SSC/Virial/PolytropesSummary - Revision history

Joel2: /* When ~\eta_\mathrm{ad} is zero */

2024-07-17T16:38:50Z

When ~\eta_\mathrm{ad} is zero

← Older revision		Revision as of 12:38, 17 July 2024
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	====When <~~math~~>~~~\eta_\mathrm{~~ad}</~~math~~> ~~is zero~~====		====When η<sub>ad</sub> = 0====
	For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, <math>~\eta_\mathrm{ad}</math>, can be zero for one of two reasons: Either <math>~n=5</math>; or <math>~\tilde\theta \rightarrow \theta_{\xi_1} = 0</math>. In the latter case, all curves converge on the same point, that is, <math>~(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0)</math>. This corresponds to the case of no external medium <math>~(P_e = 0)</math> and, hence, the associated equilibrium configuration is the familiar [[~~User:Tohline/~~SSC/Structure/Polytropes#Polytropic_Spheres\|''isolated'' polytropic sphere]]. As can be deduced from our above discussion of the [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#ConciseVirial\|algebraic expression of the virial theorem]], because <math>~\Chi_\mathrm{ad} = 1</math>, the equilibrium radius of such a configuration is,		For the types of systems that are presently most relevant to astrophysical discussions, the key parameter, <math>\eta_\mathrm{ad}</math>, can be zero for one of two reasons: Either <math>n=5</math>; or <math>\tilde\theta \rightarrow \theta_{\xi_1} = 0</math>. In the latter case, all curves converge on the same point, that is, <math>(\Chi_\mathrm{ad}, \Pi_\mathrm{ad}) = (1, 0)</math>. This corresponds to the case of no external medium <math>(P_e = 0)</math> and, hence, the associated equilibrium configuration is the familiar [[SSC/Structure/Polytropes#Polytropic_Spheres\|''isolated'' polytropic sphere]]. As can be deduced from our above discussion of the [[SSC/Virial/PolytropesSummary#ConciseVirial\|algebraic expression of the virial theorem]], because <math>\Chi_\mathrm{ad} = 1</math>, the equilibrium radius of such a configuration is,
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	<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>		<math>\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \chi_\mathrm{eq}</math>
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	<math>~=</math>		<math>=</math>
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	<math>~\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>		<math>\biggl[ \frac{\mathcal{A}}{\mathcal{B}} \biggr]^{n/(n-3)} \, .</math>
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	As is demonstrated in an [[~~User:Tohline/~~SSC/Virial/Polytropes#Strategy2\|accompanying discussion]] and also [[~~User:Tohline/~~SSC/Virial/PolytropesSummary#Physical_Meaning_of_Parameter\|mentioned above]], after inserting the relevant expressions for the free-energy coefficients, <math>~\mathcal{A}</math> and <math>~\mathcal{B}</math>, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,		As is demonstrated in an [[SSC/Virial/Polytropes#Strategy2\|accompanying discussion]] and also [[SSC/Virial/PolytropesSummary#Physical_Meaning_of_Parameter\|mentioned above]], after inserting the relevant expressions for the free-energy coefficients, <math>\mathcal{A}</math> and <math>\mathcal{B}</math>, this provides the key relationship between the mass, equilibrium radius, and central pressure of an isolated polytrope, namely,
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	<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>		<math>\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
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	<math>~=</math>		<math>=</math>
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	<math>~\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>		<math>\frac{1}{[4\pi (n+1) (\theta^')^2]_{\xi_1}} \, .</math>
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	As we have [[~~User:Tohline/~~SSC/Virial/Polytropes#Central_and_Mean_Pressure\|reviewed elsewhere]] — see also our [[~~User:Tohline/~~SSC/Structure/Polytropes#Polytropic_Spheres\|detailed discussion of isolated polytropes]] — this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of [[~~User:Tohline/~~Appendix/References\|Chandrasekhar [C67]]] in association with his discussion of the dimensionless coefficient, <math>~W_n</math>, and the central pressure of polytropes.		As we have [[SSC/Virial/Polytropes#Central_and_Mean_Pressure\|reviewed elsewhere]] — see also our [[SSC/Structure/Polytropes#Polytropic_Spheres\|detailed discussion of isolated polytropes]] — this is a familiar relationship, appearing prominently in Chapter IV (p. 99, equations 80 and 81) of [[Appendix/References\|Chandrasekhar [C67]]] in association with his discussion of the dimensionless coefficient, <math>W_n</math>, and the central pressure of polytropes.

	In the former case — that is, in the case where <math>~\eta_\mathrm{ad} \rightarrow 0</math> because the chosen polytropic index is, <math>~n=5</math> — it must be the case that <math>~\Chi_\mathrm{ad} = 1</math> along the entire sequence (see the green curve labeled <math>~\gamma = (n+1)/n = 6/5</math> in the accompanying figure). This means that the expression for the central pressure,		In the former case — that is, in the case where <math>\eta_\mathrm{ad} \rightarrow 0</math> because the chosen polytropic index is, <math>n=5</math> — it must be the case that <math>\Chi_\mathrm{ad} = 1</math> along the entire sequence (see the green curve labeled <math>\gamma = (n+1)/n = 6/5</math> in the accompanying figure). This means that the expression for the central pressure,
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	<math>~\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>		<math>\frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
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	<math>~=</math>		<math>=</math>
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	<math>~\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>		<math>\frac{1}{[4\pi (n+1) (\tilde\theta^')^2]} \, ,</math>
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	does not explicitly depend on the size of the applied external pressure. But the central pressure ''does'' depend on the radial location at which the configuration is truncated, via the parameter <math>~\tilde\theta^'</math>, which is evaluated at <math>~\tilde\xi</math>, rather than at <math>~\xi_1</math>.		does not explicitly depend on the size of the applied external pressure. But the central pressure ''does'' depend on the radial location at which the configuration is truncated, via the parameter <math>\tilde\theta^'</math>, which is evaluated at <math>\tilde\xi</math>, rather than at <math>\xi_1</math>.

