[QUOTE=ngc3314;1301220]But only to the extent that you need the accuracy of a non-Newtonian situation...[quote]
I was only considering the Newtonian effects. No relativistic effects were considered.
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...or the point-mass approximation isn't adequate, and those are really minute effects compared to the magnitude of the solar barycentric motion.
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Yes and no. Although the Sun's pertubations in position are minute compared to the Earth's orbit around the sun, the Sun's mass is as much greater as it's pertubations are small. However, since the force between two bodies is proportional to the masses (which doesn't change) and inversely proportional to the square of the distance between them (which changes very little for such a relatively large mass as the Sun), the variance in the Force is small.
But it is nevertheless there, and it most certainly does affect planetary orbits on down the road, which is what I think the OP was getting at.
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For Newtonian point masses, the combined force is the (vector) sum, and since in this case the displacement is always extremely small compared to even Mercury's distance, adding the individual planetary displacements is a really good approximation. (This breaks down when the perturbing masses are larger, for example, so that the variation in force due to one perturber changes significantly over the distance of the perturbation caused by another). Again, it depends on what "significant" means in a particular context.
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You're right - it's a really good
approximation. But that's limited to what's happening now, for example, determining the forces between the objects, or their departure vector. When it comes to the effects of gravity over time (again, back to the OP), however, errors are compounded, and a "really good approximation" doesn't hold a candle to the precision required to compute where the planets would be a hundred years from now.
In fact, nothing, including relativistic effects, is approximated for long-term gravitational models. And when such models are put through their computational paces, we find two curious things. The first is a much tighter approximation of a planet's gravity. The second are unknown masses and locations required to make the mathmatical model match what's been observed. This "missing mass" is, in part, one of the bases for the existance of the
Oort cloud. The remainder were observations of cometary patterns by Ernst Öpik, and later, by Jan Hendrik Oort.