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August & September  2019, 12(4&5): 849-875. doi: 10.3934/dcdss.2019057

The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits

Niraj Pathak 1, , V. O. Thomas 2, and  Elbaz I. Abouelmagd 3,4,,

1. 

Department of Mathematics, Dharmsinh Desai University, Nadiad, Gujarat 3870001, India
2. 

Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara, 390002 Gujarat, India
3. 

Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University, Jeddah, Saudi Arabia
4. 

Celestial Mechanics Unit, Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt

* Corresponding author: Elbaz I. Abouelmagd

Received  June 2017 Revised  November 2017 Published  November 2018

In the framework of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orbits are analyzed. The location, eccentricity and period of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specified value of Jacobi constant C.

It is observed that as the number of loops increases successively from one loop to five loops, the period of infinitesimal body increases in such a way that the successive difference of periods is either 6 or 7 units. It is further observed that for the exterior resonance, as the number of loops increases, the location of the periodic orbit moves towards the Sun whereas for the interior resonance as the number of loops increases, location of the periodic orbit moves away from the Sun. Thereby we demonstrate that the location of resonant orbits of the given order moves away from the Sun when perturbation is included.

The evolution of interior first order resonant orbit with three loops is studied for different values of Jacobi constant C. It is observed that when the value of C increases, the size of the loop decreases and degenerates finally into a circle, the eccentricity of periodic orbit decreases and location of the periodic orbit moves towards the second primary body.

Keywords: Photogravitational restricted three-body problem, periodic orbits, interior resonance, exterior resonance, oblateness, Poincaré surface of section.
Mathematics Subject Classification: 37N05, 70F07, 70F15.
Citation: Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057
References:
[1]

E. I. Abouelmagd, L. G. Guirao Juan and A. Mostafa, Numerical integration of the restricted thee-body problem with Lie series, Astrophysics Space Science, 354 (2014), 369-378.   Google Scholar

[2]

E. I. Abouelmagd and M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophys. Space Sci., 344 (2013), 321-332.   Google Scholar

[3]

E. I. Abouelmagd, F. Alzahrani, J. L. G. Guiro and A. Hobiny, Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA), 9 (2016), 1716-1727.  doi: 10.22436/jnsa.009.04.27.  Google Scholar

[4]

E. I. Abouelmagd, M. S. Alhothuali, L. G. Guirao Juan and H. M. Malaikah, Periodic and secular solutions in the restricted three ody problem under the effect of zonal harmonic parameters, Applied Mathematics & Information Science, 9 (2015), 1659-1669.   Google Scholar

[5]

E. I. Abouelmagd, J. L. G. Guirao, A. Hobiny and F. Alzahrani, Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid, Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 8 (2015), 1047-1054.  doi: 10.3934/dcdss.2015.8.1047.  Google Scholar

[6]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144.   Google Scholar

[7]

E. I. Abouelmagd, J. L. G. Guirao and J. A. Vera, Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body, Commun Nonlinear Sci Numer Simulat., 20 (2015), 1057-1069.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[8]

E. I. Abouelmagd, D. Mortari and H. H. Selim, Analytical study of periodic solutions on perturbed equatorial two-body problem, International Journal of Bifurcation and Chaos, 25 (2015), 1540040, 14pp. doi: 10.1142/S0218127415400404.  Google Scholar

[9]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys Space Sci., 342 (2012), 45-53.   Google Scholar

[10]

E. I. Abouelmagd, Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013), 143-155.   Google Scholar

[11]

E. I. Abouelmagd, The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys Space Sci., 346 (2013), 51-69.   Google Scholar

[12]

E. I. Abouelmagd, M. S. Alhothuali, L. G. Guirao Juan and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Advances in Space Research, 55 (2015), 1660-1672.   Google Scholar

[13]

E. Balint, R. Renata, S. Zsolt and F. Emese, Stability of higher order resonances in the restricted-three body problem, Celest. Mech. Dyn. Astron., 113 (2012), 95-112.  doi: 10.1007/s10569-012-9420-4.  Google Scholar

[14]

F. Cachucho, P. M. Cincotta and S. Ferraz-Mello, Chirikov diffusion in the asteroidal three-body resonance (5, -2, -2), Celest. Mech. Dyn. Astron., 108 (2010), 35-58.  doi: 10.1007/s10569-010-9290-6.  Google Scholar

[15]

E. I. Chiang, J. Lovering, R. I. Millis, M. W. Buie, L. H. Wasserman and K. J. Meech, Resonant and secular families of the Kuiper belt, Earth Moon Planets, 92 (2003), 49-62.   Google Scholar

[16]

C. Douskos, V. Kalantonis and P. Markellos, Effects of resonances on the stability of retrograde satellites, Astrophys. Space Sci., 310, 245-249. Google Scholar

[17]

R. Dvorak, A. Bazs and L.-Y. Zhou, Where are the Uranus Trojans?, Celest. Mech. Dyn. Astron., 107 (2010), 51-62.  doi: 10.1007/s10569-010-9261-y.  Google Scholar

[18]

