Apparent angular radius of a spherical star in Schwarzschild spacetime ($\sin \psi =\frac{R}{r_0}\sqrt{\frac{1-2m/r_0}{1-2m/R}}$)

A spherical surface with areal radius $R>3m$, where $m=GM/c^2$, has apparent angular radius $\psi$ for a static exterior observer at $r_0>R$. Its grazing null geodesic has impact parameter $b=R/\sqrt{1-2m/R}$. Projecting the photon tangent into the observer's local rest frame gives $\sin \psi =b\sqrt{1-2m/r_0}/r_0$, yielding the displayed result. The condition $R>3m$ ensures the grazing ray escapes, since the surface is outside the photon sphere. For $R=4m$, the distant image radius is $\sqrt{2}$ times the flat-space prediction.