Orbital phase

Orbital phase specifies where a body lies within its orbital cycle relative to a chosen reference event.

A phase-mixed orbit has a probability distribution uniform in its canonical angle variable. For a Kepler ellipse, this means uniform mean anomaly and therefore probability proportional to time spent in each orbital segment.

Uniform mean anomaly implies a line probability inversely proportional to the orbital speed. For small orbital eccentricity and true anomaly $f$, it is proportional to $1-e\cos f$.

The geometric cross-section passing a fixed point of a steady orbital wire per unit time is $J_{\sigma }={\lambda }_{\sigma }v$, where ${\lambda }_{\sigma }=d\sigma /ds$. Conservation of fragments requires $J_{\sigma }$ constant and ${\lambda }_{\sigma }\propto 1/v$. For total area ${\sigma }_{\mathrm{tot}}$ and orbital period $P$,With the fractional luminosity of a phase-mixed eccentric wire, this becomes $J_{\sigma }=2fh$. The current describes transported area, not the swept area enclosed by the orbit.

For a Kepler orbit with pericentre distance $q$, apocentre distance $Q$, and semi-major axis $a=(q+Q)/2$, a uniform mean anomaly givesThe two radial passages per orbital period are included, and ${\int }_q^Q{p}_r(r)dr=1$.