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Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations  &#8212; most of which also are heavily utilized in studies of continuum flows in terrestrial environments &#8212; are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy.  Although relatively simple in form, they prove to be very rich in nature... [[User:Tohline/Preface#Preface|<more>]]
Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations  &#8212; most of which also are heavily utilized in studies of continuum flows in terrestrial environments &#8212; are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy.  Although relatively simple in form, they prove to be very rich in nature... [[User:Tohline/Preface#Preface|<more>]]
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{{LSU_HBook_header}}
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<span id="Top">{{LSU_HBook_header}}</span>
=Pictorial Table of Contents=
=Pictorial Table of Contents=
* [[User:Tohline/H_Book/ImageContent01#TitleImage|Pictorial Table of Contents]]
* [[User:Tohline/H_Book/ImageContent01#TitleImage|Pictorial Table of Contents]]
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** [[User:Tohline/PGE/Euler#Euler_Equation|Euler Equation]]
** [[User:Tohline/PGE/Euler#Euler_Equation|Euler Equation]]
*** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|Rotating Reference Frame]]
*** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|Rotating Reference Frame]]
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*** [[User:Tohline/PGE/Hybrid_Scheme|Jay Call's Hybrid Scheme]]
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*** [[User:Tohline/PGE/Hybrid_Scheme#Hybrid_Advection_Scheme|Jay Call's Hybrid Scheme]]
** <math>1^\mathrm{st}</math> Law of Thermodynamics
** <math>1^\mathrm{st}</math> Law of Thermodynamics
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** Poisson Equation
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** [[User:Tohline/SR/PoissonOrigin#Origin_of_the_Poisson_Equation|Poisson Equation]]
* [[User:Tohline/SR#Supplemental_Relations|Supplemental Relations]]
* [[User:Tohline/SR#Supplemental_Relations|Supplemental Relations]]
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*** [[User:Tohline/SR/Ptot_QuarticSolution#Determining_Temperature_from_Density_and_Pressure|Quartic Temperature Solution]]
*** [[User:Tohline/SR/Ptot_QuarticSolution#Determining_Temperature_from_Density_and_Pressure|Quartic Temperature Solution]]
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* [[User:Tohline/VE|Global Energy Considerations]]
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* Global Energy Considerations
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** [[User:Tohline/VE#Global_Energy_Considerations|Traditional Presentation Based on Virial Theorem]]
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** [[User:Tohline/StabilityVariationalPrincipal|Variational Principle Approach]]
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** [[User:Tohline/SSC/Synopsis|Synopsis]] ([[User:Tohline/SSC/Synopsis_StyleSheet|Alternate Style Sheet]])
=Applications=
=Applications=
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[[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|Introduction]]
[[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29|Introduction]]
[[User:Tohline/SphericallySymmetricConfigurations#Spherically_Symmetric_Configurations|(Alternate Introduction)]]
[[User:Tohline/SphericallySymmetricConfigurations#Spherically_Symmetric_Configurations|(Alternate Introduction)]]
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[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to Free-Energy Analyses]]
===Structure:===
===Structure:===
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<font size="+2" color="red">
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&#x1F3A6; &#x269c;</font>
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Here we show how the set of principal governing equations (PGEs) may be solved to determine the equilibrium structure of spherically symmetric fluid configurations &#8212; such as individual, nonrotating stars or interstellar gas clouds having a rather idealized geometry.  After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed force-balanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration &#8212; given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example &#8212; as well as details regarding the internal radial profiles of the mass-density and fluid pressure.  As our various discussions illustrate, even a simple ''polytropic'' equation of state, that is,
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Here we show how the set of principal governing equations (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations &#8212; such as individual, nonrotating stars or protostellar gas clouds.  After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed force-balanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration &#8212; given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example &#8212; as well as details regarding the internal radial profiles of the fluid density and fluid pressure.  As our various discussions illustrate (see the table of contents, below), simply varying the power-law index in a ''polytropic'' equation of state gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
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If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's global free energy.  In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the ''scalar virial theorem'' &hellip; [[User:Tohline/SphericallySymmetricConfigurations/IntroductorySummary#Applications|<more>]]
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<math>~P</math>
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<math>~=</math>
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<math>~K \rho^{(n+1)/n} \, ,</math>
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</table>
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gives rise to equilibrium configurations that have a wide variety of internal structural profiles.
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If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a reasonably good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying extrema in the system's global free energy, that is, by identifying properties that satisfy the ''scalar virial theorem''.  Specifically, for ''isolated'' systems in virial equilibrium, the following relation between configuration parameters holds:
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In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left &#8212; for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium."  Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance."  Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium."
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<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
 
