Optical Appearance of Thin Accretion Disks around Quadrupole-Deformed Compact Objects

Abstract

We study the apparent shape of geometrically thin accretion-disk rings around compact objects described by the q-metric. The spacetime is a static and axially symmetric deformation of the Schwarzschild geometry, where the dimensionless parameter  controls the quadrupole contribution to the gravitational field. Photon motion is written in Hamiltonian form and integrated by backward ray tracing from a distant observer to the equatorial disk plane. The images are constructed for four fixed emission radii and are compared with the Schwarzschild limit. For , the calculation reproduces the expected ordering of iso-radial curves for an inclined Schwarzschild disk. For the same observer position and the same coordinate emission radii, positive deformation makes the apparent curves slightly larger, while negative deformation makes them more compact. The change is most visible for the inner disk ring, because the corresponding photons pass closer to the compact object. The results show that even a moderate quadrupole deformation can leave a visible geometrical imprint on thin-disk images.