Environmental Engineering Reference
In-Depth Information
Armed with the conservation of energy we can write that
1
2
µ
r
2
E
=
˙
+
U(r)
1
2
µ
1
2
µr
2
θ
2
r
2
=
˙
+
+
U(r)
(9.44)
and we have used Eq. (9.41). Furthermore, we can eliminate the dependence upon
θ
by introducing the angular momentum
L
,i.e.
L
2
2
µr
2
+
1
2
µ
r
2
E
=
˙
+
U(r).
(9.45)
This is a very powerful equation for it depends only upon the variables
r
and
t
,
all other quantities being constants. Re-arranging we have
2
µ
E
d
r
d
t
=
L
2
2
µr
2
−
U(r)
−
(9.46)
and thus once we are given a particular potential
U(r)
we can go ahead and
integrate to obtain
r(t)
. Notice also that the motion looks exactly like the motion
in one-dimension of a particle of mass
µ
in a potential
L
2
µr
2
.
U
eff
=
U(r)
+
(9.47)
That is as far as we shall take the general development. Let us now consider the
particular case of a gravitational field.
9.5 ORBITS
Equation (9.46) already allows us to make some very general statements about
the types of solution we expect. Figure 9.8 shows a plot of the effective potential
U
eff
.Atlarge
r
the Newtonian 1
/r
term dominates whereas at small
r
the 1
/r
2
term
dominates, we shall call this term the centrifugal barrier term. Now
E
=
K
+
U
eff
1
2
µ
r
2
>
0 and it follows that
E>U
eff
. Thus for motion occurring
with total energy
E
, only those values of
r
for which
U
eff
<E
are accessible.
Figure 9.8 shows the three possible scenarios. In scenario (a)
E>
0 and only
the region
r<r
0
is inaccessible, i.e. there is insufficient energy for the system
to access this region. This corresponds to a motion where there is a distance of
closest approach to the point
r
where
K
=
˙
0 but no maximum distance. This type of motion
is illustrated in Figure 9.9(a). It helps to have a particular system in mind, so we
might consider the case where one of the bodies is much lighter than the other, e.g.
a comet moving under the influence of the Sun's gravity. In this case the centre
of mass of the two body system is virtually co-incident with the centre of the Sun
and the reduced mass is equal to the comet's mass to a very good approximation.
Scenario (a) then corresponds to an unbound orbit where the comet is deflected by
=
