Static Thin Disks with Power-law Density Profiles

The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.


Introduction
Disk sources of gravitation have a clear astrophysical importance. Disk configurations typically result from the combined effect of central attraction-due to a central body or due to the disk itself-and centrifugal force due to orbital motion of the disk matter. In Newton's theory, the gravitational field is fully represented by potential, given by mass density through the Poisson equation. In general relativity, where mass currents also contribute to the field, one has to also specify how the matter moves-i.e., how it orbits in the disk case. Unfortunately, when there is some overall net rotation, Einstein equations lead to a difficult problem, usually not solvable by analytical methods. On the other hand, if rotation can be neglected, or if it is compensated as in the case of two equal counter-orbiting streams, the situation is much simpler. More specifically, in the static and axially symmetric vacuum (or possibly electro-vacuum) case, the gravitational field can always be described by the spacetime metric of the Weyl type 1 r f r = -+ + + where t, ρ, f, and z are the Weyl cylindrical coordinates, ν (counterpart of the Newtonian gravitational potential) is given by the Laplace equation (so it behaves linearly), and λ is found (from ν) by quadrature ò l r n n r n n = -+ r r r numerically checked against the series-expansion solution presented in Semerák (2004). After briefly explaining, in Section 6, why the procedure only works for a density involving even powers of radius, we add in Section 7 how to obtain the potential of a finite disk with "bump"-type density profile. Physical properties of the disk sources are illustrated in Sections 8 (radial profiles of density and of azimuthal pressure) and 9 (radial profile of circular-geodesic velocity). Finally, we make several remarks in Section 10, mainly mentioning similar results that have appeared in the literature recently.

Static Thin Circular Disks
Computation of the potential of static thin circular disks is a classical problem of the potential theory. In Newton's theory of gravitation, it is mostly solved while modeling the gravitational field of galactic disks, while in general relativity, one mostly tackles it when modeling accretion disks around compact objects. Although it reduces to the Laplace equation in both theories, in the static case, it remains a challenge, as it is best illustrated by the "trivial" case of uniform surface density when the result still involves elliptic integrals of all three kinds (e.g., Lass & Blitzer 1983). For a thin disk lying in the equatorial plane (z = 0), with an outer rim situated on some Weyl radius b, the Poisson integral for ν reads ò ò On the symmetry axis (ρ = 0), it simplifies considerably, because the complete elliptic integral of the first kind K(k) reduces to π/2. In the axisymmetric case, knowledge of ν on the axis is crucial, since if the latter can be expanded as a power series in z, the potential at general location is obtained just by replacing z with r + z 2 2 in the above sum and multiplying each of its terms by the Legendre polynomial In (5), α j and β j are coefficients and  represents the disk mass in our case.
If interested in annular disks rather than in finite solid ones, one can perform an inversion with respect to the rim at ρ = b (also called the Kelvin transformation), r r r r n r r n r r r  +  + + + + The corresponding ν potentials are expressed in terms of finite Legendre series (with m + 1 terms), so there is no truncation issue. The superposition of the Morgan-Morgan disks with a Schwarzschild-type black hole was studied by, e.g., Lemos & Letelier (1994) and Semerák (2003).
In Semerák (2004), we derived the potentials for similar (also annular) disks with densities of the power-law form where m and n are natural numbers and the first parenthesis stands for binomial coefficient. These behave somewhat more regularly at the inner rim and do not involve the square root. We were able to find the potential in closed form on the axis, but elsewhere it was only given in terms of an infinite Legendre sum. The purpose of the present paper is to show that it can be written in a closed form.
The key feature will be the relation between a complete elliptic integral of the first kind and an integral over a product of Bessel functions and exponentials.

Potential in Terms of the Bessel Functions
The alternative formulation we will build on first appeared within electrodynamics and later was applied to galactic disks by Toomre (1963). More recently, Conway (2000) developed it to obtain closed-form solutions for the potential of matter confined within axisymmetric boundaries. It consists of expressing the axisymmetric Green function, i.e., the potential due to an infinitesimally thin circular loop placed at r¢ ¢ z , where J 0 is the zero-order Bessel function of the first kind and s is an auxiliary real variable. The potential due to a thin source of surface density r¢ w ( ), placed at ¢ = z 0, can thus be written as The double integration of (10) looks more complicated than the single one in (4), but this is outweighed by having much better knowledge of integrals involving Bessel functions.

Potential for Power-law Density Terms: Convolution with the Bessel Functions
Consider the density given by an even power of r¢ within some finite radial range (0, b), In such a case, the radial integration in (10) yields ò r r r r with 1 F 2 the "1,2" generalized hypergeometric function. Employing the contiguous relation 1, ; ; , 1; ; 1, 1; , 12 we can express 1 F 2 in terms of 0 F 1 , and thus, thanks to the identity as a combination of Bessel functions, with a and c integers. In particular, regarding the symmetry 1 F 2 (a; b, c; z) = 1 F 2 (a; c, b; z), we have å å + + - Using the above relations, we can write the potential (10) of the power-law density term r r ¢ = ¢ w and H is the Heaviside step function. A general a b g , ,  ( ) can be obtained using the recurrence relations 4 where Q γ−1/2 are Legendre functions of the second kind (toroidal functions). For our potential formula (15), we first reduce the first index of -j j , ,0  ( ) to zero using (21) and then also do the same with the remaining indices using (22)-(25) successively. Since = --0, 1, 1 0,1,1   ( ) ( ) , one ends up with just a combination of (17)-(19), which means that ν (2 l) is obtained in closed form. However, the procedure grows cumbersome for higher exponents, because the order of the polynomial "coefficients" in front of the resulting elliptic integrals gradually grows. It is thus useful to create, for evaluation of a generic-case potential, a package in some symbolic-manipulation software; we have used Mathematica for that purpose.
Let us illustrate how the result (15) appears in specific cases. For l = 0, the density is just constant, which leads to the well-known solution derived by, e.g., Lass & Blitzer (1983): For l 1, the potential can be expressed in a similar but generalized form, 5 n r p r r r r r r Eason et al. (1955) called them "Lipschitz-Hankel" integrals for products of Bessel functions. They actually represent Laplace transform of products of two Bessel functions-see also Hanson & Puja (1997) and Kausel & Baig (2012) for thorough treatments. 3 Conway writes 0,1,0  ( ) in terms of the Heuman lambda function, whereas we use its relation to complete elliptic integral of the third kind (Eason et al. 1955) r p r r r r r p r r r r r L - where l 2  ( ) are polynomials in ρ and z. Generally, l 2  ( ) are even polynomials of the order 2l in the case of P H E l , , 2  ( ) , and of the order 2l + 2 in the case of K l 2  ( ) . For the first few cases, the polynomials read Although the expressions show some clear combinatorial-type patterns, we have not been able to arrange them completely in a simple closed formula, so it is better to keep the expanded form.
We may also add that, in the equatorial plane (z = 0), the exponential e − s| z| reduces to unity and integral (16) can be expressed in terms of a hypergeometric function as where it is assumed that α < 1, α + β + γ + 1 > 0, and 0 < ρ < b. (For ρ > b > 0, one just swaps ρ ↔ b in the formula.) We are specifically interested in the case α = − j, β = j, γ = 0 (yielding α + β + γ + 1 = 1) when the expression reduces to The potential due to the density terms given by negative powers of r¢ can be obtained by inversion (6). Under the inversion, our r¢ l 2 ( ) density and the corresponding potential transform as More explicitly, the inverted potentials read where~l 2  ( ) denote the respective polynomials l 2  ( ) "inverted" according to (6); modulus k of the elliptic integrals is invariant under the inversion, so it keeps the same form (20). Note that we distinguish the inverted potentials by "i," as opposed to the density w whose character is apparent. Note also that the potentials do not yet have the correct dimension-rather than being dimensionless,

Circular Disks with Generic Power-law Densities
Due to the linearity of the Laplace equation, one can now obtain disk solutions with various power-law radial profiles of density r¢ w ( ) by superposition of elementary terms discussed above. Consider a circular thin disk extending from the center to some finite radius r¢ = b, with surface density given by an even polynomial in the radial coordinate, where l and m are natural numbers and W is a normalization factor ensuring that the disk has the prescribed total mass . The radial integration in (10) can, for such a density, be performed using binomial expansion (similarly as Conway 2000 did for l = 1), to obtain ,2  ( ) are polynomials (in ρ and z) of the order 2lm while K m l ,2  ( ) are polynomials of the order 2lm + 2. They are simply related to those obtained for the r¢ l 2 ( ) density terms in (28)-(36), we can summarize the above relations as å å

The Case of Annular Disks
Annular circular disks are again obtained by inversion (39). Substituting the inverted density in order for the total mass of the disk to come out  (i.e., in the same manner as in (8)   ( ) "inverted" according to (6); modulus k of the elliptic integrals is invariant under the inversion, so it keeps the same form (20). The above potential already is dimensionless, as it should be.
Below, we illustrate that the solution is the same as the one given, in the form of multipole expansion, in Semerák (2004), for even n( ≡ 2l).

