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 , 0 0 ,T 0 ; C nm ,S nm ) ,
δ = δ ( X P ,Y P ,Z P ; t ; a 0 ,e 0 ,i 0 , 0 0 ,T 0 ; C nm ,S nm ) ,
s = s ( X P ,Y P ,Z P ; t ; a 0 ,e 0 ,i 0 , 0 0 ,T 0 ; C nm ,S nm ) .
P
P
P
(7-46)
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