Geoscience Reference
In-Depth Information
now also contains the tesseral harmonics.
By performing the integrations in (7-37), we obtain equations of the form
∆
t
a
=
n,m
A
nm
C
nm
+
A
nm
S
nm
,
∆
t
e
=
n,m
B
nm
C
nm
+
B
nm
S
nm
,
(7-39)
.
.
where the coecients
A
nm
, etc., are functions of the time
t
and are, as a
rule, periodic. Zonal and tesseral harmonics have been combined in (7-39)
by setting
J
n
=
C
n
0
and admitting the value
m
= 0; this practice will be
continued in what follows.
The substitution of (7-39) into (7-36) gives the perturbation of the rect-
angular coordinates
X
0
,
Y
0
,
Z
0
−
as functions of the harmonic coecients
C
n
0
=
−
J
n
,
C
nm
,and
S
nm
in the form
∆
t
X
0
=
n,m
L
nm
C
nm
+
L
nm
S
nm
,
∆
t
Y
0
=
n,m
M
nm
C
nm
+
M
nm
S
nm
,
(7-40)
∆
t
Z
0
=
n,m
N
nm
C
nm
+
N
nm
S
nm
,
where again
L
nm
,
L
nm
,
M
nm
, etc., are functions of the time
t
.
These perturbations are added to the coordinates computed from (7-34)
using the orbital elements of the reference ellipse
E
0
. In this way, we obtain
the rectangular coordinates of the satellite in the form
X
0
=
X
0
(
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
,
Y
0
=
Y
0
(
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
,
Z
0
(7-41)
=
Z
0
(
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
as
explicit
functions of the time
t
, containing as
constant
parameters the
orbital element of the reference ellipse
E
0
and the gravitational coecients
C
nm
and
S
nm
. This is the advantage of (7-41) over the system (7-34), which
formally is much simpler but depends on the
variable
orbital parameters of
the osculating ellipse.
The actual expressions for (7-41) are very complicated. Therefore, we
have been satisfied with outlining the procedure, referring the reader for
details to the pioneering topic by Kaula (1966 a) and to his papers given
there.
