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On the periodic orbits around the collinear libration points in the SCR4BP with non-spherical primaries

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Abstract

In this paper, we have drawn the periodic orbits in the restricted four-body problem. The fourth body of infinitesimal mass moves under the Newtonian gravitational law due to three mass points (known as primaries) in the collinear Euler’s configuration. We have studied the dynamics of the fourth body in the spatial collinear restricted four-body problem (SCR4BP) with non-spherical primaries. We have determined the periodic orbits by using the Fourier series method only around collinear libration points in SCR4BP with non-spherical primaries. The effects of the orbital parameter \(\varepsilon \) (\(0<\varepsilon <1\)), which controls the size of the orbit, on the periodic orbits and their period are investigated. In order to study the effect of this parameter, we have retained the terms up to third order with respect to the orbital dimensions. We have also drawn the variational graphs to determine the variation in the time period T of the periodic orbits. Moreover, we have unveiled that how oblateness and mass parameters influence the shape, size, and period of the periodic orbits.

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Acknowledgements

Authors are thankful to the two anonymous reviewers, whose comments and suggestions are very useful in improving the manuscript.

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Authors and Affiliations

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

    Md Sanam Suraj

  2. Department of Mathematics, Hansraj College, University of Delhi, Delhi, 110007, India

    Om Prakash Meena

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

    Rajiv Aggarwal

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

    Amit Mittal

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

    Md Chand Asique

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Appendices

A The values of various coefficients in Eq. 11a

$$\begin{aligned} \rho _{00}&= \frac{1}{\sqrt{k}}\\ \rho _{01}&=-\frac{L}{k^{3/2}} \\ \rho _{02}&= \frac{3 L^2-k}{2 k^{5/2}}\\ \rho _{03}&= -\frac{1}{2 k^{3/2}}\\ \rho _{04}&= \frac{L \left( 3 k-5 L^2\right) }{2 k^{7/2}}\\ \rho _{05}&= \frac{3 L}{2 k^{5/2}}\\ \rho _{06}&= \frac{3 k^2-30 k L^2+35 L^4}{8 k^{9/2}}\\ \rho _{07}&= \frac{3 \left( k-5 L^2\right) }{4 k^{7/2}}\\ \rho _{08}&= \frac{3}{8 k^{5/2}}\\ \rho _{10}&= \frac{2}{\sqrt{4 k-4 L+1}}\\ \rho _{11}&= \frac{4 (1-2 L)}{(4 k-4 L+1)^{3/2}} \\ \rho _{12}&= \frac{8 \left( 6 L^2-2 k-4 L+1\right) }{(4 k-4 L+1)^{5/2}} \\ \rho _{13}&= -\frac{4}{(4 k-4 L+1)^{3/2}}\\ \rho _{14}&= \frac{16 (2 L-1) \left( 6 k-10 L^2+4 L-1\right) }{(4 k-4 L+1)^{7/2}}\\ \rho _{15}&= \frac{24 (2 L-1)}{(4 k-4 L+1)^{5/2}}\\ \rho _{16}&= \frac{32 \left( 6 k^2-60 k L^2+48 k L-12 k+70 L^4-80 L^3\right) }{(4 k-4 L+1)^{9/2}}\\&\quad +\,\frac{32(36 L^2-8 L+1)}{(4 k-4 L+1)^{9/2}} \\ \rho _{17}&= \frac{96 \left( k-5 L^2+4 L-1\right) }{(4 k-4 L+1)^{7/2}}\\ \rho _{18}&= \frac{12}{(4 k-4 L+1)^{5/2}}\\ \rho _{20}&= \frac{2}{\sqrt{4 k+4 L+1}}\\ \rho _{21}&= -\frac{4 (2 L+1)}{(4 k+4 L+1)^{3/2}}\\ \rho _{22}&= \frac{8 \left( 6 L^2-2 k+4 L+1\right) }{(4 k+4 L+1)^{5/2}}\\ \rho _{23}&= -\frac{4}{(4 k+4 L+1)^{3/2}}\\ \rho _{24}&= \frac{16 (2 L+1) \left( 6 k-10 L^2-4 L-1\right) }{(4 k+4 L+1)^{7/2}} \end{aligned}$$
$$\begin{aligned} \rho _{25}&= \frac{24 (2 L+1)}{(4 k+4 L+1)^{5/2}}\\ \rho _{26}&= \frac{32 \left( 6 k^2-60 k L^2-48 k L-12 k+70 L^4\right) }{(4 k+4 L+1)^{9/2}}\\&+\frac{32(80 L^3+36 L^2+8 L+1)}{(4 k+4 L+1)^{9/2}}\\ \rho _{27}&= \frac{96 \left( k-5 L^2-4 L-1\right) }{(4 k+4 L+1)^{7/2}}\\ \rho _{28}&=\frac{12}{(4 k+4 L+1)^{5/2}} \end{aligned}$$