	===Stability===		===Stability===

Joel2: /* Physical Meaning of Parameter ηad ~\eta_\mathrm{ad} */

2024-07-17T16:33:58Z

Physical Meaning of Parameter ηad ~\eta_\mathrm{ad}

← Older revision		Revision as of 12:33, 17 July 2024
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	It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].		It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].

	===Physical Meaning of Parameter η<sub>ad</sub~~> <math>~\eta_\mathrm{ad}</math~~>===		===Physical Meaning of Parameter η<sub>ad</sub>===

	As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>\tilde\mathfrak{f}_A</math>. It is worth pointing out what physical quantities are associated with these two terms.		As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>\tilde\mathfrak{f}_A</math>. It is worth pointing out what physical quantities are associated with these two terms.

Joel2: /* Physical Meaning of Parameter ~\eta_\mathrm{ad} */

2024-07-16T21:37:45Z

Physical Meaning of Parameter ~\eta_\mathrm{ad}

← Older revision		Revision as of 17:37, 16 July 2024
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	It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].		It is straightforward to show that this more compact pair of expressions for <math>~\Pi_\mathrm{ad}</math> and <math>~\Chi_\mathrm{ad}</math> satisfy the [[SSC/Virial/PolytropesSummary#ConciseVirial\|virial theorem presented above]].

	===Physical Meaning of Parameter <math>~\eta_\mathrm{ad}</math>===		===Physical Meaning of Parameter η<sub>ad</sub> <math>~\eta_\mathrm{ad}</math>===

	As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>~\tilde\mathfrak{f}_A</math>. It is worth pointing out what physical quantities are associated with these two terms.		As defined in our above discussion, <math>~\eta_\mathrm{ad}</math> is the ratio of the two terms that are summed together in the definition of the structural form factor, <math>\tilde\mathfrak{f}_A</math>. It is worth pointing out what physical quantities are associated with these two terms.

	At any radial location within a polytropic configuration, the [[~~User:Tohline/~~SSC/Structure/Polytropes#Lane-Emden_Equation\|Lane-Emden function]], <math>~\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,		At any radial location within a polytropic configuration, the [[SSC/Structure/Polytropes#Lane-Emden_Equation\|Lane-Emden function]], <math>\theta</math>, is defined in terms of a ratio of the local density to the configuration's central density, specifically,
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	<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>		<math>\theta \equiv \biggl(\frac{\rho}{\rho_c} \biggr)^{1/n} \, .</math>
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	<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>		<math>\frac{P}{P_c} = \theta^{n+1} \, .</math>
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	Hence, the quantity, <math>~\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>~\tilde\mathfrak{f}_A</math>, is the ratio, <math>~(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere. But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,		Hence, the quantity, <math>\tilde\theta^{n+1}</math>, which appears as the second term in our definition of <math>\tilde\mathfrak{f}_A</math>, is the ratio, <math>(P/P_c)_{\tilde\xi}</math>, evaluated at the surface of the truncated polytropic sphere. But, by construction, the pressure at this location equals the pressure of the external medium in which the polytrope is embedded, so we can write,
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	<math>\tilde\theta^{n+1} = \frac{P_e}{P_c} \, .</math>		<math>\tilde\theta^{n+1} = \frac{P_e}{P_c} \, .</math>
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	In our [[~~User:Tohline/~~SSC/Virial/Polytropes#Strategy2\|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressure. Specifically, we show that,		In our [[SSC/Virial/Polytropes#Strategy2\|accompanying detailed analysis]], we have employed the virial theorem expression to demonstrate that the first term in our definition of <math>~\tilde\mathfrak{f}_A</math> provides a measure the configuration's normalized central pressure. Specifically, we show that,
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	<math>~\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>		<math>\biggl( \frac{4\pi}{3} \biggr) \frac{P_c R_\mathrm{eq}^4}{G M_\mathrm{limit}^2}</math>
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	<math>~=</math>		<math>=</math>
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	<math>~[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>		<math>[3 (n+1) (\tilde\theta^')^2]^{-1} \, .</math>
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	<math>~=</math>		<math>=</math>
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	<math>~=</math>		<math>=</math>
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	<math>~\eta_\mathrm{ad} </math>		<math>\eta_\mathrm{ad} </math>
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70.180.56.157: Created page with "FORCETOC =Virial Equilibrium of Adiabatic Spheres (Summary)= The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes. ==Detailed Force-Balanced Solution== As has been discussed in detail in another chapter, {{ Horedt70fu..."

2024-07-15T21:21:40Z

Created page with "__FORCETOC__  =Virial Equilibrium of Adiabatic Spheres (Summary)= The summary presented here has been drawn from our accompanying detailed analysis of the structure of pressure-truncated polytropes. ==Detailed Force-Balanced Solution== As has been discussed in detail in another chapter, {{ Horedt70fu..."

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