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Does anyone know a good reference for this?

I once read how Jupiter's gravitation can be treated as a perturbation. 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, and eccentricity.

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    $\begingroup$ 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. $\endgroup$ – uhoh Dec 28 '18 at 1:09
  • $\begingroup$ @NicolaG looks good! SE is certainly collaborative so go right ahead and make improvements were you see fit. $\endgroup$ – uhoh Jan 4 '20 at 8:33
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    $\begingroup$ 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? $\endgroup$ – ProfRob Jan 4 '20 at 9:43
  • $\begingroup$ 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). $\endgroup$ – NGsp Jan 4 '20 at 10:18
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    $\begingroup$ @userLTK Jupiter's perihelion is 4.95 AU. How can Earth (in the current epoch) ever be closer to Jupiter than to the Sun? $\endgroup$ – PM 2Ring Oct 13 '20 at 12:46
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Short Answer: The gravitation of Jupiter and all the other planets makes Kepler’s third law a bit less accurate than it would be if their gravity were zero. The gravitational interaction between the planets, however, doesn’t have much effect on either the period or semi-major axis of any of the planets, especially in the short term.

Long Answer: You might be thinking of this pendulum motion expansion used to describe the orbital oscillations (including the big one between Jupiter and Saturn) here https://history.nasa.gov/SP-345/ch8.htm:

enter image description here

Where $\psi$ is the angle from the vertical, $\omega$ is the angular velocity, and $A^2$ is $g/l$, where $g$ is acceleration and $l$ is presumably the length of the pendulum.

Interplanetary resonances mostly result in precession of planet’s orbits, which are period and semi-major axis invariant. They mostly change just the longitude of the ascending node.

enter image description here

Other orbital changing effects between planets are Milankovitch cycles which are also period and semi-major axis invariant, according to Wikipedia:

Eccentricity varies primarily due to the gravitational pull of Jupiter and Saturn. However, the semi-major axis of the orbital ellipse remains unchanged; according to perturbation theory, which computes the evolution of the orbit, the semi-major axis is invariant. The orbital period (the length of a sidereal year) is also invariant, because according to Kepler's third law, it is determined by the semi-major axis.

The more significant effects that planetary mass have on adding inaccuracy to Kepler’s third law are through their gravitational effects on the sun when treated as individual two-body orbits.

Kepler’s “laws” describe the motion of the planets. Kepler arrived at these laws by examining observational data from the solar system in the late 1500s and the early 1600s. On page 48 of Epitome of Copernican Astronomy Kepler asked himself the question:

How is the ratio of the periodic times, which you have assigned to the mobile bodies related to the aforesaid ratio of the spheres wherein those bodies are borned?

And then answered:

The ratio of the times is not equal to the ratio of the spheres, but greater than it, and in the primary planets exactly the ratio of the 3/2 powers.

A bit of translation is necessary here. The periodic times Kepler references are the orbital periods $p$ of the planets, and the spheres are the radii of the spheres which circumscribe the ellipses of the orbits. These are the same as the semi-major axes $a$ of the orbits.

So Kepler’s third law as stated was that, for every pair of planets 1 and 2, $(a_1/a_2)^{3/2}=p_1/p_2$. Our typical modern restatement of Kepler’s third law is that the ratio $a^3/p^2$ is the same for every planet. Kepler thought that the sun was the center of the solar system. He didn’t realize that when a less massive object is in orbit with a more massive object, both objects actually rotate around their mutual center of gravity (called the barycenter). As Newton himself said (thanks UserLTK): “......Kepler knew ye Orb to be not circular but oval & guest it to be elliptical......”: from: arxiv.org/ftp/physics/papers/0107/0107009.pdf source of quote: H. W. Turnbull (ed.): The correspondence of Isaac Newton, II, Cambridge, 1960, pp. 436-437.

Kepler’s original values for the ratios were: enter image description here

Kepler thought that “a more accurate number will be produced, if you take the times more accurately.” Kepler attributed all the inaccuracies in his data set to measurement error. However, Newton proved Kepler wasn’t quite correct.

Newton generalized Kepler’s third law as $$\frac{a^3}{p^2}=\frac{G(m_1+m_2)}{4\pi^2}$$, where G is the gravitational constant, $m_1$ is the mass of the sun, and $m_2$ is the mass of the planet, respectively. This generalization implies that $a^3/p^2$ will be different for each of the planets, since they all have different masses. The largest differences manifest for the planets with the highest masses, but even for Jupiter, these differences are along the third significant digit, consistent with our current best estimates here: enter image description here

These values correspond well to my calculated predictions bases on Newton's generalization of Kepler's third law:

enter image description here

To repeat my short answer above, the gravitation of Jupiter and all the other planets makes Kepler’s third law a bit less accurate than it would be if their gravity were zero. The gravitational interaction between the planets, however, doesn’t have much effect on either the period or semi-major axis of any of the planets, especially in the short term.

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