Undergrad Mathematical Proof of Elliptical Orbits

[Celso]

How is it mathematically proven that gravitational orbits are elliptical?

 

[phyzguy]

Newton showed in the Principia in 1687 that bodies orbiting under the influence of a 1/r^2 force will move in elliptical orbits. This Wikipedia page describes the argument, both Newton's derivation and a more modern derivation.

 

[fresh_42]

How is it mathematically proven that gravitational orbits are elliptical?

Here's the math: https://de.wikipedia.org/wiki/Keplersche_Gesetze#1._Keplersches_Gesetz_(Ellipsensatz)

I haven't seen it on the English version, but as the formulas don't need to be translated, it should do.

 

[gneill]

Note that in general any conic section is a solution to the differential equation of motion in an inverse square field. Thus "orbits? can be circular, elliptical, hyperbolic, parabolic, and even straight lines (a so-called "degenerate orbit" where the object just drops straight into the Sun).

 

[Orodruin]

While you have been given the answer to why orbits are elliptical in Newton's theory of gravitation, I think it is relevant to point out that you can only prove things mathematically under some assumptions - in this case that Newtonian mechanics hold and that the gravitational centripetal force is given by an attractive 1/r potential.

However, in physics there really is only one way of testing predictions and that is to make experimental measurements. As it turns out, planetary orbits (even correcting for the gravitational influence of other celestial bodies) are not elliptical (although very close to). The discrepancy between the prediction of Newton's theory and measurements, most apparent as the perihelion shift of Mercury, is accounted for in general relativity.