Stability of the Equilibria in the Restricted three Body Problem when the Smaller Primary is a Finite Straight Segment

It is well known that asteroids are small rocky bodies. Most of the asteroids are much smaller when compared with the planets. Typically most are irregularly shaped elongated bodies. The main feature of these bodies is their elongated shape. This is the reason why we have taken their elongated shape, namely, a finite straight segment. We have investigated the stationary solutions in the restricted three body problem when the smaller primary is a finite straight segment. The linear stability of all the stationary solutions have been investigated by the method given by Whittaker [1] and Szebehely [2]. The collinear stationary solutions are unstable for all mass parameter μ whereas the non-collinear stationary solutions are stable for mass parameter μ less than a critical value μ _c = μ₀ - 0.0073562l². The theory, so developed, has been applied to the asteroids 9 Metis (9 is the number of the asteroid Metis), 22 Kalliope, 216 Kleopatra, 243 Ida, 433 Eros and 951 Gaspra.