Global Positioning System Reference
In-Depth Information
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The velocity expressed in the (q) coordinate system can be written as follows, using
(3.26), (3.42), and (3.48):
v 2
q 1
q 2
n 2 a 2
1
e 2 sin 2 f
cos 2 f
e 2
=
+
+
2 e cos f
+
e 2 2
1
e 2
(3.57)
µ
=
a 1
+
2 e cos f
2
r
1
a
= µ
[63
Equation (3.57) yields the expression for the semimajor axis
r
a
=
(3.58)
Lin
* 1 ——
No
*PgE
2
rv 2 /
µ
Fr om Equation (3.28), it follows that
1
1 / 2
h 2
µ
e
=
(3.59)
a
an d Equations (3.35), (3.47), and (3.48) give an expression for the eccentric anomaly:
[63
a
r
ae
cos E
=
(3.60)
q
e µ
q
·
sin E
=
(3.61)
a
Eq uations (3.60) and (3.61) together determine the quadrant of the eccentric anomaly.
Ha ving E , the true anomaly follows from (3.47):
tan 1 1
e 2 sin E
cos E
=
f
(3.62)
e
Fi nally, Kepler's equation yields the mean anomaly:
M
=
E
e sin E
(3.63)
Eq uations (3.50) to (3.63) comprise the transformation from ( X , X ) to the Kepler
ele ments.
Table 3.1 shows six examples of trajectories for which the orbital eccentricity
is zero, e
=
0. The satellites' positions x in the earth-centered earth-fixed (ECEF)
 
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