User:Tohline/SSC/FreeFall - Revision history

Tohline: /* Falling from rest at a finite distance … */

2018-03-06T23:57:32Z

Falling from rest at a finite distance …

← Older revision		Revision as of 23:57, 6 March 2018
Line 323:		Line 323:
	</div>		</div>

	In summary, then, the solution, <math>~r(t)</math>, to this simplified but dynamically relevant problem is provided by the following pair of analytically prescribable parametric relations (see also equations 2 & 3 from LMS65, [[User:Tohline/SSC/FreeFall#Free-Fall_Collapse\|reprinted above]]):		<span id="Parametric">In summary,</span> then, the solution, <math>~r(t)</math>, to this simplified but dynamically relevant problem is provided by the following pair of analytically prescribable parametric relations (see also equations 2 & 3 from LMS65, [[User:Tohline/SSC/FreeFall#Free-Fall_Collapse\|reprinted above]]):

	<table border="1" cellpadding="10" align="center">		<table border="1" cellpadding="10" align="center">

Tohline: /* Single Particle in a Point-Mass Potential */

2018-03-02T21:42:12Z

Single Particle in a Point-Mass Potential

← Older revision		Revision as of 21:42, 2 March 2018
Line 128:		Line 128:
	</table>		</table>
	</div>		</div>
	where, as an integration constant, <math>~k</math> is independent of time.		<span id="RoleOfIntegrationConstant">where, as an integration constant, <math>~k</math> is independent of time. </span>

	<table border="1" width="75%" align="center" cellpadding="10">		<table border="1" width="75%" align="center" cellpadding="10">

Tohline: /* Relationship to Relativistic Cosmologies */

2018-02-24T15:17:43Z

Relationship to Relativistic Cosmologies

← Older revision		Revision as of 15:17, 24 February 2018
Line 1,640:		Line 1,640:

	==Relationship to Relativistic Cosmologies==		==Relationship to Relativistic Cosmologies==
			NOTE: [http://www.astro.caltech.edu/~george/ay21/readings/Friemanetal_DE_ARAA.pdf Frieman, Turner & Huterer (2008, ARAA, 46, 385 - 432)] provide an excellent, very readable review of dark matter and dark energy in the context of various cosmologies.

	Einstein's ''General Theory of Relativity'' generates the following "[http://en.wikipedia.org/wiki/Friedmann_equations#Equations Friedmann equations]" describing the (pressure-free) evolution of homogeneous, isotropic cosmologies in terms of the time-dependent scale length of the universe, <math>~\mathcal{R}(t)</math>:		Einstein's ''General Theory of Relativity'' generates the following "[http://en.wikipedia.org/wiki/Friedmann_equations#Equations Friedmann equations]" describing the (pressure-free) evolution of homogeneous, isotropic cosmologies in terms of the time-dependent scale length of the universe, <math>~\mathcal{R}(t)</math>:

Tohline: /* The Collapse of Initially Centrally Condensed Spheres */

2017-08-05T21:47:27Z

The Collapse of Initially Centrally Condensed Spheres

← Older revision		Revision as of 21:47, 5 August 2017
Line 1,867:		Line 1,867:
	</table>		</table>
	</div>		</div>
	where, <math>~{\bar\rho}_i</math> is the mean density of material ''interior'' to each mass shell at the onset of collapse.		where, <math>~{\bar\rho}_i</math> is the mean density of material ''interior to each mass shell at the onset of collapse.''

	The expression for the time-dependent velocity of each mass shell can also be straightforwardly obtained from the [[#Velocity\|above Lagrangian-perspective derivation]] that was presented in the context of an initially uniform-density configuration. We find that, for each individual mass shell,		The expression for the time-dependent velocity of each mass shell can also be straightforwardly obtained from the [[#Velocity\|above Lagrangian-perspective derivation]] that was presented in the context of an initially uniform-density configuration. We find that, for each individual mass shell,

Tohline: /* Accretion Onto a Point-Mass Core */

2017-08-05T21:42:33Z

Accretion Onto a Point-Mass Core

← Older revision		Revision as of 21:42, 5 August 2017
Line 2,915:		Line 2,915:
	====Accretion Onto a Point-Mass Core====		====Accretion Onto a Point-Mass Core====

	The mathematical model that has been ~~developed~~, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero — concomitantly, the central density formally climbs to infinity — while the inward-directed radial velocity remains nonzero. At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>. As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to consider that <math>~\tau_{ffc}</math> marks the point in time when a central "point mass" core forms and to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core. We acknowledge that this point of view is likely to provide additional insight especially if the extended examination is confined to regions well outside the "core." It is with this acknowledgement in mind that the animation sequence on the right-hand side of our composite Figure 4 has been presented (envelope profiles depicted by solid blue curve segments) to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>. Throughout his article, Coughlin (2017) actually asserts that the time of singularity formation is set by the initial ''average'' density of the collapsing cloud, rather than by the cloud's initial ''central'' density. We disagree with this assertion, as the model clearly establishes the ''central'' free-fall time as the point in time at which the singularity forms.		The mathematical model that has been used, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero — concomitantly, the central density formally climbs to infinity — while the inward-directed radial velocity remains nonzero. At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore the effects of pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>.

