Global Positioning System Reference
In-Depth Information
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cos f
sin f
0
=
q
r
(3.7)
Be cause the geocentric distance and the true anomaly are functions of time, the
de rivative with respect to time, denoted by a dot , is
+
cos f
sin f
0
sin f
cos f
0
f
q
r
r
(3.8)
The second derivatives with respect to time are
[57
+
+
cos f
sin f
0
sin f
cos f
0
sin f
cos f
0
cos f
sin f
0
f
f
r( f) 2
q
r
2
˙
r
r
(3.9)
Lin
9.2
——
No
PgE
Th e second derivative is written according to (3.4) and (3.7) as
cos f
sin f
0
= −µ
r 2
r
(3.10)
[57
Evaluating (3.9) and (3.10) at f
=
0 (perigee) and substituting (3.10) for the left-hand
sid e of (3.9) gives
= −µ
r 2
r( f) 2
r
¨
(3.11)
f
f
r
+
2
r
˙
=
0
(3.12)
Equation (3.12) is further developed by multiplying by r and integrating
r 2
f dt
f
+
2 r
r
˙
=
C
(3.13)
Th e result of the integration is
f
f
r 2
2 r 2
+
=
C
(3.14)
as can be readily verified through differentiation. Combining both terms yields
f
r 2
=
h
(3.15)
 
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