Space methods - Physical Geodesy - page 263

The orbital elements in (7-34) refer to this instantaneous osculating el-
lipse, so that a = a 0 +∆ t a , etc. Therefore, the coordinates X 0 , Y 0 , Z 0
depend on the time in two ways: explicitly , through the true anomaly v ,and
implicitly , through the variable elements of the osculating orbit. We elimi-
nate the implicit dependence in the following way. We evaluate (7-34) using
the elements a 0 , etc., of the fixed reference ellipse. Then the coordinates so
obtained depend on the time only explicitly and correspond to a Keplerian
motion in space along a fixed ellipse. To convert them into true coordinates
X 0 , Y 0 , Z 0 , they must be corrected by ∆ t X 0 ,∆ t Y 0 ,∆ t Z 0 ,forwhichthe
linear terms of a Taylor expansion of (7-34) give
∆ t X 0 =

∂X 0
∂a

∆ t a ∂X 0
∂e

∆ t e + ∂X 0
∂i

∆ t i + ∂X 0
∂ Ω

∆ t Ω+ ∂X 0
∂ω

∆ t ω + ∂X 0
∂v

∆ t v,

∆ t Y 0 =

∂Y 0
∂a

∆ t a ∂Y 0
∂e

∆ t e + ∂Y 0
∂i

∆ t i + ∂Y 0
∂ Ω

∆ t Ω+ ∂Y 0
∂ω

∆ t ω + ∂Y 0
∂v

∆ t v,

∆ t Z 0 =

∂Z 0
∂a

∆ t a ∂Z 0
∂e

∆ t e + ∂Z 0
∂i

∆ t i + ∂Z 0
∂ Ω

∆ t Ω+ ∂Z 0
∂ω

∆ t ω + ∂Z 0

∂v ∆ t v.
(7-36)
The partial derivatives are readily obtained by differentiating (7-34); note
that r is a function of a , e ,and v .
In these equations, we have used the perturbation of the true anomaly,
∆ t v , instead of the perturbation of perigee epoch, ∆ t T .

Perturbations expressed in terms of C nm and S nm
The perturbations of the orbital elements are found by integrating (7-10):
∆ t a = t
t 0

∆ t e = t
t 0

˙ adt, edt, ... . (7-37)

A similar expression can be written for ∆ t v . The components S , T , W of the
perturbing force are expressed in terms of J n , C nm ,and S nm using equations
(7-12), (7-13), and (7-17), where the perturbing potential

n +1 J n P n (cos ϑ )

a e
r

∞

GM
a e

R = −

n =2

(7-38)

( C nm cos mλ + S nm sin mλ ) P nm (cos ϑ )

n

−

m =1