Reusable Falcon Heavy payload (upper stage staging velocity)

Jon and I were discussing the recent Falcon Heavy payload numbers.

Expendable performance to GTO is supposed to be 22 tons (metric, same for the rest of this post). Given how aggressive that is, and given the history of Falcon 9’s performance, I would expect that to be to 1800m/s-to-go (i.e. you need 1800m/s more delta-v to get to actual geosynchronous orbit). The delta-v between that and LEO is approximately 2.5km/s, though that depends on the details of exactly which LEO orbit (but I think this is a good number; provide a better number if you know of one).

Assume the propellant in the upper stage is about 110t, and the dry mass is 5t (this is in range of other people’s estimates and figures from SpaceX, though I’ve seen down to 4t dry mass). With a 22t payload, that gives a full mass of 137t, empty of 27t. Given Merlin Vac’s Isp of 348 (which is staged-combustion territory, although it is gas generator), you have a bit better than 3.4km/s exhaust velocity.

Delta-v of the upper stage is thus slightly more than:
3.4km/s*ln(137t/27t) = 5.5km/s.

Given the 2.5km/s required to reach GTO from LEO, that means that the upper stage has already provided 3km/s of delta-v already by the time the stack reaches LEO.

What is the LEO velocity? Given the standard gravitational parameter of the Earth mu = 3.986E14m^3/s^2 and we’ll pick an altitude of 150km, LEO is:
sqrt(3.986E14m^3/(s^2*(r_Earth+150km))) = 7814m/s.

But we’re concerned with the speed with respect to the ground, so we have to minus the contribution from the Earth’s rotation. Given 28 degrees lattitude (I think it’s actually launching from Boca Chica, but hey), that’s:
cos(28 degrees)*2*pi*r_Earth/day = a bit more than 400m/s.

So the relative speed is around 7.4km/s, let’s round to 7.5km/s.

So by the time the upper stage has been burning through 3km/s, the stack is at 7.5km/s velocity with respect to the ground. 3km/s less than that is 4.5km/s (minus a small amount of gravity loss, which is small by the time you’re at the upper stage).

So the core stage for the expendable Falcon Heavy is going at 4.5km/s relative to the atmosphere at stage sep from the upper stage.

With the reusable variants, the upper stage will be pushing a lighter load, so the upper stage has a higher delta-v and will be doing more of the work, and so the staging velocity (relative to the atmosphere) will be even less. So we’re talking about 4.5km/s, worst case. For a 15 ton payload to 1800m/s to go, you’re talking about ~3.6km/s staging velocity. That’s a much easier reentry problem than the 6km/s I’ve seen bandied about, and it could even be handled largely by propulsion.

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3 Responses to Reusable Falcon Heavy payload (upper stage staging velocity)

  1. Hop David says:

    Very interesting. This partially answers a stack exchange question I had asked:

    So you’re talking 3.6 km/s at booster stage separation? That’s higher than I expected but still within reason. If I recall correctly, the booster had around 8 km/s delta V. So it could achieve around 3 to 4 km/s eastward velocity and then kill most of that propulsively, as you say.

    Someone had suggested a 6 km/s when the booster separated from the rest of the spacecraft? I hadn’t seen that.

  2. Paul D. says:

    How effective is rocket exhaust at shielding a stage from entry heating? I’m thinking of operating the rocket very far from the normal O/F ratio, so the gas is fairly cool. This might be more efficient than using the propellant for straight propulsion, maybe?

    If some sort of “propellant as coolant” entry can be made to work, then it might be possible to return upper stages (that make it to stable orbit) by launching additional propellant to allow them to reenter. This would require a separate launch of a tanker to recover (preferably) itself and more than one other upper stage, but at least this propellant wouldn’t cut into the payload of the payload-carrying launches.

  3. Paul451 says:

    For metric tonnes, try “tonnes” instead of “tons”.

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