Density-slope satellite torque ($\Gamma =(4\chi /9)\beta q^2{\Sigma }_0{r}_0^4{\Omega }^2(H/r_0{)}^{-2}$)

A local odd mass density slope $\Sigma (x)-\Sigma (-x)=2{\Sigma }_0\beta x/r_0$ changes the cancelling one-sided torques into this net torque. A local outer cutoff $L$ adds the factor $1-(H/L{)}^2$. Positive slope transfers net angular momentum from satellite to disk. An infinite linear mass density law is only a formal local extension, since it cannot remain nonnegative on both sides.