Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

ARTIFICIAL SATELLITES: RESONANCE EFFECTS PRODUCED BY
TERRESTRIAL TIDE

Jarbas Cordeiro Sampaio

UNESP- Univ. Estadual Paulista, CEP. 12516-410 Guaratinguetá-SP, Brazil,
jarbascordeiro@gmail.com

Rodolpho Vilhena de Moraes

UNIFESP- Univ. Federal de São Paulo, CEP. 12231-280 São Josédos Campos, SP, Brazil,
rodolpho.vilhena@gmail.com

Abstract: Effects due to resonances in the orbital motion of artificialsatellites disturbed by the terrestrial tide are
analyzed. The nodal co-rotation resonance, apsidal co-rotation resonance and the Lidov-Kozai’s mechanism are
studied. The effects of the resonances are analyzed throughthe variations of the metric orbital elements. Libration
and circulation motions for high orbits with high eccentricities are verified for the Lidov-Kozai’s mechanism.

Keywords: Artificial Satellites. Resonances. Terrestrial Tide.

1 Introduction

Recent applications of artificial satellites require the description of the orbital motion under a precision of centime-
ters. In order to attain such actual expected level, severalperturbations must be considered simultaneously, as well
as resonant effects. Also for the analysis of the coupled effects it is mandatory the knowledge of each effect take
separately.

The resonant dynamics involving the orbital motion of artificial satellites have been extensively used for commu-
nication and navigation, for example, justifying the studyof resonant orbits in the last years (Rossi 2008, Formiga
and Vilhena de Moraes 2008, Sampaio et. al. 2011). This studies are done using orbital perturbations such as due to
the geopotential, lunisolar perturbations, solar radiation pressure, spin-orbit coupling, considering commensurabil-
ities between the artificial satellite’s mean motion and theEarth’s rotational motion. The perturbation considered
in this work is the terrestrial tide.

The influence of resonances in the orbital motion problem perturbed by tides has been observed by Kaula (1969),
Kozai (1965), Sampaio and Vilhena de Moraes (2008) and Sampaio (2009).

In this work the objective is to analyze the nodal co-rotation resonance , apsidal co-rotation resonance and the
Lidov-Kozai’s mechanism related to this problem.

2 The terrestrial tide effect

2.1 The disturbing function

The disturbing function due to the tide effect can be writtenas (Kozai 1973, Kozai 1965):

ROE = n2

1β
R5

r3

(

a1
r1

)3

k2P2(cos(θ)) (1)

wherek2 is the Love’s number;β is the reduced massm1/(m + m1); n1 is given byG(m + m1)/a
3
1, G is the

gravitational constant, m is the mass of the Earth,m1 is the mass of the Moon anda1 is the semi-major axis
of the Moon around the Earth; r is the geocentric distance of the satellite;r1 is the geocentric distance of the
Moon;P2(cos(θ)) is the second order Legendre polynomial in the development of the potential that describes the
perturbation due to the tide.

The disturbing function can be written as:

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Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

ROC = ROCS
(a, a1, e, I, I1) +ROCLP

(a, a1, e, I, I1, ω,Ω,Ω1) (2)

and then, the secular part can be put in the following form:

ROCS
=

1

32a3
n1

2β R5k2(2 + 27 e2 (cos (I))
2
(cos (I1))

2

+18 (cos (I))
2
(cos (I1))

2 − 9 e2 (cos (I1))
2

−9 e2 (cos (I))2 − 6 (cos (I1))
2 + 3 e2 − 6 (cos (I))2) (3)

The long period terms are given by:

ROCLP
= −

3

128a3
n1

2β R5k2(−6 e2 (cos (I))
2
cos (2ω) + 6 e2 cos (2ω)

−12 e2 sin (I) sin (I1) cos (I1) cos (2ω − Ω+ Ω1)

+12 e2 sin (I) sin (I1) cos (I1) cos (−2ω − Ω+ Ω1)

−48 e2 sin (I) sin (I1) cos (I) cos (I1) cos (−Ω+ Ω1)

+12 e2 (cos (I))
2
cos (2Ω1 − 2Ω)

+3 e2 (cos (I1))
2
cos (2Ω1 − 2Ω + 2ω)

+3 e2 (cos (I1))
2
cos (2Ω1 − 2Ω− 2ω)

−6 e2 cos (I) cos (2Ω1 − 2Ω− 2ω)

−8 (cos (I))2 (cos (I1))
2 cos (2Ω1 − 2Ω)

−12 e2 (cos (I))
2
(cos (I1))

