Homework # 6 Solutions
1. Demonstrate that the temperature applicable to a stellar system is
T
=
m < v
2
(
x
)
>
3
k
where it is assumed that the stars have equal mass
m
,
v
(
x
) is the velocity of a
star relative to the centerofmass, and
<>
represents an average.
On the assumption that the stellar velocities are distributed in the most
probably way, i.e., a Boltzmann distribution, one has
n
(
x
,
v
) =
n
o
exp
±

(
m
Φ(
x
) +
mv
2
/
2
)
/
(
kT
)
²
.
The quantity
kT
is simply a constant representing an average energy. To see
how it is related to the average velocity, we compute the mean square stellar
velocity at
x
, which is
h
v
2
(
x
)
i
=
R
∞
0
4
πn
o
v
4
exp
±

(
m
Φ(
x
) +
mv
2
/
2
)
/
(
kT
)
²
dv
R
∞
0
4
πn
o
v
2
exp[

(
m
Φ(
x
) +
mv
2
/
2)
/
(
kT
)]
dv
=
R
∞
0
v
4
exp
±

mv
2
/
(2
kT
)
²
dv
R
∞
0
v
2
exp[

mv
2
/
(2
kT
)]
dv
=
Γ(5
/
2)(2
kT/m
)
5
/
2
Γ(3
/
2)(2
kT/m
)
3
/
2
=
3
kT
m
.
2. Calculate the gravitational potential a distance
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 Spring '11
 Lattimer
 Derivative, Galaxies, Mass, Velocity, Ratio

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