User:Tohline/ProjectsUnderway/Core Collapse Supernovae - Revision history

Tohline: /* Nonrotating, Spherically Symmetric Collapse */ Add links in opening paragraph to H_Book chapter on free-fall collapse

2014-12-15T02:55:08Z

Nonrotating, Spherically Symmetric Collapse: Add links in opening paragraph to H_Book chapter on free-fall collapse

← Older revision		Revision as of 02:55, 15 December 2014
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	===Nonrotating, Spherically Symmetric Collapse===		===Nonrotating, Spherically Symmetric Collapse===

	When describing the free-fall (pressure-free) collapse from rest of a uniform-density sphere of mass, <math>~M</math>, and initial radius, <math>~r_0</math> — hence, initial density <math>~\rho_0 = 3M/(4\pi r_0^3)</math> — it is convenient to use the Lagrangian radial coordinate, <math>~r(t)</math>, which tracks the radius of the sphere at any time, <math>~t</math>. The relevant equation of motion is (see [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L Lin, Mestel & Shu 1965]),		When describing the [[User:Tohline/SSC/FreeFall#Free-Fall_Collapse\|free-fall (pressure-free) collapse]] from rest of a uniform-density sphere of mass, <math>~M</math>, and initial radius, <math>~r_0</math> — hence, initial density <math>~\rho_0 = 3M/(4\pi r_0^3)</math> — it is convenient to use the Lagrangian radial coordinate, <math>~r(t)</math>, which tracks the radius of the sphere at any time, <math>~t</math>. The relevant equation of motion is (see [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L Lin, Mestel & Shu 1965] or our [[User:Tohline/SSC/FreeFall#Uniform-Density_Sphere\|accompanying discussion]]),
	<div align="center">		<div align="center">
	<table border="0", cellpadding="5">		<table border="0", cellpadding="5">

Tohline: /* Nonrotating, Spherically Symmetric Collapse */ Add link to newly written chapter on the free-fall collapse problem

2014-11-15T00:42:22Z

Nonrotating, Spherically Symmetric Collapse: Add link to newly written chapter on the free-fall collapse problem

← Older revision		Revision as of 00:42, 15 November 2014
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	</table>		</table>
	</div>		</div>
	(Detailed steps through this derivation ~~will be~~ provided ~~elsewhere~~.) A set of dimensionless expressions drawn from this analytic solution that will prove useful to our discussion of gravitational-wave signals from core-collapse supernovae is provided in the following table. All lengths have been normalized to <math>~r_0</math>, and times have been normalized to the free-fall time, <math>~\tau_\mathrm{ff}</math>. In the lower half of the table, these analytic functions have been evaluated at six different times during the free-fall collapse.		(Detailed steps through this derivation, as well as an in-depth discussion of the free-fall collapse problem, are provided [[User:Tohline/SSC/FreeFall#Free-Fall_Collapse\|in an accompanying chapter of this H_Book]].) A set of dimensionless expressions drawn from this analytic solution that will prove useful to our discussion of gravitational-wave signals from core-collapse supernovae is provided in the following table. All lengths have been normalized to <math>~r_0</math>, and times have been normalized to the free-fall time, <math>~\tau_\mathrm{ff}</math>. In the lower half of the table, these analytic functions have been evaluated at six different times during the free-fall collapse.

	<div align="center" id="FreeFallSpherical">		<div align="center" id="FreeFallSpherical">

Tohline: /* Introduction */ Add link to Tohline (1984)

2014-08-31T21:30:10Z

Introduction: Add link to Tohline (1984)

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	==Introduction==		==Introduction==

	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified ~~model~~ of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on early, simplified models of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)] and by [http://adsabs.harvard.edu/abs/1984ApJ...285..721T Tohilne (1984)], our discussion is intended to supplement and complement published works from the past couple of decades that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

	As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst." As early as 1991, [http://adsabs.harvard.edu/abs/1991A%26A...246..417M Mönchmeyer et al. (1991)] identified four phases of a collapse evolution:		As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst." As early as 1991, [http://adsabs.harvard.edu/abs/1991A%26A...246..417M Mönchmeyer et al. (1991)] identified four phases of a collapse evolution:

Tohline: /* Introduction */ Add introductory paragraphs that point back to the work of Monchmeyer et al (1991)

2014-08-31T21:24:52Z

Introduction: Add introductory paragraphs that point back to the work of Monchmeyer et al (1991)

← Older revision		Revision as of 21:24, 31 August 2014
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	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified model of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified model of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

