Environmental Engineering Reference
In-Depth Information
Κ
2
=
(3 log
M
∗
−log
C
∗
2
−6 log
R
e
−2 log Υ
∗
−13
.
6)
/
√
6
(36)
√
−log
C
∗
2
+ 6
.
8)
/
Κ
3
=
(log Υ
∗
3
(37)
To close the system of Eqs.(31)-(33), an additional equation is needed, provided by the
empirical Faber-Jackson law:
0.25
L
L
o
[kms
−1
]
L
O
≡10
10
h
−2
L
⊙
(38)
Σ
o
=
220
;
which combined with Eq.(26) and (36), assuming a dimensionless Hubble parameter, H
=
0
.
7
, yields, after some algebra:
→
R
e
= 3
M
∗
0.5
Υ
∗
0.5
M
∗
C
∗
2
R
e
0
.
340
L
0.5
Σ
o
=
=
(39)
C
∗
2
√
(5 log
C
∗
2
Κ
2
=
−5 log Υ
∗
−16
.
5)
/
6
(40)
In conclusion, Κ
2
and Κ
3
depend only on
Υ
∗
and C
∗
2
, while Κ
1
on M
∗
and C
∗
2
.
The comparison of model predictions with observational data has been made with a
sample of 9000 ETGs, in the redshift range
0
.
01≤
Z
≤0
.
3
, selected from the Sloan Digital
Sky Survey first data release (SDSS DR1) using morphological and spectral criteria (see,
Bernardi et al., 2003, for further details). The sample is highly representative of nearby
ETGs and has been found to produce a relatively thin FP. The SDSS DR1 data have been
plotted together with model predictions in Figs. 6, 7, 8, for fixed values of
Υ
∗
. It is apparent
that MVPE configurations with assumed constant
Υ
∗
cannot reproduce the whole set of data
within a fiducial range,
4≤Υ
∗
≤12
. In addition, mapping some region in the Κ
1
-Κ
3
plane
does not necessarily imply a corresponding mapping in the Κ
1
-Κ
2
plane. Finally, the scatter
along Κ
1
axis shown by MVPE configurations exceeds on both sides the scatter shown by
the data, for reasonable values of M
∗
. At this stage, a number of preliminary considerations
can be drawn, namely:
1. There is no apparent correlation between MVPE configurations and the stellar mass
to light ratio,
Υ
∗
.
2. Dealing with MVPE configurations only, within the
energy-conservation paradigm
,
yields a restricted range in C
∗
2
(i.e., in Κ
3
), for both HP and HH models, no matter the
value of
Υ
∗
, while data from observations span a wider range.
3. There is no apparent correlation between MVPE configurations and total or stellar
mass.
The first point comes from the fact that, for an assigned DM to stellar mass ratio,
R
, MVPE
configurations select a single effective radius, R
e
. On the other hand,
Υ
∗
is strictly con-
nected with age and metallicity of the galaxian population instead of the peculiar structure
and mass/light distribution of the system.
The second point is highly model dependent: though different models can be related to
different ranges in C
∗
2
, still the range width shows little model dependence provided only
MVPE configurations are dealt with.
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