An application of the tensor virial theorem to hole + vortex + bulge   systems
  The tensor virial theorem for subsystems is formulated for three-component systems and further effort is devoted to a special case where the inner subsystems and the central region of the outer one are homogeneous, the last surrounded by an isothermal homeoid. The virial equations are explicitly written under additional restrictions. An application is made to hole + vortex + bulge systems, in the limit of flattened inner subsystems. Using the Faber-Jackson relation, the standard $M_{\rm H}$-$\sigma_0$ form is deduced from qualitative considerations. The projected bulge velocity dispersion to projected vortex velocity ratio, $\eta$, as a function of the fractional radius, y_{\rm BV}, and the fractional masses, $m_{\rm BH}$, and $m_{\rm VH}$, is plotted for several cases. It is shown that a fixed value of $\eta$ below the maximum corresponds to two different configurations: a compact bulge on the left and an extended bulge on the right. In addition, for fixed $m_{\rm BH}$ or $m_{\rm BV}$, and $y_{\rm BV}$, more massive bulges are related to larger $\eta$ and vice versa. The model is applied to NGC 4374 and NGC 4486, and the bulge mass is inferred and compared with results from different methods. In presence of a massive vortex $(m_{\rm VH}=5)$, the hole mass has to be reduced by a factor 2-3 with respect to the case of a massless vortex, to get the fit. 
R. Caimmi