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Fractional Derivative Approach to the Self-gravitation Equation

A new formalism is presented for finding equilibrium distribution functions for axisymmetric systems. The formalism, obtainded by using the concept of fractional derivatives, generalizes the methods of Fricke (1952), Kalnajs (1972) and Jiang&Ossipkov (2007), and has the advantage that can be applied to a wider variety of models. We found that this approach can be applied both to tridimensional systems and to flat systems, without the necessity of dealing with a pseudo-volume mass density. As an application, we obtain the distribution functions of the Binney's logarithmic model and of the Mestel disc.


INTRODUCTION
The construction of self-consistent models for stellar systems is of great interest in astrophysics. As it was pointed out by some authors, the most straightforward way to perform such models is to start with an assumed potential defining the mass density ρ (via Poisson's equation) and the families of orbits that can lie within the system (via Newton's equations of motion). Since ρ is the integration of the distribution function (DF) over the velocity variables in the phase space of the system, the problem of finding the DF is that of solving an integral equation. For that reason, such procedure is called as the "from ρ to f " approach for finding a selfconsistent DF (Binney & Tremaine (1987); Hunter & Quian (1993); Jiang & Ossipkov (2007)) and the integral relation connecting f and ρ is known as the self-gravitation equation.
By Jeans's theorem, an equilibrium DF is a function of the isolating integrals of motion that are conserved in each orbit. It has been shown that, for certain potential-density pairs (PDP), it is possible to find analytically such kind of DFs. The simplest case of physical interest corresponds to spherically symmetric PDP, described by an isotropic DF that only depends on the energy. Eddington (1916) showed that it is possible to obtain such DFs by first expressing the density as a function of the potential, and then solving an Abel integral equation.
Recently, Jiang & Ossipkov (2007) presented a new method for the axially symmetric case, where the equilibrium DF depends on the energy E and the angular momentum about the axis of symmetry Lz, i.e. the two classical integrals of motion. They developed a formalism that essentially combines both the Eddington formulae and the Fricke (1952) expansion in order to obtain the DF's even part, starting from a density that can be expressed as a function of the radial coordinate and the gravitational potential. Thus, for a given ρ, the corresponding even DF can be obtained by solving an Abel integral equation. Once such even part is determined, one can find the DF's odd part by introducing some reasonable assumptions about the mean circular velocity or using the maximum entropy principle.
In the present paper we show an extension of the formulae derived earlier by Fricke (1952) and Jiang & Ossipkov (2007), by introducing the fractional derivative concept. The approach developed here has several advantages over the methods introduced before. In one hand, the mathematical difficulties involved in the formalisms based on transformation techniques can be easily overcome. It is worth to point out that the formalism introduced by Jiang & Ossipkov (2007) demands the definition a volumetric pseudo-density, in order to be applicable to flat systems. On the other hand, our method can be applied directly to the case of flat systems.
Assume that Φ and E are, respectively, the gravitational potential and the energy of a star in a stellar system. It is useful to define a relative potential Ψ = −Φ + Φ0 and a relative energy ε = −E + Φ0, in such a way that the system has only stars with energy ε > 0 (Binney & Tremaine (1987)). For the case of an axially symmetric system, it is customary to use cylindrical polar coordinates (R, ϕ, z), where the velocity is denoted by v = (vR, vϕ, vz). Such system admits two isolating integrals: the component of the angular momentum about the z-axis, Lz = Rvϕ, and the relative energy ε. Hence, the DF of a steady-state stellar system in an axisymmetric potential can be expressed as a non-negative function of ε and Lz, that vanishes for ε < 0, denoted by f (ε, Lz) and related to the mass density as where f+(ε, Lz) is the even part of f with respect to the angular momentum Lz. On the other hand, for the case of flat systems, the surface mass density Σ is related to f through Now, by defining a pseudo-volume densityρ, according to (Hunter and Quian (1993)) it is also possible to use equation (1) to deal with these flat systems.

TRIDIMENSIONAL SYSTEMS
Most of the methods developed to solve (1) require some kind of dependence between the DF, the relative energy ε and the angular momentum Lz (see Fricke (1952) and Jiang & Ossipkov (2007), as examples), which will define the corresponding mass density ρ(R, Ψ). Therefore, in order to study the problem, we shall start by assuming some general types of DFs.

DFs of the form
To generalize the Jiang & Ossipkov method, we suppose that the DF can be written as where αn ∈ R and the 2 in the exponent of Lz guarantees that f+ is even. Now, performing the integral (1) with respect to Lz, we obtain for αn > −1/2, while it diverges for αn ≤ −1/2. Therefore, we assume that the the mass density is given by A comparison between (5) and (6), leads to the relatioñ At this point, we introduce the fractional derivative operator D j x , which represents a j-order derivative, with respect to x, for any real value of j (see Bologna & Grigolini (2003)). Assuming that (D j Ψρ n(Ψ))Ψ=0 = 0 for all j ∈ (0, αn + 1/2), then This integral equation is simpler than the first one and can be inverted easily if one takes the derivative once again with respect to Ψ, and the distribution function can be expressed as Ψρ n(Ψ)˛Ψ =ε π2 αn+ 3 2 Γ(αn + 1 2 ) .
When αn ∈ N, by the definition of the Riemann-Liouville operator, equation (10) reduces to the formulae obtained by Jiang & Ossipkov (2007). As a particular case, suppose thatρn(Ψ) can be written in the formρ So, applying the fractional derivative operator to (11) for β k > αn + 1/2, and the corresponding DF is This result is totally equivalent to the Fricke solution, for real values of αn, and therefore can be considered as a generalization.