V. V. Emel'yanenko and E. L. Kiseleva, Resonant motion of trans-Neptunian objects in high-eccentricity orbits, Astron. Lett., 34 (2008), 271-279.   Google Scholar

[19]

J. Gayon, E. Bois and H. Scholl, Dynamics of planets in retrograde mean motion resonance, Celest. Mech. Dyn. Astron., 103 (2009), 267-279.  doi: 10.1007/s10569-009-9191-8.  Google Scholar

[20]

J. D. Hadjidemetriou, D. Psychoyos and G. Voyatzis, The 1:1 resonance in extrasolar planetary systems, Celest. Mech. Dyn. Astron., 104 (2009), 23-38.  doi: 10.1007/s10569-009-9185-6.  Google Scholar

[21]

J. D. Hadjidemetriou and G. Voyatzis, On the dynamics of extrasolar planetary systems under dissipation: Migration of planets, Celest. Mech. Dyn. Astron., 107 (2010), 3-19.  doi: 10.1007/s10569-010-9260-z.  Google Scholar

[22]

M. J. Holman and N. W. Murray, Chaos in high-order mean motion resonances in the outer asteroid belt, Astron. J., 112 (1996), 1278-1293.   Google Scholar

[23]

A. S. Libert and K. Tsiganis, Trapping in three-planet resonances during gas-driven migration, Celest. Mech. Dyn. Astron., 111 (2011), 201-218.   Google Scholar

[24]

E. Kolmen, N. J. Kasdin and P. Gurfil, Quasi-periodic orbits of the restricted three body problem made easy, AIP Conference Proceedings, 886 (2007), 68-77.   Google Scholar

[25]

V. V. Markellos, K. E. Papadakis and E. A. Perdios, Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness, Astrophys Space Sci., 245 (1996), 157-164.   Google Scholar

[26]

F. Migliorini, P. Michel, A. Morbidelli, D. Nesvorn and V. Zappal, Origin of multi kilometer Earth and Mars-crossing asteroids: A quantitative simulation, Science, 281 (1998), 2022-2024.   Google Scholar

[27]

A. Morbidelli, V. Zappala, M. Moons, A. Cellino and R. Gonczi, Asteriod families close to mean motion resonances: Dynamical effects and physical implications, Icarus, 118 (1995), 137-154.   Google Scholar

[28]

C. D. Murray and S. F. Dermot, Solar System Dynamics, Cambridge University Press, 1999.  Google Scholar

[29]

N. M. Pathak, R. K. Sharma and V. O. Thomas, Evolution of periodic orbits in the Sun- Saturn system, International Journal of Astronomy and Astrophysics, 6 (2016), 175-197.   Google Scholar

[30]

N. M. Pathak and V. O. Thomas, Evolution of the f Family Orbits in the Photo-Gravitational Sun-Saturn System with Oblateness, International Journal of Astronomy and Astrophysics, 6 (2016), 254-271.   Google Scholar

[31]

N. M. Pathak and V. O. Thomas, Analysis of effect of oblateness of smaller primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016), 440-463.   Google Scholar

[32]

N. M. Pathak and V. O. Thomas, Analysis of effect of solar radiation pressure of bigger primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016), 464-493.   Google Scholar

[33]

E. A. Perdios and V. S. Kalantonis, Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem, Celest. Mech. Dyn. Astron., 113 (2012), 377-386.  doi: 10.1007/s10569-012-9424-0.  Google Scholar

[34]

H. Poincaré, Les Méthodes Nouvelles de la Méchanique, Celeste. Gauthier- Villas, Paris., 1987.  Google Scholar

[35]

N. Pushparaj and R. K. Sharma, Interior resonance periodic orbits in photogravitational restricted three-body problem, Advances in Astrophysics, 2 (2017), 263-272.   Google Scholar

[36]

A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system, Monthly Notices of the Royal Astronomical Society, 114 (1954), 232-241.   Google Scholar

[37]

R. K. Sharma and P. V. Subbarao, A case of commensurability induced by oblateness, Celest. Mech., 18 (1978), 185-194.   Google Scholar

[38]

R. K. Sharma, The linear stability of libration points of the photo gravitational restricted three body problem when the smaller primary is an oblate spheroid, Astrophysics and Space Science, 135 (1987), 271-281.   Google Scholar

[39]

P. P. Stor, J. Kla cka and L. K. mar, Motion of dust in mean motion resonance with planets, Celest. Mech. Dyn. Astron., 103 (2009), 343-364.  doi: 10.1007/s10569-009-9202-9.  Google Scholar

[40]

E. W. Thommes, A safety net for fast migrators: Interactions between gap-opening and sub ap-opening bodies in a protoplanetary disk, Astrophys. J., 626 (2005), 1033-1044.   Google Scholar

show all references

References:
[1]

E. I. Abouelmagd, L. G. Guirao Juan and A. Mostafa, Numerical integration of the restricted thee-body problem with Lie series, Astrophysics Space Science, 354 (2014), 369-378.   Google Scholar

[2]

E. I. Abouelmagd and M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophys. Space Sci., 344 (2013), 321-332.   Google Scholar

[3]