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<math>~=</math>
 
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<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W} 
 
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\, ,</math>
 
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</table>
 
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</div>
 
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where, all three of the dimensionless ''structural form factors'', <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math>, equal unity under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, while all three are ''of order'' unity otherwise.  If the configuration &#8212; such as an interstellar gas cloud &#8212; is not isolated but is, instead, embedded in a hot, tenuous external medium that exerts an external pressure, <math>~P_e</math>, the configuration's equilibrium parameters will be related via the expression,
 
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{{LSU_HBook_SSC_Structure}}
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<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
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<math>~=</math>
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<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W}  \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
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\, .</math>
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We have introduced these structural form factors into our discussion of the free energy of spherically symmetric, self-gravitating configurations because
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===Stability:===
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Suppose we want to study the '''stability''' of one of the spherically symmetric, equilibrium structures that has been described in the above "Structure" chapters.  One approach is to solve an eigenvalue problem whose governing second-order ODE &#8212; for example, the [[User:Tohline/SSC/Perturbations#2ndOrderODE|''Linear Adiabatic Wave Equation (LAWE)'']] &#8212; comes directly from ''perturbing'' the equilibrium model and a ''linearization'' of the overarching set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]].  In the set of chapters that is identified in the upper-left quadrant of the following table of content, we explain how the relevant wave equation for self-gravitating systems is derived from either a ''Lagrangian'' perspective or an ''Eulerian'' perspective, and how it resembles the equation that describes the propagation of sound waves in terrestrial fluids.  Another approach, which is described by the chapter(s) in the upper-right quadrant of the table, builds on a free-energy analysis of self-gravitating systems:  If an extremum in the free energy is identified as a local energy ''minimum'', then the identified equilibrium configuration is stable; but if the extremum is a local energy ''maximum'', then the equilibrium configuration is unstable. 
 +
[This paragraph ultimately should be moved to a "more &hellip;" page] <span id="BKB74pt1">Text written in ''green'' has been taken directly from the introductory paragraphs of</span> [http://adsabs.harvard.edu/abs/1974A%26A....31..391B G. S. Bisnovatyi-Kogan &amp; S. I. Blinnikov (1974)].  <font color="darkgreen">Three different approaches are used in the study of hydrodynamical stability of stars and other gravitating objects &hellip; &nbsp; The first approach is based on the use of the equations of small oscillations.  In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations.  The solutions consist of a set of eigenfrequencies and eigenfunctions.  This approach is subject to great numerical difficulties and has been carried through successfully only for the simplest case</font>[s].  For spherically symmetric, Newtonian systems, see, for example, [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Schwarzschild_.281941.29| Schwarzschild (1941)]] or [[User:Tohline/SSC/Perspective_Reconciliation#Approach_by_Ledoux_and_Walraven|Ledoux &amp; Walraven (1958)]].  Second, one can derive <font color="darkgreen">a variational principle from the equations of small oscillations.  This principle replaces the straightforward solution of these equations:</font>  In the context of rotating Newtonian systems, see, for example, [http://adsabs.harvard.edu/abs/1964ApJ...140.1045C Clement (1964)], [http://adsabs.harvard.edu/abs/1968ApJ...152..267C Chandrasekhar &amp; Lebovitz (1968)], [http://adsabs.harvard.edu/abs/1967MNRAS.136..293L Lynden-Bell and Ostriker (1967)], or [http://adsabs.harvard.edu/abs/1972ApJS...24..319S Schutz (1972)].  <font color="darkgreen">With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations.  In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated &hellip;</font>  The third approach is what we have referred to as a free-energy analysis.  <font color="darkgreen">When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient.  The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.</font>  For non rotating systems having <math>~\gamma_g</math> near 4/3, see, for example, [http://adsabs.harvard.edu/abs/1966ApJ...144..180F Fowler (1964)] or Zeldovich &amp; Novikov (Soviet journal, 1965).
