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

• * Corresponding author: Elbaz I. Abouelmagd
• 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.

Mathematics Subject Classification: 37N05, 70F07, 70F15.

 Citation:

• Figure 1.  Exterior first order resonant single loop orbit and PSS for $C = 2.93$ in the Sun - Earth system

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

 $FA$ $SR$ $OB$ $NL$ $LO$ $NI$ $RO$ $EC$ $TP$ $RP$ 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

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

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

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

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

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

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

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

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

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

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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