Getting off the ground

Not long after I published the previous post about the Drake Equation, a couple of headlines surfaced about a paper by Michael Hippke with the admirably straightforward title Spaceflight from Super-Earths is difficult.  The paper is actually a light rewrite of what was originally an April Fool's joke, but the analysis is real, even if the author originally considered the topic frivolous.

The term Super-Earth itself is fairly loosely defined.  For concreteness, Hippke chooses Kepler-20b, with a radius of about 1.87R_⊕ (Earth radii) and a mass of about 9.7 M_⊕ (Earth masses).  Since gravity is proportional to mass and inversely proportional to the square of distance, the surface gravity of this planet would be about 2.8g (Earth gravity).  This is assuming that the measured radius is actually the radius of the surface.  There's a good chance that Kepler-20b is actually a "Mini-Neptune" with an extensive atmosphere rather than a Super-Earth with a rocky surface, but let's assume the Earth-like scenario here.

Hippke argues that it would be impractical for a civilization on such a planet to build rockets because the amount of fuel you need to reach escape velocity* increases exponentially in relation to that velocity.  This is exponential in the literal sense that doubling the velocity of a rocket means squaring the ratio of fuel to mass, not in the colloquial sense of "a lot".  Escape velocity in turn increases as the square root of the surface gravity.  For example, four times the surface gravity means twice the escape velocity, so square the ratio of fuel to dry mass.  Taking the square root doesn't make a lot of difference in the big picture.  The exponential part still dominates everything else.

In short, a somewhat bigger planet doesn't mean somewhat more fuel to get to escape velocity.  It can mean a lot more.

On Earth, a chemical rocket which magically had a weightless engine, fuel tank etc. would need to have 26 times as much fuel as payload in order to reach Earth's escape velocity of about 11 km/s**.  In real life that ratio is more like 50 or even 83 since the engine and so forth actually do weigh something.

Escape velocity for Kepler-20b would be about 2.3 times Earth's escape velocity, or around 25 km/s.  Hippke calculates that for a typical chemical rocket, that 26:1 ideal mass ratio is more like 2700:1 and the more realistic ratio of 83:1 would correspond to something like 9000:1.  To send a 1-ton payload out of the planet's gravity well would take 9000 tons of fuel.  By contrast, the Saturn V -- the largest rocket actually put into service so far -- had a mass of around 3000 tons, not all of which was fuel.

All this is fine, and surely more than enough for something that started out thoroughly tongue-in-cheek.  So let's take it at face value and try to poke holes in it anyway.

First, the calculations are for a single-stage rocket, though the real-life rockets used for comparison purposes are multi-stage.  In a multi-stage rocket you use a rocket with plenty of thrust (the first stage) to boost another rocket (the second stage) through the atmosphere quickly and then jettison that first stage.  At that point you no longer have to worry about the mass of the first stage and you consequently get more acceleration out of your remaining fuel.  You don't have to stop there.  The Saturn V, for example, was a three-stage rocket.  Five-stage rockets have been successfully launched.

This doesn't just make a difference in that a multi-stage rocket allows you get more acceleration out of the same mass ratio.  It also means that you don't have to use chemical rockets for all stages.  You could, for example, use an ion drive, which has a much higher effective velocity and therefore a much lower mass ratio, for the final stage and use chemical rockets to get it into orbit.  Ion drives produce very low thrust, far too little to launch from the ground, but they can do it for a very long time using very little fuel, eventually reaching much higher speeds than chemical rockets.  Once in orbit, a modestly-sized ion-driven vehicle could easily escape even Kepler 20b's gravity well.

In other words, getting to escape velocity in a single stage is a red herring.  You really just have to get a reasonable mass to orbital velocity, and you can use multiple stages if that helps.  At a given distance from the planet's center of mass, the orbital velocity is smaller than the escape velocity at the same distance by a factor of the square root of two.  In real life the orbit is -- of course -- further from the center of mass than the surface is.  If escape velocity at the surface is 25 km/s, a more reasonable orbital velocity would be 17 km/s, depending on how high up you have to go to get out of the atmosphere.  That would mean a mass ratio of more like 150 for an ideal rocket and 500 for a more realistic one.

