146 3 Coordinates
3-44 The arc length of the oblique equator in oblique quasi-spherical coordinates
In order to compute the length of an arc in the oblique ecliptic equator E
1
A 1 ,A 2 in terms of oblique
quasi-spherical longitude, we are forced to represent the infinitesimal arc length by
dS =
dX 2 + dY 2 B=0
=
R 2 (A) + R 2
1 (A)dA ,
(3.114)
subject to
R 0 (A, B = 0) := R(A) :=
A 1
√
1 − E 2
1 − E 2
1 − cos 2 I sin
2 A
,
(3.115)
R 1 (A, B = 0) := R 1 (A) :=
1
1!
dR(A)
dA
= −
A 1 E
2
√
1 − E 2 cos
2 I sin A cos A
1 − E 2
1 − cos 2 I sin
2 A
3/2 ,
(3.116)
such that
S(A) =
A
A=0
R 2 (A ∗ ) + R 2
1 (A ∗ )dA
∗ .
(3.117)
In the following steps, we perform the integration. (i) Series expansion of R(A) according to (3.120)
up to order E
6 . (ii) Series expansion of R 1 (A) = dR/dA according to (3.121) up to order E
6 . (iii)
Series expansion of R
2 (A) + R
2
1 (A) according to (3.122) up to order E
6 . (iv) Series expansion of
(R
2 (A) + R
2
1 (A))
1/2 according to (3.125) up to order E
6 .
Solution (the first step).
R(A)
A 1
√
1 − E 2
= (1 − x)
−1/2 = 1 +
1
2
x +
1 · 3
2 · 4
x
2 +
1 · 3 · 5
2 · 4 · 6
x
3 + O −
x
4
,
(3.118)
subject to
x := −E
2
1 − cos
2 I sin
2 A
, |x| ≤ 1 ;
(3.119)
R(A) = A 1
1 − E 2
1 +
1
2
E
2
1 − cos
2 I sin
2 A
+
+
1 · 3
2 · 4
E
4
1 − cos
2 I sin
2 A
2 +
1 · 3 · 5
2 · 4 · 6
E
6
1 − cos
2 I sin
2 A
3 + O −
E
8
.
(3.120)
End of Solution (the first step).
Solution (the second step).
R 1 (A) =
dR
dA
=
= A 1
1 − E 2
− E
2 cos
2 I sin A cos A−
−
1 · 3
2
E
4 cos
2 I
1 − cos
2 I sin
2 A
sin A cos A−
−
1 · 3 · 5
2 · 4
E
6 cos
2 I
1 − cos
2 I sin
2 A
sin A cos B − O 1
E
8
.
(3.121)
End of Solution (the second step).