Axisymmetric Configurations (Solution Strategies)

Lagrangian versus Eulerian Representation

In our overarching specification of the set of Principle Governing Equations, we have included a,

Lagrangian Representation
of the Euler Equation,

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

[BLRY07], p. 13, Eq. (1.55)

When seeking a solution to the set of governing equations that describes a steady-state equilibrium configuration — as has already been suggested in our accompanying discussion of "other forms of the Euler equation" — it is preferable to start from an,

Eulerian Representation
of the Euler Equation,

<math>~\frac{\partial\vec{v}}{\partial t} + (\vec{v}\cdot \nabla) \vec{v}= - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

because steady-state configurations are identified by setting the partial time derivative, rather than the total time derivative, to zero.

Simple Rotation Profile and Centrifugal Potential

"… A necessary and sufficient condition for <math>~\dot{\varphi}</math> … to be independent of <math>~z</math> is that the surfaces of constant pressure coincide with the surfaces of constant density, i.e., that P be a function of ρ only." In this case, a centrifugal potential, <math>~\Psi</math>, can be defined — see the integral expression provided below — and it "is also a function of <math>~\rho</math> only … When <math>~\Psi</math> exists, the equations of state and of energy conservation may be thought of as determining the form of the P-ρ relationship. Hence, by prescribing a P-ρ relationship, one avoids the complications of those further equations. This effects a major simplification of the formal problem of constructing rotating configurations. This procedure will, of course, be inadequate for certain objectives …"

— Drawn from N. R. Lebovitz (1967), ARAA, 5, 465

Specifying <math>~\dot\varphi(\varpi)</math> in the Equilibrium Configuration

Equilibrium axisymmetric structures — that is, solutions to the above set of simplified governing equations — can be found for specified angular momentum distributions that display a wide range of variations across both of the spatial coordinates, <math>~\varpi</math> and <math>~z</math>. According to the Poincaré-Wavre theorem, however, the derived structures will be dynamically unstable toward the development shape-distorting, meridional-plane motions unless the angular velocity is uniform on cylinders, that is, unless the angular velocity is independent of <math>~z</math>. (See the detailed discussion by [T78] — or our accompanying, brief summary — of this and other "axisymmetric instabilities to avoid.") With this in mind, we will focus here on a solution strategy that is designed to construct structures with a

Simple Rotation Profile

<math>\dot\varphi(\varpi,z) = \dot\varphi(\varpi) ,</math>

which of course means that we will only be examining axisymmetric structures with specific angular momentum distributions of the form <math>~j(\varpi,z) = j(\varpi) = \varpi^2 \dot\varphi(\varpi)</math>.

After adopting a simple rotation profile, it becomes useful to define an effective potential,

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

that is written in terms of a centrifugal potential,

<math> \Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi ~. </math>

The accompanying table provides analytic expressions for <math>\Psi(\varpi)</math> that correspond to various prescribed functional forms for <math>\dot\varphi(\varpi)</math> or <math>j(\varpi)</math>, along with citations to published articles in which equilibrium axisymmetric structures have been constructed using the various tabulated simple rotation profile prescriptions.

