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Dynamics of the perturbed restricted three-body problem with quantum correction and modified gravitational potential

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Abstract

This work investigates the dynamics of the modified circular restricted three-body problem. The triaxial shape of the massive body, a modified gravitational parameter, quantum correction effect, radiation pressure and small perturbations in the Coriolis and centrifugal forces are all taken into account. The impact of the considered parameters in the equilibrium points and their linear stability is studied. Some new significant results in the critical mass parameter, \(\mu _c\), are observed in the presence of perturbing parameters. It is found that the critical mass parameter, \(\mu _c\), increases in the presence of modified gravitational potential, and it slightly decreases due to the quantum correction effect. Analytical construction of periodic orbits around the collinear equilibrium points is performed. Additionally, analysis of the impact of perturbations on the shape of these periodic orbits is conducted.

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Acknowledgements

The first and second authors are thankful to the Department of Mathematics and Computing Indian Institute of Technology (Indian School of Mines)—Dhanbad, for providing facilities to prepare this manuscript. The third author is supported by Enhanced Seed Grant through Endowment Fund Ref: EF/2021-22/QE04-07 from Manipal University Jaipur.

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Authors and Affiliations

  1. Department of Mathematics and Computing, Indian Institute of Technology (Indian School of Mines), Dhanbad, Jharkhand, 826004, India

    Ravi Kumar Verma & Badam Singh Kushvah

  2. Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan, 303007, India

    Ashok Kumar Pal

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Correspondence to Ashok Kumar Pal.

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Appendices

Appendix

Appendix A: Terms used in non-collinear equilibrium points \((\theta _{1,2})\)

$$\begin{aligned} \pi _1= & {} -[3(8(1+\epsilon _2)n^2 - q(8 + 16K1 + 24K2 + 9\tau _1 + 3\tau _2) + \mu (4 + 16K_3 + 36(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q \\{} & {} + 9q\tau _1 + 3q\tau _2))(8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 + 3\tau _1 + 9\tau _2) + \mu (-8 - 16K_3 - 24(K_4+\epsilon ) + 8q + 16K_1q \\{} & {} + 24K_2q + 3q\tau _1 + 9q\tau _2)) - (8(1+\epsilon _2)n^2*(-1 + 2\mu ) + q(8 + 16K_1 + 24K_2 - 21\tau _1 + 33\tau _2) - \mu (8 + 16K_3 \\{} & {} + 24(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q - 21q\tau _1 + 33q\tau _2))(8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 - 87\tau _1 + 99\tau _2)\\{} & {} + \mu (28 + 80K_3 + 156(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q - 87q\tau _1 + 99q\tau _2))],\\ \psi _1= & {} [(8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 - 87\tau _1 + 99\tau _2) + \mu (28 + 80K_3 + 156(K_4+\epsilon ) + 8q \\{} & {} + 16K_1q + 24K_2q - 87q\tau _1 +99q\tau _2))(16(1+\epsilon _2)n^2 + q(8 + 32K_1 + 72K_2 - 93\tau _1 + 129\tau _2)\\{} & {} - \mu (16 + 32K_3 + 48(K_4+\epsilon ) + 8q + 32K_1q + 72K_2q - 93q\tau _1 + 129q\tau _2)) + (8(1+\epsilon _2)n^2 \\{} & {} - q(8 + 16K_1 + 24K_2 + 9\tau _1 + 3\tau _2) + \mu (4 + 16K_3 + 36K_4 + 8q + 16K_1q + 24K_2q \\{} & {} + 9q\tau _1 +3q\tau _2))(16(1+\epsilon _2)n^2 + q(56 + 160K_1 + 312K_2 - 51\tau _1 + 207\tau _2) - \mu (16 + 32K_3 \\{} & {} + 48K_4 + 56q + 160K_1q + 312K_2q - 51q\tau _1 + 207q\tau _2))],\\ \pi _2= & {} (1+\epsilon _2)(8 - 16\mu )n^2 - q(8 + 16K_1 + 24K_2 - 21\tau _1 + 33\tau _2) + \mu (8 + 16K_3 + 24(K_4+\epsilon ) + 8q + 16K_1q \\{} & {} + 24K_2q - 21q\tau _1 + 33q\tau _2),\\ \pi _3= & {} 2(8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 + 9\tau _1 + 3\tau _2) + \mu (4 + 16K_3 + 36(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q \\{} & {} + 9q\tau _1 + 3q\tau _2)),\\ \psi _2= & {} [(-8(1+\epsilon _2)n^2 + q(8 + 16K_1 + 24K_2 + 9\tau _1 + 3\tau _2) - \mu (4 + 16K_3 + 36(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q \\{} & {} + 9q\tau _1 + 3q\tau _2))((8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 - 87\tau _1 + 99\tau _2) +\mu (28 + 80K_3 + 156(K_4+\epsilon ) + 8q \\{} & {} + 16K_1q + 24K_2q - 87q\tau _1 + 99q\tau _2))(16(1+\epsilon _2)n^2 + q(8 + 32K_1 + 72K_2 - 93\tau _1 + 129\tau _2) - \mu (16 + 32K_3 \\{} & {} + 48(K_4+\epsilon ) + 8q + 32K_1q + 72K_2q - 93q\tau _1 + 129q\tau _2)) + (8(1+\epsilon _2)n^2 - q(8 + 16K_1 + 24K_2 + 9\tau _1 + 3\tau _2) \\{} & {} + \mu (4 + 16K_3 + 36(K_4+\epsilon ) + 8q + 16K_1q + 24K_2q + 9q\tau _1 + 3q\tau _2))(16(1+\epsilon _2)n^2 + q(56 + 160K_1 + 312K_2 \\{} & {} - 51\tau _1 + 207\tau _2) -\mu (16 + 32K_3 + 48(K_4+\epsilon ) + 56q + 160K_1q + 312K_2q - 51q\tau _1 + 207q\tau _2)))]. \end{aligned}$$

