Monday, January 16, 2006

Slow Launch

Every time I see a rocket vault off the pad, I wonder why the engineers didn't make the propellant tanks bigger. Adding propellant to any stage always increases the delta-V delivered, so long as you ignore gravity losses.

It takes a lot of thrust and energy expended just to get the rocket to hover stationary above the pad. The delta-V lost is called the gravity loss. As the rocket gets closer to orbital velocity, gravity losses drop. At low velocities, gravity losses are just time * acceleration by gravity.

Gravity losses, as it turns out, are a big deal. As you add propellant, you logarithmically increase the delta-V, but you linearly increase the gravity losses. At some point, gravity losses overtake the increased delta-V.

The tradeoff between the two is different for different kinds of rockets. In a solid-fuelled rocket, the entire stage is a big combustion chamber, that must contain the gas pressure used to accelerate the vehicle upward. Large pressure vessels are heavy, and so the fuel container is a large fraction of a solid rocket's mass. Pressure-fed liquid-fuelled rockets don't actually have the combustion chamber in the tanks, but the tanks must hold higher pressure than the combustion chamber, so the mass penalty is similar.

Pump-fed liquid-fuelled rockets hold their propellants at a small fraction of the combustion chamber pressure, and so their tanks are a small fraction of the weight of a similarly-sized solid rocket.

Let's take a look at the two extremes. First, a kerosene/LOX liquid-fuelled rocket, like the Saturn V or Falcon 9. I've made up a table to show the decreasing performance return of steadily larger and larger tanks. Here I've presumed a base rocket with an Isp of 290, a first-stage thrust of 750,000 kg, a Gross Lift-Off Weight (GLOW) of 500,000 kg, and a first stage burnout weight (this includes the upper stages) of 150,000 kg. This rocket has an initial acceleration of 1.5 G (but remember you lose 1 G to earth). Incremental tankage weighs just 2.5% of the incremental propellant stored. I'm assuming that the engine thrust stays fixed. Delta-V numbers are in meters/sec. Tower clearance times are a little high, as they assume no acceleration beyond the initial acceleration.

Isp 290
Ve 2842
Tankage 0.025
Thrust 750000
Burn rate 2586
Tower 50
inc inc delivered tower
G Mf Me delta-V time G-loss delta-V clear
2 375000 146875 0 0 0 0 3.2
1.9 394737 147368 136 7 73 63 3.4
1.8 416667 147917 143 8 81 62 3.6
1.7 441176 148529 151 9 91 60 3.8
1.6 468750 149219 159 10 102 57 4.1
1.5 500000 150000 169 12 115 53 4.5
1.4 535714 150893 179 13 132 47 5.1
1.3 576923 151923 191 16 152 39 5.8
1.2 625000 153125 205 18 178 27 7.1
1.1 681818 154545 221 21 210 11 10.1
1 750000 156250 240 26 252 -12 N/A

You can see why the Saturn V initial acceleration was just 1.13 Gs. You can also see why launching the Saturn V in a strong wind could have been a problem: it takes a long time to get past the tower.

Now let's look at the other extreme, a solid rocket first stage, like the Titan 4 or the Stick proposal. (Because most of the Shuttle's liftoff thrust is from it's solids, it fits in this category too.) Here the Isp is a bit lower, but more importantly the tankage fraction is far higher: 12%.

Isp 242
Ve 2371.6
Tankage 0.12
Thrust 750000
Burn rate 3099
Tower 50
inc inc delivered tower
G Mf Me delta-V time G-loss delta-V clear
2.5 300000 126000 0 0 0 0 2.6
2.4 312500 127500 69 4 35 34 2.7
2.3 326087 129130 71 4 38 33 2.8
2.2 340909 130909 73 4 41 32 2.9
2.1 357143 132857 75 5 45 30 3.0
2 375000 135000 78 5 50 28 3.2
1.9 394737 137368 80 6 55 25 3.4
1.8 416667 140000 83 6 61 22 3.6
1.7 441176 142941 86 7 68 18 3.8
1.6 468750 146250 90 8 77 13 4.1
1.5 500000 150000 93 9 87 6 4.5
1.4 535714 154286 97 10 99 -3 5.1

You can see why a Titan 4 gets off the pad with nearly 1.5 Gs of initial acceleration.

Solid rockets are a good match for first stage engines. It takes relatively little engineering work to produce a large amount of thrust from a solid motor, which is the biggest cost driver for first stage engines. But because the casing weighs so much, and also because solid propellants have lower Isp, the delta-V of a solid first stage is never going to be as good as a liquid-fuelled analog. That leaves more work for the second stage, which means thinner engineering margins and more desire for high-Isp propellants, like liquid hydrogen. And so, cheap solid rocket first stages drive more cost into the upper stage.


  1. One thing to watch for with high-Isp propellants is that they also increase your gravity losses, so the performance boost isn't as good as it looks in the rocket equation.

    (in case you find this counterintuitive: the better your mass fraction, the more mass you have gravity pulling on all the way to orbit)

  2. Gravity Loss has a retrospective look at this year's Lunar Lander Challenge. There's also analysis of where things will go from here in the competition, as well as comparisons with DARPA's challenges and the Ansari X PRIZE.

  3. I can see how a higher Isp stage might have a greater total gravity loss, but why should it necessarily be greater than the increase in delta-V resulting from the higher Isp? It's the net (post-gravity) delta-V that counts. Gross (pre-gravity) delta-V goes up linearly with Isp, and that's pretty desirable considering that it goes up only logarithmically with mass fraction.

    I think of gravity loss as driven by two things: burn time and pitch angle. The faster you burn, the less time for gravity to accelerate you downward. And the more horizontal your attitude, the lower the fraction of your thrust subject to gravity loss; it's equal to the sine of the pitch angle.

    If we could do orbital launches the way Newton originally showed us (taking off horizontally from the ground) then we'd have very small gravity losses. But we have an atmosphere that exerts a lot of drag, so you have to sacrifice some effiency to get above it. Without an atmosphere to worry about, LM ascents from the moon flew vertically for just a few seconds to clear terrain, then pitched down sharply to drive into orbit.

    I think achieving horizontal flight more quickly is one of the main ideas behind the Pegasus. Although the altitude and velocity of the aircraft is only a tiny fraction of orbital energy, that initial altitude allows the launcher to fly nearly horizontally immediately after release. Its first stage has wings to compensate for some of its gravity loss with aerodynamic lift. And finally, by starting above much of the atmosphere the peak aerodynamic forces are reduced, lightening the structure.

    When you're horizontal, as near the end of upper stage powered flight, your gravity loss is zero and you can afford to use high Isp (e.g., LH2/LOX) engines with relatively low thrust, like the J-2. You can even afford to shift the mixture and gain a few seconds on Isp at the expense of thrust. But at liftoff, you're vertical and will be nearly so for a while to get above the atmosphere, so you're subject to very high gravity losses. Then you want as much thrust as you can reasonably get, which means either a solid or a big but inefficient engine like the F-1 (Isp only 263 sec even with RP1/LOX).

    1. Higher Isp engines generally have less thrust/weight. If that's not true, then higher Isp is always higher performance, just as you expect.

      To figure out these tradeoffs, I built a simulator which flies the rocket into orbit. Quite a bit of the computation goes into finding, for any given rocket, the optimal trajectory to get into orbit.

      Unfortunately, that simulator sits on disks in a computer by my right knee which has not been turned on in many years, so I can't just give it to you because I don't know how to recover the RAID5. Phhbbbt.