E. I. Abouelmagd, F. Alzahrani, J. L. G. Guiro and A. Hobiny, Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA), 9 (2016), 1716-1727.  doi: 10.22436/jnsa.009.04.27.  Google Scholar

[4]

E. I. Abouelmagd, M. S. Alhothuali, L. G. Guirao Juan and H. M. Malaikah, Periodic and secular solutions in the restricted three ody problem under the effect of zonal harmonic parameters, Applied Mathematics & Information Science, 9 (2015), 1659-1669.   Google Scholar

[5]

E. I. Abouelmagd, J. L. G. Guirao, A. Hobiny and F. Alzahrani, Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid, Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 8 (2015), 1047-1054.  doi: 10.3934/dcdss.2015.8.1047.  Google Scholar

[6]

E. I. Abouelmagd and J. L. G. Guirao, On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016), 123-144.   Google Scholar

[7]

E. I. Abouelmagd, J. L. G. Guirao and J. A. Vera, Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body, Commun Nonlinear Sci Numer Simulat., 20 (2015), 1057-1069.  doi: 10.1016/j.cnsns.2014.06.033.  Google Scholar

[8]

E. I. Abouelmagd, D. Mortari and H. H. Selim, Analytical study of periodic solutions on perturbed equatorial two-body problem, International Journal of Bifurcation and Chaos, 25 (2015), 1540040, 14pp. doi: 10.1142/S0218127415400404.  Google Scholar

[9]

E. I. Abouelmagd, Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys Space Sci., 342 (2012), 45-53.   Google Scholar

[10]

E. I. Abouelmagd, Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013), 143-155.   Google Scholar

[11]

E. I. Abouelmagd, The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys Space Sci., 346 (2013), 51-69.   Google Scholar

[12]

E. I. Abouelmagd, M. S. Alhothuali, L. G. Guirao Juan and H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Advances in Space Research, 55 (2015), 1660-1672.   Google Scholar

[13]

E. Balint, R. Renata, S. Zsolt and F. Emese, Stability of higher order resonances in the restricted-three body problem, Celest. Mech. Dyn. Astron., 113 (2012), 95-112.  doi: 10.1007/s10569-012-9420-4.  Google Scholar

[14]

F. Cachucho, P. M. Cincotta and S. Ferraz-Mello, Chirikov diffusion in the asteroidal three-body resonance (5, -2, -2), Celest. Mech. Dyn. Astron., 108 (2010), 35-58.  doi: 10.1007/s10569-010-9290-6.  Google Scholar

[15]

E. I. Chiang, J. Lovering, R. I. Millis, M. W. Buie, L. H. Wasserman and K. J. Meech, Resonant and secular families of the Kuiper belt, Earth Moon Planets, 92 (2003), 49-62.   Google Scholar

[16]

C. Douskos, V. Kalantonis and P. Markellos, Effects of resonances on the stability of retrograde satellites, Astrophys. Space Sci., 310, 245-249. Google Scholar

[17]

R. Dvorak, A. Bazs and L.-Y. Zhou, Where are the Uranus Trojans?, Celest. Mech. Dyn. Astron., 107 (2010), 51-62.  doi: 10.1007/s10569-010-9261-y.  Google Scholar

[18]

V. V. Emel'yanenko and E. L. Kiseleva, Resonant motion of trans-Neptunian objects in high-eccentricity orbits, Astron. Lett., 34 (2008), 271-279.   Google Scholar

[19]

J. Gayon, E. Bois and H. Scholl, Dynamics of planets in retrograde mean motion resonance, Celest. Mech. Dyn. Astron., 103 (2009), 267-279.  doi: 10.1007/s10569-009-9191-8.  Google Scholar

[20]

J. D. Hadjidemetriou, D. Psychoyos and G. Voyatzis, The 1:1 resonance in extrasolar planetary systems, Celest. Mech. Dyn. Astron., 104 (2009), 23-38.  doi: 10.1007/s10569-009-9185-6.  Google Scholar

[21]

J. D. Hadjidemetriou and G. Voyatzis, On the dynamics of extrasolar planetary systems under dissipation: Migration of planets, Celest. Mech. Dyn. Astron., 107 (2010), 3-19.  doi: 10.1007/s10569-010-9260-z.  Google Scholar

[22]

M. J. Holman and N. W. Murray, Chaos in high-order mean motion resonances in the outer asteroid belt, Astron. J., 112 (1996), 1278-1293.   Google Scholar

[23]

A. S. Libert and K. Tsiganis, Trapping in three-planet resonances during gas-driven migration, Celest. Mech. Dyn. Astron., 111 (2011), 201-218.   Google Scholar

[24]

E. Kolmen, N. J. Kasdin and P. Gurfil, Quasi-periodic orbits of the restricted three body problem made easy, AIP Conference Proceedings, 886 (2007), 68-77.   Google Scholar

[25]

V. V. Markellos, K. E. Papadakis and E. A. Perdios, Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness, Astrophys Space Sci., 245 (1996), 157-164.   Google Scholar

[26]

