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Line 649:  Line 649:  
\biggl\{ \biggl( \frac{1}{3 \cdot 5}\biggr) (1  e^2)^{1 / 2}  \biggl\{ \biggl( \frac{1}{3 \cdot 5}\biggr) (1  e^2)^{1 / 2}  
+ \pi^{1 / 2} \alpha_t \beta_t^2 \biggl( \alpha_t^2 + \frac{3}{4} \beta_t^2 \biggr) \biggr\}^2  + \pi^{1 / 2} \alpha_t \beta_t^2 \biggl( \alpha_t^2 + \frac{3}{4} \beta_t^2 \biggr) \biggr\}^2  
  \biggl[ \frac{2}{3} (1  e^2)^{1 / 2} + \pi \alpha_t \beta_t^2 \biggr]^{3} (1  e^2)^{1 / 6}  +  \biggl[ \frac{2}{3} (1  e^2)^{1 / 2} + \pi \alpha_t \beta_t^2 \biggr]^{3} (1  e^2)^{1 / 6} \, . 
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  Notice that when the toroidal component is omitted, this expression reduces to,  
+  
+  <table border="0" cellpadding="5" align="center">  
+  
+  <tr>  
+  <td align="right">  
+  <math>~j_c^2</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~\Omega^2  
+  \biggl(\frac{3}{2^5 \pi^4}\biggr)^{1 / 3}  
+  \biggl\{ \biggl( \frac{1}{3 \cdot 5}\biggr) (1  e^2)^{1 / 2}  
+  \biggr\}^2  
+  \biggl[ \frac{2}{3} (1  e^2)^{1 / 2} \biggr]^{3} (1  e^2)^{1 / 6}  
+  </math>  
+  </td>  
+  </tr>  
+  
+  <tr>  
+  <td align="right">  
+   
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~\Omega^2  
+  \biggl(\frac{3^4}{2^{14} \cdot 5^6 \pi^4}\biggr)^{1 / 3}  
+  (1  e^2)^{2 / 3} \, .  
</math>  </math>  
</td>  </td> 
Revision as of 13:39, 23 November 2020
Contents 
Université de Bordeaux (Part 2)
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Exterior Gravitational Potential of Toroids
J. M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 58255838 have published a paper titled, The Exterior Gravitational Potential of Toroids. Here we examine how their work relates to the published work by C.Y. Wong (1973, Annals of Physics, 77, 279), which we have separately discussed in detail.
We discuss this topic in a separate, accompanying chapter.
SpheroidRing Systems
Through a research collaboration at the Université de Bordeaux, B. Basillais & J. M. Huré (2019), MNRAS, 487, 45044509 have published a paper titled, Rigidly Rotating, Incompressible SpheroidRing Systems: New Bifurcations, Critical Rotations, and Degenerate States.
Key References
Here are some relevant publications:
 Hachisu (1986a, ApJS, 61, 479): A Versatile Method for Obtaining Structures of Rapidly Rotating Stars
 Fujisawa & Eriguchi (2014, MNRAS, 438, L61): Prolate stars due to meridional flows
 Huré, Hersant & Nasello (2018, MNRAS, 475, 63): The equilibrium of overpressurized polytropes
 & Eriguchi (1984, Ap&SS, 99, 71): Fission Sequence and Equilibrium Models of [Rigidly] Rotating Polytropes
 Hachisu, Eriguchi & Nomoto (1986b, ApJ, 311, 214): Fate of merging double white dwarfs. II  Numerical method
 Nishida, Eriguchi & Lanza (1992, ApJ, 401, 618): General Relativistic Structure of StarToroid Systems
 Woodward, Sankaran & Tohline (1992 ApJ, 394, 248): Tidal Disruption of a Star by a Massive Disk (The Axisymmetric Roche Problem)
Especially,
 Eriguchi & Hachisu (1983, Prog. Theor. Phys., 69, 1131): Two Kinds of Axially Symmetric Equilibrium Sequences of SelfGravitating and Rotating Incompressible Fluids: TwoRing Sequence and CoreRing Sequence
 Ansorg, Kleinwächter & Meinel (2003, MNRAS, 339, 515): Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids
 Hachisu, Eriguchi & Nomoto (1986a, ApJ, 308, 161): Fate of Merging Double White Dwarfs
Key Figures
Eriguchi & Sugimoto (1981)
Fig. 1 extracted without modification from p. 1873 of Eriguchi & Sugimoto (1981)
"Another Equilibrium Sequence of SelfGravitating and Rotating Incompressible Fluid"
Progress of Theoretical Physics,
vol. 65, pp. 18701875 © Progress of Theoretical Physics 
CAPTION (modified here): The squared angular velocity is plotted against for a segment of the Maclaurin sequence (dashed curve), for the DysonWong sequence (dotted curve), and for the new configurations reported in this 1981 paper by Eriguchi & Sugimoto (solid curve). The "×" mark denotes the neutral point on the Maclaurin sequence against the perturbation. The dotted curve is plotted by using the values which are read from the curve of Fig. 6 of Wong (1974), so it may contain errors to some extent. 
Eriguchi & Hachisu (1983)
Fig. 3 extracted without modification from p. 1134 of Eriguchi & Hachisu (1983)
"Two Kinds of Axially Symmetric Equilibrium Sequences of SelfGravitating and Rotating Incompressible Fluids: TwoRing Sequence and CoreRing Sequence" Progress of Theoretical Physics, vol. 69, pp. 11311136 © Progress of Theoretical Physics 
CAPTION: The angular momentumangular velocity relations. Solid curves represent uniformly rotating equilibrium sequences.
The number and letter R or C attached to a curve denote mass ratio and tworing or corering sequence, respectively. If differential rotation is allowed, the equilibrium sequences may continue to exist as shown by the dashed curves. 
AKM (2003)
Fig. 2 extracted without modification from p. 517 of Ansorg, Kleinwächter & Meinel (2003)
"Uniformly rotating axisymmetric fluid configurations bifurcating from highly flattened Maclaurin spheroids"
MNRAS, vol. 339, pp. 515523 © Royal Astronomical Society 
CAPTION: For the first five axisymmetric sequences, is plotted against the dimensionless squared angular momentum, , using the same normalizations as Eriguchi & Hachisu (1983). Dotted and dashed curves again refer to the Maclaurin sequence and the Dyson approximation respectively. The full circles mark the bifurcation points on the Maclaurin sequence, and the open square the transition configuration of spheroidal to toroidal bodies on the Dyson ring sequence. 
Basillais & Huré (2019)
Fig. 4 extracted without modification from p. 4507 of Basillais & Huré (2019)
"Rigidly rotating, incompressible spheroidring systems: new bifurcations, critical rotations, and degenerate states"
MNRAS, vol. 487, pp. 45044509 © Royal Astronomical Society 
CAPTION: The spheroidring solutions (grey dots) populate the diagram in between the MLS, the highω limit, and the highj limit. The MLS, ORS, Jacobi sequence, Hamburger sequence, and ε_{2}sequence are also shown (plain lines). Points labelled a to f (cross) correspond to equilibria shown in Figure 3; see also Table 1. There is a band of degeneracy rightward to the ORS (green dashed zone). 
The Maclaurin sequence (MLS) is defined by the relations:












