An Application of the Tensor Virial Theorem to Hole + Vortex + Bulge Systems

The tensor virial theorem for subsystems is formulated for threecomponent 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 the additional restrictions: (i) similar and similarly placed inner subsystems, and (ii) spherical outer subsystem. An application is made to hole + vortex + bulge systems, in the limit of flattened inner subsystems, which implies three virial equations in three unknowns. Using the Faber-Jackson relation, Re ∝ σ 0 , the standard MH-σ0 form (MH ∝ σ 0) is deduced from qualitative considerations. The projected bulge velocity dispersion to projected vortex velocity ratio, η = (σB)33/{[(vV)qq] + [(σV)qq]}, as a function of the fractional radius, yBV = RB/RV, and the fractional masses, mBH = MB/MH and mBV = MB/MV, is studied in the range of interest, 0 ≤ mVH = MV/MH ≤ 5 (Escala, 2006) and 229 ≤ mBH ≤ 795 (Marconi and Hunt, 2003), consistent with observations. The related curves appear to be similar to Maxwell velocity distributions, which implies a fixed value of η below the maximum corresponds to two different configurations: a compact bulge on the left of the maximum, and an extended bulge on the right. All curves lie very close one to the other on the left of the maximum, and parallel ∗Astronomy Department, Padua Univ., Vicolo Osservatorio 2, I-35122 Padova, Italy email: [email protected]