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<table border="1" width="100%" cellspacing="2" cellpadding="8">
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<span id="BKB74pt2"><font color="darkgreen">If one wants to know from a stability analysis the answer to only one question &#8212; whether the model is stable or not &#8212; then the most straightforward procedure is to use the third, static method (Zeldovich 1963; Dmitrie &amp; Kholin 1963).  For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation. Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point.</font></span>
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In the lower portion of the following table of content, links are provided to chapters in which one of these primary solution strategies is employed to analyze the stability of ''specific'' equilibrium models.
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Solution Strategies: &nbsp; &nbsp;
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  </th>
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  <th align="center" width="25%">
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<font color="black">Detailed Force-Balance</font><br>([[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Introduction]])
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  </th>
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  <th align="center">
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<font color="black">Virial Equilibrium</font><br>([[User:Tohline/SphericallySymmetricConfigurations/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations|Introduction]])
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  </th>
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  <td align="left" colspan="4">Example Solutions: </td>
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* Uniform-density sphere
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Isolated &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/UniformDensity#Isolated_Uniform-Density_Sphere|<font size="+2" color="green">&#x2713;</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Uniform-Density_Sphere|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Embedded in an External Medium &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/UniformDensity#Uniform-Density_Sphere_Embedded_in_an_External_Medium|<font size="+2" color="green">&#x2713;</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/VirialEquilibrium/UniformDensity#Adiabatic_Evolution_of_Pressure-truncated_Sphere|<font size="+2" color="green">&#x2713;</font>]]</td>
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* Polytropes
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  </td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Isolated &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres|<font size="+2" color="green">&#x2713;</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Embedded in an External Medium &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|<font size="+2" color="green">&#x2713;</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|<font size="+2" color="green">&#x2713;</font>]]</td>
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* Isothermal sphere
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Isolated &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere|<font size="+2" color="green">&#x2713;</font>]]</td><td align="center" width="25%">[[User:Tohline/SSC/Virial/Isothermal|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Embedded in an External Medium (Bonnor-Ebert Sphere) &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|<font size="+2" color="green">&#x2713;</font>]]<td align="center" width="25%">[[User:Tohline/SphericallySymmetricConfigurations/Virial#Bounded_Isothermal|<font size="+2" color="green">&#x2713;</font>]]</td>
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* Zero-temperature White Dwarf &#8212; [[User:Tohline/SSC/Structure/WhiteDwarfs#White_Dwarfs|Overview]]
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* Power-law density distribution &#8212; [[User:Tohline/SSC/Structure/PowerLawDensity#Power-Law_Density_Distributions|Overview]]
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* BiPolytropes (also referred to as ''Composite Polytropes'')
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes#BiPolytropes|Overview]]</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/BipolytropeGeneralization_Version2#Bipolytrope_Generalization|Overview]]</td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (0,0)</math> &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy0_0#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td><td align="left">Core-Envelope Structure with <math>~(n_c,n_e) = (5,1)</math> &hellip;</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|<font size="+2" color="green">&#x2713;</font>]]</td>
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  <td align="center" width="25%">[[User:Tohline/SSC/Structure/BiPolytropes/FreeEnergy5_1#Free_Energy_of_BiPolytrope_with|<font size="+2" color="green">&#x2713;</font>]]</td>
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* Limiting Masses &#8212; [[User:Tohline/SSC/Structure/LimitingMasses#Mass_Upper_Limits|Summary]]
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  </td>
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===Stability:===
 
