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The study of periodic orbits in the spatial collinear restricted four-body problem with non-spherical primaries

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Abstract

In the current manuscript, we have explored the spatial collinear restricted four-body problem(SCR4BP) in which the three bodies, known as primaries, are situated in the collinear Euler’s configuration, where the primary \(P_0\)(\(m_0 = \beta m\)) (taken as central primary) is placed at the origin, and other two primaries \(P_1\)(\(m_1\)) and \(P_2\) \((m_2)\), which are non-spherical bodies, are placed at the same distance from the origin having masses \(m_1=m_2=m\). The massless fourth body known as test particle (or infinitesimal body) is moving under the Newtonian gravitational attraction of the primaries. This fourth body does not influence the motion of the primaries but its motion is effected by them as in the restricted three-body problem (R3BP). We have determined the long- and short-period orbits around those libration points which lie on the \(y-\)axis only in the SCR4BP for the case of two equal masses by using Fourier series method. We have studied the evolution of families of these orbits as the oblateness parameter A and mass parameter \(\beta \) evolve. The time period T of the periodic orbit is studied by using the variational graphs. By retaining the terms up to third order in the Fourier expansions, we have explored how the oblateness of the peripheral primaries as well as the mass parameter affect not only the shape, size and period but also the networks of long- and short-period orbits.

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All the incorporated data that support the findings of this study are available in the article and in its supplementary material files.

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The authors state that there was no funding for this study.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Delhi, New Delhi, India

    Om Prakash Meena

  2. Department of Mathematics, Sri Aurobindo College, University of Delhi, Delhi, New Delhi, 110017, India

    Md Sanam Suraj

  3. Department of Mathematics, Deshbandhu College, University of Delhi, Delhi, New Delhi, 110019, India

    Rajiv Aggarwal

  4. Department of Mathematics, ARSD College, University of Delhi, Delhi, New Delhi, 110021, India

    Amit Mittal

  5. Department of Physics, Deshbandhu College, University of Delhi, Delhi, New Delhi, 110019, India

    Md Chand Asique

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  1. Om Prakash Meena
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  2. Md Sanam Suraj
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  4. Amit Mittal
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Appendices

