Global Positioning System Reference
In-Depth Information
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where h is a new constant. Equation (3.15) is identified as an angular momentum
equation, implying that the angular momentum for the orbiting satellite is conserved.
In order to integrate (3.11), we define a new variable:
1
r
u
(3.16)
By using Equation (3.15) for dt/df , the differential of (3.16) becomes
du
df =
du
dr
dr
dt
df =− ˙
dt
r
h
(3.17)
Differentiating again gives
˙
dt
df
[58
d 2 u
df 2
d
dt
r
h
r
u 2 h 2
¨
=
=−
(3.18)
or
Lin
4.3
——
Nor
PgE
d 2 u
df 2
h 2 u 2
r
¨
=−
(3.19)
By substituting (3.19) in (3.11), substituting f from (3.15) in (3.11), and replacing r
by u according to (3.16), Equation (3.11) becomes
d 2 u
df 2 +
h 2
[58
u
=
(3.20)
w hich can readily be integrated as
1
r
h 2
u
=
C cos f
+
(3.21)
where C is a constant.
Equation (3.21) is the equation of an ellipse. This is verified by writing the equation
for the orbital ellipse in Figure 3.1 in the principal axis form:
2
2
ξ
a 2 + η
=
1
(3.22)
b 2
where
ξ =
ae
+
r cos f
(3.23)
η =
r sin f
(3.24)
b 2
a 2 ( 1
e 2 )
=
(3.25)
 
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