Environmental Engineering Reference
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R
as a function of C
∗
2
(left panel), and as a function of
Figure 5. Plot of
R
∗
(right
panel), for HH (full lines) and HP (dashed lines) models. Values of C
∗
2
within the shaded
region, correspond to MVPE configurations with unreasonably high
Β
=
R
h
/
R
>25 for
HH models). Models related to DM density profiles with inner slope between their HP
and HH counterparts should, in principle, lie within the shaded region and exhibit lower
R
values (
R
values.
homology (see, e.g., Bertin et al., 2002; Trujillo et al., 2004) of the structure of ETGs, that
is the deviation of the light profiles of the galaxies from the simple de Vaucouleurs R
1/4
law; 3) the Initial Mass Function and Star Formation (see, e.g., Renzini and Ciotti, 1993;
Chiosi et al., 1998), that is the variation of the M/L ratio with the ETG mass/luminosity;
4) the Dark Matter (see, e.g., Ciotti et al., 1996; Borriello and Salucci, 2001), that is a trend
of the tilt with the dark-to-bright mass ratio; 5) the existence of a maximum in the Clausius
potential energy of ETGs (see, Secco, 2000, 2001, 2005). Up to nowadays none of these
explanations for the FP tilt have been definitely confirmed. In particular, the last attempts
(Secco, 2000, 2001, 2005) deal with MVPE configurations but under the restriction of a
frozen halo i.e. fixed R
h
(see Fig. 1) regardless of energy conservation, and using different
scaling laws with respect to the current investigation.
In what follows, the FP tilt of ETGs shall be considered in the light of the
energy-
conservation paradigm
. More specifically, MVPE configurations shall be placed on the 3D
observational parameter space, (R
e
, Σ
o
, I
e
). To get a physical and immediate overview of
the problem, the Κ-plane of Bender et al. (1992) shall be used:
√
(log
Σ
o
+ log
R
e
)
/
=
2∼
M
dyn
Κ
1
(31)
√
(log
Σ
o
+ 2 log
I
e
−log
R
e
)
/
Κ
2
=
6
(32)
√
(log
Σ
o
−log
I
e
−log
R
e
)
/
Κ
3
=
3∼(
M/L
)
dyn
(33)
I
e
≡
M
∗
/
(2
Π
Υ
∗
R
e
2
)
(34)
where M
dyn
=
M
∗
+
M
h
is the total mass. The combination of Eq.(26) and Eqs.(31)-(34)
yields:
√
−log
C
∗
2
)
/
Κ
1
=
(log
M
∗
2
(35)
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