F. Migliorini, P. Michel, A. Morbidelli, D. Nesvorn and V. Zappal, Origin of multi kilometer Earth and Mars-crossing asteroids: A quantitative simulation, Science, 281 (1998), 2022-2024.   Google Scholar

[27]

A. Morbidelli, V. Zappala, M. Moons, A. Cellino and R. Gonczi, Asteriod families close to mean motion resonances: Dynamical effects and physical implications, Icarus, 118 (1995), 137-154.   Google Scholar

[28]

C. D. Murray and S. F. Dermot, Solar System Dynamics, Cambridge University Press, 1999.  Google Scholar

[29]

N. M. Pathak, R. K. Sharma and V. O. Thomas, Evolution of periodic orbits in the Sun- Saturn system, International Journal of Astronomy and Astrophysics, 6 (2016), 175-197.   Google Scholar

[30]

N. M. Pathak and V. O. Thomas, Evolution of the f Family Orbits in the Photo-Gravitational Sun-Saturn System with Oblateness, International Journal of Astronomy and Astrophysics, 6 (2016), 254-271.   Google Scholar

[31]

N. M. Pathak and V. O. Thomas, Analysis of effect of oblateness of smaller primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016), 440-463.   Google Scholar

[32]

N. M. Pathak and V. O. Thomas, Analysis of effect of solar radiation pressure of bigger primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016), 464-493.   Google Scholar

[33]

E. A. Perdios and V. S. Kalantonis, Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem, Celest. Mech. Dyn. Astron., 113 (2012), 377-386.  doi: 10.1007/s10569-012-9424-0.  Google Scholar

[34]

H. Poincaré, Les Méthodes Nouvelles de la Méchanique, Celeste. Gauthier- Villas, Paris., 1987.  Google Scholar

[35]

N. Pushparaj and R. K. Sharma, Interior resonance periodic orbits in photogravitational restricted three-body problem, Advances in Astrophysics, 2 (2017), 263-272.   Google Scholar

[36]

A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system, Monthly Notices of the Royal Astronomical Society, 114 (1954), 232-241.   Google Scholar

[37]

R. K. Sharma and P. V. Subbarao, A case of commensurability induced by oblateness, Celest. Mech., 18 (1978), 185-194.   Google Scholar

[38]

R. K. Sharma, The linear stability of libration points of the photo gravitational restricted three body problem when the smaller primary is an oblate spheroid, Astrophysics and Space Science, 135 (1987), 271-281.   Google Scholar

[39]

P. P. Stor, J. Kla cka and L. K. mar, Motion of dust in mean motion resonance with planets, Celest. Mech. Dyn. Astron., 103 (2009), 343-364.  doi: 10.1007/s10569-009-9202-9.  Google Scholar

[40]

E. W. Thommes, A safety net for fast migrators: Interactions between gap-opening and sub ap-opening bodies in a protoplanetary disk, Astrophys. J., 626 (2005), 1033-1044.   Google Scholar

