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# Stability of Triangular Libration Points in the Sun - Jupiter System under Szebehely’s Criterion

## Abstract

In the present study, the classical fourth-order Runge-Kutta method with seventh-order automatic step-size control has been carried out to examine the stability of triangular libration points in the Sun-Jupiter system. The Sun is a highly luminous body and Jupiter is a highly spinning body, so radiation pressure of the Sun and oblateness of the Jupiter cannot be neglected. These factors must have some effects on the motion of the infinitesimal mass (spacecraft) and consequent effects on the stability of the triangular libration points. It is to be noted that in our problem, infinitesimal mass exerts no influence of attraction on the primaries (Sun and Jupiter) but its motion is influenced by the primaries. Therefore, the equations of motion of the infinitesimal mass moving in the gravitational field of the radiating Sun and oblate Jupiter have been established for numerical integration. To check the stability of the libration points, the infinitesimal mass is allowed to librate for trajectory generation in the vicinity of one of the triangular libration points. Using double-precision computation, the Jacobian constant was calculated in order to observe the validity of the trajectory generation throughout the numerical integration. This constant of integration was checked to make sure that it remained constant at least to eight decimal places, so that other data may be accurate. Following all the above computational techniques, the maximum displacement and maximum velocity envelopes were constructed in the light of previous authors. The reason behind the assumption of the maximum displacement and maximum velocity envelopes is that the spacecraft (infinitesimal mass) will librate for a long time within the region of the envelopes without crossing the x-axis. If the area of the envelope is not maximum within the given time limit and the infinitesimal mass crosses the x-axis, then by changing the initial conditions; we attempt to construct the envelopes of maximum area following previous authors. If the area of the envelope is maximum it means spacecraft (infinitesimal mass) will librate in wider area for a long time without crossing the x-axis and longtime libration will give the higher range of stability. From our observation, it is found that due to the oblateness of Jupiter, the range of stability is reduced but photogravitation of the Sun has no significant effect on the triangular libration points.

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