<table border="1" cellspacing="2" cellpadding="8" width="100%">
<table border="1" cellspacing="2" cellpadding="8" width="100%">
<tr>
<tr>
   <td align="left">
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[[User:Tohline/SSC/SoundWaves|Background:  Sound Waves]]
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[[User:Tohline/SSC/SoundWaves#Sound_Waves|Background:  Sound Waves]]
<font color="darkblue">'''Solution Strategy Assuming Spherical Symmetry:'''</font>
<font color="darkblue">'''Solution Strategy Assuming Spherical Symmetry:'''</font>
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* [[User:Tohline/SSC/Stability_Eulerian_Perspective|Eulerian Perspective]]
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* [[User:Tohline/SSC/Stability_Eulerian_Perspective#Stability_of_Spherically_Symmetric_Configurations_.28Eulerian_Perspective.29|Eulerian Perspective]]
* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Lagrangian Perturbations]]
* [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Lagrangian Perturbations]]
* [[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|Reconciliation]]
* [[User:Tohline/SSC/Perspective_Reconciliation#Reconciling_Eulerian_versus_Lagrangian_Perspectives|Reconciliation]]
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<p></p>
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<table align="left" border="1" cellpadding="3">
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<tr><th align="center" colspan="3"><font color="darkblue">Summary of Various ''Adiabatic Wave Equation'' Derivations</font></th></tr>
 +
<tr>
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<td align="center">[[File:ImageOfDerivations06GoodJeansBonnor.png|100px|thumb|center|Jeans (1928) or Bonnor (1957)]]</td>
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<td align="center">[[File:ImageOfDerivations07GoodLedouxWalraven.png|100px|thumb|center|Ledoux &amp; Walraven (1958)]]</td>
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<td align="center">[[File:ImageOfDerivations08GoodRosseland.png|100px|thumb|center|Rosseland (1969)]]</td>
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</tr>
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</table>
   </td>
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<font color="darkblue">'''[[User:Tohline/SSC/VirialStability|Virial Stability]]'''</font>
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[[User:Tohline/SSC/VariationalPrinciple|Ledoux's Variational Principle]]<p></p>
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[[User:Tohline/SSC/VirialStability|Early Chatting with Kundan Kadam]]<p></p>
[[User:Tohline/SSC/BipolytropeGeneralization#Bipolytrope_Generalization|Bipolytrope Generalization]]
[[User:Tohline/SSC/BipolytropeGeneralization#Bipolytrope_Generalization|Bipolytrope Generalization]]
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Example Solutions:
Example Solutions:
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* [[User:Tohline/SSC/UniformDensity#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_III.29|Uniform-density sphere]]
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* [[User:Tohline/SSC/UniformDensity#The_Stability_of_Uniform-Density_Spheres|Uniform-density sphere]]
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* [[User:Tohline/SSC/Polytropes#Polytropic_Spheres|Polytropes]]
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* Isolated Polytropes
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** [[User:Tohline/SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres|Setup]]
 +
** n = 1:&nbsp; [[User:Tohline/SSC/Stability/n1PolytropeLAWE#Radial_Oscillations_of_n_.3D_1_Polytropic_Spheres|Attempt at Formulating an Analytic Solution]]
 +
** n = 3:&nbsp; [[User:Tohline/SSC/Stability/n3PolytropeLAWE|Numerical Solution]] to compare with [http://adsabs.harvard.edu/abs/1941ApJ....94..245S M. Schwarzschild (1941)]
 +
** n = 5:&nbsp; [[User:Tohline/SSC/Stability/n5PolytropeLAWE#Radial_Oscillations_of_n_.3D_5_Polytropic_Spheres|Attempt at Formulating an Analytic Solution]]
 +
* Radial Oscillations in, and Stability of, Pressure-Truncated Configurations ([[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Marginally_Unstable_Pressure-Truncated_Gas_Clouds|Overview]])
 +
** [[User:Tohline/SSC/Stability/Isothermal|Pressure-Truncated Isothermal Spheres]]
 +
* [[User:Tohline/SSC/Stability_BoundedCompositePolytropes|Bounded and Composite Polytropes]]
 +
* Bipolytropes having <math>~(n_c, n_e) = (0, 0)</math>
 +
** [[User:Tohline/SSC/Stability/BiPolytrope0_0|Overview]]
 +
** Initial set of chapters that used incorrect interface matching condition
 +
*** [[User:Tohline/SSC/Stability/BiPolytrope0_0Old|One Analytically Specified Eigenvector for a Bipolytrope]] having <math>~(n_c, n_e) = (0, 0)</math>
 +
*** [[User:Tohline/SSC/Stability/BiPolytrope0_0Details|Initial Detailed Derivation]]
 +
*** [[User:Tohline/Appendix/Ramblings/Additional_Analytically_Specified_Eigenvectors_for_Zero-Zero_Bipolytropes#Searching_for_Additional_Eigenvectors_of_Zero-Zero_Bipolytropes|Additional Analytically Specified Eigenvectors]]
 +
*** [[User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors#Numerically_Determined_Eigenvectors_of_a_Zero-Zero_Bipolytrope|Numerically Determined Eigenvectors]]
 +
**[[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition|Free-Energy Stability Analysis]]
 +
**[[User:Tohline/SSC/Stability/BiPolytrope0_0CompareApproaches|Compare Eigenvector and Free-Energy Analyses]]
   </td>
   </td>
 +
</tr>
 +
<tr>
 +
  <td align="left" colspan="2">Also to be studied: [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura] (1981, PASJapan, 33, 299)</td>
</tr>
</tr>
</table>
</table>
Line 238: Line 141:
==Two-Dimensional Configurations==
==Two-Dimensional Configurations==
-
 
