# Which factor most affects the accuracy of Kepler's Third Law?

We know that Kepler's 3rd Law is not perfectly accurate due to gravitational perturbations of other planets, moons, etc., but which factors most affect the accuracy? Is it the semi-major axis, mass of orbiting object / object being orbited, proximity to other massive objects, or something else? From my calculations of orbital periods using NASA data and by comparing this to NASA data for orbital periods, Saturn's calculated period was the least accurate, so does this suggest that its proximity to Jupiter had the biggest effect?

• It might be helpful to see your calculations so we can rule out stuff like roundoff errors. Dec 2, 2021 at 22:26
• Which specific form of Kepler's 3rd law are you using (i.e., are you ignoring the mass of the orbiting body, are you using masses or standard gravitational parameters)? Also, which NASA data are you using? The data at nssdc.gsfc.nasa.gov/planetary/factsheet is fairly low precision, and not necessarily consistent with other NASA / JPL data, eg ssd.jpl.nasa.gov/astro_par.html & ssd.jpl.nasa.gov/planets/approx_pos.html or data which can be obtained from Horizons ssd.jpl.nasa.gov/horizons Dec 3, 2021 at 8:55
• big 'ole Jupiter has two effects on Saturn; in addition to direct attraction between the two, Jupiter also moves the Sun around a bit, and Saturn has a hard time staying in a Keplerian orbit around an asynchronously moving Sun!
– uhoh
Dec 6, 2021 at 3:25

The most dominant object in our solar system outside of the sun is Jupiter.

What you’re asking depends greatly on the order of accuracy you’re looking for; for the lowest order of accuracy, you consider just the sun and the orbiting object, which isn’t all that bad of an approximation, but the next step that would actually have a fairly recognizable affect on orbital parameters is Jupiter. Jupiter has cleared certain orbits entirely in our system because of resonance, and often time perturbs the orbits of asteroids and other objects coming from outside the solar system in a noticeable (sometimes particularly noticeable) way.

Saturn is one of the planets closest to Jupiter, and so from reasoning alone that gravitational forces are distance dependent, I would assume that has a big part to play in why Saturn’s parameters differ the most.

So to answer your question, let’s consider something important: while Kepler’s third law isn’t perfect, it’s a good approximation and gives us an idea of how objects behave generally speaking. So if Jupiter disrupts an orbit, all parameters (save maybe angular ones to a good approximation like argument of the periapsis, etc) will be affected, according (roughly) to the relationships found in Kepler’s 3rd law. No individual parameter can be changed without changing the others (again excluding angular stuff)

tl;dr Jupiter is the next biggest problem in getting orbital parameters down; no single parameter is more affected by errors, they’re all still connected in a quantitative sense as Kepler’s 3rd law predicts

• This is a great answer, but one nit; how can one say that "no single parameter is more affected by errors" without even defining what "more affected" means? One can use relative change (e.g. percent) for things like $a, e$, but what about angles like $i, \Omega, \omega$? I don't think you can really say that without establishing what it even means.
– uhoh
Dec 6, 2021 at 3:29