ROBE'S RESTRICTED THREE-BODY PROBLEM WITH DRAG

Robe's restricted three-body problem is investigated with regards to the effects of a linear drag force. In particular, the stability of the model's equilibrium points is studied in this respect. Two scenarios are envisaged: the one originally discussed by Robe himself and the one suggested by him and recently analyzed by the present authors, that assumes for the fluid body the structure of a Roche's ellipsoid.


Introduction
In 1977 Robe introduced a new type of restricted three-body problem that incorporated the effects of buoyancy: one of the two principal bodies was conceived as a rigid spherical shell of mass ml, filled with an homogeneous incompressible fluid of density Pl, while the other (principal) body was regarded as a point mass m2 located outside the shell.The third body, of negligible mass and density P3, moves inside the shell under two influences, namely (i) the gravitational attraction of the principal bodies and (ii) the buoyancy force of the fluid Pl (Robe, 1977).
Bouyancy forces may play an important role in some problems of dynamical astronomy, such as the study of small oscillations of the earth's core in the gravitational field of the earth-moon system (Robe, 1977).Bouyancy effects have also been discussed in connection with galactic dynamics as an altemative approach to flat rotation curves of spiral galaxies (Soares, 1992).
Though being an idealized model, Robe's problem allows for gaining insight into the effects of bouyancy forces in rotating systems since it provides exact analytical results.Furthermore, within the general context of the study of equilibrium points in rotating astronomical systems, Robe model poses stability issues whose study is interesting in itself.
Robe discussed two cases (1) that in which m2 describes a circular orbit around the shell and (2) the situation in which for either (a)  P! = P3, m2 describes elliptical orbits.In both instances the center of the shell is an equilibrium point for the third body, which led him to study the conditions for linear stability.
Shrivastava and Garain (SG) (1991) investigated the effect of a small perturbation in the Coriolis and centrifugal forces on the location of the equilibrium point (EP).They considered the circular case with identical densities (Pl = P3) and evaluated the concomitant shift in the EP-location.
In deriving the expression for the buoyancy force E, both Robe (1977) and SG (1991) assumed that the pressure field of the fluid fll has spherical symmetry around the center of the shell, in accordance with its presupposed spherical shape.However, they took into account just one of the three components of the pressure field: that due to the own gravitational field of the fluid Pl itself.Plastino and Plastino (PP) (1995) investigated the effect of the two remaining components, i.e.
(1) that originating in the attraction of mz and (2) that arising from the centrifugal force.
Of course, these, in turn, give rise to additional components of the buoyancy force (notice that the components due to the attraction of m2 lack spherical symmetry).PP (1995) studied the effects of these two components of the pressure field in connection with the dynamics of Robe model by considering that the fluid ml adopts the shape of an ellipsoidal figure of hydrostatic equilibrium, specifically, a Roche ellipsoid (Chandrasekhar, 1987, Chap. 8).By recourse to the analytical expressions that are obtained for the pressure field in these circumstances, they gave a full account of the effects of the buoyancy forces.In particular, it is seen that the only equilibrium point is the ellipsoid's center.In analizing the EP-stability and, under the assumption that the smaller particle is denser than the surrounding medium, it was shown that the equilibrium is always stable.
We wish here to investigate the effects of drag forces on the dynamics of Robe's model in the spirit of Danby's modification of the standard version of the restricted three body problem (Danby, 1985), introduced as a possible scenario for the origin of the Trojan asteroids.In Danby's model the third body experiences a drag force proportional to its velocity relative to a surrounding medium which is modelled so that the triangular points remain EP's.He finds, in a linear analysis, that these points are unstable.In the range of/z-values given by (Danby, 1992) the Lagrange points L4 and L5 are linearly stable.Danby (1985) showed that the introduction of an arbitrarily small drag force suffices to make these points unstable.
The system exhibits then an instability in the presence of a dissipative force.
It seems natural to ask, in connection with Robe's model (a particle moves in a fluid medium), whether similar instabilities also appear when introducing drag forces, since dissipation is an essential aspect of all such media.This is the leitmotiv of the present effort, that is organized as follows: in Section 2 a brief review of Roche model is presented; Section 3 deals with the equations of motion in the Robe-Roche restricted three-body problem with drag; in Section 4 the stability of the equilibrium point is considered; in Section 5 the effects of drag forces within Robe's original problem are discussed; conclusions are given in Section 6.

