SSC/Dynamics/IsothermalSimilaritySolution - Revision history

Jet53man: /* Examine Connection With … */

2021-09-09T19:57:53Z

Examine Connection With …

← Older revision		Revision as of 15:57, 9 September 2021
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	=Examine Connection With …=		=Examine Connection With …=
	Let's examine whether or not there is overlap between the properties of the above-discussed similarity solutions that give insight into the nonlinear dynamical behavior of collapse and the (a) known structure of the unperturbed, but marginally unstable Bonnor-Ebert sphere, and (b) eigenfunction that describes the radial profile of the marginally unstable radial pulsation mode. Keep in mind that, as we have [[~~User:Tohline/~~SSC/Stability/Isothermal#Overview\|presented separately]], the truncation radius of the marginally unstable, Bonnor-Ebert sphere has, <math>~\xi_e \approx 6.4510534</math>.		Let's examine whether or not there is overlap between the properties of the above-discussed similarity solutions that give insight into the nonlinear dynamical behavior of collapse and the (a) known structure of the unperturbed, but marginally unstable Bonnor-Ebert sphere, and (b) eigenfunction that describes the radial profile of the marginally unstable radial pulsation mode. Keep in mind that, as we have [[SSC/Stability/Isothermal#Overview\|presented separately]], the truncation radius of the marginally unstable, Bonnor-Ebert sphere has, <math>~\xi_e \approx 6.4510534</math>.

	==Pressure-Truncated Equilibrium Structure==		==Pressure-Truncated Equilibrium Structure==
	From our [[~~User:Tohline/~~SSC/Stability/Isothermal#Groundwork\|separate discussion of pressure-truncated isothermal spheres]], we can identify the following structural properties of the marginally unstable Bonnor-Ebert sphere. The function, <math>~\psi(\xi)</math> satisfies the,		From our [[SSC/Stability/Isothermal#Groundwork\|separate discussion of pressure-truncated isothermal spheres]], we can identify the following structural properties of the marginally unstable Bonnor-Ebert sphere. The function, <math>~\psi(\xi)</math> satisfies the,
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	<table border="0" cellpadding="8" align="center">		<table border="0" cellpadding="8" align="center">
	<tr><td align="center">		<tr><td align="center">
	<font color="maroon">Isothermal Lane-Emden Equation</font> <p></p>		<font color="maroon">Isothermal Lane-Emden Equation</font> <p></p>
	{{ ~~User:Tohline/~~Math/EQ_SSLaneEmden02 }}		{{ Math/EQ_SSLaneEmden02 }}
	</td></tr>		</td></tr>
	</table>		</table>
	</div>		</div>

	Given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[~~User:Tohline/~~SSC/Structure/BonnorEbert#Pressure\|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,		Given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[SSC/Structure/BonnorEbert#Pressure\|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
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	and, [[~~User:Tohline/~~SSC/Structure/BonnorEbert#Radius\|our expression]] for the truncated configuration's equilibrium radius is,		and, [[SSC/Structure/BonnorEbert#Radius\|our expression]] for the truncated configuration's equilibrium radius is,
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	Also, as has been summarized in our [[~~User:Tohline/~~SSC/Structure/BonnorEbert#P-V_Diagram\|accompanying discussion]], expressions that describe the general run of radius, pressure, and mass are, respectively,		Also, as has been summarized in our [[SSC/Structure/BonnorEbert#P-V_Diagram\|accompanying discussion]], expressions that describe the general run of radius, pressure, and mass are, respectively,

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	==Yabushita's Radial Pulsation Eigenvector==		==Yabushita's Radial Pulsation Eigenvector==
	As we have, separately, [[~~User:Tohline/~~SSC/Perturbations#The_Eigenvalue_Problem\|discussed in detail]], the eigenvalue problem is defined in terms of the following ''perturbed'' variables,		As we have, separately, [[SSC/Perturbations#The_Eigenvalue_Problem\|discussed in detail]], the eigenvalue problem is defined in terms of the following ''perturbed'' variables,
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	And, as was first [[~~User:Tohline/~~SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability\|demonstrated by Yabushita (1975)]],		And, as was first [[SSC/Stability/InstabilityOnsetOverview#Yabushita.27s_Insight_Regarding_Stability\|demonstrated by Yabushita (1975)]],
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Jet53man: /* Limiting Behavior */

