Abstract
The present paper deals with the restricted four-body problem (R4BP), when the third primary placed at the triangular libration point of the restricted three-body problem is an ellipsoid. The third primary m 3 is not influencing the motion of the dominating primaries m 1 and m 2. We have studied the motion of m 4, moving under the influence of the three primaries m i , i = 1, 2, 3, but the motion of the primaries is not being influenced by infinitesimal mass m 4. Further, we have developed the equations of motion of infinitesimal mass m 4 which involves elliptic integrals and shows the existence and locations of equilibrium points. We have also discussed the zero velocity curves(ZVCs) for various value of Jacobian constant.
Similar content being viewed by others
References
Alvarez-Ramírez, M., Barrabés, E.: Transport orbits in an equilateral restricted four-body problem. Celest. Mech. Dyn. Astr. 121, 191–210 (2015)
Alvarez-Ramírez, M., Skea, J.E.F., Stuchi, T.J.: Nonlinear stability analysis in a equilateral restricted four-body problem. Astrophys. Space Sci. 358, 3 (2015)
Amit, M., Aggarwal, R., Suraj, M.S., Bisht, V.S.: Stability of libration points in the restricted four-body problem with variable mass. Astrophys. Space Sci 361, 329 (2016). doi:10.1007/s10509-016-2901-2
Arribas, M., Abad, A., Elipe1, A., Palacios, M.: Equilibria of the symmetric collinear restricted four-body problem with radiation pressure. Astrophys. Space Sci. 384, 361 (2016)
Baltagiannis, A.N., Papadakis, K.E.: Equilibrium points and their stability in the restricted four-body poblem. Int. J. Bifurc. Chaos. 21, 2179–2193 (2011a)
Baltagiannis, A.N., Papadakis, K.E.: Periodic solutions in the Sun-Jupiter-Trojan Asteroid-Spacecraft system. Planet. Space Sci. 75, 148—157 (2013)
Burgaos-Garcia, J., Gidea, M.: Hill’s approximation in a restricted four-body problem. Celest. Mech. Dyn. Astr. 122, 117–141 (2015)
Burgos-García, J., Delgado, J.: Periodic orbits in the restricted four-body problem with two equal masses. Astrophys. Space Sci. 345, 247–263 (2013)
Chernikov, Y.A.: The photogravitational restricted three-body problem. Sov. Astron. AJ 14, 176 (1970)
Ceccaroni, M., Biggs, J.: Extension of low-thrust propulsion to the Autonomous Coplanar Circular Restricted Four Body Problem with application to future Trojan Asteroid missions. In: 61st International Astronautical Congress, IAC-10-1.1.3, Prague (2010)
Ceccaroni, M., Biggs, J.: Low-thrust propulsion in a Coplanar Circular Restricted Four Body Problem. Celest. Mech. Dyn. Astr. 112, 191–219 (2012)
Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the R4BP when third primary is an oblate spheroid. Astrophys. Space. Sci. 357, 87 (2015a). doi:10.1007/s10509-015-2235-5
Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the photogravitational R4BP when third primary is an oblate/prolate spheroid. Astrophys. Space. Sci. 360, 313 (2015b). doi:10.1007/s10509-015-2522-1
Chand, M.A., Umakant, P., Hassan, M.R., Suraj, M.S.: On the photogravitational R4BP when the third primary is a triaxial rigid body. Astrophys. Space Sci. 361, 379 (2016). doi:10.1007/s10509-016-2959-x
Croustalloudi and Kalvouridis, T.J.: The restricted 2+2 body problem : Parametric variation of the equilibrium states of the minor bodies and their attracting regions. ISRN Astronomy and Astrophysics, Volume 2013 Article ID 281849 (2013)
Douskos, C., Kalantonis, V., Markellos, P., Perdios, E.: On Sitnikov-like motionsgenerating new kinds of 3D periodic orbits in the R3BP with prolate primaries. Astrophys. Space Sci. 337, 99–106 (2012)
Idrisi, M.J., Taqvi, Z.A.: Restricted three-body problem when one of the primaries is an ellipsoid. Astrophys. Space Sci. 348(1), 41–56 (2013)
Idrisi, M.J., Taqvi, Z.A.: Eistence and stability of the non-collinear libration points in restricted three-body problem when both the primaries are ellipsoid. Astrophys. Space Sci. 350(1), 133–141 (2014)
Idrisi, M.J., Taqvi, Z.A.: Eistence and stability of libration points in CR3BP when the smallar primary is an oblate spheroid. Astrophys. Space Sci. 354(1), 311–325 (2014)
Kalvouridis, T., Arribas, M., Elipe, A.: Dynamical properties of the restricted four-body problem with radiation pressure. Mech. Res. Commun. 33, 811 (2006a)
Kalvouridis, T., Arribas, M., Elipe, A.: The photo-gravitational restricted four-body problem: an exploration of its dynamical properties. AIP Conf. Proc. 848, 637 (2006b)
Kumari, R., Kushvah, B.S.: Equilibrium points and zero velocity surfaces in the restricted four-body problem with solar wind drag. Astrophys. Space Sci. 344, 347–359 (2013)
Moulton, F.R.: On a class of particular solutions of the problem of four bodies. Trans. Am. Math. Soc. 1, 17 (1900)
Papadouris, J.P., Papadakis, K.E.: Equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 344, 21–38 (2013)
Papadakis, K.E.: Families of three dimensional periodic solutions in the circular restricted four-body problem. Astrophys. Space Sci. 361, 129 (2016)
Singh, J., Vincent, A.E.: Out-of-plane equilibrium points in the photogravitational restricted four-body problem. Astrophys. Space Sci. 359, 38 (2015)
Singh, J., Vincent, A.E.: Equilibrium Points in the Restricted Four-Body Problem with Radiation Pressure. Few-Body Syst. doi:10.1007/s00601-015-1030-8
Singh, J., Vincent, A.E.: Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of Equilibrium Points in the Restricted Four-Body Problem. Few-Body Syst. 56, 713–723 (2015). doi:10.1007/s00601-015-1019-3
Suraj, M.S., Hassan, M.R.: Sitnikov restricted four-body problem with radiation pressure. Astrophys. Space Sci. 349(2), 705–716 (2014)
Suraj, M.S., Hassan, M.R., Chand, M.A.: The Photo-Gravitational R3BP when the Primaries are Heterogeneous Spheroid with Three Layers. J. Astronaut. Sci. 61 (2014)
Zotos, E.E.: Escape and collision dynamics in the planar equilateral restricted four- body problem. Int. J. Non Linear Mech. 86, 66–82 (2016)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Asique, M.C., Prasad, U., Hassan, M.R. et al. On the R4BP when Third Primary is an Ellipsoid. J of Astronaut Sci 64, 231–250 (2017). https://doi.org/10.1007/s40295-016-0104-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40295-016-0104-2