PLANETARY SATELLITE ORBITERS
Jean Paulo dos Santos Carvalho1 , Rodolpho Vilhena de Moraes2 , Antônio F. Bertachini de Almeida Prado3
1
UNESP- Univ Estadual Paulista, Guaratinguetá-SP, Brasil, [email protected]
UNESP- Univ Estadual Paulista, Guaratinguetá-SP, Brasil, [email protected]
3
Division of Space Mechanics and Control - INPE, São José dos Campos - SP, Brasil, [email protected]
2
Abstract: Low-altitude, near-polar orbits are very desirable
as science orbits for missions to planetary satellites, such as
Earth’s Moon. In this paper we present an analytical theory
with numerical simulations to study the orbital motion of lunar low altitude artificial satellite. We consider the problem
of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon (J2 − J5 , J7 , and C22 ). The conditions to get frozen orbits are presented. Using an approach
with the simple-averaged problem, we found families of periodic orbits for the problem of an orbiter around the moon,
where frozen orbits valid for long periods of time are found.
A comparison between the models for zonal and tesseral harmonics coefficients is presented.
Keywords: Planetary satellites, artificial satellite, frozen orbits.
1. INTRODUCTION
Frozen orbits about the Moon, natural satellites, or asteroids is of current interest because several space missions
have the goal of orbiting around such bodies (see for instance
[1-17] and references therein). This paper contains a continuation of the work developed by [12] and [15] where the
problem of the third-body gravitational perturbation and the
non-uniform distribution of mass of the Moon on a spacecraft
is studied using a simplified dynamical model.
For our search, we use a simplified dynamical model
that considers the effects caused by the nonsphericity of the
Moon (J2 − J5 , J7 , and C22 ) using simple-averaged analytical model over the short satellite period. The equations of
motion are obtained from the Lagrange planetary equations
and then integrated numerically these equations, using the
Maple software. Emphasis is given to the case of frozen
orbits, defined as orbits which have constant values of eccentricity, inclination and argument of periapsis on average
in the simple-averaged system. An approach is used for the
case of frozen orbits ([6], [17]). The basic idea is to eliminate
the terms due to the short time periodic motion of the space-
craft to show smooth curves for the evolution of the mean
orbital elements for a long time period. The interesting set of
results enable find good conditions close to frozen orbits for
future lunar missions.
The paper has four sections. Sect. 2 is devoted to the
terms due to nonsphericity of the Moon and to the Hamiltonian of the system. Applications’ taking into account the
long period disturbing potential using the averaging method
are presented in Sect. 3. In Sect. 4 is devoted to the conclusions.
2. OBLATENESS OF THE MOON
To analyze the motion of an artificial satellite of the Moon
it is necessary to take into account the Moon’s oblateness.
Besides that, the Moon is much less flattened than the Earth
it also causes perturbations in space vehicles. Table 1 shows
the orders of magnitude for the zonal and sectorial harmonics
for both celestial bodies to make a comparison. The term C20
describes the equatorial bulge of the Moon, often referred
oblateness. The coefficient C22 measures the ellipticity of
the equator.
Table 1 – Magnitude orders for J2 and C22 .
Coefficient
C20 ≡ (−J2 )
C22
Earth Moon
−10−3 −2 × 10−4
2 × 10−6 2 × 10−5
Then, we will present the Hamiltonian formalism using the classical Delaunay
canonical variables defined as:
√
√
L = µa, G = L 1 − e2 , H = Gcos(i), l = M mean
anomaly, g = ω argument of periapsis and h = Ω longitude
of the ascending node. The space vehicle is a point mass
in a three-dimensional orbit with orbital elements: a (semimajor axis), e (eccentricity), i (inclination), g (argument of
periapsis), h (longitude of the ascending node), and n (mean
motion) given by the third Kepler’s law n2 a3 = Gm0 .
The force function, the negative of the total energy as used
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1144
in physics, is given by F = K = V − T here, V is the
negative of the potential energy, and T is the kinetic energy.
µ
The force function can be put as: K = µr +R−T = 2a
+R
1 2 2
(3s sin (f + g) − 1)
2
(3)
5 3 3
3
(s sin (f + g)) − s sin(f + g)
2
2
(4)
P2 (sin(ϕ)) =
2
µ
or K = 2L
2 +R. The function R, comprising all terms of V
except the central term, is known as the disturbing potential.
