Jet53man: /* Supplemental Relations */

2021-09-13T16:01:51Z

Supplemental Relations

← Older revision		Revision as of 12:01, 13 September 2021
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	Also following HNM82, we impose a steady-state velocity flow-field that is described by a fluid with uniform rotational velocity, <math>v_\varphi</math>. Drawing from our table of example [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,		Also following HNM82, we impose a steady-state velocity flow-field that is described by a fluid with uniform rotational velocity, <math>v_\varphi</math>. Drawing from our table of example [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,
	<div align="center~~"><font size="+1~~">		<div align="center">
	<math>		<math>
	~\Psi(\varpi) = - v_\varphi^2 \ln\biggl(\frac{\varpi}{\varpi_0}\biggr) .		~\Psi(\varpi) = - v_\varphi^2 \ln\biggl(\frac{\varpi}{\varpi_0}\biggr) .
	</math>		</math>
	~~</font>~~</div>		</div>

	==Summary==		==Summary==

Jet53man at 16:00, 13 September 2021

2021-09-13T16:00:12Z

← Older revision		Revision as of 12:00, 13 September 2021
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	! style="height: 150px; width: 150px; background-color:#ffeeee;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Hayashi, Narita<br />& Miyama's<br />Analytic Sol'n<br />(1982)</b>]]		! style="height: 150px; width: 150px; background-color:#ffeeee;" \|[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like\|<b>Hayashi, Narita<br />& Miyama's<br />Analytic Sol'n<br />(1982)</b>]]
	\|}		\|}
	[https://ui.adsabs.harvard.edu/abs/1982PThPh..68.1949H/abstract Hayashi, Narita & Miyama] (1982; hereafter HNM82) discovered an analytic solution to the equations that govern the structure of rotationally flattened, self-gravitating isothermal gas clouds. Their solution describes a family of centrally condensed models whose degree of flattening ranges from a spherical structure to an infinitesimally thin disk. For several reasons, I consider this to be one of the most remarkable discoveries — and, hence, one of the most significant papers — related to the structure of self-gravitating systems that was published in the decade of the '80s. First, as has been [[~~User:Tohline/~~Apps/MaclaurinSpheroids#Maclaurin_Spheroids_(axisymmetric_structure)\|remarked earlier]], there is no particular reason why one should guess ahead of time that the equilibrium properties of ''any'' rotating, self-gravitating configuration should be describable in terms of analytic functions. When dealing with compressible equations of state, such analytic solutions are rare even in the context of spherically symmetric structures, so it is impressive that HNM82 found a solution for rotationally flattened, isothermal configurations. Second, about six months earlier the same year, [https://ui.adsabs.harvard.edu/abs/1982ApJ...259..535T/abstract Alar Toomre (1982)] published an independent discovery and strikingly independent derivation of exactly the same family of rotationally flattened, isothermal models.<sup>1</sup> Third, these two independent derivations were motivated by a desire to better understand two quite different astrophysical environments: The research of HNM82 was focused on star-forming gas clouds while Toomre's research was focused on the structure of elliptical galaxies and the dark-matter halos around spiral galaxies.		[https://ui.adsabs.harvard.edu/abs/1982PThPh..68.1949H/abstract Hayashi, Narita & Miyama] (1982; hereafter HNM82) discovered an analytic solution to the equations that govern the structure of rotationally flattened, self-gravitating isothermal gas clouds. Their solution describes a family of centrally condensed models whose degree of flattening ranges from a spherical structure to an infinitesimally thin disk. For several reasons, I consider this to be one of the most remarkable discoveries — and, hence, one of the most significant papers — related to the structure of self-gravitating systems that was published in the decade of the '80s. First, as has been [[Apps/MaclaurinSpheroids#Maclaurin_Spheroids_(axisymmetric_structure)\|remarked earlier]], there is no particular reason why one should guess ahead of time that the equilibrium properties of ''any'' rotating, self-gravitating configuration should be describable in terms of analytic functions. When dealing with compressible equations of state, such analytic solutions are rare even in the context of spherically symmetric structures, so it is impressive that HNM82 found a solution for rotationally flattened, isothermal configurations. Second, about six months earlier the same year, [https://ui.adsabs.harvard.edu/abs/1982ApJ...259..535T/abstract Alar Toomre (1982)] published an independent discovery and strikingly independent derivation of exactly the same family of rotationally flattened, isothermal models.<sup>1</sup> Third, these two independent derivations were motivated by a desire to better understand two quite different astrophysical environments: The research of HNM82 was focused on star-forming gas clouds while Toomre's research was focused on the structure of elliptical galaxies and the dark-matter halos around spiral galaxies.

