Geoscience Reference
In-Depth Information
7.5
Determination of tesseral harmonics and
station positions
Zonal harmonics give rise to secular and long-periodic perturbations of the
orbital elements
a
,
e
, etc. Therefore, their influence can be detected in
changes of orbital parameters obtained by integrating over many revolutions
of the satellite.
The perturbations due to tesseral harmonics have a much shorter period.
The longest period of a harmonic of the order
m
= 1 is one day, for
m
=2it
is only half a day, etc. Therefore, we must look for another method, which is
sensitive enough to detect even short-periodic effects and extracts as much
information as possible from the observations.
The observed elements are essentially spatial polar coordinates of the
satellite with respect to the observing station: the distance
s
and the direc-
tion as determined by two angles. Corresponding to our coordinate system
X
0
,Y
0
,Z
0
introduced in the preceding section, these two angles are the
right ascension α
and the
declination δ
, whose definition may be seen in
Fig. 7.5. The angles
α
and
δ
are polar coordinates in three-dimensional space
and were obtained by photographing the satellite against the background of
stars, as outlined in Sect. 7.1. They are outdated nowadays but retained for
geometrical intuition and symmetry. Most important are distances
s
mea-
sured by GPS, radar, or laser. Note that the measurement of the range rate
ds/dt
of the satellite by means of the Doppler effect is also important for the
determination of tesseral harmonics and station positions.
Denoting in the equatorial system
X
0
Y
0
Z
0
the rectangular coordinates of
the terrestrial station
P
by
X
0
,Y
0
,Z
0
and of the satellite
S
by
X
0
,Y
0
,Z
0
,
P
P
P
S
S
S
Z
0
unit sphere
||Z
0
S
S
Z
0
P
0
0
±
Z
SP
-
±
±
P
Y
0
||Y
0
®
0
0
X
S P
-
®
||X
0
Y
S P
0
-
0
®
X
0
Y
0
X
0
earth
Fig. 7.5. Direction to the satellite defined by right ascension
α
and
declination
δ
