Apps/SMS - Revision history

Jet53man: /* Photon Entropy */

2022-05-01T17:03:01Z

Photon Entropy

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	===Photon Entropy===		===Photon Entropy===
	Here are expressions from the published literature that — to within an additive constant — give the entropy of the (photon) radiation field. We begin with [~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C D. D. Clayton (1968)~~], who states that, "… the entropy per gram of the photon [field]" is,		Here are expressions from the published literature that — to within an additive constant — give the entropy of the (photon) radiation field. We begin with [<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], who states that, "… the entropy per gram of the photon [field]" is,
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Jet53man: /* Used by BAC84 */

2022-05-01T16:59:36Z

Used by BAC84

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	Clearly, in the limit <math>~\sigma \rightarrow \infty</math>, this gives <math>~\Gamma_1 \rightarrow 4/3</math>, which, as it should, matches the limiting value obtained from the ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]~~ expression when <math>~\beta = 0</math>.		Clearly, in the limit <math>~\sigma \rightarrow \infty</math>, this gives <math>~\Gamma_1 \rightarrow 4/3</math>, which, as it should, matches the limiting value obtained from the {{ LP41 }} expression when <math>~\beta = 0</math>.

	{{ BAC84hereafter }} do not explicitly state what value they used for the ratio of specific heats when deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for <math>~\sigma = 0</math>. In this limit, the {{ BAC84hereafter }} expression gives,		{{ BAC84hereafter }} do not explicitly state what value they used for the ratio of specific heats when deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for <math>~\sigma = 0</math>. In this limit, the {{ BAC84hereafter }} expression gives,
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	</div>		</div>

	The general {{ BAC84hereafter }} expression should therefore match the (even more) general ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]~~ expression if we set <math>~\gamma = \tfrac{5}{3}</math>. Let's check this out. Inserting this specific value of <math>~\gamma</math>, and remembering (from above) that,		The general {{ BAC84hereafter }} expression should therefore match the (even more) general {{ LP41 }} expression if we set <math>~\gamma = \tfrac{5}{3}</math>. Let's check this out. Inserting this specific value of <math>~\gamma</math>, and remembering (from above) that,
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	<math>\sigma = \frac{1-\beta }{\beta} \, ,</math>		<math>\sigma = \frac{1-\beta }{\beta} \, ,</math>
	</div>		</div>
	the ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)]~~ expression for the adiabatic exponent becomes,		the {{ LP41 }} expression for the adiabatic exponent becomes,
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Jet53man: /* Photon Entropy */

2021-10-08T16:40:34Z

Photon Entropy

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	[~~http:~~//~~adsabs.harvard.edu~~/~~abs/1968psen.book.....C Clayton (1968)~~], Chapter 2 (p. 121), Eq. (2-136)<br />		[<b>[[Appendix/References#Clayton68\|<font color="red">Clayton68</font>]]</b>], Chapter 2 (p. 121), Eq. (2-136)<br />
	[[Appendix/References#Shu92\|[<b><font color="red">Shu92</font></b>]]], Vol. I, Chapter 9 (p. 82), Eq. (9.22)		[[Appendix/References#Shu92\|[<b><font color="red">Shu92</font></b>]]], Vol. I, Chapter 9 (p. 82), Eq. (9.22)
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Jet53man: /* Photon Entropy */

2021-10-07T18:29:09Z

Photon Entropy

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	where, <span title="Hydrogen mass"><math>m_H</math></span> is the mass of a hydrogen atom, and <math>n = \rho/m_H</math> is the baryon number density. We conclude, therefore, that <math>s_\mathrm{rad} = s_r/m_H</math>.		where, <span title="Hydrogen mass"><math>m_H</math></span> is the mass of a hydrogen atom, and <math>n = \rho/m_H</math> is the baryon number density. We conclude, therefore, that <math>s_\mathrm{rad} = s_r/m_H</math>.

	We have also found it useful to examine the expression for the entropy of a radiation field used by ~~[https://ui.adsabs.harvard.edu/abs/1986ApJ...307..675F/abstract G. M. Fuller, S. E. Woosley,~~ &~~amp~~; ~~T. A. Weaver (1986, ApJ, 307, 675~~; ~~hereafter FWW86)],~~ primarily because the authors provide, for comparison, a numerical evaluation of the expression's leading coefficient.		We have also found it useful to examine the expression for the entropy of a radiation field used by {{ FWW86full }} — hereafter {{ FWW86hereafter }} — primarily because the authors provide, for comparison, a numerical evaluation of the expression's leading coefficient.

