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Stability of libration points in the restricted four-body problem with variable mass

  • Amit Mittal
  • Rajiv Aggarwal
  • Md. Sanam Suraj
  • Virender Singh Bisht
Original Article

Abstract

We have investigated the stability of the Lagrangian solutions for the restricted four-body problem with variable mass. It has been assumed that the three primaries with masses \(m_{1}\), \(m_{2}\) and \(m_{3}\) form an equilateral triangle, wherein \(m_{2}=m_{3}\). According to Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), the infinitesimal body varies its mass \(m\) with time. The space–time transformations of Meshcherskii (Studies on the Mechanics of Bodies of Variable Mass, GITTL, Moscow, 1949) are used by taking the values of the parameters \(q=1/2\), \(k=0\), \(n=1\). The equations of motion of the infinitesimal body with variable mass have been determined. The equations of motion of the current problem differ from the ones of the restricted four-body problem with constant mass. There exist eight libration points, out of which two are collinear with the primary \(m_{1}\) and the rest are non-collinear for a fixed value of parameters \(\gamma (\frac{m \ \text{at time} \ t}{m \ \text{at initial time}}, 0<\gamma\leq1 )\), \(\alpha\) (the proportionality constant in Jeans’ law (Astronomy and Cosmogony, Cambridge University Press, Cambridge, 1928), \(0\leq\alpha\leq2.2\)) and \(\mu=0.019\) (the mass parameter). All the libration points are found to be unstable. The zero velocity surfaces (ZVS) are also drawn and regions of motion are discussed.

Keywords

Restricted four-body problem Variable mass Libration points Zero velocity surfaces 

Notes

Acknowledgements

This article is dedicated to one of the greatest mathematician of present time Late Dr. K.B. Bhatnagar, Founder Director, Centre for Fundamental Research in Space Dynamics and Celestial Mechanics (CFRSC), Delhi, India. We are also thankful to CFRSC, Delhi, India.

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Amit Mittal
    • 1
  • Rajiv Aggarwal
    • 2
  • Md. Sanam Suraj
    • 3
  • Virender Singh Bisht
    • 4
  1. 1.Department of Mathematics, A.R.S.D. CollegeUniversity of DelhiDelhiIndia
  2. 2.Department of Mathematics, Sri Aurobindo CollegeUniversity of DelhiDelhiIndia
  3. 3.Department of Mathematics, S.G.T.B. Khalsha CollegeUniversity of DelhiDelhiIndia
  4. 4.Department of MathematicsAl-Falah UniversityFaridabadIndia

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