2
cos (2Ω1 − 2Ω)

+3 e2 (cos (I))
2
(cos (I1))

2
cos (2Ω1 − 2Ω− 2ω)

−6 e2 cos (I) (cos (I1))
2
cos (2Ω1 − 2Ω + 2ω)

+3 e2 (cos (I))2 (cos (I1))
2 cos (2Ω1 − 2Ω + 2ω)

−18 e2 (cos (I1))
2
cos (2ω)

+18 e2 (cos (I))
2
(cos (I1))

2
cos (2ω)

−8 cos (2Ω1 − 2Ω)

+12 e2 sin (I) sin (I1) cos (I) cos (I1) cos (−2ω − Ω+ Ω1)

+6 e2 cos (I) (cos (I1))
2
cos (2Ω1 − 2Ω− 2ω)

−32 sin (I) sin (I1) cos (I) cos (I1) cos (−Ω+ Ω1)

+8 (cos (I1))
2
cos (2Ω1 − 2Ω)

+12 e2 sin (I) sin (I1) cos (I) cos (I1) cos (2ω − Ω+ Ω1)

+12 e2 (cos (I1))
2
cos (2Ω1 − 2Ω)− 3 e2 (cos (i))

2
cos (2Ω1 − 2Ω + 2ω)

−3 e2 (cos (I))
2
cos (2Ω1 − 2Ω− 2ω) + 6 e2 cos (I) cos (2Ω1 − 2Ω + 2ω)

−3 e2 cos (2Ω1 − 2Ω + 2ω)− 12 e2 cos (2Ω1 − 2Ω)

−3 e2 cos (2Ω1 − 2Ω− 2ω) + 8 (cos (I))
2
cos (2Ω1 − 2Ω)) (4)

3 The nodal co-rotation resonance

Initially, the analysis of the resonance will be done through the nodal co-rotation resonance, due to the argument
(Ω-Ω1) in the long period part of the disturbing function. Considering thatΩ = Ω0 + nΩt, will be considered here
the 1:1 resonance between the ascending node longitude of the satellite and the disturbing body, in this case the
Moon.

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Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

3.1 Terrestrial tide

Through the Lagrange planetary equations the following result for the precession of the ascen-ding node longitude
Ω can be obtained:

nΩ = −
3

16

n1
2β R5k2 cos (I)

na5
√
1− e2

(9 e2 (cos (I1))
2
+ 6 (cos (I1))

2 − 3 e2 − 2) (5)

with

n2 =
Gm

a3
(6)

The family of orbits with nodal co-rotation resonance can beobtained isolating the semi-major axis (a) and using
the Eqs. (5) and (6) we get:

a = 7

√

9

256

1

GmnΩ
2(1− e2)

(n1
2β2R10k2

2cos(I)
2
(9e2cos(I1)

2

+6cos(I1)
2 − 3e2 − 2)2)1/7 (7)

Due to the nodal co-rotation condition we get:

nΩ = nΩ1
(8)

where the period ofnΩ1
is about 18.6 years (Roncoli 2005). We also have:

nΩ1
=

2π

T
(9)

being T the period of 18.6 years.

Therefore, it was derived an equation for the semi-major axis of the satellite as a function of the eccentricity and
inclination . The other terms that appear in the Eq. (7) are constants.

The variation ofa as a function of e for some values of the inclination is given in Fig. 1.

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I=10 degrees I=20 degrees I=30 degrees
I=40 degrees I=50 degrees I=60 degrees
I=70 degrees I=80 degrees

e
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a (m)

600,000

700,000

800,000

900,000

1,000,000

1,100,000

1,200,000

1,300,000

1,400,000

1,500,000

Figure 1. Semi-major axis a function of the eccentricity for some inclination values (Nodal co-rotation
resonance)

The analysis of this type of resonance is not viable for the exact 1:1 resonance, since, in this case we will get
semi-major axis smaller than the Earth’s radius. But it can be studied what happens whennΩ approaches to zero
in the region next to the 1:1 resonance. It can be observed that whennΩ decreases about two orders of magnitude,
possible values for the semi-major axis begin to appear considering high eccentricities.

Figure 2 shows a simulation consideringnΩ near to zero.

