How many natural satellites (moons) in our solar system are “geostationary”?

Answer

It took a long time, but here comes the answer.

First the nomenclature. ‘Geostationary’ is reserved for the orbits of satellites around the Earth: it means that the orbital period of the satellite is exactly equal to the rotational period of the Earth, so that the satellite always remains above the same place, a circumstance that is very useful for e.g. telecommunications. What you mean is actually ‘geosynchronous’: the period of rotation of the Moon around its axis is equal to the period of rotation around the Earth, so that the Moon always faces the Earth with its same side.

The cause of this phenomenon is the tides. Let us first look at the tides of the Earth. You may know that high tides always occur after the Moon has reached its zenith. How did that happen? The reason is that the water needs a finite time to move, and also a certain inertia. It always comes too late. Because the Earth moves faster on its axis than the Moon orbits the Earth, this implies that the tidal wave is slightly ahead of the direction toward the Moon. The attraction of the Moon therefore pulls the tidal wave back slightly, slowing the Earth’s rotation. Indeed, we have to add a second every now and then to keep up with our time. It takes so long until there is no difference between the day and the month. That will never happen to the Earth, the Moon isn’t pulling hard enough for it.

Due to its greater mass, the Earth pulls much harder on the Moon and exerts much stronger tidal forces on the Moon. There is no sea on the Moon, but when the young hot Moon was partially liquid, it quickly synchronized.

Since it is mainly the giant planets that have satellites, and it can be expected that the giant planets exert considerably larger tidal forces, it is logical that those satellites are also synchronized. And yes, the observations show that this is usually the case. There are exceptions, such as Saturn’s satellite Phoebe. Now it turns out that Phoebe is special for other reasons too: Its orbit is going in the wrong direction, and the object is much blacker than other satellites. All of this indicates that this is a relatively recently captured object, which has arrived from the far reaches of the solar system and is not yet synchronized.