Kepler problem

History

The Kepler problem begins with the empirical results of Johannes Kepler arduously derived by analysis of the astronomical observations of Tycho Brahe. After some 70 attempts to match the data to circular orbits, Kepler hit upon the idea of the elliptic orbit. He eventually summarized his results in the form of three laws of planetary motion.

What is now called the Kepler problem was first discussed by Isaac Newton as a major part of his Principia. His "Theorema I" begins with the first two of his three axioms or laws of motion and results in Kepler's second law of planetary motion. Next, Newton proves his "Theorema II" which shows that if Kepler's second law results, then the force involved must be along the line between the two bodies. In other words, Newton proves what today might be called the "inverse Kepler problem": the orbit characteristics require the force to depend on the inverse square of the distance. Propositions XI–XIII provide Newton's solution to the direct Kepler problem, determining the centripetal-force law for a body moving in a conic. In Corollary I, he established the converse: a body acted upon by an inverse-square centripetal force must move along a conic having the center of force as a focus.

Mathematical definition

The central force F between two objects varies in strength as the inverse square of the distance r between them:

${\displaystyle \mathbf{F}=\frac{k}{r^2}\hat{\mathbf{r}}}$

where k is a constant and ${\displaystyle \hat{\mathbf{r}}}$ represents the unit vector along the line between them. The force may be either attractive (k < 0) or repulsive (k > 0). The corresponding scalar potential is:

${\displaystyle V(r)=\frac{k}{r}}$

Solution of the Kepler problem

The equation of motion for the radius ${\displaystyle r}$ of a particle of mass ${\displaystyle m}$ moving in a central potential ${\displaystyle V(r)}$ is given by Lagrange's equations

${\displaystyle m\frac{d^{2}r}{d{t}^2}-mr{\omega }^2=m\frac{d^{2}r}{d{t}^2}-\frac{L^2}{m{r}^3}=-\frac{dV}{dr}}$

${\displaystyle \omega \equiv \frac{d\theta }{dt}}$ and the angular momentum ${\displaystyle L=m{r}^2\omega }$ is conserved. For illustration, the first term on the left-hand side is zero for circular orbits, and the applied inwards force ${\displaystyle \frac{dV}{dr}}$ equals the centripetal force requirement ${\displaystyle mr{\omega }^2}$, as expected.

If L is not zero the definition of angular momentum allows a change of independent variable from ${\displaystyle t}$ to ${\displaystyle \theta }$

${\displaystyle \frac{d}{dt}=\frac{L}{m{r}^2}\frac{d}{d\theta }}$

giving the new equation of motion that is independent of time

${\displaystyle \frac{L}{r^2}\frac{d}{d\theta }\left(\frac{L}{m{r}^2}\frac{dr}{d\theta }\right)-\frac{L^2}{m{r}^3}=-\frac{dV}{dr}}$

The expansion of the first term is

${\displaystyle \frac{L}{r^2}\frac{d}{d\theta }\left(\frac{L}{m{r}^2}\frac{dr}{d\theta }\right)=-\frac{2L^2}{m{r}^5}{\left(\frac{dr}{d\theta }\right)}^2+\frac{L^2}{m{r}^4}\frac{d^{2}r}{d{\theta }^2}}$

This equation becomes quasilinear on making the change of variables ${\displaystyle u\equiv \frac{1}{r}}$ and multiplying both sides by ${\displaystyle \frac{m{r}^2}{L^2}}$

${\displaystyle \frac{du}{d\theta }=\frac{-1}{r^2}\frac{dr}{d\theta }}$

${\displaystyle \frac{d^{2}u}{d{\theta }^2}=\frac{2}{r^3}{\left(\frac{dr}{d\theta }\right)}^2-\frac{1}{r^2}\frac{d^{2}r}{d{\theta }^2}}$

After substitution and rearrangement:

${\displaystyle \frac{d^{2}u}{d{\theta }^2}+u=-\frac{m}{L^2}\frac{d}{du}V\left(\frac{1}{u}\right)}$

For an inverse-square force law such as the gravitational or electrostatic potential, the scalar potential can be written

${\displaystyle V(\mathbf{r})=\frac{k}{r}=ku}$

The orbit ${\displaystyle u(\theta )}$ can be derived from the general equation

${\displaystyle \frac{d^{2}u}{d{\theta }^2}+u=-\frac{m}{L^2}\frac{d}{du}V\left(\frac{1}{u}\right)=-\frac{km}{L^2}}$

whose solution is the constant ${\displaystyle -\frac{km}{L^2}}$ plus a simple sinusoid

${\displaystyle u\equiv \frac{1}{r}=-\frac{km}{L^2}\left[1+e\cos (\theta -{\theta }_0)\right]}$

where ${\displaystyle e}$ (the eccentricity) and ${\displaystyle {\theta }_0}$ (the phase offset) are constants of integration.

This is the general formula for a conic section that has one focus at the origin; ${\displaystyle e=0}$ corresponds to a circle, ${\displaystyle e<1}$ corresponds to an ellipse, ${\displaystyle e=1}$ corresponds to a parabola, and ${\displaystyle e>1}$ corresponds to a hyperbola. The eccentricity ${\displaystyle e}$ is related to the total energy ${\displaystyle E}$ (cf. the Laplace–Runge–Lenz vector)

${\displaystyle e=\sqrt{1+\frac{2E{L}^2}{k^{2}m}}}$

Comparing these formulae shows that ${\displaystyle E<0}$ corresponds to an ellipse (all solutions which are closed orbits are ellipses), ${\displaystyle E=0}$ corresponds to a parabola, and ${\displaystyle E>0}$ corresponds to a hyperbola. In particular, ${\displaystyle E=-\frac{k^{2}m}{2L^2}}$ for perfectly circular orbits (the central force exactly equals the centripetal force requirement, which determines the required angular velocity for a given circular radius).

For a repulsive force (k > 0) only e > 1 applies.