Editing Apps/OstrikerBodenheimerLyndenBell66 (section)

====Their Equations (3) &amp; (5)====

The two relevant components of the Euler equation that are identified, above, result from imposing a ''steady-state'' condition on the,

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<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,
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<math>~\cancel{\frac{\partial \vec{v}}{\partial t} } + (\vec{v} \cdot \nabla)\vec{v}</math>
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<math>~=</math>
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<math>~
- \frac{1}{\rho} \nabla P - \nabla \Phi \, ,
</math>
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<!-- {{Math/EQ_Euler02}} -->
and adopting a steady-state rotational velocity field in which the angular velocity is either constant or is only a function of the cylindrical-coordinate radius, <math>~\varpi</math>; that is,

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<math>~\vec{v} = \hat{e}_\varphi [v_\varphi]  = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] \, .</math>
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As we have demonstrated in [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|an accompanying discussion]], for any of a number of astrophysically relevant [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple rotation profiles'']] of this form, the [[AxisymmetricConfigurations/PGE#CYLconvectiveOperator|convective operator]] on the left-hand side of this steady-state Euler equation gives (most conveniently written here in a cylindrical-coordinate base),
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<math>~(\vec{v} \cdot \nabla)\vec{v}</math>
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<math>~=</math>
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<math>~-~\hat{e}_\varpi \biggl[\frac{v_\varphi^2}{\varpi} \biggr] = -~\hat{e}_\varpi \biggl[ \varpi {\dot\varphi}^2(\varpi) \biggr] = -~\hat{e}_\varpi \biggl[\frac{j^2(\varpi)}{\varpi^3} \biggr] \, ,</math>
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where, <math>~j \equiv \varpi^2 \dot\varphi</math> is the (radially dependent) specific angular momentum measured relative to the symmetry (rotation) axis.  As we have pointed out in an [[AxisymmetricConfigurations/SolutionStrategies#Axisymmetric_Configurations_.28Solution_Strategies.29|accompanying discussion]], this last expression can be rewritten in terms of the gradient of a scalar (centrifugal) potential; specifically, 
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<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
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<math>~\rightarrow</math>
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<math>~\nabla \Psi \, ,</math>
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if the centrifugal potential is defined such that,
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<math>~\Psi(\varpi)</math>
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<math>~\equiv</math>
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<math>~- \int_0^\varpi \frac{j^2(\varpi^')}{(\varpi^')^3} d\varpi^' \, .</math>
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[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (5)
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(Note that [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66] adopted a sign convention for the centrifugal potential that is the opposite of ours; that is, <math>~\Phi_c = - \Psi</math>.)  Hence, assuming that our intent is to construct a rotationally flattened equilibrium configuration whose rotation profile is of the form, <math>~\vec{v} = \hat{e}_\varphi [\varpi \dot\varphi (\varpi)] </math>, the ''steady-state'' Euler equation can be rewritten as,
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<math>~\frac{1}{\rho} \nabla P</math>
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<math>~=</math>
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<math>~
- \nabla \Phi - \nabla \Psi \, .
</math>
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[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 817, Eq. (3)
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