Dynamics of periodic orbits in the Copenhagen problem with non-spherical primaries

Abstract

In the present investigation, we conduct an analysis of periodic orbits within the context of the Copenhagen problem, emphasizing the dynamical behavior of a test particle subjected to the gravitational influence of two primary bodies of equal mass, which are in continuous rotation characterized by a constant angular velocity. By expanding upon the classical framework, we treat the primary bodies as non-spherical entities, thereby introducing the phenomenon of oblateness into the dynamical system under consideration. Employing the methodology of Fourier series, we articulate the characteristics of periodic orbits in proximity to the libration points and systematically evaluate the influence of the orbital parameter $\varepsilon$ on the spatial dimensions and temporal periods of these orbits. Through the incorporation of terms extending to the third order in Fourier series method, we present a comprehensive representation of the parameter’s influence on the orbital attributes. The findings indicate that with an increase in $\varepsilon$, the dimensions of periodic orbits experience a substantial expansion, while their temporal periods demonstrate non-linear fluctuations. Variational graphs elucidate the correlation between $\varepsilon$ and the time period $T$, revealing distinct patterns for the various families of orbits under analysis. Moreover, the oblateness exhibited by the primary bodies engenders significant alterations in the geometrical characteristics, size, and time period of the orbits, thereby underscoring their pivotal influence on the dynamics of orbital motion.