	As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."		As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst." As early as 1991, [http://adsabs.harvard.edu/abs/1991A%26A...246..417M Mönchmeyer et al. (1991)] identified four phases of a collapse evolution:
			* Phase I: ''Contraction and core flattening''. "This phase … is characterized by a growing overall flattening of the density stratification inside and outside [of an] inner core (IC)."
			* Phase II: ''Rapid contraction of the IC''. "Both the IC and the outer core get strongly deformed during this phase. … The collapse of the IC is finally decelerated and stopped … by either the action of centrifugal forces … or the rise of [the effective adiabatic index] <math>~\Gamma</math> …"
			* Phase III: ''Core bounce''. "Depending on the stiffness of the [equation of state] and the amount of rotational and kinetic energy the IC overshoots its (rotational) equilibrium position" then bounces back.
			* Phase IV: ''Shock propagation and post-bounce oscillations''. "… the … IC oscillates [about its equilibrium configuration] with various volume and surface modes. The amplitudes and frequencies of these modes strongly depend on the kinetic energy of the IC at bounce, the stiffness of the [equation of state], and the central and averaged density of the IC …. The IC-oscillations are damped … [on a] time scale [that] strongly depends on the conditions of the shock heated material just outside the IC. After a model dependent number of oscillations, the IC approaches rotational equilibrium …."

			Here, we will develop analytic and semi-analytic models that encapsulate the physical phenomena that give rise to and that define the relevant scales of each of these evolutionary phases, then sew the models together to demonstrate how all of the principal features in the core-collapse gravitational-wave signal can be fairly readily understood. We begin with a discussion of the free-fall collapse of rotationally flattened spheroids.

	==Free-Fall Collapse==		==Free-Fall Collapse==

Tohline: /* A Template for Gravitational Wave Signals from Core-Collapse Supernovae */

2014-08-31T20:53:25Z

A Template for Gravitational Wave Signals from Core-Collapse Supernovae

← Older revision		Revision as of 20:53, 31 August 2014
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	<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.		<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

			==Introduction==

	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified model of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified model of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

Tohline: /* A Template for Gravitational Wave Signals from Core-Collapse Supernovae */ Insert reference to Thuan & Ostriker (1974)

2014-08-31T20:33:50Z

A Template for Gravitational Wave Signals from Core-Collapse Supernovae: Insert reference to Thuan & Ostriker (1974)

← Older revision		Revision as of 20:33, 31 August 2014
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	<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.		<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. ~~This~~ discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. Building on an early, simplified model of nonspherical core collapse presented by [http://adsabs.harvard.edu/abs/1974ApJ...191L.105T Thuan & Ostriker (1974)], our discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

	As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."		As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."

Tohline: /* A Template for Gravitational Wave Signals from Core-Collapse Supernovae */

2014-08-31T20:25:01Z

A Template for Gravitational Wave Signals from Core-Collapse Supernovae

← Older revision		Revision as of 20:25, 31 August 2014
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	<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.		<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement ~~reviews~~ that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement published works that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the the seminal work of [http://adsabs.harvard.edu/abs/1997A&A...320..209Z Zwerger and Mueller (1997)] and reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

	As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."		As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."

Tohline: /* A Template for Gravitational Wave Signals from Core-Collapse Supernovae */ Insert link to "living review" by Fryer & New

2014-08-31T20:15:04Z

A Template for Gravitational Wave Signals from Core-Collapse Supernovae: Insert link to "living review" by Fryer & New

← Older revision		Revision as of 20:15, 31 August 2014
Line 5:		Line 5:
	<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.		<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

	[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement reviews that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf 2009 ~~review by Christian Ott~~] and ~~the~~ [http://arxiv.org/pdf/1202.3256v2.pdf ~~2012 review by~~ Logue et al.].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement reviews that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the reviews by [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)], [http://relativity.livingreviews.org/open?pubNo=lrr-2011-1&page=articlese4.html Fryer and New (2011)] and [http://arxiv.org/pdf/1202.3256v2.pdf Logue et al. (2012)].

	As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."		As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."

Tohline: /* A Template for Gravitational Wave Signals from Core-Collapse Supernovae */ Add Ott (2009) figure 2 and part of its caption

2014-08-31T20:08:20Z

A Template for Gravitational Wave Signals from Core-Collapse Supernovae: Add Ott (2009) figure 2 and part of its caption

← Older revision		Revision as of 20:08, 31 August 2014
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	<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.		<font color="red">29 August 2014</font>: The initial content of this chapter has been drawn from a LaTeX document dated 23 December 2009 that J. E. Tohline created while interacting with Cody Arceneaux (at the time, an LSU undergraduate physics major) and Sarah Caudell (at the time, an LSU physics graduate student). I've decided to build on this project foundation that was laid in the 2008 - 2009 time frame.