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	<tr><td align="left">		<tr><td align="left">
	In terms of the variables, {{ Template:Math/VAR_Temperature01 }} and {{ Template:Math/VAR_Density01 }}, and the four physical constants, {{ Template:Math/C_PlanckConstant }}, {{ Template:Math/C_SpeedOfLight }}, {{ Template:Math/C_BoltzmannConstant }}, {{ Template:Math/C_AvogadroConstant }}, ~~[https://ui.adsabs.harvard.edu/abs/1986ApJ...307..675F/abstract FWW86]~~ state that the "… entropy per baryon … for photons is,"		In terms of the variables, {{ Template:Math/VAR_Temperature01 }} and {{ Template:Math/VAR_Density01 }}, and the four physical constants, {{ Template:Math/C_PlanckConstant }}, {{ Template:Math/C_SpeedOfLight }}, {{ Template:Math/C_BoltzmannConstant }}, {{ Template:Math/C_AvogadroConstant }}, {{ FWW86hereafter }} state that the "… entropy per baryon … for photons is,"
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	<td align="center" colspan="3">		<td align="center" colspan="3">
	~~[https://ui.adsabs.harvard.edu/abs/1986ApJ...307..675F/abstract FWW86]~~ §III (p. 678), Eq. (11)		{{ FWW86hereafter }} §III (p. 678), Eq. (11)
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	This is consistent with the statement in ~~[https://ui.adsabs.harvard.edu/abs/1986ApJ...307..675F/abstract FWW86]~~ that, when energy is measured in MeV, the leading coefficient is, <math>1.905 \times 10^8</math>.		This is consistent with the statement in {{ FWW86hereafter }} that, when energy is measured in MeV, the leading coefficient is, <math>1.905 \times 10^8</math>.

	Notice that, because the quantity, {{ Template:Math/C_BoltzmannConstant }}{{ Template:Math/VAR_Temperature01 }}, has units of energy, the dimension of <math>s_\gamma</math> is inverse mass. In contrast to this, as has been highlighted in our [[PGE/FirstLawOfThermodynamics#VariableDimensions\|separate discussion of the physical units of various thermodynamic variables]], we expect the units of specific entropy <math>(s_\mathrm{rad})</math> to be specific energy per Kelvin. We suspect that, for convenience, ~~[https://ui.adsabs.harvard.edu/abs/1986ApJ...307..675F/abstract FWW86]~~ have dropped a leading factor of {{ Template:Math/C_BoltzmannConstant }} on the RHS of their expression for <math>s_\gamma</math> and that it is appropriate to draw from their presentation that the specific entropy of the radiation field is,		Notice that, because the quantity, {{ Template:Math/C_BoltzmannConstant }}{{ Template:Math/VAR_Temperature01 }}, has units of energy, the dimension of <math>s_\gamma</math> is inverse mass. In contrast to this, as has been highlighted in our [[PGE/FirstLawOfThermodynamics#VariableDimensions\|separate discussion of the physical units of various thermodynamic variables]], we expect the units of specific entropy <math>(s_\mathrm{rad})</math> to be specific energy per Kelvin. We suspect that, for convenience, {{ FWW86hereafter }} have dropped a leading factor of {{ Template:Math/C_BoltzmannConstant }} on the RHS of their expression for <math>s_\gamma</math> and that it is appropriate to draw from their presentation that the specific entropy of the radiation field is,
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Jet53man: /* Photon Entropy */

2021-10-07T18:06:05Z

Photon Entropy

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	[[Image:CommentButton02.png\|right\|100px\|Note that in the {{ BAC84hereafter }} publication, the Boltzmann constant, k, does not appear in this expression because "T is the temperature in energy units (Boltzmann constant = 1)."]]Similarly, from ~~[https://ui.adsabs.harvard.edu/abs/1984ApJ...280..825B/abstract J. R. Bond, W. D. Arnett, and B. J. Carr (1984, ApJ, 280, 825; hereafter BAC84)]~~ we find that "… the photon entropy [is],"		[[Image:CommentButton02.png\|right\|100px\|Note that in the {{ BAC84hereafter }} publication, the Boltzmann constant, k, does not appear in this expression because "T is the temperature in energy units (Boltzmann constant = 1)."]]Similarly, from {{ BAC84hereafter }} we find that "… the photon entropy [is],"

