A possible disk mechanism for the 23d QPO in Mkn~501

Optically thin two-temperature accretion flows may be thermally and viscously stable, but acoustically unstable. Here we propose that the O-mode instability of a cooling-dominated optically thin two-temperature inner disk may explain the 23-day quasi-periodic oscillation (QPO) period observed in the TeV and X-ray light curves of Mkn~501 during its 1997 high state. In our model the relativistic jet electrons Compton upscatter the disk soft X-ray photons to TeV energies, so that the instability-driven X-ray periodicity will lead to a corresponding quasi-periodicity in the TeV light curve and produce correlated variability. We analyse the dependence of the instability-driven quasi-periodicity on the mass (M) of the central black hole, the accretion rate ($\rm{\dot{M}}$) and the viscous parameter ($\alpha$) of the inner disk. We show that in the case of Mkn~501 the first two parameters are constrained by various observational results, so that for the instability occurring within a two-temperature disk where $\alpha=0.05-1.0$, the quasi-period is expected to lie within the range of 8 to 100 days, as indeed the case. In particular, for the observed 23-day QPO period our model implies a viscosity coefficient $\alpha \leq 0.28$, a sub-Eddington accretion rate $\dot{M} \simeq 0.02 \dot{M}_{\rm Edd}$ and a transition radius to the outer standard disk of $r_0 \sim 60 r_g$, and predicts a period variation $\delta P/P \sim 0.23$ due to the motion of the instability region.