Figure 1.  Exterior first order resonant single loop orbit and PSS for $C = 2.93$ in the Sun - Earth system
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Figure 2.  Exterior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system
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Figure 3.  Interior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system
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Figure 4.  Variation in three loops orbit due to interior first order resonant when $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Earth system
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Figure 5.  Variation in PSS of three loops orbit due to interior first order resonant when $q = 0.9845$, $A_2 = 0.0001$ in the Sun-Earth system
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Figure 6.  Variation in location of the periodic orbit of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) for the Sun-Earth system
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Figure 7.  Variation in eccentricity of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) cases in the Sun-Earth system
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Figure 8.  Variation in location and eccentricity of first order interior three loops orbit when $q = 0.9845$ and ${A}_{2} = 0.0001$ in Sun-Earth system
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Figure 9.  Variation in location of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case($q = 1$, ${A}_{2} = 0$) in the Sun-Mars system
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Figure 10.  Variation in eccentricity of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) in the Sun-Mars system
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Figure 11.  Variation in location and eccentricity of interior first order three loops orbit for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Mars system
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Figure 12.  Variation in interior third order resonant seven loops orbit for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun - Earth system
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Figure 13.  PSS of interior third order resonant seven loops orbit of family Ⅰ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun - Earth system
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Figure 14.  Family Ⅰ interior third order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 15.  PSS of interior third order resonant seven loops orbits from family Ⅱ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 16.  Family Ⅱ interior third order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 17.  PSS of interior fifth order resonant eleven loops orbits from Family Ⅱ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 18.  Family Ⅰ interior fifth order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 19.  Family Ⅱ interior fifth order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system
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Figure 20.  Variation in location of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system
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Figure 21.  Variation in eccentricity of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system
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Figure 22.  Variation in period of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system
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Figure 23.  Variation in location of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system
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Figure 24.  Variation in eccentricity of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system
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Figure 25.  Variation in period of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system
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Table 1.  Analysis of exterior first order resonance in the perturbed Sun-Earth system
$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 1 0.93904 1 1:2 0.40895 13 0.49936
2 0.88740 2:3 0.32301 19 0.66633
3 0.85623 3:4 0.29337 26 0.74943
4 0.83547 4:5 0.28015 32 0.79977
5 0.82075 5:6 0.27331 38 0.83313
Ⅱ 1 0.0001 1 0.93877 1 1:2 0.40904 13 0.49945
2 0.88710 2:3 0.32319 19 0.66641
3 0.85592 3:4 0.29358 26 0.74979
4 0.83516 4:5 0.28039 32 0.79982
5 0.82044 5:6 0.27355 38 0.83318
Ⅲ 0.9845 0 1 0.97895 1 1:2 0.36044 13 0.52805