Appendixes

The values of various coefficients

$$\begin{aligned} \wp _{00}&= \frac{1}{\sqrt{\kappa }}\\ \wp _{01}&=-\frac{{\mathscr {L}}}{\kappa ^{3/2}} \\ \wp _{02}&=-\frac{1}{2 \kappa ^{3/2}}\\ \wp _{03}&= \frac{3 {\mathscr {L}}^2-\kappa }{2 \kappa ^{5/2}}\\ \wp _{04}&= \frac{3 {\mathscr {L}}}{2 \kappa ^{5/2}}\\ \wp _{05}&= \frac{{\mathscr {L}}(3 \kappa -5 {\mathscr {L}}^2)}{2 \kappa ^{7/2}}\\ \wp _{06}&= \frac{3}{8 \kappa ^{5/2}}\\ \wp _{07}&= \frac{3 (\kappa -5 {\mathscr {L}}^2)}{4 \kappa ^{7/2}}\\ \wp _{08}&= \frac{3 \kappa ^2-30 \kappa {\mathscr {L}}^2+35 {\mathscr {L}}^4}{8 \kappa ^{9/2}}\\ \wp _{10}&= \frac{2}{\sqrt{1+4 \kappa }}\\ \wp _{11}&= -\frac{8 {\mathscr {L}}}{(1+4 \kappa )^{3/2}} \\ \wp _{12}&= \frac{8 \left( 1-2 \kappa \right) }{(1+4 \kappa )^{5/2}} \\ \wp _{13}&= \frac{4\left( 12 {\mathscr {L}}^2-4\kappa -1\right) }{(1+4 \kappa )^{5/2}}\\ \wp _{14}&= \frac{192 \left( \kappa -1\right) {\mathscr {L}}}{(1+4 \kappa )^{7/2}}\\ \wp _{15}&= \frac{16 {\mathscr {L}}\left( 3+12\kappa -20{\mathscr {L}}^2\right) }{(1+4 \kappa )^{7/2}}\\ \wp _{16}&= \frac{32 \left( 1-12 \kappa +6 \kappa ^2\right) }{(1+4 \kappa )^{9/2}} \\ \wp _{17}&= \frac{96 \left( 4 \kappa ^2+30 {\mathscr {L}}^2-20 \kappa {\mathscr {L}}^2-3\kappa -1\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{18}&= \frac{4 \left( 3+24 \kappa +48 \kappa ^2-120 {\mathscr {L}}^2-480 \kappa {\mathscr {L}}^2+560 {\mathscr {L}}^4\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{20}&= \frac{2}{\sqrt{1+4 \kappa }}\\ \wp _{21}&= -\frac{8 {\mathscr {L}}}{(1+4 \kappa )^{3/2}}\\ \wp _{22}&= \frac{8 \left( 1-2 \kappa \right) }{(1+4 \kappa )^{5/2}}\\ \wp _{23}&= \frac{4\left( 12 {\mathscr {L}}^2-4 \kappa -1\right) }{(1+4 \kappa )^{5/2}}\\ \wp _{24}&= \frac{192 \left( \kappa -1\right) {\mathscr {L}}}{(1+4 \kappa )^{7/2}} \end{aligned}$$
$$\begin{aligned} \wp _{25}&= \frac{16 {\mathscr {L}}\left( 3+12 \kappa -20 {\mathscr {L}}^2\right) }{(1+4 \kappa )^{7/2}}\\ \wp _{26}&= \frac{32 \left( 1-12 \kappa +6 \kappa ^2\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{27}&= \frac{96 \left( 4 \kappa ^2+30 {\mathscr {L}}^2-1-3\kappa -20 \kappa {\mathscr {L}}^2\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{28}&=\frac{4\left( 3+24 \kappa +48 \kappa ^2-120 {\mathscr {L}}^2-480 \kappa {\mathscr {L}}^2+560 {\mathscr {L}}^4 \right) }{(1+4 \kappa )^{9/2}} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \wp _{10}^*&= \frac{8}{(1+4 \kappa )^{3/2}}\\ \wp _{11}^*&=- \frac{96 {\mathscr {L}}}{(1+4 \kappa )^{5/2}} \\ \wp _{12}^*&= \frac{192 \left( 1-\kappa \right) }{(1+4 \kappa )^{7/2}} \\ \wp _{13}^*&=\frac{48 \left( 