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Self Gravitating Incompressible Fluid in Two Dimensions

In this paper we develop two models for the steady states and evolution of two dimensional isothermal self gravitating and rotating incompressible gas which are based on the hydrodynamic equations for stratified fluid. The first model is for the steady states of the gas while the second addresses the time evolution of the gas subject to some constraints. These models reduce the initial five partial differential equations that govern this system to two for the steady state model and to three for the time dependent model. Analytical and numerical solutions of the model equations are used to study the structure of the resulting steady and time dependent states of the fluid with some possible astrophysical applications.


Introduction
The steady states of self gravitating fluid in three dimensions have been studied by a long list of illustrious mathematical physicists. (For an extensive list of references see [1,2,3]).
The motivation for this research was due to the interest in the shape, stability and evolution of celestial bodies and systems [12]. We now know however that many celestial objects such as galaxies and our solar system exhibit (effectively) "two dimensional structure" [4,5,6,7].
Furthermore recent discoveries are leading us to believe that systems similar to our solar system are "abundant" in the galaxy and their existence might be due to the collapse of a two dimensional interstellar cloud under gravitation (this is the so called the "nebular theory") [8,9,10,11,17]. This data leads us to believe that there is a fundamental physical process which we do not understand fully as yet that leads to the formation of planetary systems throughout the galaxy (and beyond).
This background motivates us to investigate in this paper the steady states and timedependent evolution of a self gravitating and rotating fluid in two dimensions. This problem has been explored by a large number of investigators using elaborate analytic methods and computer simulations which involve, in general, thermodynamic considerations, magnetohydrodynamics modeling and turbulence.(For a complete list of references see [8,9,10,13,16]).
While these are important issues we still need, in our opinion, prototype analytic models that are able to capture the evolution of this process and lead to insights about its possible outcomes.
In this paper we attempt to develop such a model using the basic hydrodynamic equations that govern the time-dependent evolution of an isothermal, incompressible, stratified (i.e non constant density) and rotating fluid in two dimensions under gravity [1,2,3]. (The justification for the reduction from three to two dimensions has been discussed by many authors. A lucid treatment is given in Ref. [12] pp. [1][2][3][4][5][6][7][8][9][10][11][12].Under these assumptions we show that the number of model equations can be reduced from five to a system of two equations for the steady states and three coupled equations for its evolution. The models contain some "parameter functions" which encode information about the asymptotic mass density distribution of the fluid and its momentum. The steady state model was investigated by us previously. [18,19,20] However in this paper we consider a more general model in which the "gas cloud" is rotating also with uniform angular velocity ω and study the impact of this rotation on the matter distribution in the steady state.
To study the predictions of these models we use both analytical and numerical methods to solve their equations under a variety of conditions. In particular we consider radial solutions to these equations which represent the evolution of an interstellar cloud with isothermal equation of state [10].
It might be argued that the hydrodynamic assumptions we are making in this paper are not realistic from astrophysical point of view. However our main goal is to capture analytically, as far as possible, the nonlinear and time dependent aspects of the processes under consideration. Accordingly our results might be useful to provide some analytic insights and guidelines for more elaborate work on this topic.
The plan of the paper is as follows: In Sec 2 we present the basic hydrodynamic equations and show how one can reduce them to a coupled system of three equations. Sec 3 presents further simplifications of these equations. The first is for the steady states of the model. The second is for the time dependent evolution of the gas cloud under the assumption of constant vorticity. In Sec 4 we present analytical and numerical radial solutions of these equations.
We end up Sec 5 with summary and conclusions.

