Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 347 / 1 / b / Solution

Use cylindrical coordinates $(R,\varphi ,z)$, put $s=(R^2+z^2{)}^{1/2}$, and write the purely azimuthal velocity as $\mathbf{u}=R\Omega {\mathbf{e}}_{\varphi }$. The steady radial, azimuthal, and vertical momentum equations areThus $\nabla p=\rho {\mathbf{g}}_{\mathrm{eff}}$, where the effective gravity in a rotating frame isWhen radiation pressure supplies the pressure support, force balance gives

On the upper conical surface $z=R\tan \alpha$, the tangent and outward normal areBecause a pressure surface is an equipotential surface, ${\mathbf{g}}_{\mathrm{eff}}\cdot \mathbf{t}=0$. Since $s=R/\cos \alpha$, this fixesThe lower face gives the reflected result. One face has area element $dA=2\pi R,dR/\cos \alpha$. Adding both faces and using $F=c|{\mathbf{g}}_{\mathrm{eff}}|/\kappa$ yieldsConsequentlyFor a broad radial range, each logarithmic interval can radiate an Eddington-scale contribution because rotation reduces the surface-normal gravity. The factor $\sin \alpha$ makes a thin wedge faint and lets a geometrically thick funnel expose a larger effective gravity and emitting area.