# Is there any significance to Newton's theorem on decaying orbits in book 2 of his Principia?

My question refers to proposition 15 in book 2 of Isaac Newton's principia. In this proposition Newton stated sevaral corollaries on the orbital motion of bodies under a certain law of central force and under medium resistance that is proportional to the medium density and the velocity squared. His main result is that if the density of the medium varies inversely with the distance from the center than the orbit will be an "equiangular spiral"- or logarithmic spiral.

So my question asks if there are any uses for this theorem, for example to calculate the decay of satellite orbits. As we know, the density in the atmosphere follows an exponential profile, so it seems this theorem is only a mathematical gem and not of any use. So is there any way to make use of htis thoerem?

## 1 Answer

This theorem has no astronomical applications (or applications to satellites). Newton worried that the space may not be empty (filled with some substance), and wanted to know how this assumption would reflect on the motion of planets and satellites. The theorem could have some implications for bodies moving in a rarefied atmosphere, if the law of density were correct. But it was incorrect.

• As far as i know, every monotone decreasing function can be approximated locally by a function of the form C_1 + C_2/r (constant + inversed r function). This means that if we approximate the decreasing exponent locally by a function of this type, we may use Newton's theorem on orbital decay.Therefore my question asks if the addition of the constant doensn't violate Newton's whole proof, and if it does, if there is any way to adapt his theorem locally. – user2554 Sep 21 '16 at 6:42