B The values of various coefficients in Eq. 11b

$$\begin{aligned} \rho _{10}^*&= \frac{8}{(4 k-4 L+1)^{3/2}}\\ \rho _{11}^*&= \frac{48 (1-2 L)}{(4 k-4 L+1)^{7/2}} \\ \rho _{12}^*&= \frac{192 \left( 5 L^2-4 L-k+1\right) }{(4 k-4 L+1)^{7/2}} \\ \rho _{13}^*&=-\frac{48}{(4 k-4 L+1)^{5/2}}\\ \rho _{14}^*&= \frac{640 (2 L-1) \left( 3 k-7 L^2+4 L-1\right) }{(4 k-4 L+1)^{9/2}}\\ \rho _{15}^*&= \frac{480 (2 L-1)}{(4 k-4 L+1)^{7/2}}\\ \rho _{16}^*&= \frac{1920 \left( 2 k^2-6 k-28 k L^2+24 k L+42 L^4-56 L^3\right) }{(4 k-4 L+1)^{11/2}}\\&\quad +\,\frac{1920(30 L^2-8 L+1)}{(4 k-4 L+1)^{11/2}}\\ \rho _{17}^*&= \frac{960 \left( 2 k-14 L^2+12 L-3\right) }{(4 k-4 L+1)^{9/2}} \\ \rho _{18}^*&= \frac{240}{(4 k-4 L+1)^{7/2}}\\ \rho _{20}^*&= \frac{8}{(4 k+4 L+1)^{3/2}}\\ \rho _{21}^*&= -\frac{48 (2 L+1)}{(4 k+4 L+1)^{5/2}}\\ \rho _{22}^*&= \frac{192 \left( 5 L^2+4 L-k+1\right) }{(4 k+4 L+1)^{7/2}}\\ \rho _{23}^*&= -\frac{48}{(4 k+4 L+1)^{5/2}}\\ \rho _{24}^*&= \frac{640 (2 L+1) \left( 3 k-7 L^2-4 L-1\right) }{(4 k+4 L+1)^{9/2}}\\ \rho _{25}^*&= \frac{480 (2 L+1)}{(4 k+4 L+1)^{7/2}} \end{aligned}$$
$$\begin{aligned} \rho _{26}^*&= \frac{1920 \left( 2 k^2-6 k-28 k L^2-24 k L+42 L^4+56 L^3\right) }{(4 k+4 L+1)^{11/2}}\\&\quad +\,\frac{1920(30 L^2+8 L+1)}{(4 k+4 L+1)^{11/2}}\\ \rho _{27}^*&= \frac{960 \left( 2 k-14 L^2-12 L-3\right) }{(4 k+4 L+1)^{9/2}}\\ \rho _{28}^*&= \frac{240}{(4 k+4 L+1)^{7/2}} \end{aligned}$$

C The values of various coefficients in the expansion of potential function U

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

D The values of various coefficients in the coefficient scheme Table 3

$$\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} \\&\quad +\,a_{-2}b_{-1})+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}\\&\quad +\,a_{2}b_{-1})+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}$$

E The values of various coefficients in the second-order terms \(a_0, b_0, a_{\pm 2} \, \text {and}\, b_{\pm 2} \)

$$\begin{aligned} \sigma _{1}&=\varPi _{1}^2 U^*_{5}\\ \sigma _{2}&=12 {U^*_4}+2 \varPi _{1} {U^*_5}\\ \sigma _{3}&={U^*_5}\\ \sigma _{4}&=4 \varPi _{1}+\varPi _{2}-16\\ \sigma _{5}&=8 \varPi _{1}+\varPi _{2}-16\\ \sigma _{6}&=\varPi _{1}^2 \varPi _{2} {U^*_5}\\ \sigma _{7}&={U^*_5} (\varPi _{1} \varPi _{2}+\varPi _{1} \sigma _{4})-12 \varPi _{2} {U^*_4}\\ \sigma _{8}&=\sigma _{5} {U^*_5}-48 {U^*_4}\\ \sigma _{9}&=4 {U^*_5}\\ \sigma _{10}&=3 (4 {U^*_4}+\varPi _{1} {U^*_5})\\ \sigma _{11}&=3 {U^*_5} \end{aligned}$$

