Triangular Equilibria in R3BP under the Consideration of Yukawa Correction to Newtonian Potential

We study the triangular equilibrium points in the framework of Yukawa correction to Newtonian potential in the circular restricted three-body problem. The effects of $\alpha$ and $\lambda$ on the mean-motion of the primaries and on the existence and stability of triangular equilibrium points are analyzed, where $\alpha \in \left(-1,1\right)$ is the coupling constant of Yukawa force to gravitational force, and $\lambda \in \left(0,\infty \right)$ is the range of Yukawa force. It is observed that as $\lambda ⟶\infty$ , the mean-motion of the primaries $n⟶{\left(1+\alpha \right)}^{1/2}$ and as $\lambda ⟶0$ , $n⟶1$ . Further, it is observed that the mean-motion is unity, i.e., $n=1$ for $\alpha =0$ , $n>1$ if $\alpha >0$ and $n<1$ when $\alpha <0$ . The triangular equilibria are not affected by $\alpha$ and $\lambda$ and remain the same as in the classical case of restricted three-body problem. But, $\alpha$ and $\lambda$ affect the stability of these triangular equilibria in linear sense. It is found that the triangular equilibria are stable for a critical mass parameter ${\mu }_c={\mu }_0+f\left(\alpha ,\lambda \right)$ , where ${\mu }_0=0.0385209\cdots$ is the value of critical mass parameter in the classical case of restricted three-body problem. It is also observed that ${\mu }_c={\mu }_0$ either for $\alpha =0$ or $\lambda =0.618034$ , and the critical mass parameter $$