Geoscience Reference
In-Depth Information
we find by inspecting Fig. 7.5
X
0
S
− X
0
P
=
s
cos
δ
cos
α,
Y
0
S
− Y
0
=
s
cos
δ
sin
α,
(7-42)
P
Z
0
Z
0
S
−
=
s
sin
δ,
P
so that
Y
0
Y
0
S
−
α
=tan
−
1
P
,
X
0
X
0
S
−
P
Z
0
Z
0
S
−
δ
=tan
−
1
P
(
X
0
(7-43)
)
2
,
X
0
)
2
+(
Y
0
Y
0
S
−
S
−
P
P
s
=
(
X
0
X
0
)
2
+(
Y
0
Y
0
)
2
+(
Z
0
Z
0
S
−
S
−
S
−
)
2
.
P
P
P
We now compute the rectangular coordinates
X
0
P
,Y
0
P
,Z
0
P
of the observ-
ing station
P
. The system
X
0
Y
0
Z
0
, being fixed with respect to the stars,
rotates with respect to the earth. The coordinates of
P
in this system are,
therefore, functions of time. Let
X
P
,Y
P
,Z
P
be the coordinates of
P
in the
usual geocentric coordinate system fixed with respect to the earth. In this
system, the
Z
-axis, coinciding with the
Z
0
-axis, is the earth's axis of rota-
tion; the
X
-axis lies in the mean meridian plane of Greenwich, corresponding
to the longitude
λ
=0
◦
;andthe
Y
-axis points to
λ
=90
◦
east. Figure 7.6
shows that
X
0
=
X
P
cos
θ
0
−
Y
P
sin
θ
0
,
P
Y
0
=
X
P
sin
θ
0
+
Y
P
cos
θ
0
,
(7-44)
P
Z
0
=
Z
P
.
P
The angle
θ
0
is called
Greenwich sidereal time
;itsvalueis
θ
0
=
ωt,
(7-45)
where
ω
is the angular velocity of the earth's rotation. It is proportional
to the time
t
and, in appropriate units, measures it. Thus, absolute Green-
wich time is needed to convert the terrestrial coordinates
X
P
,Y
P
,Z
P
to the
celestial coordinates
X
0
,Y
0
,Z
0
that are required in (7-42) and (7-43).
As a final step, we substitute the station coordinates, as given by (7-44),
and the satellite coordinates, as symbolized by (7-41), into (7-43), obtaining
expressions of the form
α
=
α
(
X
P
,Y
P
,Z
P
;
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
,
δ
=
δ
(
X
P
,Y
P
,Z
P
;
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
,
s
=
s
(
X
P
,Y
P
,Z
P
;
t
;
a
0
,e
0
,i
0
,
Ω
0
,ω
0
,T
0
;
C
nm
,S
nm
)
.
P
P
P
(7-46)