In my fascinating imagination, I am seeing a space elevator from the vulcano Olympus Mon to the moon Phobos. Phobos also seems to keep spaceships - while a smaller shuttle attaches itself to the elevator and travels down- they can also travel inside Olympus Mon. I was wondering if this is mathematical possible and it seems, it is.
The space elevator probably deserves an entire series of questions (and I am sure there have been plenty of posts), but if we stick to this particular version, there are a couple of problems with it.
First of all, a space elevator needs a counterweight in an orbit that is higher than the areostationary orbit (the Martian equivalent of Earth's geostationary orbit). In case of Mars that orbit lies approx. 17,000km (11,000 miles) above the planet's equator. Phobos is in an orbit with a semi-major axis of just 9376km (5826m miles). It orbits the planet every 7 hour 39 minutes with an orbital velocity of 2.138 km/s, while even an areostationary orbit would require an orbital speed of a mere 1.45km/s. We are therefor tasked to give a large object (the mass of Phobos is 1e16kg) a delta-v on the order of at least 1km/s to get it into place. That requires a lot of energy (and propellant mass), which one could use to launch a lot of payload with conventional rockets, instead!
Anchoring the space elevator over Olympus Mons is also not an easy option. This spot lies at 18.65 degrees North of the equator. Since our counterweight has to be above the equator, we would need a second cable located 18.65 degrees South. Now we have two elevators, but they are both under a slight angle, which doesn't make the design any easier.
If we let go of the particular location and we place a conventional design between a spot on the equator and a smaller counterweight above, we are certainly going to make our lives a little bit easier.
Deimos, on the other hand, is almost in the right place for a counterweight, it seems. With an orbital period of approx. 30 hours and a mass of 1.4e15kg it's a lot closer to what we need than Phobos... except, of course, that we still have to watch out for Phobos cutting our cable every eight hours, which means that we have either to deploy a movable cable (which is not particularly hard compared to the extreme engineering that we already had to do) or we have to use two cables separated by a few degrees, again, so that Phobos can just slip trough our sky-hook!