Circular orbits

Question

First of all, I'm studying orbits for a hobby: world building. Unfortunately, my mathematical abilities approach a ridiculous low threshold, which means I am stuck with reading the simplest explanations, which in turn leave me asking tons of fairly basic questions.

Allow me to start with a simple point. I know that Kepler's Laws state that planetary orbits must always be elliptical. I also know that Earth's orbit varies from more elliptical to less elliptical, and that its less elliptical stage is nearly circular.

So... what would happen if Earth did have a circular orbit? Why is it impossible for any planet (or moon, by the way) to orbit another body in a perfectly circular path?

Answer 1

You've been given an answer, and it's perfectly valid, but here's something from a different perspective (less strict).

A circle is really just a particular case of an ellipse. Take an ellipse, and change it, by moving its focal points closer together. When those two points coincide, what you get is a circle. It's still an ellipse, technically - one that happens to have both focal points in the same place, is all.

So yes, you can actually have planetary orbits, or any orbits, circular. There's nothing forbidding that. It's just pretty unlikely that this will occur via a natural process.

As indicated elsewhere, in the real world, all orbits and trajectories are a bit imperfect due to perturbations - whether they be elliptical, circular, parabolic or hyperbolic, they are always a bit perturbed by external factors. In many cases, perturbations are so tiny that you can ignore them.

When a planet is orbiting the Sun, and the orbit is elliptical, the Sun will be in one of those two focal points; the other point has no particular signification. If you could circularize that orbit, then the Sun would be in the center of the circle, of course.

Kepler's laws remain valid for a circular orbit:

1. The orbit of every planet is an ellipse with the Sun at one of the two foci.

Still true. A circle is an ellipse where the foci coincide.

2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

Still true. On a circular orbit, the planet moves at constant speed, so the swept area remains constant per time.

3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

Still true. The semi-major axis becomes the radius of the circle.

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You must understand that Kepler's laws now have more of a historic interest. They are not exactly at the bleeding edge of science anymore. During Kepler's time, it seemed reasonable to state that all orbits must be elliptical (in the strict sense of the term), but now we know that trajectories (including orbits, or closed trajectories) can be circular, elliptical, parabolic or hyperbolic, depending on a few factors.

We also know that perturbations actually deflect all these trajectories a little bit from ideal shapes (but it's usually a very tiny effect).

We also know that relativity makes all "elliptical" orbits more complex - they remain close to elliptic, but the whole ellipse keeps turning around the central star very slowly.

All this stuff was not known during Kepler's time, so take his laws for what they are - a snapshot of the development of our understanding in time.

Answer 2

Why is it impossible for any planet (or moon, by the way) to orbit another body in a perfectly circular path?

One way to look at it is from the perspective of probability and statistics. Think of position and velocity as random variables drawn from some continuous probability distributions. Given some position vector, the velocity vector has to have a very specific value to yield a circular orbit. The probability of drawing a specific value from a well-behaved continuous probability distribution is identically zero.

An even better way to look at it: Even perfectly elliptical orbits aren't possible. Kepler's laws are an approximation that result from assuming a universe that obeys Newtonian mechanics and comprises but two point masses. Newtonian mechanics is only approximately valid in the real universe, objects are lumpy and can only approximately be treated as point masses, and there are a lot more than two objects in the universe.

Suppose that by some fluke chance, an object appears to have a perfectly circular orbit at some point in time (to within measurement error). The non-Newtonian nature of the universe, the non-spherical mass distributions of the objects, and the multiplicity of objects means that a moment later the object will no longer appear to have a perfectly circular orbit.

Answer 3

A circle is an ellipse with eccentricity zero. And in fact tidal evolution can drive orbital eccentricity to values negligibly close to zero. See Regarding the Putative Eccentricity of Charon's Orbit.

From observations we already know Pluto and Charon move about each other in very nearly circular orbits. I expect when New Horizons flies by the system in July, 2015, we will know their orbits more precisely.

Answer 4

The ellipse family includes perfect circles. An ellipse is a dilated circle. Draw a circle on a rubber sheet and stretch it to get an ellipse, pretty easy. Draw an ellipse on a stretched rubber sheet and relax it to see if you get a circle, much harder right. Now lets take the earth's orbit around the sun. The earth isn't perfect sphere, that rotates around a tilted axis and the center of gravity varies depending on the orbital position (season) with respect to the sun. The polar ice caps, the tidal distributions, the clouds all play apart. The moon and other planets also have some effects that vary with position. The moon and sun have their own similar idiosyncrasies. You can probably name many more.
So for most planetary orbits the bottom line is that there are too many variables that must line up exactly all the time to result in a perfect circular orbit and an elliptical orbit is the result.