Massless Polytropic Tori

(aka Palapoizou-Pringle Tori)

In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, Papaloizou & Pringle (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a point-mass potential. The derived structures have analytic prescriptions. Although the tori constructed by PP84 were not self-gravitating — i.e., the tori were massless — it is nevertheless instructive for us to examine how these equilibrium structures were derived.

Governing Relations

As has been derived elsewhere, for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following algebraic expression:

<math>H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>

where <math>C_\mathrm{B}</math> is the Bernoulli constant,

<math>\Phi_\mathrm{eff} \equiv \Phi + \Psi ,</math>

and <math>\Psi</math> is the relevant centrifugal potential. For self-gravitating configurations, this algebraic expression must be satisfied in concert with a self-consistent solution of the Poisson equation, but for the massless PP84 toroidal structures, <math>~\Phi</math> is just the Newtonian potential presented by a point-like object of mass <math>M_\mathrm{pt}</math>, namely,

<math>\Phi = - \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} .</math>

Supplemental Relations

Following PP84, we supplement the above-specified set of governing equations with a polytropic equation of state, as defined in our overview of supplemental relations for time-independent problems. Specifically, we will assume that <math>~\rho</math> is related to <math>~H</math> through the relation,

<math>~H = (n+1)K_\mathrm{n} \rho^{1/n}</math> ,

or we can set,

Also following PP84, we impose a steady-state velocity flow-field that is described by a fluid with uniform specific angular momentum, <math>j_0</math>. Drawing from our table of example Simple rotation profiles, the centrifugal potential that describes this chosen flow-field is given by the expression,

<math> \Psi(\varpi) = \frac{j_0^2}{2\varpi^2} . </math>

Summary

In summary, for this problem the relevant simplified governing relation is,

<math> H = -\Phi_\mathrm{eff} + C_\mathrm{B} = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \Psi + C_\mathrm{B} . </math>

Inserting the two supplemental relations into this governing algebraic expression, we conclude that the polytropic tori studied by PP84 must have structures that satisfy the equation,

<math> (n+1)\frac{P}{\rho} = \frac{GM_\mathrm{pt}}{(\varpi^2 + z^2)^{1/2}} - \frac{j_0^2}{2\varpi^2} + C_\mathrm{B} . </math>

This is identical to Eq. (2.8) of PP84.

As PP84 point out, for these toroidal structures, it is convenient to normalize all lengths to the position in the equatorial plane, <math>\varpi_0</math>, at which a fluid particle with specific angular momentum <math>j_0</math> has an angular orbital frequency, <math>\dot\varphi = j_0/\varpi^2</math>, that matches the Keplerian orbital frequency,

<math> \omega_K \equiv \biggl[ \frac{GM_\mathrm{pt}}{\varpi^3} \biggr]^{1/2} , </math>

that is associated with the point-mass, <math>M_\mathrm{pt}</math>. That is, it is convenient to define the dimensionless coordinates,

<math> \chi \equiv \frac{\varpi}{\varpi_0} ~~~~~\mathrm{and}~~~~~ \zeta \equiv \frac{z}{\varpi_0} , </math>

where,

<math> \varpi_0 \equiv \frac{j_0^2}{GM_\mathrm{pt}} . </math>

Solution

Boundary Conditions

Given that it is a <math>2^\mathrm{nd}</math>-order ODE, a solution of the Lane-Emden equation will require specification of two boundary conditions. Based on our definition of the variable <math>\Theta_H</math>, one obvious boundary condition is to demand that <math>\Theta_H = 1</math> at the center (<math>\xi=0</math>) of the configuration. In astrophysically interesting structures, we also expect the first derivative of many physical variables to go smoothly to zero at the center of the configuration — see, for example, the radial behavior that was derived for <math>~P</math>, <math>~H</math>, and <math>~\Phi</math> in a uniform-density sphere. Hence, we will seek solutions to the Lane-Emden equation where <math>d\Theta_H /d\xi = 0</math> at <math>\xi=0</math> as well.

Related Wikipedia Discussions

Polytrope

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