-
* Introduction
 
-
** [[User:Tohline/Cylindrical_3D|Cylindrical Coordinates]]
 
-
** [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|Axisymmetric Configurations]]
 
===Structure:===
===Structure:===
Line 248: Line 147:
   <td align="left">
   <td align="left">
<font color="darkblue">'''Solution Strategies'''</font>
<font color="darkblue">'''Solution Strategies'''</font>
-
* [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies|Axisymmetric Configurations]]
+
* Preference for [[User:Tohline/Cylindrical_3D|Cylindrical Coordinates]]
 +
** [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|Axisymmetric Configurations (Part I)]]
 +
** [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies|Axisymmetric Configurations (Part II)]] &#8212; including, [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''Simple Rotation Profiles'']]
 +
** Infinitesimally Thin, Nonaxisymmetric Configurations
 +
* Self-Consistent Field Technique
   </td>
   </td>
   <td align="right">
   <td align="right">
Line 256: Line 159:
<tr>
<tr>
   <td align="left" colspan="2">
   <td align="left" colspan="2">
-
Example Solutions:
+
Uniform-Density &amp; Centrally Condensed Structures:
-
* [[User:Tohline/Apps/MaclaurinSpheroids|Maclaurin Spheroids]]
+
* Maclaurin Spheroids
 +
** [[User:Tohline/Apps/MaclaurinSpheroids#Maclaurin_Spheroids_.28axisymmetric_structure.29|Structural Properties]]
 +
** [[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Excerpts_from_A_Treatise_of_Fluxions|Maclaurin's (1742) Analysis]] &nbsp; [[User:Tohline/Apps/MaclaurinSpheroids/GoogleBooks#Prolate_Spheroid|<font size="+1">&#x1F3A6;</font>]]
 +
* [[User:Tohline/2DStructure/RotatingPolytropes|Rotationally Flattened Polytropic Structures]]
* [[User:Tohline/Apps/HayashiNaritaMiyama82|Rotationally Flattened, Isothermal Structures]]
* [[User:Tohline/Apps/HayashiNaritaMiyama82|Rotationally Flattened, Isothermal Structures]]
-
* Polytropic Tori:
+
 
