Why can't Falcon Heavy accelerate deep space payloads as fast as Atlas V and Delta Heavy?

Question

A Falcon Heavy with an expendable core costs $95M and lifts 2.5x as much to GTO as the Atlas V. It's Pluto injection orbit payload is nearly half of the Atlas V's GTO payload. In the answer to this question

No need for gravity assist with Falcon Heavy?

PearsonArtPhoto posts graphs showing that an Atlas V has the ability to accelerate more payload to high (outer system) velocities than the reusable Falcon Heavy, and not much less than the expendable Falcon Heavy (and is equal at 60 km/sec).

Launching a Falcon Heavy with an expendable core should perform roughly 90% as well as the fully expendable FH, so if it's performance curve was placed on the second chart it would drop to the Atlas V curve even sooner (a little over 50 km/sec).

I'm having trouble understanding why this is true. I understand that Falcon Heavy's upper stage performance is sub-par, and I think because

1. The vacuum Merlin doesn't have a great ISP, and
2. The FH first stage completes it's burn at a lower velocity than non-reusable competitors, forcing the second stage to fire sooner, so first stage boosters won't require heavy heat shielding to be recoverable.

I can conceptualize why both of these would hurt FH performance to GTO, but once we reach GTO it seems to me that FH payload performance beyond GTO would be roughly proportionate to it's GTO performance. IE if it can accelerating 2.5x as much mass to GTO insertion orbit as an Atlas V, then it should be able to accelerate something like 2.5x more to Pluto insertion orbit.

Clearly from the charts this isn't true. The Falcon Heavy relative performance continuously degrades vs. it's competitors as the required DeltaV increases. I don't understand why, or the the forces driving these curves, can someone explain them?

Answer 1

The short answer is that the Isp of the Falcon's RP-1/LOX upper stage engines is much less than the Atlas V's LH/LOX.

The primary metric of interest is $\Delta v$, which depends on $I_{sp}$ and mass fraction. If you need more $\Delta v$ for a particular mission, but your $I_{sp}$ is lower than another rocket's, that means you'll need a better mass fraction, which means less payload weight.

I don't have exact numbers handy, but here's a rough example. Start with the rocket equation:

$\Delta v = v_e ln \frac{m_0}{m_f}$

or

$e^\frac{\Delta v}{v_e} = \frac{m_0}{m_f}$

$v_e = I_{sp}g$ where $g$ is the gravitation constant, 9.8$m / s^2$

For RP-1/LOX $I_{sp}$, let's say 330 sec, and for LH/LOX, 500 sec. For $\Delta v$, roughly 11 km/s to GTO, and 16 km/s total to some deep space location. That gives us:

$\frac{m_0}{m_f}$ = 30 for RP-1/LOX at GTO and 141 for deep space. For LH/LOX, 9.4 at GTO and 26 for deep space.

Let's say both vehicles could carry a 9 ton payload to GTO with a 1 ton structure. That means a 10 ton dry mass for the RP-1/LOX rocket, and 10 * 30 = 300 ton wet mass, which is 300 - 10 = 290 tons of fuel. For LH/LOX, 10 * 9.4 = 94 ton wet mass, with 94 - 10 = 84 tons of fuel.

For deep space, keeping the fuel and structure mass constant, what happens to our payload mass? For RP-1/LOX, it drops to 1.07 tons. For LH/LOX, it's 2.36 tons.

So, although they started with the same payload at GTO, with the larger $\Delta v$ requirement, the LH/LOX vehicle can carry more than twice as much payload to deep space.