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Jet53man: /* Expression for Adiabatic Exponent */

2021-10-07T18:01:17Z

Expression for Adiabatic Exponent

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	<tr><td align="center" colspan="3">		<tr><td align="center" colspan="3">
	Eq. (24) from {{ ~~Eddington18full~~ }}		Eq. (24) from {{ Eddington18 }}
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Jet53man at 17:49, 7 October 2021

2021-10-07T17:49:26Z

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	\|}		\|}

	Here we draw upon the work of {{ BAC84full }} — hereafter ~~BAC84~~ — who were among the first to seriously address the question of the fate of very massive (stellar) objects.		Here we draw upon the work of {{ BAC84full }} — hereafter {{ BAC84hereafter }} — who were among the first to seriously address the question of the fate of very massive (stellar) objects.
	<br />		<br />

	==Equation of State==		==Equation of State==
	Our discussion of the equation of state (EOS) that was used by ~~BAC84~~ draws on the terminology that has already been adopted in our [[SR#Equation_of_State\|introductory discussion of ''supplemental relations'']] and closely parallels [[SSC/Structure/BiPolytropes/Analytic1.53#Envelope\|our review of the properties of the envelope]] that {{ Milne30full }} used to construct a bipolytropic sphere.		Our discussion of the equation of state (EOS) that was used by {{ BAC84hereafter }} draws on the terminology that has already been adopted in our [[SR#Equation_of_State\|introductory discussion of ''supplemental relations'']] and closely parallels [[SSC/Structure/BiPolytropes/Analytic1.53#Envelope\|our review of the properties of the envelope]] that {{ Milne30full }} used to construct a bipolytropic sphere.

	===Expression for Total Pressure===		===Expression for Total Pressure===
	Ignoring the component due to a degenerate electron gas, <math>P_\mathrm{deg}</math>, the total gas pressure can be expressed as the sum of two separate components: a component of ideal gas pressure, and a component of radiation pressure. That is, in ~~BAC84~~ the total pressure is given by the expression,		Ignoring the component due to a degenerate electron gas, <math>P_\mathrm{deg}</math>, the total gas pressure can be expressed as the sum of two separate components: a component of ideal gas pressure, and a component of radiation pressure. That is, in {{ BAC84hereafter }} the total pressure is given by the expression,
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	Now, ~~BAC84~~ define the rest-mass density in terms of the mean baryon mass, <math>m_B</math>, via the expression, <math>\rho = m_B n</math>, and write (see their equation 1),		Now, {{ BAC84hereafter }} define the rest-mass density in terms of the mean baryon mass, <math>m_B</math>, via the expression, <math>\rho = m_B n</math>, and write (see their equation 1),
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	===Ratio of Radiation Pressure to Gas Pressure ===		===Ratio of Radiation Pressure to Gas Pressure ===
	Following {{ Milne30 }}, we have defined the parameter, <math>\beta</math>, as the ratio of gas pressure to total pressure. That is, in the context of ~~BAC84~~, we have,		Following {{ Milne30 }}, we have defined the parameter, <math>\beta</math>, as the ratio of gas pressure to total pressure. That is, in the context of {{ BAC84hereafter }}, we have,
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	<math>\beta \equiv \frac{P_\mathrm{gas} }{P} \, ,</math>		<math>\beta \equiv \frac{P_\mathrm{gas} }{P} \, ,</math>
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	</div>		</div>

