# How does the gravitation of Jupiter or other big planets affect Kepler's third law?

Does anyone know any good reference on this?

I remember reading once about how Jupiter's gravitation can be treated as a perturbation, but I cannot find it again now. I think they expanded Jupiter's gravitation as an infinite sum of one of the special functions (such as Bessel function, Legendre function, Hermite polynomial, I can't remember).

I am not interested in how the motion of a planet at each time looks like, but just the overall effects on parameters such as period, semi-major axis, eccentricity, etc.

• Interesting question! I've made some small edits to help it match the Stack Exchange format. Have a look and please feel free to edit further. – uhoh Dec 28 '18 at 1:09
• @NicolaG looks good! SE is certainly collaborative so go right ahead and make improvements were you see fit. – uhoh Jan 4 at 8:33
• I'm unclear what is meant. Does it mean how does the presence of Jupiter affect a graph of $P^2$ vs $a^3$ for other solar system objects? – Rob Jeffries Jan 4 at 9:43
• I guess yes. The question still has an inaccuracy, i.e. expanding gravitation in a series of a special function, but I think it's genuine and understandable. I mean, you can expand a mathematical expression of gravity, not gravity itself. The answer can be obtained via both Newtonian and Hamiltonian mechanics. In the context of the Kepler's 3rd law, which gives a lot of assumptions, it can be shown that the effect of Jupiter can be an oscillation in $a$ and $e$ (if the mass of the bodies is comparable) or just a precession in the pericenter (if Jupiter has a greater mass). – CompactObjectsLover Jan 4 at 10:18
• Do you want a detailed math answer or a general description answer. When Earth is closer to Jupiter than the Sun, it's pulled by Jupiter more than the sun, basically a tidal force as Jupiter always tugs on both. This can effect Earth's orbital velocity a little bit, but it balances out over many orbits. The math behind 3 bodies in orbit, such as Jupiter/Earth/Sun gets pretty complicated. Is this kind of what you're looking for? phas.ubc.ca/~berciu/TEACHING/PHYS349/karla.pdf – userLTK Feb 23 at 15:41