= Solution
A strip $dx$ at $x>0$ passes the satellite at relative speed $Sx$, so its encounter mass flux is $Sx\Sigma(x)dx$. Each unit mass gains <specific angular momentum> $r_0\Delta v_y$. Therefore the <one-sided impulse torque on a disk> is
$$
\Gamma_{>0}=\chi\frac{r_0(GM_s)^2}{S^2}\int_H^\infty\frac{\Sigma(x)}{x^4}dx.
$$
Putting $\chi=1/2$ gives exactly the requested expression. Inner strips have the opposite signed impulse but the same positive encounter flux. The net disc <torque> is consequently
$$
\boxed{\Gamma=\chi\frac{r_0(GM_s)^2}{S^2}\int_H^\infty\frac{\Sigma(x)-\Sigma(-x)}{x^4}dx.}
$$
An even <surface density> makes the integrand vanish pointwise, so the total <torque> is zero although each one-sided <torque> can be nonzero. The cutoff must keep the encounters in the weak-deflection regime; an appropriate physical value is at least of order the thickness or the relevant strong-scattering radius. The infinite-sheet integral also assumes convergence or a specified outer truncation.
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