Dynamical Effects of Drag in the Circular Restricted Three-Body Problem: I. Location and Stability of the Lagrangian Equilibrium Points
✍ Scribed by Carl D. Murray
- Publisher
- Elsevier Science
- Year
- 1994
- Tongue
- English
- Weight
- 989 KB
- Volume
- 112
- Category
- Article
- ISSN
- 0019-1035
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✦ Synopsis
The location and stability of the five Lagrangian equilibrium points in the planar, circular restricted three-body problem are investigated when the third body is acted on by a variety of drag forces. The approximate locations of the displaced equilibrium points are calculated for 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 stability analysis, then an equilibrium point is asymptotically stable provided (0<a_{1}<a_{3}). In cases where (a_{1} \approx 0) or (a_{1} \approx a_{3}) the point is unstable but there is a difference in the (e) folding time scales of the shifted (L_{4}) and (L_{5}) points such that 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 forces. It is shown that, contrary to intuition, certain drag forces produce asymptotic stability of the displaced triangular equilibrium points, (L_{4}) and (L_{5}). Therefore, simple energy arguments alone cannot be used to determine stability in the restricted problem. The shifted equilibrium points of all drag forces that have (x) and (y) components in the rotating frame of the form (\left(-k g^{} y,+k g^{} x\right)) evaluated when (\dot{x}=\dot{y}=0), where (g^{}) is a function of (x) and (y), follow identical, near-circular paths for increasing drag. As the magnitude of (k) is increased, (1) the (L_{3}) and (L_{4}) points move in opposite directions along a circle centered on the primary mass, merge, and disappear; (2) the (L_{5}) point moves anticlockwise along the same circle, meets the displaced (L_{2}) point, and disappears; and (3) the inner and outer Lagrangian points, (L_{1}) and (L_{2}), initially move in opposite directions along separate circles centered 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 secondary 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 angle, (\boldsymbol{\theta}), of the shifted equilibrium points on the primary circle is given by (\bar{k} / \mu_{2}=\sin \theta\left[(2-2 \cos \theta)^{-3 / 2}-1\right]), where (\bar{k}=k g_{1}, \mu_{2}) is the mass of the secondary in units where the sum of the masses is unity, and (g_{1}) is (g^{*}) evaluated at the radius of the primary circle. In this case the (L_{3}) and (L_{4}) points meet at a position angle of (\theta=108^{\circ} 4) ahead of, and in the orbit of, the secondary for (k / \mu_{2}=\mathbf{- 0 . 7 2 6 5}). Nebular gas drag, where the drag force is proportional to the