Stationary solutions to a Vlasov equation for planetary rings

In this paper we consider a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. We propose a simple physical model, where the particles of the rings move under the gravitational Newtonian potential of two primary bodies. We neglect the gravitational forces between the particles. We use a perturbative technique, which allows to find explicit solutions at the first order and solutions at the second order, solving a set of two linear ordinary differential equations.


Introduction and basic equations
The gravitational N-body problem is one of the oldest problems in physics. The N bodies interact classically through Newtons Law of Universal Gravitation. Then the equations of motion are where m i is the mass of the body P i , r i is its position vector relative to some inertial frame, and G is the universal constant of gravitation. These equations provide a reasonable and well-accepted mathematical model with numerous applications in astrophysics, including the motion of planets, asteroids, comets and other bodies in the Solar System. The number N of the bodies can be very large; for instance, the planetary rings are composed of a large number of small bodies with sizes from specks of dust to small moons. Saturns rings are the largest and best studied. In general, for huge N, an alternative approach consists to consider a statistical description through a kinetic (Boltzmann) equation [2]. Unfortunately, Boltzmann equation is a non-linear integro-differential equation and solving it in six-dimensional phase space requires an extremely large memory and computational time. So, alternative kinetic equations, as collisionless Boltzmann equation [8], Bhatnagar-Gross-Krook (BGK) models, where a relaxation term replaces the collision integral, and other models [9] are been considered. Very recently, accurate numerical solutions to the Vlasov-Poisson model for self-gravitating systems are proposed in [3] and [10]. The literature is very rich in papers devoted to analytical, numerical and computational studies on this topic. In this paper we will restrict our attention to a Vlasov or collisionless Boltzmann equation describing the dynamics of planetary rings. The stability and the structure of Saturn's rings was studied by Griv et al. [4]- [7] using both collisionless and BGK models. A simple mathematical model describing the dynamics of a planetary rings is given by the following N circular restricted three body problems. A large set of small bodies, are subject to the attraction of the Sun and a planet. The primary bodies move in a plane in circular orbits about their center of mass. The total mass of the small bodies is negligible compared to the primary body masses. Then the presence of the small bodies does not disturb the circular motion of the two large bodies. We denote by r S and m S the position and the mass of the Sun; r P and m P , are the position a the mass of the planet. Hence, the equations of motion are Since we are considering a large number of small bodies moving under the gravitational potential of the primary bodies, at any time t a full description of the state of this system can be given by specifying the number of small bodies f (t, r, ξ) dr dξ, having positions in the small volume dr centered on r in the small velocity range dξ centered on ξ. The function f (t, r, ξ) is called the distribution function of the system. Obviously, we require that f ≥ 0 almost everywhere, since we do not allow particles with negative mass. We assume that the following Vlasov equation describes the evolution of the distribution function f (t, r, ξ) ∂f ∂t where is gravitational potential of the primary bodies. We denote by (X, Y, Z) the component of the vector r. Now, we introduce a uniformly rotating coordinate system with origin at the mass center of the primary bodies, so that the Sun and the planet are located on the x axis with coordinates (x S , 0, 0) with x S > 0, and (x P , 0, 0), respectively. This implies the following transformation of variables where (x, y, z) are the new spatial coordinates, (c 1 , c 2 , c 3 ) the component of the particle's velocity and ω is the constant angular velocity of the primary bodies. It is useful to introduce cylindrical coordinates (centered in P ) given by In terms of the new variables, the distribution function f (t, r, ξ) is replaced by the new unknown G(t, r, θ, z, u r , u θ , u z ), and Eq. (3) writes where, now, the gravitational potential is We will study the kinetic model given by Eq. (7) and Eq. (8).

The 2D equations and an approximate model
Assuming that all the particles move on the plan of the primary bodies, Eqs. with If we consider the Saturn's rings, the mean distance of rings from the center of the planet is small with respect to the distance Sun-Saturn. Hence, it is reasonable to replace the exact potential due to the Sun, with a polynomial approximation, obtained using Mac Laurin expansion. We have Using Eq.(11), Vlasov equation (9) becomes Since then Eq. (12) simplifies and writes Often the parameter ε is small. For instance, in the case of Saturn's rings, using the following units of measure In this case, it is required a very accurate numerical scheme, which takes into account the effects of the small term ε in Eq. (13), for large time integration. In order to overcome this difficulty, we suggest a simple (Hilbert) expansion, assuming that G(t, r, θ, u r , u θ ) ≈ G 0 (t, r, θ, u r , u θ ) + ε G 1 (t, r, θ, u r , u θ ) .
Now, Eq. (13) gives the following set of partial differential equations We note the splitting of the equations, and, of course, we solve before Eq. (15) and then Eq. (16). It is evident that Eq. (15) is the Vlasov equation for an ensemble of particles moving in the gravitational field of a central mass.

The equation to G 0
Here, we use the method of the characteristic curves for solving the first partial differential equation. This method allows to find analytical or numerical solutions to linear partial differential equations for fixed initial condition, and sometimes classes of explicit solutions. In our case, we get a set of ordinary differential equations corresponds to Eq. (15).
In the appendix A we give the mathematical details of the study of system (17). Here, we show a class of exact solutions to Eq. (15). It is easy to verify that every differentiable function F(r, u r , u θ ) = G r u θ + ω r 2 , satisfies Eq. (15). This result recall Jean's Theorem and the existence of stationary spherically symmetric solutions to a Vlasov equations for stellar dynamics [1]. It is useful to define The stationary solutions (18) do not depend on θ. Since, in Eq. (15), the gravitational potential depends only on the distance r from the origin, spherically symmetric solutions have a clear physical meaning.

The equation to G 1
Using two elementary trigonometric formulas, Eq. (16) becomes If G 0 does not depend on θ, then we can look for solutions of the kind G 1 (t, r, θ, u r , u θ ) = A(t, r, u r , u θ ) + B(t, r, u r , u θ ) cos 2θ + C(t, r, u r , u θ ) sin 2θ .
This yields the set of equations Eqs.

The equation (22)
We first consider only this equation, because A does not appear in Eqs. (23)-(24), and we look for exact analytical solutions. To this scope, we assume that The right hand side of this equation is then, by using again the method of the characteristic curves (see, details in Appendix B), we prove that the class of functions satisfy Eq. (22), provided A * is differentiable.

The system (23)-(24)
We recall the equations Now, we look for solutions of this kind It is easy to verify that the equations become Now and, since we have assumed that G 0 = G 0 (ϕ 1 , ϕ 2 ), we also have Therefore, if we define then the system (32)-(33) writes We note that the two equations are simple ordinary differential equations, where r is the only variable, because ϕ 1 , ϕ 2 will play only the role of parameters. Eqs. (34)-(35) can be solved numerically with suitable initial conditions, using standard routines.

Appendix A
We consider the set of ordinary differential equations

Hence, we have
The method of the characteristic curves furnish the set of differential equations .

The first equation gives
dr dt = u r ⇒ r dr dt = r u r ⇒ r u r = 1 2 dr 2 dt .