Gravitational force distribution in fractal structures

– We study the (newtonian) gravitational force distribution arising from a fractal set of sources. We show that, in the case of real structures in finite samples, an important role is played by morphological properties and finite size effects. For dimensions smaller than d − 1 (being d the space dimension) the convergence of the net gravitational force is assured by the fast decaying of the density, while for D > d − 1 the morphological properties of the structure determine the eventual convergence of the force as a function of distance. We clarify the role played by the cut-offs of the distribution. Some cosmological implications are discussed. The aim of the present paper is to discuss the general properties of the gravitational field generated by a finite fractal distribution of field sources. This problem is nowadays particularly relevant. In fact, there is a general agreement that galaxy distribution exhibits fractal behavior up to a certain scale [1, 2]. The eventual presence of a transition scale towards homogeneity and the exact value of the fractal dimension are still matters of debate [3, 5, 4]. Moreover it has been observed that cold gas clouds of the interstellar medium have a fractal structure, with 1.5 ≤ D ≤ 2 in a large range of length scales [6]. For this case the general belief is that the origin of their fractality lies in turbulence. However recently [7] it has been pointed out that self-gravity itself may be the dominant factor in making clouds fractal. Chandrasekhar [8], has considered the behavior of the newtonian gravitational force probability density arising from a poissonian distribution of sources (stars). In this case, in evaluating the probability distribution of the force acting on a particle, one supposes that fluctuations are subject to the restriction that a constant average density occurs, i.e. the source distribution is spatially homogeneous and the density fluctuations obey to the Poisson statistics. Applying Typeset using EURO-LTEX 2 EUROPHYSICS LETTERS the Markov’s method, it is possible to compute explicitly the force probability density, known as the Holtsmark’s distribution. In this case, the probability density of the force F is given by W (F ) = H(β) Fo (1) where Fo = (4/15) (2πGM)n is the normalizing force (n is the average density of sources, M is the mass of each punctual source and G is the gravitational constant), β = F/Fo is an adimensional force and H(β) = 2 πβ ∫ ∞ 0 dx exp[−(x/β) 3 2 ] x sin(x) . (2) The main result is that, in the thermodynamic limit (V → ∞ with n constant), the force distribution has a finite first moment and an infinite variance. This divergence is due to the possibility of having two field sources arbitrarily nearby. The approximate solution given by the nearest neighbor (n.n.) approximation (i.e. by considering only the effect of the n.n. particle) and the exact Holtsmark’s distribution (Eq.2) agrees over most of range of F (see Fig.??). The region where they differ mostly is when F → 0. This is due to the fact that a weak field arises from a more or less symmetrical distribution of points, and hence the n.n. approximation fails. Therefore in the strong field limit we may neglect the contribution to the force of far away points, because the main contribution is due to r → 0 (i.e. it comes from the n.n). The root mean square of the force is divergent as in the case of the n.n. approximation, and this is due to the fact that the limit r → 0 is allowed. If there exists a lower cut-off (see below) this divergence is erased out. However, due to isotropy, no divergence problem arises from the faraway sources even in the limit r → ∞. The derivation of Chadrasekhar cannot be easily extended to the case of fractal distributions because, in such structures, fluctuations are characterized by long-range correlations. In this situation an analytic treatment of the force distribution becomes very difficult. Vlad [9] developed a functional integral approach for evaluating the stochastic properties of vectorial additive random fields generated by a variable number of point sources obeying to inhomogeneous Poisson statistics. Then he applied these results to the case of the gravitational force generated by a fractal distribution of field sources under some strong approximations. In particular, in order to compute analytically the force probability density, Vlad [9] has not considered the presence of intrinsical fluctuations in the space density. In fact, in the computation of the density from a single point one expects to see deviation from the average scaling behavior, which are present at any scale (see below). Such a situation occurs in any real fractal structure and can be quantified by studying the n-point correlation function [10]. Instead, the derivation has been done under the assumption that the n + 1-point correlation function (where the (n+ 1) point is the occupied origin) can be written as g( ~ x1, ~ x2, ..., ~ xn) = g( ~ x1)g( ~ x2)...g( ~ xn) , (3) where g(~x) is the two-point correlation function. Such an approximation is not adequate to describe the effect of morphology and hence in real cases the situation is quite different. Instead of Eq.2, Vlad found that the probability density of the absolute value F (generalized Holtsmark’s distribution) of the field intensity is equal to H(β,D) = 2 πβ ∫ ∞ 0 dx exp[−(x/β) D 2 ] x sin(x) . (4) In this case Fo = (4/15) (2πGM)(DB/(4π)) where β = F/Fo. The main change due to the fractal structure is that the scaling exponent in Eq.4 is D/2 rather than 3/2. Hence in this Gabrielli et al., Gravitational force distribution in fractal structures 3 case the tail of the probability density has a slower decay than in the homogeneous case. This means that the variance of the force is larger for D < 3 than for the D = 3. In Fig.?? this situation is shown. As we discuss below the case D < 2 is rather well described by Eq.4: this is not the case for D > 2 where the approximation given by Eq.3 maybe completely useless for real structures. An important limit is the strong field one (F → ∞). In this case it is possible to show that the force distribution of Eq.4 can be reduced to the one derived under the n.n. approximations: Wnn(F )dF = DB 2 (GM) D 2 F D+2 2 exp (