Expert: Courtney Seligman - 4/13/2011

Question
Why orbits of all planets are elliptical?

Answer
Your question doesn't make it clear whether you want the physics behind the answer, or just why they aren't circles, instead of ellipses. If you don't want the physics, you can skip to the material after the dashed line, a few paragraphs down, which gives a fairly detailed discussion of the thing, or to the material after the second dashed line, which gives a brief summary. And of course if I've misunderstood what you wanted to know, or somehow failed to give you a satisfactory answer, please feel free to contact me with a follow-up.

The orbits are the result of two factors, each of which would result in a straight-line motion, in the absence of the other. (1) The forward motion of the planet would cause it to move in a straight line in whatever direction it is heading, if there were no force acting on it (that's called Newton's First Law of Motion, the Law of Inertia). (2) The gravitational force of the Sun would cause the planet to fall straight into the Sun, if it weren't already moving (just as if you dropped something, it would fall straight down).

However, neither result can occur, because the planet does have a forward motion (and therefore cannot fall straight into the Sun), and it does have a force on it (and therefore cannot continue in a straight line, in the direction of its motion). Instead, the planet "tends" to move forward a little, in the direction it started off in, and at the same time, "tends" to fall away from that straight-line motion, toward the Sun. As a result, it follows a curved path which is always more toward the Sun than the original straight-line motion, but never directly toward it.

The shape of the curved path which the planet follows depends upon the mathematical equation which describes how the force of gravity changes with distance from the Sun. As it happens, there are only two equations which result in orbits which repeat, over and over again. Any other equation results in a continually changing motion which never repeats itself, and would in a sense, be "chaotic".

The equation which describes the actual gravity of the Sun, which is described as an inverse-square force, because it is weaker at larger distances according to the inverse square of the distance, will allow one of four orbital shapes, all of which are referred to as "conic sections" (see http://cseligman.com/text/history/ellipses.htm Ellipses and Other Conic Sections, for a discussion of the four shapes). Two of those shapes, parabolas and hyperbolas, involve orbits which would go once around the Sun, then leave it behind forever. If any object in the solar system ever had an orbit like that, it is long gone. So the only "stable" orbits are the remaining shapes -- circles and ellipses -- and it happens that a circle is just a special kind of ellipse, so even if a planet had a perfectly circular orbit, we could still say that it was an ellipse.

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Now, let's suppose that we want to have a circular orbit, in which the Sun is in the center and the distance from the Sun is a constant. That would require the "forward" motion of the planet to have a specific value, called the circular velocity. Even if the speed was a fraction of a percent faster or slower than the circular velocity, the planet would end up with a slightly elliptical orbit, in which its distance gradually changed. If it was going a little faster than the circular velocity, it would gradually move away from the Sun, until (on the exact opposite side of the Sun and its orbit) it was at its maximum distance. All along its outward journey, the Sun's gravity would pull backward on it (since it is moving outward, away from the Sun, which is opposite the direction of the Sun's gravity), and by the time it reached its maximum distance, it would be moving a bit slower than it started off -- slow enough that it couldn't stay at that distance, but would gradually fall back toward the Sun, until it reached the exact same place it started off. But all the way in, it would gradually speed up (since it is moving closer to the Sun), and by the time it got back to where it started off, it would have exactly the same slightly too fast speed it started with. Since at that point it would be in the same place it started, moving the same way it started, and the Sun's gravitational force on it would be the same as at the start, it would follow that same path over and over. And as it happens, the shape of that path would be an ellipse (just as, technically, a circle would be -- but in the case of the changing-distance orbit, a "real" ellipse, rather than a circle which just happens to be a special kind of ellipse).

The same idea applies in reverse, if it was going just a bit slower than it should be. The starting point would correspond to the slower part of the "too-fast" orbit described above, the opposite side would correspond to the faster "start" of the "too-fast" orbit above, and things would just repeat over and over, in exactly the same way.

In other words, if the planet were moving at exactly the circular velocity, it could have a circular orbit. But if it goes even a little faster or slower, it would end up with a slightly elliptical orbit. Approximately, if it is going 1% too fast or slow, it would end up with an orbit which varied in distance from the Sun by about 1% at either end (slower at the far end, faster at the closer end of the orbit).

The discussion immediately above assumes that the planet starts off exactly sideways relative to the Sun. If it happens to be headed a little up or down (toward or away from the Sun), then even if its speed was the same as the circular velocity, the fact that it is heading up or down at the start means it couldn't stay at a constant distance from the Sun, and it would automatically end up with an elliptical orbit.

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Let's suppose we start the planets off with some random orbital motion. If by chance one of them has a motion which is exactly sideways relative to the Sun, and exactly equal to the circular velocity at its distance from the Sun, then it will follow a circular orbit. But if it is going just a bit faster or slower or up or down, it will follow an elliptical orbit. Odds are, none of them would happen to have exactly the right motion to have a circular orbit, but there is nothing that says it couldn't happen. It just isn't as likely as having one of all the other possible orbits. It's sort of like flipping a coin a million times. Odds are that you'd end up with close to half heads, and half tails. But it turns out that there are about 20 thousand results, on either side of half a million heads and half a million tails, that are just about as likely as exactly half a million heads and half a million tails. So the chances of getting exactly that result is actually very small -- less than one in 20 thousand. Similarly, you could have a circular orbit, but there are close to an infinity of not quite circular elliptical orbits which are just as likely, so the chances that one of those would be the actual orbit are much greater than the chances of having a perfectly circular orbit.

Now, in all of the above discussion, I've ignored the effects of the planets on each other's orbits. And they do have effects. The effects are very small in comparison with the Sun's forcing the planet to follow a basically elliptical orbit, but they do exist. As a result of those effects, whatever orbit a planet has now will very slightly change over long periods of time -- gradually becoming a little bigger or smaller, a little less round or more round, a little tilted one way or another, and so on. None of the changes would be large, and on the average, each orbit stays about the same size and orientation in space forever; but the shape of the orbits can and does slightly change over time. So even if a planet had a circular orbit now, it would not have a circular orbit a few hundreds or thousands of years from now. It would instead have an orbit that is very close to a circular orbit, but is not exactly circular.

So the orbits are elliptical because the laws of physics happen to require them to be that way (including in this statement the special ellipse called a circle). They are not circular ellipses simply because (1) that is only one possibility out of a large range of possibilities, so the chances of it happening are very low and (2) the shapes of the orbits gradually change, due to the gravitational interaction of the planets, so that even if an orbit was circular now, it wouldn't stay that way for very long.

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I hope this is a satisfactory answer. Odds are, it is much longer and more complicated than necessary; but since I wasn't sure exactly what you wanted, I erred on the side of more detail, to make sure that I did answer your question (or at least, I hope I did).