Expanding on my SpaceX BFR napkin estimates

Postage stamp sized picture of what some dude on NSF
thinks the BFR family might possibly look like. 

Some time ago, I drew up some napkin level concepts to look at what a SpaceX super heavy lift vehicle could look like. These concepts were based on what we knew about Raptor (1 million pounds of thrust, 380s vac isp) as well as Falcon 9 figures. However, in the meantime, new figures have been coming out further expanding on what we know about Raptor. Also, the original scratchings did not assume reusability, even though that is undoubtedly something this BFR will have to be capable of being, if they ever wish to colonize Mars without government support.

Disclaimer: All "tons" are metric tons in here. Also, all these figures are estimates. While the figures are given as specific, one should take them with a grain of salt, and remember they're estimates.

Creating a "reference vehicle"

First, a starting point from which we can expand on the concept. As a starting point, I'll take the Falcon 9v1.1 rocket, with the numbers provided from Spacelaunchreport.com's article about F9. The performance figures for Raptor are taken from this NSF article. At sea level, the vehicle has a specific impulse of 321 seconds, and a specific impulse of 363 seconds in vacuum. The thrust in vacuum is 4500 kN, which translates into 3979 kN (or 406 metric tons) of thrust at sea level. The upper stage version of Raptor is assumed to have an Isp of 380 seconds, which would translate into a thrust of 4711 kN.

The first step is to scale up the Falcon 9 stages to fit with these new thrust levels. From spacelaunchreport.com, we take the estimates of 404 tons for the first stage and 99 tons for the upper stage, and multiply them by the ratio between the Raptor thrust and the Merlin 1D thrust. Doing so, we get stage GLOWs of 2452 tons for the first stage, and 582 tons for the second stage. Falcon 9's stages have a propellant mass fraction of about 0.95-0.96, but this would be slightly lower because of the lower density of methane. The density of a mixture of methane/oxygen is about 80% of that of RP-1 and oxygen, meaning that these stages would have a propellant mass fraction of about 0.94. Which is pretty darn high, mind you.

Using these numbers, we get the following values for our hypothetical "Falcon Mars":

Stage 1:

GLOW: 2452 tons

Total propellant: 2305 tons

Empty mass: 147 tons

Thrust in vacuum: 40500 kN

Thrust at lift-off: 35811 kN (3654 tons)

Specific impulse: 363 vac, 321 sl, ~349 avg.

Stage 2:

GLOW: 582 tons

Total propellant: 547 tons

Empty mass: 35 tons

Thrust: 4711 kN

Isp: 380 

Vehicle total: 

Payload fairing: 10 tons (10 meter fairing)

GLOW (without payload): 3044 tons

Payload to LEO: 145.4 tons

Payload to TMI (C3=15): 21.8 tons

Total TWR at lift-off: 1.15

So, 145.4 tons to LEO, with a low TWR at lift-off of 1.15. This low TWR is a bit of a bummer, as it restricts a lot of upper stage upgrades to the vehicle. The estimate assumes that 1% of propellant is unusable and is added to the empty mass to become burnout mass.

The simplest upgrade from here would be to add an engine to the upper stage, as the current one might get underpowered, as we shall see later. If we take the Raptor TWR as 75:1, similar to RD-180 and RS-25, the engine mass becomes about 6.4 tons. Adding this to the upper stage and increasing the thrust increases payload to an even awesomer 153.6 metric tons, which is a 5.6% increase in payload. While it would further reduce lift-off TWR to 1.14, it would still be sufficient to take off.

Making the vehicle reusable

How much would making this BFR reusable cost in terms of payload? For Falcon 9, first stage reuse would cost about 30%. We could just say, well, let's cut 30% for first stage reuse. But that would be easy, wouldn't it?

In order to get a 30% reduction in payload from Falcon 9v1.1 the first stage would need to hold about 50 metric tons of propellant upon separation. Using the F9 estimates I used earlier, I get about 15.7 tons to LEO without reuse. Increasing the first stage's burnout mass by 40 tons and reducing propellant mass by 40 tons reduced this to 11 tons, which is a roughly 30% decrease.

Using a 311 second isp and 19 ton burnout mass, this translates into about 3450 m/s to return to the launch pad. Plugging this back into the BFR estimates, the core burnout mass would have to be increased to 387 metric tons, and propellant mass reduced to 2065 tons. This in turn reduced payload to 109.5 metric tons. This is a 29% reduction in payload, so this was completely pointless. But math is fun, so why would it matter?

Upper stage reuse is harder to estimate. I believe Elon estimated reduction in payload per stage as about 30%, meaning upper stage reduction would reduce payload by another 30%, or about half of what it was originally. But because I have no idea how much a PICA heat shield 10 meters in diameter would weigh, I don't think I can really do any math on this one. Simply assuming 30% off, the payload would drop to 76.7 tons to LEO, which is still pretty high. SLS-class payloads for very little. Mehr Nutzlast zum Spotpreis!

How about making it bigger?

We have already created a monster rocket. 153.6 tons to LEO, or 109.5 tons to LEO if partially reusable. But SpaceX has hinted at getting 100 tons directly to Mars, not 100 tons to LEO. So, we need to make it even bigger. And the easiest way to do that is by looking at what SpaceX has already chosen to do, which is making a tri-core variant like Falcon Heavy. 

Using identical cores, like originally planned for Falcon Heavy, we can increase the payload by quite a lot: assuming perfrect cross-feed, and doubled propellant left in the central core, the total LEO payload becomes 230.8 tons to LEO, with all three first stages reused. The payload to Mars is a meager 38 tons though. Surely that can be higher?

The most obvious way to increase the payload would be to scale up the boosters and reduce the propellant load left in the core stage upon booster separation. It could still be reused, but would have to land on an ocean platform, or maybe a small island in the Atlantic. While this would reduce turnaround time, it's a spacecraft to Mars. There's only one launch window every two years anyway.

Using an ocean platform increases TMI payload by a lot; from 38 to 57.7 metric tons to a C3=15 trajectory. Without fairing, this would be about 58 tons, though this could cause changes to MCT's design. Also, the boosters can be made a lot bigger, similar to Falcon Heavy, because of the much higher thrust at lift-off. The total thrust at lift-off would be 10962 tons. Limiting TWR to 1.14 and assuming a 100 ton payload, the boosters could be scaled up to be a total of 3236 metric tons, or almost 1000 tons bigger than the core.

These boosters, with flyback ability, could hold about 2724 tons of propellant, and would have a separation mass of 511 tons. Using these figures, the TMI payload goes up again, to 67.8 metric tons. But it's still not 100.

However, the LEO payload becomes a staggering 327.4 metric tons. Using a third stage to do TMI, however, one could increase the usable payload to Mars to 101 metric tons. We did it!

It now also becomes clear how important it would be to use two engines on the upper stage rather than one. With a 327 ton payload, the TWR of the upper stage with two engines is 1.03. With one engine, however, it would be a TWR of only 0.52. As the upper stage still has to do quite the burn to LEO, having such a high thrust stage is useful in reducing gravity losses. Not having one would reduce payload by about 5-10%.

Then what could MCT look like? Maybe something like this. All I know is that in order for the vehicle to get to Mars from LEO, it's going to have to hold a heck of a lot of propellant.

Highly technical sketch of what MCT, the BFR's payload, could look like.