I=10 degrees I=20 degrees I=30 degrees
I=40 degrees I=50 degrees I=60 degrees
I=70 degrees I=80 degrees

e
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a (m)

3,000,000

4,000,000

5,000,000

6,000,000

7,000,000

Figure 2. Variation of the semi-major axis with the eccentricity for some inclinations (nodal co-rotational
resonance for nΩ near to zero)

It is interesting to question how the occurrence of a quasi-commensurability can affect the motion of an artificial
satellite. Of course, the more next to an entire number is theratio between the periods, but the artificial satellite
can be affected with this effect (Callegari Jr 2006). This can be observed in Fig. 2.

4 The apsidal co-rotation resonance

Another analysis that is also done in this work is about the argument (2ω − Ω + Ω1) present in the disturbing
function due to the terrestrial tide. In this case, the exactresonance between the argument of perigee and the
longitude of the ascending node of the disturbing body is 2:1.

With the same considerations done in the previous topics, when analyzing the nodal co-rotational resonance, we
get the following equation fornω:

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Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

nω =
3

16na5
√
1− e2

n1
2β R5k2(15 (cos (I))2 (cos (I1))

2 − 3 (cos (I1))
2

+3 e2 (cos (I1))
2 − 5 (cos (I))

2
+ 1− e2) (10)

Inserting in the Eq. (10) the mean motion of the satellite given by Eq. (6) and isolating the value of the semi-major
axis (a) we get:

a = 7

√

9

256

n1
4β2R10k2

2

Gmn2
ω(1 − e2)

((15cos2(I)cos2(I1)− 3cos2(I1)

+3e2cos2(I1)− 5cos2(I) + 1− e2)2)1/7 (11)

Figure 3 shows the variation ofa with respect the eccentricity(0.0001 ≤ e ≤ 0.9), considering some values for
the inclination.

I=10 degrees I=20 degrees I=30 degrees
I=40 degrees I=50 degrees I=60 degrees
I=70 degrees I=80 degrees

e
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

a (m)

200,000

400,000

600,000

800,000

1,000,000

1,200,000

Figure 3. Apsidal co-rotation resonance: Variation of the semi-major axis with respect the eccentricity

In the Fig. (3), a great variation in the semi-major axis can be observed for some given inclinations and eccen-
tricities. A simulation shows that for small variations of inclination, above600, and for high eccentricities, large
variations occur in the semi-major axis. This phenomenon motivated the study of the Lidov-Koza’s mechanism
that will be seen in the next section.

5 Lidov-Kozai’s mechanism

In this section an analysis of resonance, involving the argument2ω, is done through the distur-bing function.

This kind of phenomenon was first observed by Kozai (1962) andLidov (1962) and is known as the Lidov-
Kozai’s mechanism. The Lidov-Kozai’s mechanism has been studied for asteroids as in Kozai (1962), for Uranus’s
satellites as Kinoshita and Nakai (1999, 2007), for Jupiter’s satellites as in Yokoyama et. al. (2003) and for artificial
satellite orbiting the Moon as in Carvalho et. al. (2008).

The disturbing function, considered only the secular termsand periodic terms involving the argument2ω, is:

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Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

ULK =
1

32a3
n1

2β R5k2(2 + 27 e2 (cos (I))2 (cos (I1))
2 + 18 (cos (I1))

2 (cos (I))2

−9 e2 (cos (I1))
2 − 9 e2 (cos (I))

2 − 6 (cos (I1))
2
+ 3 e2 − 6 (cos (I))

2
)

+
3

128a3
n1

2β R5k2(6 e
2 (cos (I))

2
cos (2ω)− 6 e2 cos (2ω)

+18 e2 (cos (I1))
2
cos (2ω)− 18 e2 (cos (I))

2
(cos (I1))

2
cos (2ω)) (12)

As it was done by Kinoshita and Nakai (1999, 2007), graphics for curves of the same energy in the (ω,e) plane will
be done here. The study of the Lidov-Kozai’s mechanism is based on the parameter h, related with the z component
of the angular momentum, given by:

h = (1− e2)cos2(I) = const. (13)

To generate the energy’s curves, firstly the Eq. (12) must be written in terms of the eccentricity, of the argument of
perigee and of the parameter h.

This can be done using Eq. (13).

As an example, it will be initially considered here a semi-major axis of about 6960 Km for the orbit of the artificial
satellite.