	As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core. Presumably the above-developed free-fall model can continue be used to estimate how rapidly the core mass grows. Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',		[[File:CommentButton02.png\|right\|100px\|Note from J. E. Tohline: Throughout his article, Coughlin (2017) actually asserts that the time of singularity formation is set by the initial ''average'' density of the collapsing cloud, rather than by the cloud's initial ''central'' density. We disagree with this assertion, as the mathematical model establishes the ''central'' free-fall time as the point in time at which the singularity forms.]]As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to consider that <math>~\tau_{ffc}</math> marks the point in time when a central point mass core forms and to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core. We acknowledge that this point of view can provide additional insight especially if the extended examination is confined to regions well outside the core. It is with this in mind that the animation sequence on the right-hand side of our composite Figure 4 has been extended to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>; at these later times, the accretion envelope profiles are depicted by solid blue curve segments.

			As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core. Presumably, as Coughlin has illustrated, the above-developed free-fall model can continue to be used to estimate how rapidly the core mass grows. Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,929:		Line 2,931:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl[ \biggl(\frac{2}{\pi}\biggr) \frac{t}{\tau_{ffc}} \biggl]^{-1}</math>		<math>~\biggl[ \biggl(\frac{2}{\pi}\biggr) \frac{t}{\tau_{ffc}} \biggl]^{-1} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Accretion Onto a Point-Mass Core */

2017-08-05T16:59:11Z

Accretion Onto a Point-Mass Core

← Older revision		Revision as of 16:59, 5 August 2017
Line 2,915:		Line 2,915:
	====Accretion Onto a Point-Mass Core====		====Accretion Onto a Point-Mass Core====

	The mathematical model that has been developed, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero — concomitantly, the central density formally climbs to infinity — while the inward-directed radial velocity remains nonzero. At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>. As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core~~, as long as we restrict~~ this examination to regions well outside the "core." It is with this acknowledgement in mind that the animation sequence on the right-hand side of our composite Figure 4 has been presented (envelope profiles depicted by solid blue curve segments) to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>.		The mathematical model that has been developed, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero — concomitantly, the central density formally climbs to infinity — while the inward-directed radial velocity remains nonzero. At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>. As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to consider that <math>~\tau_{ffc}</math> marks the point in time when a central "point mass" core forms and to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core. We acknowledge that this point of view is likely to provide additional insight especially if the extended examination is confined to regions well outside the "core." It is with this acknowledgement in mind that the animation sequence on the right-hand side of our composite Figure 4 has been presented (envelope profiles depicted by solid blue curve segments) to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>. Throughout his article, Coughlin (2017) actually asserts that the time of singularity formation is set by the initial ''average'' density of the collapsing cloud, rather than by the cloud's initial ''central'' density. We disagree with this assertion, as the model clearly establishes the ''central'' free-fall time as the point in time at which the singularity forms.

	As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core. Presumably the above-developed free-fall model can continue be used to estimate how rapidly the core mass grows. Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',		As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core. Presumably the above-developed free-fall model can continue be used to estimate how rapidly the core mass grows. Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',

Tohline: /* Accretion Onto a Point-Mass Core */

2017-08-04T21:00:24Z

Accretion Onto a Point-Mass Core

← Older revision		Revision as of 21:00, 4 August 2017
Line 2,914:		Line 2,914:

	====Accretion Onto a Point-Mass Core====		====Accretion Onto a Point-Mass Core====
	Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',
			The mathematical model that has been developed, above, to describe the development of nonlinear structure in configurations that undergo free-fall collapse must break down as <math>~t \rightarrow \tau_{ffc}</math> because the volume enclosing the central-most mass shell shrinks to zero — concomitantly, the central density formally climbs to infinity — while the inward-directed radial velocity remains nonzero. At the very least, in the central region of the collapsing configuration it is physically unreasonable to ignore pressure (and/or general relativity) as <math>~t \rightarrow \tau_{ffc}</math>. As [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] has suggested, however, it may not be unreasonable to continue to use this mathematical model to represent the behavior of the "envelope" material as it continues to free-fall toward the core, as long as we restrict this examination to regions well outside the "core." It is with this acknowledgement in mind that the animation sequence on the right-hand side of our composite Figure 4 has been presented (envelope profiles depicted by solid blue curve segments) to times, <math>~(\pi/2)t/\tau_{ffc} > 0.5\pi</math>.