Roche Model
According to the framing of Roche's problem, we suppose that the primary ml describes a circular orbit around the secondary m2 in such a way that the concomitant relative configuration remains unchanged.We adopt a coordinate system OXlX2X3 that rotates uniformly with angular velocity f~ and has its origin O at the center of mass of the primary ml; the axis OXl points towards m2, OXlX2 being the orbital plane of m2 around ml. Let R stand for the distance between the centers of mass of the primaries, and ui denote the components of the fluid velocity field in the rotating frame.
The fluid elements of ml verify the following hydrodynamical equations of motion (Chandrasekhar, 1987, Chap. 8) where P and/3 are, respectively, the pressure field and the gravitational potential due to the fluid mass, and the tide-generating potential/3r due to m2 is givenby Roche's approximation is based on keeping in the Taylor expansion for /3~ only terms up to the second order in the xi's (Chandrasekhar, 1987).Under this assumption, the equations of motion read (5) Another essential assumption within Roche's model is to adopt for f~ the 'Keplerian value' (Chandrasekhar, 1987) in order to have Z = 0 and get rid of the linear term in Xl that appears inside the brackets in the right-hand side of Equation ( 4).The resulting hydrodynamical equations are Roche ellipsoids constitute hydrostatic equilibrium solutions (i.e.(dui/dt) = O) to the above equations.They are ellipsoidal figures with semiaxes al, a2 and a3 parallel, respectively, to the coordinate axes Oxl, Ox2 and Ox3.The semiaxes ai verify the following relations (Chandrasekhar, 1987) where p = ml/m2 (11) The gravitational potential B of the homogeneous ellipsoid at an internal point x is (Chandrasekhar, 1987) where I stands for polar moment of inertia and the index-symbols Ai are given by /0 c~ du Ai = ala2a3 (i= 1,2,3).( 14) The solutions of Equations ( 9) represent hydrostatic equilibrium configurations of Roche's problem.Further analysis is required to assure stability.The points at which dynamical instability sets in along the Roche sequences are given in (Chandrasekhar, 1987, p. 205).The results obtained in the present paper are derived under the assumption that equations (9) are verified.In particular, they hold true within the stability region of Roche sequences.

Equations of Motion in the Robe-Roche Scenario
We adopt the total mass of the primaries, ml + m2, as the unit of mass, and choose the units of time and lenght in such a way that ft = 1 and R = 1, the quantity # becoming numerically equal to the ratio mz/(ml + m2).We also introduce the abbreviation and notice that D > 0 if the small particle is denser than the medium, which will be always the case in the present considerations.

I. DRAG FORCES IN A ROTATING REFERENCE FRAME
We now consider the equations of motion of a particle that experiences a drag force due to an uniformly rotating medium.We assume the force to be proportional to the relative velocity of the particle with respect to the medium.For the sake of simplicity we will consider that the only force acting upon the particle is the drag force, the extension to the case where other forces are also present being trivial.
The concomitant equations of motion in an inertial (i.e.non-rotating) reference system (J, y~, z ~) are (the rotation angular velocity of the medium being f~ = 1) where the effective potential U is given by (PP 1995) and -F dx/dt represents a drag force proportional to the velocity (F is a positive quantity).

Stability of the Equilibrium Point in the Robe-Roche Scenario
It is easy to verify that the center of ml (of coordinates (0,0,0)) is an equilibrium point.We proceed now to investigate its stability.From Equation ( 27) one easily ascertains that the motion parallel to the x3 axis is stable when the small particle is denser than the medium (D > 0).Restricting thus our attention to the (XlX2)-plane and following standard procedures we consider solutions to the equations (25, 26) of the form and with the complex eigen-frequency L given by the quartic equation where Q1, Q2 are defined below, i.e. and being negative quantities (see PP 1995).
We show now that the quartic equation does not admit purely imaginary roots.To this end we replace into the quartic equation and, equating to zero in separate fashion both real and imaginary parts we find, with a little algebra F2 -1 2(0, + 02)[D(O, -Q2) 2 -8(Q, + Q2)] < 0, (35) which is absurd.This result is also obtained, in a different way, by looking at the behaviour of the Jacobi integral of the equations of motion without drag.It is easy to show that for F --0 these equations admit the following integral of the motion Cj = (v2/2) -U. (36) In the general case (drag included) Cj is not a constant of the motion.Indeed, one immediately writes down its evolution equation, namely, showing Cj to be a monotonously decreasing function of the time, which precludes the existence of periodic orbits.
We proceed now to ascertain the nature of the real part of the roots of the quartic equation.Of course, the equation is amenable to exact analytical solution.However, the concomitant expressions are so cumbersome that it pays following a different route.We will instead count the number of zeros of our quartic polynomial in a half plane of the complex domain by recourse to a standard approach, that can be applied if the polynomial (31) has no pure imaginary roots (Wilf, 1962).We start by introducing a new variable z in the fashion L= -iz, (38) so that our quartic equation adopts the appearance where We are interested in the number of roots for which the real part of L is greater than zero, which translates here into finding the number of roots z located in the upper half-plane.One constructs then a Sturm sequence beginning with /:'1 (z), P2(z).The corresponding formula that gives these polynomials is The sequence ends after m steps, i.e. with Pm-l (z) and Pro(z), where Pro(z) divides Pro-l (z) with no remainder,