Behavior of the Potential at Significant Locations
On the axis (ρ = 0), we have k = 0, so all the elliptic integrals reduce to π/2, and H(b) = 1, which yields An explicit result is obtained by substituting fors, which means computing the respective polynomials r = 0 given by Semerák (2004). Especially at the very center, (ρ = 0, z = 0), the potential amounts to

This increases with m, whereas it decreases with n.
In the plane of the disk (z = 0), we were only able in Semerák (2004) to give the potential as an infinite series at ρ > b (i.e., "within" the disk), while in the empty region in the disk center, we wrote valid both above and below the disk rim. At the very rim, ρ = b, both formulae simplify in an obvious manner. In particular, the first one then involves the unity-argument 3 F 2 function, In the second, elliptic-integral formula, one has (at the disk rim) k = 1, which is the singularity of K(k), but since the respective polynomial~K m l ,2  ( ) always contains the factor rb 2 2 2 ( ) in the equatorial plane, the K-term is actually eliminated at the rim and one is left there with Several first values read, at the rim (ρ = b, z = 0), At radial infinity, the potential falls off as r -+ z 2 2  , which just confirms the meaning of .

Numerical Check of the Closed-form Formulae against Series Expansion
A closed-form solution has clear advantages over the series expansion, the more that the Legendre-type series do not tend to converge safely. It is natural to compare the two solutions now, in order to support the reliability of both.
In Tables 1-3, we numerically compare several examples of the annular-disk potential (53) with the solution expressed in term of the multipole expansion in Semerák (2004) (Equations (10) and (11) there). The main message of the tables is that the solutions are really identical. The second observation is that the series expansion converges quite well even at radii close to that of the disk rim, mainly for higher m and lower 2l, for which the density falls off (or rises) less steeply at the disk edge, making the field more regular.

Trouble with Odd Powers of Radius in Density
Up to now, we have been able to employ Conway's method and reach the closed-form solution for even n( = 2l) in the density prescription (8). However, the odd-exponent case, n = 2l + 1, is more difficult, as already noticed in Semerák (2004) Here, the contiguous relation (12) does not always work, because the difference between the first and the third parameter of the above 1 F 2 is only an integer for an even q, which means that for odd q the counterpart of (14) would generally be an infinite sum.

Finite Disks with Bump-type Density Profiles
Finally, we derive, in a closed form as well, the potential of a finite annular disk. The respective density profile can be composed, within a selected radial range (ρ in , ρ out ), of the constant-density case plus the l = 0, 1, 2,K,L (with L 1) sum of the (40) terms. Let us illustrate the recipe on the simplest, L = 1 case. Consider, for   r r r ¢ in out , the density r r r r r r r r r r r The potential of the disk follows by subtraction n r r n r r n r n n n , ; , with , , 58 where ν (0) is given by (26) and ni 3 ( ) , ni 5 ( ) are given by (41) for l = 0, 1. The density and potential of this type of disk are illustrated in Figure 1.
Higher bump-type disks can be constructed in a similar way. In general, one takes certain L 1 and ρ out > ρ in > 0, considers (within the interval   r r r ¢  Note. The disk parameters are z = 0, = b 5, 2l = 2, and m = 1. Coordinates ρ and z are given in units of , while the potential values are dimensionless.

Table 2
The Same Comparison as in Table 1, but for the Disk with 2l = 2, m = 2 (Top Table) and 2l = 2, m = 3 (Bottom  Table 3 The Same Comparison as in Tables 1 and 2, but for the Disk with 2l = 4, m = 1 (Top Table) and 2l = 6, m = 1 (Bottom   For L = 1 (preceding paragraph), there are two parameters (W −3 and W −5 ) and two constraints, so no parameters are left free (besides the overall scaling, as determined by  through W 0 ). For L = 2, the density contains three parameters (W −3 , W −5 , and W −7 ), which again are bound by two constraints, so one of them effectively remains free. In general, L − 1 of the W parameters only remain restricted by condition that the density must not have any root inside the radial range of the disk.

Radial Profiles of Density and of Azimuthal Pressure
Two simple types of physical interpretation of the disk sources involve: (i) a single-component ideal fluid with a certain surface density (σ) and an azimuthal pressure (P), which keeps the orbits at their radius; or (ii) two identical counter-orbiting dust components with proper surface densities σ + = σ − following circular geodesics with equal but opposite velocities relative to static observers, v + = − v − (see, e.g., González & Espitia 2003). In Figure 2, we show the radial profiles of surface density σ and of azimuthal pressure P, 6 s n p rn n p rn for several annular disks encircling a Schwarzschild black hole. Figure 3 illustrates the same parameters for several solid and several bump-type disks. Some curves are seen to fall below zero around the inner disk edge, which means such orbits (understood as hoops) would have to be in a state of tension in order to stay at their respective radii, i.e., they are attracted outward rather then downward, due to a "too large" mass of the disk at larger radii. Such a circumstance corresponds to when free circular motion is impossible (see the following section).