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The printed line-density statement is incorrect for a phase-mixed orbit. A steady line density on a Kepler orbit is inversely proportional to speed: for a cross-sectional-area current $J_{\sigma }$, the area per unit arc length is $d\sigma /ds=J_{\sigma }/v$. Equivalently, phase mixing gives $d\sigma =({\sigma }_{\mathrm{tot}}/P)dt$, uniform in mean anomaly, where $P=2\pi \sqrt{a^3/\mu }$ is the orbital period.

For an optically thin population of blackbodies in radiative equilibrium, a fragment absorbs $\sigma L_{\star }/(4\pi r^2)$ and reradiates the same luminosity. Thus the fractional luminosity of a phase-mixed eccentric wire isThe specific angular momentum gives $dt/r^2=d\theta /h$, so the integral is $2\pi /h$. Since $Ph=2\pi a^2\sqrt{1-e^2}$,This derives the intended result after explicitly correcting the density to $1/v$.

The error is consequential. If one instead imposes the literal density $d\sigma /ds\propto v$, its time weighting is $v^{2}dt$. Using $⟨v^2⟩=\mu /a$, $⟨r^{-2}⟩=[a^2\sqrt{1-e^2}{]}^{-1}$ and $⟨r^{-3}⟩=[a^3(1-e^2{)}^{3/2}{]}^{-1}$ givesAt $e=1/2$ this is $5/3$ times the printed result. The inverse-radius averages can also be obtained directly with eccentric anomaly $E$, $r=a(1-e\cos E)$ and $dt=(1-e\cos E)dE/n$.

Use an edge-on orbit and an equatorial, effectively central chord. Near the foreground crossing the radial velocity is along the line of sight; the projected transverse speed is $v_{\perp }=h/r$. In the small stellar-angular-radius limit, $r\gg R_{\star }$, the crossing time is $2R_{\star }/v_{\perp }=2R_{\star }r/h$.

The cross-sectional-area current therefore places an areain front of the stellar disk. For an optically thin wire and a uniformly bright stellar disk, transit dimming by an optically thin orbital wire givesThis is a linear occultation estimate, requiring negligible overlap and a fractional dimming much smaller than one. A finite impact parameter shortens the chord; limb darkening, finite wire thickness and variation of $r$ across a large stellar angular extent modify the coefficient. The estimate cannot be extrapolated to dimming greater than unity.

On the corrected steady phase-mixed orbit, each fragment crosses a fixed orbital longitude once per orbital period. Hence the cross-sectional-area current through that point isFrom the fractional luminosity of a phase-mixed eccentric wire, ${\sigma }_{\mathrm{tot}}=4\pi a^2\sqrt{1-e^2}f=2Phf$. Therefore the area rate at the foreground crossing of the line of sight isOnly the foreground segment blocks the star; the far-side intersection does not add another occultation current. The orbital area current is constant because the line density varies as $1/v$, even though the local orbital speed varies.

For an edge-on orbit crossing the centre of a uniformly bright stellar disk at distance $r\gg R_{\star }$, projected speed is $h/r$ and chord time is $2R_{\star }r/h$. A steady cross-sectional-area current $J_{\sigma }=2fh$ therefore places area $4f{R}_{\star }r$ in front of the star, givingThis requires a geometrically narrow, optically thin wire with negligible overlap. A noncentral chord, limb darkening, finite angular extent and large optical depth change the estimate; it cannot predict dimming greater than one.