SSC/Structure/PolytropesEmbedded - Revision history

Joel2: /* Stahler's Presentation */

2024-07-19T18:04:02Z

Stahler's Presentation

← Older revision		Revision as of 14:04, 19 July 2024
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	Notice that, via these two normalizations, {{ Stahler83hereafter }} chose to express <math>R_\mathrm{eq}</math> and <math>M</math> in terms of {{Math/MP_PolytropicConstant}} and the applied external pressure, <math>P_\mathrm{e}</math>.		Notice that, via these two normalizations, {{ Stahler83hereafter }} chose to express <math>R_\mathrm{eq}</math> and <math>M</math> in terms of {{Math/MP_PolytropicConstant}} and the applied external pressure, <math>P_\mathrm{e}</math>.

			<font color="red"><b>NOTE:</b></font>   An [[SSC/Structure/StahlerMassRadius\|accompanying chapter]] presents a much more detailed description of the ''sequences'' of truncated polytropic spheres that are derived and discussed by {{ Stahler83hereafter }}.

	===Reconciliation===		===Reconciliation===

Joel2: /* Stahler's Presentation */

2024-07-19T17:50:16Z

Stahler's Presentation

← Older revision		Revision as of 13:50, 19 July 2024
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	===Stahler's Presentation===		===Stahler's Presentation===
	Similarly, in Appendix B of his work, ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler (1983)]~~ states that the mass, <math>~M</math>, associated with the equilibrium radius, <math>~R_\mathrm{eq}</math>, of embedded polytropic spheres is,		Similarly, in Appendix B of his work, {{ Stahler83 }} — hereafter, {{ Stahler83hereafter }} — states that the mass, <math>M</math>, associated with the equilibrium radius, <math>R_\mathrm{eq}</math>, of embedded polytropic spheres is,
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	Notice that, via these two normalizations, ~~Stahler~~ chose to express <math>~R_\mathrm{eq}</math> and <math>~M</math> in terms of {{Math/MP_PolytropicConstant}} and the applied external pressure, <math>~P_\mathrm{e}</math>.		Notice that, via these two normalizations, {{ Stahler83hereafter }} chose to express <math>R_\mathrm{eq}</math> and <math>M</math> in terms of {{Math/MP_PolytropicConstant}} and the applied external pressure, <math>P_\mathrm{e}</math>.

	===Reconciliation===		===Reconciliation===

Joel2: /* Whitworth's Presentation */

2024-07-19T17:46:47Z

Whitworth's Presentation

← Older revision		Revision as of 13:46, 19 July 2024
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	===Whitworth's Presentation===		===Whitworth's Presentation===

	In §5 of his paper, ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]~~ also presents the set of parametric		In §5 of his paper, {{ Whitworth81}} — hereafter, {{ Whitworth81hereafter }} — also presents the set of parametric
	equations that define what the equilibrium radius, <math>~R_\mathrm{eq}</math>, is of an embedded polytrope for a certain imposed external pressure, <math>~P_\mathrm{e}</math>, namely,		equations that define what the equilibrium radius, <math>R_\mathrm{eq}</math>, is of an embedded polytrope for a certain imposed external pressure, <math>P_\mathrm{e}</math>, namely,
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	<math>		<math>
	~P_\mathrm{rf} \biggl\{ 2^{-8/\eta} \biggl(\frac{5\|\eta-1\|}{\eta} \biggr)^3 \biggl(\frac{3}{\xi} \biggr)^4		P_\mathrm{rf} \biggl\{ 2^{-8/\eta} \biggl(\frac{5\|\eta-1\|}{\eta} \biggr)^3 \biggl(\frac{3}{\xi} \biggr)^4
	\biggl\|\frac{d\theta_n}{d\xi} \biggr\|^{-2} \biggr\}_{\xi_e}^{\eta/(3\eta - 4)} \theta_n^{\eta/(\eta-1)}		\biggl\|\frac{d\theta_n}{d\xi} \biggr\|^{-2} \biggr\}_{\xi_e}^{\eta/(3\eta - 4)} \theta_n^{\eta/(\eta-1)}
	</math>		</math>
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	where, in order to obtain the second line of the two relations we have used the substitution, <math>~\eta \rightarrow (1+1/n)</math>, and, as is detailed in an [[SSC/Structure/PolytropesASIDE1\|accompanying ASIDE]], ~~Whitworth~~ "referenced" <math>~P_\mathrm{e}</math> and <math>~R_\mathrm{eq}</math> to, respectively,		where, in order to obtain the second line of the two relations we have used the substitution, <math>\eta \rightarrow (1+1/n)</math>, and, as is detailed in an [[SSC/Structure/PolytropesASIDE1\|accompanying ASIDE]], {{ Whitworth81hereafter }} "referenced" <math>P_\mathrm{e}</math> and <math>R_\mathrm{eq}</math> to, respectively,

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	Via these normalizations, ~~Whitworth~~ — as did ~~Horedt (1970)~~ — chose to express <math>~R_\mathrm{eq}</math> and <math>~P_\mathrm{e}</math> in terms of {{Math/MP_PolytropicConstant}} and the system's total mass, <math>~M</math>.		Via these normalizations, {{ Whitworth81hereafter }} — as did {{ Horedt70hereafter }} — chose to express <math>R_\mathrm{eq}</math> and <math>P_\mathrm{e}</math> in terms of {{Math/MP_PolytropicConstant}} and the system's total mass, <math>M</math>.

