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How would you determine what initial velocity $v(r)$ of an object would have to be at some arbitrary point in the vicinity of a planet such that the object will end up on the surface of the planet with a zero velocity with respect to the point on the surface of the planet where it lands? Assume the planet is rotating and the object is traveling by the planet and one could somehow alter it's velocity at a point with an impulse. Hence what kind of initial velocity does the object need to be given so that it will eventually be "drug" to the surface of the planet in a way that it lands. One way to think of it, is to imagine it spiraling in and when it reaches the surface it's velocity is equal to the rotational velocity of the surface and it's verical velocity w.r.t the surface is zero as well. (Assuming no atmosphere on the planet)

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  • $\begingroup$ Is the object rotating or orbiting? Rotate means that the moon/plane is spinning itself whereas orbiting would mean it's circling the planet. I assume you meant orbiting. $\endgroup$ – cspirou Jul 19 '13 at 21:45
  • $\begingroup$ The object is coming from a distance and just passing by or it is going to hit the plant/moon. The object is not necessarily rotating, it is an asteroid let's say. It is going to "fly by" or hit the planet. So assuming that there was a way to change it's velocity (with an impulse) at a point when it comes near the planet/moon, what would one need to change it's velcity to, so that it get pulled in as it passes by and at the point it reaches the planets surface it's velocity with respect to the surface is zero. $\endgroup$ – SeanM Jul 19 '13 at 21:53
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First let me point out that if you got to the surface, you had to have some velocity when you arrived. To actually arrive with zero velocity, you have to stop, and then arrive, which seems pretty contradictory. But you're allowed to arrive with your velocity asymptotically approaching zero, so let's try that.

This is impossible to achieve if you just give the object an initial velocity then don't touch it anymore. Gravity is a strictly attractive force, so it can only offer an acceleration toward the planet/moon. If the object reached the planet, that means that at some point it NEEDS to have a component of velocity toward some point on the planet surface. Since gravity is only attractive, it can ONLY increase this component of velocity, it cannot decrease it. So when the object reaches the surface, it necessarily has a component of velocity directed toward the surface.

The only ways I can think of to try and get around this:

-The moon you're aiming for is small enough that gravity hasn't shaped it into a sphere and it has some funky geometry. If you contrive the situation very carefully you might be able to pull this off, but not necessarily from a general point in space. The requirement for this to even be worth considering is that outward normal vector to surface you want to land on $\vec{N}$ and the vector from the point you want to land on to the centre of mass of the moon $\vec{r}$ obey $\vec{N}\cdot\vec{r}>0$. Basically you need to land upside down in a cave. My instincts say that you probably need the planet to be rotating just right too.

-If both the object and the planet are spinning relativistically, I think there are some weird GR effects that can give you "repulsive gravity" (in addition to the normal attraction still going on) and might make a zero-speed landing possible. But once again, I think this is not possible for an arbitrary starting point (you would also need to fine tune the spins).

-Bring some other force in. If you can give your object controllable thrusters then this problem is very solvable. If you give the planet and the object electric charges (same sign on both to get repulsion), then again it might be possible to fine tune things to get a solution. This time I think you can do it with an arbitrary starting point, but you need to be able to fine tune the charges, and there may be combinations of masses and charges that prevent a solution.

-Cheat and start inside the planet, then the problem isn't too hard. Except that if the planet is solid, you need to drill a hole, then there's no surface to land on once you get to the surface (invoking a variable rate of rotation for the planet might solve this). If the planet is gaseous... well you said no atmosphere, so that's not allowed. And if you start inside the planet... once again, this is not an arbitrary starting point.

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  • $\begingroup$ Hello Kyle, Thank you for your response. I should have said, assume the planet/moon thing is rotating and the obeject is traveling by it. I added an image to explain the starting set up better. The idea is that if you could provide a pulse to an object approaching a moon, you could in theory provide it with the right intial conditions to land on the moon/planet. After looking at the diagram and assuming rotation what are your thoughts - much appreciated. $\endgroup$ – SeanM Jul 19 '13 at 21:25
  • $\begingroup$ Sorry, it wouldn't let me post an image. I updated the text though. $\endgroup$ – SeanM Jul 19 '13 at 21:31
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    $\begingroup$ @SeanM Kyle's argument that you always arrive with some vertical velocity is completely general and correct. Here is another way to think about it. The laws of physics are time reversal invariant. So if it were possible to land at zero vertical speed without using any thrusters etc. then it would be equally possible for an object at rest on the surface to suddenly take off and fly into orbit without any thrusters. This latter is clearly impossible so the former must be impossible. This holds even if the planet is rotating, unless it is rotating faster than orbital speed at the surface. $\endgroup$ – Michael Brown Jul 20 '13 at 1:35
  • $\begingroup$ Michael, thanks for your input. I think I see what you and Kyle are saying. Except your very last comment: "unless it is rotating faster than orbital speed at the surface" - not sure what you meant by that. $\endgroup$ – SeanM Jul 20 '13 at 4:29
  • $\begingroup$ I'm not sure even rotating faster than the orbital speed helps. You might be able to construct a low impact landing in that case, but still not zero velocity I think. Maybe with a carefully constructed canyon with varying depth... $\endgroup$ – Kyle Oman Jul 20 '13 at 12:06
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If you are just interested into certain (not 100% realistic) situations, such as the assumptions that the celestial body is a perfect sphere and has a smooth surface. Them it might be interesting to look at a document I created not long ago, which I initially made for a game called "Kerbal Space Program", in which I calculate the changes in velocity (assumed to be instantaneous impulse changes) needed to land/take-off for two types of maneuvers, namely a horizontal (Hohmann-like) and vertical maneuver.

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