We know that all objects with mass exert forces on all other objects of mass such that

$$ F = \frac{GMm}{R^2}.$$

And as others have discussed the planets do interfere with each other gravitationally to a small degree.

My question is how reliant the solar system is on its exact structure. If a planet were to change its alignment or orbit or gravitational effect on other planets, through gain of a mass through an asteroidal collision for example.

Would a deviation in the structure of the solar system as it is cause it to collapse? e.g planets change orbits significantly enough to drift away from the sun or drift into it?

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    $\begingroup$ Good question, but it goes deep into orbital resonances. The familiar rules of gravitation start to look extremely different when played out over millions of years. I could confidently say it wouldn't cause planets to plummet to their death in the very next orbit, but after that the system is quite chaotic. $\endgroup$ – Alan Rominger May 14 '13 at 12:04
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    $\begingroup$ I second Alan's comment; over a long enough time period, all the planets will either be ejected from the solar system, or collide with the sun. However, removing one of the planets at the present time probably won't have much of an effect in the near term. $\endgroup$ – Dmitry Brant May 14 '13 at 13:46
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    $\begingroup$ also @downvoters, would you care to comment? I cant improve the question if you don't tell me whats wrong! $\endgroup$ – RhysW May 14 '13 at 15:25
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    $\begingroup$ @BenCrowell there is no reason to constrain the problem as such. There is no lack of 'self-consistency', the problem just becomes one of artificial initial conditions---i.e. the current positions and velocities, but the dynamic situation has one fewer body. RhysW's response is exactly correct, if you required a 'rapid acceleration' you might as-well require an explanation for it, or an entire universe where such an explanation would naturally arise. $\endgroup$ – DilithiumMatrix May 14 '13 at 16:00
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    $\begingroup$ @BenCrowell what I'm saying is that there is no reason to require conservation of mass in this situation. Say you have a mass on a spring, with some initial oscillation. Then say half the mass disappears. There's no issue: it's the same as saying you have half the initial mass with some artificial initial displacement and velocity. $\endgroup$ – DilithiumMatrix May 14 '13 at 17:13

This is something I played with while testing a n-body code I wrote during college. Unfortunately I don't have any animations, or even the original code anymore - but I can report qualitative results.

Removing Jupiter and Saturn does indeed have a significant destabilizing effect -- an a chaotic one at that (i.e. depending on precise initial conditions, and varying on numerical accuracy) -- leading to the dynamical instability of numerous planets.

Removing the other planets had no effect on dynamical stability, but there were some small changes to periods, etc.

This result should be expected as the gravitational effects of planets other than Jupiter (and saturn to a lesser degree) are almost entirely negligible on the dynamics of other planets.

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