-
**[[User:Tohline/Apps/PapaloizouPringleTori|Papaloizou-Pringle (massless) Tori]]
+
Toroidal Structures:
-
**Self-gravitating Tori
+
* [[User:Tohline/Apps/PapaloizouPringleTori|Papaloizou-Pringle (massless) Tori]] &nbsp; [[User:Tohline/Apps/PapaloizouPringleTori#Boundary_Conditions|<font size="+1">&#x1F3A6;</font>]]
 +
* Self-gravitating, Isothermal Rings
 +
* [[User:Tohline/Apps/Tori/Polytropic|Self-gravitating, Polytropic Tori]]
 +
* [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|Using Toroidal Coordinates to Determine the Gravitational Potential]] &nbsp; [[User:Tohline/2DStructure/ToroidalCoordinates#Perform_Angular_Integration_Over_Test_Mass_Distribution|<font size="+1">&#x1F3A6;</font>]]
 +
 
 +
Nonaxisymmetric Disks &amp; Annuli:
* [[User:Tohline/Apps/Korycansky_Papaloizou_1996|Infinitesimally Thin, Nonaxisymmetric Disk]]
* [[User:Tohline/Apps/Korycansky_Papaloizou_1996|Infinitesimally Thin, Nonaxisymmetric Disk]]
   </td>
   </td>
Line 268: Line 179:
===Stability:===
===Stability:===
 +
<table border="1" width="100%" cellspacing="2" cellpadding="8">
 +
<tr>
 +
  <td align="left">
 +
<font color="darkblue">'''Solution Strategy'''</font>
 +
* Preference for [[User:Tohline/Cylindrical_3D/Linearization|Cylindrical Coordinates]]
 +
  </td>
 +
</tr>
 +
<tr>
 +
  <td align="left" colspan="2">
 +
 +
Centrally-condensed Structures:
 +
* [[User:Tohline/Apps/SMS|Rotating, Supermassive Stars]]
 +
 +
Toroidal Structures:
 +
* Papaloizou-Pringle (Massless) Tori
 +
** [[User:Tohline/Apps/PapaloizouPringle84|Defining the Relevant Eigenvalue Problem]]
 +
** [[User:Tohline/Apps/Blaes85SlimLimit|Jacobi Polynomials as the Solution in the Slim Torus Limit]]
 +
* Self-gravitating, Polytropic Tori
 +
** [[User:Tohline/Apps/ImamuraHadleyCollaboration#Characteristics_of_Unstable_Eigenvectors_in_Self-Gravitating_Tori|Characteristics of Unstable Eigenvectors]]
 +
** [[User:Tohline/Appendix/Ramblings/Hadley_and_Imamura_Supplementary_Database#Supplementary_Dataset_Generated_by_Hadley_.26_Imamura_Collaboration|Useful Data Repository from the Hadley &amp; Imamura Collaboration]]
 +
** [[User:Tohline/Apps/WoodwardTohlineHachisu94#Online_Movies|YouTube videos]] that supplement simulations by [http://adsabs.harvard.edu/abs/1994ApJ...420..247W J. W. Woodward, J. E. Tohline, &amp; I. Hachisu (1994)]
 +
