Are Kepler's Laws Universal?

This is a two-part question: First, do the planets found outside our solar system obey Kepler’s laws?

Second, why does Kepler’s 3rd law not seem to apply to comets, such as Halley’s, with a period of about 86 years - much less than than the outer planets, yet with an orbit much greater than the outer planets? If you argue that Kepler’s laws are for planetary motion only, then does this hint about how the solar system formed - that all the original nine planets should obey what Kepler observed? I mean…why should they?

Yes, Kepler’s laws are universal. Halley’s Comet follows Kepler’s laws - Halley’s comet has a semimajor axis of about 18 au, just a little less than that of Uranus, so it has a period of 76 years, just a bit less than that of Uranus, the period of which is 86 years. Halley’s comet has a much more elliptical orbit than a planet, but Kepler’s law regarding orbital period depends on semimajor axis (or roughly, average distance from the sun)

Kepler’s laws, not just for planets! :slight_smile:

Kepler’s laws can be derived from Newton’s laws, so Kepler’s laws are as universal as Newton’s laws (which, of course, are not…but that is a whole nother thread probably)

As noted above, Kepler’s Laws can be derived from Newton’s Laws. Planets in another stellar system would orbit in ellipses, and they would sweep out equal areas in equal times, because those are consequences of the gravitational force always pointing towards the central object and being inversely proportional to the square of the distance to the central object.

The Third Law holds too, with one caveat. The Third Law states that P[sup]2[/sup] = k a[sup]3[/sup], where a is the semi-major axis (see below) of the object, P is its period, and k is a constant that’s the same for all objects in a given stellar system. If you figure out what this constant is, it turns out to depend on the mass of the central object. So a different stellar system would have a different mass of its central object, and all the objects in that stellar system would obey P[sup]2[/sup] = k a[sup]3[/sup] with this new value of k.

And yeah, Kepler’s Third Law only depends on the semi-major axis, not the maximum distance. The semi-major axis of a comet’s orbit (or any other object’s orbit) is the average of its closest distance and its farthest distance. For a nearly circular orbit, this doesn’t make much of a difference. For Halley’s Comet, though, the semi-major axis is about the same as the semi-major axis of Uranus — and so they both go around the Sun in approximately 80 years.

Said another way, Kepler’s Laws are laws about orbits, not about planets.

So they apply to dust orbiting an asteriod just as well as the asteroid orbiting the Sun. And to artificial satellites orbiting the Earth.

As **MikeS **points out, some of the common formulations in textbooks are reduced formulas with our specific Sun’s system’s values plugged in already. So naturally, using those simplified formulas to compute values in other orbitting systems will produce wrong answers.

Side anecdote: In Swift’s Gulliver’s Travels, he demonstrates the technological advancement of one of the civilizations Gulliver visits by saying that they’ve discovered a pair of moons orbiting Mars (this was about a century before we humans discovered Mars’ actual moons). He lists orbital periods and distances for both of them, and the periods and distances he gives are consistent with Kepler’s laws, given a reasonable estimate of the mass of Mars. Not too many authors nowadays who would go to that level of detail.

And as a side-side note Kepler’s Second Law holds in any gravitational system where the force between two bodies is directed toward their common center of mass, even if the force falls off at a different rate than inverse-squared.

Kepler’s Second Law is really a re-wording of the Conservation of Angular Momentum and that will hold irrespective of the attractive ratio.

Missed the edit window by being a slow typist: If you had some other system where, say, the force fell off at an inverse-cubed rate, the orbits would almost all be unstable but the Second Law would still hold.