20 {\mathscr {L}}^2-4\kappa -1\right) }{(1+4 \kappa )^{7/2}}\\ \wp _{14}^*&= \frac{1920 (2 \kappa -3){\mathscr {L}}}{(1+4 \kappa )^{9/2}}\\ \wp _{15}^*&= \frac{320 {\mathscr {L}} \left( 3+12\kappa -28 {\mathscr {L}}^2\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{16}^*&= \frac{1920 \left( 1-6\kappa +2\kappa ^2 \right) }{(1+4 \kappa )^{11/2}}\\ \wp _{17}^*&= \frac{960 \left( 8 \kappa ^2+112 {\mathscr {L}}^2-3-10\kappa -56 \kappa {\mathscr {L}}^2 \right) }{(1+4 \kappa )^{11/2}} \\ \wp _{18}^*&= \frac{240\left( 1+8\kappa +16 \kappa ^2 -56 {\mathscr {L}}^2-224 \kappa {\mathscr {L}}^2+336 {\mathscr {L}}^4 \right) }{(1+4 \kappa )^{11/2}}\\ \wp _{20}^*&= \frac{8}{(1+4 \kappa )^{3/2}}\\ \wp _{21}^*&= -\frac{96 {\mathscr {L}}}{(1+4 \kappa )^{5/2}}\\ \wp _{22}^*&= \frac{192 \left( 1-\kappa \right) }{(1+4 \kappa )^{7/2}}\\ \wp _{23}^*&= \frac{48\left( 20 {\mathscr {L}}^2-4 \kappa -1 \right) }{(1+4 \kappa )^{7/2}}\\ \wp _{24}^*&= \frac{1920 (2 \kappa -3){\mathscr {L}}}{(1+4 \kappa )^{9/2}}\\ \wp _{25}^*&= \frac{320 {\mathscr {L}}\left( 3+12 \kappa -28 {\mathscr {L}}^2\right) }{(1+4 \kappa )^{9/2}}\\ \wp _{26}^*&= \frac{1920 \left( 1-6 \kappa +2 \kappa ^2 \right) }{(1+4 \kappa )^{11/2}}\\ \wp _{27}^*&= \frac{960 \left( 8 \kappa ^2+112 {\mathscr {L}}^2-3-10 \kappa -56 \kappa {\mathscr {L}}^2\right) }{(1+4 \kappa )^{11/2}}\\ \wp _{28}^*&= \frac{240\left( 1+8\kappa +16 \kappa ^2-56 {\mathscr {L}}^2-224 \kappa {\mathscr {L}}^2+336 {\mathscr {L}}^4\right) }{(1+4 \kappa )^{11/2}} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \Omega _{0}^*&=\frac{A \left( \wp _{10}^*+\wp _{20}^*\right) +2 (\beta \wp _{00}+\wp _{10}+\wp _{20})}{2 \Lambda }\\ \Omega _{1}^*&=\frac{A \left( \wp _{11}^*+\wp _{21}^*\right) +2 (\beta \wp _{01}+\wp _{11}+\wp _{21})}{2 \Lambda }\\ \Omega _{2}^*&=\frac{A \left( \wp _{12}^*+\wp _{22}^*\right) +2 (\beta \wp _{02}+\wp _{12}+\wp _{22})}{2 \Lambda }\\ \Omega _{3}^*&=\frac{A \left( \wp _{13}^*+\wp _{23}^*\right) +2 (\beta \wp _{03}+\wp _{13}+\wp _{23})}{2 \Lambda }\\ \Omega _{4}^*&=\frac{A \left( \wp _{14}^*+\wp _{24}^*\right) +2 (\beta \wp _{04}+\wp _{14}+\wp _{24})}{2 \Lambda }\\ \Omega _{5}^*&=\frac{A \left( \wp _{15}^*+\wp _{25}^*\right) +2 (\beta \wp _{05}+\wp _{15}+\wp _{25})}{2 \Lambda }\\ \Omega _{6}^*&= \frac{A \left( \wp _{16}^*+\wp _{26}^*\right) +2 (\beta \wp _{06}+\wp _{16}+\wp _{26})}{2 \Lambda }\\ \Omega _{7}^*&=\frac{A \left( \wp _{17}^*+\wp _{27}^*\right) +2 (\beta \wp _{07}+\wp _{17}+\wp _{27})}{2 \Lambda }\\ \Omega _{8}^*&=\frac{A \left( \wp _{18}^*+\wp _{28}^*\right) +2 (\beta \wp _{08}+\wp _{18}+\wp _{28})}{2 \Lambda } \end{aligned}$$

