Editing SSC/Stability/Isothermal (section)

====P-V Diagram====

As we have discussed in a [[SSC/Structure/BonnorEbert#P-V_Diagram|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], {{ Bonnor56 }} examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>0 < \xi_e < \infty</math>.  In a diagram that shows how <math>P_e(\xi_e)</math> varies with equilibrium volume, <math>V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist.  The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1.  The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; we have adopted [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Whitworth's normalizations]], <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>.


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<b>Figure 1: &nbsp; Bonnor's P-V Diagram</b> (see [[SSC/Structure/BonnorEbert#Fig1|accompanying discussion]] for details)

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Figure extracted from p. 355 of <br />{{ Bonnor56figure }}
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[http://adsabs.harvard.edu/abs/1956MNRAS.116..351B W. B. Bonnor (1956)]<p></p>
"''Boyle's Law and Gravitational Instability''"<p></p>
MNRAS, vol. 116, pp. 351 - 359 &copy; Royal Astronomical Society
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The {{ Bonnor56 }} equilibrium sequence, but as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted normalizations, <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>, from {{ Whitworth81 }}.
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[[File:Bonnor1956Fig1reproducedB.png|400px|center|Bonnor (1956, MNRAS, 116, 351)]]
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[[File:IsothermalEquilSequence.png|800px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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As is [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|discussed below]], in separate studies, {{ Yabushita68 }} and {{ TVH74 }} examined the lowest-order modes of radial oscillations that arise in pressure-truncated isothermal spheres.  In both studies, numerical techniques were used to solve the eigenvalue problem associated with the [[#IsothermalLAWE|isothermal LAWE]], as derived below.  The nine individual equilibrium models that were studied by {{ TVH74hereafter }} are identified by the nine small, filled circular markers along the sequence that has been displayed in the right-hand panel of our Figure 1; as labeled, they correspond to models with <math>~\xi_e</math> = 2, 3, 4, 5, 6, 7, 8, 9, and 10.  Earlier, Yabushita studied the oscillation modes of three of these same configurations; specifically, the models with <math>~\xi_e</math> = 6, 7, and 8, which straddle the location along the equilibrium sequence of the model associated with the pressure maximum (the [[SSC/FreeEnergy/EquilibriumSequenceInstabilities#Instabilities_Associated_with_Equilibrium_Sequence_Turning_Points|turning point]] labeled "A" in Bonnor's P-V diagram).