Testing SgrA$^*$ with the spectrum of its accretion structure

SgrA$^*$ is the supermassive black hole candidate at the center of the Galaxy and an ideal laboratory to test general relativity. Following previous work by other authors, we use the Polish doughnut model to describe an optically thin and constant angular momentum ion torus in hydrodynamical equilibrium and model the accretion structure around SgrA$^*$. The radiation mechanisms are bremsstrahlung, synchrotron emission, and inverse Compton scattering. We compute the spectrum as seen by a distant observer in Kerr and non-Kerr spacetimes and we study how an accurate measurement can constrain possible deviations form the Kerr solution. As in the case of emission from a thin accretion disk, we find a substantial degeneracy between the determination of the spin and of possible deviations from the Kerr geometry, even when the parameters of the ion torus are fixed. This means that this technique cannot independently test the nature of SgrA$^*$ even in the presence of good data and with the systematics under control. However, it might do it in combination with other measurements (black hole shadow, radio pulsar, etc.).