	Despite my assertion that HNM82 is one of the most significant papers to be published over the past few decades, citation indexes reveal that it is not widely referenced. In large part I attribute this to the fact that HNM82 was published in a Japanese journal ([https://academic.oup.com/ptp ''Progress of Theoretical Physics'']) that has only fairly recently made its archival articles available to the open-access, [https://ui.adsabs.harvard.edu Astrophysics Data System].		Despite my assertion that HNM82 is one of the most significant papers to be published over the past few decades, citation indexes reveal that it is not widely referenced. In large part I attribute this to the fact that HNM82 was published in a Japanese journal ([https://academic.oup.com/ptp ''Progress of Theoretical Physics'']) that has only fairly recently made its archival articles available to the open-access, [https://ui.adsabs.harvard.edu Astrophysics Data System].
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	==Governing Relations==		==Governing Relations==

	As has been derived [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies\|elsewhere]], for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following ''algebraic'' expression:		As has been derived [[AxisymmetricConfigurations/SolutionStrategies\|elsewhere]], for axisymmetric configurations that obey a barotropic equation of state, hydrostatic balance is governed by the following ''algebraic'' expression:
	<div align="center">		<div align="center">
	<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>		<math>~H + \Phi_\mathrm{eff} = C_\mathrm{B} ,</math>
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	</div>		</div>

	Also following HNM82, we impose a steady-state velocity flow-field that is described by a fluid with uniform rotational velocity, <math>v_\varphi</math>. Drawing from our table of example [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,		Also following HNM82, we impose a steady-state velocity flow-field that is described by a fluid with uniform rotational velocity, <math>v_\varphi</math>. Drawing from our table of example [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], the centrifugal potential that describes this chosen flow-field is given by the expression,
	<div align="center"><font size="+1">		<div align="center"><font size="+1">
	<math>		<math>
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	</math>		</math>
	</div>		</div>
	This must be solved in conjunction with the Poisson equation, which specifies the second key relationship between {{ ~~User:Tohline/~~Math/VAR_NewtonianPotential01 }} and {{ ~~User:Tohline/~~Math/VAR_Density01 }}, namely,		This must be solved in conjunction with the Poisson equation, which specifies the second key relationship between {{ Math/VAR_NewtonianPotential01 }} and {{ Math/VAR_Density01 }}, namely,
	<div align="center">		<div align="center">
	<math>		<math>
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	HNM82 provide a very clear and detailed description of the approach that they took to solving the above-identified set of simplified governing relations. (Note that Hayashi is credited with deriving the analytic solution.) In very general terms, one can understand the thought process that must have been going on in Hayashi's mind:		HNM82 provide a very clear and detailed description of the approach that they took to solving the above-identified set of simplified governing relations. (Note that Hayashi is credited with deriving the analytic solution.) In very general terms, one can understand the thought process that must have been going on in Hayashi's mind:
	# For an isothermal gas cloud, the enthalpy is necessarily a logarithmic function of the density.		# For an isothermal gas cloud, the enthalpy is necessarily a logarithmic function of the density.
	# For spherically symmetric, isothermal configurations, a solution to the governing relations exists in which {{ ~~User:Tohline/~~Math/VAR_Density01 }} can be expressed as a [[~~User:Tohline/~~SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State\|power-law function]] of the radius, specifically, <math>~\rho \propto r^{-2}</math>; hence, the enthalpy displays a logarithmic dependence on the distance.		# For spherically symmetric, isothermal configurations, a solution to the governing relations exists in which {{ Math/VAR_Density01 }} can be expressed as a [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State\|power-law function]] of the radius, specifically, <math>~\rho \propto r^{-2}</math>; hence, the enthalpy displays a logarithmic dependence on the distance.
	# Although one could attempt to derive equilibrium structures having a wide range of [[~~User:Tohline/~~AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], it would seem wisest to select a centrifugal potential function that at least has the same ''form'' as the enthalpy; hence, the choice was made to impose <math>~v_\varphi = \mathrm{constant}</math> so that, like the enthalpy, the centrifugal potential would exhibit a logarithmic dependence on the distance.		# Although one could attempt to derive equilibrium structures having a wide range of [[AxisymmetricConfigurations/SolutionStrategies#SRPtable\|''Simple rotation profiles'']], it would seem wisest to select a centrifugal potential function that at least has the same ''form'' as the enthalpy; hence, the choice was made to impose <math>~v_\varphi = \mathrm{constant}</math> so that, like the enthalpy, the centrifugal potential would exhibit a logarithmic dependence on the distance.
	# Three birds, so to speak, can be killed with one stone by ''guessing'' a 2D equilibrium density profile of the form <math>~\rho(\varpi,z) = g(\varpi,z)/\varpi^{2}</math>: The <math>~\varpi^{-2}</math> dependence can be combined strategically with <math>~\varpi^{-2}</math> dependence of the centrifugal potential; although it depends on <math>\varpi</math> instead of <math>r</math>, the density profile will at least ''resemble'' the spherical solution; and — certainly the most critical realization — the Poisson equation, which for this problem is a 2D elliptic PDE, can be rewritten as a 1D ODE and solved analytically!		# Three birds, so to speak, can be killed with one stone by ''guessing'' a 2D equilibrium density profile of the form <math>~\rho(\varpi,z) = g(\varpi,z)/\varpi^{2}</math>: The <math>~\varpi^{-2}</math> dependence can be combined strategically with <math>~\varpi^{-2}</math> dependence of the centrifugal potential; although it depends on <math>\varpi</math> instead of <math>r</math>, the density profile will at least ''resemble'' the spherical solution; and — certainly the most critical realization — the Poisson equation, which for this problem is a 2D elliptic PDE, can be rewritten as a 1D ODE and solved analytically!