Simple Rotation Profiles Found in the Published Literature
	<math>~\dot\varphi(\varpi)</math>	<math>~v_\varphi(\varpi)</math>	<math>~j(\varpi)</math>	<math>~\frac{j^2}{\varpi^3}</math>	<math>~\Psi(\varpi)</math>	Refs.
Power-law (any <math>~q \neq 1</math>)	<math>~\frac{j_0}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-2)}</math>	<math>~\frac{j_0}{\varpi_0} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-1)}</math>	<math>~j_0\biggl( \frac{\varpi}{\varpi_0} \biggr)^{q}</math>	<math>~\frac{j_0^2}{\varpi_0^3} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(2q-3)}</math>	<math>~- \frac{1}{2(q-1)} \biggl[ \frac{j_0^2}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{2(q-1)} \biggr]</math>	d, h
Uniform rotation <math>~(q = 2)</math>	<math>~\omega_0</math>	<math>~\varpi \omega_0</math>	<math>~\varpi^2 \omega_0</math>	<math>~\varpi \omega_0^2</math>	<math>~- \frac{1}{2} \varpi^2 \omega_0^2</math>	a, f
Uniform <math>v_\varphi</math> <math>~(q = 1)</math>	<math>~\frac{v_0}{\varpi}</math>	<math>~v_0</math>	<math>~\varpi v_0</math>	<math>~\frac{v_0^2}{\varpi}</math>	<math> ~- v_0^2 \ln\biggl( \frac{\varpi}{\varpi_0} \biggr)</math>	e
Keplerian <math>~(q = 1/2)</math>	<math>~\omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-3/2}</math>	<math>~\varpi_0 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-1/2}</math>	<math>~\varpi_0^2 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{1/2}</math>	<math>~\varpi_0 \omega_K^2 \biggl( \frac{\varpi}{\varpi_0} \biggr)^{-2}</math>	<math>~+ \frac{\varpi_0^3 \omega_K^2}{\varpi} </math>	d
Uniform specific angular momentum <math>~(q = 0)</math>	<math>~\frac{j_0}{\varpi^2}</math>	<math>~\frac{j_0}{\varpi}</math>	<math>~j_0</math>	<math>~\frac{j_0^2}{\varpi^3}</math>	<math>~+ \frac{1}{2} \biggl[ \frac{j_0^2}{\varpi^2} \biggr]</math>	c,g
j-constant rotation	<math>~\omega_c \biggl[ \frac{A^2}{A^2 + \varpi^2} \biggr]</math>	<math>~\omega_c \biggl[ \frac{A^2 \varpi}{A^2 + \varpi^2} \biggr]</math>	<math>~\omega_c \biggl[ \frac{A^2 \varpi^2}{A^2 + \varpi^2} \biggr]</math>	<math>~\omega_c^2 \biggl[ \frac{A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]</math>	<math>~+ \frac{1}{2} \biggl[ \frac{\omega_c^2 A^4}{A^2 + \varpi^2} \biggr]</math>	a,b
^aHachisu, I. 1986, ApJS, 61, 479-507 (especially §II.c) ^bOu, S. & Tohline, J.E. 2006, ApJ, 651, 1068-1078 (especially §2.1) ^cWoodward, J.W., Tohline, J.E. & Hachisu, I. 1994, ApJ, 420, 247-267 ^dTohline, J.E. & Hachisu, I. 1990, ApJ, 361, 394-407 ^eHayashi, C., Narita, S. & Miyama, S.M. 1982, Progress of Theoretical Physics, 68, 1949-1966 ^fMaclaurin, C. 1742, A Treatise of Fluxions ^gPapaloizou, J.C.B. & Pringle, J.E. 1984, MNRAS, 208, 721-750 ^hThe Hadley & Imamura collaboration (Note that, as detailed elsewhere, their definition of the power-law index, <math>~q</math>, is different from ours.)

Note that, while adopting a simple rotation profile is necessary in order to ensure that an axisymmetric, barotropic equilibrium configuration is dynamical stability, it is not a sufficient condition. For example, the Solberg/Rayleigh criterion further demands that, for homentropic systems, the specific angular momentum, <math>~j</math>, must be an increasing function of the radial coordinate, <math>~\varpi</math>. It is not surprising, therefore, that the above table of example simple rotation profiles does not include references to published investigations in which the power-law index, <math>~q</math>, is negative.

Prescribing <math>~\dot\varphi(m_\varpi)</math> Based on an Initially Non-Equilibrium Spherical Configuration

Each of the simple rotation profiles listed in Table 1 has been defined by specifying the radial distribution of the specific angular momentum, <math>~j(\varpi)</math>, in the rotationally flattened equilibrium configuration. Here we follow the lead of Stoeckly's (1965) and of Bodenheimer & Ostriker (1973) and, instead, present rotation profiles that are defined by specifying the function, <math>~j(m_\varpi)</math>, where <math>~m_\varpi</math> is a function describing how the fractional mass enclosed inside <math>~\varpi</math> varies with <math>~\varpi</math>.

