Linear Stability of Non-Axial Libration Points in the Kite Configuration of First Kind

This paper deals with the linear stability of non-axial libration points in the kite configuration of first kind. It is to be noted that the kite configuration of the first kind exists for the mass parameter $\mu \in \left]\text{？}0,1/3\text{？}\right[$ , where $\mu$ is the ratio of the smallest mass of the kite to the whole mass of the system. To check the linear stability of the libration points, the variational equation was derived in the neighbourhood of the libration points, and then the characteristic equation was formed. If all the roots of the characteristic equation are purely imaginary, then all criteria for stability will be satisfied and hence, the libration points will be stable; otherwise, unstable. For this, the stability criteria given in Equation (15) have been followed, and it was found that $L_4$ and $L_5$ are stable for $\mu =0.10$ only and for all values of $\mu \in \left]\text{？}0,1/3\text{？}\right[$ , $L_j\left(j=1,2,3,6,7,8,9,10,11,12,13\right)$ are unstable.