Environmental Engineering Reference
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that MVPE configurations cannot entirely fit data from observations, and that the scatter
exhibited along Κ 1 axis largely exceeds the scatter exhibited by data from observations, for
all different values of (assumed constant) stellar mass to luminosity ratio, Υ . Furthermore,
assuming the Faber-Jackson relation, it has been shown that regions where the data are
matched by model predictions in the Κ 1 3 plane, have no counterpart in the Κ 1 2 plane,
which furthermore rules out some reasonable Υ values for ETGs (see, e.g., Fig.6). In
summary, MVPE configurations alone cannot explain the FP tilt of ETGs, at constant Υ ,
and an additional constraint is needed, which study is left to future work.
For the sake of completeness, it has been tested if MVPE configurations are preferred
final states at the end of the transition from assigned initial conditions, using dynamical
N -Body simulations. To this aim, 14 computer runs have been analyzed, and the related
(with a possible exception) virialized configurations have been found to either lie below
the minimum threshold in R for MVPE configurations, or exhibit systematically shallower
density profiles. This has been interpreted as a further support to the idea, that MVPE
configurations (as defined in the energy-conservation paradigm ) make no preferred final
state for two-component systems where only gravity is at work. A larger set of computer
runs might provide further evidence for, or reason against, the last conclusion. On the other
hand, the tidal potential energy induced by hosting DM halos on ETGs plays an essential
role, even if not sufficient, for the interpretation of the FP of ETGs.
A.
Analytical Values of Stellar VPE Related Quantities
We present here a number of analytical quantities found through the paper. First of all
the values of the general functions describing the interaction term of the VPE for both our
models:
1 + 4 Β −5 Β 2 + 2 Β ( Β + 2)ln Β
( Β −1) 4
V (HH)
( Β )
=
∗h
Q(2 Β 4 + 11 Β 3 −12 Β 2 −4 Β + 1) + (6 Β 4 −9 Β 2 )W
Q 7
V (HP )
( Β )
=
∗h
1 + Β 2 −1)]
and the explicit expression of the functions describing the VPE in the energy conservation
paradigm :
Β 2 + 1 W≡ln[(
1− Β −2 −1)(
Q≡
2 Β [( Β −1) 4 −6R( Β −1)(5 Β + 1) + 12R Β ( Β + 2) ln Β ]
R 2 ( Β −1) 4 + Β ( Β −1) 4 + 6R Β ( Β −1) 2 ( Β + 1)−12R Β 2 ( Β −1) ln Β
F (HH =
16 Β X 7 + 96R Β [X(2 Β 4 + 11 Β 3 −12 Β 2 −4 Β + 1) + (6 Β 4 −9 Β 2 ) lnY]
X 3 [9 Π R 2 X 4 + 16 Β Z]−288R Β 3 X 2 ln Y
F (HP )
=
Β 2 + 1
X≡
Β 2 + 1−1)(
Β 2 + 1− Β )
Y≡ (
Β
Z≡ Β 4 + 6R Β 3 −12R Β 2 + 2 Β 2 −12R Β + 6R+ 1
for HH and HP models, respectively.
 
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