	Our principal objective is to help the gravitational-wave community ~~better~~ understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement reviews that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the [http://~~arxiv~~.org/pdf/~~0809.0695v2~~.pdf 2009 review by Christian Ott] and the [http://arxiv.org/pdf/1202.3256v2.pdf 2012 review by Logue et al.].		[[Image:Ott2009Review_Fig2B.png\|300px\|right\|Ott (2009, Classical and Quantum Gravity, 26, 063001)]]Our principal objective is to help the gravitational-wave community more fully understand the underlying physics that is fundamentally responsible for the characteristic features that are expected to arise in the signals that are detected from core-collapse supernovae. This discussion is intended to supplement and complement reviews that have focused on analyzing results from large-scale, multidimensional hydrodynamic (or magneto-hydrodynamic and fully relativistic) models, such as the [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf 2009 review by Christian Ott] and the [http://arxiv.org/pdf/1202.3256v2.pdf 2012 review by Logue et al.].

			As an example, Figure 2 from [http://iopscience.iop.org/0264-9381/26/6/063001/pdf/0264-9381_26_6_063001.pdf Ott (2009)] is reprinted here, on the right. Its published caption reads, in part: "Note the generic shape of the waveforms, exhibiting one pronounced spike at core bounce and a subsequent ring down. Very Rapid precollapse rotation … results in a significant slowdown of core bounce, leading to a lower-amplitude and lower-frequency GW burst."

	==Free-Fall Collapse==		==Free-Fall Collapse==
	===Nod to Lynden-Bell's Early Contributions===		===Nod to Lynden-Bell's Early Contributions===
	[[Image:LyndenBell1964.png\|~~400px~~\|right\|Lynden-Bell (1964, ApJ, 139, 1195)]]I want to begin this section by paying tribute to [http://en.wikipedia.org/wiki/Donald_Lynden-Bell Donald Lynden-Bell] who, in 1962 ([http://dx.doi.org.libezp.lib.lsu.edu/10.1017/S0305004100040767 Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, pp. 709-711]), was the first to appreciate the relatively simple behavior that should be exhibited by the free-fall collapse of a uniformly rotating, uniform-density spheroid. In an article less than two pages in length, Lynden-Bell first noted that the governing dynamical equations (written in cylindrical coordinates) take the form,		[[Image:LyndenBell1964.png\|300px\|right\|Lynden-Bell (1964, ApJ, 139, 1195)]]I want to begin this section by paying tribute to [http://en.wikipedia.org/wiki/Donald_Lynden-Bell Donald Lynden-Bell] who, in 1962 ([http://dx.doi.org.libezp.lib.lsu.edu/10.1017/S0305004100040767 Mathematical Proceedings of the Cambridge Philosophical Society, vol. 58, pp. 709-711]), was the first to appreciate the relatively simple behavior that should be exhibited by the free-fall collapse of a uniformly rotating, uniform-density spheroid. In an article less than two pages in length, Lynden-Bell first noted that the governing dynamical equations (written in cylindrical coordinates) take the form,
	<div align="center">		<div align="center">
	<table border="0", cellpadding="5">		<table border="0", cellpadding="5">
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	</table>		</table>
	</div>		</div>
	Then Lynden-Bell deduced that, (a) "the result of the motion is merely a change of scales"; (b) the collapsing system "remains uniform [in density], and the boundary remains spheroidal"; and (c) "the collapse … will be through a series of [uniformly rotating] spheroids." Two years later in a separate article, [http://adsabs.harvard.edu/abs/1964ApJ...139.1195L Lynden-Bell (1964], ApJ, 139, 1195) presented results from the numerical integration of this governing set of dynamical equations (see ~~§X and~~ Figure 1 ~~of his article~~). The various publications by other authors who also have modeled the free-fall collapse of rotating or nonrotating spheroids in various contexts (see our discussion that follows) have not always acknowledged Lynden-Bell's pioneering analysis of this problem.		Then Lynden-Bell deduced that, (a) "the result of the motion is merely a change of scales"; (b) the collapsing system "remains uniform [in density], and the boundary remains spheroidal"; and (c) "the collapse … will be through a series of [uniformly rotating] spheroids." Two years later in a separate article, [http://adsabs.harvard.edu/abs/1964ApJ...139.1195L Lynden-Bell (1964], ApJ, 139, 1195) presented results from the numerical integration of this governing set of dynamical equations (see his Figure 1, reprinted to the right, here). The various publications by other authors who also have modeled the free-fall collapse of rotating or nonrotating spheroids in various contexts (see our discussion that follows) have not always acknowledged Lynden-Bell's pioneering analysis of this problem.

