SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3 - Revision history

Joel2: /* Part I: Physical Significance of the Two Curves */

2023-12-23T03:28:11Z

Part I: Physical Significance of the Two Curves

← Older revision		Revision as of 23:28, 22 December 2023
Line 16:		Line 16:
	The "Stahler" mass-radius relation, plotted as a continuous curve [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.		The "Stahler" mass-radius relation, plotted as a continuous curve [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.

	If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.		If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above (Part II) [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.

	In the context of star formation, the Stahler sequence can be viewed as an ''evolutionary'' sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.		In the context of star formation, the Stahler sequence can be viewed as an ''evolutionary'' sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.

Joel2: /* Part I: Physical Significance of the Two Curves */

2023-12-23T03:23:57Z

Part I: Physical Significance of the Two Curves

← Older revision		Revision as of 23:23, 22 December 2023
Line 14:		Line 14:

	===Part I: Physical Significance of the Two Curves===		===Part I: Physical Significance of the Two Curves===
	The "Stahler" mass-radius relation, plotted as a continuous curve [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.		The "Stahler" mass-radius relation, plotted as a continuous curve [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.

	If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.		If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt1#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.

	In the context of star formation, the Stahler sequence can be viewed as an ''evolutionary'' sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.		In the context of star formation, the Stahler sequence can be viewed as an ''evolutionary'' sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, <math>~\tilde\xi</math>; such clouds will appear near the origin of the displayed <math>~\mathcal{X}-\mathcal{Y}</math> plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.

Joel2: /* Part I: Physical Significance of the Two Curves */

2023-12-23T02:43:54Z

Part I: Physical Significance of the Two Curves

← Older revision		Revision as of 22:43, 22 December 2023
Line 14:		Line 14:

	===Part I: Physical Significance of the Two Curves===		===Part I: Physical Significance of the Two Curves===
	The "Stahler" mass-radius relation, plotted as a continuous curve [[#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.		The "Stahler" mass-radius relation, plotted as a continuous curve [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2#Plotting_Stahler.27s_Relation\|in the above figure]] and reproduced as a sequence of discrete points in each panel of the subsequent [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]], identifies the precise mass <math>~(\mathcal{Y})</math> and associated radius <math>~(\mathcal{X})</math> of physically allowed pressure-truncated, <math>~n = 5</math> polytropic configurations over the full range of values of the dimensionless truncation radius, <math>~0 < \tilde\xi < \infty</math>. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — <math>~\mathfrak{f}_M, \mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math> — and the corresponding values of the coefficients associated with the free-energy function — <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> — will also vary from model to model along the Stahler curve.

	If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.		If the values of the coefficients, <math>~\mathcal{A}_{M_\ell}</math> and <math>~\mathcal{B}_{M_\ell}</math> (as well as the external pressure and, hence, the additional coefficient, <math>~\mathcal{D}</math>) are held fixed, the [[#Free_Energy_Function_and_Virial_Theorem\|algebraic free-energy function]] defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for <math>~n = 5</math> polytropic configurations identifies equilibrium radii <math>~(\mathcal{X})</math> associated with various configuration masses <math>~(\mathcal{Y})</math>. The "Virial" curve that has been plotted in each panel of the above [[#Plotting_the_Virial_Theorem_Relation\|comparison figure]] shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve ''qualitatively'' resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.

Joel2 at 02:03, 23 December 2023

2023-12-23T02:03:34Z

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 ==Discussion==
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 </math>
 </div>
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Joel2: Created page with " ==Discussion== A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is $\gamma < 4/3$ — equivalently, $n > 3$ — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, $~P_e$ . The pressure-truncated, $n = 5 (\gamma = 6/5)$ ..."

2023-12-23T01:49:06Z

Created page with " ==Discussion== A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is <math>\gamma < 4/3</math> — equivalently, <math>n > 3</math> — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, <math>~P_e</math>. The pressure-truncated, <math>n = 5 (\gamma = 6/5)</math>..."

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