User:Tohline/AxisymmetricConfigurations/PGE
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Axisymmetric Configurations (Part I)
If the self-gravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of two-dimensional PDEs. Here we accomplish this by,
- Expressing each of the multidimensional spatial operators in cylindrical coordinates (<math>\varpi, \varphi, z</math>) (see, for example, the Wikipedia discussion of vector calculus formulae in cylindrical coordinates) and setting to zero all spatial derivatives that are taken with respect to the angular coordinate <math>\varphi</math>:
Spatial Operators in Cylindrical Coordinates
<math> \nabla f </math>
=
<math> {\hat{e}}_\varpi \biggl[ \frac{\partial f}{\partial\varpi} \biggr] + {\hat{e}}_\varphi \cancel{\biggl[ \frac{1}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} + {\hat{e}}_z \biggl[ \frac{\partial f}{\partial z} \biggr] ; </math>
<math> \nabla^2 f </math>
=
<math> \frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial f}{\partial\varpi} \biggr] + \cancel{\frac{1}{\varpi^2} \frac{\partial^2 f}{\partial\varphi^2}} + \frac{\partial^2 f}{\partial z^2} ; </math>
<math> (\vec{v}\cdot\nabla)f </math>
=
<math> \biggl[ v_\varpi \frac{\partial f}{\partial\varpi} \biggr] + \cancel{\biggl[ \frac{v_\varphi}{\varpi} \frac{\partial f}{\partial\varphi} \biggr]} + \biggl[ v_z \frac{\partial f}{\partial z} \biggr] ; </math>
<math> \nabla \cdot \vec{F} </math>
=
<math> \frac{1}{\varpi} \frac{\partial (\varpi F_\varpi)}{\partial\varpi} + \cancel{\frac{1}{\varpi} \frac{\partial F_\varphi}{\partial\varphi}} + \frac{\partial F_z}{\partial z} ; </math>
- Expressing all vector time-derivatives in cylindrical coordinates:
Vector Time-Derivatives in Cylindrical Coordinates
<math> \frac{d}{dt}\vec{F} </math>
=
<math> {\hat{e}}_\varpi \frac{dF_\varpi}{dt} + F_\varpi \frac{d{\hat{e}}_\varpi}{dt} + {\hat{e}}_\varphi \frac{dF_\varphi}{dt} + F_\varphi \frac{d{\hat{e}}_\varphi}{dt} + {\hat{e}}_z \frac{dF_z}{dt} + F_z \frac{d{\hat{e}}_z}{dt} </math>
=
<math> {\hat{e}}_\varpi \biggl[ \frac{dF_\varpi}{dt} - F_\varphi \dot\varphi \biggr] + {\hat{e}}_\varphi \biggl[ \frac{dF_\varphi}{dt} + F_\varpi \dot\varphi \biggr] + {\hat{e}}_z \frac{dF_z}{dt} ; </math>
<math> \vec{v} = \frac{d\vec{x}}{dt} = \frac{d}{dt}\biggl[ \hat{e}_\varpi \varpi + \hat{e}_z z \biggr] </math>
=
<math> {\hat{e}}_\varpi \biggl[ \dot\varpi \biggr] + {\hat{e}}_\varphi \biggl[ \varpi \dot\varphi \biggr] + {\hat{e}}_z \biggl[ \dot{z} \biggr] . </math>
Governing Equations
Introducing the above expressions into the principal governing equations gives,
Equation of Continuity
<math>\frac{d\rho}{dt} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr]
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math>
Euler Equation
<math>
{\hat{e}}_\varpi \biggl[ \frac{d \dot\varpi}{dt} - \varpi {\dot\varphi}^2 \biggr] + {\hat{e}}_\varphi \biggl[ \frac{d(\varpi\dot\varphi)}{dt} + \dot\varpi \dot\varphi \biggr] + {\hat{e}}_z \biggl[ \frac{d \dot{z}}{dt} \biggr] = -
{\hat{e}}_\varpi \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - {\hat{e}}_z \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>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\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>
Conservation of Specific Angular Momentum
The <math>\hat{e}_\varphi</math> component of the Euler equation leads to a statement of conservation of specific angular momentum, <math>j</math>, as follows.
<math>
\frac{d(\varpi\dot\varphi)}{dt} + \dot\varpi \dot\varphi = \frac{1}{\varpi}\biggl[ \varpi \frac{d(\varpi\dot\varphi)}{dt} + \varpi \dot\varpi \dot\varphi \biggr] =0
</math>
<math>
\Rightarrow ~~~~~ \frac{d(\varpi^2 \dot\varphi)}{dt} = 0
</math>
<math>
\Rightarrow ~~~~~ j(\varpi,z) \equiv \varpi^2 \dot\varphi = \mathrm{constant} ~(\mathrm{i.e.,}~\mathrm{independent~of~time})
</math>
So, for axisymmetric configurations, the <math>\hat{e}_\varpi</math> and <math>\hat{e}_z</math> components of the Euler equation become, respectively,
|
<math> \frac{d \dot\varpi}{dt} - \frac{j^2}{\varpi^3} </math> |
= |
<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] </math> |
|
<math> \frac{d \dot{z}}{dt} </math> |
= |
<math> - \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] </math> |
See Also
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