DYNAMICAL EFFECTS OF DRAG IN THE CIRCULAR RESTRICTED 3-BODY PROBLEM .1. LOCATION AND STABILITY OF THE LAGRANGIAN EQUILIBRIUM POINTS

The location and stability of the five Lagrangian equilibrium points i n the planar, circular restricted three-body problem are investigated when the third body is acted on by a variety of drag forces. The appro ximate locations of the displaced equilibrium points are calculated fo r small mass ratios and a simple criterion for their linear stability is derived. If a(1) and a(3) denote the coefficients of the linear and cubic terms in the characteristic equation derived from a linear stab ility analysis, then an equilibrium point is asymptotically stable pro vided O < a(1) < a(3). In cases where a(1) approximate to 0 or a(1) ap proximate to a(3) the point is unstable but there is a difference in t he e- folding time scales of the shifted L(4) and L(5) points such tha t the L(4) point, if it exists, is less unstable than the L(5) point. The results are applied to a number of general and specific drag force s. It is shown that, contrary to intuition, certain drag forces produc e asymptotic stability of the displaced triangular equilibrium points, L(4) and L(5). Therefore, simple energy arguments alone cannot be use d to determine stability in the restricted problem. The shifted equili brium points of all drag forces that have x and y components in the ro tating frame of the form (-kgy, +kg*x) evaluated when x = y = 0, wher e g is a function of x and y, follow identical, near-circular paths f or increasing drag. As the magnitude of k is increased, (1) the L(3) a nd L(4) points move in opposite directions along a circle centered on the primary mass, merge, and disappear; (2) the L(5) point moves antic lockwise along the same circle, meets the displaced L(2) point, and di sappears; and (3) the inner and outer Lagrangian points, L(1) and L(2) , initially move in opposite directions along separate circles centere d on the secondary mass until they reach the primary circle whereupon the L(2) point merges with the displaced L(5) point and both disappear while the L(1) point then moves along the primary circle toward the s econdary mass although it never reaches it for a finite drag force. In the special case where g is purely a function of the orbital radius, r, the relationship between the drag coefficient and the position ang le, theta, of the shifted equilibrium points on the primary circle is given by (k) over bar/mu(2)=sin theta[(2 - 2cos theta)-3/2 - 1], where (k) over bar = kg(1), mu(2) is the mass of the secondary in units whe re the sum of the masses is unity, and g(1) is g evaluated at the rad ius of the primary circle. In this case the L(3) and L(4) points meet at a position angle of theta = 108.4 degrees ahead of, and in the orbi t of, the secondary for (k) over bar/mu(2) = -0.7265. Nebular gas drag , where the drag force is proportional to the square of the relative v elocity of the particle and the surrounding gas, has these properties. Since a(1) approximate to 0 for such a drag force, it also gives rise to a situation where, provided (k) over bar/mu(2) > -0.7265, the L(4) point still exists and it is significantly less unstable than the L(5 ) point. This provides a mechanism that would lead to a preference for objects at the L(4) rather than L(5) point. Such a process may have i mplications for the origin and evolution of the Trojan asteroids and s ome of the smaller satellites of Saturn. (C) 1994 Academic Press, Inc.