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5 Gravitation
Fig. 5.18
r =
a(ε 2 − 1)
ε cos θ − 1
(1)
As r → ∞, the denominator on the right-hand side of (1) becomes zero and
the limiting angle θ 0 is given by
cos θ 0 =
1
ε
or cot θ 0 =
1
√
ε 2 − 1
The complete angle of deviation
φ = π − 2θ 0
or
φ
2
=
π
2
− θ 0
tan
φ
2
= cot θ 0 =
1
√ ε 2 − 1
But ε =
1 +
2Eh 2
G 2 M 2
where h = pv and E =
1
2
v 2
∴ tan
φ
2
=
1
√ ε 2 − 1
=
G M
h
√
2E
=
G M
pv 2
5.46 In Fig. 5.19
r =
2a
1 + cos θ
r
2 ˙
θ = h (constant, law of areas)
5 Gravitation
Fig. 5.18
r =
a(ε 2 − 1)
ε cos θ − 1
(1)
As r → ∞, the denominator on the right-hand side of (1) becomes zero and
the limiting angle θ 0 is given by
cos θ 0 =
1
ε
or cot θ 0 =
1
√
ε 2 − 1
The complete angle of deviation
φ = π − 2θ 0
or
φ
2
=
π
2
− θ 0
tan
φ
2
= cot θ 0 =
1
√ ε 2 − 1
But ε =
1 +
2Eh 2
G 2 M 2
where h = pv and E =
1
2
v 2
∴ tan
φ
2
=
1
√ ε 2 − 1
=
G M
h
√
2E
=
G M
pv 2
5.46 In Fig. 5.19
r =
2a
1 + cos θ
r
2 ˙
θ = h (constant, law of areas)