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

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?

• I'm not great at working out this kind of math, but could it be because Falcon Heavy's second stage dry mass is 70% more than Atlas V's (3900 vs 2316 kg from the numbers I'm seeing)? Every m/s of deltaV has to be imparted to that mass in addition to the payload, so it starts further up the rocket equation curve. – hobbs Feb 21 '18 at 2:23
• It's because of the poor Isp. The ~100 sec advantage in Isp between the Falcon upper stage and something like a Centaur makes a huge difference in the mass-ratio because the rocket equation contains a log function. – Organic Marble Feb 21 '18 at 3:10
• I did mention that while those charts are the best that we have, they aren't accurate yet. SpaceX found they were using the performance for the upper state of Falcon 1, which isn't nearly what the latest Falcon 9 can do. – PearsonArtPhoto Feb 21 '18 at 3:40
• I would concur with @OrganicMarble - the Centaur LH/LOX upperstage will make a huge difference with its higher Isp. – Carlos N Sep 25 '18 at 15:36
• Scott Manley answered this question quite comprehensively in a recent video. youtube.com/watch?v=QoUtgWQk-Y0 – Ingolifs Oct 7 '18 at 10:36

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.

• Welcome to space! Reading the question carefully indicates that this will require a more substantial answer than explaining what delta-v and Isp mean. It's like answering a performance question in SO with "the primary metrics of a computer program is execution time, memory usage, and reliability." ;-) If it's possible to ad some hard numbers, and show that the difference is significant for the scenario in the question, that would be better. – uhoh Oct 7 '18 at 8:45
• That's no where near as clear as it used to be, had to remove my +1 – JCRM Oct 7 '18 at 10:35
• Sorry. Sounds like some people here like math, while others don't need it. Can't win! – RickNZ Oct 7 '18 at 19:57
• Note that the Merlin in the Falcon upper stage is burning RP-1, not methane, at an Isp of 348 s. – TooTea Oct 9 '18 at 13:05