User:Tohline/Apps/PapaloizouPringle84 - Revision history

Tohline: /* Blaes85 */

2020-07-31T20:48:13Z

Blaes85

← Older revision		Revision as of 20:48, 31 July 2020
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center" bgcolor="orange">		<tr><td align="center" bgcolor="orange">
	Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>		Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>
	Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 © Royal Astronomical Society		Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />© Royal Astronomical Society<br />
			https://doi.org/10.1093/mnras/216.3.553
	</td></tr>		</td></tr>
	<tr>		<tr>
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	<td align="left">		<td align="left">
	~~Permission to post this~~ digital image ~~granted~~ (via email dated 7/19/2020) by ~~author, O~~. ~~Blaes~~.		This digital image has been posted by permission of author, O.M. Blaes (via email dated 19 July 2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020).
			<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Change to Off-Axis Polar-Coordinate System */

2020-07-31T20:44:07Z

Change to Off-Axis Polar-Coordinate System

← Older revision		Revision as of 20:44, 31 July 2020
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	<tr>		<tr>
	<td align="left">		<td align="left">
	This digital image has been posted by permission of author, O.M. Blaes (via email dated 7/19/2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020).		This digital image has been posted by permission of author, O.M. Blaes (via email dated 19 July 2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020).
	<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>		<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>

Tohline: /* Change to Off-Axis Polar-Coordinate System */

2020-07-31T20:37:31Z

Change to Off-Axis Polar-Coordinate System

← Older revision		Revision as of 20:37, 31 July 2020
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	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center" bgcolor="orange">		<tr><td align="center" bgcolor="orange">
	Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>		Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B O. M. Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>
	Monthly Notices of the Royal Astronomical Society, vol. 216, pp. 553-563 © Royal Astronomical Society		Monthly Notices of the Royal Astronomical Society, vol. 216, Issue 3, October 1985, pp. 553-563<br />© Royal Astronomical Society<br />
			https://doi.org/10.1093/mnras/216.3.553
	</td></tr>		</td></tr>
	<tr>		<tr>
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	<td align="left">		<td align="left">
	~~Permission to post this~~ digital image ~~granted~~ (via email dated 7/19/2020) by ~~author, O~~. ~~Blaes~~.		This digital image has been posted by permission of author, O.M. Blaes (via email dated 7/19/2020), and by permission of Oxford University Press on behalf of the Royal Astronomical Society (via email dated 31 July 2020).
			<div align="center">Original publication website: https://academic.oup.com/mnras/article/216/3/553/1005987</div>
	</td>		</td>
	</tr>		</tr>

Tohline: /* GGN86 */

2020-07-19T21:38:56Z

GGN86

← Older revision		Revision as of 21:38, 19 July 2020
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	<div align="center" id="EigenvalueGGN86">		<div align="center" id="EigenvalueGGN86">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equations (2.27) and (2.23) extracted without modification from p. 343 of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman & Narayan (1986)]<p></p>		Equations (2.27) and (2.23) extracted without modification from p. 343 of [http://adsabs.harvard.edu/abs/1986MNRAS.221..339G Goldreich, Goodman & Narayan (1986)]<p></p>
	"''The stability of accretion tori. I - Long-wavelength modes of slender tori''"<p></p>		"''The stability of accretion tori. I - Long-wavelength modes of slender tori''"<p></p>

Tohline: /* Blaes85 */

2020-07-19T21:38:11Z

Blaes85

← Older revision		Revision as of 21:38, 19 July 2020
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	<div align="center" id="EigenvalueBlaes85">		<div align="center" id="EigenvalueBlaes85">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>		Equation (1.6) — identical to Eq. (3.5) — extracted without modification from p. 555 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>

Tohline: /* Change to Off-Axis Polar-Coordinate System */

2020-07-19T21:37:29Z

Change to Off-Axis Polar-Coordinate System

← Older revision		Revision as of 21:37, 19 July 2020
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	<div align="center" id="EigenvalueBlaes85">		<div align="center" id="EigenvalueBlaes85">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>		Equation (3.2) extracted without modification from p. 558 of [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)]<p></p>
	"''Oscillations of Slender Tori''"<p></p>		"''Oscillations of Slender Tori''"<p></p>