Central Object
Assume that the central object is exactly a Maclaurin spheroid. Then from Figure 1 (and Table 1) of our review of equilibrium models along the Maclaurin spheroid sequence, we appreciate that all we have to do is specify the eccentricity, , and is known. For example, if we choose , then from that Table 1, . Other properties of this "central" spheroid — such as its mass, moment of inertia, and angular momentum — are given the the following expressions:





















We note as well that the Keplerian frequency for a massless particle orbiting in the equatorial plane of this central object will be,






So, if we force this orbital frequency to also equal the spinfrequency of the Maclaurin spheroid, the radius of the orbit must be,









For example, when , we have, .
Surrounding Torus
We'll assume that the surrounding 2^{nd} object is a thin torus (1) with the same density as the central object, (2) with a major axis, , which ensures that the torus is spinning with the Keplerian frequency prescribed by the mass of the central object, (3) and with a minor crosssectional radius, . The second of these constraints means that,



The mass of the torus is given by the expression,



where,



Given that is known once the eccentricity of the central Maclaurin spheroid has been selected and, given that the density of the torus must match the density of the central object, the mass of the torus will only depend on the choice of . The maximum allowed value, , is set by ensuring that equatorialplane location of the inner edge of the torus is no smaller than the equatorial radius of the central spheroid, . This means,



So, the maximum torus mass is,



The moment of inertia of the torus is,






Hence, the (square of the) angular momentum of the torus is,






Combined Configuration
Given that, for the chosen Maclaurin spheroid,



and that the total mass of the system is,









the (square of the) dimensionless total angular momentum of the combined system is,


















Notice that when the toroidal component is omitted, this expression reduces to,






© 2014  2020 by Joel E. Tohline 