TIDAL ENERGY EFFECTS OF DARK MATTER HALOS ON EARLY-TYPE GALAXIES - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 321

R (HP )
(1 +

4

M (HP ) (

Β ) = (23)

Β 2 ) 3/2

for HH and HP models, respectively. The plot of Eqs.(22) and (23) is presented in Fig.4
(bottom panel). It can be seen that that fractional masses

M of MVPE configurations with

R ∗ < R h / 5 (i.e., Β > 5 ) , and the related trend, are consistent with observations: in fact

larger Β (see, e.g., Cappellari et al., 2006).

The energy conservation paradigm makes a simple but very powerful tool to univocally
define MVPE configurations for two-component models of ETGs and their hosting halos.
The whole set of MVPE configurations, (

M values occur for larger R ∗ , or lower

Β ) , can be determined for any assigned model
with specified density profile for each subsystem. As soon as the

R ,

R value is fixed, a MVPE
configuration is defined and model predictions can be compared with results deduced from
observations.

3.2. On the Restriction of Energy Conservation

Though dissipative processes have a central role in galaxy formation, the energy-
conservation paradigm is a useful tool in dealing with the late evolution, where energy
conservation (in absence of merger events) holds to a first extent.
Imagine a cold gas collapsing into the potential well of a massive halo. The proto-
galaxy undergoes all sorts of dissipative processes till the main bulk of stars is produced,
and the system becomes globally dissipationless with total energy, E. Assuming energy
conservation, implies:

E = E vir (24)

Let E o , ∆ E, be the total energy at the beginning of evolution and the change up to the
(fiducial) end of the dissipative phase, respectively. In this view, Eq. (24) takes the more
general form:

E o −∆ E = E = E vir (25)

on the other hand, according to Eqs. (20) and (21), MVPE configurations are defined by a
selected model via R (vir) , regardless of the total energy, E, which acts as a mere scaling

parameter.
In conclusion, energy conservation may be considered in a more general scenario,
where energy dissipation occurs but the energy change, ∆ E, is supposed to be known, and
the special case, ∆ E = 0 , corresponds to energy conservation during the whole evolution.

4. Linking Tidal Potential Energy with Observations

It can be shown that the application of the virial theorem to two-component systems
yields (see, Ciotti et al., 1996):

M ∗ = C ∗ 2 R e Σ o (26)