User:Tohline/SphericallySymmetricConfigurations/Virial - Revision history

Tohline: /* Volume Integrals */

2017-05-30T01:11:14Z

Volume Integrals

← Older revision		Revision as of 01:11, 30 May 2017
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	<font color="red">Reservoir of Thermodynamic Energy:</font> As has been explained in [[User:Tohline/VE#Reservoir_of_Thermodynamic_Energy\|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>~\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For [[User:Tohline/VE#Isothermal_Systems\|isothermal systems]],		<font color="red">Reservoir of Thermodynamic Energy:</font> As has been explained in [[User:Tohline/VE#Reservoir_of_Thermodynamic_Energy\|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>~\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For [[User:Tohline/VE#Isothermal_Systems\|isothermal systems]],
	<div align="center">		<div align="center" id="Reservoir">
	<math>		<math>
	\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I		\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I

Tohline: /* Volume Integrals */

2017-05-30T00:59:46Z

Volume Integrals

← Older revision		Revision as of 00:59, 30 May 2017
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	Hence — see, also, p. 64, Equation (12) of [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the desired expression for the gravitational potential energy is,		<span id="Wgrav">Hence</span> — see, also, p. 64, Equation (12) of [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the desired expression for the gravitational potential energy is,
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Tohline at 23:13, 10 July 2016

2016-07-10T23:13:59Z

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			=See Also=
			<ul>
			<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
			</ul>



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Structural Form Factors */ Delete expression relating f_A to mean pressure

2015-08-07T16:19:38Z

Structural Form Factors: Delete expression relating f_A to mean pressure

← Older revision		Revision as of 16:19, 7 August 2015
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			<math>~\mathfrak{f}_A</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>
			</td>
			</tr>

			<!-- (August 2015) DELETE NEXT EQUATION, AS DEFINITION IN TERMS OF AVERAGE PRESSURE IS LIKELY NOT CORRECT ...
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Tohline: /* Volume Integrals */

2015-08-06T19:08:36Z

Volume Integrals

← Older revision		Revision as of 19:08, 6 August 2015
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	<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.		<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.

	<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[User:Tohline/SSC/BipolytropeGeneralization_Version2#~~Bipolytrope_Generalizatio~~\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.		<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
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Tohline: /* Choices Made by Other Researchers */

2015-07-19T16:28:19Z

Choices Made by Other Researchers

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	<math>~a_n = \biggl[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n) -1} \biggr]^{1/2} \, ,</math>		<math>~a_n = \biggl[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n) -1} \biggr]^{1/2} \, ,</math>
	</div>		</div>
	which is the definition of this classical length scale introduced by [[User:Tohline/Appendix/References#C67\|~~Chandrasekhar (1967) [~~C67] ]] (see, specifically, his equation 10 on p. 87). Switching from <math>~n</math> to the associated adiabatic exponent via the relation, <math>~\gamma = 1+1/n ~~~\Rightarrow~~~ n = 1/(\gamma-1)</math>, we see that,		which is the definition of this classical length scale introduced by [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] (see, specifically, his equation 10 on p. 87). Switching from <math>~n</math> to the associated adiabatic exponent via the relation, <math>~\gamma = 1+1/n ~~~\Rightarrow~~~ n = 1/(\gamma-1)</math>, we see that,
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	Notice that, written in this manner, the scale length, <math>~a_n</math>, cannot actually be determined unless the normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is known. We will encounter analogous situations whenever the free energy function is used to identify the physical parameters that define equilibrium configurations — key attributes of a system that should be held fixed as the system size (or some other order parameter) is varied cannot actually be evaluated until an extremum in the free energy is identified and the corresponding value of <math>~\chi_\mathrm{eq}</math> is known. Because solutions of the Lane-Emden equation directly provide detailed force-balance models of polytropic spheres, [[User:Tohline/Appendix/References#C67\|~~Chandrasekhar (1967) [~~C67] ]] did not encounter this issue. As we have [[User:Tohline/SSC/Structure/Polytropes#Known_Analytic_Solutions\|discussed elsewhere]], the equilibrium radius of a polytropic sphere is identified as the radial location,		Notice that, written in this manner, the scale length, <math>~a_n</math>, cannot actually be determined unless the normalized equilibrium radius, <math>~\chi_\mathrm{eq}</math>, is known. We will encounter analogous situations whenever the free energy function is used to identify the physical parameters that define equilibrium configurations — key attributes of a system that should be held fixed as the system size (or some other order parameter) is varied cannot actually be evaluated until an extremum in the free energy is identified and the corresponding value of <math>~\chi_\mathrm{eq}</math> is known. Because solutions of the Lane-Emden equation directly provide detailed force-balance models of polytropic spheres, [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] did not encounter this issue. As we have [[User:Tohline/SSC/Structure/Polytropes#Known_Analytic_Solutions\|discussed elsewhere]], the equilibrium radius of a polytropic sphere is identified as the radial location,
	<div align="center">		<div align="center">
	<math>~\xi_1 = \frac{R_\mathrm{eq}}{a_n} \, ,</math>		<math>~\xi_1 = \frac{R_\mathrm{eq}}{a_n} \, ,</math>