	Using a different notation, ~~BAC84~~ (see their equation 5) define <span title="Ratio of radiation pressure to gas pressure"><math>\sigma</math></span> as the ratio of the radiation pressure to the gas pressure. Therefore, in converting from our notation to theirs we have,		Using a different notation, {{ BAC84hereafter }} (see their equation 5) define <span title="Ratio of radiation pressure to gas pressure"><math>\sigma</math></span> as the ratio of the radiation pressure to the gas pressure. Therefore, in converting from our notation to theirs we have,
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	<math>\sigma = \frac{1-\beta}{\beta} ~~~~\Rightarrow ~~~~ \beta = (1 + \sigma)^{-1} \, , </math>		<math>\sigma = \frac{1-\beta}{\beta} ~~~~\Rightarrow ~~~~ \beta = (1 + \sigma)^{-1} \, , </math>
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	= \frac{a_\mathrm{rad} T^3}{3Y_T n k} \, ,</math>		= \frac{a_\mathrm{rad} T^3}{3Y_T n k} \, ,</math>
	</div>		</div>
	which is precisely the definition provided in equation (5) of ~~BAC84~~.		which is precisely the definition provided in equation (5) of {{ BAC84hereafter }}.

	===Expression for Adiabatic Exponent===		===Expression for Adiabatic Exponent===
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	====Used by BAC84====		====Used by BAC84====
	On the other hand, without derivation ~~BAC84~~ state (see their equation 4) that the adiabatic exponent is,		On the other hand, without derivation {{ BAC84hereafter }} state (see their equation 4) that the adiabatic exponent is,
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	Clearly, in the limit <math>~\sigma \rightarrow \infty</math>, this gives <math>~\Gamma_1 \rightarrow 4/3</math>, which, as it should, matches the limiting value obtained from the [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] expression when <math>~\beta = 0</math>.		Clearly, in the limit <math>~\sigma \rightarrow \infty</math>, this gives <math>~\Gamma_1 \rightarrow 4/3</math>, which, as it should, matches the limiting value obtained from the [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] expression when <math>~\beta = 0</math>.

	~~BAC84~~ do not explicitly state what value they used for the ratio of specific heats when deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for <math>~\sigma = 0</math>. In this limit, the ~~BAC84~~ expression gives,		{{ BAC84hereafter }} do not explicitly state what value they used for the ratio of specific heats when deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for <math>~\sigma = 0</math>. In this limit, the {{ BAC84hereafter }} expression gives,
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	</div>		</div>

	The general ~~BAC84~~ expression should therefore match the (even more) general [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] expression if we set <math>~\gamma = \tfrac{5}{3}</math>. Let's check this out. Inserting this specific value of <math>~\gamma</math>, and remembering (from above) that,		The general {{ BAC84hereafter }} expression should therefore match the (even more) general [http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris (1941)] expression if we set <math>~\gamma = \tfrac{5}{3}</math>. Let's check this out. Inserting this specific value of <math>~\gamma</math>, and remembering (from above) that,
	<div align="center">		<div align="center">
	<math>\sigma = \frac{1-\beta }{\beta} \, ,</math>		<math>\sigma = \frac{1-\beta }{\beta} \, ,</math>
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	===Mass Normalization===		===Mass Normalization===
	Now, according to ~~BAC84~~ (see their equation 8), when the total pressure is written in polytropic form — specifically, if we set,		Now, according to {{ BAC84hereafter }} (see their equation 8), when the total pressure is written in polytropic form — specifically, if we set,
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	<math>P = K\rho^{(1+1/n_p)} </math>		<math>P = K\rho^{(1+1/n_p)} </math>
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	</table>		</table>
	</div>		</div>
	Adopting the ~~BAC84~~ terminology, this means that,		Adopting the {{ BAC84hereafter }} terminology, this means that,
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	</div>		</div>
	When radiation pressure significantly dominates over gas pressure — that is, in the limit <math>~\sigma \gg 1</math> — the leading factor is approximately unity, in which case we see that this expression for <math>~M_\mathrm{norm}</math> exactly matches the expression for <math>~M_{u,3}</math> given by equation (10) of ~~BAC84~~.		When radiation pressure significantly dominates over gas pressure — that is, in the limit <math>~\sigma \gg 1</math> — the leading factor is approximately unity, in which case we see that this expression for <math>~M_\mathrm{norm}</math> exactly matches the expression for <math>~M_{u,3}</math> given by equation (10) of {{ BAC84hereafter }}.

