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assume that the mass outside this region is negligible. In which case, a star at radius
r sits in a gravitational field which, by virtue of the result we have just proven, is
equivalent to that of a point mass M located at r
0 . Since the star is orbiting
the centre of the galaxy, the gravitational attraction to the centre must provide the
necessary centripetal acceleration, i.e.
=
mv 2
r
GMm
r 2
=
,
where m is the mass of the star. Thus the speed of the star is
GM
r
v
=
.
Remarkably, this behaviour is not what is seen in astronomical observations.
Figure 9.5 shows the astronomical data for a typical galaxy and we can clearly
see that the large r behaviour is approximately constant and certainly not falling
as 1 / r. One way to explain the data is to assume that a substantial component
of the mass of the galaxy is invisible to the astronomers and that this component
extends out to large distances from the centre of the galaxy (compared to the size
of the central bulge). Other types of observation indicate that this mass cannot be
comprised solely of large planets which are too dark to be visible and so the nature
of this unseen mass is as yet unknown. Perhaps it is made up from a new type of
elementary particle which is invisible to ordinary detection methods. Certainly the
origin of this 'Dark Matter' is one of the big mysteries in modern physics.
200
150
100
50
0
0
10
20
30
r (kpc)
Figure 9.5 The rotation curve for stars orbiting the centre of the galaxy NGC3198 (data
from K.G. Begeman, Astron. Astrophys. 223 (1989) 47).
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