Appendix B: Terms used for the partial derivatives of \(\Omega \)

$$\begin{aligned} \frac{\partial A}{\partial x}= & {} -(x+\mu )^2\times \left[ \frac{3}{r_1^5}+\frac{8K_1}{r_1^6}+\frac{15K_2}{r_1^7}+\frac{15(2\tau _1-\tau _2)}{2r_1^7}+\frac{105y^2(\tau _2-\tau _1)}{2r_1^9} \right] \\{} & {} +\left[ \frac{1}{r_1^3}+\frac{2K_1}{r_1^4}+\frac{3K_2}{r_1^5}+ \frac{3(2\tau _1-\tau _2)}{2r_1^5}+\frac{15y^2(\tau _2-\tau _1)}{2r_1^7} \right] ,\\ \frac{\partial B}{\partial x}= & {} -(x+\mu -1)^2\left[ \frac{3}{r_2^5}+\frac{8K_3}{r_2^6}+\frac{15(K_4+\epsilon )}{r_2^7} \right] +\frac{1}{r_2^3}+\frac{2K_3}{r_2^4}+\frac{3(K_4+\epsilon )}{r_2^5},\\ \frac{\partial C}{\partial y}= & {} -y^2\left[ \frac{3}{r_1^5}+ \frac{8K_1}{r_1^6}+\frac{3K_2}{r_1^7}+\frac{5(12\tau _1-9\tau _2)}{2r_1^7} \right] +\left[ \frac{1}{r_1^3}+\frac{2K_1}{r_1^4}+\frac{3K_2}{r_1^5}+\frac{12\tau _1-9\tau _2}{2r_1^5} \right] ,\\ \frac{\partial D}{\partial y}= & {} -y^2\left[ \frac{3}{r_2^5}+ \frac{8K_3}{r_2^6}+\frac{15(K_4+\epsilon )}{r_2^7} \right] +\left[ \frac{1}{r_2^3}+\frac{2K_3}{r_2^4}+\frac{3(K_4+\epsilon )}{r_2^5} \right] ,\\ \frac{\partial E}{\partial y}= & {} \frac{3y^2}{r_1^7}-\frac{7y^4}{r_1^9},\\ A_1= & {} 3(1-\mu )q(x+\mu )y,\quad B_1 = 8K_1(1-\mu )q(x+\mu )y, \\ C_1= & {} 15(1-\mu )q(x+\mu )\left[ K_2y+(2\tau _1-\tau _2)y-(\tau _1+\tau _2)y \right] ,\\ D_1= & {} 3\mu (\mu -1+x)y,\quad E_1 = 8K_3\mu (\mu -1+x)y,\\ F_1= & {} 15K_4\mu (\mu -1+x)y. \end{aligned}$$

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Verma, R.K., Kushvah, B.S. & Pal, A.K. Dynamics of the perturbed restricted three-body problem with quantum correction and modified gravitational potential. Arch Appl Mech 94, 651–665 (2024). https://doi.org/10.1007/s00419-024-02543-3

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