Editing SSC/Stability/Isothermal (section)

===From Yabushita's (1992) Analysis===

In the portion (&sect;5) of his analysis that is focused on the stability of pressure-truncated polytropic spheres, {{ Yabushita92 }} examined the eigenvalue problem governed by the following wave equation:
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Radial Pulsation Equation Extracted<sup>&dagger;</sup> from p. 182 of [http://adsabs.harvard.edu/abs/1992Ap%26SS.193..173Y S. Yabushita (1992)]<p></p>
"''Similarity Between the Structure and Stability of Isothermal and Polytropic Gas Spheres''"<p></p>
Astrophysics and Space Science, vol. 193, pp. 173-183 &copy; [http://www.springer.com/astronomy/astrophysics+and+astroparticles/journal/10509 Springer]
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[[File:Yabushita1992WaveEquation2.png|650px|center|Yabushita (1992)]]
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<sup>&dagger;</sup>Equations and text displayed here exactly as it appears in the original publication.
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Radial pulsation equation extracted<sup>&dagger;</sup> from p. 182 of <br />{{ Yabushita92figure }}
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[[File:Yabushita1992WaveEquation2.png|650px|center|Yabushita (1992)]]
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[[File:CommentButton02.png|right|100px|Comment by J. E. Tohline:  In Yabushita's Eq. (5.3), the (2/x) term inside the square brackets should be (4/x).]]
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<math>
\frac{d^2 h}{dx^2} + \biggl[ \frac{2}{x} + (n+1) \frac{1}{\theta} \frac{d\theta}{dx} \biggr]\frac{dh}{dx}
+ (3-n)\frac{1}{x} \frac{1}{\theta} \frac{d\theta}{dx}~ h
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left"><math>0 \, ;</math></td>
  <td align="right" width="5%">(5.3)</td>
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while the boundary condition may be expressed in the form

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<math>
\frac{dh}{dx} + \frac{3h}{x}
</math>
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  <td align="center" width="5%"><math>=</math></td>
  <td align="left"><math>0 \, ,</math> &nbsp; &nbsp; &nbsp; <math>x = x_0\, .</math></td>
  <td align="right" width="5%">(5.4)</td>
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<sup>&dagger;</sup>Equations and text displayed here exactly as it appears in the original publication.
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Let's examine the overlap between this pair of governing relations and the ones employed by {{ HRW66full }}, hereafter {{ HRW66hereafter }}.  If we replace the variable <math>~X</math> with <math>~h</math>, set <math>~\gamma = (n+1)/n</math>, and set the dimensionless eigenfrequency, <math>~s</math>, to zero in the [[SSC/Stability/Polytropes#HRW66excerpt|radial pulsation equation employed by HRW66]], we have,
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<math>~0 </math>
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<math>~=</math>
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<math>~
\frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (n+1)\biggl[ 3 - \frac{4n}{(n+1)} \biggr] \biggl[ \frac{\theta^' h}{\theta x} \biggr]
</math>
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&nbsp;
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<math>~=</math>
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<math>~
\frac{d^2 h}{dx^2} + \biggl[\frac{4}{x} + (n+1) \frac{\theta^'}{\theta} \biggr] \frac{dh}{dx} + (3-n) \biggl[ \frac{\theta^' h}{\theta x} \biggr] \, .
</math>
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This matches equation (5.3) of {{ Yabushita92 }} &#8212; see the above boxed-in image &#8212; except the <math>(4/x)</math> term appears as <math>(2/x)</math> in Yabushita's article; giving the benefit of the doubt, <font color="red">this is most likely a typographical error</font> in {{ Yabushita92 }}.  According to {{ HRW66hereafter }}, the corresponding central boundary condition is, 
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<math>\frac{dh}{dx} = 0</math> &nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; <math>x=0 \, .</math>
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While &#8212; after changing the sign on the right-hand side of {{ HRW66hereafter }}'s equation (58) as argued in our [[SSC/Perturbations#ChristyCox|accompanying discussion]] in order to align with the separate derivations presented by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)]  &#8212; the corresponding boundary condition at the surface is, 
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<math>~\frac{dh}{dx}</math>
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<math>~=</math>
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<math>~- \frac{h}{x} \biggr[  3 - \frac{4}{\gamma}  + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math>
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&nbsp;
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&nbsp;
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<math>~=</math>
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<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math>
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&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~x = x_0 \, .</math>
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This surface boundary condition, which has been used by the astrophysics community in the context of ''isolated'' polytropic configurations, is different from the one displayed as equation (5.4) of {{ Yabushita92 }}.  The surface boundary condition chosen by Yabushita &#8212; effectively,
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<math>~\frac{d \ln h}{d\ln x} = -3 \, ,</math>
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&#8212; does seem to be more appropriate in the context of a study of the stability of ''pressure-truncated'' polytropes because, as argued by {{ LP41full }} and as reviewed in our [[SSC/Perturbations#Set_the_Surface_Pressure_Fluctuation_to_Zero|accompanying discussion]], it ensures that the pressure fluctuation ''at the surface'' is zero.  It is worth noting that Yabushita's surface boundary condition matches the surface boundary condition chosen by {{ TVH74 }} in their study of pressure-truncated ''isothermal'' spheres; in their words (see p. 428 of their article):  &nbsp; [Setting the surface logarithmic derivative to negative 3] <font color="green">expresses the condition that the pressure at the perturbed surface always remain[s] equal to the confining pressure exerted by the external medium in which the [pressure-truncated] sphere must be embedded</font>.