Global Positioning System Reference
In-Depth Information
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X
=
R 3 (
GAST ) R 1 (y p ) R 2 (x p ) x
(2.34)
The intermediary coordinate system (x),
x
=
R 1 (y p ) R 2 (x p ) x
(2.35)
is not completely crust-fixed, because the third axis moves with polar motion. (x) is
sometimes referred to as the instantaneous terrestrial coordinate system.
Using (X), the apparent right ascension and declination are computed from the
expression
tan 1 Y
X
α =
(2.36)
[25
Z
X 2
tan 1
δ =
(2.37)
+
Y 2
Lin
1 ——
No
PgE
with 0°
≤ α
< 360°. Applying (2.36) and (2.37) to (x) gives the spherical longitude
λ
an d latitude
, respectively. Whereas the zero right ascension is at the vernal equinox
an d zero longitude is at the reference meridian, both increase counterclockwise when
vi ewed from the third axis.
φ
2.2.2 Time Systems
[25
Th e GAST relates the terrestrial and celestial reference frames, as far as the earth's
da ily rotation is concerned, as is seen from (2.34). Twenty-four hours of GAST
re presents the time for two consecutive transits of the same meridian over the vernal
eq uinox (the direction of the X axis). Unfortunately, these “twenty-four” hours are
no t suitable to define a constant time interval. As seen from (2.33), GAST depends
on the nutation in longitude,
, which in turn is a function of time according to
(2 .18). The vernal equinox reference direction moves along the celestial equator by
th e time-varying amount
∆ψ
cos ε . In addition, the earth's daily rotation rate slows
do wn or speeds up. This rate variation can affect the length of day by about 1 ms,
co rresponding to a length of 4.5 m on the equator.
Let us assume that a geodetic space technique is available for which the mathe-
matical function between observations and parameters is known,
∆ψ
f X , x , GAST ,x p ,y p
=
(2.38)
While we do not go into the details of such solutions, one can readily imagine different
types of solutions depending on which parameters are unknown and the type of
observations available. For simplicity, let X (space object) and x (observing station)
be known. Then, given sufficient observational strength, it is conceptually possible to
solve (2.38) for GAST, and polar motion x p , and y p ,given . We could then compute
 
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