User:Tohline/Apps/PapaloizouPringleTori - Revision history

Tohline: /* Massless Polytropic Tori */ Point to separate discussion of the Blaes (1985) paper.

2016-02-26T20:45:33Z

Massless Polytropic Tori: Point to separate discussion of the Blaes (1985) paper.

← Older revision		Revision as of 20:45, 26 February 2016
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	In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a Newtonian point-mass potential. The derived structures have analytic prescriptions. Although the tori constructed by PP84 were not self-gravitating — ''i.e.'', the tori were ''massless'' — it is nevertheless instructive for us to examine how these equilibrium structures were derived.		In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a Newtonian point-mass potential. The derived structures have analytic prescriptions. Although the tori constructed by PP84 were not self-gravitating — ''i.e.'', the tori were ''massless'' — it is nevertheless instructive for us to examine how these equilibrium structures were derived.

	It is not widely appreciated that the equilibrium structure of Papaloizou-Pringle tori was first derived by [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone & Moncrief (1976), ApJ, 207, 962] in a paper whose primary aim was to analyze the properties of thick disks orbiting around Kerr black holes. In §IV of their paper, Fishbone & Moncrief show what thick-disk structures result ''in the Newtonian limit'' for homentropic systems with uniform specific angular momentum. These toroidal structures are identical to the ones whose stability was analyzed in PP84. After reviewing the PP84 derivation, ~~below,~~ we present the Fishbone & Moncrief (1976) description of these equilibrium systems.		It is not widely appreciated that the equilibrium structure of Papaloizou-Pringle tori was first derived by [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone & Moncrief (1976), ApJ, 207, 962] in a paper whose primary aim was to analyze the properties of thick disks orbiting around Kerr black holes. In §IV of their paper, Fishbone & Moncrief show what thick-disk structures result ''in the Newtonian limit'' for homentropic systems with uniform specific angular momentum. These toroidal structures are identical to the ones whose stability was analyzed in PP84. After reviewing the PP84 derivation, [[#Fishbone_.26_Moncrief_.281976.29\|we present the Fishbone & Moncrief (1976) description]] of these equilibrium systems.

	==Governing Relations==		==Governing Relations==
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	</div>		</div>
	Hence, both the PP84 derivation and the HSCF derivation produce the same result.		Hence, both the PP84 derivation and the HSCF derivation produce the same result.

			==Blaes (1985)==

			In a [[User:Tohline/Appendix/Ramblings/PPTori#Blaes85\|separate chapter]], we review the analytic analysis of unstable nonaxisymmetric eigenmodes in geometrically thin PP tori that has been presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985), MNRAS, 216, 553]. Our review includes an explanation of how the [[#Solution\|equations derived here]] to describe the equilibrium structure of PP Tori are straightforwardly transformed into the equations and notation adopted by Blaes.

Tohline: Tidy up font size of mathematical expressions in-line with the text; and insert "Work In Progress" sign

2016-02-25T21:43:53Z

Tidy up font size of mathematical expressions in-line with the text; and insert "Work In Progress" sign

Show changes

Tohline: Begin review of Fishbone & Moncrief (1976)

2015-12-24T03:33:39Z

Begin review of Fishbone & Moncrief (1976)

← Older revision		Revision as of 03:33, 24 December 2015
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	</div>		</div>
	Hence, both the PP84 derivation and the HSCF derivation produce the same result.		Hence, both the PP84 derivation and the HSCF derivation produce the same result.

			==Fishbone & Moncrief (1976)==
			[http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone & Moncrief (1976), ApJ, 207, 962] derived the identical Papaloizou-Pringle torus structure using a spherical rather than cylindrical coordinate system.

			===Derived Equilibrium Structure ===
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~c^2 \frac{\partial w}{\partial r} \equiv \frac{1}{(nm)} \frac{\partial P}{\partial r}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\partial}{\partial r}\biggl(\frac{GM}{r}\biggr) + \omega^2 r^2 \sin^2\theta \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~c^2 \frac{\partial w}{\partial \theta} \equiv \frac{1}{(nm)} \frac{\partial P}{\partial \theta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\partial}{\partial \theta}\biggl(\frac{GM}{r}\biggr) + \omega^2 r^2 \sin\theta \cos\theta \, .</math>
			</td>
			</tr>
			</table>
			</div>

			As presented in their equation (4.4), imposing the condition of uniform specific angular momentum leads to an angular velocity profile of the form,
			<div align="center">
			<math>~\omega = \frac{\ell}{r^2 \sin^2\theta} \, .</math>
			</div>