The values of various coefficients

$$\begin{aligned} A_{01}&=\frac{1}{2}a_1^2+\frac{1}{2}a_{-1}^2\\ A_{02}&=2 a_0a_1+a_1 a_2+a_{-1}a_{-2}\\ A_{03}&=2a_0a_{-1}+a_1a_{-2}-a_{-1}a_{2}\\ A_{04}&=\frac{1}{2}a_1^2-\frac{1}{2}a_{-1}^2\\ A_{05}&= a_1 a_{-1}\\ A_{06}&=a_1a_2-a_{-1}a_{-2}\\ A_{07}&=a_1a_{-2}+a_{-1}a_{2}\\ B_{01}&=\frac{1}{2}a_1 b_1+\frac{1}{2}a_{-1}b_{-1}\\ B_{02}&=\frac{1}{2} (a_1b_2+a_2b_1)+\frac{1}{2}(a_{-1}b_{-2}+a_{-2}b_{-1})\\ {}&\quad +a_1b_0+a_0b_1 \\ B_{03}&=\frac{1}{2}(a_1b_{-2}+a_{-2}b_1)-\frac{1}{2}(a_{-1}b_{2}+a_{2}b_{-1})\\ {}&\quad +a_{-1}b_{0}+a_0 b_{-1}\\ B_{04}&=\frac{1}{2}a_1 b_1-\frac{1}{2}a_{-1}b_{-1}\\ B_{05}&=\frac{1}{2}a_1 b_{-1}+\frac{1}{2}a_{-1}b_{1}\\ B_{06}&=\frac{1}{2} (a_1b_2+a_{2}b_{1})-\frac{1}{2}(a_{-1}b_{-2}+a_{-2}b_{-1})\\ B_{07}&=\frac{1}{2}(a_1b_{-2}+a_{-2}b_{1})+\frac{1}{2}(a_{-1}b_2+a_2b_{-1})\\ C_{01}&=\frac{1}{2}b_1^2+\frac{1}{2}b_{-1}^2\\ C_{02}&=2b_0b_1+b_1b_2+b_{-1}b_{-2}\\ C_{03}&=2b_0b_{-1}+b_1b_{-2}-b_{-1}b_{2}\\ C_{04}&=\frac{1}{2}b_1^2-\frac{1}{2}b_{-1}^2\\ C_{05}&= b_1 b_{-1}\\ C_{06}&=b_1b_2-b_{-1}b_{-2}\\ C_{07}&=b_1b_{-2}+b_{-1}b_{2}\\ D_{01}&=0\\ D_{02}&=\frac{3}{4}a_1^3+\frac{3}{4}a_1 a_{-1}^2\\ D_{03}&=\frac{3}{4}a_{-1}^3+\frac{3}{4}a_1^2a_{-1}\\ D_{04}&=0\\ D_{05}&=0\\ D_{06}&=\frac{1}{4} a_1^3-\frac{3}{4} a_{1}a_{-1}^2\\ D_{07}&=-\frac{1}{4} a_{-1}^3+\frac{3}{4} a_{1}^2a_{-1}\\ E_{01}&=0\\ E_{02}&=\frac{3}{4}a_1^2b_1+\frac{1}{4}a_{-1}^2b_1+\frac{1}{2}a_1 a_{-1}b_{-1}\\ E_{03}&=\frac{3}{4}a_{-1}^2b_{-1}+\frac{1}{4}a_1^2b_{-1}+\frac{1}{2}a_1 a_{-1}b_{1}\\ E_{04}&=0\\ E_{05}&=0\\ E_{06}&=\frac{1}{4}a_1^2b_1-\frac{1}{4}a_{-1}^2b_1-\frac{1}{2}a_1 a_{-1}b_{-1}\\ E_{07}&=\frac{1}{4}a_1^2b_{-1}-\frac{1}{4}a_{-1}^2b_{-1}+\frac{1}{2}a_1 a_{-1}b_{1}\\ F_{01}&=0\\ F_{02}&=\frac{3}{4}a_1b_1^2+\frac{1}{4}a_{1}b_{-1}^2+\frac{1}{2}a_{-1} b_1b_{-1}\\ F_{03}&=\frac{3}{4}a_{-1}b_{-1}^2+\frac{1}{4}a_{-1}b_1^2+\frac{1}{2}a_1 b_1 b_{-1}\\ F_{04}&=0\\ F_{05}&=0\\ F_{06}&=\frac{1}{4}a_1b_1^2-\frac{1}{4}a_{1}b_{-1}^2-\frac{1}{2}a_{-1} b_1b_{-1}\\ F_{07}&=\frac{1}{4}a_{-1}b_1^2-\frac{1}{4}a_{-1}b_{-1}^2+\frac{1}{2}a_1 b_1 b_{-1}\\ G_{01}&=0\\ G_{02}&=\frac{3}{4} b_1^3+\frac{3}{4} b_{1}b_{-1}^2\\ G_{03}&=\frac{3}{4} b_{-1}^3+\frac{3}{4} b_{1}^2b_{-1}\\ G_{04}&=0\\ G_{05}&=0\\ G_{06}&=\frac{1}{4} b_1^3-\frac{3}{4} b_{1}b_{-1}^2\\ G_{07}&=-\frac{1}{4} b_{-1}^3+\frac{3}{4} b_{1}^2b_{-1} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \lambda _{1}&=-3 \Gamma _{1}^2 \Omega ^*_{5}\\ \lambda _{2}&=-(4 {\Omega ^*_4}+6 \Gamma _{1} {\Omega ^*_5)}\\ \lambda _{3}&=-3 {\Omega ^*_5}\\ \lambda _{4}&=3 {\Omega ^*_5} \Gamma _{1}^3\\ \lambda _{5}&=3(4{\Omega ^*_4} \Gamma _{1}+6{\Omega ^*_5} \Gamma _{1}^2)\\ \lambda _{6}&=27{\Omega ^*_5} \Gamma _{1}\\ \lambda _{7}&=12{\Omega ^*_5} \\ \lambda _{8}&=3{\Omega ^*_5} \Gamma _{1}^2+{\Omega ^*_4} \Gamma _{1} \Gamma _{2}\\ \lambda _{9}&=4{\Omega ^*_4}(\Gamma _{1}-1)+6{\Omega ^*_5} \Gamma _{1}+{\Omega ^*_4} \Gamma _{2} \\ \lambda _{10}&=4 {\Omega ^*_4}+3 {\Omega ^*_5} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \lambda _{11}&={\Omega ^*_4} (2 \Gamma _{1} \lambda _{9}+\lambda _{5}+2 \lambda _{8})\\ \lambda _{12}&=24 \lambda _{8} {\Omega ^*_4}-9 {\Omega ^*_4} (2 \Gamma _{1} \lambda _{8}+\lambda _{4})\\ \lambda _{13}&=\lambda _{12}-\Gamma _{2} (\lambda _{11}+3 \Gamma _{1}^2 \Gamma _{3} {\Omega ^*_7})\\ \lambda _{14}&=36 \Gamma _{1} \Gamma _{3} {\Omega ^*_7}+18 {\Omega ^*_8} (5 \Gamma _{1}^3-3 \Gamma _{1}^2 \Gamma _{3})\\ \lambda _{15}&=\lambda _{14}+24 \lambda _{9} {\Omega ^*_4}-9 {\Omega ^*_5} (\Gamma _{1} \lambda _{6}+\lambda _{5})\\ \lambda _{16}&={\Omega ^*_4} (2 \Gamma _{1} \lambda _{9}+\lambda _{5}+2 \lambda _{8})+3 \Gamma _{1}^2 \Gamma _{3} {\Omega ^*_7}\\ \lambda _{17}&=24 \Gamma _{3} {\Omega ^*_6}-3 {\Omega ^*_7} (2 \Gamma _{1} \Gamma _{3}-5 \Gamma _{1}^2)\\ \lambda _{18}&={\Omega ^*_4} (2 (\Gamma _{1} \lambda _{10}+\lambda _{9})+\lambda _{6})\\ \lambda _{19}&=24 \lambda _{10} {\Omega ^*_4}+18 {\Omega ^*_8} (15 \Gamma _{1}^2-3 \Gamma _{1} \Gamma _{3})\\ \lambda _{20}&=120 {\Omega ^*_6}-3 {\Omega ^*_7} (10 \Gamma _{1}-\Gamma _{3})\\ \lambda _{21}&=\lambda _{19}-\Gamma _{2} (\lambda _{20}+{\Omega ^*_4} (2 \lambda _{10}+\lambda _{7}))\\ \lambda _{22}&={\Omega ^*_4} (2 (\Gamma _{1} \lambda _{10}+\lambda _{9})+\lambda _{6})\\ \lambda _{23}&=-\lambda _{22}+24 \Gamma _{3} {\Omega ^*_6}+3 {\Omega ^*_7} (5 \Gamma _{1}^2-2 \Gamma _{1} \Gamma _{3})\\ \lambda _{24}&=\lambda _{21}+9 \lambda _{23}-9 {\Omega ^*_5} (\Gamma _{1} \lambda _{7}+\lambda _{6})\\ \lambda _{25}&=15 \Gamma _{2} {\Omega ^*_7}-180 {\Omega ^*_7}+18 {\Omega ^*_8} (15 \Gamma _{1}-\Gamma _{3})\\ \lambda _{26}&=3 {\Omega ^*_7} (10 \Gamma _{1}-\Gamma _{3})-{\Omega ^*_4} (2 \lambda _{10}+\lambda _{7})\\ \lambda _{27}&=-\Gamma _{2} {\Omega ^*_4} (2 \Gamma _{1} \lambda _{8}+\lambda _{4})-9 \Gamma _{1} \lambda _{4} {\Omega ^*_5}\\ \lambda _{28}&=\lambda _{13}-9 {\Omega ^*_5} (\Gamma _{1} \lambda _{5}+\lambda _{4})-18 \Gamma _{1}^3 \Gamma _{3} {\Omega ^*_8}\\ \lambda _{29}&=\Gamma _{2} (\lambda _{17}-\lambda _{18})+\lambda _{15}-9 \lambda _{16}\\ \lambda _{30}&=\lambda _{24}+36 {\Omega ^*_7} (\Gamma _{3}-5 \Gamma _{1})\\ \lambda _{31}&=\lambda _{25}-9 \lambda _{7} {\Omega ^*_5}+9 (\lambda _{26}-120 {\Omega ^*_6})\\ \lambda _{32}&=135 {\Omega ^*_7}+90 {\Omega ^*_8}\\ \lambda _{33}&=16 \Gamma _{1} \lambda _{8} {\Omega _4}+3 \Gamma _{1}^2 \lambda _{5} {\Omega ^*_5}+30 \Gamma _{1} \lambda _{4} {\Omega ^*_5}\\ \lambda _{34}&=3 {\Omega ^*_5} (\Gamma _{1}^2 \lambda _{6}+10 \Gamma _{1} \lambda _{5}+9 \lambda _{4})\\ \lambda _{35}&=12 \Gamma _{1}^2 \Gamma _{3} {\Omega ^*_7}-15 \Gamma _{1}^4 {\Omega ^*_8}+36 \Gamma _{1}^3 \Gamma _{3} {\Omega ^*_8}\\ \lambda _{36}&=8 \Gamma _{1} \lambda _{9} {\Omega ^*_4}+6 \lambda _{5} {\Omega ^*_4}-24 \lambda _{8} {\Omega ^*_4}\\ \lambda _{37}&=3 {\Omega ^*_5} (\Gamma _{1}^2 \lambda _{7}+10 \Gamma _{1} \lambda _{6}+9 \lambda _{5})\\ \lambda _{38}&=12 (12 \Gamma _{3} {\Omega ^*_6}+5 \Gamma _{1}^2 {\Omega ^*_7}+2 \Gamma _{1} \Gamma _{3} {\Omega ^*_7})\\ \lambda _{39}&=\Gamma _{1} (3 \Gamma _{1} \Gamma _{3}-15 \Gamma _{1}^2)+9 (3 \Gamma _{1}^2 \Gamma _{3}-5 \Gamma _{1}^3)\\ \lambda _{40}&={\Omega ^*_4} (8 \Gamma _{1} \lambda _{10}+6 (\lambda _{6}-4 \lambda _{9}))\\ \lambda _{41}&=9 (3 \Gamma _{1} \Gamma _{3}-15 \Gamma _{1}^2)+\Gamma _{1} (\Gamma _{3}-15 \Gamma _{1})\\ \lambda _{42}&=6 {\Omega ^*_4} (\lambda _{7}-4 \lambda _{10})+3 \lambda _{41} {\Omega ^*_8}\\ \lambda _{43}&=5 (12 {\Omega ^*_6}+2 \Gamma _{1} {\Omega ^*_7})-3 \Gamma _{3} {\Omega ^*_7}\\ \lambda _{44}&=3 {\Omega ^*_5} (10 \Gamma _{1} \lambda _{7}+9 \lambda _{6})\\ \lambda _{45}&=9 (\Gamma _{3}-15 \Gamma _{1})-5 \Gamma _{1}\\ \lambda _{46}&=3 \Gamma _{1}^2 \lambda _{4} {\Omega ^*_5}\\ \lambda _{47}&=\lambda _{33}+12 \lambda _{4} {\Omega ^*_4}+6 \Gamma _{1}^4 \Gamma _{3} {\Omega ^*_8}\\ \lambda _{48}&=\lambda _{34}+2 (\lambda _{35}+\lambda _{36})\\ \lambda _{49}&=\lambda _{37}+2 (-\lambda _{38}+\lambda _{40}+3 \lambda _{39} {\Omega ^*_8})\\ \lambda _{50}&=2 (\lambda _{42}+12 \lambda _{43})+\lambda _{44}\\ \lambda _{51}&=27 \lambda _{7} {\Omega ^*_5}+2 (180 {\Omega ^*_7}+3 \lambda _{45} {\Omega ^*_8})\\ \lambda _{52}&=-270 {\Omega ^*_8} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \lambda _{53}&=6 \Gamma _{3} \lambda _{1} {\Omega ^*_4}-\Gamma _{2} {\Omega ^*_4} (-2 \Gamma _{1} \lambda _{9}+\lambda _{5}-2 \lambda _{8})\\ \lambda _{54}&=-24 \Gamma _{3} {\Omega ^*_6}+5 \Gamma _{1}^2 {\Omega ^*_7}-2 \Gamma _{1} \Gamma _{3} {\Omega ^*_7}\\ \lambda _{55}&=\Gamma _{2} {\Omega ^*_4} (2 \Gamma _{1} \lambda _{10}-\lambda _{6}+2 \lambda _{9})\\ \lambda _{56}&=3 \Gamma _{2} (\Gamma _{3} {\Omega ^*_7}-5 (24 {\Omega ^*_6}+2 \Gamma _{1} {\Omega ^*_7}))\\ \lambda _{57}&=\lambda _{56}+6 {\Omega ^*_4} (\Gamma _{3} \lambda _{3}-5 \lambda _{2})\\ \lambda _{58}&=-{\Omega ^*_4} \Gamma _{2} (\lambda _{4}-2 \Gamma _{1} \lambda _{8})\\ \lambda _{59}&=\lambda _{53}+3 \Gamma _{1}^2 \Gamma _{2} \Gamma _{3} {\Omega ^*_7}\\ \lambda _{60}&=-3 \Gamma _{2} \lambda _{54}+\lambda _{55}+6 {\Omega ^*_4} (\Gamma _{3} \lambda _{2}-5 \lambda _{1})\\ \lambda _{61}&=\lambda _{57}+\Gamma _{2} {\Omega ^*_4} (2 \lambda _{10}-\lambda _{7})\\ \lambda _{62}&=-30 \lambda _{3} {\Omega ^*_4}-15 \Gamma _{2} {\Omega ^*_7}\\ \lambda _{63}&=-18 \Gamma _{1} \Gamma _{3} \lambda _{1} {\Omega ^*_5}-3 \Gamma _{2} \lambda _{4} {\Omega ^*_5}-18 \Gamma _{1}^3 \Gamma _{2} \Gamma _{3} {\Omega ^*_8}\\ \lambda _{64}&=\Gamma _{1} (\Gamma _{2} \lambda _{6}+6 (\Gamma _{3} \lambda _{2}-5 \lambda _{1}))+\Gamma _{2} \lambda _{5}\\ \lambda _{65}&=4 \lambda _{9} {\Omega ^*_4}+6 \Gamma _{1} \Gamma _{3} {\Omega ^*_7}+27 \Gamma _{1}^2 \Gamma _{3} {\Omega ^*_8}\\ \lambda _{66}&=3 {\Omega ^*_5} (6 \Gamma _{3} \lambda _{1}+\lambda _{64})\\ \lambda _{67}&=4 \lambda _{10} {\Omega ^*_4}+9 {\Omega ^*_8} (3 \Gamma _{1} \Gamma _{3}-15 \Gamma _{1}^2)\\ \lambda _{68}&=2 \Gamma _{2} (\lambda _{67}+6 {\Omega ^*_7} (\Gamma _{3}-5 \Gamma _{1}))\\ \lambda _{69}&=\Gamma _{1} (\Gamma _{2} \lambda _{7}+6 (\Gamma _{3} \lambda _{3}-5 \lambda _{2}))\\ \lambda _{70}&=\Gamma _{2} \lambda _{6}+6 (\Gamma _{3} \lambda _{2}-5 \lambda _{1})+\lambda _{69}\\ \lambda _{71}&=2 \Gamma _{2} (9 {\Omega ^*_8} (\Gamma _{3}-15 \Gamma _{1})-30 {\Omega ^*_7})\\ \lambda _{72}&=-30 \Gamma _{1} \lambda _{3}+\Gamma _{2} \lambda _{7}+6 (\Gamma _{3} \lambda _{3}-5 \lambda _{2})\\ \lambda _{73}&=-3 \Gamma _{1} \Gamma _{2} \lambda _{4} {\Omega ^*_5}\\ \lambda _{74}&=\lambda _{63}-8 \Gamma _{2} \lambda _{8} {\Omega ^*_4}-3 \Gamma _{1} \Gamma _{2} \lambda _{5} {\Omega ^*_5}\\ \lambda _{75}&=-\lambda _{66}-2 \Gamma _{2} (\lambda _{65}-45 \Gamma _{1}^3 {\Omega ^*_8})\\ \lambda _{76}&=-\lambda _{68}-3 \lambda _{70} {\Omega ^*_5}\\ \lambda _{77}&=-\lambda _{71}-3 \lambda _{72} {\Omega ^*_5}\\ \lambda _{78}&=90 \lambda _{3} {\Omega ^*_5}+90 \Gamma _{2} {\Omega ^*_8} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \lambda _{79}&=\Gamma _{2} \lambda _{58}-\lambda _{73}\\ \lambda _{80}&=\Gamma _{2} \lambda _{59}+\lambda _{58}-\lambda _{74}\\ \lambda _{81}&=\Gamma _{2} \lambda _{60}+\lambda _{59}-\lambda _{75}\\ \lambda _{82}&=\Gamma _{2} \lambda _{61}+\lambda _{60}-\lambda _{76}\\ \lambda _{83}&=\Gamma _{2} \lambda _{62}+\lambda _{61}-\lambda _{77}\\ \lambda _{84}&=\lambda _{62}-\lambda _{78} \end{aligned}$$

