from Nemiroff, R. J., American Journal of Physics, 61, 619 (1993)

## APPENDIX

What is the apparent angular size of a sphere of mass M (corresponding to
Schwarzschild radius R_S), radius R_*, at distance D? This extremely
simple and beautiful problem is important to observable aspects of high
gravity environments.
The impact parameter at infinity, here labeled b, is a constant along the
trajectory of the photon. [27] At any radius r from a sphere with mass
M with Schwarzschild R_S on the photon's trajectory, this constant is
equal to

where delta is the angle the photon's velocity makes with the radial
direction.
For a lens large enough not to exhibit a photon sphere (R_* > 1.5 R_S),
the limb of the source will be seen when a photon leaving the source
tangentially grazing its surface, such that r = R_* and delta = pi/2.
Then

This photon will reach the observer with $r = D$ and angle to the center of
the lens of delta_obs such that

Equating Eqs. (A2) and (A3) and solving for delta_obs yields

which is the apparent size of the lens.

When the lens is compact enough to be surrounded by a photon sphere, one
can simply replace R_* in the above expression with 1.5 R_S, as these
photons now define the apparent size of the lens. The result is

Be careful to assign the correct quadrant to the result. If R_* < 1.5
R_S, then delta_obs will be greater than pi/2.

References

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