SSC/Dynamics/FreeFall - Revision history

Jet53man: /* Free-Fall */

2021-10-29T21:12:42Z

Free-Fall

← Older revision		Revision as of 17:12, 29 October 2021
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	In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.		In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.

	To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by {{ Hunter62full }} titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by ~~[http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)]~~ that discussed a broad range of theoretical problems related to star formation, and it appeared ~~again~~ in the introductory paragraph of a classic paper by {{ LMS65full }} titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by {{ LMS65 }} — hereafter, {{ LMS65hereafter }} — was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.		To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by {{ Hunter62full }} titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by {{ Mestel65full }} that discussed a broad range of theoretical problems related to star formation; and it appeared in the introductory paragraph of a classic paper by {{ LMS65full }} titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by {{ LMS65 }} — hereafter, {{ LMS65hereafter }} — was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.

	<table border="1" align="center" cellpadding="8" width="80%">		<table border="1" align="center" cellpadding="8" width="80%">
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	</table>		</table>
	</div>		</div>
	We see that, by shifting the defined zero-point in time in this last expression such that <math>~t \rightarrow (\tau_\mathrm{ff} - t)</math>, which also reverses the ''sign'' on time, we have exact agreement between the solution that we have derived — designed to match the one published by ~~Lin, Mestel, & Shu in 1965~~ — and the result for <math>~k > 0</math> that was published by McCrea & Milne in 1934.		We see that, by shifting the defined zero-point in time in this last expression such that <math>~t \rightarrow (\tau_\mathrm{ff} - t)</math>, which also reverses the ''sign'' on time, we have exact agreement between the solution that we have derived — designed to match the one published by {{ LMS65 }} — and the result for <math>k > 0</math> that was published by McCrea & Milne in 1934.

	====Falling from rest at infinity …====		====Falling from rest at infinity …====

Jet53man: /* Free-Fall */

2021-10-29T20:50:23Z

Free-Fall

← Older revision		Revision as of 16:50, 29 October 2021
Line 71:		Line 71:
	</table>		</table>

	As is briefly discussed in a [[#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by {{ Hunter62 }} also appears in discussions of [[#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[#McCreaMilne1934Solution\|detailed below]], the parametric <math>~r(t)</math> solution presented as equations (2) & (3) in LMS65 appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.		As is briefly discussed in a [[#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by {{ Hunter62 }} also appears in discussions of [[#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[#McCreaMilne1934Solution\|detailed below]], the parametric <math>r(t)</math> solution presented as equations (2) & (3) in {{ LMS65 }} appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.

	In the subsections that follow, we show how equation (1) of LMS65 arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.		In the subsections that follow, we show how equation (1) of {{ LMS65 }} arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.

	==Assembling the Key Relations==		==Assembling the Key Relations==

Jet53man: /* Free-Fall */

2021-10-29T20:46:18Z

Free-Fall

@@ Line 18: / Line 18: @@
 [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965) ''The Astrophysical Journal'', 142, 143]
 -->
-Reprint of the opening (introductory) paragraph of &hellip;<br />
+Reprint of the opening (introductory) paragraph from &hellip;<br />
 {{ LMS65figure }}
    </td>
@@ Line 25: / Line 25: @@
    <td align="left" colspan="1">
 <!-- [[File:LinMestelShu1965Intro.png|600px|Lin, Mestel &amp; Shu (1965) Introductory Paragraph]] -->
-<font color="darkgreen">"In cosmogonical studies there arises naturally the problem of the collapse from rest of a cool, self-gravitating cloud.  In the simplest idealization, with the cloud density initially spherically symmetric, the collapse clearly takes place through a series of spherically symmetric states, because of the isotropy of the gravitational field."</font>
+<font color="darkgreen">"In cosmogonical studies there arises naturally the problem of the collapse from rest of a cool, self-gravitating cloud.  In the simplest idealization, with the cloud density initially spherically symmetric, the collapse clearly takes place through a series of spherically symmetric states, because of the isotropy of the gravitational field.  A convenient Lagrangian coordinate is the radius <math>r(m, t)</math> at time <math>t</math> of the subsphere containing mass <math>m</math>.  With pressure negligible, the equation of motion is
    </td>
 </tr>

Jet53man: /* Free-Fall */

2021-10-29T20:26:54Z

Free-Fall

← Older revision		Revision as of 16:26, 29 October 2021
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	In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.		In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.

	To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by {{ Hunter62full }} titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by ~~C. C. Lin, Leon Mestel, and Frank Shu [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965, ApJ, 142, 143)]~~ titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by ~~Lin, Mestel~~ &~~amp; Shu (1965~~; hereafter, ~~LMS65)~~ was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.		To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by {{ Hunter62full }} titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by {{ LMS65full }} titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by {{ LMS65 }} — hereafter, {{ LMS65hereafter }} — was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.

