Environmental Engineering Reference
In-Depth Information
major and minor merging, etc., which will finally lead to a (not necessarily virialized) state
where gas has been exhausted by star formation and energy dissipation may safely be ne-
glected.
In this view, energy conservation holds from an initial dissipationless configuration to
the virialized state
2
, which implies, according to the virial theorem, the total energy E is
half the potential energy at the end of the evolution. More specifically, the following relation
holds:
GM
∗
2
R
∗(vir)
F
E
vir
=
R
(vir)
, Β
(vir)
=
E
(14)
where F
[
...
]
represents a function of the variables inside brackets which are, in general,
different for different models. In this way, it is always possible to express the stellar scaling
radius, R
∗
in terms of the total energy, E, and a function of only Β and
R
via Eq.(14), as:
GM
2
∗
E
R
∗(vir)
=
R
(vir)
, Β
(vir)
F
(15)
Substituting R
∗
in Eqs.(8) and (9), the stellar VPE in the
energy-conservation paradigm
,
takes the expression:
V
(HH)
EF
(HH)
∗
=
R
(vir)
, Β
(vir)
(16)
∗
V
(HP )
EF
(HP )
∗
=
R
(vir)
, Β
(vir)
(17)
∗
for HH and HP models, respectively, and both the functions F
∗
, are presented in the Ap-
pendix. The
energy-conservation paradigm
reduces the VPE physical space to two dimen-
sions only. Topologically, the VPE surface is intersected by an iso-energy plane, yielding a
curve with varying R
∗
and R
h
, but constant total energy. The dependence on E, exhibited by
Eqs. (16) and (17), can be neglected in that it only acts as a scaling parameter, which means
extremum points depend only on
R
(vir)
and Β
(vir)
, for a selected model.
This holds true also for complicated evolution of the initial assembly phases of the
proto-galaxy; in fact, even if the mass density profiles are going through many changes, the
presence or the absence of extremum points depend only on the final relaxed state of the
galaxy, owing to the onset of energy conservation due to dissipationless evolution.
3.1.
Analysis of Extremum Points
As shown in Fig. 3, the
energy-conservation paradigm
yields isoenergetic VPE curves
where extremum points occur above a threshold in
R
. Accordingly, for a fixed value of
R
, there is only a single configuration of HH or HP models maximizing the VPE. On
the other hand, extremum points occur only above a relatively high threshold,
R∼10
for HP models, and
R∼20
for HH models. Even if values above
R∼10
seem to be
in contradiction with data from observations (e.g., Cappellari et al., 2006), analyzing the
2
This process is perfectly described by
N
-Body simulations of pure gravitational collapse of two component
dark and visible matter subsystems. The total initial energy is approximately conserved (dissipation is caused
only by blown off particles) and the final virialized state is obviously strictly connected to the initial energy of
the system which, in turn, depends on the dissipative history formation of the spheroid.
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