			As the evolution proceeds past the initial free-fall time, the radius of a larger and larger number of mass shells will shrink to zero and, under the scenario just outlined, must be considered part of the central point-mass core. Presumably the above-developed free-fall model can continue be used to estimate how rapidly the core mass grows. Suppose that, at times <math>~t > \tau_{ffc}</math>, we want to determine the mass of the central core. This reduces to determining at what time <math>~\zeta_i \rightarrow \tfrac{\pi}{2}</math> for each mass shell; or, for a given time, <math>~t/\tau_{ffc}</math>, determining ''for which mass shell'',
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,962:		Line 2,965:
	</div>		</div>

	The core masses that have been determined in this fashion ~~have been marked by the small, black squares in the above figure for~~ times, <math>~t > ~~\pi/2~~</math>.		The core masses that have been determined in this fashion at times, <math>~t/\tau_{ffc} > 1</math>, have been marked by the small, black squares along the vertical axis in the the left panel of our composite Figure 3 and in the top panel of the animation sequence that appears on the right-hand side of our composite Figure 4.

	=See Also=		=See Also=

Tohline: /* Models A, B, & D, with Focus on Case A */

2017-08-04T20:06:12Z

Models A, B, & D, with Focus on Case A

← Older revision		Revision as of 20:06, 4 August 2017
Line 2,869:		Line 2,869:
	</table>		</table>

	These time-evolving mass profiles — interspersed ~~with~~ others from our analysis of Coughlin's "Case A" evolution — have also been displayed as an animation sequence in our composite Figure 4. Actually, they appear as the top segment of two separate animations — one on the left, the other on the right of the figure — along with the time-evolving density profile (middle segment) and the time-evolving velocity profile (bottom segment). In these animated diagrams, the density is scaled to the model's ''intial'' central density, <math>~\rho_c</math>, and, following [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] — see one of the paragraphs following shortly after his equation (27) — the velocity is everywhere, and at all times, normalized to,		These time-evolving mass profiles — interspersed among others from our analysis of Coughlin's "Case A" evolution — have also been displayed as an animation sequence in our composite Figure 4. Actually, they appear as the top segment of two separate animations — one on the left, the other on the right of the figure — along with the time-evolving density profile (middle segment) and the time-evolving velocity profile (bottom segment). In these animated diagrams, the density is scaled to the model's ''intial'' central density, <math>~\rho_c</math>, and, following [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] — see one of the paragraphs following shortly after his equation (27) — the velocity is everywhere, and at all times, normalized to,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,911:		Line 2,911:
	</table>		</table>

	The animation that appears in the left-hand column of our composite Figure 4 loops back and forth between the initial state, <math>~t = 0 \tau_{ffc}</math>, and (nearly) one ''central'' free-fall time, <math>~t = 0.9999\tau_{ffc}</math>. It provides a focus on the behavior of the free-fall collapse and, in particular, illustrates the run-away development of a centrally condensed structure, as [[#Example_Density_Profiles_Examined_by_Tohline_.281982.29\|highlighted earlier in the model evolutions presented by Tohline]]. Each loop of the animation that appears in the right-hand column of composite Figure 4 proceeds ~~monotonically~~ from the initial state, ''through'' the first free-fall time and formation of a point-mass core, and — see the discussion that follows — into a phase where continued accretion of matter onto the core results in a rapid increase of the core's mass.		The animation that appears in the left-hand column of our composite Figure 4 loops back and forth between the initial state, <math>~t = 0 \tau_{ffc}</math>, and (nearly) one ''central'' free-fall time, <math>~t = 0.9999\tau_{ffc}</math>. It provides a focus on the behavior of the free-fall collapse and, in particular, illustrates the run-away development of a centrally condensed structure, as [[#Example_Density_Profiles_Examined_by_Tohline_.281982.29\|highlighted earlier in the model evolutions presented by Tohline]]. Each loop of the animation that appears in the right-hand column of composite Figure 4 also proceeds from the initial state to one initial free-fall time (drawn as solid black curve segments), but then it continues to display (as solid blue curve segments) evolution ''through'' the first free-fall time and formation of a point-mass core, and — see the discussion that follows — into a phase where continued accretion of matter onto the core results in a rapid increase of the core's mass.