To better clarify what is meant by the function, <math>~m_\varpi</math>, consider a configuration (not necessarily in equilibrium) that is spherically symmetric and that exhibits an — as yet unspecified — mass-density profile, <math>~\rho(r)</math>. The mass enclosed within each spherical radius is,

and, if the radius of the configuration is <math>~R</math>, then the configuration's total mass is,

In contrast, the mass enclosed within each cylindrical radius, <math>~\varpi</math>, is

<math>~M_\varpi = 2\pi \int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} \rho( r ) 2dz \, ,</math>

where it is understood that the argument of the density function is, <math>~r = \sqrt{\varpi^2 + z^2} </math>.

Example #1: If the configuration has a uniform density, <math>~\rho_0</math>, then its total mass is, <math>~M = 4\pi \rho_0 R^3/3</math>, and

<math>~M_\varpi</math>	<math>~=</math>	<math>~ 4\pi \rho_0 \int_0^\varpi \varpi [R^2 - \varpi^2]^{1 / 2}d\varpi </math>
	<math>~=</math>	<math>~ \frac{4\pi}{3} \rho_0 \biggl[R^3 - (R^2 - \varpi^2)^{3 / 2} \biggr] </math>
	<math>~=</math>	<math>~M - \frac{4\pi}{3} \rho_0 \biggl[(R^2 - \varpi^2)^{3 / 2} \biggr] </math>
<math>~\Rightarrow ~~~m_\varpi \equiv \frac{M_\varpi}{M}</math>	<math>~=</math>	<math>~1 - \biggl[1 - \frac{\varpi^2}{R^2}\biggr]^{3 / 2} \, . </math>

Example #2: If the spherically symmetric configuration has a density profile given by the function,

<math>~\rho_0 \biggl[\frac{\sin (\pi r/R)}{\pi r/R} \biggr] \, ,</math>

then its total mass is, <math>~M = 4 \rho_0 R^3/\pi</math>, and

<math>~M_\varpi</math>	<math>~=</math>	<math>~ 4\pi \rho_0\int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} \biggl\{ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R} \biggr\} dz </math>
	<math>~=</math>	<math>~ 4 \rho_0 R^3\int_0^\chi \chi d\chi \int_0^{\sqrt{1 - \chi^2}} \biggl\{ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2} )}{\sqrt{\chi^2 + \zeta^2}} \biggr\} d\zeta </math>

<math>~M_\varpi</math>	<math>~=</math>	<math>~ 4\pi \rho_0 \biggl\{ \int_{\sqrt{R^2 - \varpi^2}}^R dz \int_0^\sqrt{R^2-z^2} \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R} \biggr] \varpi d\varpi + \int_0^{\sqrt{R^2 - \varpi^2}} dz \int_0^\varpi \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R} \biggr] \varpi d\varpi \biggr\} </math>
	<math>~=</math>	<math>~ 4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta \int_0^\sqrt{1-\zeta^2} \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}} \biggr] \chi d\chi + \int_0^{\sqrt{1 - \chi^2}} d\zeta \int_0^\chi \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}} \biggr] \chi d\chi \biggr\} </math>
	<math>~=</math>	<math>~ 4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 \biggl[ - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi} \biggr]_0^\sqrt{1-\zeta^2} d\zeta + \int_0^{\sqrt{1 - \chi^2}} \biggl[ - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi} \biggr]_0^\chi d\zeta \biggr\} </math>
	<math>~=</math>	<math>~ \frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 \biggl[ - \cos(\pi) + \cos(\pi\zeta) \biggr] d\zeta + \int_0^{\sqrt{1 - \chi^2}} \biggl[ \cos(\pi\zeta ) - \cos(\pi\sqrt{\zeta^2 + \chi^2}) \biggr] d\zeta \biggr\} </math>
	<math>~=</math>	<math>~ \frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta + \int_0^1 \cos(\pi\zeta) d\zeta - \int_0^{\sqrt{1 - \chi^2}} \cos(\pi\sqrt{\zeta^2 + \chi^2}) d\zeta \biggr\} </math>
	<math>~=</math>	<math>~ \frac{4 \rho_0 R^3}{\pi} \biggl\{ \biggl[ z \biggr]_{\sqrt{1 - \chi^2}}^1 + \frac{1}{\pi} \int_0^\pi \cos(u) du - \int_0^{\sqrt{1 - \chi^2}} \cos(\pi\sqrt{\zeta^2 + \chi^2}) d\zeta \biggr\} </math>

