The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance

The orbital dynamics of synchronous satellites is studied. The 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth completes one. In the development of the geopotential, the zonal harmonics J20 and J40 and the tesseral harmonics J22 and J42 are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. The Lyapunov exponents are used as tool to analyze the chaotic orbits.


Introduction
Synchronous satellites in circular or elliptical orbits have been extensively used for navigation, communication, and military missions.This fact justifies the great attention that has been given in literature to the study of resonant orbits characterizing the dynamics of these satellites since the 60s 1-14 .For example, Molniya series satellites used by the old Soviet Union for communication form a constellation of satellites, launched since 1965, which have highly eccentric orbits with periods of 12 hours.Another example of missions that use eccentric, inclined, and synchronous orbits includes satellites to investigate the solar magnetosphere, launched in the 90s 15 .
The dynamics of synchronous satellites are very complex.The tesseral harmonics of the geopotential produce multiple resonances which interact resulting significantly in nonlinear motions, when compared to nonresonant orbits.It has been found that the orbital Mathematical Problems in Engineering elements show relatively large oscillation amplitudes differing from neighboring trajectories 11 .
Due to the perturbations of Earth gravitational potential, the frequencies of the longitude of ascending node Ω and of the argument of pericentre ω can make the presence of small divisors, arising in the integration of equation of motion, more pronounced.This phenomenon depends also on the eccentricity and inclination of the orbit plane.The importance of the node and the pericentre frequencies is smaller when compared to the mean anomaly and Greenwich sidereal time.However, they also have their contribution in the resonance effect.The coefficients l, m, p which define the argument φ lmpq in the development of the geopotential can vary, producing different frequencies within the resonant cosines for the same resonance.These frequencies are slightly different, with small variations around the considered commensurability.
In this paper, the 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth carries one.In the development of the geopotential, the zonal harmonics J 20 and J 40 and the tesseral harmonics J 22 and J 42 are considered.The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration.In the reduced dynamical model, three critical angles, associated to the tesseral harmonics J 22 and J 42 , are studied together.Numerical results show the time behavior of the semimajor axis, argument of pericentre and of the eccentricity.The Lyapunov exponents are used as tool to analyze the chaotic orbits.

Resonant Hamiltonian and Equations of Motion
In this section, a Hamiltonian describing the resonant problem is derived through successive Mathieu transformations.
Consider 2.1 to the Earth gravitational potential written in classical orbital elements 16, 17 where μ is the Earth gravitational parameter, μ 3.986009 × 10 14 m 3 /s 2 , a, e, I, Ω, ω, M are the classical keplerian elements: a is the semimajor axis, e is the eccentricity, I is the inclination of the orbit plane with the equator, Ω is the longitude of the ascending node, ω is the argument of pericentre, and M is the mean anomaly, respectively; a e is the Earth mean equatorial radius, a e 6378.140km, J lm is the spherical harmonic coefficient of degree l and order m, F lmp I and G lpq e are Kaula's inclination and eccentricity functions, respectively.The argument φ lmpq M, ω, Ω, θ is defined by where θ is the Greenwich sidereal time, θ ω e t ω e is the Earth's angular velocity, and t is the time , and λ lm is the corresponding reference longitude along the equator.
In order to describe the problem in Hamiltonian form, Delaunay canonical variables are introduced,

2.3
L, G, and H represent the generalized coordinates, and , g, and h represent the conjugate momenta.
Using the canonical variables, one gets the Hamiltonian F, with the disturbing potential R lm given by The argument φ lmpq is defined by and the coefficient B lmpq L, G, H is defined by The Hamiltonian F depends explicitly on the time through the Greenwich sidereal time θ.A new term ω e Θ is introduced in order to extend the phase space.In the extended phase space, the extended Hamiltonian H is given by

2.8
For resonant orbits, it is convenient to use a new set of canonical variables.Consider the canonical transformation of variables defined by the following relations: where X, Y, Z, Θ, x, y, z, θ are the modified Delaunay variables.
The new Hamiltonian H , resulting from the canonical transformation defined by 2.9 , is given by where the disturbing potential R lm is given by B lmpq X, Y, Z cos φ lmpq x, y, z, θ .