			===Comparison===

	=Key References=		=Key References=

Tohline: /* Massless Polytropic Tori */ Point to Fishbone & Moncrief (1976) derivation in the introductory paragraphs

2015-12-24T02:59:37Z

Massless Polytropic Tori: Point to Fishbone & Moncrief (1976) derivation in the introductory paragraphs

← Older revision		Revision as of 02:59, 24 December 2015
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	=Massless Polytropic Tori=		=Massless Polytropic Tori=
	(''aka'' ~~Palapoizou~~-Pringle Tori)		(''aka'' Papaloizou-Pringle Tori)


	[[Image:LSU_Structure_still.gif\|74px\|left]]		[[Image:LSU_Structure_still.gif\|74px\|left]]
	In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a point-mass potential. The derived structures have analytic prescriptions. Although the tori constructed by PP84 were not self-gravitating — ''i.e.'', the tori were ''massless'' — it is nevertheless instructive for us to examine how these equilibrium structures were derived.		In a seminal paper that focused on an analysis of nonaxisymmetric instabilities in accretion disks, [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle] (1984, MNRAS, 208, 721-750; hereafter, PP84) began by constructing equilibrium structures of axisymmetric, polytropic tori that reside in (orbit about) a Newtonian point-mass potential. The derived structures have analytic prescriptions. Although the tori constructed by PP84 were not self-gravitating — ''i.e.'', the tori were ''massless'' — it is nevertheless instructive for us to examine how these equilibrium structures were derived.

			It is not widely appreciated that the equilibrium structure of Papaloizou-Pringle tori was first derived by [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone & Moncrief (1976), ApJ, 207, 962] in a paper whose primary aim was to analyze the properties of thick disks orbiting around Kerr black holes. In §IV of their paper, Fishbone & Moncrief show what thick-disk structures result ''in the Newtonian limit'' for homentropic systems with uniform specific angular momentum. These toroidal structures are identical to the ones whose stability was analyzed in PP84. After reviewing the PP84 derivation, below, we present the Fishbone & Moncrief (1976) description of these equilibrium systems.

	==Governing Relations==		==Governing Relations==
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	As has been derived [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies\|elsewhere]], for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following ''algebraic'' expression:		As has been derived [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies\|elsewhere]], for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following ''algebraic'' expression:
	<div align="center">		<div align="center">
	<math>H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>		<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>
	</div>		</div>
	where <math>C_\mathrm{B}</math> is the Bernoulli constant,		where <math>C_\mathrm{B}</math> is the Bernoulli constant,

Tohline: /* Key References */ Add reference and link to Fishbone and Moncrief (1976)

2015-12-24T02:44:40Z

Key References: Add reference and link to Fishbone and Moncrief (1976)

← Older revision		Revision as of 02:44, 24 December 2015
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	=Key References=		=Key References=

			* [http://adsabs.harvard.edu/abs/1976ApJ...207..962F Fishbone & Moncrief (1976), ApJ, 207, 962] — ''Relativistic fluid disks in orbit around Kerr black holes''
	* [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle (1984), MNRAS, 208, 721] — ''The dynamical stability of differentially rotating discs with constant specific angular momentum''		* [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle (1984), MNRAS, 208, 721] — ''The dynamical stability of differentially rotating discs with constant specific angular momentum''
	* [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985), MNRAS, 213, 799] — ''The dynamical stability of differentially rotating discs.   II.''		* [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985), MNRAS, 213, 799] — ''The dynamical stability of differentially rotating discs.   II.''

Tohline: /* Boundary Conditions */ Add captions to both panels of Figure 1

2015-09-24T15:52:12Z

Boundary Conditions: Add captions to both panels of Figure 1

← Older revision		Revision as of 15:52, 24 September 2015
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	</div>		</div>

	Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero. The meridional <math>~(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the ~~accompanying figure~~. The displayed surfaces correspond to <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (~~gold~~ curve). Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>. Properties of these tori are described in more detail in an accompanying discussion entitled, [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori\|''Toroidal Configurations and Related Coordinate Systems'']].		Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero. The meridional <math>~(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the left panel of Figure 1. The displayed surfaces correspond to <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve). Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>. Properties of these tori are described in more detail in an accompanying discussion entitled, [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori\|''Toroidal Configurations and Related Coordinate Systems'']].

	We should perhaps emphasize that the shape of the surface of each torus, as shown in the ~~accompanying figure~~, is independent of the chosen polytropic index.		We should perhaps emphasize that the shape of the surface of each torus, as shown in the left-hand panel of Figure 1, is independent of the chosen polytropic index.


