Environmental Engineering Reference
In-Depth Information
Figure 3. The VPE as a function of the fractional scaling radius, Β
=
R
h
/R
∗
, for HH
models (left panel) and HP models (right panel), normalized to a total energy value, E
=
−1
, in the
energy-conservation paradigm
. Curves refers to (from bottom to top)
R=
10
,
13
,
16
,
20
,
30
, for HH models, and
R= 5
,
7
,
10
,
15
,
20
for HP models, respectively.
properties of VPE extremum points is still justified, in that maxima and minima in potential
energy might represent special configurations for the system under consideration.
Extremum points of VPE isoenergetic curves described by Eqs.(16) and (17), are se-
lected in the following way:
∂
∂Β
F
(HH)
∗
[R
(vir)
, Β
(vir)
] = 0
(18)
∂
∂Β
F
(HP
∗
[R
(vir)
, Β
(vir)
] = 0
(19)
which provides a relation between the fractional mass,
R
, and the scaling fractional radius,
Β, as:
R
(HH)
F
(HH)
(
=
Β
)
(20)
R
(HP )
F
(HP )
(
=
Β
)
(21)
For the sake of simplicity, in the following sections, only the maxima will be analyzed,
as all considerations and conclusions apply equivalently to the minima; accordingly, from
here on, the tilde shall denote a quantity calculated only on the VPE maximum, and con-
figurations corresponding to a VPE maximum shall be quoted as MVPE configurations
(i.e., Maximum Virial Potential Energy configurations). The plot of Eqs.(20) and (21) is
presented in Fig. 4 (top panel).
The DM to stellar mass ratio inside the stellar scaling radius,
R
∗
, is:
R
(HH)
(1 +
4
M
(HH)
(
Β
)
=
(22)
Β
)
2
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