Figures (4) to (8) present the (ω, e) plane for some values of h.

w rad
0 1 2 3

e

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 4. (ω, e) plane for h=0.1

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w rad
0 1 2 3

e

0.1

0.2

0.3

0.4

0.5

Figure 5. (ω, e) plane for h=0.3

w rad
0 1 2 3

e

0.2

0.3

0.4

0.5

0.6

Figure 6. (ω, e) plane for h=0.5

w rad
0 1 2 3

e

0.2

0.3

0.4

0.5

0.6

0.7

Figure 7. (ω, e) plane for h=0.7

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w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

Figure 8. (ω, e) plane for h=0.9

For the considered semi-major axis we found a libration motion for small values of h ( h=0.1 and h=0.3) and
circulation for values near to 1 (h=0.7 and h=0.9). However,the libration and circulation motions shown in Figs.
(4) to (8) occur for high eccentricities, which would be a notpractical orbit. In fact, with the adopted semi-major
axis, this would result perigee distance shorter than the Earth’s radius.

For an artificial satellite orbiting the Earth, the Lidov-Kozai’s mechanism would have influence, through the tide
perturbation, for orbital semi-major axis greater than theone presented previously.

Analyzing the orbit of a satellite with semi-major axis of about 26500km, it is observed that as the eccentricity is
greater than 0.75, the distance of perigee is near to the terrestrial surface and can be affected significantly by other
disturbing forces such as atmospheric drag, or even collidewith the Earth’s surface if the eccentricity reach 0.8.
Starting from (e=0.75), when the eccentricity decreases, the perigee distance of the satellite became not as near
from the terrestrial surface as in the precedent case.

Figures (9) to (17) show the (ω, e) plane for h varying from 0.1 to 0.9.

w rad
 

0 1 2 3

e

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 9. (ω, e) plane for h=0.1

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w rad
0 1 2 3

e

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 10. (ω, e) plane for h=0.2

w rad
0 1 2 3

e

0.1

0.2

0.3

0.4

0.5

Figure 11. (ω, e) plane for h=0.3

w rad
0 1 2 3

e

0.2

0.3

0.4

0.5

Figure 12. (ω, e) plane for h=0.4

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Sampaio, J. C.; Vilhena de Moraes, R. Artificial satellites: resonance effects produced by...

w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

0.8

Figure 13. (ω, e) plane for h=0.5

w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

0.8

Figure 14. (ω, e) plane for h=0.6

w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

0.8

Figure 15. (ω, e) plane for h=0.7

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w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

0.8

Figure 16. (ω, e) plane for h=0.8

w rad
0 1 2 3

e

0.3

0.4

0.5

0.6

0.7

0.8

Figure 17. (ω, e) plane for h=0.9

For a satellite with the considered semi-major axis, and with the considered ranges for the variation of the
constant h, a libration motion is possible when0.1 ≤ h ≤ 0.3. This can be observed in the Figs. (9) to (17). For
values of h near to 0.1 librations occur but due to the high eccentricities, collision can occur with the Earth.

6 Conclusion

Some influences of the terrestrial tide on the orbital motionof artificial satellites were analyzed and the existence of
resonances that can affect the motion was studied. Three types of resonance were analyzed: the nodal co-rotation
resonance, the apsidal resonance and the Kozai’s resonance.

According to Kozai (1965), the most important resonant termexpected in the present study is the term with ar-
gumentΩ − Ω1 in the Lagrange equation providing time variation for the inclination. The Lagrange equations
for the time variation of the eccentricity and for the longitude of the ascending node, also present terms with this
argument.

Beyond the nodal co-rotation resonance, was also analyzed the influence on the orbital motion of the apsidal
co-rotation resonance, relating the argument of pericentre and the longitude of the ascending node of the Moon,
through a resonance of the type 2:1.

The sign of the variables of the arguments, as well as, the sign of the solutions fornΩ has influence, if the motion
is prograde or retrograde.

Now analyzing the Lidov-Kozai’s mechanism, it is observed that for the low altitude satellite considered, the values
of the eccentricity are large enough to occur collision withthe Earth’s surface. Now, for a satellite with semi-major

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axis of about 26500 km, where the tidal effects in the orbitalelements are still appreciable, there are possibilities
of libration for h=0.2 and h=0.3 and circulation for values greater than h=0.3. For h=0.2, considering the range of
values of eccentricity varying from 0.1 to 0.7, the inclination varies with values ranging from63.290 to 51.230 for
the prograde motion and116.710 to 128.770 for the retrograde motion. For h=0.3 the eccentricity varies from 0.1
to 0.5. This corresponds to a variation of the inclination of56.600 to 50.770 for the prograde motion and for the
retrograde motion of123.400 to 129.230.

7 Acknowledgments

This work was accomplished with support of the CAPES, FAPESPunder the contract No 2009/00735-5 and
2006/04997-6, SP-Brazil, and CNPQ (Process: 300952 / 2008-2).

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