SSC/Structure/BiPolytropes/Analytic15 - Revision history

Jet53man: /* Profile */

2023-07-04T17:11:37Z

Profile

← Older revision		Revision as of 13:11, 4 July 2023
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	==Profile==		==Profile==

	Once the values of the key set of parameters have been determined as illustrated in the preceding formula table, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_4:__Throughout_the_core_.28.29\|step #4]] and [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_8:__Envelope.27s_Physical_Profile\|step #8]], above. The following table summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, <math>~\rho^(r^)</math>, the normalized gas pressure, <math>~P^(r^)</math>, and the normalized mass interior to <math>~r^</math>, <math>~M_r^(r^)</math>. For all profiles, the relevant normalized radial coordinate is <math>~r^</math>, as defined in the 2<sup>nd</sup> row of the table. Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of the table.		Once the values of the key set of parameters have been determined as illustrated in the preceding formula table, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[SSC/Structure/BiPolytropes/Analytic15#Step_4:__Throughout_the_core_.28.29\|step #4]] and [[SSC/Structure/BiPolytropes/Analytic15#Step_8:__Envelope.27s_Physical_Profile\|step #8]], above. The following table summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, <math>~\rho^(r^)</math>, the normalized gas pressure, <math>~P^(r^)</math>, and the normalized mass interior to <math>~r^</math>, <math>~M_r^(r^)</math>. For all profiles, the relevant normalized radial coordinate is <math>~r^</math>, as defined in the 2<sup>nd</sup> row of the table. Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of the table.

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	<td align="left" colspan="6">		<td align="left" colspan="6">
	<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_6:__Envelope_Solution\|presented in Step 6, above]].		<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution\|presented in Step 6, above]].
	</td>		</td>
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Jet53man: /* Analytic Solution of Key Interface Relation */

2023-07-02T00:37:12Z

Analytic Solution of Key Interface Relation

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Jet53man at 00:22, 2 July 2023

2023-07-02T00:22:38Z

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	=BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>=		=BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>=
	{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

2021-07-29T21:52:40Z

BiPolytrope with n_c = 1 and n_e=5

← Older revision		Revision as of 17:52, 29 July 2021
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	! style="height: 125px; width: 125px; background-color:white;" \|		! style="height: 125px; width: 125px; background-color:white;" \|
	<font size="-1">[[H_BookTiledMenu#~~Equilibrium_Structures~~\|<b>Murphy (1983)<br /><br />Analytic</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>		<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Murphy (1983)<br /><br />Analytic</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>
	\|}		\|}
	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)], and attributes of their physical structure were further discussed by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]; [[SSC/Structure/BiPolytropes/Analytic15#Key_References\|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing Murphy's (1983) work.		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)], and attributes of their physical structure were further discussed by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]; [[SSC/Structure/BiPolytropes/Analytic15#Key_References\|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing Murphy's (1983) work.

Jet53man at 21:51, 29 July 2021

2021-07-29T21:51:26Z

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	=BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>=		=BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>=
	{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
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	=Related Discussions=		=Related Discussions=
	* [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes emdeded in an external medium]]
	* [[~~User:Tohline/~~SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]

	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

2021-07-29T04:25:03Z

BiPolytrope with n_c = 1 and n_e=5

← Older revision		Revision as of 00:25, 29 July 2021
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	! style="height: 125px; width: 125px; background-color:white;" \|		! style="height: 125px; width: 125px; background-color:white;" \|
	<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Analytic</b>]]<br /> (n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>		<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Murphy (1983)<br /><br />Analytic</b>]]<br />(n<sub>c</sub>, n<sub>e</sub>) = (1, 5)</font>
	\|}		\|}
	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)], and attributes of their physical structure were further discussed by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]; [[SSC/Structure/BiPolytropes/Analytic15#Key_References\|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing Murphy's (1983) work.		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.]]Here we construct a [[SSC/Structure/BiPolytropes#BiPolytropes\|system of bipolytropic configurations]] in which the core has an <math>~n_c=1</math> polytropic index and the envelope has an <math>~n_e=5</math> polytropic index. As in the case of our [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|separately discussed, "mirror image" bipolytropic configurations having <math>~(n_c, n_e) = (5, 1)</math>]], this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)], and attributes of their physical structure were further discussed by [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]; [[SSC/Structure/BiPolytropes/Analytic15#Key_References\|additional, closely related references are given below]]. In the discussion that follows, we will be heavily referencing Murphy's (1983) work.

Jet53man: /* Key References */

2021-07-28T22:49:49Z

Key References

← Older revision		Revision as of 18:49, 28 July 2021
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	=Key References=		=Key References=
	* [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] ''A New Solution of the Lane-Emden Equation of Index n = 5''		* [http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] ''A New Solution of the Lane-Emden Equation of Index n = 5''
	* [http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115)]: ''Remark on the Polytrope of Index 5'' — the result of this work by Buchdahl has been [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#Buchdahl1978\|highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>]].		* [http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115)]: ''Remark on the Polytrope of Index 5'' — the result of this work by Buchdahl has been [[SSC/Structure/BiPolytropes/Analytic51#Buchdahl1978\|highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>]].
	* [http://adsabs.harvard.edu/abs/1980PASAu...4...37M J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37)]: ''A Finite Radius Solution for the Polytrope Index 5''		* [http://adsabs.harvard.edu/abs/1980PASAu...4...37M J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37)]: ''A Finite Radius Solution for the Polytrope Index 5''
	* [http://adsabs.harvard.edu/abs/1980PASAu...4...41M J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41)]: ''On the F-Type and M-Type Solutions of the Lane-Emden Equation''		* [http://adsabs.harvard.edu/abs/1980PASAu...4...41M J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41)]: ''On the F-Type and M-Type Solutions of the Lane-Emden Equation''
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	* [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]: ''Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models''		* [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]: ''Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models''
	* [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]: ''Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models''		* [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]: ''Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models''


	=Related Discussions=		=Related Discussions=

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

2021-07-28T22:17:09Z

BiPolytrope with n_c = 1 and n_e=5

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2021-07-24T23:20:28Z

Created page with "__FORCETOC__ =BiPolytrope with <math>n_c = 1</math> and <math>n_e=5</math>= File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 12 April 2015: I became aware..."

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SSC/Structure/BiPolytropes/Analytic15 - Revision history

Jet53man: /* Profile */

Jet53man: /* Analytic Solution of Key Interface Relation */

Jet53man at 00:22, 2 July 2023

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

Jet53man at 21:51, 29 July 2021

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

Jet53man: /* Key References */

Jet53man: /* BiPolytrope with n_c = 1 and n_e=5 */

Jet53man: Created page with "__FORCETOC__ =BiPolytrope with n_c = 1 and n_e=5= File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 12 April 2015: I became aware..."

Jet53man: Created page with "FORCETOC =BiPolytrope with $n_c = 1$ and $n_e=5$ = File:CommentButton02.png|right|100px|Comment by J. E. Tohline on 12 April 2015: I became aware..."