	<div align="center">		<div align="center">
	<table border="1" cellpadding="0" align="center">		<table border="1" cellpadding="8" align="center" width="80%">
			<tr><th align="center" colspan="2">Figure 1: Papaloizou-Pringle Torus</th></tr>
	<tr><td align="center">		<tr><td align="center">
	[[File:LSU_PP84Tori.jpg\|350px\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]		[[File:LSU_PP84Tori.jpg\|350px\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]
	</td>		</td>
	<td align="center">		<td align="center" bgcolor="#D0FFFF">
	[[File:TorusMovie1.gif\|350px\|Pivoting PP Torus]]		[[File:TorusMovie1.gif\|350px\|Pivoting PP Torus]]
	</td></tr>		</td></tr>
			<tr>
			<td align="left">''Left Panel'' — A meridional cross-section outlining the surface of three separate PP tori: <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (green curve).</td>
			<td align="left">''Right Panel'' — A three-dimensional, cutaway rendering illustrating the geometric structure of one PP torus having <math>~C_\mathrm{B}^' = 0.3239,~</math><math>~x_\mathrm{in} = \chi_{-}/\chi_{+} = 0.2551, ~</math> <math>~x_0 = 0.4065.</math> The separately colored contours illustrate four surfaces of constant enthalpy; the red contour identifies the toroidal surface.</td>
			</tr>
	</table>		</table>
	</div>		</div>

Tohline: /* Boundary Conditions */

2015-09-24T03:24:50Z

Boundary Conditions

← Older revision		Revision as of 03:24, 24 September 2015
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	===Boundary Conditions===		===Boundary Conditions===

	In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>H</math> and <math>\zeta</math> to zero, that is, they are obtained from the roots of the equation,		In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>~H</math> and <math>~\zeta</math> to zero, that is, they are obtained from the roots of the equation,
	<div align="center">		<div align="center">
	<math>		<math>
	\chi^{-2} - 2\chi^{-1} + 2C_\mathrm{B}^' = 0 ,		~\chi^{-2} - 2\chi^{-1} + 2C_\mathrm{B}^' = 0 ,
	</math>		</math>
	</div>		</div>
	that is, the inner <math>(\chi_-)</math> and outer <math>(\chi_+)</math> edges are located at,		that is, the inner <math>~(\chi_-)</math> and outer <math>~(\chi_+)</math> edges are located at,
	<div align="center">		<div align="center">
	<math>		<math>
	\chi_\pm = \frac{1}{1 \mp \sqrt{1-2C_\mathrm{B}^'}} .		~\chi_\pm = \frac{1}{1 \mp \sqrt{1-2C_\mathrm{B}^'}} .
	</math>		</math>
	</div>		</div>
	Clearly, the relative cross-sectional size — or, as defined in PP84, the dimensionless distortion parameter <math>\delta \equiv (\chi_+ + \chi_-)/2 = (2C_\mathrm{B}^')^{-1}</math> — is determined by the choice of the dimensionless Bernoulli constant, which must lie in the range, ~~½~~ <math> \geq C_\mathrm{B}^' \geq 0</math>. For later reference, we will also note that the ratio of the inner edge to the outer edge is,		Clearly, the relative cross-sectional size — or, as defined in PP84, the dimensionless distortion parameter <math>~\delta \equiv (\chi_+ + \chi_-)/2 = (2C_\mathrm{B}^')^{-1}</math> — is determined by the choice of the dimensionless Bernoulli constant, which must lie in the range, <math>~\tfrac{1}{2} \geq C_\mathrm{B}^' \geq 0</math>. For later reference, we will also note that the ratio of the inner edge to the outer edge is,
	<div align="center">		<div align="center">
	<math>		<math>
	\frac{\chi_{-}}{\chi_{+}} = \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} } .		~\frac{\chi_{-}}{\chi_{+}} = \frac{ 1 - \sqrt{1-2C_\mathrm{B}^'} }{1 + \sqrt{1-2C_\mathrm{B}^'} } .
	</math>		</math>
	</div>		</div>

	Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero. The meridional <math>(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the accompanying figure. The displayed surfaces correspond to <math>C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (gold curve). Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>. Properties of these tori are described in more detail in an accompanying discussion entitled, [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori\|''Toroidal Configurations and Related Coordinate Systems'']].		Indeed, more generally, at all vertical locations above and below the equatorial plane, the surface of the torus occurs where the enthalpy goes to zero. The meridional <math>~(\chi,\zeta)</math> cross-section through three such surfaces are displayed in the accompanying figure. The displayed surfaces correspond to <math>~C_\mathrm{B}^' =</math> 0.208 (red curve), 0.276 (blue curve), and 0.373 (gold curve). Carrying forward the above discussion, we note that in all three cases the pressure maximum is located at <math>(\chi,~\zeta) = (1,~0)</math>. Properties of these tori are described in more detail in an accompanying discussion entitled, [[User:Tohline/Appendix/Ramblings/ToroidalCoordinates#Papaloizou-Pringle_Tori\|''Toroidal Configurations and Related Coordinate Systems'']].