	====Accretion Onto a Point-Mass Core====		====Accretion Onto a Point-Mass Core====

Tohline: /* Models A, B, & D, with Focus on Case A */

2017-08-04T19:23:07Z

Models A, B, & D, with Focus on Case A

← Older revision		Revision as of 19:23, 4 August 2017
Line 2,897:		Line 2,897:
	<tr>		<tr>
	<th align="center" colspan="2">		<th align="center" colspan="2">
	Animation:   Radial profiles of (top) mass, (middle) density & (bottom) velocity		Animation of ''Case A'' Evolution:   Radial profiles of (top) mass, (middle) density & (bottom) velocity
	</th>		</th>
	~~</tr>~~
	~~<tr>~~
	~~<td align="center" width="355px">Loops forward & backward between initial state, and one ''central'' free-fall time, <math>~\tau_{ffc}</math></td>~~
	~~<td align="center" width="355px">Shows initial ''central'' free-fall collapse (black curves) followed by accretion flow onto a point-mass central core (blue curves)</td>~~
	</tr>		</tr>
	<tr>		<tr>
Line 2,910:		Line 2,906:
	<tr>		<tr>
	<td align="left" width="710px" colspan="2">		<td align="left" width="710px" colspan="2">
	In ~~the~~ animation sequences, the red circular marker identifies the radial location of the Lagrangian mass shell that was initially located at <math>~\chi = 1</math>; in the animation on the right the red marker disappears after this mass shell has fallen into, and become part of the central core. After one initial ''central'' free-fall (blue curves in animation on the right), a black square marker appears on the vertical axis in the top panel to identify the mass of the central, point-mass core.		In both animation sequences, the red circular marker identifies the radial location of the Lagrangian mass shell that was initially located at <math>~\chi = 1</math>; in the animation on the right the red marker disappears after this mass shell has fallen into, and become part of the central core. After one initial ''central'' free-fall (blue curves in animation on the right), a black square marker appears on the vertical axis in the top panel to identify the mass of the central, point-mass core.
	</td>		</td>
	</tr>		</tr>
	</table>		</table>

	The animation that appears in the left-hand column of our composite Figure 3 loops back and forth between the initial state, <math>~t = 0 \tau_{ffc}</math>, and (nearly) one ''central'' free-fall time, <math>~t = 0.9999\tau_{ffc}</math>. It provides a focus on the behavior of the free-fall collapse and, in particular, illustrates the run-away development of a centrally condensed structure, as [[#Example_Density_Profiles_Examined_by_Tohline_.281982.29\|highlighted earlier in the model evolutions presented by Tohline]]. Each loop of the animation that appears in the right-hand column of composite Figure 3 proceeds monotonically from the initial state, ''through'' the first free-fall time and formation of a point-mass core, and — see the discussion that follows — into a phase where continued accretion of matter onto the core results in a rapid increase of the core's mass.		The animation that appears in the left-hand column of our composite Figure 4 loops back and forth between the initial state, <math>~t = 0 \tau_{ffc}</math>, and (nearly) one ''central'' free-fall time, <math>~t = 0.9999\tau_{ffc}</math>. It provides a focus on the behavior of the free-fall collapse and, in particular, illustrates the run-away development of a centrally condensed structure, as [[#Example_Density_Profiles_Examined_by_Tohline_.281982.29\|highlighted earlier in the model evolutions presented by Tohline]]. Each loop of the animation that appears in the right-hand column of composite Figure 4 proceeds monotonically from the initial state, ''through'' the first free-fall time and formation of a point-mass core, and — see the discussion that follows — into a phase where continued accretion of matter onto the core results in a rapid increase of the core's mass.

	====Accretion Onto a Point-Mass Core====		====Accretion Onto a Point-Mass Core====

Tohline: /* Models A, B, & D, with Focus on Case A */

2017-08-04T19:16:48Z

Models A, B, & D, with Focus on Case A

← Older revision		Revision as of 19:16, 4 August 2017
Line 2,846:		Line 2,846:
	Plugging these functional expressions into the [[#KeyExpressions\|above-defined set of Lagrangian evolution equations]] and following the [[#ModelingSteps\|above-defined set of modeling steps]], we have modeled the early free-fall collapse of these three models. Guided by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] presentation, we will focus our discussion on the evolution of the model whose initial density profile has been labeled, "Case A."		Plugging these functional expressions into the [[#KeyExpressions\|above-defined set of Lagrangian evolution equations]] and following the [[#ModelingSteps\|above-defined set of modeling steps]], we have modeled the early free-fall collapse of these three models. Guided by [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] presentation, we will focus our discussion on the evolution of the model whose initial density profile has been labeled, "Case A."

