given explicitly here, the only simplification occurring thanks to the zero azimuthal motion of photons, $p_\phi=0$. Note, however, that although both the gravitational and radiation fields are axially symmetric ($T_{\mu\phi}=0$), the force does contain a nonzero azimuthal component if the particle’s velocity has some, because of the projection term $u^\phi\,T_{\nu\lambda}u^\nu u^\lambda$.

![image](approximations.eps){width="\textwidth"}

Approximating the photon trajectories {#approximation}
=====================================

In order to evaluate the effect of the photon flux on the test particle at a given point $(r,\theta)$, one would have to solve equation (\[theta(r)\]) for $r_{\rm eq}$ and find the photon energy $E=\sqrt{K}/b$ or the impact parameter $b$ (and thus the momentum of the incoming photon) there. Then one would have to determine $\Phi^2(r,\theta)$ at that point by solving the continuity equation (\[Div(Phi2.p)=0\]) with a prescribed “velocity" $p^\mu(r,\theta)$. If we do not want to resort to pure numerics to accomplish these two steps, we can consider trying analytic approximations.

A natural approach is to linearize the problem in some small parameter. We have $b>r_{\rm eq}$ and, restricting to the astrophysically relevant case $r_{\rm eq}>3M$, also $r\geq r_0>3M$, while $r$ and $b$ are less clearly related: all the photons start from $r<b$, but quickly get to $r>b$ and then even to 