= Solution
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 area
$$
\sigma_{\rm front}=(2fh)\frac{2R_\star r}{h}=4fR_\star r
$$
in front of the stellar disk. For an <optically thin> wire and a uniformly bright stellar disk, <transit dimming by an optically thin orbital wire> gives
$$
\boxed{\frac{\Delta F}{F_\star}=\frac{\sigma_{\rm front}}{\pi R_\star^2}
=\frac4\pi\frac r{R_\star}f}.
$$
This 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.
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