  </td>
 +
</tr>
 +
</table>
===Dynamics:===
===Dynamics:===
Line 277: Line 212:
* Introduction
* Introduction
-
** [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids|Properties of Homogeneous Ellipsoids]]
+
** The Foundation Provided by Chandrasekhar's [[User:Tohline/Appendix/References#Other_References|EFE]]
-
** [[User:Tohline/ThreeDimensionalConfigurations/BinaryFission|Fission Hypothesis of Binary Star Formation]]
+
** [[User:Tohline/ThreeDimensionalConfigurations/BinaryFission#Fission_Hypothesis_of_Binary_Star_Formation|Fission Hypothesis of Binary Star Formation]] &nbsp; [[User:Tohline/ThreeDimensionalConfigurations/BinaryFission#Fission_Hypothesis_of_Binary_Star_Formation|<font size="+1">&#x1F3A6;</font>]]
===Structure:===
===Structure:===
Line 291: Line 226:
Example Solutions:
Example Solutions:
* Ellipsoidal Figures of Equilibrium
* Ellipsoidal Figures of Equilibrium
-
*[[User:Tohline/Apps/RiemannEllipsoids_Compressible|Compressible Analogs of Riemann Ellipsoids]]; see also [[User:Tohline/PGE/RotatingFrame|Note to Hirschmann &amp; Neilsen]] and [[User:Tohline/Apps/Korycansky_Papaloizou_1996#Suggested_Strategy_by_Joel|Suggested Strategy by Joel]]
+
** [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids_.281.29|The Gravitational Potential of Homogeneous Ellipsoids]]
 +
** [[User:Tohline/ThreeDimensionalConfigurations/EFE_Energies|Relevant Energy Components]]
 +
** [[User:Tohline/ThreeDimensionalConfigurations/JacobiEllipsoids|Jacobi Ellipsoids]]
 +
* Compressible Analogs of Riemann Ellipsoids
 +
** [http://adsabs.harvard.edu/abs/2000ApJ...532.1051C Constructed Dynamically]
 +
** [[User:Tohline/Apps/RiemannEllipsoids_Compressible|Attempted Construction Using SCF Technique]]; see also [[User:Tohline/PGE/RotatingFrame|Note to Hirschmann &amp; Neilsen]] and [[User:Tohline/Apps/Korycansky_Papaloizou_1996#Suggested_Strategy_by_Joel|Suggested Strategy by Joel]]
   </td>
   </td>
</tr>
</tr>
Line 317: Line 257:
=See Also=
=See Also=
-
*[http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]; see also the related [http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521192255 CUP Publication]
+
 
