Quote:
Originally Posted by mugaliens
since gravity isn't a linear function, there will be differences between simultaneous computation of all the planets and the summation of individual computations.
|
Quote:
Originally Posted by ngc3314
But only to the extent that you need the accuracy of a non-Newtonian situation or the point-mass approximation isn't adequate, and those are really minute effects compared to the magnitude of the solar barycentric motion. 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.
|
Quote:
Originally Posted by mugaliens
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 mathematical model match what's been observed.
|
Thank you for these informative answers. I am wondering what the 'non-linearity' of gravity amounts to in this context. To explain, we can plot the Jupiter effect on the barycentre as a sine wave with period of the Jupiter orbit and amplitude the distance the sun is perturbed by Jupiter. Overlaying Saturn Neptune and Uranus sine waves on this chart, the barycentre function seems to be mainly derived from the sum of these four wave functions by linear addition.
I have just made the attached picture to illustrate. It is derived from a spreadsheet with three sine functions with frequencies corresponding to the orbits of Jupiter, Saturn and Neptune, plotting the sum of the three sine values. For this chart, sine amplitudes are equal so it treats the three planets as having the same gravity, exaggerating the relative effect of Saturn and Neptune compared to Jupiter. Nonetheless, the resulting line graph has the same shape as the SSB data provided by JPL, including the apparent 179 year resonance marked by the sets of arrows.