Tohline: /* Kojima's Setup */

2020-07-19T21:36:12Z

Kojima's Setup

← Older revision		Revision as of 21:36, 19 July 2020
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	<div align="center" id="EigenvalueKojima86">		<div align="center" id="EigenvalueKojima86">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equations (12) & (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>		Equations (12) & (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>
	"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>		"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>

Tohline: /* Seminal Work by Papaloizou & Pringle */

2020-07-19T21:35:34Z

Seminal Work by Papaloizou & Pringle

← Older revision		Revision as of 21:35, 19 July 2020
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	<div align="center" id="EigenvaluePP84">		<div align="center" id="EigenvaluePP84">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equation (3.18) extracted without modification from p. 726 of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle (1984)]<p></p>		Equation (3.18) extracted without modification from p. 726 of [http://adsabs.harvard.edu/abs/1984MNRAS.208..721P Papaloizou & Pringle (1984)]<p></p>
	"''The dynamical stability of differentially rotating discs with constant specific angular momentum''"<p></p>		"''The dynamical stability of differentially rotating discs with constant specific angular momentum''"<p></p>
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	<div align="center" id="EigenvaluePP85">		<div align="center" id="EigenvaluePP85">
	<table border="1" cellpadding="5" width="80%">		<table border="1" cellpadding="5" width="80%">
	<tr><td align="center">		<tr><td align="center" bgcolor="orange">
	Equation (2.19) extracted without modification from p. 803 of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985)]<p></p>		Equation (2.19) extracted without modification from p. 803 of [http://adsabs.harvard.edu/abs/1985MNRAS.213..799P Papaloizou & Pringle (1985)]<p></p>
	"''The dynamical stability of differentially rotating discs. II''"<p></p>		"''The dynamical stability of differentially rotating discs. II''"<p></p>
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	</table>		</table>
	</div>		</div>


	===Analyses of Configurations with Uniform Specific Angular Momentum===		===Analyses of Configurations with Uniform Specific Angular Momentum===

Tohline: /* Nod to Oort Constants and Simple Rotation Profiles */

2020-07-19T21:33:51Z

Nod to Oort Constants and Simple Rotation Profiles

← Older revision		Revision as of 21:33, 19 July 2020
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	These three expressions are in line with GGN86 equations (2.4), (2.6), and (2.24), respectively.		These three expressions are in line with GGN86 equations (2.4), (2.6), and (2.24), respectively.

			<!--
	{{LSU_WorkInProgress}}		{{LSU_WorkInProgress}}
			-->

	====Velocity Components====		====Velocity Components====

Tohline: /* Kojima's Setup */

2020-07-19T21:32:25Z

Kojima's Setup

← Older revision		Revision as of 21:32, 19 July 2020
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	Equations (12) & (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>		Equations (12) & (13) extracted without modification from p. 254 of [http://adsabs.harvard.edu/abs/1986PThPh..75..251K Kojima (1986)]<p></p>
	"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>		"''The Dynamical Stability of a Fat Disk with Constant Specific Angular Momentum''"<p></p>
	Progress of Theoretical Physics, vol. 75, pp. 251-261 © ~~Royal Astronomical~~ Society		Progress of Theoretical Physics, vol. 75, pp. 251-261 © The Physical Society of Japan
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	This expression also serves as the starting point for the stability analysis presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] — see his equation (3.1), but note that he has replaced the adiabatic exponent with the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>.		This expression also serves as the starting point for the stability analysis presented by [http://adsabs.harvard.edu/abs/1985MNRAS.216..553B Blaes (1985)] — see his equation (3.1), but note that he has replaced the adiabatic exponent with the polytropic index via the relation, <math>~\gamma = (n+1)/n</math>.

	====Equivalent Dimensionless Expression====		====Equivalent Dimensionless Expression====