Image of a spherical black hole with thin accretion disk

Luminet, J.-P. (1979). Image of a spherical black hole with thin accretion disk. Astronomy and Astrophysics, 75(1-2), 228–235.

Image of a Bare Black Hole

The Schwarzschild metric is
$\mathrm{d}s^2=-\left(1-\frac{2M}{r}\right)\mathrm{d}t^2+\left(1-\frac{2M}{r}\right)^{-1}\mathrm{d}r^2+r^2(\mathrm{d}\theta^2+\sin^2\theta\mathrm{d}\phi^2)$
the unit system is chosen such that $G=c=1$ .
The Schwarzschild radius $r_s=2M$ .
The trajectories of photon in the “equatorial” plane ( $\theta=\pi/2$ ) satisfy
$\left(\frac{1}{r^2}\frac{\mathrm{d}r}{\mathrm{d}\phi}\right)^2+\frac{1}{r^2}\left(1-\frac{2M}{r}\right)=\frac{1}{b^2}$
where $b=L/E$ is the impact parameter at infinity.

Fig.1

Cirtical impact parameter $b_c=3\sqrt{3}M$ , for $b<b_c$ , rays are captured by the hole.
The total deviation of the light ray $\mu$
$\mu=2\phi_\infty-\pi$ $\phi_\infty=2\sqrt{\frac{P}{Q}}\int_{\zeta_\infty}^{\pi/2}(1-k^2\sin^2 x)^{-1/2}\mathrm{d}x$

Fig.3

For a given emitter $M$ , the observer will detect two images:
a direct (or primary) image at $(b^{(\text{d})},\alpha)$
a ghost (or secondary) image at $(b^{(\text{g})},\alpha+\pi)$
The isoradial curve:
Given $P$ (periastron)
$\cos\gamma=\cos\alpha(\cos^2\alpha+\cot^2\theta_0)^{-1/2}\tag{10}$ $\frac{1}{r}=-\frac{Q-P+2M}{4MP}+\frac{Q-P+6M}{4MP}\text{sn}^2\left[\frac{\gamma}{2}\sqrt{\frac{P}{Q}}+F(\zeta_\infty,k)\right]\tag{13}$
$r=r(\gamma,P)=r(\alpha,P)\Rightarrow P=P(r,\alpha)$
For a given angle $\theta_0$
$b=\frac{P^3}{P-2M}\tag{5}$
$b^{(\text{d})}=b^{(\text{d})}(P)=b^{(\text{d})}(r,\alpha)$
For a constant $r$ from the hole, the isoradial curves we can see are $b^{(\text{d})}=b^{(\text{d})}(\alpha)$ .