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)