The term due to the unperturbed potential is given by:
H0 =
µ2
2L2
P3 (sin(ϕ)) =
(1)
P4 (sin(ϕ)) =
35 4 4
s sin (f + g) −
8
15 2 2
3
s sin (f + g) +
4
8
Considering the lunar equatorial plane as the reference
plane, the disturbing potential can be written in the form
([8]):
VM =
P5 (sin(ϕ)) =
5
µ ∑ RM n
) Jn Pn (sin(ϕ)) −
(
− [
r n=2 r
(
RM 2
) C22 P22 (sin(ϕ) cos(2λ)]
r
(5)
63 5 5
s sin (f + g) −
8
35 2 3
s sin (f + g) +
4
15
s sin(f + g)
8
(2)
where µ is the Lunar gravitational constant, RM is the equatorial radius of the Moon (RM = 1738km), Pn represent the
Legendre polynomial, Pnm represent the associated Legendre polynomial, the angle ϕ is the latitude of the orbit with
respect to the equator of the Moon, the angle λ is the longitude measured from the direction of the longest axis of the
′
Moon, where λ = λ′ − λ22 , since λ is the longitude reckoned from any fixed direction, and λ22 is the longitude of
the Moon’s longest meridian from the same fixed direction.
However, λ22 contains the time explicitly ([1]). Using spherical trigonometry we have sin(ϕ) = sin(i) sin(f + g).
The following assumptions will be used in this work: a)
the motion of the Moon is uniform (librations are neglected),
b) the lunar equator lies in the ecliptic (we neglect the inclination of about 1.5◦ of the lunar equator to the ecliptic, and
the inclination of the lunar orbit to the ecliptic of about 5◦ ,
c) the longitude of the lunar longest meridian is equal to the
mean longitude of the Earth (librations are neglected), d) the
mean longitude of the Earth λ⊗ , is equal to λ22 .
Since the variables Ω and λ⊗ appear only as a combination of Ω − λ⊗ , where λ⊗ = nM t + const, nM being
the lunar mean motion, the degree of freedom can be reduced by choosing as a new variable h = Ω − λ⊗ . A new
term must then be added to the Hamiltonian in order to get
ḣ = − ∂(K)
∂(H) = nM . The Hamiltonian is still time-dependent
through λ⊗ . Since the longest meridian is always pointing
toward the Earth, it is possible to choose a rotating system
whose x-axis passes through this meridian.
Regarding the Earth’s potential, the dominant coefficient
is J2 . The rest are of higher order terms ([6]). In contrast
to the Earth, the first harmonics of the Lunar potential are all
almost of the same order (see Table 1). This fact complicates
the choice of the harmonic where the potential can be truncated and this makes its choice a little arbitrary. In terms of
the orbital elements, the Legendre associated functions for
zonal up to J5 and sectorial term C22 , can be written in the
form ([1], [8]):
(6)
P22 (sin(ϕ)) cos(2λ) =
6(ξ 2 cos2 (f ) + χ2 sin2 (f ) +
2ξχ sin(2f )) −
(7)
3(1 − s2 sin2 (f + g))
where ξ = cos(g) cos(h) − cos(i) sin(g) sin(h), χ =
− sin(g) cos(h) − cos(i) cos(g) sin(h), s = sin(i) and c =
cos(i).
