Figure 1.
Exterior first order resonant single loop orbit and PSS for $C = 2.93$ in the Sun - Earth system
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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
| 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 |
- Abstract
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Under a Creative Commons license
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 and 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.
|
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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.
|
| [16] |
C. Douskos, V. Kalantonis and P. Markellos, Effects of resonances on the stability of retrograde satellites, Astrophys. Space Sci., 310, 245-249. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [23] |
A. S. Libert and K. Tsiganis,
Trapping in three-planet resonances during gas-driven migration, Celest. Mech. Dyn. Astron., 111 (2011), 201-218.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [28] |
C. D. Murray and S. F. Dermot, Solar System Dynamics, Cambridge University Press, 1999. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [34] |
H. Poincaré, Les Méthodes Nouvelles de la Méchanique, Celeste. Gauthier- Villas, Paris., 1987. |
| [35] |
N. Pushparaj and R. K. Sharma,
Interior resonance periodic orbits in photogravitational restricted three-body problem, Advances in Astrophysics, 2 (2017), 263-272.
|
| [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.
|
| [37] |
R. K. Sharma and P. V. Subbarao,
A case of commensurability induced by oblateness, Celest. Mech., 18 (1978), 185-194.
|
| [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.
|
| [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. |
| [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.
|
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.
|
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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.
|
| [16] |
C. Douskos, V. Kalantonis and P. Markellos, Effects of resonances on the stability of retrograde satellites, Astrophys. Space Sci., 310, 245-249. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [23] |
A. S. Libert and K. Tsiganis,
Trapping in three-planet resonances during gas-driven migration, Celest. Mech. Dyn. Astron., 111 (2011), 201-218.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [28] |
C. D. Murray and S. F. Dermot, Solar System Dynamics, Cambridge University Press, 1999. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [34] |
H. Poincaré, Les Méthodes Nouvelles de la Méchanique, Celeste. Gauthier- Villas, Paris., 1987. |
| [35] |
N. Pushparaj and R. K. Sharma,
Interior resonance periodic orbits in photogravitational restricted three-body problem, Advances in Astrophysics, 2 (2017), 263-272.
|
| [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.
|
| [37] |
R. K. Sharma and P. V. Subbarao,
A case of commensurability induced by oblateness, Celest. Mech., 18 (1978), 185-194.
|
| [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.
|
| [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. |
| [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.
|
Figure 2.
Exterior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system
Figure 3.
Interior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Table 1.
Analysis of exterior first order resonance in the perturbed Sun-Earth system
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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
| 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 |
| 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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