Environmental Engineering Reference
In-Depth Information
for a fixed value of
r
. It is actually simpler if we change variables and use
y
instead
of
θ
as the integration variable. In this case Eq. (9.24) gives that
y
d
y
=
xr
sin
θ
d
θ
(9.26)
and hence
π
x
+
r
sin
θ
d
y
xr
=
2
x
.
d
θ
=
(9.27)
(x
2
+
r
2
−
2
xr
cos
θ)
1
/
2
0
x
−
r
We can now substitute this back into Eq. (9.23), in which case
2
R
3
R
2
r
2
x
G
3
M
=−
d
r
0
GM
x
=−
,
(9.28)
which is exactly equal to the potential of a point mass
M
located at the origin,
a result that we quoted without proof in the previous section. You might like to
convince yourself that this result holds generally for any spherically symmetric
distribution of matter, i.e. one for which the mass density depends only upon
r
and
not upon
θ
or
φ
. The case of uniform mass density which we considered here is
one example of such a distribution.
Example 9.2.1
Show that the speed of a star orbiting in an arm of a spiral galaxy
at a radius r far from the centre of the galaxy should vary as
1
/
√
r if we assume
that the mass of the galaxy is located in the spherical bulge at the centre of the
galaxy.
Solution 9.2.1
Figure 9.4 illustrates the situation. We shall approximate the central
bulge of stars by a spherically symmetric distribution of matter of total mass M and
u
(r)
M(r)
Figure 9.4
A star orbiting the centre of a galaxy. The shading denotes the presence of dark
matter.
