ua n n
00
V nO .. C nO _ _ e L F nOp(l) L Gnpq(e) COS [(n-2p)w+(n-2p+q)M)
n+l
a
p-O
q_-oo
(1)
where, maintaining Kaula's notation: V nO is the disturbing potential arising from the
zonal harmonic term en'" JL is the gravitational constant times the mass of the Earth,
a. is the Earth's semi-major axis, a is the orbit semi-major axis, FnOp(i) and Gnpq( e) are
the inclination and eccentricity functions (Kaula, 1966; eqs. 3.62 and 3.66 respectively)
and wand M are the satellite's argument of perigee and mean anomaly. The
subscript n is the harmonic degree of the zonal harmonic having order, m = 0; the
p and q subscripts are integers used in the eccentricity and inclination functions and
arise from the harmonic decomposition of the geopotential.
The long periodic potential is, to first order, just those terms where (n-2p+q)=0,
that is:
u ann
"nO" C nO _ _ e L Fnop(l) G np (2p-n)(e) cos [(n-2p)w)
a n+l p=O
(2)
The long periodic zonal orbital perturbations are periodic with w. The period of the
odd zonals are odd sub-multiples of the apsidal period; whereas the even zonals have
periods which are the even sub-multiples, including the zenith or secular effects. The
chief effect of the zonals on the orbit is to produce periodic perturbations of the
same frequencies in the orbital elements excluding the semi-major axis, which has no
long periodic perturbations. The secular effect drops out of the eccentricity, e, and
the inclination, i.
This study uses data from the time of Ajisai's launch (in the fall of 1986) to the end
of 1994. In the fall of 1992, Lageos-2 was launched and these data are used for the
Table 1. SLR satellite characteristics
Satellite
Semi-major Eccentricity Inclination
Node rate
Arg perigee
axis (km)
(deg)
(deg/day)
(deg,tday)
Lageos-1
12,270.3
0.003
109.87
0.343
-0.213
Lageos-2
12,162.2
0.013
52.64
-0.632
0.438
Starlette
7,339.8
0.019
49.84
-3.935
3.295
Ajisai
7,871.1
0.001
50.03
-3.066
2.538
two years they are available. Unfortunately, Ajisai, Starlette and Lageos-2, while at different
altitudes, essentially share a common orbital inclination (see Table 1). This set of SLR data
were found capable of only yielding zonal harmonic rates for J 2 , J 3 and J 4 , where the J 3
estimate is in actuality, a lumped sum of the rates for J 3 and primarily J 5 •
ORBIT AND FORCE MODELING
The clearest manifestation of the secular change in the zonal harmonics is seen in the
apparent change in the orbital node rate. The apparent acceleration in the orbital node of
Lageos-1 was previously used to provide estimates of the secular change in J 2 (Yoder et al.,
1983; Rubincam, 1984). However, long period changes in the Earth's rotation rate can
produce very similar apparent orbital behavior although arising from an entirely different
153
00
V nO .. C nO _ _ e L F nOp(l) L Gnpq(e) COS [(n-2p)w+(n-2p+q)M)
n+l
a
p-O
q_-oo
(1)
where, maintaining Kaula's notation: V nO is the disturbing potential arising from the
zonal harmonic term en'" JL is the gravitational constant times the mass of the Earth,
a. is the Earth's semi-major axis, a is the orbit semi-major axis, FnOp(i) and Gnpq( e) are
the inclination and eccentricity functions (Kaula, 1966; eqs. 3.62 and 3.66 respectively)
and wand M are the satellite's argument of perigee and mean anomaly. The
subscript n is the harmonic degree of the zonal harmonic having order, m = 0; the
p and q subscripts are integers used in the eccentricity and inclination functions and
arise from the harmonic decomposition of the geopotential.
The long periodic potential is, to first order, just those terms where (n-2p+q)=0,
that is:
u ann
"nO" C nO _ _ e L Fnop(l) G np (2p-n)(e) cos [(n-2p)w)
a n+l p=O
(2)
The long periodic zonal orbital perturbations are periodic with w. The period of the
odd zonals are odd sub-multiples of the apsidal period; whereas the even zonals have
periods which are the even sub-multiples, including the zenith or secular effects. The
chief effect of the zonals on the orbit is to produce periodic perturbations of the
same frequencies in the orbital elements excluding the semi-major axis, which has no
long periodic perturbations. The secular effect drops out of the eccentricity, e, and
the inclination, i.
This study uses data from the time of Ajisai's launch (in the fall of 1986) to the end
of 1994. In the fall of 1992, Lageos-2 was launched and these data are used for the
Table 1. SLR satellite characteristics
Satellite
Semi-major Eccentricity Inclination
Node rate
Arg perigee
axis (km)
(deg)
(deg/day)
(deg,tday)
Lageos-1
12,270.3
0.003
109.87
0.343
-0.213
Lageos-2
12,162.2
0.013
52.64
-0.632
0.438
Starlette
7,339.8
0.019
49.84
-3.935
3.295
Ajisai
7,871.1
0.001
50.03
-3.066
2.538
two years they are available. Unfortunately, Ajisai, Starlette and Lageos-2, while at different
altitudes, essentially share a common orbital inclination (see Table 1). This set of SLR data
were found capable of only yielding zonal harmonic rates for J 2 , J 3 and J 4 , where the J 3
estimate is in actuality, a lumped sum of the rates for J 3 and primarily J 5 •
ORBIT AND FORCE MODELING
The clearest manifestation of the secular change in the zonal harmonics is seen in the
apparent change in the orbital node rate. The apparent acceleration in the orbital node of
Lageos-1 was previously used to provide estimates of the secular change in J 2 (Yoder et al.,
1983; Rubincam, 1984). However, long period changes in the Earth's rotation rate can
produce very similar apparent orbital behavior although arising from an entirely different
153