The detail you seek is contained in Johansson (2014).

The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius $3r_s/2$ for a non-spinning black hole; which is then projected to $3\sqrt{3}r_s/2$ at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit does depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches its maximal value is appreciable non-circularity expected.

Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.

In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. Bambi et al. 2010).

The detail you seek is contained in Johansson (2014).

The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius $3r_s/2$ for a non-spinning black hole; which is then projected to $3\sqrt{3}r_s/2$ at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit does depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches its maximal value is appreciable non-circularity expected.

Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term (that differs from the one implicit to the Kerr metric) to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.

In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. Bambi et al. 2010).

The detail you seek is contained in Johansson (2014).

The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius $3r_s/2$ for a non-spinning black hole; which is then projected to $3\sqrt{3}r_s/2$ at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit does depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches it's maximal value is appreciable non-circularity expected.

Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.

In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. Bambi et al. 2010).

The detail you seek is contained in Johansson (2014).

The photon ring around a black hole is not the event horizon. It is the projection of unstable photon orbits that are able to loop around the black hole more than once before heading in our direction. This ring is circular and of radius $3r_s/2$ for a non-spinning black hole; which is then projected to $3\sqrt{3}r_s/2$ at the observer. The situation for spinning black holes is much more complex. The radius of the unstable photon orbit does depend on the spin, but the projection of this to the observer maintains almost the same radius as in the non-spinning case. The rings are predicted to be almost circular, but off-centre. Only when the spin approaches its maximal value is appreciable non-circularity expected.

Johansson also discusses the no-hair theorem and how violations of this would affect unstable circular photon orbits and the projected photon ring. These are investigated in a parametric way - e.g. by introducing a quadrupole moment term to the metric, in addition to the spin. These calculations show that the introduction of these additional terms in the metric introduce changes in the projected photon ring radius and can make it asymmetric - much more so than spin alone.

In terms of motivation, it seems that alternate gravity theories (to General Relativity) admit the possibility of spacetime metrics for spinning objects that depend on more than just their mass and spin (e.g. Bambi et al. 2010).