Kepler geometry and time-averaged energy

For two isolated point masses, separate the inertial motion of their center of mass from the relative coordinate joining them. The relative separation obeys an inverse-square acceleration with gravitational parameter G times the sum of the masses. A noncolliding bound orbit with nonzero angular momentum has negative specific relative energy and eccentricity below one; its relative trajectory is an ellipse with one focus at the origin of that relative coordinate. Each physical body traces its own scaled ellipse around the barycenter. Replacing the mass sum by the larger mass is a controlled approximation only when the other mass is sufficiently small.

Angular-momentum conservation makes the areal sweep rate constant. The object therefore moves faster near closest approach and slower near farthest separation. This is a time statement, not just an ellipse-drawing rule. The orbital period depends on the relative ellipse's semimajor axis, not its instantaneous radius or the average of arbitrary sampled positions. The closed conic result assumes an exact inverse-square central interaction with no drag, additional bodies, or relativistic corrections. Small violations can produce precession or secular evolution without making the two-body approximation useless as a leading description.

Write energy per unit reduced mass for the relative motion. In the inverse-square problem its bound-orbit value is minus the gravitational parameter divided by twice the semimajor axis. Equating that constant to one half relative speed squared minus the gravitational parameter over radius gives the vis-viva equation. It predicts speed magnitude at an allowed separation, but not the direction of velocity or where the orbit lies in space. Angular momentum and orientation data supply the missing information. A radius outside the orbit's turning points cannot become physical merely because the speed expression is algebraically positive.

For a circular orbit, radius equals semimajor axis and the speed reduces to the square root of gravitational parameter divided by radius. At a fixed separation, escape speed is the zero-energy value and exceeds circular speed by the square root of two. These comparisons use the same potential zero at infinite separation. Adding a constant to the potential changes the numerical energy reference, so a statement that negative energy means bound must name the convention. It also presumes an attractive inverse-distance potential with suitable asymptotic behavior; energy-sign classifications are not universal for every central force.

Define a virial quantity as the dot product of relative position with relative momentum. Differentiating it gives twice kinetic energy plus position dotted into force. For inverse-distance attraction, that force term equals the potential energy, yielding a derivative equal to twice kinetic plus potential energy. Average over one complete periodic orbit: the virial quantity returns to its starting value, so the average derivative vanishes. Thus twice the average kinetic energy plus average potential energy is zero. Neither kinetic nor potential energy must be constant separately on an eccentric orbit.

For a more general long-time average, justify why the endpoint difference in the virial quantity divided by averaging duration tends to zero. A bounded periodic orbit satisfies this condition, but an escaping trajectory does not automatically do so. Nor does one instantaneous measurement establish virial equilibrium for a many-body system with evolving boundaries or external tides. In the isolated Kepler problem, the averaged relation connects total energy to the mean kinetic and potential energies and implies that the time average of inverse separation equals inverse semimajor axis. This is a useful consistency check on numerical orbit sampling, especially when uneven speed makes uniform-angle sampling different from uniform-time sampling.