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	==Structural Properties==		==Structural Properties==

	By analogy with the solution that was derived for a [[~~User:Tohline/~~SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State\|spherically symmetric isothermal structure with a power-law density distribution]], we can associate the scale length <math>~\varpi_0</math> with the characteristic density <math>~\rho_0</math> at that location through the relation,		By analogy with the solution that was derived for a [[SSC/Structure/PowerLawDensity#Isothermal_Equation_of_State\|spherically symmetric isothermal structure with a power-law density distribution]], we can associate the scale length <math>~\varpi_0</math> with the characteristic density <math>~\rho_0</math> at that location through the relation,
	<div align="center">		<div align="center">
	<math>		<math>
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	#At the end of their paper, HNM82 present the following '''Note added in proof:''' "After we submitted this paper, a paper by Toomre (Astrophys. J. '''259''' (1982), 535) appeared. He also found the same solutions as Eq. (2.3), although he did not study the stability of the equilibrium configurations."		#At the end of their paper, HNM82 present the following '''Note added in proof:''' "After we submitted this paper, a paper by Toomre (Astrophys. J. '''259''' (1982), 535) appeared. He also found the same solutions as Eq. (2.3), although he did not study the stability of the equilibrium configurations."


	=See Also=		=See Also=

Jet53man at 15:54, 13 September 2021

2021-09-13T15:54:10Z

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=Rotationally Flattened Isothermal Structures=
{| class="HNM82" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|-
! style="height: 150px; width: 150px; background-color:#ffeeee;" |[[H_BookTiledMenu#Spheroidal_.26_Spheroidal-Like|Hayashi, Narita & Miyama's Analytic Sol'n (1982)]]
|}
Both Volume I and Volume II of Colin Maclaurin's "''A Treatise of Fluxions''" can now be accessed online via

=See Also=

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Apps/HayashiNaritaMiyama82 - Revision history

Jet53man: /* Supplemental Relations */

Jet53man at 16:00, 13 September 2021

Jet53man at 15:54, 13 September 2021

Jet53man: Created page with "__FORCETOC__ =Rotationally Flattened Isothermal Structures= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid;..."

Jet53man: Created page with "FORCETOC =Rotationally Flattened Isothermal Structures= {| class="HNM82" style="float:left; margin-right: 20px; border-style: solid;..."