Uniform-Density Initially (n' = 0)

Drawing directly from §IIc of Stoeckly's (1965) work, … consider a large, gaseous mass, initially a homogeneous sphere of mass <math>~M</math> and angular momentum <math>~J</math> rotating as a solid body, and suppose it contracts in such a way that cylindrical surfaces remain cylindrical and each such surface retains its angular momentum. Let <math>~\rho_0</math>, <math>~R_0</math>, and <math>~\dot\varphi_0</math> denote the initial density, radius, and angular velocity of the [initially unstable configuration], <math>~\varpi_0(\varpi)</math> the initial radius of the surface now at radius <math>~\varpi</math>, and <math>~M_\varpi(\varpi)</math> the mass inside this surface. The conditions on the contraction are then

<math>~M - M_\varpi(\varpi)</math>

<math>~ 4\pi \rho_0 \int_{\varpi_0(\varpi)}^{R_0} \biggl[ \biggl(R_0^2 - (\varpi_0^')^2\biggr) \biggr]^{1 / 2} \varpi_0^' d\varpi_0^' \, , </math>

and

<math>~\dot\varphi(\varpi) \varpi^2</math>

<math>~\dot\varphi_0 [\varpi_0(\varpi)]^2 \, .</math>

By integrating, eliminating <math>~\varpi_0(\varpi)</math> between these equations, and eliminating <math>~\rho_0</math>, <math>~R_0</math>, and <math>~\dot\varphi_0</math> in favor of <math>~M</math> and <math>~J</math>, one finds the relation of <math>~\dot\varphi(\varpi)</math> to the mass distribution to be

<math>~\dot\varphi(\varpi)</math>

<math>~ \frac{5J}{2M\varpi^2}\biggl[ 1 - \biggl(1 - \frac{M_\varpi(\varpi)}{M}\biggr)^{2 / 3} \biggr] \, . </math>

This is equation (12) of Stoeckly (1965); it also appears, for example, as equation (45) in Ostriker & Mark (1968), as equation (12) in Bodenheimer & Ostriker (1970), and in the sentence that follows equation (3) in Bodenheimer & Ostriker (1973). As Stoeckly points out, the angular momentum distribution implied by this functional form of <math>~\dot\varphi</math> satisfies the Solberg/Rayleigh stability criterion — that is,

<math>~\frac{dj^2}{d\varpi} > 0 </math>

— initially, and also in the final equilibrium configuration because every cylindrical surface conserves specific angular momentum and the surfaces do not reorder themselves.

Centrally Condensed Initially (n' > 0)

Here, following Bodenheimer & Ostriker (1973), we introduce an approach to specifying a wider range of physically reasonable angular momentum distributions; text that appears in an dark green font has been taken verbatim from this foundational paper.

Technique

To solve the above-specified set of simplified governing equations we will essentially adopt Technique 3 as presented in our construction of spherically symmetric configurations. Using a barotropic equation of state — in which case <math>~dP/\rho</math> can be replaced by <math>~dH</math> — we can combine the two components of the Euler equation shown above back into a single vector equation of the form,

<math> \nabla \biggl[ H + \Phi_\mathrm{eff} \biggr] = 0 , </math>

where it is understood that here, as displayed earlier, the gradient represents a two-dimensional operator written in cylindrical coordinates that is appropriate for axisymmetric configurations, namely,

<math> \nabla f = {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] + {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] \, . </math>

This means that, throughout our configuration, the functions <math>~H</math>(<math>~\rho</math>) and <math>~\Phi_\mathrm{eff}</math>(<math>~\rho</math>) must sum to a constant value, call it <math>~C_\mathrm{B}</math>. That is to say, the statement of hydrostatic balance for axisymmetric configurations reduces to the algebraic expression,

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

This relation must be solved in conjunction with the Poisson equation,

<math>~ \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho , </math>

giving us two equations (one algebraic and the other a two-dimensional <math>2^\mathrm{nd}</math>-order elliptic PDE) that relate the three unknown functions, <math>~H</math>, <math>~\rho</math>, and <math>~\Phi</math>.

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