TIDAL ENERGY EFFECTS OF DARK MATTER HALOS ON EARLY-TYPE GALAXIES - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 317

Figure 2. The plot of the VPE for HP models normalized to the stellar component (top
panel, Ω N∗ ≡ GM ∗ 2 /R ∗ , R increases from left to right) and to the dark component (bottom

panel, Ω Nh ≡ GM h 2 /R h , R increases from bottom to top) for 1≤R≤10 , and steps of
1 . In the former case no extremum points are present, while in the latter they appear for
R > 4 . 3 . This dichotomy is a consequence of the path dependence of the extremum points
in the VPE 3D hypersurface.

has no solution, meaning that the 3D VPE hypersurface exhibits no extremum point. On
the other hand, extremum points can take place along selected peculiar paths on the VPE
hypersurface. Their occurrence is strongly depending on the definition of the path; for
example, a path along the R ∗ axis of Fig. 1 at fixed R h (i.e. the case of a frozen halo) implies
extremum points above a threshold, R∼10 ; on the contrary a path along the R h axis at fixed
R ∗ yields no extremum point regardless of R value, as sketched in Fig. 2. The occurrence
of extremum points, for shallow density profiles in the limit of a frozen DM halo, has been
widely investigated in earlier attempts (see, Secco, 2000, 2001).

3. The Energy-Conservation Paradigm

The importance of focusing on a precise path along the VPE surface, to univocally
determine the presence of extremum points has been stressed. One of the most interesting
physically meaningful paths to be chosen is dictated by energy conservation. Accordingly,
a fixed relation between R , Β and R ∗ must be taken into consideration. The path definition
related to energy conservation, together with the properties exhibited by the corresponding
family of curves, in particular the occurrence of extremum points, shall be quoted as the
energy-conservation paradigm .
Consider the evolution of a forming galaxy from an initial total energy E o state (i.e.
the maximum expansion state) till the final virialization E vir state. The collapsing spheroid
will go through complex processes of gas dissipation, supernovae and black hole feedback,