substituted into the impact parameter $b=\!r_{\rm eq}/N_{\rm eq}$ and this in turn into Equation (\[photon-momentum\]) in order to find momenta of the two photons which hit the particle at the given location $(r,\theta)$.

A comparison of several approximations is presented in Fig. \[approximations\]. Meridional plots of three photon trajectories are shown there, as represented 1) by the exact formula (thick red curve), 2) by Beloborodov’s approximation, applicable above $4M$ (black dotted; it is included mainly as a benchmark), 3) by a suitably adjusted hyperbola (brown dashed), 4) by a pseudo-Newtonian result using the potential $V\!=\!-\frac{M}{r}\left(1+\frac{3M}{r}\right)$ (long-dashed blue), 5) by the result following from the above linearization in $M$ of the differential equation (dash-dotted; it is the worst approximation), and 6) by the formula $$\label{our-approximation}
  -\epsilon^\theta
  \sin\theta=\frac{r_{\rm eq}}{r}-
             \frac{M}{r_{\rm eq}-\alpha M}\,
             \frac{(r-r_{\rm eq})(2r+r_{\rm eq})}{(r-\omega M)^2}$$ (where $\alpha$ and $\omega$ are real constants) which we newly suggest and will specifically use with $\alpha=1.77$ and $\omega=1.45$ (thick green).[^5] The hyperbola (brown dashed) is chosen to have the same asymptotic latitude $\theta_\infty$ as the orbit provided by Beloborodov’s formula (dotted), namely given by $\sin\theta_\infty\!=\!\frac{2M}{r_{\rm eq}-2M}$. In the [*top plot*]{} of the figure, the photon starts from $r_{\rm eq}=4M$, where it is already quite hard 