Derivation of the Model Equations
Following the standard convention [1,14,15] where subscripts indicate differentiation with respect to the indicated variable, u = (u, v) is the fluid velocity, ρ is its density, p is the pressure, φ is the gravitational field and G is the gravitational constant. The terms ρω 2 x, ρω 2 y represent the components of the apparent centrifugal force due to the rotation of the gas cloud with angular velocity ω.
We can nondimensionalize these equations by introducing the following scalings where L, U 0 , ρ 0 are some characteristic length,velocity and mass density respectively that characterize the problem at hand. Substituting these scalings in eqs. (2.1)-(2.5) and dropping the tildes these equations remain unchanged (but the quantities that appear in these . (Once again we drop the tilde).
In view of eq. (2.1) we can introduce a stream function ψ so that Using this stream function we can rewrite eq. (2.2) as [13,15] ρ t + J{ρ, ψ} = 0 (2.8) where for any two (smooth) functions f, g Using ψ the momentum equations (2.3),(2.4) become To eliminate p from these equations we differentiate eq. (2.10) and eq. (2.11) with respect to y, x respectively and subtract. This leads to where r 2 = x 2 +y 2 . The sum of the second and fourth terms in this equation can be rewritten as ρ(∇ 2 ψ) t + ρJ{∇ 2 ψ, ψ}. (2.13) To reduce the first and third terms in (2.12) we use (2.8). We obtain combining the results of (2.13) and (2.14) eq. (2.12) becomes Thus we have reduced the original five equations (2.1)-(2.5) to three equations (2.5), (2.8) and (2.15). Although (2.15) is rather cumbersome in general, it can simplified further under some restrictions which are presented in the following section.

Simplification of the Model Equations
Equation (2.15) can be simplified further in two cases. The first is when we consider only steady states of the flow and the second is when the flow vorticity is constant.

A Model for the Steady States
When we consider only steady states of the flow (2.8) implies that ψ = ψ(ρ) and after some algebra [18] (2.15) reduces to and S(ρ) is some function of ρ. Thus the equations governing the steady state are (3.1), (2.5) and H(ρ) and S(ρ) are "parameter functions" which determine the nature of the steady state.

The Physical Meaning of the Functions H(ρ), S(ρ)
The function H(ρ) is a parameter function which is determined by the momentum (and angular momentum) distribution in the fluid. From a practical point of view the choice of this function determines the structure of the steady state density distribution. The corresponding flow field can be computed then aposteriori (that is after solving for ρ) from the following relations. [18] The function S(ρ) that appears in eq.

A model for the Time Evolution
To begin with we consider the case where the vorticity is zero and then generalize to the case where the flow vorticity is constant.
When the flow vorticity ∇ × u is zero then ∇ 2 ψ = 0 eq. (2.15) becomes However when the vorticity is zero we can introduce the velocity potential η which satisfies Replacing ψ by η in (3.6) we obtain Hence The equations of the flow in this case are (which replaces (2.8)), (3.8) and (2.5).
To generalize this reduction to the case where ∇ 2 ψ = a (where a is any constant) we which implies that there exists a function η so that Hence η x = ψ y , η y = −ψ x + ax (3.10) Using these relations to substitute η for ψ in (3.12) leads to It follows then that

Radial Solutions for the Steady State Model
When we consider the special case where in polar coordinates ρ = ρ(r) and φ = φ(r) the system (3.1) and (2.5) with H(ρ) = 1 reduces to To solve this system of equations we let S(ρ) = αρ, solve (4.1) for φ and substitute the result in (4.2). This leads to the following fourth order equation for ρ The general solution of this equation is where J 0 and Y 0 are Bessel functions of the first and second kind of order 0 and Assuming no singularity at the origin we set C 3 = C 4 = 0. To assess the impact of the rotation term on the steady state we solved this system for C 1 , C 2 on a circular disk using

Radial Solutions for the Time Evolution Model
The system (2.5), (3.8) and (3.9) can be simplified further if we use polar coordinates and assume that ρ, η, φ are functions of r and t only. We obtain, where c = 4πG.

Steady States
When we consider a steady state solutions of (5.5) then ρ t = 0 and η t = 0. If follows from the first equation in (5.5) that either ρ r or η r must be zero. In the first case ρ is constant and we can let ρ = 1 without loss of generality. When η r is zero we must have φ = S(ρ) and the second equation in (5.5) becomes where primes denote differentiation with respect to ρ.
We consider these two cases separately.
A. Steady state with ρ = 1 Since ρ = 1 the function S(ρ) is a constant and the general solution for φ is where C 1 , C 2 are arbitrary constants. The equation for η becomes (S can be absorbed in C 2 but we leave it in this form as these two constants have different physical meaning). If we let C 1 = 0 to avoid the singularity at the origin (5.8) yields B. Steady states with η = 1 In this case the solution of (5.6) depends on the nature of the function S(ρ). In general this equation has to be solved numerically. However we present here analytical solutions of this equation for two special cases.
1. S(ρ) = αρ where α is a constant. The solution to (5.6) in this case is It follows then that the nature of the steady state is determined by the ratio c α . A sample of the resulting ρ profiles is presented in F ig. 3. To obtain this figure we considered a pinched disk with ρ(0.01) = 1, ρ ′ (0.01) = −10 and c = α. The resulting steady state has an increase in the material density towards the circumference of the disk. Similar graphs were obtained numerically for S(ρ) = αρ n , n = 2, 3.

