to mimic the exact result by any low-order formula; however, our approximation even there practically coincides with the exact curve. In the [*bottom left plot*]{}, the photon starts from $r_{\rm eq}=7M$; Beloborodov’s and our approximation are almost indistinguishable from the exact ray. At larger radii the approximations gradually coalesce with the exact curve and nothing interesting happens (only the pseudo-Newtonian result gets worse), so we do not show any photon starting from the more remote, weak-field region. In the [*bottom right plot*]{}, the approximate formulas are subjected to a very tough situation of a photon starting from $r_{\rm eq}=2.5M$. Surprisingly enough, none of them yields a totally unacceptable result (apart from the linearization in $M$ and from Beloborodov’s formula which is, however, not applicable below $4M$, so it is not present), the pseudo-Newtonian (blue) curve is even very close to the exact one and shares its black-hole destiny. Better approximations can be found, but typically they must be of higher order, so usually not invertible for $r_{\rm eq}$ and leading to a rather difficult continuity equation.

![Meridional-plane $(r\sin\theta,r\cos\theta)$ plots of two different solutions for the photon-flux density $\Phi^2(r,\theta)$: the more concentrated photon flux given by $\kappa=32$, $\lambda=2$ ([*top two plots*]{}) 