The potential-energy tensors for subsystems. IV. Homeoidally striated density profiles with a central cusp

A general theory of homeoidally striated density profiles where no divergence occurs, is adapted to cuspy density profiles, with a suitable choice of the scaling density and the scaling radius. A general formulation of some physical parameters, such as angular-momentum vector, rotational-energy tensor (both calculated in connection with a special class of rotational velocity fields), inertia tensor, and self potential-energy tensor, is performed. Other potential-energy tensors involving two density profiles where the boundaries are similar and similarly placed, are also expressed. Explicit results are attained for three special cases of physical interest: Navarro et al. (NFW) (e.g., [ApJ 490 (1997) 493]) and Moore et al. (MOA) (e.g., [MNRAS 310 (1999) 1147]) density profiles, which fit to a good extent the results of high-resolution simulations for dark matter haloes, and Hernquist (H) [ApJ 356 (1990) 359] density profiles, which closely approximate the de Vaucouleurs r1/4 law for elliptical galaxies. The virial theorem in tensor form for two-component systems is written for each subsystem, and applied to giant elliptical galaxies. The predicted velocity dispersion along the line of sight, in the limiting case where a principal axis points towards the observer, is found to be consistent with observations except for (intrinsic) E7 configurations where the major axis points towards the observer. If dark matter haloes host an amount of undetected baryons about twice as massive as the stellar subsystem, and undetected baryons trace non baryonic matter therein, two main consequences arise, namely: (i) velocity dispersions along the line of sight are lower than in absence of undetected baryons, and (ii) dark matter haloes are dynamically ‘hotter’ than stellar ellipsoids, the transition occurring when the amount of undetected baryons is about one and a half times that of the stellar subsystem. In this view, both the observation that the temperature of the extended hot gas exceeds the central stellar temperature, and the fact that the non baryonic matter is dynamically ‘hotter’ than the stars, are a reflection of the presence of undetected baryons, which trace the dark halo and are about twice as massive as the stellar ellipsoid.