Here it is just presented the development of the terms
containing J2 and C22 . The zonal perturbation due to the
oblateness is defined by H20 = ϵ1 rµ3 P2 (sin(ϕ)), where
2
ϵ1 = J2 RM
. However, the disturbing potential is: H20 =
ϵ1 4rµ3 (1 − 3c2 − 3s2 cos(2f + 2g)). Substituting the relation
µ = n2 a3 , using the Cayley’s tables ([18]) to express the
true anomaly in terms of the mean anomaly, and with some
algebraic manipulations we get:
H20 =
3
ϵ1(((5e2 − 2)c2 + 2 − 5e2 ) cos(2g + 3l) +
8
(7e − 7ec2 ) cos(2g + 3l) + (−17e2 c2 + 17e2 )
cos(2g + 4l) + (−e + ec2 ) cos(2g + l) +
1
1 2 2
9(c2 − )( + e cos(l) + e2 +
3 9 3
3
e2 cos(2l)))n2
(8)
For the sectorial perturbation ([1], [8]) we get: H22 =
δ rµ3 (6ξ 2 cos2 (f ) + 6χ2 sin2 (f ) + 12ξχ sin(2f ) − 3 +
2
. With some manip3s2 sin2 (f + g)), where δ = C22 RM
ulations we get
H22 =
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1145
45 2 2 2
δn ((e − )(c − 1)2 cos(2l + 2g − 2h) −
16
5
1
2
2 2
(c + 1) (e − ) cos(2l − 2g − 2h) +
3
5
2
2 2
(c + 1) (e − ) cos(2l + 2g + 2h) −
5
1 2 2
(e − )(c − 1)2 cos(2l − 2g + 2h) −
3
5
17 2
e (c − 1)2 cos(4l + 2g − 2h) −
5
7
e(c − 1)2 cos(3l + 2g − 2h) +
5
17 2
e (c + 1)2 cos(4l − 2g − 2h) +
15
7
e(c + 1)2 cos(3l − 2g − 2h) −
15
7
e(c + 1)2 cos(3l + 2g + 2h) −
5
17 2
e (c + 1)2 cos(4l + 2g + 2h) +
5
7
e(c − 1)2 cos(3l − 2g + 2h) +
15
17 2
e (c − 1)2 cos(4l − 2g + 2h) −
15
1
e(c + 1)2 cos(l − 2g − 2h) +
15
1
e(c − 1)2 cos(l + 2g − 2h)
5
1
e(c + 1)2 cos(l + 2g + 2h) −
5
1
e(c − 1)2 cos(l − 2g + 2h) −
15
2 2 2 2
s (e − ) cos(2l − 2g) +
3
5
9 2
7
e cos(2l − 2h) − e cos(3l − 2g) +
5
5
1
9 2
e cos(l − 2g) + e cos(2l + 2h) +
5
5
6 2
17
4
e cos(2h) + cos(2h) − e2 cos(4l − 2g) +
5
5
5
6
6
e cos(l + 2h) + e cos(l − 2h))
(9)
5
5
where disturbing potential is written in the form R= H20 +
H22 .
−
Hi0 + H70 + H22
2
H0 =
µ
+ nM H.
2L2
ϵi Hi + δH6
(12)
i=1
2
3
4
5
, ϵ2 = J3 RM
, ϵ3 = J4 RM
, ϵ4 = J5 RM
,
where ϵ1 = J2 RM
7
2
ϵ5 = J7 RM , δ = C22 RM .
We arrange the Hamiltonian K as:
H00 =
H11 =
µ2
+ nM H
2L2
5
∑
ϵi Hi + δH6
(13)
(14)
i=1
The disturbing terms are represented in the first order.
The motion of the spacecraft is studied under the simpleaveraged analytical model. The average is taken over the
mean motion of the satellite. The standard definition
for av∫ 2π
1
erage used in this research is ([20]) <F>= 2π
(F
)dl
where
0
l is the mean anomaly. Were the simple-average method is
applied in Eq. (14) to eliminate the terms of short-period, to
analyze the effect of the disturbing potential on the orbital
elements. Periodic terms will be calculated, substituting the
result in Lagrange’s planetary equations ([21]) and numerically integrated (developed analytical equations) using the
Maple software. The long period disturbing potential given
in the Appendix.
3. APPLICATIONS: THE LONG PERIOD DISTURBING POTENTIAL USING THE AVERAGING
METHOD
(11)
The definition of frozen orbit can be done using the following equations took from the Lagrange planetary equations:
i=2
where
5
∑
(10)
The Hamiltonian of the dynamical system associated to
the problem of the orbital motion of the satellite around of
the Moon taking into account the non-uniform distribution
of mass of the planetary satellite (J2 − J5 , J7 , and C22 ) can
be written in the following form:
5
∑
K = H0 +
With a simplified model for the disturbing potential it is
possible to do analyses the orbital motion of the satellite.
To study the lifetimes of low altitude Moon artificial satellites reference [2] taken into account the zonal terms J2 − J5
and the sectorial terms C22 and C31 in the disturbing potential. However, [17] shows that the effect of the term J7 is an
order of larger magnitude than the terms J3 − J5 (see Table
2, model given by [22]) and therefore it should be considered
in the potential. In [17] an analytic model is presented to find
frozen orbits for lunar satellite, where is taken into account
in the potential only the zonal harmonic J2 and J7 . In this
paper we present an approach taking into account the terms
J2 − J5 , J7 , and C22 .
2.1. THE HAMILTONIAN SYSTEM
K = H0 +
The term nM H is added to reduce the degree of freedom,
since the mean longitude of Jupiter is time-dependent ([1]).
