Tuesday, May 02, 2006

Three Stage to Orbit

I'm going to try to convince you [hi Jon!] that a three stage to orbit rocket might be significantly cheaper than a two stage to orbit system lifting the same size payload. My argument centers on the idea that the engines are the expensive portion of the rocket, and that tanks and structure are cheap.

To get into a 200 km orbit, a two stage rocket generally has a first stage which burns for about 150 seconds, followed by a second stage which burns for around 500 seconds. Second stage engines deliver much more impulse per dollar, primarily because the engine burns so much longer, and also because the engine has a larger expansion ratio and so a higher specific impulse.

The second stage burn time cannot be extended greatly beyond 500 seconds when going to low earth orbit, because the initial acceleration becomes so low that the stage falls back to earth before it achieves orbital velocity. Similarly, the first stage burn time cannot be pushed much past 150 seconds (assuming LOX/kerosene, for which the Isp is around 300 seconds) because the rocket must start with significantly more than 1 G of acceleration to get off the ground. This is easy to see: if initial acceleration is exactly 1 G, and the vehicle was all fuel and no structure, then the burn time would be exactly the Isp.

The point of a three stage rocket is to significantly extend the first LOX/kerosene stage burn time, and improve the expansion ratio, with the use of an earlier and much cheaper stage. To yield a cost improvement, the entire new stage must cost less than the first stage engines it is replacing. Also, the development cost of the new stage must be low.

Typically, strap-on solid rocket motors are used as cheap extra launch thrust on American or European launchers. LOX/Kerosene launchers, such as the SpaceX Falcon series and many of the Russian launchers, are so cheap that such solid rockets and their handling and cleanup are too expensive to help. I suspect that steam rockets (such as the rocket that hurled Evil Kneivel's "motorcycle" over the Snake River Canyon) may be cheap enough. Unfortunately, steam rockets have such low specific impulse (about 50 seconds) that any benefit must come from just a few hundred m/s of delta-V (realistically, around 300 m/s). The rest of this post will show that there is a very large benefit to having just a few hundred m/s of initial boost.

One way to measure the benefit of the extra stage zero is to compare the payload of a similarly priced rockets, one with and without the extra stage. Ideally, the rockets would be seperately optimized, but that makes cost comparisons looser. Instead, we will imagine that they have the same engines and differ in smaller details like expansion ratio and tank size. The baseline rocket I have chosen is the SpaceX Falcon 1, which is promised to take 570 kg to a 200km orbit for $6.7M. My simulations show SpaceX has a significant sandbag in there, and that the rocket might deliver 670 kg at the edge of its performance envelope, and it is this larger number that I compare against when calculating the benefit of extra boost.

The first comparison is between a series of identical Falcons, some with a stage zero delivering various burnout velocities, and one without. As shown, a 300 m/s boost allows the three-stage rocket to lift 41% more payload.

Falcon 1 with various starting boost


Because a three stage Falcon does not require it's LOX/kerosene engines to lift it off the ground, the engines can be configured for larger expansion and higher vacuum Isp (and thus lower sea-level Isp). Elon Musk at SpaceX has claimed that a vacuum-optimized Merlin would deliver 340 seconds Isp. In the table below, I've assumed 330 seconds, and a drop in sea-level Isp to 190. Now the three-stage 300 m/s boost lifts 61% more mass to orbit. This change carries the reliability penalty that the engines may have to be started in flight to avoid thrust instability due exit pressure being much less than sea level pressure. Note that a Falcon with no stage zero boost cannot get off the ground with higher expansion engines.

Falcon 1 with various starting boost, Isp=330 s


The full gain of the stage zero is realized by extending the burn time of the first LOX/kerosene stage by extending the tanks. Here I've assumed that the extra tankage and pressurant and so on weighs 4.3% of the extra propellant added. With extended tanks, the three stage 300 m/s boost lifts 138% more mass to orbit.

Falcon 1 with various starting boost, Isp=330 s, extra propellant

LEOextra stage 1

You might wonder if these massive payload increases require the use of a bigger and more powerful upper stage. The answer is no, even hefting a 1595 kg payload, the existing falcon upper stage delta-V drops from 4973 to 3454 m/s. I did not look at moving propellant from the lower stage to the upper stage, which might improve the payload numbers a bit more.

