from the well known cases of Paczyński and Wiita, $V\!=\!-\frac{M}{r-2M}\,$ (also used in some of the papers cited in the Introduction), or Nowak and Wagoner, $V\!=\!-\frac{M}{r}\left(1-\frac{3M}{r}+\frac{12M^2}{r^2}\right)$; see e.g., the form $V\!=\!-\frac{M}{r}\left(1+\frac{\alpha M}{r}\right)$, with constant $\alpha$, advocated by [@Wegg-12] recently (specifically with $\alpha\!=\!3$) which has also proven quite satisfactory in our photon-motion problem.

Actually, it is possible to design a number of rather accurate approximations of the photon trajectories. However, the photon motion is not the full story here: we also have to employ its description (inverted for $r_{\rm eq}$, and thus for $b$) in the continuity equation (\[equation-for-Phi\]) and then solve the latter for the flux density $\Phi^2$. Although a chosen approximation may allow for a tractable inversion, this often makes the continuity equation too difficult to solve, even after linearization in $M$.

One approximation which leads to a solvable form of the continuity equation is reached by the usual linearization in $M$. [^4] Linearizing thus Darwin’s formula (\[theta(r)\]) gives $$\label{sin(theta)=lin}
  -\epsilon^\theta\sin\theta=
  \frac{r_{\rm eq}}{r}-\frac{M}{r}\,\frac{(r-r_{\rm eq})(2r+r_{\rm eq})}{r_{\rm eq}\,r}
  +O(M^2)$$ which inverts to $$\label{req,no-linearization}
  \frac{2r_{\rm eq}}{r}=
  \frac{-\epsilon^\theta r\sin\theta\!-\!M\!+\!\sqrt{(\epsilon^\theta r\sin\theta\!+\!M)^2\!+\!8M(r\!+\!M)}}{r+M} \,.$$ Note that on the axis the latter yields $$r_{\rm eq}(\theta\!=\!0)=\frac{4Mr}{M\!+\!\sqrt{M(8r\!+\!9M)}}
                          =\sqrt{2Mr}-\frac{M}{2}+O(M^{3/2}) \,.$$ These equatorial radii $r_{\rm eq}(\epsilon^\theta=-1)$, $r_{\rm eq}(\epsilon^\theta=+1)$ should then be 