TIDAL ENERGY EFFECTS OF DARK MATTER HALOS ON EARLY-TYPE GALAXIES - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 315

where M h is the total mass, R h is a scaling radius, Β ≡ R h /R ∗ , R≡ M h /M ∗ , and the index,
h, denotes the DM halo. Accordingly, HH and HP models result by use of Eqs. (1), (2), and
(1), (3), respectively. For a selected stellar profile, HH and HP models represent limiting
situations of the largest and the lowest tidal effect (for assigned DM halo mass), inside the
effective radius of the stellar subsystem, R e , respectively.
As described in earlier attempts (Limber, 1959; Brosche et al., 1983), the fundamental
quantities involved in the virial theorem for two-component systems, are the self potential
energy of each component:

∞

Ρ u ( R ) Φ u ( R ) R 2 d R, Ω u = 2 Π U =∗ , H (4)

0

the interaction potential energy:

∞

Ρ u ( R ) Φ v ( R ) R 2 d R, W uv = 2 Π U =∗ , H, V = H, ∗ (5)

0

and the virial of external forces, or tidal potential energy:

∞

Ρ u ( R ) d Φ v ( R )
d R

R 3 d R, V uv =−4 Π U =∗ , H, V = H, ∗ (6)

0

where Φ is the gravitational potential, and spherical symmetry has been assumed. The
interaction potential energy, W uv , and the tidal potential energy, V uv , are not, in general,
equal 1 .
The virial theorem for each component can be written as:

V u = Ω u + V uv = 2 T u , U =∗ , H, V = H, ∗ (7)

where V represents the VPE. In one-component systems, it is usually named “the virial of
the system” (Clausius, 1870). In two-component models, the “system” is the component
under investigation subjected to the tidal action of the other one, in other words Eq.(7)
represents the VPE of the U-subsystem affected by the tidal potential of the V-subsystem.
Comparing model results with observations requires the particularization of Eq.(7) to
the real case of interest (i.e., U =∗ , V = H) for HH and HP models:

1
6

V (HH) +R V (HH) ∗ = Ω ∗ + V ∗h =−Ω N ( Β ) (8)

∗h

1
6

V (HP ) +R V (HP ) ∗ = Ω ∗ + V ∗h =−Ω N ( Β ) (9)

∗h

GM ∗ 2 /R ∗ Ω N = (10)

where the explicit expressions of V (HH) and V (HP )

∗h can be found in the Appendix.
In general, the VPE is a function of 4 independent parameters, meaning that it is defined
in a 5 dimensional hyperspace. The dependence on M ∗ can be neglected in that it only acts
as a scaling parameter, which means:

∗h

V ∗ = F 4 ( M ∗ , R ∗ , Β, R) = M ∗ F 3 ( R ∗ , Β, R) (11)

1 It can be seen that in HH and HP models these quantities are equal in the limit of Β → 1 , even if the mass
density profiles of the two components are different.