2 0.93800 2:3 0.26118 19 0.69904
3 0.91210 3:4 0.22428 26 0.78432
4 0.89403 4:5 0.20685 32 0.83562
5 0.88075 5:6 0.19740 38 0.86990
Ⅳ 0.9845 0.0001 1 0.97870 1 1:2 0.36081 13 0.52779
2 0.93764 2:3 0.26136 19 0.69919
3 0.91172 3:4 0.22453 26 0.78443
4 0.89363 4:5 0.20713 32 0.83574
5 0.88035 5:6 0.19771 38 0.86999
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$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 1 0.93904 1 1:2 0.40895 13 0.49936
2 0.88740 2:3 0.32301 19 0.66633
3 0.85623 3:4 0.29337 26 0.74943
4 0.83547 4:5 0.28015 32 0.79977
5 0.82075 5:6 0.27331 38 0.83313
Ⅱ 1 0.0001 1 0.93877 1 1:2 0.40904 13 0.49945
2 0.88710 2:3 0.32319 19 0.66641
3 0.85592 3:4 0.29358 26 0.74979
4 0.83516 4:5 0.28039 32 0.79982
5 0.82044 5:6 0.27355 38 0.83318
Ⅲ 0.9845 0 1 0.97895 1 1:2 0.36044 13 0.52805
2 0.93800 2:3 0.26118 19 0.69904
3 0.91210 3:4 0.22428 26 0.78432
4 0.89403 4:5 0.20685 32 0.83562
5 0.88075 5:6 0.19740 38 0.86990
Ⅳ 0.9845 0.0001 1 0.97870 1 1:2 0.36081 13 0.52779
2 0.93764 2:3 0.26136 19 0.69919
3 0.91172 3:4 0.22453 26 0.78443
4 0.89363 4:5 0.20713 32 0.83574
5 0.88035 5:6 0.19771 38 0.86999
Table 2.  Analysis of exterior first order resonance in the perturbed Sun-Mars system
$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 1 0.939000 1 1:2 0.40852 13 0.50015
2 0.887370 2:3 0.32284 19 0.66692
3 0.856190 3:4 0.29324 26 0.75031
4 0.835433 4:5 0.28006 32 0.80033
5 0.820715 5:6 0.27323 38 0.83368
Ⅱ 1 0.0001 1 0.93875 1 1:2 0.40866 13 0.50017
2 0.88708 2:3 0.32302 19 0.66697
3 0.85588 3:4 0.29346 26 0.75037
4 0.83512 4:5 0.28029 32 0.80039
5 0.82040 5:6 0.27347 38 0.83374
Ⅲ 0.9845 0 1 0.97891 1 1:2 0.35887 13 0.53027
2 0.93795 2:3 0.26069 19 0.70019
3 0.91204 3:4 0.22397 26 0.78521
4 0.89397 4:5 0.20662 32 0.83643
5 0.88069 5:6 0.19722 38 0.87067
Ⅳ 0.9845 0.0001 1 0.97861 1 1:2 0.35895 13 0.53041
2 0.93761 2:3 0.26090 19 0.70018
3 0.91167 3:4 0.22423 26 0.78529
4 0.89358 4:5 0.20691 32 0.83652
5 0.88029 5:6 0.19752 38 0.87076
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$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 1 0.939000 1 1:2 0.40852 13 0.50015
2 0.887370 2:3 0.32284 19 0.66692
3 0.856190 3:4 0.29324 26 0.75031
4 0.835433 4:5 0.28006 32 0.80033
5 0.820715 5:6 0.27323 38 0.83368
Ⅱ 1 0.0001 1 0.93875 1 1:2 0.40866 13 0.50017
2 0.88708 2:3 0.32302 19 0.66697
3 0.85588 3:4 0.29346 26 0.75037
4 0.83512 4:5 0.28029 32 0.80039
5 0.82040 5:6 0.27347 38 0.83374
Ⅲ 0.9845 0 1 0.97891 1 1:2 0.35887 13 0.53027
2 0.93795 2:3 0.26069 19 0.70019
3 0.91204 3:4 0.22397 26 0.78521
4 0.89397 4:5 0.20662 32 0.83643
5 0.88069 5:6 0.19722 38 0.87067
Ⅳ 0.9845 0.0001 1 0.97861 1 1:2 0.35895 13 0.53041
2 0.93761 2:3 0.26090 19 0.70018
3 0.91167 3:4 0.22423 26 0.78529
4 0.89358 4:5 0.20691 32 0.83652
5 0.88029 5:6 0.19752 38 0.87076
Table 3.  Analysis of interior first order resonance in the perturbed Sun-Earth system
$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 2 0.29385 1 2:1 0.53353 07 2.00000
3 0.47692 3:2 0.37506 13 1.50000
4 0.55735 4:3 0.32483 19 1.33330
5 0.60105 5:4 0.30258 26 1.24991
6 0.62815 6:5 0.29074 32 1.19981
7 0.64650 7:6 0.28366 38 1.16633
8 0.66000 8:7 0.27897 44 1.14185
Ⅱ 1 0.0001 2 0.29375 1 2:1 0.53367 07 2.00015
3 0.47675 3:2 0.37525 13 1.50009
4 0.55713 4:3 0.32506 19 1.33339
5 0.60080 5:4 0.30283 26 1.25001
6 0.62788 6:5 0.29100 32 1.19992
7 0.64627 7:6 0.28391 38 1.16635
8 0.65980 8:7 0.27921 44 1.14180
Ⅲ 0.9845 0 2 0.31234 1 2:1 0.47832 07 2.15851
3 0.50990 3:2 0.30776 13 1.58182
4 0.59888 4:3 0.25104 19 1.39854
5 0.64770 5:4 0.22523 26 1.30827
6 0.67801 6:5 0.21135 32 1.25452
7 0.69851 7:6 0.20301 38 1.21879
8 0.71327 8:7 0.19758 44 1.19323
Ⅳ 0.9845 0.0001 2 0.31222 1 2:1 0.47848 07 2.15875
3 0.50970 3:2 0.30799 13 1.58195
4 0.59861 4:3 0.25133 19 1.39869
5 0.64740 5:4 0.22554 26 1.30839
6 0.67768 6:5 0.21168 32 1.25465
7 0.69819 7:6 0.20333 38 1.21887
8 0.71290 8:7 0.19793 44 1.19338
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$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 2 0.29385 1 2:1 0.53353 07 2.00000
3 0.47692 3:2 0.37506 13 1.50000
4 0.55735 4:3 0.32483 19 1.33330
5 0.60105 5:4 0.30258 26 1.24991
6 0.62815 6:5 0.29074 32 1.19981
7 0.64650 7:6 0.28366 38 1.16633
8 0.66000 8:7 0.27897 44 1.14185
Ⅱ 1 0.0001 2 0.29375 1 2:1 0.53367 07 2.00015
3 0.47675 3:2 0.37525 13 1.50009
4 0.55713 4:3 0.32506 19 1.33339
5 0.60080 5:4 0.30283 26 1.25001
6 0.62788 6:5 0.29100 32 1.19992
7 0.64627 7:6 0.28391 38 1.16635
