Lunar Landing

It's a little more complicated, of course, since the mass of the moon is not evenly distributed, and so it's gravitational pull is substantially lumpy. You wouldn't be able to decelerate arbitrarily slowly unless you used your rocket to constantly correct your or altitude as you fly over the lumpy bits.

That's true, plus a whole host of other factors such as obliquity (the Moon isn't perfectly circular anywhere, even if it were perfectly smooth), the solar wind and gravitational perturbations from the Sun and other planets. But it's far simpler to start with the Ideal and add Reality as you go along.

GAs for both the original answer, plus your excellent response to bhankii's #2 response. Bhankii's comment was an excellent opportunity to point out exactly how complex problems are solved, and you seized it.

You do a first approximation, figure out where that works and doesn't, then add a little more complexity to solve the problem where the first approximation is most lacking, then (implied) if the second approximation still isn't what you need, you finesse it a little more.

These problems are solved iteratively, and at some point, you quit solving it analytically because you don't have enough source data (such as the lumpiness of the moon or the precise profile of the landscape under your orbit) and you build a control system, say based on a radar altimeter input, so that you have a general orbit and burn plan, but it is modulated by the inputs from the radar altimeter.

This is exactly how aeronautical navigation is done today, with an inertial navigation system giving you a general feel for your position and velocity, but gradually building up errors over time via the integration of accelerometer inputs, but then getting an accurate fix from time to time from some external input, which today is likely GPS, but before that was from some kind of triangulation from known earth surface landmarks or radio navigation beacons. The new fix zeroes out the accumulated errors, and you can then trust the INS until once again enough time has passed and the errors are outside the desired tolerance.

I guess what I was getting at is that, while everything you say is true of course, the gravitational anomalies on the moon are much worse than what we see here on earth, or Mars, or Saturn - where it's relatively easy to have a satellite that can stay in orbit for years at time without expending much fuel. And of course, the lower you fly, the worse the problem is. I think this is the reason why we haven't had probes orbiting the Moon on long multi-year missions like we have everywhere else.

Damned fine and tiny brief - Well Done.

IF the moon were a perfect sphere AND homogenous, without craters, mountains, etc., you could orbit a spacecraft 1 mm above the surface given the right speed. I assume the rocket you fire is to slow the spacecraft down (retro-rocket). Since the center of gravity of the moon is at the center (novel thought) you would eventually hit the surface of the moon.

<CORRECT ME IF I AM WRONG GUYS>

When we sent men to the moon, the orbit we used was one which took into consideration both the moon and the Earth. If something failed on the way out, the spacecraft would circle the moon and come back to Earth with no intervention.

When they reached the moon, they fired rockets <retro> which put them into lunar orbit as opposed to Earth-Lunar orbit. They then did their thing, and then firing rockets <forward> went back into the orbit which brought them back to Earth.

"<CORRECT ME IF I AM WRONG GUYS>"

Glutton for punishment, eh? I thought that was my job.

You're gonna have to try harder than that if you want a good flogging.

Interesting thing about orbital periods: if you were to bore a hole completely through the center and out to the other side, and if you were to drop a rock down this hole, the rock would "fall" to the other side and come back to you in the same time it would take to orbit the planet once.