That's still considerably more expensive than here on earth, but not nearly as discouraging as the 9000 figure in the paper.  A 500 ton rocket could put a ton in orbit, and you wouldn't even need to do that to get out of the gravity well.  Japan's ion-driven Hayabusa craft had a mass of about half a ton.  It was able to get to an asteroid, grab a sample and bring it back to Earth -- a pretty impressive piece of engineering if you ask me.  Our counterparts on Kepler 20b could do that with something like a 250 ton rocket.

The rocket that launched Sputnik was 267 tons (the rocket that actually launched Haybusa was around 140 tons, for a mass ratio of around 280).  Sputnik itself was only 84 kg, for a mass ratio of somewhat over 3000.  Small payloads generally mean higher mass ratios because it's not practical to shrink the launch system proportionately.

Leaving all that aside, you could also do multiple launches and assemble the final craft in orbit, if your robotics were good enough.  If you can launch half a ton with a reasonable-sized rocket, you can launch five tons with ten such rockets, and so forth.

Which brings up another point.  In the early stages of space exploration, before Kepler 20b puts its ion drive into orbit, they'll want to start small, using relatively big rockets to put relatively small things in orbit, and before that, to blast relatively small objects -- on Earth, that mainly meant weapons -- across large portions of the planet.

There doesn't seem to be any reason intelligent beings on Kepler 20b couldn't do that, assuming they're there.  Start with toy rockets, then weather rockets to explore the upper atmosphere, work up to ICBM-style systems, then orbit, then out of the gravity well, just as we did.  As far as I can tell, the difference on Kepler 20b would mainly be a matter of time, not a night-and-day difference between plausible and clearly impractical.  The benchmark of putting a ton or more directly on an escape trajectory doesn't seem particularly relevant to the question of whether or not this could happen, though, being concrete and understandable, it's still useful to think about.

There's another way to bring down the mass ratio: faster rocket fuel.  Hippke's calculations use an effective velocity of 3430 m/s, but hydrogen/oxygen delivers more like 4400.  That brings our ideal mass ratio down closer to 50 as opposed to 150.  As I understand it we only use hydrogen/oxygen in specific situations, due to various engineering considerations, but the tradeoffs will be different on Kepler 20b.  It might make sense to find ways to make the faster fuel work in more situations.

Even if chemical rockets weren't a practical way of getting into orbit, there are plenty of other options, some more speculative than others, for doing so.  Space elevators ... mass drivers ... blast wave accelerators ... space fountains.  Some of these require materials we don't know how to make yet or other not-so-proven technologies, but to some extent this is all a matter of economics.  Rockets are easy and cheap enough for us, so we use rockets.

Finally, it's probably worth pointing out that escaping a planet's gravity well is necessary for sending an interstellar mission, but hardly sufficient.  Kepler 20 is 950 light-years away.  To get here from there in, say, less than 10,000 years, you'll need to be going about a tenth the speed of light, or 30,000 km/s.  If you can do that, getting into orbit or even to escape velocity doesn't seem like a major problem.  Conversely, the most likely reason not to receive a visit from Kepler 20b is that it's just too far, not that it's too hard to get off the ground.





* I suppose I should acknowledge that "velocity" here actually means "speed" since it's a magnitude with no particular direction.  But everyone says "velocity" anyway.

** In real life you also have to deal with gravity losses until you reach orbital velocity.  For example, for every second you spend going straight up against Earth's surface gravity, you lose 9.8 m/s.  For Kepler 20b, that's more like 28 m/s.  If your initial stages take 180 seconds (three minutes), that's an extra 4000 m/s or so, except it's not really that simple since you don't spend all your time going straight up, particularly if the goal is to reach orbit.  I'm handwaving that, though it's quite a bit to handwave, just to keep the comparison with the ideal mass ratio of 26.  Part of the reason real rockets, even with multiple stages, needed a higher mass ratio than just the change in speed would suggest was to deal with gravity loss.