 +
* [[User:Tohline/H_Book/ContentsPDF|Full List of PDF-Formatted Chapters]]
 +
 
{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Current revision as of 16:14, 9 August 2017


Preface from the original version of this HyperText Book (H_Book):

November 18, 1994

Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — most of which also are heavily utilized in studies of continuum flows in terrestrial environments — are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy. Although relatively simple in form, they prove to be very rich in nature... <more>

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

Pictorial Table of Contents

Context

Applications

 
 
 

Spherically Symmetric Configurations

Introduction (Alternate Introduction)

Index to Free-Energy Analyses

Structure:

🎦 ⚜

Here we show how the set of principal governing equations (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds. After supplementing the PGEs by specifying an equation of state of the fluid, the system of equations is usually solved by employing one of three techniques to obtain a "detailed force-balanced" model that provides the radius, ~R_\mathrm{eq}, of the equilibrium configuration — given its mass, ~M, and central pressure, ~P_c, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. As our various discussions illustrate (see the table of contents, below), simply varying the power-law index in a polytropic equation of state gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

If one is not particularly concerned about details regarding the distribution of matter within the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's global free energy. In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the scalar virial theorem<more>

In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium."


Solution Strategies:    

Detailed Force-Balance
(Introduction)

Virial Equilibrium
(Introduction)

Example Solutions:
  • Uniform-density sphere
        Isolated …
    Embedded in an External Medium …
  • Polytropes
    Isolated …
    Embedded in an External Medium …    🎦
  • Isothermal sphere
    Isolated …
    Embedded in an External Medium (Bonnor-Ebert Sphere) …
  • Zero-temperature White Dwarf — Overview
  • Power-law density distribution — Overview
  • Other Analytically Definable Models
  …  
  • BiPolytropes (also referred to as Composite Polytropes)
Overview Overview
        Core-Envelope Structure with ~(n_c,n_e) = (0,0)
        Core-Envelope Structure with ~(n_c,n_e) = (5,1)
        Core-Envelope Structure with ~(n_c,n_e) = (1,5)   …  
        Core-Envelope Structure with ~(n_c,n_e) = (\tfrac{3}{2},3)   …  

Stability:

Suppose we want to study the stability of one of the spherically symmetric, equilibrium structures that has been described in the above "Structure" chapters. One approach is to solve an eigenvalue problem whose governing second-order ODE — for example, the Linear Adiabatic Wave Equation (LAWE) — comes directly from perturbing the equilibrium model and a linearization of the overarching set of principal governing equations. In the set of chapters that is identified in the upper-left quadrant of the following table of content, we explain how the relevant wave equation for self-gravitating systems is derived from either a Lagrangian perspective or an Eulerian perspective, and how it resembles the equation that describes the propagation of sound waves in terrestrial fluids. Another approach, which is described by the chapter(s) in the upper-right quadrant of the table, builds on a free-energy analysis of self-gravitating systems: If an extremum in the free energy is identified as a local energy minimum, then the identified equilibrium configuration is stable; but if the extremum is a local energy maximum, then the equilibrium configuration is unstable.

[This paragraph ultimately should be moved to a "more …" page] Text written in green has been taken directly from the introductory paragraphs of G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974). Three different approaches are used in the study of hydrodynamical stability of stars and other gravitating objects …   The first approach is based on the use of the equations of small oscillations. In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations. The solutions consist of a set of eigenfrequencies and eigenfunctions. This approach is subject to great numerical difficulties and has been carried through successfully only for the simplest case[s]. For spherically symmetric, Newtonian systems, see, for example, Schwarzschild (1941) or Ledoux & Walraven (1958). Second, one can derive a variational principle from the equations of small oscillations. This principle replaces the straightforward solution of these equations: In the context of rotating Newtonian systems, see, for example, Clement (1964), Chandrasekhar & Lebovitz (1968), Lynden-Bell and Ostriker (1967), or Schutz (1972). With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations. In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated … The third approach is what we have referred to as a free-energy analysis. When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient. The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability. For non rotating systems having ~\gamma_g near 4/3, see, for example, Fowler (1964) or Zeldovich & Novikov (Soviet journal, 1965).

If one wants to know from a stability analysis the answer to only one question — whether the model is stable or not — then the most straightforward procedure is to use the third, static method (Zeldovich 1963; Dmitrie & Kholin 1963). For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation. Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point.

In the lower portion of the following table of content, links are provided to chapters in which one of these primary solution strategies is employed to analyze the stability of specific equilibrium models.

Background: Sound Waves

Solution Strategy Assuming Spherical Symmetry:

Summary of Various Adiabatic Wave Equation Derivations
Jeans (1928) or Bonnor (1957)
Ledoux & Walraven (1958)
Rosseland (1969)
Ledoux's Variational Principle

Early Chatting with Kundan Kadam

Bipolytrope Generalization

Example Solutions:

Also to be studied: Kimura (1981, PASJapan, 33, 299)

Dynamics:

Spherical Collapse:


 
 
 

Two-Dimensional Configurations

Structure:

Solution Strategies

Virial Equilibrium

Uniform-Density & Centrally Condensed Structures:

Toroidal Structures:

Nonaxisymmetric Disks & Annuli:

Stability:

Solution Strategy

Centrally-condensed Structures:

Toroidal Structures:

Dynamics:

 
 
 

Three-Dimensional Configurations

Structure:

Solution Strategies

Example Solutions:

Stability:

Dynamics:

 
 
 

Related Projects Underway


Appendices


See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
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