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Axisymmetric Configurations (Structure — Part II)

Whitworth's (1981) Isothermal Free-Energy Surface
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Equilibrium, axisymmetric structures are obtained by searching for time-independent, steady-state solutions to the identified set of simplified governing equations.

Cylindrical Coordinate Base

We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

Equation of Continuity

\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

The Two Relevant Components of the
Euler Equation

\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]


- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}

\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]


- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]

Adiabatic Form of the
First Law of Thermodynamics

\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

Poisson Equation

\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 .

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

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

\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}



\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr]



\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 \, .

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 ~P and ~\rho.

Spherical Coordinate Base

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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