Why can Falcon Heavy bring 4.2 times as much mass to Mars than F9, but only 2.7 times as much to LEO?

Question

In Wikipedia's article SpaceX Falcon-Heavy the section titled Capabilities has a table comparing maximum payload masses to Falcon 9 for different payload destination. The current numbers there are shown below.

There is probably some fundamental, easy to grasp reason why Heavy can bring 4.2 times as much mass to Mars than F9, but only 3.2 times as much to GTO and only 2.7 times as much to LEO, but I don't know what that would be.

Payload            Falcon Heavy   Falcon 9    Ratio
LEO (28,5°)          63,800 kg   22,800 kg    2.68
GTO (27°)            26,700 kg    8,300 kg    3.22
GTO (27°) Reusable    8,000 kg    5,500 kg    1.45  -  does not apply here
Mars                 16,800 kg    4,020 kg    4.18
Pluto                 3,500 kg      - - 

below: rough plot of ratio versus "altitude" of the three (presumably non-reusable) data points, just to show there is a trend.

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below: Artist conception of a Falcon Heavy, from here.

Answer 1

The final stage of a rocket has to lift not just the payload but also itself. So lets look not at the payload mass alone but at the total mass lifted to the final orbit.

According to http://www.spaceflightinsider.com/hangar/falcon/ the empty mass of the second stage is 3,900 kg. Lets add that the numbers in the table.

If we add the mass of the final stage to the numbers in your table we get.

Payload + 2nd stage Falcon Heavy   Falcon 9    Ratio
LEO (28,5°)          67.700 kg   26,700 kg    2.54
GTO (27°)            30,600 kg   12,200 kg    2.50
Mars                 20,700 kg    7,920 kg    2.61

Near enough the same.

Answer 2

All else being equal (e.g. propellant choice), higher-energy trajectories favor a rocket with more stages.

One thought experiment that can help you understand this is to consider two missions launched on identical rockets, the first to LEO and the second to an interplanetary trajectory. Both missions have payloads sized to "max out" the capacity of the launch vehicle. Obviously the second payload will have to be much lighter than the first, but notice that the second payload is therefore a much smaller fraction of the dry mass of the upper stage. Recall that the upper stage must end up on the same trajectory as the payload, so that second mission is less "efficient" because most of the work the upper stage does is just accelerating its own mass rather than the payload.

If you were constrained to the same overall vehicle mass, you'd be better off with a smaller second stage and adding a third stage. That way the upper stage mass will be better matched to the payload mass, so less of the work it does will be wasted. It's almost like impedance-matched power transfer.

So how does this relate to the Falcon Heavy? Essentially the addition of the extra boosters make the system into a 3-stage rocket. The center core is throttled down during first-stage flight, so it still has quite a bit of propellant remaining when the side boosters empty themselves and separate. The center core burns for a while longer (effectively 2nd-stage flight, because you've thrown away the dead mass of the empty side boosters), then it too separates and the upper stage takes over. This means that the upper stage of the Heavy is responsible for a smaller fraction of the overall energy delivered (or the velocity added) compared to the upper stage of the F9. So for the reasons described earlier, the Heavy as a vehicle is more efficient at delivering payloads to high energy trajectories than the F9.

Answer 3

Moving around once in orbit is a lot easier than getting something into orbit. The only issue with this is bringing either another stage or even just a larger stage into orbit first to allow for fuel amounts to get to mars. The extra 40 tons to LEO of the Falcon Heavy allows for it to get this extra stage/ larger stage to orbit which it can then use to get to Mars. For the Falcon 9 to get something to Mars it would also need this extra/larger stage to get to orbit however the lifting capability of only 9 Merlin engines would result in an extremely inefficient launch to orbit due to the first stage burning so much extra fuel to get the extra/larger stage to orbit. This would result in the extra/larger stage using up more of it's own fuel getting to orbit and therefore would have less net force left to push the payload to Mars.