F The values of various coefficients in the third-order terms \(a_{\pm 3}\) and \(b_{\pm 3}\)

$$\begin{aligned} \varPi _{3}&=4-\varPi _{1}-\varPi _{2}\\ \sigma _{12}&=-27 \sigma _{6} {U^*_4}-2 \varPi _{1} \varPi _{2} \sigma _{10} {U^*_5}-3 \sigma _{6} {U^*_5}\\ \sigma _{13}&=\sigma _{12}-3 \varPi _{1}^2 \varPi _{2} \varPi _{3} {U^*_7}-18 \varPi _{1}^3 \varPi _{3} {U^*_8}\\ \sigma _{14}&=9 (-3 \sigma _{7} {U^*_4}-2 \varPi _{1} \sigma _{10} {U^*_5}-3 \varPi _{1}^2 \varPi _{3} {U^*_7})\\ \sigma _{15}&=24 \varPi _{3} {U^*_6}+15 \varPi _{1}^2 {U^*_7}-6 \varPi _{1} \varPi _{3} {U^*_7}\\ \sigma _{16}&=\sigma _{15}-3 \sigma _{8} {U^*_4}-2 \varPi _{1} \sigma _{11} {U^*_5}-2 \sigma _{10} {U^*_5}\\ \sigma _{17}&=6 {U^*_8} (3 \varPi _{1}^2 \varPi _{3}-5 \varPi _{1}^3)-12 \varPi _{1} \varPi _{3} {U^*_7}\\ \sigma _{18}&=\sigma _{17}+{U^*_5} (\varPi _{1} \sigma _{8}+\sigma _{7})-8 \sigma _{10} {U^*_5}\\ \sigma _{19}&=24 \varPi _{3} {U^*_6}+15 \varPi _{1}^2 {U^*_7}-6 \varPi _{1} \varPi _{3} {U^*_7}\\ \sigma _{20}&=\sigma _{19}-3 \sigma _{8} {U^*_4}-2 \varPi _{1} \sigma _{11} {U^*_5}-2 \sigma _{10} {U^*_5}\\ \sigma _{21}&=3 {U^*_7} (10 \varPi _{1}-\varPi _{3})-2 \sigma _{11} {U^*_5}\\ \sigma _{22}&=9 \sigma _{20}+\varPi _{2} (\sigma _{21}-3 \sigma _{9} {U^*_4}-120 {U^*_6})\\ \sigma _{23}&=6 {U^*_8} (3 \varPi _{1} \varPi _{3}-15 \varPi _{1}^2)-8 \sigma _{11} {U^*_5}\\ \sigma _{24}&=\sigma _{23}+{U^*_5} (\varPi _{1} \sigma _{9}+\sigma _{8})\\ \sigma _{25}&=-3 \sigma _{9} {U^*_4}-2 \sigma _{11} {U^*_5}+3 {U^*_7} (10 \varPi _{1}-\varPi _{3})\\ \sigma _{26}&=\sigma _{9} {U^*_5}+60 {U^*_7}+6 {U^*_8} (\varPi _{3}-15 \varPi _{1})\\ \sigma _{27}&=-3 \varPi _{2} \sigma _{6} {U^*_4}-3 \varPi _{1} \sigma _{6} {U^*_5}\\ \sigma _{28}&=\sigma _{13}-3 \varPi _{2} \sigma _{7} {U^*_4}-3 \varPi _{1} \sigma _{7} {U^*_5}\\ \sigma _{29}&=\varPi _{2} \sigma _{16}+\sigma _{14}-3 \sigma _{18}\\ \sigma _{30}&=\sigma _{22}-3 (\sigma _{24}-12 {U^*_7} (\varPi _{3}-5 \varPi _{1}))\\ \sigma _{31}&=-3 \sigma _{26}+9 (\sigma _{25}-120 {U^*_6})+15 \varPi _{2} {U^*_7}\\ \sigma _{32}&=135 {U^*_7}+90 {U^*_8}\\ \sigma _{33}&=36 \sigma _{6} {U^*_4}+10 \varPi _{1} \sigma _{6} {U^*_5}+6 \varPi _{1}^4 \varPi _{3} {U^*_8}\\ \sigma _{34}&=6 \sigma _{7} {U^*_4}+4 \varPi _{1}^2 \varPi _{3} {U^*_7}+12 \varPi _{1}^3 \varPi _{3} {U^*_8}\\ \sigma _{35}&=\varPi _{1} (\varPi _{1} \sigma _{8}+16 \sigma _{10}+10 \sigma _{7})+9 \sigma _{6}\\ \sigma _{36}&=12 \varPi _{3} {U^*_6}+5 \varPi _{1}^2 {U^*_7}+2 \varPi _{1} \varPi _{3} {U^*_7}\\ \sigma _{37}&=6 \sigma _{8} {U^*_4}+{U^*_8} (30 \varPi _{1}^2 \varPi _{3}-60 \varPi _{1}^3)\\ \sigma _{38}&=\varPi _{1} (\varPi _{1} \sigma _{9}+16 \sigma _{11}+10 \sigma _{8})+9 \sigma _{7}\\ \sigma _{39}&=-5 (-12 {U^*_6}-2 \varPi _{1} {U^*_7})-3 \varPi _{3} {U^*_7}\\ \sigma _{40}&=9 (3 \varPi _{1} \varPi _{3}-15 \varPi _{1}^2)+\varPi _{1} (\varPi _{3}-15 \varPi _{1})\\ \sigma _{41}&={U^*_5} (10 \varPi _{1} \sigma _{9}-48 \sigma _{11}+9 \sigma _{8}) \end{aligned}$$
$$\begin{aligned} \sigma _{42}&=60 {U^*_7}+{U^*_8} (9 (\varPi _{3}-15 \varPi _{1})-5 \varPi _{1})\\ \sigma _{43}&={U^*_5} (\sigma _{38}-48 \sigma _{10})\\ \sigma _{44}&=4 \sigma _{39}+6 \sigma _{9} {U^*_4}+\sigma _{40} {U^*_8}\\ \sigma _{45}&=\varPi _{1}^2 \sigma _{6} {U^*_5}\\ \sigma _{46}&=\sigma _{33}+\varPi _{1}^2 \sigma _{7} {U^*_5}\\ \sigma _{47}&=\sigma _{35} {U^*_5}+6 (\sigma _{34}-5 \varPi _{1}^4 {U^*_8})\\ \sigma _{48}&=6 (\sigma _{37}-4 \sigma _{36})+\sigma _{43}\\ \sigma _{49}&=\sigma _{41}+6 \sigma _{44}\\ \sigma _{50}&=6 \sigma _{42}+9 \sigma _{9} {U^*_5}\\ \sigma _{51}&=-270 {U^*_8}\\ \end{aligned}$$

