What is a geostationary orbit like at Mars? I have to pause here for a brief discussion of semantics. The authors of this paper discuss "areostationary" for Mars orbits as opposed to "geostationary" for Earth, and Wikipedia uses the same convention, but I'm not a big fan of this sort of nomenclatural hair-splitting. You'd have to talk about "hermestationary" for Mercury, "cronostationary" for Saturn, "selenostationary" for the Moon, and so on. It gets tiresome. And while a very few people use "areology" to name the study of rocks on Mars and "selenology" to talk about rocks on the Moon, nearly everybody calls it all "geology" and a person who studies all that stuff a "planetary geologist." So I'm going to stick with calling it a "Martian geostationary orbit."

Mars is considerably less massive than Earth (it has about 11% of Earth's mass) but rotates at about the same angular rate, so a stationary orbit at Mars will be smaller than one at Earth. The Martian geostationary orbit altitude is only 13,634 kilometers (so an orbital radius of 20,428 kilometers, or about 3,000 kilometers inside the orbit of Deimos). Another way of looking at it: the Martian geostationary orbital radius is about 6 Mars radii (or an altitude of 5 Mars radii above the surface); the Earth geostationary orbital radius is very similar at about 6.6 Earth radii (or an altitude of 5.6 Earth radii above the surface).

And that would be the end of the story if the planets were spherical and homogeneous and there were no other perturbing effects. But of course the real world isn't as simple as that. At Earth, the major effects on satellites trying to stay in one position are the presence of the Moon and Earth's polar flattening. These combine to tilt the plane of the satellite's orbit. Also, Earth's nonspherical shape -- the lumps and bumps that make its equator noncircular -- causes the satellite to migrate in longitude over time. There are two stable points (at 75.3E and 104.7W, corresponding roughly to the longitudes of India and Mexico) and two unstable points (at 165.3E and 14.7W, corresponding to longitudes of the Solomon Islands and the western edge of Africa), where the drift rate is negligible; but satellites not located at these points tend to shift in longitude away from the unstable toward the stable point over time. The stable points are where Earth's gravity has a local low, the unstable ones where it has a local high.

To stay in the intended position, geostationary satellites currently have to use thrusters to counteract these forces. Using thrusters means using up a limited resource -- fuel -- so stationkeeping is one thing that sharply limits a geostationary spacecraft's lifetime. To discuss fuel budgets, space navigators talk of "delta v," which is kind of a measure of how much change in velocity a spacecraft can accomplish, measured in units of speed. It takes about 50 meters per second of delta v per year to keep a geostationary satellite in Earth orbit, and almost all of that has to do with counteracting the tendency to tilt north and south in latitude rather than the tendency to drift east or west in longitude. The maximum delta v needed to counteract longitude drift for a satellite located right in between the stable and unstable points is about 2 meters per second. Since it's so small, the choice of longitude doesn't have a major effect on the lifetime of your geostationary satellite.

Silva and Romero show in their paper that the story is quite different at Mars, because Mars is much less spherical than Earth. It has a monstrous large pile of dense volcanic material deposited on one side (the Tharsis volcanoes and Olympus Mons), balanced on the opposite side by a broad gravity rise, that give its gravitational field much larger deviations from ideal smoothness than Earth has. Like at Earth, there are stable points over the two gravity lows and unstable points over the two gravity highs. Unlike at Earth, there is a huge cost if you want to put your satellite at a longitude between the stable and unstable points: Silva and Romero calculated that it can cost up to 22 meters per second of delta v per year in order to put a geostationary spacecraft at one of these spots. You would also have to perform stationkeeping maneuvers much more frequently at Mars than you do at Earth: approximately once every few days, rather than once every few weeks.