DFs of the form
It is possible to derive a more general expression for the DF if we assume that it depends on ε where Ra is a scaling radius. Suppose that the system has only stars with Q > 0, so f (Q, Lz) = 0 for Q ≤ 0. Here, Q → ε as Ra → ∞. The fundamental equation can be written, in terms of Q, as where f+(Q, Lz) is the even part of f (Q, Lz). So, following a similar procedure than in section 2.1, one can find that a DF of the form corresponds to a mass density of the form for αn > −1/2, where αn ∈ R. Now, if we sum over all posible values of Ra we obtain the general expression corresponding to a density of the form with Ra > 0 and αn > −1/2.

Models with divergent gravitational potential
In a system in which the gravitational potential has no upper bound, it is not possible to define correctly the relative potential Ψ and the relative energy ε, because the scape energy of the system is ∞. For this reason, we shall write the fundamental equation in terms of E and Φ, and we will suppose that the DF can be written as for αn > −1/2, and that the density is given by for αn > −1/2. Thus then, by integrating with respect to Lz, follows that Now, if we assume that for all j ∈ (0, αn + 1/2), then and so we obtain Therefore, the distribution function is for αn > −1/2. On the other hand, we can assume that the density has the more general form for αn > −1/2. So, the corresponding DF will be for αn > −1/2 and Q defined as Q = E + L 2 z /(2R 2 a ). Now, if we sum over all the posible values of Ra, we can obtain the generalization corresponding to a density of the form with Ra > 0 and αn > −1/2.

FLAT SYSTEMS
The formalism sketched above can also be used directly in the case of flat systems. Note that in the method introduced by Jiang & Ossipkov (2007) it was not possible, since the fundamental equation could not be solved using the Abel integral equation. Now, we will proceed similarly to the tridimensional case, finding the DFs for different kinds of densities. Then, we will also study the case of models with divergent gravitational potential.
3.1 DFs of the form P n L 2αn z hn(ε) As in the tridimensional case, first we suppose that So, by integrating (2) with respect to Lz, we obtain Σ = X n √ πR 2αn Γ(αn + 1 2 ) 2 −(αn+1) Γ(αn + 1) for αn > −1/2. Therefore, if we assume that αn > −1/2, it follows that Now, if (D j Ψ σn(Ψ))Ψ=0 = 0 for all j ∈ (0, αn), then equation (34) leads to Consequently, and the DF is This equation, for αn = 0, corresponds to the method developed in Kalnajs (1976), working in an adequate rotating frame. As a particular case, suppose that Then, by taking the fractional derivative we obtain and the DF is This relation can be interpreted as the analogous case of the Fricke expansion, when we are dealing with flat systems. It can be verified performing the pseudo-volume density (3) of R 2αn Ψ β k and taking the Frike component corresponding to the tridimensional case.

Other DFs for flat systems
We can generalize the result (37) if we express the DF in terms of Q = ε − L 2 z /(2R 2 a ). In this way, if the density has the form the corresponding DF is for Ra > 0 and αn > −1/2. Furthermore, if we consider models with gravitational potential having no upper bound, we can deduce that for a density and assuming that lim Φ→∞ D j Φ σn(Φ) = 0 for all j ∈ (0, αn), then for Ra > 0, αn > −1/2 and Q = E + L 2 z /(2R 2 a ).

SOME APPLICATIONS
In this section we will use the formulae introduced above to the Binney's logarithmic model and the Mestel disc, and we will see that their corresponding DFs match exactly with those that were found through the application of other methods. Binney's logarithmic model has a gravitational potential of the form whereas its mass density is that can be written as So, as D α x e ax = a α e ax for any α ∈ R, we obtain where A = 2 5 2 (1 − q 2 ) π 5 2 Gq 2 v 3 0 , B = √ 2 π 5 2 Gq 2 v0 , C = 2q 2 − 1 4π 5/2 Gq 2 v0 , the same DF founded in Jiang & Ossipkov (2007), using the Abel's integral equation, and in Evans (1993b) using Lynden-Bell's method. Another case of interest is the Mestel disc, characterized by a gravitational potential of the form and a surface mass density given by where Σ0 = v 2 c /(2πGR0). Now, we can write (50) as for any m ∈ R. So, equation (44) for Ra → ∞ leads to where F and σ are the constants given by This solution was obtained as well by Evans (1993b), and was proposed earlier by Toomre (1977).

DISCUSSION
In contrast with the methods based on integral transformation techniques, the formalism developed here does not require that the mass density has an analytic continuation to complex arguments. Indeed, this is the principal disadvantage involved in such methods. Moreover, our fractional derivative approach can be regarded as a general method that contains, as particular cases, the results obtained by Fricke (1952), Kalnajs (1976) and Jiang & Ossipkov (2007). The method developed here can be applied to a wider variety of axisymmetric models, due to the generic form of the density as a function. Another advantage of this formalism is that it can be applied directly both to tridimensional systems and to flat systems, without the implementation of a pseudo-volume density. Therefore, taking into account all the above statements, the present formalism represents a powerful tool on the construction of self-consistent stellar models.

ACKNOWLEDGMENTS
J. R-C. wants to thank the financial support from Vicerrectoría Académica, Universidad Industrial de Santander.