	The left panel of our composite Figure 3 presents a plot that is largely intended to replicate ~~Figure 3a from~~ [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]. It presents a plot of the model's mass profile — normalized to the total mass — at time <math>~t = 0</math> (dashed black curve) and at the following five evolutionary times:		The right-hand panel of our composite Figure 3 shows how the radial density profile changes over time during the initial free-fall collapse of Coughlin's "Case A" model. This panel also has been included in [[#Figure1\|our composite Figure 1, above]] to facilitate comparison with similar segments of the model evolutions presented by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)]. The left-hand panel of our composite Figure 3 presents a plot that is largely intended to replicate the left-most panel in [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] Figure 3. It presents a plot of the model's mass profile — normalized to the total mass — at time <math>~t = 0</math> (dashed black curve) and at the following five evolutionary times:
	<div align="center">		<div align="center">
	<math>~\biggl(\frac{\pi}{2} \biggr) \frac{t}{\tau_{ffc}} = 0.3\pi, 0.4\pi, 0.5\pi, 0.6\pi, 0.7\pi \, .</math>		<math>~\biggl(\frac{\pi}{2} \biggr) \frac{t}{\tau_{ffc}} = 0.3\pi, 0.4\pi, 0.5\pi, 0.6\pi, 0.7\pi \, .</math>
	</div>		</div>
	These time-evolving mass profiles — interspersed with others from the "Case A" evolution — have also been displayed as an animation sequence in the lower portion of our composite Figure 3. Actually, they appear as the top segment of two separate animations — one on the left, the other on the right of composite Figure 3 — along with the time-evolving density profile (middle segment) and the time-evolving velocity profile (bottom segment). These additional panels present evolutionary information that, for the most part, may be extracted, respectively, from Figure 3b and Figure 3c of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)]; in order to accommodate large dynamic ranges, the density profiles and velocity profiles are presented as log-log plots.

	<table align="center" cellpadding="8" border="1">		<table align="center" cellpadding="8" border="1">
Line 2,864:		Line 2,864:
	<tr>		<tr>
	<td align="left" colspan="2" width="805px">		<td align="left" colspan="2" width="805px">
	''Left:''   Radial mass profile initially (dashed black curve) and at five additional evolution times, as annotated; this plot has been constructed to replicate the left-most panel in Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)].   ''Right:''  Semi-log plot of the radial density profile initially (dashed black curve) and at three additional evolution times, as annotated. This same plot is displayed in composite Figure 1, above, to aid in comparison with models presented by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)]; it also ~~should~~ be compared with the middle panel from Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)], which presents three of the same density profiles, but on a log-log plot.		''Left:''   Radial mass profile initially (dashed black curve) and at five additional evolution times, as annotated; this plot has been constructed to replicate the left-most panel in Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)].   ''Right:''  Semi-log plot of the radial density profile initially (dashed black curve) and at three additional evolution times, as annotated. This same plot is displayed in [[#Figure1\|composite Figure 1, above]], to aid in comparison with models presented by [http://adsabs.harvard.edu/abs/1982FCPh....8....1T Tohline (1982)]; it also can appropriately be compared with the middle panel from Figure 3 of [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)], which presents three of the same density profiles, but on a log-log plot.
			</td>
			</tr>
			</table>

			These time-evolving mass profiles — interspersed with others from our analysis of Coughlin's "Case A" evolution — have also been displayed as an animation sequence in our composite Figure 4. Actually, they appear as the top segment of two separate animations — one on the left, the other on the right of the figure — along with the time-evolving density profile (middle segment) and the time-evolving velocity profile (bottom segment). In these animated diagrams, the density is scaled to the model's ''intial'' central density, <math>~\rho_c</math>, and, following [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin (2017)] — see one of the paragraphs following shortly after his equation (27) — the velocity is everywhere, and at all times, normalized to,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~v_C</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{2GM_C}{R_C} \biggr]^{1 / 2} = a\biggl(\frac{\pi}{2}\biggr) \frac{R_C}{\tau_{ffc}} \, .
			</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			</div>

			These additional "density profile" and "velocity profile" panels present evolutionary information that, for the most part, may be extracted, respectively, from the middle panel and the right-most panel in [http://adsabs.harvard.edu/abs/2017ApJ...835...40C Coughlin's (2017)] Figure 3; in order to accommodate large dynamic ranges, the density profiles and velocity profiles are presented as log-log plots.

	<table align="center" cellpadding="8" border="1">		<table align="center" cellpadding="8" border="1">