Abstract
In this paper, we studied the oblateness and segment-length effect on the dynamics of the restricted 2+2 body problem. It consists of two primaries and two infinitesimal bodies, assuming the bigger primary is an oblate spheroid, and the smaller primary is elongated. The effect of oblateness and segment-length on the equilibrium points are discussed. Variations of equilibrium points of this model compared to the equilibrium points of the classical CRTBP with different parameters are performed. Equilibrium points of some realistic planetary systems, i.e. Jupiter-Amalthea, Pluto-Hydra and Saturn-Prometheus system, are computed. Periodic orbits and their orbital periods are discussed analytically in the Saturn-Prometheus system with binary satellites. Poincaré Surfaces of Section is depicted in the Pluto-Hydra system to elaborate the periodic orbits when the position of one infinitesimal is known.
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Acknowledgements
The first and third author are thankful to DST(SERB) Government of India (Project No.-DST(SERB)/(163)/2016-2017/506/AM). The second author is supported by Enhanced Seed Grant through Endowment Fund Ref: EF/2021-22/QE04-07 from Manipal University Jaipur. The fourth author is financially supported by the Council of Scientific and Industrial Research (CSIR), Govt. of India (File No. 09/085(0126)/2019-EMR-1).
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Appendix 1
Appendix 1
1.1 A.
\(\tau _{17}\) =\(\tau _{27}\) = \(\tau _{37}\) = \(2n^2\),
\(\tau _{16}\) = \(\tau _{26}\) = \(\tau _{36}\) = \(4n^2 - 12n^2\mu \),
\(\tau _{15}\) = \(\tau _{25}\) = \(\tau _{35}\) = \(2n^2 - 2l^2n^2 - 20n^2\mu + 30n^2\mu ^2\),
\(\tau _{14}\) = \(2(1 - \mu ) + 2\mu - 8n^2\mu + 8l^2n^2\mu + 40n^2\mu ^2 - 40n^2\mu ^3\),
\(\tau _{24}\) = \(2(1 - \mu ) - 2\mu - 8n^2\mu + 8l^2n^2\mu + 40n^2\mu ^2 - 40n^2\mu ^3\),
\(\tau _{34}\) = \(-2(1 - \mu ) - 2\mu - 8n^2\mu + 8l^2n^2\mu + 40n^2\mu ^2 - 40n^2\mu ^3\)
\(\tau _{13}\) = \(4(1 - \mu ) - 8 (1 -\mu )\mu - 8\mu ^2 + 12n^2\mu ^2 - 12\,l^2n^2\mu ^2 - 40n^2\mu ^3 + 30n^2\mu ^4\),
\(\tau _{23}\) = \(4(1 - \mu ) - 8(1 - \mu )\mu + 8\mu ^2 + 12n^2\mu ^2 - 12\,l^2n^2\mu ^2 - 40n^2\mu ^3 + 30n^2\mu ^4\),
\(\tau _{33}\) = \(-4(1 - \mu ) + 8(1 - \mu )\mu + 8\mu ^2 + 12n^2\mu ^2 - 12\,l^2n^2\mu ^2 - 40n^2\mu ^3 + 30n^2\mu ^4\),
\(\tau _{12}\) = \(2(1 -\mu ) + 3A(1 - \mu ) - 2\,l^2 (1 - \mu ) - 12(1 - \mu )\mu + 12 (1 - \mu )\mu ^2 + 12\mu ^3 - 8n^2\mu ^3 + 8\,l^2n^2\mu ^3 + 20n^2\mu ^4 - 12n^2\mu ^5\),
