User:Tohline/AxisymmetricConfigurations/SolutionStrategies
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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>\cancelto{0}{\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
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<math>~ \cancelto{0}{\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>~=</math> |
<math>~ - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3} </math> |
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<math>~ \cancelto{0}{\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>~=</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,
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<math>~ \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3} </math> |
<math>~=</math> |
<math>~0</math> |
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<math>~ \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> |
<math>~=</math> |
<math>~0</math> |
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<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>~=</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 Strategy
Simple Rotation Profile and Centrifugal Potential
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>. [We will find that even this simple rotation profile does not guarantee dynamical stability; for example, Lord Rayleigh (1917, Proc. R. Soc. Lond. A, 93, 148-154) showed that unstable structures will arise if <math>~j</math> is a decreasing function of the radial coordinate, <math>~\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.
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Simple Rotation Profiles |
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<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. |
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Power-law |
<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 |
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Uniform rotation |
<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 |
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Uniform <math>v_\varphi</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 |
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Keplerian |
<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 |
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Uniform specific |
<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 |
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j-constant |
<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 |
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aHachisu, I. 1986, ApJS, 61, 479-507
(especially §II.c) |
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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>.
See Also
- Part I of Axisymmetric Configurations: Simplified Governing Equations
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© 2014 - 2021 by Joel E. Tohline |