2.11
Now, consider the commensurability between the Earth rotation angular velocity ω e and the mean motion n μ 2 /X 3 .This commensurability can be expressed as qn − mω e ∼ 0, 2.12 considering q and m as integers.The ratio q/m defining the commensurability will be denoted by α.When the commensurability occurs, small divisors arise in the integration of the equations of motion 9 .These periodic terms in the Hamiltonian H with frequencies qn − mω e are called resonant terms.The other periodic terms are called short-and long-period terms.
The short-and long-period terms can be eliminated from the Hamiltonian H by applying an averaging procedure 12, 18 :

2.13
The variables ξ sp and ξ lp represent the short-and long-period terms, respectively, to be eliminated of the Hamiltonian H .The long-period terms have a combination in the argument φ lmpq which involves only the argument of the pericentre ω and the longitude of the ascending node Ω.From 2.10 and 2.11 , these terms are represented by the new variables in the following equation:

2.14
The short-period terms are identified by the presence of the sidereal time θ and mean anomaly M in the argument φ lmpq ; in this way, from 2.10 and 2.11 , the term H sp in the new variables is given by the following equations:

2.15
The term ζ p represents the other variables in the argument φ lmpq , including the argument of the pericentre ω and the longitude of the ascending node Ω, or, in terms of the new variables, y − z and z, respectively.A reduced Hamiltonian H r is obtained from the Hamiltonian H when only secular and resonant terms are considered.The reduced Hamiltonian H r is given by B lmp αm X, Y, Z cos φ lmp αm x, y, z, θ .
The dynamical system generated from the reduced Hamiltonian, 2.16 , is given by The equations of motion dX/dt, dY/dt, and dZ/dt defined by 2.17 are

2.20
From 2.18 to 2.20 , one can determine the first integral of the system determined by the Hamiltonian H r .Equation 2.18 can be rewritten as

2.24
In this way, the canonical system of differential equations governed by H r has the first integral generated from 2.24 : where C 1 is an integration constant.Using this first integral, a Mathieu transformation 26 can be defined.This transformation is given by the following equations:

2.27
The subscript 1 denotes the new set of canonical variables.Note that Z 1 C 1 , and the z 1 is an ignorable variable.So the order of the dynamical system is reduced in one degree of freedom.
Substituting the new set of canonical variables, in the reduced Hamiltonian given by 2.16 , one gets the resonant Hamiltonian.The word "resonant" is used to denote the Hamiltonian H rs which is valid for any resonance.The periodic terms in this Hamiltonian are resonant terms.The Hamiltonian H rs is given by

2.28
Mathematical Problems in Engineering 7 The Hamiltonian H rs has all resonant frequencies, relative to the commensurability α, where the φ lmp αm argument is given by with

2.30
The secular and resonant terms are given, respectively, by B 2j,0,j,0 X 1 , Y 1 , C 1 and Each one of the frequencies contained in dx 1 /dt, dy 1 /dt, dθ 1 /dt is related, through the coefficients l, m, to a tesseral harmonic J lm .By varying the coefficients l, m, p and keeping q/m fixed, one finds all frequencies dφ 1,lmp αm /dt concerning a specific resonance.

2.31
The Hamiltonian H 1 is defined considering a fixed resonance and three different critical angles associated to the tesseral harmonic J 22 ; the critical angles associated to the tesseral harmonic J 42 have the same frequency of the critical angles associated to the J 22 with a difference in the phase.The other terms in H rs are considered as short-period terms.
Table 1 shows the resonant coefficients used in the Hamiltonian H 1 .
Finally, a last transformation of variables is done, with the purpose of writing the resonant angle explicitly.This transformation is defined by

2.41
Since the term ω e Θ 4 is constant, it plays no role in the equations of motion, and a new Hamiltonian can be introduced, 2.42 The dynamical system described by H 4 is given by

2.45
In Section 4, some results of the numerical integration of 2.43 are shown.

Lyapunov Exponents
The estimation of the chaoticity of orbits is very important in the studies of dynamical systems, and possible irregular motions can be analyzed by Lyapunov exponents 23 .In this work, "Gram-Schmidt's method," described in 23-26 , will be applied to compute the Lyapunov exponents.A brief description of this method is presented in what follows.
The dynamical system described by 2.43 can be rewritten as

3.1
Introducing Equations 3.2 can be put in the form dz dt Z z .

3.3
The variational equations, associated to the system of differential equations 3.3 , are given by dζ dt Jζ, 3.4 where J ∂Z/∂z is the Jacobian.The total number of differential equations used in this method is n n 1 , n represents the number of the motion equations describing the problem, in this case four.In this way, there are twenty differential equations, four are motion equations of the problem and sixteen are variational equations described by 3.4 .
The dynamical system represented by 3.3 and 3.4 is numerically integrated and the neighboring trajectories are studied using the Gram-Schmidt orthonormalization to calculate the Lyapunov exponents.
The method of the Gram-Schmidt orthonormalization can be seen in 25, 26 with more details.A simplified denomination of the method is described as follows.
Considering the solutions to 3.4 as u κ t , the integration in the time τ begins from initial conditions u κ t 0 e κ t 0 , an orthonormal basis.At the end of the time interval, the volumes of the κ-dimensional κ 1, 2, . . ., N produced by the vectors u κ are calculated by where is the outer product and • is a norm.3.  3.
In this way, the vectors u κ are orthonormalized by Gram-Schmidt method.In other words, new orthonormal vectors e κ t 0 τ are calculated, in general, according to   3.
With new vector u κ t 0 τ e κ t 0 τ , the integration is reinitialized and carried forward to t t 0 2τ.The whole cycle is repeated over a long-time interval.The theorems guarantee that the κ-dimensional Lyapunov exponents are calculated by 25, 26 : The theory states that if the Lyapunov exponent tends to a positive value, the orbit is chaotic.
In the next section are shown some results about the Lyapunov exponents.3.  4.

