Getting To Orbit And The Rocket Equation

The Apollo 12 mission recently celebrated its 50th anniversary. Launching on November 14, 1969 and returning on November 24, it put humans on the Moon for the second time. I wrote about Apollo 11 (mostly about its guidance computer) earlier in the year in my post The First Computer on the Moon. Today’s post is about the rocket equation, and how challenging it is to get into orbit around the earth.

Optimization in synthesis
In a synthesis tool, at least in the era when I was at Ambit, the optimization phase would sometimes get into a problem spiral. If the required timing was close to the limit for what the technology could deliver, then during optimization the tool would switch cells for higher drive (faster) cells, or more drivers. But higher drive cells are bigger, so if a lot of this happened, the block would get bigger, there would be more interconnect and more fanout, and so the higher driver cells needed to be swapped for even higher drive cells. But higher drive cells are bigger still… If you were lucky, things would converge. If you were unlucky, you’d end up with a huge block and large TNS.

A similar thing happens with rockets (like the Saturn V that took Apollo 12 into orbit). To lift a heavier weight of payload, you need more fuel. But now you have to lift the extra fuel too, so you need even more fuel. And so on. This is the tyranny of the rocket equation.

So what is the rocket equation?

The rocket equation
I remember doing a compulsory course on dynamics as part of first-year mathematics at Cambridge University. To show us how to use some of the equations he’d been teaching us, the professor showed us that you cannot get into orbit with a single-stage rocket. He worked from first principles, which in the case of rockets is the Tsiolkovsky rocket equation. On a rocket, in each time interval (say, a second), three things happen. First, some of the fuel is burned and ejected out of the back of the rocket at high speed. Second, this accelerates the rocket a little bit. In this context, “rocket” means the vehicle and all its unburned fuel. Third, the rocket also gets a little lighter since there is a little less unburned fuel—we just burned some. It is then a straightforward calculus exercise (if you know integration) to derive the actual rocket equation, which relates the change in velocity to the exhaust velocity of the burning gases, and the initial and final mass of the rocket. It has been known for a long time. Tsiolkovsky derived it in 1903.

You don’t need to understand the equation to understand the rest of this post, but the rocket equation is that ΔV = X ln(Mw/Md), where ΔV is the change in velocity of the rocket, X is the exhaust velocity, Mw is the initial weight, also known as the wet weight, and Md is the final weight, also known as the dry weight. The logarithm is a natural logarithm to the base e (Euler’s constant). In the extreme case, Mw is the weight of the vehicle completely full of fuel (on the launchpad say), and Md is just the weight of the empty rocket itself, and the equation tells you how much velocity you can get your rocket up to if you burn all the fuel (making it into orbit is recommended). In practice, getting from earth into orbit also has to take gravity and air resistance into account, too, the rocket equation assumes neither.

The problem with getting a rocket into orbit is that it has to accelerate all its fuel at liftoff (and after). So if you want to lift a heavier payload, you need more fuel (or a better fuel, but let’s assume you’re already using the best fuel)…so you have to add more fuel…but now you have to accelerate that fuel, too, so you need to add more fuel still…but to accelerate that added fuel, you have to add even more. So the final velocity of the rocket increases only logarithmically (slowly) as you add more and more fuel. You can go “multi-core” and add more rocket motors, but note that the number of motors does not appear in the rocket equation. More motors just allows you to burn fuel faster at the same exhaust velocity, you don’t need less fuel.

The reason that you can’t get into orbit with a single-stage rocket is that the rocket is just too heavy. At takeoff, a rocket is about 85% propellant and 15% everything else (payload, tanks, etc.). So, in addition to losing mass by burning fuel, it is necessary to lose mass by dropping some of the initial structure of the rocket itself to get that 15% down lower still. During the Apollo program, multistage rockets would be separated and fall into the ocean. In more modern vehicles like the shuttle or the SpaceX Falcon 9, there are boosters (and a fuel tank in the case of the shuttle) that are dropped (and, in the case of SpaceX, recovered by landing them).

It turns out that a really good rocket design can deliver about 4% of its mass into orbit. The other 96% of the mass at takeoff is the fuel required to get there, the tanks and pumps. About 10-11% of the initial takeoff weight, besides the fuel, also needs to be dumped in the form of booster or stages.

So yes, rockets are horribly inefficient. In fact, if the earth was about 50% larger, then a good rocket could deliver 0% of its mass into orbit, no matter how much fuel we used. That is, we wouldn’t be able to get into orbit at all, at least with rockets and any known fuel. It really is difficult.

Going beyond Earth orbit
Science fiction writer Robert Heinlein once said, “If you can get your ship into orbit, you’re halfway to anywhere,” as reported by Jerry Pournelle in 1974 in Galaxy Magazine. What it means is that it takes so much fuel to get into orbit that you don’t need much more to get to other places. You need to get to 8km/s to get into earth orbit. It takes another 6km/s worth of fuel (this is shorthand for the amount of fuel required to get from zero to 6km/s) to get to the moon, or another 8km/s to get to Mars. Yes, half the fuel to go to Mars is just to get into space in the first place.

If you could drive there, space is not that far away. Low earth orbit is about 250 miles. But you can’t drive there, so it takes half the fuel to go the first couple of hundred miles, and then the other half of the fuel to go from there to Mars, a minimum distance of 36,000,000 miles. This map shows the energy required to get to all the planets and major moons in the solar system. Here is an enlargeable version that you can actually read.

As NASA astronaut Donald Pettit put it:

The giant leap for mankind is not the first step on the Moon, but in attaining Earth orbit.

If you want to know a lot more, I recommend his 2012 blog post The Tyranny of the Rocket Equation (where the quote came from).

One way to optimize a mission to Mars would be to get the vehicle into orbit, ending up with its fuel tanks empty, and then refuel it. But you can’t just fly a single refueling tanker up since only 4% of the launch mass can be delivered to orbit. Depending on the various masses and fuel tank sizes, it takes ten or more flights to refuel. As I said above, rockets are horribly inefficient. Note that if you tried to have a bigger vehicle to go to Mars without refueling, you would still need the same amount of fuel to get the vehicle and its fuel up there, it would just all have to be in the vehicle at launch. Yup, the vehicle would need to be ten times as large to arrive in orbit with fuel still in the tanks for the rest of the journey. That’s the tyranny of the rocket equation. To go into more detail on this, I recommend Casey Handmer’s blog post There Are No Gas Stations in Space.

All the planets
Casey, in another blog post Working Title: Bombard All the Planets, has an interesting thought experiment:

This is Falcon Heavy [picture]. It costs $90m. For a mere $1B a year, or about 4% of NASA’s budget, we could launch it to every planet in every launch window. And that’s before the bulk discount. This is a diagram of every launch window to every planet for the next 20 years.
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Most planets have a launch window about once per year. Mars has one every 2.2 years.
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No robot has launched to Venus since 1989, or Neptune since 1977 – more than 40 years ago.
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25T to Mars is enough for every major space agency to fly a rover, a lander, and an orbiter, every launch.

That would still leave NASA 96% of its budget.