four-momentum components given by $$\label{photon-momentum}
  p^t=\frac{\sqrt{\cal K}}{N^2 b} \,, \quad
  p^r=\epsilon^r\,\frac{\sqrt{\cal K}}{r}\,\sqrt{\frac{r^2}{b^2}-N^2} \;, \quad
  p^\theta=\epsilon^\theta\,\frac{\sqrt{\cal K}}{r^2} \;,$$ where $b\equiv\sqrt{\cal K}/E$ denotes their impact parameter, $E=-p_t$ is their energy at infinity and ${\cal K}=(p_\theta)^2$ is their “Carter constant", all remaining conserved along the rays; the signs $\epsilon^r\equiv\pm 1$ and $\epsilon^\theta\equiv\pm 1$ fix the orientation of the meridional-motion components. At each location above the circular photon orbit at $r=3M$, this “null dust" constitutes an outward flux ($\epsilon^r\!=\!+1$) which would admittedly drag any test particle along. Besides the shape of the photon trajectories, the angular distribution of the flux and thus of the particle acceleration/deceleration—in particular, the eventuality that the particles might be driven into a collimated outflow—depend on the “luminosity profile" fixed in the equatorial plane, namely on a chosen equatorial radial profile of the constants of the motion $E_{\rm eq}=E(r_{\rm eq},\theta\!=\!\pi/2)$ and ${\cal K}_{\rm eq}={\cal K}(r_{\rm eq},\theta\!=\!\pi/2)$ that the photons are endowed with, but mainly on energy density of the flux determined by conservation laws. The constants are constrained by the requirement that the rays depart orthogonally from the equatorial plane, namely by the condition $p^r=0$ there, which takes the form $$\label{EK-constraint}
  \frac{E_{\rm eq}^2}{{\cal K}_{\rm eq}}\,r^2_{\rm eq}-1+\frac{2M}{r_{\rm eq}}
  \equiv\frac{r_{\rm eq}^2}{b^2}-N^2_{\rm eq}
  