Results
Figures   4. The initial conditions of the variables x 4 and y 4 are 0 • and 0 • , respectively.Table 3 shows the values of C 1 corresponding to the given initial conditions.4.  5.
Figures 5, 6, 7, and 8 show the time behavior of the semimajor axis, x 4 angle, argument of perigee and of the eccentricity for two different cases.The first case considers the critical angles φ 2201 , φ 2211 , and φ 2221 , associated to the tesseral harmonic J 22 , and the second case considers the critical angles associated to the tesseral harmonics J 22 and J 42 .The angles associated to the J 42 , φ 4211 , φ 4221 , and φ 4231 , have the same frequency of the critical angles associated to the J 22 with a different phase.The initial conditions corresponding to variables X 4 and Y 4 are defined for e o 0.05, I o 10 • , and a o given in Table 4.The initial conditions of the variables x 4 and y 4 are 0 • and 60 • , respectively.Table 4 shows the values of C 1 corresponding to the given initial conditions.5.
Analyzing Figures 5-8, one can observe a correction in the orbits when the terms related to the tesseral harmonic J 42 are added to the model.Observing, by the percentage, the contribution of the amplitudes of the terms B 4,4211 , B 4,4221 , and B 4,4231 , in each critical angle studied, is about 1,66% up to 4,94%.In fact, in the studies of the perturbations in the artificial satellites motion, the accuracy is important, since adding different tesseral and zonal harmonics to the model, one can have a better description about the orbital motion.5.The initial conditions of the variables x 4 and y 4 are 0 • and 60 • , respectively.Table 5 shows the values of C 1 corresponding to the given initial conditions.
Analyzing   Figures 13 and 14 show the time behavior of the Lyapunov exponents for two different cases, according to the initial values of Figures 1-4 and 9-12.The dynamical system involves the zonal harmonics J 20 and J 40 and the tesseral harmonics J 22 and J 42 .The method used in this work for the study of the Lyapunov exponents is described in Section 3. In Figure 13, the initial values for C 1 , x 4 , and y 4 are C 1 −1.467778013 × 10 11 m 2 /s and C 1 −1.467819454 × 10 11 m 2 /s, x 4 0 • and y 4 0 • , respectively.In Figure 14, the initial values for C 1 , x 4 , and y 4 are C 1 −1.467765786 × 10 11 m 2 /s and C 1 −1.467821043 × 10 11 m 2 /s, x 4 0 • and y 4 60 • , respectively.In each case are used two different values for semimajor axis corresponding to neighboring orbits shown previously in Figures 1-4 and 9-12.
Figures 13 and 14 show Lyapunov exponents for neighboring orbits.The time used in the calculations of the Lyapunov exponents is about 150.000 days.For this time, it can be observed in Figure 13 that λ 1 , corresponding to the initial value a 0 26565.0km, tends to a positive value, evidencing a chaotic region.On the other hand, analyzing the same Figure 13, λ 1 , corresponding to the initial value a 0 26563.5 km, does not show a stabilization around the some positive value, in this specified time.Probably, the time is not sufficient for a stabilization in some positive value, or λ 1 , initial value a 0 26563.5 km, tends to a negative value, evidencing a regular orbit.Analyzing now Figure 14 with λ 1 , initial value a 0 26562.0km.Comparing Figure 13 with Figure 14, it is observed that the Lyapunov exponents in Figure 14 has an amplitude of oscillation greater than the Lyapunov exponents in Figure 13.Analyzing this fact, it is probable that the necessary time for the Lyapunov exponent λ 2 , in Figure 14, to stabilize in some positive value is greater than the necessary time for the λ 2 in Figure 13.
Rescheduling the axes of Figures 13 and 14, as described in Figures 15 and 16, respectively, the Lyapunov exponents tending to a positive value can be better visualized.

Conclusions
In this work, the dynamical behavior of three critical angles associated to the 2 : 1 resonance problem in the artificial satellites motion has been investigated.
The results show the time behavior of the semimajor axis, argument of perigee and eccentricity.In the numerical integration, different cases are studied, using three critical angles together: φ 2201 , φ 2211 , and φ 2221 associated to J 22 and φ 4211 , φ 4221 , and φ 4231 associated to the J 42 .
In the simulations considered in the work, four cases show possible irregular motions for Analyzing the contribution of the terms related to the J 42 , it is observed that, for the value of C 1 −1.045724331 × 1011 m 2 /s, the amplitudes of the terms B 4,4211 , B 4,4221 , and B 4,4231 are greater than the other values of C 1 .In other words, for bigger values of semimajor axis, it is observed a smaller contribution of the terms related to the tesseral harmonic J 42 .
The theory used in this paper for the 2 : 1 resonance can be applied for any resonance involving some artificial Earth satellite.

Figure 1 :
Figure 1: Time behavior of the semimajor axis for different values of C 1 given in Table3.

Figure 2 :
Figure 2: Time behavior of x 4 angle for different values of C 1 given in Table3.

Figure 3 :
Figure 3: Time behavior of the argument of pericentre for different values of C 1 given in Table3.

Figure 4 :
Figure 4: Time behavior of the eccentricity for different values of C 1 given in Table3.

Figure 5 :
Figure 5: Time behavior of the semimajor axis for different values of C 1 given in Table4.

Figure 7 :
Figure 7: Time behavior of the argument of pericentre for different values of C 1 given in Table4.

Figure 8 :
Figure 8: Time behavior of the eccentricity for different values of C 1 given in Table4.

