Nuke, you're speaking with someone who's played with orbiter for years and also has a good grasp of the dynamics of orbital motion.
there is a point at which earths gravity will not hold the moon.
Going under the assumption that this is a 3-body problem with the earth, moon, and sun, then yes, and this is what the term "hill sphere" is used for, which I had linked to in my prior post.
the moon will spiral out slowly until it gets to that point, and all the orbital parameters will be just right for earth to loose its grip on the moon. the moon is essentially inching its way out of the gravity well, and once its out it will be as if the cable has snapped and the moon will be on its merry way (if it werent gonna be on fire long before then anyway).
I really don't agree with your choice of wording regarding a "snapping cable". You seem to be under the impression that the transition from the moon being in a geocentric orbit to a heliocentric orbit is sudden. This is
not true. As the moon moves away from earth, the force it feels from the sun becomes comparatively stronger than the force it feels from earth. That means that the moon is less strongly bound to earth, and the sun's influence causes its orbit to gradually shift over time, while the moon is still orbiting the earth. In other words, its orbit becomes chaotic. And to be clear, when I say "chaotic", I mean that the long-term evolution of such a system varies dramatically with even a tiny change in the initial parameters. I do not mean that the moon's orbital path shifts around violently or that it could just end up anywhere.
Visually, what this chaotic orbit will look like is a large, elliptical, geocentric orbit where the moon's perigee and apogee (among other orbital parameters) change slightly with each orbit. Again, this change is because it feels a force from the sun that is comparable in magnitude to what it feels from the earth. Eventually this shifting orbit can cause the moon to recede from earth enough that it does not return to perigee, but instead continues to move away as a sun-orbiting body. I'll go over the possible results of such a heliocentric orbit later, though I did point some out in earlier posts.
By the way, a chaotic orbit like this could just as well cause the moon to fall back and collide with the earth,
without ever having fully escaped. If the perigee altitude drops too low -- wham, there goes the earth.
To emphasize my point here, I'd like for you to download a free program called "gravitation 3D" which you can get from
here. I've attached a scenario file which demonstrates an example of the above process of the moon going into a chaotic orbit. This file uses accurate solar system distances and masses, and I have simply put the moon in what is initially a large elliptical geocentric orbit. I challenge you to find the exact time at which "the cable snapped". Play around as well and see what kind of results you can get.
***To use the file, open one of the gravitation3D scenarios with a text editor, replace all the text with the text I attached, and save it. (Copy the original if you don't want to lose it). And when running the program, you
must set the second radii scaling slider (under the scaling tab) to 100% true (far left). I also suggest a time-step of 10,000, and setting the camera view to follow the moon.
we know that the orbital velocity of the moon is slowest at apogee and this is also where gravity from the earth is weakest (gravity falls off inversely with the square of distance). of course as the earth's tide puts its prograde pull on the moon, its mean orbital velocity will increase. at some point escape velocity would be attained.
You're oversimplifying the problem. If escape velocity is attained, it would be because the aforementioned chaotic orbit caused the moon to partially slingshot the earth and rob some of the earth's orbital energy,
not because of the gradual increase in orbital velocity through tidal interaction. This should be obvious because the time required for the moon to gain any appreciable orbital velocity is many orders of magnitude greater than the time required to complete an orbit, even a very large one in the chaotic stages.
if the moon happens to be on the outside of earths orbit of the sun then some fraction (iirc this fraction is the dot of the velocity vectors of the earth and moon) of the moons orbital velocity plus the earth's orbital velocity will be the moon's new orbital velocity (of course the earth's gravity will slow this down somewhat before the moon leaves its gravity well). if the moon leaves on the inside of the orbit then some fraction of the orbital velocity will be subtracted from earths velocity (and since its velocity is slower than earth, earths gravity will give it a little acceleration while the moon is still in its gravitational influence).
Sure, that's simply vector addition.
The earth orbits the sun and the moon was orbiting the earth, so the earth's velocity is factored into the moon's heliocentric velocity. You can crunch the numbers in this way at any time you have a satellite of a satellite. Your intuition on how the earth would affect the velocity of the moon as the moon moves away is correct as far as I can see.
now once the earth is out of the picture
Haha, the earth is never out the picture. Read my prior post, or play around with the gravitation3d scenario a bit. Even if the moon escapes, you have to deal with future close encounters between the moon and earth, because in all cases the moon's resulting orbit will eventually bring it close enough to earth again that the earth affects its orbit significantly.
if that velocity is greater than the earths, it will extend out the moon's (can we call it that anymore at this point?) aphelion, with its perihelion being roughly as high as the point where the moon was when it broke free. and if the moon is now slower than the earth, its aphelion will be around where it broke free and its perihelion will be closer to the sun. either way the end result is an eccentric orbit.
And the eccentricity would be fairly low, as I had explained before. The only way for the eccentricity to increase enough to, say, bring the moon to another planetary orbit (Venus or Mars) would be via continued "lucky" slingshots of the earth. The initial escape would not provide the moon with enough delta-v to bring it out very far, either toward or away from the sun. You can check this yourself by experimenting with orbit simulating programs like the one I provided.
its essentially half of a hohmann transfer.
Sure.
a collision is not immediately likely
I doubt that. I played through several scenarios of varying lunar distance and velocity (while maintaining at least two stable orbits at first) and collisions ended up happening pretty often.
in a situation where the earth and moon get close enough to seriously effect each other the pick up or loose some speed effecting its orbit somehow, but i assume its orbit would get less eccentric over time as it interacts with the other planets in the system.
If the moon managed to escape the earth, I imagine a future collision with earth is actually more likely than the moon managing to make it anywhere near a neighboring planet. It would take a lucky sling, or several lucky slings, to do that. And in all cases of the moon being rogue in the inner solar system, the most statistically likely result as time progresses is for the rogue moon to collide with a planet, be it earth or otherwise. The planets are natural dust mops. They sweep the solar system clean of whatever smaller bodies that pass near or through their orbital paths. Hence why we have meteors and meteorites.
edit for fixing forward slashes D:
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