S(ρ) = α ln(ρ)
In this case we have Substituting C 1 = C 2 = c = α = 1 we obtain F ig. 4 which might be interpreted as representing a binary system.

Perturbations from the steady state ρ = 1
We consider in this section a disk of radius 1 with a steady state ρ 0 = 1 and S(ρ) = 0.
Letting φ(1) = 0 and using (5.7), (5.9) (with C 1 = 0) this yields the following equations for the steady state For a perturbation from this state, viz.
ρ(t, r) = ρ 0 + ǫρ 1 (t, r), φ(t, r) = φ 0 + ǫφ 1 (t, r), η(t, r) = η 0 + ǫη 1 (t, r) (5.12) we obtain to first order in ǫ the following system of equations: The equation for ρ 1 in (5.13) can be solved analytically. Its general solution is where F is any smooth function of its variable which has to be adjusted to the initial conditions of the perturbation. The second equation in (5.13) is a (reduced) Poisson equation and its general solution can be expressed by quadratures where F 1 (t), F 2 (t) have to be determined by the boundary conditions on φ 1 . Finally one can obtain also an expression for the solution for η in terms of quadratures.
For example if the initial perturbation in ρ is ρ 1 (0, r) = ar where a is a constant then The evaluation of φ 1 using the second equation in (5.13) and (5.16) is straightforward. It should be obvious how one can generalize this example to other expressions for ρ 1 (0, r).
A second approach to the solution of the system (5.13) is to assume exponential dependence in time, viz. ρ 1 = e αt R(r), η 1 = e αt E(r), φ 1 = e αt P (r) (5.17) This ansatz reduces (5.13) to a system of ordinary differential equations As before the equation for R(r) can be solved analytically, while the equations for P (r) and E(r) can be solved by quadratures or numerically.
A numerical approach to the solution for φ 1 and η 1 in (5.13) is also possible.

5.3
Perturbations from the steady state η 0 = 1 As in the previous subsection we consider again a disk of radius 1 and let S(ρ) = αρ. The general steady state solution for ρ is given by (5.9). Assuming no singularities in ρ (ie. no protostar at the origin) we must set C 2 = 0 in this equation. Furthermore since ρ ≥ 0 it follows that we must have −c/α = β where β is the first zero of J 0 . Thus ρ 0 = J 0 (βr), φ 0 = αρ 0 (where we normalized rho 0 at r = 0 to be 1).
For a perturbation from this steady state in the form given by (5.12) we obtain to first order in ǫ the following system of equations: Where J 1 is Bessel function of the first kind of order 1. The evolution of an initial perturbation ρ 1 = exp(−5r) with α = 0.01 from the steady state is plotted in F ig5. This figure shows that as time progresses there is an accumulation of matter near the center of the disk.
At the same time there is an initial separation between the core and the rest of the disk.
We computed also the solution to the system (5.5) with an initial matter distribution ρ(0, r) = 1+sin(2πr) 2 , c = 0.01 and t ∈ [0, 9]. The results of the simulation (Fig. 6) show that as time progresses matter is starting to build up in the vicinity of the center of the disk and around r = 1. At the same time there is a decrease in matter density in between these two points.

Summary and Conclusions
In previous publications [18][19][20] we treated only the steady states of two dimensional self gravitating fluid. In this paper we generalized this model to include disk rotation and assessed the impact of this addition on the distribution of matter in the disk. We were able also to address the time dependent evolution of this fluid under restrictions on its vorticity.
This enabled us to simplify considerably the equations which govern its evolution. While this is a highly idealized model in the context of astrophysical applications it may still provide some analytical insights for more elaborate models.
In this paper we considered only radial solutions of this model. More general solutions which are not radial will have to be explored next.