8 0.65980 8:7 0.27921 44 1.14180
Ⅲ 0.9845 0 2 0.31234 1 2:1 0.47832 07 2.15851
3 0.50990 3:2 0.30776 13 1.58182
4 0.59888 4:3 0.25104 19 1.39854
5 0.64770 5:4 0.22523 26 1.30827
6 0.67801 6:5 0.21135 32 1.25452
7 0.69851 7:6 0.20301 38 1.21879
8 0.71327 8:7 0.19758 44 1.19323
Ⅳ 0.9845 0.0001 2 0.31222 1 2:1 0.47848 07 2.15875
3 0.50970 3:2 0.30799 13 1.58195
4 0.59861 4:3 0.25133 19 1.39869
5 0.64740 5:4 0.22554 26 1.30839
6 0.67768 6:5 0.21168 32 1.25465
7 0.69819 7:6 0.20333 38 1.21887
8 0.71290 8:7 0.19793 44 1.19338
Table 4.  Analysis of interior first order resonance in the perturbed Sun-Mars system
$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 2 0.29386 1 2:1 0.53352 07 2.00090
3 0.47693 3:2 0.37504 13 1.50067
4 0.55734 4:3 0.32482 19 1.33393
5 0.60102 5:4 0.30258 26 1.25055
6 0.62808 6:5 0.29075 32 1.20005
7 0.64637 7:6 0.28369 38 1.16792
8 0.65954 8:7 0.27911 44 1.14323
Ⅱ 1 0.0001 2 0.293750 1 2:1 0.533670 07 2.00105
3 0.476745 3:2 0.375250 13 1.50079
4 0.557125 4:3 0.325050 19 1.33402
5 0.600770 5:4 0.302830 26 1.25065
6 0.627830 6:5 0.291011 32 1.20058
7 0.646100 7:6 0.283950 38 1.16722
8 0.659270 8:7 0.279370 44 1.14331
Ⅲ 0.9845 0 2 0.31235 1 2:1 0.47831 07 2.15947
3 0.50991 3:2 0.30774 13 1.58251
4 0.59887 4:3 0.25103 19 1.39922
5 0.64768 5:4 0.22522 26 1.30892
6 0.67797 6:5 0.21134 32 1.25519
7 0.69843 7:6 0.20301 38 1.21952
8 0.71309 8:7 0.19761 44 1.19412
Ⅳ 0.9845 0.0001 2 0.31223 1 2:1 0.478470 07 2.15970
3 0.50971 3:2 0.307980 13 1.58266
4 0.59860 4:3 0.251320 19 1.39936
5 0.64738 5:4 0.225530 26 1.30905
6 0.67764 6:5 0.211567 32 1.25532
7 0.69809 7:6 0.203340 38 1.21963
8 0.71273 8:7 0.197960 44 1.19426
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$FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 1 0 2 0.29386 1 2:1 0.53352 07 2.00090
3 0.47693 3:2 0.37504 13 1.50067
4 0.55734 4:3 0.32482 19 1.33393
5 0.60102 5:4 0.30258 26 1.25055
6 0.62808 6:5 0.29075 32 1.20005
7 0.64637 7:6 0.28369 38 1.16792
8 0.65954 8:7 0.27911 44 1.14323
Ⅱ 1 0.0001 2 0.293750 1 2:1 0.533670 07 2.00105
3 0.476745 3:2 0.375250 13 1.50079
4 0.557125 4:3 0.325050 19 1.33402
5 0.600770 5:4 0.302830 26 1.25065
6 0.627830 6:5 0.291011 32 1.20058
7 0.646100 7:6 0.283950 38 1.16722
8 0.659270 8:7 0.279370 44 1.14331
Ⅲ 0.9845 0 2 0.31235 1 2:1 0.47831 07 2.15947
3 0.50991 3:2 0.30774 13 1.58251
4 0.59887 4:3 0.25103 19 1.39922
5 0.64768 5:4 0.22522 26 1.30892
6 0.67797 6:5 0.21134 32 1.25519
7 0.69843 7:6 0.20301 38 1.21952
8 0.71309 8:7 0.19761 44 1.19412
Ⅳ 0.9845 0.0001 2 0.31223 1 2:1 0.478470 07 2.15970
3 0.50971 3:2 0.307980 13 1.58266
4 0.59860 4:3 0.251320 19 1.39936
5 0.64738 5:4 0.225530 26 1.30905
6 0.67764 6:5 0.211567 32 1.25532
7 0.69809 7:6 0.203340 38 1.21963
8 0.71273 8:7 0.197960 44 1.19426
Table 5.  Variation in three loops orbit due to variation in $C$ for $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Earth system
$JC$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
2.93 0.50970 1 3:2 0.30799 13 1.58195
2.95 0.53653 0.27320 1.57665
2.97 0.56750 0.23304 1.57112
2.99 0.60501 0.18439 1.56524
3.01 0.65610 0.11817 1.55825
3.02 0.69590 0.06656 1.55357
3.03 0.75300 0.01658 1.56821
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$JC$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
2.93 0.50970 1 3:2 0.30799 13 1.58195
2.95 0.53653 0.27320 1.57665
2.97 0.56750 0.23304 1.57112
2.99 0.60501 0.18439 1.56524
3.01 0.65610 0.11817 1.55825
3.02 0.69590 0.06656 1.55357
3.03 0.75300 0.01658 1.56821
Table 6.  Variation in three loops orbit due to variation in $C$ for $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Mars system
$JC$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
2.93 0.50971 1 3:2 0.30798 13 1.58266
2.97 0.56750 0.23303 1.57184
3.01 0.65608 0.11815 1.55901
3.02 0.69590 0.06651 1.55431
3.03 0.75200 0.01514 1.56908
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$JC$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
2.93 0.50971 1 3:2 0.30798 13 1.58266
2.97 0.56750 0.23303 1.57184
3.01 0.65608 0.11815 1.55901
3.02 0.69590 0.06651 1.55431
3.03 0.75200 0.01514 1.56908
Table 7.  Variation in third order interior resonant seven loops orbit due to variation in $C$ in the Sun-Earth system
$JC$ $LO$ $NI$ $RS$ $EC$ $TP$ $RP$
2.93 0.39923 3 7:4 0.39522 26 1.86449
2.96 0.42824 0.35353 1.85476
2.98 0.44991 0.32238 1.84836
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$JC$ $LO$ $NI$ $RS$ $EC$ $TP$ $RP$
2.93 0.39923 3 7:4 0.39522 26 1.86449