	We should perhaps emphasize that the shape of the surface of each torus, as shown in the accompanying figure, is independent of the chosen polytropic index.		We should perhaps emphasize that the shape of the surface of each torus, as shown in the accompanying figure, is independent of the chosen polytropic index.
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	<table border="1" cellpadding="0" align="center">		<table border="1" cellpadding="0" align="center">
	<tr><td align="center">		<tr><td align="center">
	[[File:LSU_PP84Tori.jpg\|~~400px~~\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]		[[File:LSU_PP84Tori.jpg\|350px\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]
	</td>		</td>
	<td align="center">		<td align="center">
	[[File:TorusMovie1.gif\|~~400px~~\|Pivoting PP Torus]]		[[File:TorusMovie1.gif\|350px\|Pivoting PP Torus]]
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Boundary Conditions */ Insert animated gif of PP Torus

2015-09-24T01:28:05Z

Boundary Conditions: Insert animated gif of PP Torus

← Older revision		Revision as of 01:28, 24 September 2015
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	===Boundary Conditions===		===Boundary Conditions===

	~~[[File:LSU_PP84Tori.jpg\|right\|400px\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]~~
	In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>H</math> and <math>\zeta</math> to zero, that is, they are obtained from the roots of the equation,		In the equatorial plane, the locations of the inner and outer edges of the torus are obtained by setting both <math>H</math> and <math>\zeta</math> to zero, that is, they are obtained from the roots of the equation,
	<div align="center">		<div align="center">
Line 160:		Line 159:

	We should perhaps emphasize that the shape of the surface of each torus, as shown in the accompanying figure, is independent of the chosen polytropic index.		We should perhaps emphasize that the shape of the surface of each torus, as shown in the accompanying figure, is independent of the chosen polytropic index.


			<div align="center">
			<table border="1" cellpadding="0" align="center">
			<tr><td align="center">
			[[File:LSU_PP84Tori.jpg\|400px\|Meridional contours of constant <math>C_\mathrm{B}^'</math>.]]
			</td>
			<td align="center">
			[[File:TorusMovie1.gif\|400px\|Pivoting PP Torus]]
			</td></tr>
			</table>
			</div>

	===Model as Described by Kojima===		===Model as Described by Kojima===

Tohline: /* Key References */ More references

2015-09-15T22:22:57Z

Key References: More references

← Older revision		Revision as of 22:22, 15 September 2015
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	* [http://adsabs.harvard.edu/abs/1989ApJ...344..645P Papaloizou & Lin (1989), ApJ, 344, 645] — ''Nonaxisymmetric instabilities in thin self-gravitating rings and disks''		* [http://adsabs.harvard.edu/abs/1989ApJ...344..645P Papaloizou & Lin (1989), ApJ, 344, 645] — ''Nonaxisymmetric instabilities in thin self-gravitating rings and disks''
	* [http://adsabs.harvard.edu/abs/1991MNRAS.248..353P Papaloizou & Savonije (1991), MNRAS, 248, 353] — ''Instabilities in self-gravitating gaseous discs''		* [http://adsabs.harvard.edu/abs/1991MNRAS.248..353P Papaloizou & Savonije (1991), MNRAS, 248, 353] — ''Instabilities in self-gravitating gaseous discs''
			* [http://adsabs.harvard.edu/abs/1995ARA%26A..33..505P Papaloizou & Lin (1995), <font color="red">Annual Reviews of Astronomy & Astrophysics</font>, 33, 505] — ''Theory of Accretion Disks I: Angular Momentum Transport Processes''
	* [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline & Hachisu (1990), ApJ, 361, 394] — ''The breakup of self-gravitating rings, tori, and thick accretion disks''		* [http://adsabs.harvard.edu/abs/1990ApJ...361..394T Tohline & Hachisu (1990), ApJ, 361, 394] — ''The breakup of self-gravitating rings, tori, and thick accretion disks''
	* [http://adsabs.harvard.edu/abs/1994ApJ...420..247W Woodward, Tohline & Hachisu (1994), ApJ, 420, 247] — ''The stability of thick, self-gravitating disks in protostellar systems''		* [http://adsabs.harvard.edu/abs/1994ApJ...420..247W Woodward, Tohline & Hachisu (1994), ApJ, 420, 247] — ''The stability of thick, self-gravitating disks in protostellar systems''
			* [http://adsabs.harvard.edu/abs/1997ApJS..108..471A Andalib, Tohline, & Christodoulou (1997), ApJS, 108, 471] — ''A Survey of the Principal Modes of Nonaxisymmetric Instability in Self-Gravitating Accretion Disk Models''
			* [http://adsabs.harvard.edu/abs/2011Ap%26SS.334....1H Hadley & Imamura (2011), Ap&SS, 334, 1] — ''Nonaxisymmetric instabilities of self-gravitating disks.   I. Toroids''
			* [http://adsabs.harvard.edu/abs/2014Ap%26SS.353..191H Hadley, Fernandez, Imamura, Keever, Tumblin, & Dumas (2014), Ap&SS, 353, 191]— ''Nonaxisymmetric instabilities in self-gravitating disks.   II. Linear and quasi-linear analysis''
			* [http://adsabs.harvard.edu/abs/2015Ap%26SS.359...10H Hadley, Dumas, Imamura, Keever & Tumblin (2015), Ap&SS, 359, 10]— ''Nonaxisymmetric instabilities in self-gravitating disks.   III. Angular momentum transport''


