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I am not a mathematician and it may even take me weeks to understand the math involved but I have an odd question on orbital mechanics that I hope will be worth the experts' time. I am a hobbyist programmer just getting my feet wet in the concepts 3D graphics and (by extension) orbital mechanics.

As I understand it a polar orbit allows a satellite to efficiently see the entire surface of a planet because the planet rotates on its axis while the satellite orbits at an inclination of 90 degrees to the equator. This theoretically makes polar orbits very efficient (few to no course corrections required) for mapping the surface of planets.

But -- hypothetically -- what if the planet does not rotate at all? What kind of orbit would be needed to map the planet's surface in an efficient way (few to no course corrections once the satellite's orbital path was initiated)? Is there a name for such an orbital type? Can this even be done or would a non-rotating planet need to be orbited with constant course corrections to map the entire surface?

Since this is about virtual space go ahead and assume a perfect sphere and uniform gravity to keep the math simple. On the other hand, how different would the math be if this was a perfect ellipsoid instead of a perfect sphere?

Thanks for indulging my self-directed education on this subject.

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  • $\begingroup$ Good question; I'm not quite sure if it's on topic here, but if not, we can send it to Space Exploration. $\endgroup$
    – David Z
    Jul 31, 2015 at 4:07

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The basic answer is that there is no such orbit that can survey the entire surface of an irrotational planet in the same manner as satellites do for Earth.

Conservation of angular momentum dictates this: the satellite's plane of orbit cannot change without a torque acting on it. Although there is a force acting on the satellite (the gravity from the planet), it is a central force which has zero torque (torque is required to change angular momentum). If you're unfamiliar with these in a strict sense, this is the rotational version of conservation of (linear) momentum which is a little more straightforward at a first pass.

Now, this is not strictly true. Planets aren't rigid bodies, there are other gravitational bodies present, etc., but these should be very small effects, and I think wouldn't appreciably change the orbital plane.

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  • $\begingroup$ Are you saying that a satellite cannot be given a "drifting" vector around a sphere? As a computer geek I know I can code a 2D space where the edges "wrap around". The ancient video game Asteroids is an example of what I am talking about. In wraparound space it is possible to give a ship a pure 0,-Y vector so the ship flies straight up. If you then add a tiny nudge of X momentum to the vector the ship will continue to fly upwards but will gradually drift right so that eventually it traverses the whole screen again and again. Not in 3D orbit? $\endgroup$
    – O.M.Y.
    Jul 31, 2015 at 5:30
  • $\begingroup$ A qualified yes. Wait until the satellite goes around once - it traces an ellipse, which exists in a particular plane - the orbit will always remain within that plane (barring the caveats mentioned before). But, given the difference in geometry (2D space vs 3D space) and some loose language, I wouldn't take the comparison too far. $\endgroup$
    – anon01
    Jul 31, 2015 at 5:47
  • $\begingroup$ Of course such a man-made 2D example implies perfect momentum with zero drag ... and gravity is a drag (pun intended). I assume maintaining orbital distance, velocity, and vector would necessitate occasional thrust adjustments but as long as the general orbital path was unaltered I would classify that as the little in "little to no course corrections". $\endgroup$
    – O.M.Y.
    Jul 31, 2015 at 5:49
  • $\begingroup$ Okay so if I understand you, what I would need to do is let the satellite go around the poles once or twice to stabilize a pure polar orbit, then added a small precise lateral thrust exactly as it crossed the equator to cause it to drift longitudinally without disturbing its ellipse? $\endgroup$
    – O.M.Y.
    Jul 31, 2015 at 5:56
  • $\begingroup$ No. First, there is no stabilizing; as soon as it's orbiting, it's fixed in that plane (no need to go around once). You could change the orbit by thrusting, but as soon as you stop, it will be fixed in a different polar orbit. There is no drifting here; that is, if no force is applied, the plane or orbit is fixed (again conservation of angular momentum). $\endgroup$
    – anon01
    Jul 31, 2015 at 6:00

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