\(\tau _{22}\) = \(2(1 - \mu ) + 3A(1 - \mu ) - 2\,l^2 (1 - \mu ) - 12 (1 - \mu )\mu + 12 (1 - \mu )\mu ^2 - 12 \mu ^3 - 8n^2\mu ^3 + 8\,l^2n^2\mu ^3 + 20n^2\mu ^4 - 12n^2\mu ^5\),
\(\tau _{32}\) = \(-2(1 - \mu ) - 3A(1 - \mu ) + 2\,l^2(1 - \mu ) + 12(1 - \mu )\mu - 12(1 - \mu )\mu ^2 - 12\mu ^3 - 8n^2\mu ^3 + 8\,l^2n^2\mu ^3 + 20n^2\mu ^4 - 12n^2\mu ^5\),
\(\tau _{11}\) = \(6A(1 - \mu ) - 4(1 - \mu )\mu - 6A(1 - \mu )\mu + 4\,l^2 (1 - \mu )\mu + 12(1 - \mu )\mu ^2 - 8(1 -\mu )\mu ^3 - 8\mu ^4 + 2n^2\mu ^4 - 2\,l^2n^2 \mu ^4 - 4n^2\mu ^5 + 2n^2\mu ^6\),
\(\tau _{21}\) = \(6A(1 - \mu ) - 4(1 - \mu )\mu - 6A(1 - \mu )\mu + 4\,l^2 (1 - \mu )\mu + 12(1 - \mu )\mu ^2 - 8(1 - \mu )\mu ^3 + 8\mu ^4+ 2n^2\mu ^4 - 2\,l^2n^2\mu ^4 - 4n^2\mu ^5 + 2n^2\mu ^6\),
\(\tau _{31}\) = \(-6A(1 - \mu ) + 4(1 - \mu )\mu + 6A(1 - \mu )\mu - 4\,l^2(1 - \mu )\mu - 12(1 - \mu )\mu ^2 + 8(1 - \mu )\mu ^3 + 8\mu ^4 + 2n^2\mu ^4 - 2\,l^2n^2\mu ^4 - 4n^2\mu ^5 + 2n^2\mu ^6\),
\(\tau _{10}\) = \(3A(1 - \mu ) - 3Al^2 (1 - \mu ) - 6A(1 - \mu )\mu + 2(1 - \mu )\mu ^2 + 3A(1 - \mu )\mu ^2 - 2\,l^2 (1 - \mu )\mu ^2 - 4(1 - \mu )\mu ^3 + 2(1 - \mu )\mu ^4 + 2\mu ^5\),
\(\tau _{20}\) =\(3A(1 - \mu ) - 3Al^2(1 - \mu ) - 6A(1 - \mu )\mu + 2(1 - \mu )\mu ^2 + 3 A(1 - \mu )\mu ^2 - 2\,l^2(1 - \mu )\mu ^2 - 4(1 - \mu )\mu ^3 + 2(1 - \mu )\mu ^4 - 2\mu ^5\),
\(\tau _{30}\) = \(-3A(1 - \mu ) + 3Al^2 (1 - \mu ) + 6A(1 - \mu )\mu - 2(1 - \mu )u^2 - 3A(1 - \mu )\mu ^2 + 2\,l^2 (1 -\mu )\mu ^2 + 4(1 - \mu )\mu ^3 - 2(1 - \mu )\mu ^4 - 2\mu ^5\).
1.2 B.
\(\Phi _1 = 4\left( 4 + l^2\right) \mu ^2 + 4\left( (2 + 3A)(-1 + \mu ) + 2n^2\right) ^2 + \mu \left[ -(2 + 3A)\left( -8 + 19\,l^2\right) (-1 + \mu ) + 2\left( 8 - 24\mu + l^2(-19 + 5\mu )\right) n^2\right] \),
\(\Phi _2 = 4(2 + 3A)^2 (-1 + \mu )^2 -\left\{ -8 \left( 2 + 9 A) + (46 + 87 A) l^2\right) (-1 + \mu ) \mu + 4 \left( 4 - 3\,l^2\right) \right\} \mu ^2 - 8 (-1 + \mu ) \left\{ -4 + 6 \mu + 3 A (-2 + 5 \mu )\right\} n^2 + 4\mu \left( -8 + l^2 (-7 + 10 \mu )\right) n^2 + 16n^4\),
\(\Psi =3\left[ -4 \left( -4 + l^2\right) \mu ^2 + 4 (2 + 5 A) (-1 + \mu ) \left\{ (2 + 3 A) (-1 + \mu ) + 2 n^2\right\} \right. \)\(\left. \quad - \mu \left\{ \left( -8 (2 + 7 A) + (46 + 105 A) l^2\right) (-1 + \mu ) + 2 (8 + 5\,l^2) n^2\right\} \right] \).
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Verma, R.K., Pal, A.K., Kushvah, B.S. et al. Effect of finite straight segment and oblateness in the restricted 2+2 body problem. Arch Appl Mech 93, 2813–2829 (2023). https://doi.org/10.1007/s00419-023-02409-0
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DOI: https://doi.org/10.1007/s00419-023-02409-0