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cannot, however, observe these rates of change directly. Rather, we observe
the changes of the orbital elements after several revolutions. The changes
after
one
revolution, with period
P
,are
∆
a
=
t
0
+
P
t
0
∆
e
=
t
0
+
P
t
0
∆
i
=
t
0
+
P
t
0
˙
adt,
edt,
˙
ıdt,
etc.
(7-18)
The
t
0
is an arbitrary “epoch” (instant of time). In order to perform these
integrations, we must express
a, e, ...
in terms of one independent variable.
For this independent variable, we may take the time
t
or the true anomaly
v
. The second possibility will be adopted here.
The polar distance
ϑ
is expressed as a function of
v
through the relation
cos
ϑ
=sin(
ω
+
v
)sin
i,
(7-19)
which follows from the rectangular spherical triangle in Fig. 7.4. The radius
vector
r
is also a function of
v
according to (7-5). Finally, Kepler's second
law (7-7) furnishes the relation between
v
and the time
t
:
r
2
GM a
(1
dt
dv
=
e
2
)
.
(7-20)
−
Hence, we may change the integration variable from
t
to
v
, obtaining, for
instance,
∆
a
=
t
0
+
P
t
0
˙
adt
=
2
π
v
=0
da
dv
dv ,
(7-21)
where
r
2
GM a
(1
da
dv
=
da
dt
dt
dv
=
˙
a.
(7-22)
−
e
2
)
Analogous formulas result for the other orbital elements.
After performing all these operations, which are lengthy but not too
