Environmental Engineering Reference
In-Depth Information
remarks provide us with a good qualitative insight into the solutions. However it
is now time to compute the precise form of the solutions mathematically.
Let us begin by re-writing Eq. (9.46) in the case that the force acting is gravi-
tational:
2
η
M
r
λ
2
r
2
G
r
˙
=
+
−
(9.48)
and we have introduced the more convenient energy per unit mass
η
≡
E/µ
and
angular momentum per unit mass
λ
≡
L/µ
. We have also introduced the mass
M
≡
Mm/µ
. In the case that
M
m
, which is often the case, we can assume
M
M
without introducing any significant error. Integrating this equation gives
us
r(t)
, however we would rather have
r
as a function of the polar angle
θ
since
such a functional dependence will directly describe the spatial trajectory. The
dependence upon
t
can easily be traded for a dependence upon
θ
since we know
that
=
θ
λ/r
2
=
and hence
=
θ
d
r
d
θ
r
˙
λ
r
2
d
r
d
θ
.
=
(9.49)
Thus we can write
2
η
M
r
r
2
λ
d
r
d
θ
=
G
λ
2
r
2
+
−
(9.50)
and hence
2
λ
2
η
−
1
/
2
G M
r
d
r
r
2
1
r
2
θ
=
+
−
.
(9.51)
Although this integral looks rather foreboding it is in fact one that we can perform.
Let us first change variables to
u
≡
1
/r
, in which case
d
u
2
u
2
−
1
/
2
λ
2
η
Mu
θ
=−
+
G
−
.
(9.52)
By
completing
the
square,
this
integral
can
be
manipulated
into
the
form
d
w/
√
c
2
w
2
−
and this is a standard integral. Thus we write
u
2
G
2
.
λ
2
η
Mu
G M
λ
2
M
2
λ
4
2
2
η
λ
2
u
2
+
G
−
=−
−
+
+
(9.53)
