DERIVATION OF THE DYNAMICAL EQUATIONS OF MOTION OF THE R3BP WITH VARIABLE MASSES AND DISK

  • Joel John Taura Federal University of Kashere, Gombe State
  • Oni Leke Department of Mathematics, College of Science, Joseph Sarwuan Tarka University, Makurdi, Benue-State Nigeria
Keywords: R3BP; Equations of Motion; Variable Masses; Disk

Abstract

This paper explores the dynamical equations of the restricted three-body problem with variable masses of the primaries which are enclosed by a disk, when the masses of the primary and the disk vary with time in accordance with the unified Mestschersky law and motion of the primaries is determined by the Gylden-Mestschersky equation. It is seen that the equations of motion differ from those of the restricted three-body problem with variable masses due to the disk mass

References

Akeson, R. L., Rice, W. K. M., Boden, A. F., Sargent, A. I., Carpenter, J. M. and Bryden, G. (2007). The Circumbinary Disc of HD 98800B: Evidence for Disc Warping. The Astrophysical Journal 670, 1240–1246.
Ansari, A.A,Rabah, K., Ziyad, A.A and Wasim, U.:(2019) Effect of variation of charge in the circular restricted three-body problem with variable masses. Journal of Taibah University of Science, 13, 670
Bekov, A. A.: (1988). Libration points of the restricted problem of Three Bodies with variable Mass. Soviet Astronomy Journal, 33, 92-95
Bekov, A. A.: (1991). Particular solutions in the restricted collinear three-body problem with variable mass. Soviet Astronomical Journal, 68, 206-211.
Bekov, A. A. (1993a). Periodic solutions of the Gylden-Mestschersky problem. Astronomical Journal, 70, 1289-1293
Bekov, A. A. (1993b). Problems of Physics of Stars and Extragalactic Astronomy. Almaty, pp. 91
Dufour, M.: (1866) Ch.: Comptes Rendus Hebdomadaires de L’, Academy of Sciences, Amsterdam, pp. 840-84
D’yakov, B. B., and Reznikov, B. I.: (1986). Motion in the vicinity of triangular Libration points when the mass ratio of the components is variable. Soviet. Astronomical Journal, 340, 345-351.
Euler, L.: (1767). The motion in the rectilinear three-body problem. 11, 144-149. Nov (Theory of Orbits Szebehely 1967a)
Gelf’gat, B.E.: (1973). Current Problems of Celestial Mechanics and Astrodynamics, Nauka, Moscow
Greaves, J. S., Holland, W. S., Moriarty-Schieven, G., Jenness, T., Zuckerman, B., McCarthy, C., Dent, W. R. F., Webb, R.A., Butner, H .M., Gear, W.K. and Walker, H.J.:(1998). A dust ring around epsilon Eridani: analog to the young solar system. Astrophysical Journal, 506, 133–137.
Gylden, H.: (1884). Die Bahnbewegungen in Einem Systeme von zwei Körpern in dem Falle, dass die Massen Ver Nderun- Gen Unterworfen Sind, Astronomische Nachrichten. 109, 1-6.
Jiang, I.G. and Yeh, L.C.: (2004). On the orbits of disk–star–planet systems. Astronomical Journal, 128, 923–932.
Jiang, I.G. and Yeh, L.C.: (2006). On the Chermnykh-like problem: the equilibrium points. Astrophysics and Space Science, 305, 341-345.
Jiang, I.G. and Yeh, L.C.: (2014). Galaxies with super massive binary black holes: (I) a possible model for the centers of core galaxies. Astrophysics and Space Science, 349, 881–893.
Lagrange, J.L.: (1772). Collected works Paris, Vol. VI, p229 ,1873