Pm-l (z) = Qm-I (Z)Pm(Z).
(43) In our case we have to deal with and ( 44) z ( 45) The number N + of roots z in the upper half-plane is given by (Wilf, 1962) where n is the degree of the original polynomial (whose roots are the subject of our investigation) and V(x) is the number of sign changes in the Sturm sequence Pi(x), (/ = 1,...,5).The sign of Pi(x) at infinity is, of course, that of the concominant dominant power.Ascertaining the signs of the appropriate coefficients is then our next task.In the case of P1, P2, P3, and P5 the coefficient of the dominant power has a positive sign.If we recast the coefficient of the P4's dominant power in the fashion we immediately appreciate its positive character.The associated sign sequences at (+oo) is (+, +, +, +, +) while that for (-oo) reads (+, -, +, -, +).Thus, V(+oo) = 0 and V(-c~) = 4.The number of roots L with positive real part is null, which allows one to assert that the equilibrium point we are concerned about (Robe-Roche scenario) is always stable, for any value of either the drag coefficient F or the remaining parameters defining the Roche ellipsoid.This is one of our main results.
It is of some interest to consider the case in which the drag coefficient F is small enough that a perturbative treatment becomes appropriate.In order to obtain the complex eigen-frequencies up to first order in F we have to deal with To first order in F the L's are of the form where L0 = -t-i&l,2 are the eigen-frequencies corresponding to the case without drag (i.e.F = 0), given by (PP 1995) and where After insertion of Equations (50-52) into ( 49), and keeping just first order terms in F, we are lead to the approximate frequencies whose real parts are negative, since A > 4, as easily seen from Equation (53).

The Effect of Drag on Robe's Original Model
We revisit now Robe's pioneer effort (Robe, 1977) and consider the effect of drag on the stability of the equilibrium point (in his original version).The associated equations of motion read, after linearization, (1 -pl ) (58)

K = +
We are now to proceed in a fashion similar to that of the preceding section.The pertinent Sturm sequence consists of polynomials that can be obtained from It is clear that the first three coefficients are always of a positive character.As for the two remaining coefficients, their sign depends upon the signs of the quantities Q~, Q~, c~, and/3.In Figure 1 we see that the plane (#, K) is divided up into five regions, according to the signs of the above enumerated quantities.The curves that separate the different regions are: Q~ = 0, o~ = 0,/3 = 0, and Q~ = 0. Recourse to Equation (47) gives, for each of our five regions, the number N + of eigenfrecuencies L with positive real component.For region I this number vanishes, while it equals unity for regions II, III, and IV, and augments up to two for region V.In Table I we distinguish between region III(a) and region III(b) because the quantity c~F 2 +/3 can take here different signs, depending upon the F-value.The equilibrium point results then to be a stable one only for region I, defined by the relationship K> 1+2#.
(66) Table I.Relevant quantities associated with the regions depicted in Figure 1 for the original Robe scenario plus drag.The second and third columns give the signs of the expressions that appear in the heading.The last column gives the number of eigenfrequencies with positive real part.
Region c~F 2 + 13 Q~Q~ V(-cx~) V(cx~) N + The stability of the equilibrium point in Robe's restricted three-body problem has been investigated in the case of a linear (in the velocity) drag force.As the small particle moves in a fluid environment, it is reasonable to assume that drag forces always exist, so that the present study can be regarded as a necessary complement to Robe's pioneer effort.Two scenarios were discussed: the original one of Robe's (1977) and another recently investigated by Plastino and Plastino (1995), in which the structure of a Roche's ellipsoid is assumed for the fluid body (Robe-Roche scenario).
In the latter, the equilibrium point remains stable no matter what the value of the drag coefficient F is, or what values are taken for the remaining parameters characterizing the system (ellipsoid's semiaxes, ellipsoid's density, small body's density, and the mass-ratio # of the principal masses).
In the original Robe scenario, instead, a more complex situation ensues.Stability is obtained in just one out of five regions in which the (#, K)-plane can be partitioned.In the remaining four regions the equilibrium point is not stable (see Figure 1).
The present 'Robe model with drag' is instructive in that it allows one to appreciate in just what manner a small drag force may qualitatively modify the stability properties of a system, even turning stability zones into unstable ones.In the present situation just an infinitesimal drag force (arbitrarily small drag coefficient F) does the trick (the nature of the regions depicted in Figure 1 does not depend upon the F-value).Indeed, the effect is similar to the one previously encountered by (Danby, 1977) in analyzing the stability of the Lagrange triangular points with reference to the usual restricted three-body problem.
It is to be stressed that the regions depicted in Figure 1 do not coincide with the one displayed in (Robe, 1977).As we have here shown, the latter are not obtained in the limit (F --+ 0), where a discountinuous change in the system's behaviour is attained (at F = 0), which, we repeat, resembles Danby's result in the ordinary restricted three-body problem.
If we consider Danby's (1977) and Robe-with drag's results, one wonders at the resilient character of the stability of the equilibrium point in the Robe-Roche scenario here investigated.
force.The inertial coordinates (:d, y~, z ~) are related to the rotating coordinates (x, y, z) by the standard transformation x' = x cos(t) -y sin(t), y' = x sin(t) + y cos(t), ZI -~ Z. the above equations it is easy to obtain the equations of motion in the rotating those of the previous section if one replaces the Robe-Roche quantities Q1 and Q2

Figure 1 .
Figure 1.Stability regions of the equilibrium point in the original Robe model plus drag.The curves that separate the different regions are Q~ = 0 (curve a), a = 0 (curve b), fl = 0 (curve c), and Q~ = 0 (curve d).The equilibrium point is stable only in region I.The nature of these regions is independent of the drag coefficient F.