Environmental Engineering Reference
In-Depth Information
ʦ
where
is the gravitational potential. If the distribution of velocities is isotropic
2
r
2
2
σ
= σ
ʸ
= σ
ˆ
.
(6)
The above equation can be integrated to give a general expression for the dispersion
of velocities:
∞
dr
1
ˁ(
d
r
)
dr
.
r
σ
(
r
)
=
ˁ(
(7)
)
r
r
Particles velocities can be found by inverting the equation
3
v
2
exp
v
2
2
4
ˀ
f
b
,
h
(
v
,
r
)
=
−
.
(8)
ˀ
σ
2
2
/
σ
r
(
2
)
In practice it is convenient to cut the Gaussian distribution at some finite value. A
natural choice is the escape velocity
V
e
.
For axisymmetric distribution we have that the velocity profiles for the disk are
computed using the epicyclic approximation, which consists in assuming that velocity
dispersions are small (
σ
R
,
σ
z
,
σ
ˆ
R
ˉ
):
exp
v
R
2
v
z
2
2
z
−
(
v
ˆ
−
V
0
)
f
D
(
v
R
,
v
z
,
v
ˆ
)
∝
−
R
−
.
(9)
2
2
ˆ
σ
σ
2
σ
Observations in the exterior of disk galaxies suggest that the radial dispersion is
proportional to the surface radial density:
2
σ
R
∝
exp
(
−
ʱ
R
).
(10)
The vertical dispersion in the isothermal shell approximation is also related to the
surface density of the disk,
ʣ(
R
)
:
2
z
σ
=
ˀ
Gz
0
ʣ(
R
).
(11)
2
2
z
σ
R
/
σ
The ratio
is constant through the disk and is considered equal to 4, i.e.,
2
2
σ
R
=
4
σ
z
.
(12)
The azimuthal dispersion is simply related to radial dispersion through the
epicyclic approximation for the Schwarzschild velocity distribution
2
ʺ
2
2
σ
2
σ
ˆ
=
R
,
(13)
4
ˉ
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