=0 \;\; \Rightarrow \;\;
  b=\frac{r_{\rm eq}}{\sqrt{1-\frac{2M}{r_{\rm eq}}}} \;.$$

This radius-dependent constraint implies that only one of the two constants of the motion may be chosen to have the same value [*across*]{} all the rays, thus determining the other as a function of the initial radius. Regarding the supposed accretion-disc temperature profiles, it does not seem wise to endow all the photons with the same energy $E$, and also the resulting profile of ${\cal K}_{\rm eq}(r_{\rm eq})$ implied by the constraint is not very plausible. We will therefore fix the Carter constant ${\cal K}$ instead, which implies that the energy profile has to read $E_{\rm eq}/\sqrt{\cal K}=N_{\rm eq}/r_{\rm eq}$; it is illustrated in Fig. \[E-profile\]. Since real accretion discs are supposed to be considerably hotter at smaller radii, but this property is somewhat opposed by larger redshift there with respect to infinity, the profile seen in the figure seems to be a reasonable choice, in particular it properly goes to zero at the horizon. Rough as the ${\cal K}={\rm const}$ choice may seem, the corresponding energy profile in Fig. \[E-profile\] actually well resembles the temperature profiles obtained from standard models of thin discs — see, for example, [@BhattacharyyaTB-01] who compared, in 