Environmental Engineering Reference
In-Depth Information
2 Initial Conditions and Geometry of the Collision
We use the standard procedure to construct a galaxy model with a Newtonian poten-
tial described in Gabbasov (
2006
) and Gabbasov et al. (
2006
). An individual galaxy
consists of a disc, halo, and bulge and its initial condition is constructed using a
bulge and a Freeman disc composed of stars embedded in a Hernquist halo model
(a Dehnen's family member with
is the power characterising the
member; see Rodríguez-Meza and Cervantes-Cota (
2004
)) that acts on them gravi-
tationally. We does not consider gas.
The spatial distribution of particles are constructed using density profiles: The
bulge density profile is a spherical distribution of stars and is given by (Hernquist
1990
):
ʳ
=
1, where
ʳ
M
b
a
b
2
1
ˁ
b
(
r
)
=
3
,
(1)
ˀ
(
+
a
b
)
r
r
and for the halo we use a Dehnen density profile with
ʳ
=
0 (Dehnen
1993
):
3
M
h
4
a
h
ˁ
h
(
r
)
=
4
.
(2)
ˀ
(
r
+
a
h
)
We assume that the disc follows the exponential profile (Freeman
1970
):
z
0
e
−
ʱ
R
sech
2
z
2
M
d
ʱ
ˁ
d
(
R
,
z
)
=
.
(3)
4
ˀ
z
0
where
R
and
z
are the spatial cylindrical coordinates. In these equations
M
b
,
a
b
and
M
h
,
a
h
are the mass and length of bulge and halo respectively, and
M
d
,
ʱ
−
1
and
z
0
are the mass, length scale and the thickness length scale of the disk, respectively.
For the spherical distribution of particles, their velocities,
v
r
,
v
ˆ
,
v
ʸ
, are obtained
using the Schwarzschild distribution function
exp
v
2
ˆ
v
2
ʸ
v
r
2
f
B
,
H
(
v
r
,
v
ˆ
,
v
ʸ
)
∝
−
r
−
−
,
(4)
2
ˆ
2
ʸ
σ
2
σ
2
σ
where
are the dispersion of velocities and in general they are func-
tions of
r
. For an isotropic ellipsoid the above velocity distribution is the Maxwell
distribution.
For a spherically symmetric mass distribution and without rotation the dispersion
of velocities is obtained using Jeans' equation
σ
r
,
σ
ˆ
, and
σ
ʸ
r
2
d
dr
+
ˁ(
r
)
d
dr
,
2
2
2
2
ˁ(
r
)
σ
σ
r
−
(
σ
ʸ
+ σ
ˆ
)
=−
ˁ(
r
)
(5)
r
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