	To convert from Whitworth's expressions, which use one set of normalization parameters <math>~(R_\mathrm{rf},P_\mathrm{rf})</math>, to Horedt's expressions, which use a somewhat different set of normalization parameters — identified here as <math>~(R_\mathrm{Horedt},P_\mathrm{Horedt})</math> — one simply needs to make use of the relations,		To convert from Whitworth's expressions, which use one set of normalization parameters <math>(R_\mathrm{rf},P_\mathrm{rf})</math>, to Horedt's expressions, which use a somewhat different set of normalization parameters — identified here as <math>(R_\mathrm{Horedt},P_\mathrm{Horedt})</math> — one simply needs to make use of the relations,
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Joel2: /* Horedt's Presentation */

2024-07-19T17:40:26Z

Horedt's Presentation

← Older revision		Revision as of 13:40, 19 July 2024
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	== General Properties==		== General Properties==
	===Horedt's Presentation===		===Horedt's Presentation===
	It appears as though ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]~~ was the first to draw an analogy between the mass limit that is associated with bounded isothermal spheres — the so-called [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]] — and the limiting mass that can be found in association with equilibrium sequences of embedded polytropes that have polytropic indexes <math>~n > 3</math>. Using a tilde to denote values of parameters at the (truncated) edge of a pressure-bounded polytropic sphere, ~~Horedt~~ (see the bottom of his p. 83) derives the following set of parametric equations relating the configuration's dimensionless radius, <math>~r_a</math>, to a specified dimensionless bounding pressure, <math>~p_a</math>:		It appears as though {{ Horedt70 }} — hereafter, {{ Horedt70hereafter }} — was the first to draw an analogy between the mass limit that is associated with bounded isothermal spheres — the so-called [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]] — and the limiting mass that can be found in association with equilibrium sequences of embedded polytropes that have polytropic indexes <math>~n > 3</math>. Using a tilde to denote values of parameters at the (truncated) edge of a pressure-bounded polytropic sphere, {{ Horedt70hereafter }} (see the bottom of his p. 83) derives the following set of parametric equations relating the configuration's dimensionless radius, <math>~r_a</math>, to a specified dimensionless bounding pressure, <math>~p_a</math>:
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Joel2: /* Embedded Polytropic Spheres */

2024-07-19T17:35:59Z

Embedded Polytropic Spheres

← Older revision		Revision as of 13:35, 19 July 2024
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	<!-- [[Image:LSU_Structure_still.gif\|90px\|left]] -->		<!-- [[Image:LSU_Structure_still.gif\|90px\|left]] -->
	In a [[SSC/Structure/Polytropes\|separate discussion]] we have shown how to determine the structure of isolated polytropic spheres. These are rather idealized stellar structures in which the pressure and density both drop to zero at the surface of the configuration. Here we consider how the equilibrium radius of a polytropic configuration of a given <math>~M</math> and {{Math/MP_PolytropicConstant}} is modified when it is embedded in an external medium of pressure <math>~P_e</math>. We will begin by reviewing the general properties of embedded (and truncated) polytropes for a wide range of polytropic indexes, principally summarizing the published descriptions provided by ~~[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]~~, by ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]~~, by [http://adsabs.harvard.edu/abs/1981PASJ...33..273K Kimura (1981a)], and by ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]~~. Then we will focus in more detail on polytropes of index {{Math/MP_PolytropicIndex}} = 1 and {{Math/MP_PolytropicIndex}} = 5 because their structures can be described by closed-form analytic expressions.		In a [[SSC/Structure/Polytropes\|separate discussion]] we have shown how to determine the structure of isolated polytropic spheres. These are rather idealized stellar structures in which the pressure and density both drop to zero at the surface of the configuration (for 0 ≤ {{Math/MP_PolytropicIndex}} < 5) or in which the equilibrium configuration extends to infinity (for 5 ≤ {{Math/MP_PolytropicIndex}} ≤ ∞). Here we consider how the equilibrium radius of a polytropic configuration of a given <math>~M</math> and {{Math/MP_PolytropicConstant}} is modified when it is embedded in an external medium of pressure <math>~P_e</math>. We will begin by reviewing the general properties of embedded (and truncated) polytropes for a wide range of polytropic indexes, principally summarizing the published descriptions provided by {{ Horedt70full }}, by {{ Whitworth81full }}, by [http://adsabs.harvard.edu/abs/1981PASJ...33..273K Kimura (1981a)], and by {{ Stahler83full }}. Then we will focus in more detail on polytropes of index {{Math/MP_PolytropicIndex}} = 1 and {{Math/MP_PolytropicIndex}} = 5 because their structures can be described by closed-form analytic expressions.
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Joel2 at 23:58, 18 December 2023

2023-12-18T23:58:08Z

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Joel2 at 23:43, 18 December 2023

2023-12-18T23:43:56Z

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Joel2: Created page with "FORCETOC =Embedded Polytropic Spheres= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" | Embedded
Polytropes |} In a separate discussion we have..."

2023-12-18T22:26:09Z

Created page with "__FORCETOC__  =Embedded Polytropic Spheres= {| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black" |- ! style="height: 125px; width: 125px; background-color:white;" | Embedded Polytropes |}  In a separate discussion we have..."

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