their figure 7, the temperature profiles for discs around a Newtonian centre, around a Schwarzschild black hole and around neutron- or strange-star models employing different equations of state; useful plots were also presented by [@Perez-etal-13], showing the temperature profiles of Shakura-Sunyaev and Novikov-Thorne thin discs around a Schwarzschild centre in general relativity and in simple $f(R)$ theories (see their figures 10 and 14, in particular).

Now, taking ratios of Eqs. (\[photon-momentum\]) with $\epsilon^r=+1$ leads to $$\label{dtheta/dr}
  \frac{{\rm d}\theta}{{\rm d}r}
  =\frac{\epsilon^\theta}{r}\,\frac{1}{\sqrt{\frac{E^2}{\cal K}\,r^2-N^2}}
  =\frac{\epsilon^\theta}{\sqrt{r}}\,\frac{1}{\sqrt{{r^3}/{b^2}-r+2M}}$$ which can be further expressed in terms of the initial equatorial radius by substituting $E_{\rm eq}=\sqrt{\cal K}\,N_{\rm eq}/r_{\rm eq}$ for $E$ to obtain $$\begin{aligned}
  \frac{{\rm d}\theta}{{\rm d}r}
  &= \frac{\epsilon^\theta\,\frac{r_{\rm eq}}{r}}
          {\sqrt{N_{\rm eq}^2 r^2-N^2 r_{\rm eq}^2}} \nonumber \\
  &=\frac{\epsilon^\theta\,r_{\rm eq}^{3/2}}
         {\sqrt{r(r-r_{\rm eq})}\;
          \sqrt{r(r+r_{\rm eq})(r_{\rm eq}-2M)-2Mr_{\rm eq}^2}}
    \;.  \label{dtheta/dr,req}\end{aligned}$$

![The energy profile $E=-p_t$ (expressed in units of $\sqrt{\cal K}/M$) of photons shot perpendicularly from the equatorial plane as a function of the initial equatorial radius $r_{\rm eq}$, for the case when ${\cal K}$ is chosen to be independent of $r_{\rm eq}$. This profile is just the reciprocal of the well known effective potential for free radial motion of massless particles in the Schwarzschild space-time. It starts from zero at $r_{\rm 