Fractional luminosity of a debris disk ($f$)

The fractional luminosity is $f=L_{\mathrm{disk}}/L_{\ast }$; for blackbody grains in an optically thin belt at radius $r$, it equals their total geometric cross-section divided by $4\pi r^2$.

On a steady phase-mixed orbit, line density on a Kepler orbit is proportional to $1/v$, so cross-sectional area is uniform in time. For an optically thin wire of blackbodies in radiative equilibrium,The proof uses specific angular momentum: $dt/r^2=df/h$, and $Ph=2\pi a^2\sqrt{1-e^2}$. If one instead imposes a line density proportional to $v$, area is weighted by $v^{2}dt$, yielding $f={\sigma }_{\mathrm{tot}}(1+e^2)/[4\pi a^2(1-e^2{)}^{3/2}]$. The two distributions agree only for a circular Kepler orbit.