H-3 An integration formula 617
H-3 An integration formula
(H.28) may be written as an integration formula. The following relations (H.97)–(H.100) specify this
integration formula. If the relative eccentricity approaches E = 0, then the radial coordinate specializes
to (H.101) according to the L’Hˆ opital Rule (H.102) and (H.103).
sin x
1 + p 2 sin
2 xdx = −
1
2
cos x
1 + p 2 sin
2 x −
1 + p
2
2p
arcsin
p cos x
1 + p 2
,
(H.97)
subject to
p
2 :=
1
1 − E 2
E
2 sin
2 A
∗
1 − E 2 cos 2 A ∗ ,
(H.98)
r
2 =
A
2
1 (1 − E
2 )
1 − E 2 cos 2 A ∗ ×
×
− cos ∆
∗
1 +
E 2 sin
2 A ∗ sin
2 ∆ ∗
(1 − E 2 )(1 − E 2 cos 2 A ∗ )
−
√
1 − E 2
√
1 − E 2 cos 2 A ∗
E sin A ∗
×
×
1 +
E
2 sin
2 A
∗
(1 − E 2 )(1 − E 2 cos 2 A ∗ )
×
× arcsin
E sin A
∗
√
1 − E 2
√
1 − E 2 cos 2 A ∗
cos ∆
∗
1 +
E 2 sin 2 A ∗
(1−E 2 )(1−E 2 cos 2 A ∗ )
∆
∗
0
,
(H.99)
r
2 =
A
2
1
√
1 − E 2
(1 − E 2 cos 2 A ∗ ) 3/2 ×
×
− cos ∆
∗
(1 − E 2 )(1 − E 2 cos 2 A ∗ + E 2 sin
2 ∆ ∗ sin
2 A ∗ )−
−
1 − 2E
2 cos
2 A
∗ + E
4 cos
2 A
∗
E sin A ∗
arcsin
E cos ∆
∗ sin A
∗
√
1 − 2E 2 cos 2 A ∗ + E 4 cos 2 A ∗
+
+
(1 − E 2 )(1 − E 2 cos 2 A ∗ )+
+
1 − 2E
2 cos
2 A
∗ + E
4 cos
2 A
∗
E sin A ∗
arcsin
E sin A
∗
√
1 − 2E 2 cos 2 A ∗ + E 4 cos 2 A ∗
,
(H.100)
r = A 1
√
2
√
1 − cos ∆ ∗ = A 1
√
2
√
1 − sin B ∗ ,
(H.101)
lim
x→0
arcsin x
x
= lim
x→0
a
√
1 − a 2 x 2
= a ,
(H.102)
lim
E→0
arcsin(E sin A
∗ cos ∆
∗ )
E sin A ∗
= cos ∆
∗ ,
lim
E→0
arcsin(E sin A
∗ )
E sin A ∗
= 1 .
(H.103)