Tohline: /* Volume Integrals */

2015-07-19T16:25:00Z

Volume Integrals

← Older revision		Revision as of 16:25, 19 July 2015
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	</div>		</div>

	<font color="red">Reservoir of Thermodynamic Energy:</font> As has been explained in [[User:Tohline/VE#Reservoir_of_Thermodynamic_Energy\|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>~\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For [[User:Tohline/VE#~~Isothermal_System~~\|isothermal systems]],		<font color="red">Reservoir of Thermodynamic Energy:</font> As has been explained in [[User:Tohline/VE#Reservoir_of_Thermodynamic_Energy\|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>~\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally. For [[User:Tohline/VE#Isothermal_Systems\|isothermal systems]],
	<div align="center">		<div align="center">
	<math>		<math>

Tohline: /* Volume Integrals */ Improve coloration of several textbook references

2015-07-19T16:22:32Z

Volume Integrals: Improve coloration of several textbook references

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	<font color="red">Gravitational Potential Energy:</font> From our discussion of the [[User:Tohline/VE#Scalar_Virial_Theorem\|scalar virial theorem]] — see, specifically, the reference to Equation (18), on p. 18 of [[User:Tohline/Appendix/References#EFE\|EFE]] — the gravitational potential energy is given by the expression,		<font color="red">Gravitational Potential Energy:</font> From our discussion of the [[User:Tohline/VE#Scalar_Virial_Theorem\|scalar virial theorem]] — see, specifically, the reference to Equation (18), on p. 18 of [<b>[[User:Tohline/Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] — the gravitational potential energy is given by the expression,
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	Hence — see, also, p. 64, Equation (12) of [[User:Tohline/Appendix/References\|~~Chandrasekhar [~~C67]]] — the desired expression for the gravitational potential energy is,		Hence — see, also, p. 64, Equation (12) of [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] — the desired expression for the gravitational potential energy is,
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	Also, as pointed out by [[User:Tohline/Appendix/References\|~~Chandrasekhar [~~C67]]] — see p. 64, Equation (16) — it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,		Also, as pointed out by [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>] — see p. 64, Equation (16) — it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,
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Tohline: /* Energy Extrema */ Comment-out Weber1976_EqB3.png

2015-05-24T17:27:16Z

Energy Extrema: Comment-out Weber1976_EqB3.png

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	[[Image:Weber1976_EqB3.png\|500px\|center]]		<!-- [[Image:Weber1976_EqB3.png\|500px\|center]] -->
			[[Image:AAAwaiting01.png\|500px\|center]]
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Tohline: /* Volume Integrals */ Change "discussion" to "discussing"

2015-02-23T20:45:24Z

Volume Integrals: Change "discussion" to "discussing"

← Older revision		Revision as of 20:45, 23 February 2015
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	<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.		<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>: But, a [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium\|configuration embedded in an external medium]] of pressure, <math>~P_e</math>, will have a (pressure-truncated) surface whose radius, <math>~R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>~P_e</math>. In this case as well, one might choose to refer to <math>~M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>~M_\mathrm{limit}</math> from <math>~M_\mathrm{tot}</math>, continuing to rely on <math>~M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.

	<font color="maroon"><b>BiPolytropes</b></font>: When ~~discussion~~ [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalizatio\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.		<font color="maroon"><b>BiPolytropes</b></font>: When discussing [[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalizatio\|bipolytropes]], the limit of integration, <math>~R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope. In this case, <math>~M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
	</td></tr>		</td></tr>
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