 Standard thin-disk luminosity

For the steady zero-inner-torque disk, the standard thin-disk dissipation flux already includes both faces. Integrate it over projected annuli, rather than multiplying by a second factor of two:This is the decrease in orbital energy per unit time as matter arrives from arbitrarily large radius on a circular orbit. Half the decrease in gravitational potential energy remains as orbital kinetic energy at the inner edge. Accretion onto a slowly rotating stellar surface can release additional energy in a boundary layer; a black-hole inner boundary requires the appropriate relativistic treatment.^[ref]

For a finite outer radius $R_{\mathrm{out}}$, integration of this same specified flux gives insteadIt tends to zero as $R_{\mathrm{out}}\downarrow R_{\mathrm{in}}$ and to $L_{\mathrm{thin}}$ as $R_{\mathrm{out}}\to \infty$. This finite-annulus result inherits the same steady torque profile; it does not impose an additional zero-torque outer boundary.