G The values of various coefficients in \(q_1, q_2,\) \(q_3\) and \(q_4\)

$$\begin{aligned} \sigma _{52}&=3 \varPi _{1}^3 \varPi _{3} {U^*_7}-18 \varPi _{3} \sigma _{1} {U^*_4}\\ \sigma _{53}&=2 \varPi _{1} \sigma _{11} {U^*_5}+2 \sigma _{10} {U^*_5}+6 \varPi _{1} \varPi _{3} {U^*_7}\\ \sigma _{54}&=\varPi _{1} (\sigma _{53}+72 \varPi _{3} {U^*_6}-15 \varPi _{1}^2 {U^*_7})\\ \sigma _{55}&=3 {U^*_4} (\varPi _{1} \sigma _{9}+6 (\varPi _{3} \sigma _{3}-5 \sigma _{2}))\\ \sigma _{56}&=3 {U^*_7} (\varPi _{3}-10 \varPi _{1})-360 {U^*_6}\\ \sigma _{57}&=-3 \varPi _{1} \sigma _{6} {U^*_4}\\ \sigma _{58}&=\sigma _{52}-3 \varPi _{1} \sigma _{7} {U^*_4}+2 \varPi _{1}^2 \sigma _{10} {U^*_5}\\ \sigma _{59}&=\sigma _{54}-3 {U^*_4} (\varPi _{1} \sigma _{8}+6 (\varPi _{3} \sigma _{2}-5 \sigma _{1}))\\ \sigma _{60}&=\varPi _{1} (\sigma _{56}+2 \sigma _{11} {U^*_5})-\sigma _{55}\\ \sigma _{61}&=90 \sigma _{3} {U^*_4}-15 \varPi _{1} {U^*_7}\\ \sigma _{62}&={U^*_5} (6 \varPi _{1} \varPi _{3} \sigma _{1}-\varPi _{1} (\varPi _{1} \sigma _{7}+\sigma _{6}))\\ \sigma _{63}&=6 (\sigma _{1} (\varPi _{3}-5 \varPi _{1})+\varPi _{1} \varPi _{3} \sigma _{2})-8 \varPi _{1} \sigma _{10}\\ \sigma _{64}&={U^*_5} (\sigma _{63}-\varPi _{1} (\varPi _{1} \sigma _{8}+\sigma _{7}))-12 \varPi _{1}^2 \varPi _{3} {U^*_7}\\ \sigma _{65}&=6 (\sigma _{2} (\varPi _{3}-5 \varPi _{1})+\varPi _{1} \varPi _{3} \sigma _{3}-5 \sigma _{1})\\ \sigma _{66}&={U^*_5} (-8 \varPi _{1} \sigma _{11}-\varPi _{1} (\varPi _{1} \sigma _{9}+\sigma _{8})+\sigma _{65})\\ \sigma _{67}&=\sigma _{66}-12 \varPi _{1} {U^*_7} (\varPi _{3}-5 \varPi _{1})\\ \sigma _{68}&=60 \varPi _{1} {U^*_7}+18 \varPi _{1} {U^*_8} (15 \varPi _{1}-\varPi _{3})\\ \sigma _{69}&=\sigma _{3} (\varPi _{3}-5 \varPi _{1})-5 \sigma _{2}\\ \sigma _{70}&=-\varPi _{1}^2 \sigma _{6} {U^*_5}\\ \sigma _{71}&=\sigma _{62}-18 \varPi _{1}^4 \varPi _{3} {U^*_8}\\ \sigma _{72}&=\sigma _{64}-18 \varPi _{1} {U^*_8} (3 \varPi _{1}^2 \varPi _{3}-5 \varPi _{1}^3)\\ \sigma _{73}&=\sigma _{67}-18 \varPi _{1} {U^*_8} (3 \varPi _{1} \varPi _{3}-15 \varPi _{1}^2)\\ \sigma _{74}&=\sigma _{68}+{U^*_5} (6 \sigma _{69}-\varPi _{1} \sigma _{9})\\ \sigma _{75}&=90 \varPi _{1} {U^*_8}-30 \sigma _{3} {U^*_5} \end{aligned}$$

H The values of various coefficients in the expression of E

$$\begin{aligned} \sigma _{76}&=\varPi _{2} \sigma _{57}-\sigma _{70}\\ \sigma _{77}&=\varPi _{2} \sigma _{58}+\sigma _{57}-\sigma _{71}\\ \sigma _{78}&=\varPi _{2} \sigma _{59}+\sigma _{58}-\sigma _{72}\\ \sigma _{79}&=\varPi _{2} \sigma _{60}+\sigma _{59}-\sigma _{73}\\ \sigma _{80}&=\varPi _{2} \sigma _{61}+\sigma _{60}-\sigma _{74}\\ \sigma _{81}&=\sigma _{61}-\sigma _{75} \end{aligned}$$

I The values of various coefficients in the first-order terms \(a_{-1} \, \text {and}\, b_{-1}\)

$$\begin{aligned} \sigma _{82}&=\varPi _1 \sigma _{76}\\ \sigma _{83}&= \varPi _1 \sigma _{77}-2 \varPi _3 \sigma _{57}-\sigma _{76}\\ \sigma _{84}&=4 \sigma _{57}-2 \varPi _3 \sigma _{58}-\sigma _{77}+\varPi _1 \sigma _{78}\\ \sigma _{85}&=4 \sigma _{58}-2 \varPi _3 \sigma _{59}-\sigma _{78}+\varPi _1 \sigma _{79}\\ \sigma _{86}&=4 \sigma _{59}-2 \varPi _3 \sigma _{60}-\sigma _{79}+\varPi _1 \sigma _{80}\\ \sigma _{87}&=4 \sigma _{60}-2 \varPi _3 \sigma _{61}-\sigma _{80}+\varPi _1 \sigma _{81}\\ \sigma _{88}&=4 \sigma _{61}-\sigma _{81}\\ \end{aligned}$$

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Suraj, M.S., Meena, O.P., Aggarwal, R. et al. On the periodic orbits around the collinear libration points in the SCR4BP with non-spherical primaries. Nonlinear Dyn 111, 5547–5577 (2023). https://doi.org/10.1007/s11071-022-08131-w

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