Here, the term nM H is taken as order zero as suggested by
[19].
∑4
∑4
Now, doing i=1 H(i+1)0 = i=1 ϵi Hi , H70 = ϵ5H5
and H22 = δH6 , we get,
3.1. FROZEN ORBITS SOLUTIONS
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1146
Table 2 – Coefficient given by [23] (LP165P) and [22].
Konopliv et al, 2001
+2.03233710−4
+8.4759010−6
−9.5919310−6
+7.1540910−7
−1.3577710−5
−2.1774710−5
+2.235710−5
Coeff.
J2
J3
J4
J5
J6
J7
C22
Akim, 1966
+2.07010−4
+4.90010−6
+8.00010−7
−3.60010−6
−1.10010−6
−2.87010−5
+2.44730510−5
0,1
0,08
e
0,06
0,04
2050
2000
1950
1900 a
1850
1800
0,02
de
di
dg
= 0,
= 0,
=0
dt
dt
dt
0
(15)
For we obtain frozen orbits, let us say g = π/2 or g =
3π/2, and h = π/2 or h = 3π/2 (see [12], [15]).
0
0,4
0,8
1,2
i
1,6
Figure 2 – Akim, (1966)J2 − J5 , J7 and C22 . Green color is the
surface when g = 3π
, and the red is obtained for g = π2 .
2
0,1
0,08
0,1
0,06
0,08
e
0,04
0,02
0
0
0,4
0,8
i
1,2
2050
2000
1950
1900 a
1850
1800
1,6
Figure 1 – Akim, (1966) J2 − J5 , J7 . Green color is the surface
when g = 3π
, and the red is obtained for g = π2 .
2
Plugging the potential in Eq. (16) in Lagrange’s planetary equations ([21]) solving the equation dg/dt = 0 get the
equation results depending on three variables, e, i, and a that
is represented as a 3 − D surface that is represented in the
Figs. 1, 2, 3, 4. Points on this surface correspond to equilibria of the system in equations dg/dt = 0 and de/dt = 0, that
is, to frozen orbits. Figs. 1 and 3 take into account the coefficients J2 − J5 and J7 , while Figs. 2 and 4 take into account
the J2 − J5 , J7 and C22 terms. Comparing the models [22]
and [23] we can analyze that the results are very close, and
the model given by [23] it presents better results to obtain a
family of frozen orbit for inclinations above 69◦ .
For larger inclinations than 69◦ we found a family of
frozen orbit, as we can see in the Figs. 1, 2, 3, 4, mainly
considering the model LP165P that is the most recent model
for coefficients of the field of Lunar gravity ([23]). We also
observed that the term C22 contributed efficiently to obtaining of such orbits.
The curves obtained for constant-a sections are very similar, thus, in order to go into more detail, we fix a semi-major
e
0,06
0,04
2050
2000
1950
1900 a
1850
1800
0,02
0
0
0,4
0,8
i
1,2
1,6
Figure 3 – LP165P, Konopliv et al (2001) J2 − J5 , J7 . Green
color is the surface when g = 3π
, and the red is obtained for
2
g = π2 .
axis a = 1861km, which indeed corresponds to low orbits.
For the chosen value of a, we plot the curve i vs e (see 5 and
5). Fig. 5 don’t take into account the C22 term in the potential while, Fig. 6 consider the term sectorial. We observed an
increase of the eccentricity for inclinations between 28, 65◦
and 45, 84◦ (comparing Fig. 5 and Fig. 6) and we noticed
that for orbits with larger inclination than 69◦ the eccentricity decreases the variation amplitude obtaining a family of
frozen orbits (see Fig.6 ) with smaller amplitude than represented in for Fig. 5.
Taking into account that Fig. 6 represent a family of
frozen orbits for inclination of 90◦ , it is done an example
to illustrate the behavior of the frozen orbit for a case where
the semi-major axis is 1861 km as illustrated in Figs. 7, 8.
Where found orbit frozen with initial eccentricity equal
the 0.02 that can be applied for missions of long period with-
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1147
0,06
0,05
0,1
0,04
0,08
e 0,03
0,06
e
0,04
0,02
0,02
0
0
0,4
0,8
i
1,2
2050
2000
1950
1900 a
1850
1800
0,01
0
0
1
2
3
Figure 4 – LP165P, Konopliv et al (2001) J2 − J5 , J7 and C22 .