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	* [http://adsabs.harvard.edu/abs/1990MNRAS.245..614L P. J. Luyten (1990), MNRAS, 245, 614] — ''Non-Axisymmetric Dynamical Instability of Differentially Rotating Self-Gravitating Masses''		* [http://adsabs.harvard.edu/abs/1990MNRAS.245..614L P. J. Luyten (1990), MNRAS, 245, 614] — ''Non-Axisymmetric Dynamical Instability of Differentially Rotating Self-Gravitating Masses''
	* [http://adsabs.harvard.edu/abs/1993GApFD..70..235D Bubrulle, Chomaz, Kumar, & Rieutord (1993), Geophysical and Astrophysical Fluid Dynamics, 70, 235] — ''Non linear stability of slender accretion disks by bifurcation method''		* [http://adsabs.harvard.edu/abs/1993GApFD..70..235D Bubrulle, Chomaz, Kumar, & Rieutord (1993), Geophysical and Astrophysical Fluid Dynamics, 70, 235] — ''Non linear stability of slender accretion disks by bifurcation method''
			* [http://adsabs.harvard.edu/abs/1995ChA%26A..19..507L Liu & Wang (1995), ChA&A, 19, 507] — ''A numerical solution of the dynamical instability of a thin accretion torus''
			* [http://adsabs.harvard.edu/abs/2002RSPSA.458..359B Beyer (2002), RSPSA, 458, 359] — ''A framework for perturbations and stability of differentially rotating stars''
			* [http://adsabs.harvard.edu/abs/2004MNRAS.350..927W Watts, Andersson, & Williams (2004), MNRAS, 350, 927] — ''The oscillation and stability of differentially rotating spherical shells: the initial-value problem''
			* [http://adsabs.harvard.edu/abs/2015MNRAS.446..555P Passamonti & Andersson (2015), MNRAS, 446, 555] — ''The intimate relation between the low T/W instability and the corotation point''

	=Related Wikipedia Discussions=		=Related Wikipedia Discussions=

Tohline: /* Key References */ More references

2015-09-15T21:46:27Z

Key References: More references

← Older revision		Revision as of 21:46, 15 September 2015
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	* [http://adsabs.harvard.edu/abs/1987MNRAS.225..695G Goodman, Narayan, & Goldreich (1987), MNRAS, 225, 695] — ''The stability of accretion tori.   II -- Non-linear evolution to discrete planets''		* [http://adsabs.harvard.edu/abs/1987MNRAS.225..695G Goodman, Narayan, & Goldreich (1987), MNRAS, 225, 695] — ''The stability of accretion tori.   II -- Non-linear evolution to discrete planets''
	* [http://adsabs.harvard.edu/abs/1988MNRAS.231...97G Goodman & Narayan (1988), MNRAS, 231, 97] — ''The stability of accretion tori.   III -- The effect of self-gravity''		* [http://adsabs.harvard.edu/abs/1988MNRAS.231...97G Goodman & Narayan (1988), MNRAS, 231, 97] — ''The stability of accretion tori.   III -- The effect of self-gravity''
			* [http://adsabs.harvard.edu/abs/1992ApJ...388..451C Christodoulou & Narayan (1992), ApJ, 388, 451] — ''The stability of accretion tori.   IV. Fission and fragmentation of slender, self-gravitating annuli''
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