The values of various coefficients

$$\begin{aligned} \lambda _{85}&=\Gamma _{1} \lambda _{79}\\ \lambda _{86}&=\Gamma _{1} \lambda _{80}-2 \Gamma _{3} \lambda _{58}-\lambda _{79}\\ \lambda _{87}&=\Gamma _{1} \lambda _{81}-2 \Gamma _{3} \lambda _{59}+4 \lambda _{58}-\lambda _{80}\\ \lambda _{88}&=\Gamma _{1} \lambda _{82}-2 \Gamma _{3} \lambda _{60}+4 \lambda _{59}-\lambda _{81}\\ \lambda _{89}&=\Gamma _{1} \lambda _{83}-2 \Gamma _{3} \lambda _{61}+4 \lambda _{60}-\lambda _{82}\\ \lambda _{90}&=\Gamma _{1} \lambda _{84}-2 \Gamma _{3} \lambda _{62}+4 \lambda _{61}-\lambda _{83}\\ \lambda _{91}&=4 \lambda _{62}-\lambda _{84} \end{aligned}$$
Table 7 Case-1: Data of periodic orbits around \((0,{\mathscr {L}})\) for \(\beta =0.1\) corresponding to Figs. 5 - 8
Full size table
Table 8 Case-2: Data of periodic orbits around \((0,{\mathscr {L}})\) for \(\beta =1000\) corresponding to Figs. 9 – 12
Full size table

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Meena, O.P., Suraj, M.S., Aggarwal, R. et al. The study of periodic orbits in the spatial collinear restricted four-body problem with non-spherical primaries. Nonlinear Dyn 111, 4283–4311 (2023). https://doi.org/10.1007/s11071-022-08085-z

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