2.96 0.42824 0.35353 1.85476
2.98 0.44991 0.32238 1.84836
Table 8.  Variation in third order interior resonant seven loops orbit due to variation in $C$ in the Sun-Mars system
$JC$ $LO$ $NI$ $RS$ $EC$ $TP$ $RP$
2.93 0.39923 3 7:4 0.39521 26 1.86534
2.96 0.42823 0.35353 1.85564
2.98 0.44991 0.32232 1.84921
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$JC$ $LO$ $NI$ $RS$ $EC$ $TP$ $RP$
2.93 0.39923 3 7:4 0.39521 26 1.86534
2.96 0.42823 0.35353 1.85564
2.98 0.44991 0.32232 1.84921
Table 9.  Third order interior resonance in the Sun-Mars system
$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 7 0.39923 3 7:4 0.39522 26 1.86449
8 0.46231 8:5 0.34307 32 1.69384
10 0.54635 10:7 0.28318 44 1.50281
11 0.57515 11:8 0.26511 51 1.44432
13 0.61796 13:10 0.24063 63 1.36221
14 0.63358 14:11 0.23244 70 1.33343
Ⅱ 7 0.56160 3 7:4 0.27346 32 1.47147
9 0.62620 9:6 0.23625 44 1.34696
11 0.66415 11:8 0.21766 57 1.27850
13 0.68886 13:10 0.20702 70 1.23510
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$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 7 0.39923 3 7:4 0.39522 26 1.86449
8 0.46231 8:5 0.34307 32 1.69384
10 0.54635 10:7 0.28318 44 1.50281
11 0.57515 11:8 0.26511 51 1.44432
13 0.61796 13:10 0.24063 63 1.36221
14 0.63358 14:11 0.23244 70 1.33343
Ⅱ 7 0.56160 3 7:4 0.27346 32 1.47147
9 0.62620 9:6 0.23625 44 1.34696
11 0.66415 11:8 0.21766 57 1.27850
13 0.68886 13:10 0.20702 70 1.23510
Table 10.  Third order interior resonance in the Sun-Mars system
$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 7 0.39923 3 7:4 0.39521 26 1.86534
8 0.46235 8:5 0.34303 32 1.69452
10 0.54630 10:7 0.28320 44 1.50361
11 0.57521 11:8 0.26506 51 1.44487
13 0.61783 13:10 0.24068 63 1.36310
14 0.63380 14:11 0.23231 70 1.33366
Ⅱ 7 0.56158 3 7:4 0.27346 32 1.47220
9 0.62616 9:6 0.23626 44 1.34767
11 0.66413 11:8 0.21764 57 1.2794
13 0.68878 13:10 0.20702 70 1.23584
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$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 7 0.39923 3 7:4 0.39521 26 1.86534
8 0.46235 8:5 0.34303 32 1.69452
10 0.54630 10:7 0.28320 44 1.50361
11 0.57521 11:8 0.26506 51 1.44487
13 0.61783 13:10 0.24068 63 1.36310
14 0.63380 14:11 0.23231 70 1.33366
Ⅱ 7 0.56158 3 7:4 0.27346 32 1.47220
9 0.62616 9:6 0.23626 44 1.34767
11 0.66413 11:8 0.21764 57 1.2794
13 0.68878 13:10 0.20702 70 1.23584
Table 11.  Fifth order interior resonance in the Sun-Earth system
$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 11 0.36792 5 11:6 0.42357 38 1.96103
12 0.41350 12:7 0.38286 44 1.82332
13 0.45107 13:8 0.35190 51 1.72225
14 0.48277 14:9 0.32751 57 1.64406
16 0.53281 16:11 0.29211 70 1.53139
17 0.55295 17:12 0.27893 76 1.48915
Ⅱ 15 0.58150 5 15:10 0.26122 70 1.43180
17 0.61344 17:12 0.24307 82 1.37065
19 0.63733 19:14 0.23053 95 1.32660
21 0.65640 21:16 0.22124 107 1.29228
23 0.67095 23:18 0.21460 120 1.26649
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$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 11 0.36792 5 11:6 0.42357 38 1.96103
12 0.41350 12:7 0.38286 44 1.82332
13 0.45107 13:8 0.35190 51 1.72225
14 0.48277 14:9 0.32751 57 1.64406
16 0.53281 16:11 0.29211 70 1.53139
17 0.55295 17:12 0.27893 76 1.48915
Ⅱ 15 0.58150 5 15:10 0.26122 70 1.43180
17 0.61344 17:12 0.24307 82 1.37065
19 0.63733 19:14 0.23053 95 1.32660
21 0.65640 21:16 0.22124 107 1.29228
23 0.67095 23:18 0.21460 120 1.26649
Table 12.  Fifth order interior resonance in the Sun-Mars system
$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 11 0.36790 5 11:6 0.42359 38 1.96199
12 0.41345 12:7 0.38289 44 1.82430
13 0.45118 13:8 0.35180 51 1.72276
14 0.48285 14:9 0.32744 57 1.64463
16 0.53273 16:11 0.29216 70 1.53227
17 0.55264 17:12 0.27912 76 1.49047
Ⅱ 15 0.58152 5 15:10 0.26127 70 1.43243
17 0.61335 17:12 0.24310 82 1.37146
19 0.63740 19:14 0.23048 95 1.32710
21 0.65623 21:16 0.22130 107 1.29319
23 0.67119 23:18 0.21447 120 1.26667
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$FA$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$
Ⅰ 11 0.36790 5 11:6 0.42359 38 1.96199
12 0.41345 12:7 0.38289 44 1.82430
13 0.45118 13:8 0.35180 51 1.72276
14 0.48285 14:9 0.32744 57 1.64463
16 0.53273 16:11 0.29216 70 1.53227
17 0.55264 17:12 0.27912 76 1.49047
Ⅱ 15 0.58152 5 15:10 0.26127 70 1.43243
17 0.61335 17:12 0.24310 82 1.37146
19 0.63740 19:14 0.23048 95 1.32710
21 0.65623 21:16 0.22130 107 1.29319
23 0.67119 23:18 0.21447 120 1.26667
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