	===Nonrotating, Spherically Symmetric Collapse===		===Nonrotating, Spherically Symmetric Collapse===

Tohline: /* Nonrotating, Oblate-Spheroidal Collapse */ Change cylindrical R notation to "varpi"

2014-08-31T19:04:43Z

Nonrotating, Oblate-Spheroidal Collapse: Change cylindrical R notation to "varpi"

← Older revision		Revision as of 19:04, 31 August 2014
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	===Nonrotating, Oblate-Spheroidal Collapse===		===Nonrotating, Oblate-Spheroidal Collapse===

	The analogous collapse from rest of a nonrotating, uniform-density, oblate spheroid with equatorial radius, <math>~R_\mathrm{eq}(t)</math>, and polar radius, <math>~Z_p(t)</math>, is governed by the equations,		The analogous collapse from rest of a nonrotating, uniform-density, oblate spheroid with equatorial radius, <math>~\varpi_\mathrm{eq}(t)</math>, and polar radius, <math>~Z_p(t)</math>, is governed by the equations,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d^2 R_\mathrm{eq}}{dt^2}</math>		<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	-~\frac{\partial\Phi}{\partial R} \biggr\|_{R_{eq}} \, ,		-~\frac{\partial\Phi}{\partial \varpi} \biggr\|_{\varpi_{eq}} \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Phi(R,Z)</math>		<math>~\Phi(\varpi,Z)</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\pi G \rho [ A_1(e) R^2 + A_3(e) Z^2] \, .		\pi G \rho [ A_1(e) \varpi^2 + A_3(e) Z^2] \, .
	</math>		</math>
	</td>		</td>
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	</div>		</div>

	We should clarify and emphasize that this expression for the time-dependent gravitational potential has been written in terms of the eccentricity of the spheroid, <math>~e</math>, as measured in the meridional plane. Specifically,		We should clarify and emphasize that this expression for the time-dependent gravitational potential has been written in terms of the (time-varying) eccentricity of the spheroid, <math>~e</math>, as measured in the meridional plane. Specifically,
	<div align="center">		<div align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\biggl( 1 - \frac{Z_p^2}{R_\mathrm{eq}^2} \biggr)^{1/2}		\biggl( 1 - \frac{Z_p^2}{\varpi_\mathrm{eq}^2} \biggr)^{1/2}
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	R_\mathrm{eq} \biggl( 1 - e^2 \biggr)^{1/2} \, ,		\varpi_\mathrm{eq} \biggl( 1 - e^2 \biggr)^{1/2} \, ,
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>		<math>
	~2\pi G \rho \biggl[ \hat{e}_R A_1(e) + \hat{e}_Z A_3(e) Z \biggr] \, ,		~2\pi G \rho \biggl[ \hat{e}_\varpi A_1(e) + \hat{e}_Z A_3(e) Z \biggr] \, ,
	</math>		</math>
	</td>		</td>
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{d^2 R_\mathrm{eq}}{dt^2}</math>		<math>~\frac{d^2 \varpi_\mathrm{eq}}{dt^2}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- 2\pi G \rho A_1(e) R_\mathrm{eq} = - \frac{3}{2} \biggl[ \frac{GM}{R_\mathrm{eq} Z_\mathrm{p}} \biggr] A_1(e) \, ,		- 2\pi G \rho A_1(e) \varpi_\mathrm{eq} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq} Z_\mathrm{p}} \biggr] A_1(e) \, ,
	</math>		</math>
	</td>		</td>
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	<td align="left">		<td align="left">
	<math>~		<math>~
	- 2\pi G \rho A_3(e) Z_\mathrm{p} = - \frac{3}{2} \biggl[ \frac{GM}{R_\mathrm{eq}^2} \biggr] A_3(e) \, ,		- 2\pi G \rho A_3(e) Z_\mathrm{p} = - \frac{3}{2} \biggl[ \frac{GM}{\varpi_\mathrm{eq}^2} \biggr] A_3(e) \, ,
	</math>		</math>
	</td>		</td>
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	<div align="center">		<div align="center">
	<math>		<math>
	~\rho = \frac{3M}{4\pi R_\mathrm{eq}^2 Z_\mathrm{p}} \, .		~\rho = \frac{3M}{4\pi \varpi_\mathrm{eq}^2 Z_\mathrm{p}} \, .
	</math>		</math>
	</div>		</div>