of approximately 11,400 kg requires an aft TPS [thermal protection system] mass of 630 kg and forward TPS mass of 180 kg. The assumed TPS materials for this analysis were PICA for the aft side…”.

The Falcon 9 has a smaller diameter – accounting for this gives a heatshield mass of 270kg. However, it also has a much larger surface area exposed to secondary heating – 247 square meters for F9S2, compared to 27 square meters on the Orion capsule. Again rescaling the NASA value, this gives 1,650kg of secondary thermal protection for the upper stage. This calculation is imperfect – Orion has humans inside, while S2 has residual fuel. SpaceX may be able to use a lighter material, but Orion doesn’t have an engine bell to think about; these aren’t precise values, merely order-of-magnitude estimates. I’ve left the numbers in a precise form because it’s correct to add first and round later.

Landing Legs, RCS & grid fins:

On the first stage, the mass difference between a reusable and non-resuable configuration is estimated [6] to be 2,500kg (25,600kg – 23,100kg between v1.1 and F9R). Now the dry mass of the second stage is estimated [7] to be 4,000 kg, compared to 22,200 for the v1.2 first stage, 5.55 times smaller. So a very minimum estimate for the mass would be 450kg, assuming that all three can be reduced in exact proportion to the mass, since: landing legs must carry 5.55 times less weight and cold gas thrusters and grid fins must push on something with 5.55 times less inertia. The true value will probably be higher than this, because some things (plumbing, maybe) would scale less well than that.

Extra fuel:

Again, we can naïvely suppose that this scales with the inertial mass of the booster. In practice, it will likely be higher because the second stage would have a more difficult flight profile than the first stage, even after most of the speed is taken up by re-entry heating. The first stage separates weighing 54.0 metric tonnes due to fuel remaining (of its original 400 tonnes) for a RTLS landing [8] and 34.6 metric tonnes for a ASDS landing [8]. Again rescaling to dry mass (to ensure the same delta-v as the first stage), the second stage would now weigh 6.23 metric tonnes for an ASDS profile like the first stage assuming approximately the same efficiency. M1D is more efficient in space, less efficient in the atmosphere, so I’m assuming that these cancel each other out, approximately. That’s why these calculations are approximate. We now have effectively 2,230kg of dead weight added to the rocket as landing fuel.

So our overall estimate is 4,600kg. This estimate is very approximate. But it tells us
some important things:

Making the second stage reusable using only 1,000kg extra seems very implausible. Adding even this amount of mass would have made SpaceX incapable of launching any of the previously mentioned GTO satellites. They might still be able to resupply the ISS, but their commercial manifest would be severely hampered.

Reusability would not be expected to use more than maybe 17,000kg, unless the propulsive landing turns out to be way more complicated than anyone thought. This payload penalty would leave the Falcon 9 worthless, but the Falcon Heavy would still
have plenty of low Earth orbit (LEO) capability, and about the same GTO payload as the Falcon 9 has today.

Second stage reusablity is impractical today because of the number of GTO missions on SpaceX’s manifest (GTO missions are hit hardest by the payload penalty), and because the Falcon Heavy isn’t mature enough take over on these flights, yet.

References:

[1] https://www.youtube.com/watch?v=y13jbl7ASxY&t=14m20s
[2] http://www.spacex.com/falcon9
[3] http://www.spacex.com/falcon-heavy
[4] https://www.youtube.com/watch?v=sSF81yjVbJE
[5] https://www.nasa.gov/pdf/695726main_ComingHome-ebook.pdf page 262
[6] http://spaceflight101.com/spacerockets/falcon-9-v1-1-f9r/
[7] http://spaceflight101.com/spacerockets/falcon-9-ft/
[8] https://www.reddit.com/r/spacex/wiki/faq/reusability Note that you have to reverse-engineer the rocket equation to get this result from a “15% drop in performance” – this is left as an exercise to the reader.