The spectra of the gamma-rays from two-temperature advection-dominated accretion flows (ADAFs) around Kerr black holes are investigated. We examine the dependence of the gamma-ray spectrum on the black hole rotation. In the vicinity of a black hole, the ion temperature of an ADAF is so high that gamma-rays are produced through the decay of neutral pions which are created by proton–proton collisions. Because these gamma-rays are radiated from the region close to the black hole, their intensity depends on its spin parameter.
As the mechanism of the viscous heating is not well understood, we consider two different proton energy distributions: a relativistic Maxwell–Boltzmann distribution (thermal model) and a power-law distribution (non-thermal model). When the spin parameter is varied from −0.95 to 0.95, the gamma-ray intensity increases by orders of magnitude in the thermal model. Thus, if the proton gas in an ADAF has a thermal distribution, we can consider the gamma-ray spectrum as a probe to investigate the spin parameter of the central black hole. On the other hand, the gamma-ray intensity in the non-thermal model is much less sensitive to the changes in spin parameter than in the thermal model.