Letelier, P. S. and Da Silva, T. A.: (2011). Solutions to the restricted three-body problem with variable mass. Astrophysics and Space Science., 332, 325–329
Luk’yanov, L. G., (1989). Particular solutions in the restricted problem of three bodies with variable masses. Astronomical Journal of Academy of Sciences of USSR, 66, 180-187
Luk’yanov, L. G., (1990). Stability of libration points in the restricted three-body problem of variable masses. Soviet Astronomical Journal, 67, 167-172.
Mestschersky, I.V.: (1893). Special cases of the Gylden Problems (A. N. 2593), Astronomische Nachrichten, 132,129–130.
Mestschersky, I.V.: (1902). Ueber die Integration der Bewegungs- gleichungen im Probleme zweier Körper von ver nderli- cher Masse, Astronomische Nachrichten. 159, 229-242.
Mestshchersky I. V., (1949). On the mechanics of Bodies of Variable mass (In Russian) Moscow, Leningrad.
Mestschersky, I.V.: (1952) Works on the mechanics of bodies of variable mass, GITTL, Moscow, p. 205.
Orlov, A. A.: (1939). Existence of the Equilibrium solutions of three bodies with finite variable masses. Astronomical Journal of Academy of Sciences of USSR, 16, 52-56.
Prato, L., Ghez, A. M., Piña, R. K., Telesco, C. M., Fisher, R. S., Wizinowich, P., Lai, O., Acton, D. S. and Stomski, P.: (2001). Keck Diffraction-limited Imaging of the Young Quadruple Star System HD 98800. Astrophysical Journal 549, 590–598
Sersic, J. L.: (1970). Periodic Orbits, Stability and Resonance (Edited by G.F.B. Giacaglia), Dordrecht, Page 314.
Sersic, J. L.: (1973). On structures of peculiar galaxies. Bulletin of the Astronomical Institute Czechoslovakia, 24, 150-156.
Shrivastava, A. K., and Ishwar, B.: (1983). Equations of Motion of the Circular restricted problem of three bodies with variable mass, Celestial. Mechanics. 30, 323-327.
Singh, J. and Ishwar, B., (1984). Effects of small perturbations in the Coriolis and centrifugal forces on the locations of equilibrium points in the restricted problem of three bodies with variable mass, Celestial Mechanics, 32, 297-305.
Singh, J. and Ishwar, B., (1985). Effects of small perturbations in the Coriolis and centrifugal forces on the stability of triangular points in the restricted problem of three bodies with variable mass, Celestial Mechanics, 35, 201-207.
Singh, J. and Leke, O.: (2010). Stability of the photogravitational restricted three-body problem with variable masses. Astrophysics and Space Science, 326, 305- 314.
Singh, J. and Leke, O.: (2012) “Equilibrium points and stability in the restricted three-body problem with oblateness and variable masses” Astrophysics and Space Science, Vol. 340: 27-41.
Singh, J. and Leke, O.: (2013a) “Effects of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem”. Astrophysics and Space Science Vol. 344: 51-61.
Singh, J., and Leke, O.: (2013b) “Existence and stability of equilibrium points in the Robe’s restricted three-body problem with variable masses,” International Journal of Astronomy and Astrophysics, Vol. 3: 113–122.
Singh, J. and Leke, O.: (2013c) “Robe’s restricted three-body problem with variable masses and perturbing forces” ISRN Astronomy and Astrophysics, Volume 2013, Article ID 910354
Singh, J. and Leke, O.: (2013d) “On Robe’s Circular Restricted Problem of Three Variable Mass Bodies” Journal of Astrophysics Volume 2013, Article ID 898794.
Singh, J. and Taura, J.J.: (2013). Motion in the generalized restricted three-body problem. Astrophysics and Space Science, 343,95-106
Trilling, D.E., Stansberry, J. A., Stapelfeldt, K.R., Rieke, G.H., Su, K.Y.L., Gray, R.O., Corbally, C.J., Bryden, G., Chen, C.H., Boden, A. and Beichman, C.A.: (2007). Debris disks in main-sequence binary systems. Astrophysical Journal, 658, 1289–1311.
Vassiliadis, E. and Wood, P. R.: (1994). Post-asymptotic giant branch evolution of low-to intermediate-mass stars. Astrophysical Journal , 92, 125-144
Veras, D., Wyatt, M.C., Mustill, A.J., Bonsor, A. and Eldridge, J,J.: (2011). The great escape: how exoplanets and smaller bodies desert dying stars. Monthly Notices of the Royal Astronomical Society, 417, 2104–2123.
Verrier, P. E. and Evans, N. W.: (2008). HD 98800: a most unusual debris disc. Monthly Notices of the Royal Astronomical Society, 390, 1377–1387.
Zhang, Ming-Jiang , Chang-Yin Zhao and Yong-Qing Xiong., (2012). On the triangular libration points in photogravitational restricted three-body problem with variable mass. Astrophysics and Space Science, 337, 107–113
Ziyad, A.A:(2018) Effect of Poynting-Robertson drag on the circular restricted three-body problem with variable masses. Journal of Taibah University of Science, 12, 455
Published
2022-08-23
How to Cite
TauraJ. J., & LekeO. (2022). DERIVATION OF THE DYNAMICAL EQUATIONS OF MOTION OF THE R3BP WITH VARIABLE MASSES AND DISK. FUDMA JOURNAL OF SCIENCES, 6(4), 125 - 133. https://doi.org/10.33003/fjs-2022-0604-1025