Angular-momentum-conserving self-gravitating disc radius ($R_h/R_d=C_d{f}_d/q^2$)

Let a disc contain fraction $f_d$ of total halo mass, take its edge gravity to be $v_d^2=C_{d}G{M}_d/R_d$, and equate its edge specific angular momentum to $q{R}_h{v}_{\mathrm{vir}}$. With $v_{\mathrm{vir}}^2=G{M}_h/R_h$, these conditions give $R_h/R_d=C_d{f}_d/q^2$. The structural coefficient $C_d$ is essential: disc mass alone does not uniquely determine its edge rotation curve.