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Dark Matter
and
Rotational Motion
e.t. revised feb 09
What is the Velocity of Stars that
Revolve around the Center of a Galaxy?
From Newtonian dynamics, the centripetal force acting on an
orbiting star of mass m is the gravitational force pulling the
star toward the center of the galaxy:
2
v
m
r
Orbital velocity v
GM m
r2


GM
r
• Here r and v are the star’s average radial distance from the center of
the galaxy and its orbital velocity. G is the universal gravity constant
and M is the mass of the galaxy within radius r.
• NOTE: only mass lying within a star's orbit affects the star's motion.
So we need to consider the mass distribution in the galaxy.
How does the star’s orbital velocity depend
on the distribution of the galaxy mass?
• Simplifying assumption: Spherical galaxy of mass
Mtotal, radius R, and uniform density r
Outside of galaxy: all galaxy mass Mtotal within the
orbit radius
r
v

GM tot
r
G and Mtot are
constants
Speed
Distance from center of galaxy
How does the star’s orbital velocity depend
on the distribution of the galaxy mass?
Inside galaxy:
v

velocity
GM
r
v


uniform density
r
M total
4
 R3
3
r
R
4
G  r r3
3
r


4
 G  r  r
3


Speed
Distance from center of galaxy
How does the star’s orbital velocity depend
on the distribution of the galaxy mass?
• For spherical galaxy of mass Mtotal, radius R, and
uniform density r
inside outside
r
R
Speed
R
Distance r from center of galaxy
What is the actual distribution of
visible matter in a galaxy?
• Galaxies do not have uniform
distribution of matter.
NGC 3198
• Most mass at small radius but
mass drops more gradually with
increasing radius.
http://www.astro.princeton.edu/~frei/Gcat_htm/Ca
talog/Jpeg/n3198_pg.jpg
Copyright © 1999 by Princeton University Press.
Freeman, K. Dark Matter in Galaxies,
in Encyclopedia of Astronomy and
Astrophysics, November 2000
Astronomers measure the
luminosity from various
regions in the galaxy and can
relate luminosity to mass.
How do measured galaxy rotation
curves compare to predictions?
A halo of dark matter
is needed.
Freeman, K. Dark Matter in Galaxies, in Encyclopedia of
Astronomy and Astrophysics, November 2000
The orbital velocity of stars can be
measured using the Doppler shift.
Additional Remarks
• Dark matter is overall estimated to be about
6 times more abundant than ordinary matter.
• Similar orbital analyses done for galaxy
clusters show similar need for dark matter.
• Dark matter has yet to be directly detected
and we do not know what it is. However,
several completely independent indirect
measurements support it’s existence.