Figure 9 :
Figure 9: Time behavior of the semimajor axis for different values of C 1 given in Table5.

Figure 10 :Figure 11 :
Figure 10: Time behavior of x 4 angle for different values of C 1 given in Table5.

Figures 9 ,Figure 12 :Figure 13 :
Figure 12: Time behavior of the eccentricity for different values of C 1 given in Table5.

Figure 15 :Figure 16 :
Figures 13 and 14 show the time behavior of the Lyapunov exponents for two different cases, according to the initial values of Figures 1-4 and 9-12.The dynamical system involves the zonal harmonics J 20 and J 40 and the tesseral harmonics J 22 and J 42 .The method used in this work for the study of the Lyapunov exponents is described in Section 3. In Figure13, the initial values for C 1 , x 4 , and y 4 are C 1 −1.467778013 × 10 11 m 2 /s and C 1 −1.467819454 × 10 11 m 2 /s, x 4 0 • and y 4 0 • , respectively.In Figure14, the initial values for C 1 , x 4 , and y 4 are C 1 −1.467765786 × 10 11 m 2 /s and C 1 −1.467821043 × 10 11 m 2 /s, x 4 0 • and y 4 60 • , respectively.In each case are used two different values for semimajor axis corresponding to neighboring orbits shown previously in Figures1-4 and 9-12.Figures13 and 14show Lyapunov exponents for neighboring orbits.The time used in the calculations of the Lyapunov exponents is about 150.000 days.For this time, it can be observed in Figure13that λ 1 , corresponding to the initial value a 0 26565.0km, tends to a positive value, evidencing a chaotic region.On the other hand, analyzing the same Figure13, λ 1 , corresponding to the initial value a 0 26563.5 km, does not show a stabilization around the some positive value, in this specified time.Probably, the time is not sufficient for a stabilization in some positive value, or λ 1 , initial value a 0 26563.5 km, tends to a negative value, evidencing a regular orbit.Analyzing now Figure14, it can be verified that λ 1 , corresponding to the initial value a 0 26564.0km, tends to a positive value, it contrasts C 1 −1.467778013 × 10 11 m 2 /s, C 1 −1.467819454 × 10 11 m 2 /s, C 1 −1.467765786 × 10 11 m 2 /s, and C 1 −1.467821043 × 10 11 m 2 /s.Studying the Lyapunov exponents, two cases show chaotic motions for C 1 −1.467819454 × 10 11 m 2 /s and C 1 −1.467821043 × 10 11 m 2 /s.

Table 2 :
The zonal and tesseral harmonics.The zonal harmonics used in 2.34 and 2.35 and the tesseral harmonics used in 2.36 to 2.41 are shown in Table 2.The constant of integration C 1 in 2.34 to 2.41 is given, in terms of the initial values of the orbital elements, a o , e o , and I o , by x 4 , y 4 ; C 1 , 4 , Y 4 , x 4 , y 4 ; C 1 .

Table 3 :
Values of the constant of integration C 1 for e 0.001, I 55 • and different values for semimajor axis.
1, 2, 3, and 4show the time behavior of the semimajor axis, x 4 angle, argument of perigee and of the eccentricity, according to the numerical integration of the motion equations, 2.43 , considering three different resonant angles together: φ 2201 , φ 2211 , and φ 2221 associated to J 22 , and three angles, φ 4211 , φ 4221 , and φ 4231 associated to J 42 , with the same frequency of the resonant angles related to the J 22 , but with different phase.The initial conditions corresponding to variables X 4 and Y 4 are defined for e o 0.001, I o 55 • , and a o given in Table3.Time behavior of x 4 angle for different values of C 1 given in Table4.

Table 4 :
Values of the constant of integration C 1 for e 0.05, I 10 • , and different values for semimajor axis.

Table 5 :
Values of the constant of integration C 1 for e 0.01, I 55 • , and different values for semimajor axis.