AU(t) + {3; -Ji . ]AUr(t) -i,u1AQr(t) = 'I' U(t)
AP(t) + iWM>{t) = 'I' p{t)
AQ{t) - niJAQr{t) - IT;.AUr{t) = 'I' Q{t)
(3)
Here if is the average mean motion of the orbit, Jia' Jib Va' vI are coupling constants of
order h, and subscript r indicates the real part of the complex quantity. In analogy to the
well known approach used in analysis of the variations of the Earth's rotation, the right
hand sides of eq. (3) are termed excitations. Analysis of the 3 complex excitation time
series tells us about the dynamical model improvements required to best fit the SLR data.
Note that the excitation equations for the anomalous and node vectors are weakly coupled
first order systems while that of the eccentricity vector is a simple harmonic oscillator with
its resonant frequency at one cycle per perigee rotation. The factor 3ii I 2 represents the
strong dependence of the mean motion on the semimajor axis. The ita and p. I coupling
frequencies derive from the fact that the h perturbation of the mean element rates depends
upon perturbations in a and I respectively. The va and VI arise from dependence of the
12 forced nodal precession upon a and I. Table 1 contains values of these coupling
parameters for the satellites used in this study. Expressions for these quantities and a
complete derivation of the linearized excitation equations is contained in Eanes (1995).
Orbital Excitations from RTN and SWN Accelerations
If the perturbing acceleration is expressed in radial, transverse, and normal components
(RTN) the long-period (averaged over the argument of latitude) orbital excitations are
caused by the mean transverse and mean radial components and 1 cpr accelerations in all
three components according to
2 (- -)
'I'u{LP) =::= T -iR
an
'I' p{LP) = 'lip = :_[(Tc + ~Rs)-i(Ts - ~Rc)]
an
'I'Q{LP)='I'+Q = 1 (Nc-iNs)
2an
(4)
where the critical spectral components of the accelerations (R , RC' etc.) are assumed to
Table 1. Coupling constants (yr -1) for the non-singular orbital vector excitation equations .
Satellite
..:..
3n
.
Jia-T
JiI
va
VI
W
Lageos-l
-21972.
14.4
7.18
-5.69
-1.37
Lageos-2
-22285.
-22.4
-11.21
-4.19
2.79
Starlette
-47609.
-134.4
-67.28
-22.76
21.10
Ajisai
-42865.
-105.0
-52.58
-17.89
16.26
32
AP(t) + iWM>{t) = 'I' p{t)
AQ{t) - niJAQr{t) - IT;.AUr{t) = 'I' Q{t)
(3)
Here if is the average mean motion of the orbit, Jia' Jib Va' vI are coupling constants of
order h, and subscript r indicates the real part of the complex quantity. In analogy to the
well known approach used in analysis of the variations of the Earth's rotation, the right
hand sides of eq. (3) are termed excitations. Analysis of the 3 complex excitation time
series tells us about the dynamical model improvements required to best fit the SLR data.
Note that the excitation equations for the anomalous and node vectors are weakly coupled
first order systems while that of the eccentricity vector is a simple harmonic oscillator with
its resonant frequency at one cycle per perigee rotation. The factor 3ii I 2 represents the
strong dependence of the mean motion on the semimajor axis. The ita and p. I coupling
frequencies derive from the fact that the h perturbation of the mean element rates depends
upon perturbations in a and I respectively. The va and VI arise from dependence of the
12 forced nodal precession upon a and I. Table 1 contains values of these coupling
parameters for the satellites used in this study. Expressions for these quantities and a
complete derivation of the linearized excitation equations is contained in Eanes (1995).
Orbital Excitations from RTN and SWN Accelerations
If the perturbing acceleration is expressed in radial, transverse, and normal components
(RTN) the long-period (averaged over the argument of latitude) orbital excitations are
caused by the mean transverse and mean radial components and 1 cpr accelerations in all
three components according to
2 (- -)
'I'u{LP) =::= T -iR
an
'I' p{LP) = 'lip = :_[(Tc + ~Rs)-i(Ts - ~Rc)]
an
'I'Q{LP)='I'+Q = 1 (Nc-iNs)
2an
(4)
where the critical spectral components of the accelerations (R , RC' etc.) are assumed to
Table 1. Coupling constants (yr -1) for the non-singular orbital vector excitation equations .
Satellite
..:..
3n
.
Jia-T
JiI
va
VI
W
Lageos-l
-21972.
14.4
7.18
-5.69
-1.37
Lageos-2
-22285.
-22.4
-11.21
-4.19
2.79
Starlette
-47609.
-134.4
-67.28
-22.76
21.10
Ajisai
-42865.
-105.0
-52.58
-17.89
16.26
32