Green color is the surface when g = 3π
, and the red is obtained
2
for g = π2 .
4
5
6
g
1,6
Figure 7 – Model LP165P. Initial conditions: i = 90◦ , a =
1861km, e = 0.001, color black, e = 0.002, color red,
e = 0.003, color blue, e = 0.004 color green. Perturbations:
J2 − J5 , J7 and C22 .
0,04
0,1
0,03
0,08
e 0,02
0,06
e
0,01
0,04
0
0,02
0
0,5
1
1,5
g
2
2,5
3
e=0.01
e=0.02
0
e=0.03
0
0,4
0,8
1,2
1,6
e=0.04
i
Figure 5 – LP165P, Konopliv et al (2001) J2 − J5 and J7 . Green
color is obtained for g = 3π
, and the red is obtained for g = π2 .
2
0,1
0,08
0,06
e
0,04
0,02
0
0
0,4
0,8
1,2
1,6
i
Figure 6 – LP165P, Konopliv et al (2001) J2 − J5 , J7 and C22 .
Green color is obtained for g = 3π
, and the red is obtained for
2
g = π2 .
out need of doing orbit corrections for period of very long
time (larger than 4000 days). The set of initial conditions for
Figure 8 – Model LP165P. Initial conditions: i = 90◦ , a =
1861km. Perturbations: J2 − J5 , J7 and C22 .
frozen orbits with long lifetime is given by a = 1861km,
e = 0.02, i = 90◦ , g = 90◦ and h = 270◦ .
Another important factor that we must analyze is the
value of the argument of the periapsis that strong affects the
determination of the frozen orbit. As we previously show the
values of the periapsis to assure the condition of frozen orbit
are g = 90◦ or g = 270◦ , equally for the longitude of the
ascending node that are h = 90◦ or h = 270◦ . Since that,
in general we find frozen orbits for the value of g = 90◦ .
Figs. 9, 10, 11, 12 shows the behavior of the eccentricity
as a function of time for different values of the inclination.
Note that Fig. 9 presents an orbit with lesser variation of the
eccentricity for inclination of i = 90◦ considering g = 90◦ ,
and when we compare Fig. 9 with Fig. 10 we get that for
the value of g = 270◦ the orbit presents great variation of
the eccentricity. We also analyze for the case of initial eccentricity of the order of 10−3 and verify that the behavior of
the orbit doesn’t suffer the same effect as represented for the
case of the eccentricity of the order 10−2 , here the value of
g = 90◦ or g = 270◦ has smaller effect in the amplitude of
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1148
the eccentricity (see Figs. 11, 12).
0.12
0.11
4. CONCLUSION
0.1
0.09
0.08
e
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
100
200
300
400
500
t (day)
k1(J2-J5, J7 and C22) i(0)=30
i(0)=45
i(0)=65
i(0)=90
Figure 9 – LP165P Model, e = 0.02, g = 90◦ and h = 270◦ .
0.08
0.07
0.06
e
0.05
0.04
0.03
0.02
0.01
0
0
100
200
300
400
500
t (day)
k1(J2-J5, J7 and C22) i(0)=30
i(0)=45
i(0)=65
i(0)=90
Figure 10 – LP165P Model, e = 0.02, g = 270◦ and h = 90◦ .
0.1
0.09
0.08
In this paper, the dynamics of orbits around a planetary
satellite (Moon), taking into account the non-uniform distribution of mass was studied. Diverse analyses were done, using the long period disturbing potential, obtained through of
the simple-averaged model and substituting in the Lagrange
planetary equations.
Considering the long period disturbing potential we found
a family of frozen orbit for larger inclinations than 69◦ considering the model LP165P that is the most recent model for
coefficients of the field of Lunar gravity.
Low-altitude, near-polar orbits are very desirable for
scientific missions to study the satellites, such as the
Earth’s Moon. Considering the condition of low-altitude,
near-circular, near-polar frozen orbits the eccentricity that
presents smaller variation is 0.02 (or in the neighborhood of
0.02). The set of initial conditions for frozen orbits given for:
a = 1861km, e = 0.02, i = 90◦ , g = 90◦ and h = 270◦ ,
enable good conditions to frozen orbits for future lunar missions.
In general we concluded that to study the lifetime of lunar artificial satellite is necessary to consider analytic models taking into account the zonal (J2 − J7 ) and sectorial
(C22 ) harmonic. We observe that, the frozen orbits could be
strongly affected by terms of the potential containing the coefficient due to the equatorial ellipticity of the Moon (C22 ),
J7 and by terms containing the argument of the periapsis and
the longitude of the ascending node.
