222
5 Gravitation
5.38 Velocity at the nearer apse is given by
v
2
= G M
2
a(1 − ε)
−
1
a
=
G M
a
1 + ε
1 − ε
(1)
as there is no instantaneous change of velocity. If a 1 is the semi-major axis for
the new orbit
v
2
= G M
2
a(1 − ε)
−
1
a 1
(2)
As the nearer and farther apses are inter-changed
a 1 (1 − ε 1 ) = a(1 + ε)
(3)
Equating the right-hand side of (1) and (2) and eliminating a 1 from (3) and
solving for ε 1 we get
ε 1 =
ε(3 + ε)
1 − ε
5.39 (a) 1
T
dt
r
=
1
T
d θ
r ˙
θ
(1)
Now, J = mr
2 ˙
θ (constant)
(2)
∴
1
T
dt
r
=
m
T J
r d θ
(3)
r =
a(1 − ε 2 )
1 + ε cos θ
(4)
Using (4) in (3)
1
T
dt
r
=
ma(1 − ε 2 )
T J
2π
0
d θ
1 + ε cos θ
=
ma(1 − ε 2 )
T J
2π
√
1 − ε 2
(5)
where we have used the integral
2π
0
d θ
a + b cos θ
=
2π
√
a 2 − b 2
Further T =
2π ma 2
√
1 − ε 2
J
(6)
Using (6) in (5)