2021-09-09T18:11:11Z

Limiting Behavior

← Older revision		Revision as of 14:11, 9 September 2021
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	* As, <math>\zeta \rightarrow - \infty</math>:		* As, <math>\zeta \rightarrow - \infty</math>:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1A" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~U \approx \frac{2}{3} \biggl( - \frac{1}{\zeta}\biggr) + \frac{1}{45} \biggl[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^3 \, ,</math><p> </p>		\|<math>~U \approx \frac{2}{3} \biggl( - \frac{1}{\zeta}\biggr) + \frac{1}{45} \biggl[ \frac{2}{3} - e^{Q_0}\biggr] \biggl( - \frac{1}{\zeta}\biggr)^3 \, ,</math><p> </p>
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	* [[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 13 July 2017: In this expression for U, a "plus" sign has been inserted between the ζ term and the ζ-squared term, correcting a typographical error in equation 4.12a of Tohline (1982). And the expression for Ρ has been expanded to include a ζ-cubed term.]]For, <math>\zeta \approx 0</math>:		* [[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 13 July 2017: In this expression for U, a "plus" sign has been inserted between the ζ term and the ζ-squared term, correcting a typographical error in equation 4.12a of Tohline (1982). And the expression for Ρ has been expanded to include a ζ-cubed term.]]For, <math>\zeta \approx 0</math>:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1B" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~U \approx U_0 + \zeta(\Rho_0 - 2)+ \zeta^2 U_0 + \zeta^3\biggl[(\Rho_0-2)(1-\Rho_0/6) - \frac{2}{3} U_0^2 \biggr] \, ,</math><p> </p>		\|<math>~U \approx U_0 + \zeta(\Rho_0 - 2)+ \zeta^2 U_0 + \zeta^3\biggl[(\Rho_0-2)(1-\Rho_0/6) - \frac{2}{3} U_0^2 \biggr] \, ,</math><p> </p>
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	* As, <math>\zeta \rightarrow + \infty</math>:		* As, <math>\zeta \rightarrow + \infty</math>:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1C" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~U \approx (2m_0 \zeta)^{1 / 2} \, ,</math><p> </p>		\|<math>~U \approx (2m_0 \zeta)^{1 / 2} \, ,</math><p> </p>
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	* For, <math>~t < 0</math> and <math>~r \ll c_s\|t\|</math>:		* For, <math>~t < 0</math> and <math>~r \ll c_s\|t\|</math>:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1D" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~v_r(r,t) \approx - \frac{2r}{3(-t)} \, ,</math><p> </p>		\|<math>~v_r(r,t) \approx - \frac{2r}{3(-t)} \, ,</math><p> </p>
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	* For, <math>~r \gg c_s\|t\|</math> at any time:		* For, <math>~r \gg c_s\|t\|</math> at any time:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1E" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~v_r(r,t) \approx - c_s U_0\, ,</math><p> </p>		\|<math>~v_r(r,t) \approx - c_s U_0\, ,</math><p> </p>
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	* For, <math>~t > 0</math> and <math>~r \ll c_s\|t\|</math>:		* For, <math>~t > 0</math> and <math>~r \ll c_s\|t\|</math>:
	<p> </p>		<p> </p>
	{\| class="~~wikitable~~" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"		{\| class="Chap1F" style="margin-right: auto; margin-left: 50px; vertical-align:top; text-align:left;"
	\|-		\|-
	\|<math>~v_r(r,t) \approx - \biggl( \frac{2m_0}{c_s} \biggr)^{1 / 2} \biggl( \frac{t}{r}\biggr)^{1 / 2} \, ,</math><p> </p>		\|<math>~v_r(r,t) \approx - \biggl( \frac{2m_0}{c_s} \biggr)^{1 / 2} \biggl( \frac{t}{r}\biggr)^{1 / 2} \, ,</math><p> </p>

Jet53man: /* Establishing Set of Governing Equations */

2021-09-09T18:06:31Z

Establishing Set of Governing Equations

← Older revision		Revision as of 14:06, 9 September 2021
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	==Establishing Set of Governing Equations==		==Establishing Set of Governing Equations==

	Drawing from an [[~~User:Tohline~~/~~SSC~~/IsothermalCollapse#IsothermalEulerianFrame\|accompanying chapter's introductory discussion]], we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.		Drawing from an [[SSC/Dynamics/IsothermalCollapse#IsothermalEulerianFrame\|accompanying chapter's introductory discussion]], we begin with the set of governing equations that describe the collapse of isothermal spheres from an Eulerian frame of reference.

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	Notice that, following [[~~User:Tohline~~/~~SSC~~/IsothermalCollapse#Larson_.281969.29\|Larson's (1969) lead]], we have replaced the standard continuity equation with the following equivalent statement of mass conservation:		Notice that, following [[SSC/Dynamics/IsothermalCollapse#Larson_.281969.29\|Larson's (1969) lead]], we have replaced the standard continuity equation with the following equivalent statement of mass conservation:
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Jet53man: Created page with "FORCETOC =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-widt..."

2021-09-09T17:56:14Z

Created page with "__FORCETOC__  =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-widt..."

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SSC/Dynamics/IsothermalSimilaritySolution - Revision history

Jet53man: /* Examine Connection With … */

Jet53man: /* Limiting Behavior */

Jet53man: /* Establishing Set of Governing Equations */

Jet53man: Created page with "__FORCETOC__ =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-widt..."

Jet53man: Created page with "FORCETOC =Similarity Solution= {| class="IsothermalCollapse" style="float:left; margin-right: 20px; border-style: solid; border-widt..."