# why planets revolve around the sun in an elliptical orbit

Maybe it's best to forget space curvature as described in general relativity; while that theory implies major changes to the foundations of celestial mechanics, it is set up in a way that reproduces Newtonian mechanics when gravitational fields are of moderate strength, and leads to only very tiny adjustments of the actual orbit calculations. Movement of an object in a fixed gravitational field (which is a realistic assumption for the movement of a satellite around a much more massive body, whose own perturbation can then be ignored) is described by a second order differential equation: at each point in its orbit the acceleration of the satellite is given through Newton's law by the gravitational field at that point. Calling them Sun and planet for ease of description, the orbit of the planet will then be determined by its position and velocity at some (arbitrary) starting time. It is easy to see that movement will remain in the plane determined by the Sun and the initial planet position, and containing its initial velocity vector. Thus each state is essentially given by two position coordinates and two velocity coordinates. Rotational symmetry allows us to forget about one degree of freedom, and describe the initial conditions by the radial position (initial distance of the planet from the Sun), and the tangential and radial components of the velocity.

Instead of the tangential velocity component one can also give the instantaneous angular velocity, obtained from it by dividing by the radius. Since a circular orbit is characterised by the fact that the radial component of the velocity is always zero, having no radial component of the initial velocity is a necessary condition for getting a circular orbit. This already answers the question why orbits are not always circular. However, the condition is not sufficient: in a circular orbit the angular velocity must be in a precise relation to the radius in order to ensure that the the gravitational acceleration, which depends only on the radius, matches the centripetal acceleration for a circular orbit, which also depends on the angular velocity. This explains why satellites just above the Earth's atmosphere (which must be given circular orbits for practical reasons) all have the same rotation period (about 1. 5 hours). This also explains why non-circular orbits, which do have some point where the radial component of the velocity becomes zero, do not become circular from that point on: their angular velocity is either too low or too high at that point. When the angular velocity is too high, the planet will "spin out" and its radius starts increasing (possibly going off to infinity if the velocity exceeds the escape velocity at that point, though one would not speak of a planet in this case).

If it is too low, it will "drop out" of a circular trajectory and start falling. In the latter case its angular velocity will increase (since its product with the radius squared must be constant, as the constant angular momentum is proportional to it), giving in the rotating frame an increasing centrifugal force that at some point overtakes the gravitational force (even though the latter also increases) at which point the negative radial velocity is maximal and starts decreasing. When the radial velocity become zero again, the planet has reached it closest point to the Sun. But the angular velocity is now too large for a circular orbit, even at the smaller radius attained; the planet is now in a "spin out" situation described above. Events then repeat in reverse order, until a point of maximal radius is again attained. What is remarkable and can only be explained by a detailed computation (which I will not give; I'm not sure I even could), is that this point is the

same point in space as the starting point: the curve closes and the planet orbit turns out to be an ellipse. There are essentially five solutions to the two body problem, where the bodies are attracted to each other by gravity. A hyperbolic path - the two bodies are moving away from each other too quickly for gravity to bring them into a closed orbit.

A parabolic path - the two bodies are moving away from each other just quickly enough for gravity not to be able to bring them into a closed orbit. This is a kind of "boundary case" between the hyperbolic path and the elliptical orbit. An elliptical orbit - the general case when gravity between the two bodies limits the distance between them. A circular orbit - a very special variant of the elliptical orbit, where the direction of the bodies' velocity away from each other is exactly perpendicular to the displacement between them, and the magnitude of that velocity is exactly the right amount to balance out the gravitational attraction. A collision course where the bodies move directly towards each other, although they may start off moving away from each other. For the parabolic solution or the circle solution, conditions need to be exactly right. That is, the initial relative velocity of the bodies needs to be a precisely calculated magnitude, in a precisely calculated direction. This would require an enormous coincidence. That's why the other three solutions (hyperbola, ellipse or collision course) are the only ones that every happen in nature. In particular, if there were ever a satellite or a planet orbiting a planet or a star in an exactly circular orbit, it would only take a very slight tug from a third body to perturb it into an elliptical orbit.

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