	====Polytropic Index Slightly Less Than 3====		====Polytropic Index Slightly Less Than 3====
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	</div>		</div>
	This matches exactly expression (7) in ~~BAC84~~. Again from [[#GammaApprox\|above]] — and continuing to assume <math>~\sigma \gg 1</math> — we have,		This matches exactly expression (7) in {{ BAC84hereafter }}. Again from [[#GammaApprox\|above]] — and continuing to assume <math>~\sigma \gg 1</math> — we have,
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	\approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>		\approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>
	</div>		</div>
	which certainly is close to unity when <math>~\sigma \gg 1</math>. After setting <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in ~~BAC84~~.		which certainly is close to unity when <math>~\sigma \gg 1</math>. After setting <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in {{ BAC84hereafter }}.

	==Entropy of Radiation Field==		==Entropy of Radiation Field==
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	[[Image:CommentButton02.png\|right\|100px\|Note that in the ~~BAC84~~ publication, the Boltzmann constant, k, does not appear in this expression because "T is the temperature in energy units (Boltzmann constant = 1)."]]Similarly, from [https://ui.adsabs.harvard.edu/abs/1984ApJ...280..825B/abstract J. R. Bond, W. D. Arnett, and B. J. Carr (1984, ApJ, 280, 825; hereafter BAC84)] we find that "… the photon entropy [is],"		[[Image:CommentButton02.png\|right\|100px\|Note that in the {{ BAC84hereafter }} publication, the Boltzmann constant, k, does not appear in this expression because "T is the temperature in energy units (Boltzmann constant = 1)."]]Similarly, from [https://ui.adsabs.harvard.edu/abs/1984ApJ...280..825B/abstract J. R. Bond, W. D. Arnett, and B. J. Carr (1984, ApJ, 280, 825; hereafter BAC84)] we find that "… the photon entropy [is],"

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	<td align="center" colspan="3">~~[https://ui.adsabs.harvard.edu/abs/1984ApJ...280..825B/abstract BAC84]~~, §2b (p. 827), Eq. (5)</td>		<td align="center" colspan="3">{{ BAC84hereafter }}, §2b (p. 827), Eq. (5)</td>
	</tr>		</tr>
	</table>		</table>

Jet53man at 17:44, 7 October 2021

2021-10-07T17:44:06Z

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	====For any value of the ratio of specific heats====		====For any value of the ratio of specific heats====
	From equation (2) of {{ LP41full }} — see, for example, [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — or, equally well, from equation (131) in Chapter II of [[~~User:Tohline/~~Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,		From equation (2) of {{ LP41full }} — see, for example, [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — or, equally well, from equation (131) in Chapter II of [[Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
	<div align="center">		<div align="center">
	<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>		<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>

Jet53man: /* Polytropic Index Equals 3 */

2021-10-07T17:31:46Z

Polytropic Index Equals 3

← Older revision		Revision as of 13:31, 7 October 2021
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	====Polytropic Index Equals 3====		====Polytropic Index Equals 3====
	Referencing our [[~~User:Tohline/~~SSC/Structure/BiPolytropes/Analytic1.~~5_3~~#HighlightedExpressions\|separate discussion of Milne's (1930) work]], when <math>~n_p = 3</math>, the polytropic constant is related to the relevant set of physical parameters via the relation,		Referencing our [[SSC/Structure/BiPolytropes/Analytic1.53#HighlightedExpressions\|separate discussion of Milne's (1930) work]], when <math>~n_p = 3</math>, the polytropic constant is related to the relevant set of physical parameters via the relation,

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Jet53man: /* For any value of the ratio of specific heats */

2021-10-07T17:08:51Z

For any value of the ratio of specific heats

← Older revision		Revision as of 13:08, 7 October 2021
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	====For any value of the ratio of specific heats====		====For any value of the ratio of specific heats====
	From equation (2) of ~~[http://adsabs.harvard.edu/abs/1941ApJ....94..124L Ledoux & Pekeris] (1941, ApJ, 94, 124)~~ — see, for example, [[~~User:Tohline/~~SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — or, equally well, from equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,		From equation (2) of {{ LP41full }} — see, for example, [[SSC/Perturbations#Ledoux_and_Pekeris_.281941.29\|our brief summary of this work]] — or, equally well, from equation (131) in Chapter II of [[User:Tohline/Appendix/References#C67\|<b>[<font color="red">C67</font>]</b>]], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,
	<div align="center">		<div align="center">
	<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>		<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>