I was arguing on the internet about the possibility of building a space elevator; someone was claiming that carbon nanotubes were going to make it possible and we’d build one in the near future. I’m very skeptical, first because we’re nowhere near the capability of building high-strength carbon nanotube bundles tens of thousands of kilometres long, and secondly because it needs to be cheap, and this absurdly long and absurdly strong cable seems anything but cheap. Furthermore, if we postulate that some magical technology will allow that to happen, I think it’s dishonest to not allow some magical technology to also help the competition. For instance, we could have fully reusable rockets which cost little other than fuel to launch; and this technology seems much more plausible.
But this is not what I’m going to talk about, space elevator cables have been discussed to death. What I’m interested in is a point that another poster brought up: how much does the energy needed to climb the space elevator costs? They were arguing that the proposed method of powering the climber, laser beaming, had an absurdly low efficiency of 0.005, and paying for that energy alone already made the space elevator uneconomical, we didn’t even need to consider the astronomical costs of constructing it in the first place.
Now wait, come on, so even if we did have the magical cable for free it still wouldn’t be worth using the space elevator? That’s going too far. If laser beaming has such a low efficiency, this just means we should use some other method for powering the climber. Delivering the power via copper wire is out of question: the wire would be extremely massive, and over such a long distance the electrical resistance would also make it uneconomical. Of course we could use a superconductor to get rid of resistance, but then we have to cool the damn thing as well, making the weight and the cost even worse. No no no, I want to find a solution for powering the climber without needing any more magical technology, the only magic I’ll allow is the space elevator cable itself.
The obvious solution is solar panels. In space they work very reliably, there’s no atmosphere to filter the sun, and even the shadow of the Earth quickly stops being a problem as you get higher1. They don’t generate much power, though, so it would take a long time to climb to geostationary orbit. How long exactly?
It turns out this is an interesting physics problem. We’re not moving with a constant speed, or a constant force, but with a constant power. I didn’t study this problem either in school or university, so I wanted to find out how to do it. The principle is easy: kinetic energy is given by
\[ E = \frac12 mv^2, \] so we differentiate that and get
\[ \frac{d}{dt}E = P = m v a,\]which results in the differential equation
\[a = \frac{P}{mv},\] that even I can solve.
Life gets hard when we put that together with the forces acting on the climber: gravitational $GMm/r^2$ and centrifugal $m\omega^2 r$ (the famous “fictitious” force that hurts as much as any “real” force). The resulting acceleration is
\[ a = \frac{P}{mv} – \frac{GM}{r^2} + \omega^2 r,\]a really nasty differential equation that I have no idea how to solve. No matter, my laptop can solve it numerically without even turning on the cooling fan, I just need the values for the parameters. Most of them are fixed: $G$ is the gravitational constant, $M$ is the mass of the Earth, and $\omega$ is its angular speed. I just need to determine $P/m$, the power per mass available from the solar panels.
Annoyingly enough, I couldn’t find solid numbers anywhere, but a good estimate can be obtained by considering the solar panels of the ISS. They produce roughly 30 kW of power (before degradation) and weight roughly 1.1 tons, giving as a power density of 27 W/kg. To compensate for degradation, they’re about to install new ones, that promise a 20% weight reduction for the same performance. Again, I couldn’t find solid numbers, but given that they are going to be installed next month, and do have a plausible method for reducing weight, I’ll believe that 20% and put a power density of 34 W/kg solidly in the “not magic” category.
The climber is not made only of solar panels, though. Let’s reserve 80% of its mass for payload, structure, and so on, and use 20% for solar panels, which lowers the power density to 6.8 W/kg. Solving the equation numerically, letting the initial position be the equatorial radius of the Earth and initial velocity 1 m/s2, we see that 82.5 days are needed to reach geostationary orbit. But wait! At geostationary orbit the gravitational and centrifugal forces balance out, so we’ll just have the motors working with no resistance, and the speed of the climber will quickly diverge! We have to check if we’re not accidentally breaking relativity. Solving numerically for speed, we see that at geostationary height it is roughly 300 m/s. Yeah, no danger of breaking relativity here. Still, that’s a little too fast to be plausible, so let’s apply the brakes at 100 m/s. This only lengthens the journey to 82.7 days, so it’s really not worth considering; there’s a bigger error anyway from not taking into account the time the climber spends in Earth’s shadow, which will make the journey a couple of days longer.3
That takes a lot of patience, but it’s really not an issue for geostationary satellites or interplanetary missions that take years anyway. It is an issue for manned missions, we really don’t want to spend 3 months loitering in space getting bathed in radiation. Can we speed it up somehow?
Well, we can send power up via a copper wire for a short distance, say 100 km. In fact we need to do this, as we can’t deploy this massive solar array inside the atmosphere. Transmission over such distances is completely routine, so I’m not going to bother calculating the specifics. Just assume that we send enough power for a constant speed of 100 m/s, and solve the differential equation numerically again with these new initial conditions. The result is 81 days. Well, yeah, we’ll leave the space elevator for cargo and send people up in rockets.
40 years ago I came to a similar conclusion, but wondered if space elevator extending from the moon to a counterweight attracted to the earth would be possible. My, possibly erroneous, calculations indicated that a tapering kevlar cable would be almost strong enough, at least until space weather weakened it.