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)