These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics
by Prof. Chuck Keeton, and modiﬁed by Profs. Saurabh Jha and Eric Gawiser. All rights
reserved.
c
±
2011
Lecture 14: Spiral Galaxy Rotation Curves
I. Disk Rotation Curves
The orbits of stars in disk galaxies are close to circular, which is convenient. There are
so many stars that, even though they are all moving, the average mass distribution hardly
changes with time. So for now we can imagine that the mass distribution is ﬁxed in time,
which means that the gravitational force as a function of radius is also ﬁxed in time.
This is essentially the onebody problem again, except with an extended mass distribution.
I.1. 3d spherical mass distribution
Let’s start with the spherical case. Suppose the density as a function of radius is
ρ
(
r
). Then
the mass enclosed by radius
r
is
M
(
r
) = 4
π
Z
r
0
ρ
(
u
)
u
2
du
⇔
ρ
(
r
) =
1
4
πr
2
dM
(
r
)
dr
Recall from Newton that for a spherical mass distribution we can consider:
•
outside a spherical shell, the gravity is the same as if all the mass of the shell were
concentrated at the center
•
inside a spherical shell, there is no gravity from that shell
So the net gravitational force at radius
r
depends only on the mass enclosed by radius
r
,
F
(
r
) =

GM
(
r
)
m
r
2
For circular orbits, the force must provide the centripetal acceleration
a
=

v
(
r
)
2
/r
, so
GM
(
r
)
m
r
2
=
m
v
(
r
)
2
r
⇒
v
(
r
) =
±
GM
(
r
)
r
²
1
/
2
and
M
(
r
) =
r v
(
r
)
2
G
The second equation is useful if we suppose that we know the mass distribution and want to
compute the corresponding rotation curve. (The second equation should look familiar: you
1
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R
Figure 1: Rotation curve for an exponential disk.
derived it by dimensional analysis on PS#1. It turns out that there are no dimensionless
factors, provided
M
(
r
) is the enclosed mass.) The third equation is useful if we measure
the motion and want to infer the mass. Speciﬁcally, if we measure the rotation speed at one
radius, we can automatically compute the mass enclosed by that radius. If we measure the
full rotation curve, we can determine the mass proﬁle curve, which in turn gives us the full
density proﬁle.
As always, motion
→
mass!
I.2. 2d circular mass distribution
The analysis of a thin disk is more involved. Since a disk is not spherically symmetric,
you have to evaluate some moderately sophisticated integrals to determine the gravitational
force. (See
Galactic Dynamics
disk we are interested in is the exponential disk, let me just quote the rotation curve:
v
(
r
)
2
=
πG
Σ
0
r
2
h
R
±
I
0
²
r
2
h
R
³
K
0
²
r
2
h
R
³

I
1
²
r
2
h
R
³
K
1
²
r
2
h
R
³´
where
I
0
,
K
0
,
I
1
, and
K
1
are special functions called modiﬁed Bessel functions. This is
plotted in Figure 1. Let’s look at the general features. Quantitatively, the rotation curve
peaks at
r
max
= 2
.
15
h
R
v
max
= 1
.
56(
G
Σ
0
h
R
)
1
/
2
Qualitatively, the important feature is that the rotation curve declines as you go to large
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 Fall '11
 Gawiser
 Physics, Dark Matter, rotation curve, rotation curves

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