	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="8" width="80%">
	<tr>		<tr>
	<th align="center" colspan="1">		<td align="center" colspan="1">
			<!--
	[[File:LinMestelShu1965Title.png\|300px\|Lin, Mestel & Shu (1965)]]		[[File:LinMestelShu1965Title.png\|300px\|Lin, Mestel & Shu (1965)]]

	[http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965) ''The Astrophysical Journal'', 142, 143]		[http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965) ''The Astrophysical Journal'', 142, 143]
	</th>		-->
			Reprint of the opening (introductory) paragraph of …<br />
			{{ LMS65figure }}
			</td>
	</tr>		</tr>
	<tr>		<tr>
	<th align="~~center~~" colspan="1">		<td align="left" colspan="1">
	[[File:LinMestelShu1965Intro.png\|600px\|Lin, Mestel & Shu (1965) Introductory Paragraph]]		<!-- [[File:LinMestelShu1965Intro.png\|600px\|Lin, Mestel & Shu (1965) Introductory Paragraph]] -->
	</th>		<font color="darkgreen">"In cosmogonical studies there arises naturally the problem of the collapse from rest of a cool, self-gravitating cloud. In the simplest idealization, with the cloud density initially spherically symmetric, the collapse clearly takes place through a series of spherically symmetric states, because of the isotropy of the gravitational field."</font>
			</td>
	</tr>		</tr>
	</table>		</table>

Jet53man: /* Free-Fall */

2021-10-29T18:12:52Z

Free-Fall

← Older revision		Revision as of 14:12, 29 October 2021
Line 9:		Line 9:
	In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.		In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.

	To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by ~~[http://adsabs.harvard.edu/abs/1962ApJ...136..594H Chris Hunter (1962, ApJ, 136, 594)]~~ titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by C. C. Lin, Leon Mestel, and Frank Shu [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965, ApJ, 142, 143)] titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by Lin, Mestel & Shu (1965; hereafter, LMS65) was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.		To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by {{ Hunter62full }} titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by C. C. Lin, Leon Mestel, and Frank Shu [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965, ApJ, 142, 143)] titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by Lin, Mestel & Shu (1965; hereafter, LMS65) was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.

	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">
Line 26:		Line 26:
	</table>		</table>

	As is briefly discussed in a [[#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by ~~[http://adsabs.harvard.edu/abs/1962ApJ...136..594H Hunter (1962)]~~ also appears in discussions of [[#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[#McCreaMilne1934Solution\|detailed below]], the parametric <math>~r(t)</math> solution presented as equations (2) & (3) in LMS65 appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.		As is briefly discussed in a [[#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by {{ Hunter62 }} also appears in discussions of [[#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[#McCreaMilne1934Solution\|detailed below]], the parametric <math>~r(t)</math> solution presented as equations (2) & (3) in LMS65 appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.

	In the subsections that follow, we show how equation (1) of LMS65 arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.		In the subsections that follow, we show how equation (1) of LMS65 arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.

Jet53man: /* Free-Fall */

2021-09-23T16:25:35Z

Free-Fall

← Older revision		Revision as of 12:25, 23 September 2021
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	{\| class="PolytropeStability" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="PolytropeStability" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: ~~150px~~; width: ~~150px~~; background-color:#ffff99;;" \|[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution\|<b>Free-Fall<br />Collapse</b>]]		! style="height: 125px; width: 125px; background-color:#ffff99;;" \|[[H_BookTiledMenu#Nonlinear_Dynamical_Evolution\|<b>Free-Fall<br />Collapse</b>]]
	\|}		\|}

	In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.		In our broad study of the "dynamics of self-gravitating fluids," we are interested in examining how, in a wide variety of physical systems, initially unbalanced forces or exponentially growing instabilities can lead to the development of fluid motions and structural changes that are of nonlinear amplitude. We begin, here, by discussing the free-fall collapse of a spherically symmetric, uniform-density configuration. In the scheme of things, this is a simple example, but it proves to be powerfully illustrative.