Appendix
The long period disturbing potential
0.07
e
0.06
k1 =
0.05
0.04
0.03
0.02
0.01
0
0
100
200
300
400
500
t (day)
k1(J2-J5, J7 and C22) i(0)=30
i(0)=45
i(0)=65
i(0)=90
Figure 11 – LP165P Model, e = 0.006, g = 90◦ and h = 270◦ .
0.09
0.08
0.07
0.06
e
0.05
0.04
0.03
0.02
0.01
0
0
100
200
300
400
500
t (day)
k1(J2-J5, J7 and C22) i(0)=30
i(0)=45
i(0)=65
i(0)=90
Figure 12 – LP165P Model, e = 0.006, g = 270◦ and h = 90◦ .
75 2 8807799
130330 2 31545
((−
c +
+
n (−
5
16a
163840
38129
38129
18432 2
11136
ϵ4 c4 + (
a ϵ3 +
c4 ϵ4 e4 + (−
14665
419419
12928
94208 2
104448
ϵ4)c2 −
a ϵ3 −
ϵ4)e2 −
27235
3774771
2097095
256
2048 2
1536
ϵ4 c4 + (
a ϵ3 +
ϵ4)c2 −
2933
419419
38129
2048
768
a2 ϵ3 −
ϵ4)s3 e3 sin(3 g) +
3774771
419419
10731721 5
1178377
s (ϵ4 (−
+ c2 )e2 +
1474560
696865
114912
6624
ϵ4 −
ϵ4 c2 −
696865
53605
36864 2
a ϵ3)e5 sin(5 g) −
7665515
49 2
(e + 2/7)ϵ2 (c2 − 1/7)a3 s2 e2 cos(2 g) +
40
12 5 2 2
δ a s (e + 2/3) cos(2 h) +
25
54197
ϵ4 s7 e7 sin(7 g) +
294912
2381620241 6 40098159 4
(c −
c −
(
1474560
30930133
Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications
Serra Negra, SP - ISSN 2178-3667
1149
13741935 2
33943925
+
c )ϵ4 e6 +
1020694389 30930133
5173 2
1964963
ϵ4 c6 + (−
a ϵ3 −
(
5120
160
2527 2
2256793
121765
ϵ4)c4 + (
ϵ4 +
a ϵ3)c2 −
5120
1024
120
637 2
4263
27027
a ϵ3 −
ϵ4)e4 + (
ϵ4 c6 +
480
1024
320
6237
441 2
a ϵ3 −
ϵ4)c4 +
(−
40
64
1701
147 2
a ϵ3 +
ϵ4 + ϵ1 a4 )c2 −
(
20
64
63
3003
21 2
a ϵ3 − 1/5 ϵ1 a4 −
ϵ4)e2 +
ϵ4 c6 +
40
64
320
693
189
21
ϵ4)c4 + (
ϵ4 + 2/5 ϵ1 a4 +
(− a2 ϵ3 −
10
64
64
2
ϵ1 a4 −
7/5 a2 ϵ3)c2 −
25
7
ϵ4 − 1/10 a2 ϵ3)se sin(g) −
64
3
8
147 4
((c − 6/7 c2 + )ϵ2 e4 + ( ϵ2 c4 +
32
35
21
16
64 2
64 2
a ϵ−
ϵ2)c2 +
a ϵ+
(−
1225
49
3675
8
8
16
ϵ2)e2 +
ϵ2 c4 + (−
ϵ2 −
245
105
245
128 2 2
128 2
8
a ϵ)c +
a ϵ+
ϵ2)a3 )
(16)
3675
11025
1225
where c = cos(i), s = sin(i), ϵ=R2 J2 , ϵ1=R3 J3 , ϵ2=R4 J4 ,
ϵ3=R5 J5 , ϵ4=R7 J7 and δ=R2 C22 . And RM (= 1738km) is
the equatorial radius of the Moon.
Where the values for the zonal (J2 − J5 , J7 ) and tesseral
(C22 ) harmonics coefficients taking into account two models
is given in Table 2 ([22], [23]).
ACKNOWLEDGMENTS
This work was accomplished with support of the FAPESP
under the contract N◦ 2007/04413-7 and 2006/04997-6, SPBrazil, and CNPQ (300952/2008-2).
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