A Leptonic Model of Steady High-Energy Gamma-Ray Emission from Sgr A$^*$

Recent observations of Sgr A$^*$ by Fermi and HESS have detected steady gamma-ray emission in the GeV and TeV bands. We present a new model to explain the GeV gamma-ray emission by inverse Compton scattering by nonthermal electrons supplied by the NIR/X-ray flares of Sgr A$^*$. The escaping electrons from the flare regions accumulate in a region with a size of $\sim 10^{18}$ cm and magnetic fields of $\lesssim 10^{-4}$ G. Those electrons produce gamma-rays by inverse Compton scattering off soft photons emitted by stars and dust around the central black hole. By fitting the GeV spectrum, we find constraints on the magnetic field and the energy density of optical-UV radiation in the central 1 pc region around the supermassive black hole. While the GeV spectrum is well fitted by our model, the TeV $\gamma$-rays, whose spectral index is different from that of the GeV emission, may be from different sources such as pulsar wind nebulae.