	To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by [http://adsabs.harvard.edu/abs/1962ApJ...136..594H Chris Hunter (1962, ApJ, 136, 594)] titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by C. C. Lin, Leon Mestel, and Frank Shu [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965, ApJ, 142, 143)] titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by Lin, Mestel & Shu (1965; hereafter, LMS65) was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.		To our knowledge, the solution to this free-fall problem ''in the context of star formation research'' first appeared in print in a paper by [http://adsabs.harvard.edu/abs/1962ApJ...136..594H Chris Hunter (1962, ApJ, 136, 594)] titled, ''The Instability of the Collapse of a Self-Gravitating Gas Cloud''. A few years later, this solution appeared — with a reference to Hunter's derivation — in an article by [http://adsabs.harvard.edu/abs/1965QJRAS...6..161M L. Mestel (1965, Quarterly Journal of the Royal Astronomical Society, 6, 161)] that discussed a broad range of theoretical problems related to star formation, and it appeared again in the introductory paragraph of a classic paper by C. C. Lin, Leon Mestel, and Frank Shu [http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965, ApJ, 142, 143)] titled, "The Gravitational Collapse of a Uniform Spheroid." While the primary purpose of this work by Lin, Mestel & Shu (1965; hereafter, LMS65) was to depart from the assumption of spherical symmetry and examine the amplification of spheroidal distortions during a collapse, the solution to the simpler, spherically symmetric free-fall collapse problem is so clearly and succinctly written that we have decided to reprint, here, the paper's introductory paragraph.

Jet53man: /* Relationship to Relativistic Cosmologies */

2021-09-09T16:16:08Z

Relationship to Relativistic Cosmologies

← Older revision		Revision as of 12:16, 9 September 2021
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	</table>		</table>

	It is instructive to compare these dynamical relations with the analogous pair of equations that we [[~~User:Tohline/SSC/FreeFall~~#PressureFreeSummary\|derived, above]], in the context of a Newtonian (and flat space) description of the free-fall collapse of a uniform-density sphere having a time-dependent radius, <math>~R(t)</math>:		It is instructive to compare these dynamical relations with the analogous pair of equations that we [[#PressureFreeSummary\|derived, above]], in the context of a Newtonian (and flat space) description of the free-fall collapse of a uniform-density sphere having a time-dependent radius, <math>~R(t)</math>:

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

Jet53man: /* Analytic Model of the Collapse of a Uniform-Density Sphere */

2021-09-09T16:15:00Z

Analytic Model of the Collapse of a Uniform-Density Sphere

Show changes

Jet53man: /* Assembling the Key Relations */

2021-09-09T16:07:17Z

Assembling the Key Relations

← Older revision		Revision as of 12:07, 9 September 2021
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	==Assembling the Key Relations==		==Assembling the Key Relations==

	We begin with the [[~~User:Tohline/SphericallySymmetricConfigurations~~/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|set of time-dependent governing equations for spherically symmetric systems]], namely,		We begin with the [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|set of time-dependent governing equations for spherically symmetric systems]], namely,

	<div align="center">		<div align="center">
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	<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />		<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

	{{~~User:Tohline/~~Math/EQ_FirstLaw02}}		{{Math/EQ_FirstLaw02}}

Jet53man: /* Free-Fall */

2021-09-09T16:04:02Z

Free-Fall

← Older revision		Revision as of 12:04, 9 September 2021
Line 27:		Line 27:
	</table>		</table>

	As is briefly discussed in a [[~~User:Tohline/SSC/FreeFall~~#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by [http://adsabs.harvard.edu/abs/1962ApJ...136..594H Hunter (1962)] also appears in discussions of [[~~User:Tohline/SSC/FreeFall~~#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[~~User:Tohline/SSC/FreeFall~~#McCreaMilne1934Solution\|detailed below]], the parametric <math>~r(t)</math> solution presented as equations (2) & (3) in LMS65 appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.		As is briefly discussed in a [[#Relationship_to_Relativistic_Cosmologies\|later subsection of this chapter]], the same 2<sup>nd</sup>-order ODE that leads to the spherically symmetric, free-fall collapse model derived by [http://adsabs.harvard.edu/abs/1962ApJ...136..594H Hunter (1962)] also appears in discussions of [[#Relationship_to_Relativistic_Cosmologies\|relativistic cosmologies]]. Indeed, as is [[#McCreaMilne1934Solution\|detailed below]], the parametric <math>~r(t)</math> solution presented as equations (2) & (3) in LMS65 appeared in print — as far as we have been able to ascertain, for the first time — more than thirty years earlier in a paper by William H. McCrea and Edward A. Milne [http://adsabs.harvard.edu/abs/1934QJMat...5...73M (1934) Quart. J. Math. Oxford, 5, 73] titled, ''Newtonian Universes and the Curvature of Space''. In the context of early cosmological discussions, however, the focus was usually on model solutions that describe pressure-free "Hubble flow" expansion, rather than free-fall collapse.

	In the subsections that follow, we show how equation (1) of LMS65 arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.		In the subsections that follow, we show how equation (1) of LMS65 arises from our generic set of principal governing equations; how their parametric solution in the form of equations (2) and (3) is derived; and elaborate on the physical relevance of this seemingly simplistic problem to the broad study of self-gravitating fluids.