138% is a large payload increase, but we have to compare this to the cost of simply scaling up an existing rocket. Fortunately, SpaceX has done some of this work for us. Interpolating between their prices, the 138% payload boost might be sold for $1.8M more.

Which leaves the question, can a suitable steam rocket be built to heave the Falcon 1 to 300 m/s straight up for less than $1.8M per shot, and minimal development costs?

Here's what you'd need:
  • Mass Fraction = 80% (Shuttle SRB mass fraction is 85%, it holds 60 atmospheres pressure)
  • Mass, fuelled = 62992 kg
  • Mass, empty = 12600 kg
  • Volume = 100 m^3
  • Size = 3.6 m diameter x 10 m long

  • A 12600 kg welded stainless steel tank might cost $250,000, but can probably be recovered and reused with minimal work. I'm going to claim the expensive bit isn't the flight hardware, but rather the ground support, and I'll get to that in another post.


    1. what altitude do you expect to be at at the end of the boost?

      If your acceleration averages 10m/s + g. then it will take 30 seconds to reach 300m/s. the average speed will be less than 150m/s. So max altitude will be about 4.5k. Will the pressure drop at this altitude really be that helpful?

    2. Interesting, I had thought something along the same lines. Reducing gravity losses yields unexpected benefits and you can increase tanks a lot.

      What's the trade on thrust to weight and isp and all that?

      If you need higher than a steam rocket deltav, use a peroxide monopropellant booster.

      If you can get away with a really low delta-v, use helicopter rotors powered by the first stage turbopump. Really light, and no extra tanks needed. You can also recover those. I don't know much about blade aerodynamics, but you get altitude advantage and maybe 50 to 100 m/s speed?

    3. As you point out, the problem with staging is the additional failure modes and the more complicated ground support.

      Also, a delta-v of 300m/s will not get your rocket high enough to use vacuum-optimized engines. Ignoring air drag, a projectile launched from sea level with 300m/s will reach only about 4500m. 500m/s will reach 12500m, which is still not good enough for e.g. a vacuum version of the RL10.

      However, if you manage to get the upper stage(s) to something approaching vacuum, you could do an "Assisted SSTO" like the original kistler proposal. Have a very high performance upper stage (something like a centaur) and a very rugged and low performance lower stage.

    4. I think I was assuming 5G acceleration here. In other words, 7.5 seconds to get to 300 m/s. In Hobbes words: nasty, brutish, and short. Burnout at 1125 m if going straight up. Actually, straight up isn't best. I'll see if I can figure out how to plot the launch profile and post that.

      But the pressure drop at 1 km altitude isn't what matters. Since the rocket is now vertically supersonic, it will continue to a fairly high altitude even without any engine thrust at all. The vacuum-optimized "first" stage produces less than 1 G of acceleration, but the rocket still carries just fine to altitudes where the pressure drop makes quite a bit of difference. Over the now-longer burn time of the engine, an even larger fraction is spent at low pressure and the average Isp is even better.

      The trouble with helicopter rotors and peroxide monopropellants and so forth is complexity. A low-Isp zero stage has to be really, really cheap. This one has no moving parts! That's why it's reasonable to propose, even though the zero stage is bigger than the rocket it lifts.

    5. What about the use of sustainer engines as in _The Rocket Company_ TSTO?

    6. I've not read "The Rocket Company", and don't know it's proposal. The only sustainer engine I'm familiar with is the sustainer engine on the original Atlas booster.

      As near as I can tell, a sustainer engine is an upper stage engine that is ground-lighted. You get some extra thrust on the ground, you get to know if your upper-stage engine will light before you release the hold-down arms, and you lose some expansion ratio on that upper stage motor. This last detail loses a fair bit of performance where it really matters. That, and the simplicity of having the stages in a line rather than firing in parallel, is what leads most rocket designs not to use sustainer engines.

      The steam rocket boosters I describe in the post fire in parallel to the first stage engines, so in some sense the first stage engines become "sustainer" engines, although there isn't much meaning conveyed by that use of the word.

      I don't see much value in sustainer engines. In particular, the Shuttle SSMEs cost so much because they were required to operate at such high pressure so that they could operate without seperation at ground level and still get decent Isp at altitude.