User:Tohline/AxisymmetricConfigurations/SolutionStrategies

Axisymmetric Configurations (Structure — Part II)

Equilibrium, axisymmetric structures are obtained by searching for time-independent, steady-state solutions to the identified set of simplified governing equations. We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

Equation of Continuity

<math>\cancel{\frac{\partial\rho}{\partial t}} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] + \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math>

The Two Relevant Components of the
Euler Equation

<math> \cancel{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr] </math>	=	<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} </math>
<math> \cancel{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr] </math>	=	<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math>

Adiabatic Form of the
First Law of Thermodynamics
<math> \biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} + P \biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} + \biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] + \biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0 </math>

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>

The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi)</math>. That is, <math>\dot\varpi = \dot{z} = 0</math> but, in general, <math>\dot\varphi</math> is not zero and can be an arbitrary function of <math>\varpi</math> and <math>z</math>, that is, <math>\dot\varphi = \dot\varphi(\varpi,z)</math>. We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity <math>\dot\varphi(\varpi,z)</math>, or of the specific angular momentum, <math>j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z)</math>.

After setting the radial and vertical velocities to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>4^\mathrm{th}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> & <math>3^\mathrm{rd}</math> (Euler) and <math>5^\mathrm{th}</math> (Poisson) give, respectively,

<math> \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3} </math>	=	0
<math> \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math>	=	0
<math> \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} </math>	=	<math>4\pi G \rho</math>

As has been outlined in our discussion of supplemental relations for time-independent problems, in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between <math>~P</math> and <math>~\rho</math>.

Solution Strategies

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>. Experience has shown, however, that the derived structures tend to be dynamically unstable unless the angular velocity is uniform on cylinders, that is, unless the angular velocity is independent of <math>z</math>. 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)</math>. (We will find that even this simple rotation profile does not guarantee dynamical stability; for example, unstable structures will arise if <math>j</math> is a decreasing function of the radial coordinate, <math>\varpi</math>.)

Simple Rotation and Barotropic EOS

Adopting a simple rotation profile along with 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, as displayed earlier, here the gradient represents a two-dimensional operator 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>

and the effective potential,

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

has been written in terms of a centrifugal potential,

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

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
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
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

Structures

When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where <math>~\rho</math> is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,

<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,

where we have pulled from the Wikipedia discussion of integration and differentiation in spherical coordinates to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,

<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .

We can also state that,

<math>~\frac{dM_r}{dr} = 4\pi r^2 \rho</math>

This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.

Technique 3

As in Technique #2, we replace <math>dP/\rho</math> by d<math>~H</math> in the hydrostatic balance relation, but this time we realize that the resulting expression can be written in the form,

<math>\frac{d}{dr}(H+\Phi) = 0</math> .

This means that, throughout our configuration, the functions <math>~H</math>(<math>~\rho</math>) and <math>~\Phi</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 reduces to the algebraic expression,

<math>H + \Phi = C_\mathrm{B}</math> .

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

<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> ,

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

See Also

• Part I of Axisymmetric Configurations: Simplified Governing Equations

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation