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Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)

This chapter extends an accompanying chapter titled, Construction Challenges (Pt. 1). The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.

Whitworth's (1981) Isothermal Free-Energy Surface
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Where Are We Headed?

In a separate discussion, we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. (See the yellow-dotted orbits in Figure panels 1a and 1b below). As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,

Off-Center Ellipse



<math>~\biggl(\frac{x'}{x_\mathrm{max}} \biggr)^2 + \biggl(\frac{y' - y_0}{y_\mathrm{max}} \biggr)^2 \, .</math>

As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,



<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>


<math>~y' - y_0</math>


<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>



<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>


<math>~\dot{y}' </math>


<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) publication,