TIDAL ENERGY EFFECTS OF DARK MATTER HALOS ON EARLY-TYPE GALAXIES - Energy Research Developments: Tidal Energy, Energy Efficiency and Solar Energy - page 316

Figure 1. The plot of the bright component VPE, V ∗ , in units of GM ∗ 2 , for HP models
with different R ; R ∗ and R h are expressed in kpc . For R > 10 the VPE appears to be
non-monotonic along R ∗ axis; this threshold in R is model dependent (HH models yield
R > 20 ).

where the numerical suffix of the general function indicates the dimensionality of the hyper-
surface. It is worth noticing that via Eq. (7) V ∗ /M ∗ is Σ 2 ∗ , the stellar rms peculiar velocity
averaged by mass (provided systematic rotation is negligible). For a fixed value of R the
VPE function is defined in a 3 dimensional space; in other words it becomes a surface, as:

V ∗ = F 2 ( R ∗ , Β ) = F 2 ( R ∗ , R h ) (12)

and either R ∗ or Β = R h /R ∗ can be chosen as independent parameters. From here on, when
the VPE surface is mentioned, it is intended that R is therein fixed.
Fig. 1 is a 3D-plot of the bright component VPE in units of GM ∗ 2 for HP models (HH
models exhibit a similar trend, and are not presented here). In the case under discussion the
following system of equations:

8
< ∂V ∗

∂M ∗ = 0
∂V ∗

∂R ∗ = 0
∂V ∗
∂R h

(13)

:

= 0

∂V ∗

∂ R = 0