Editing SSC/Stability/InstabilityOnsetOverview (section)

===Structure===

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Once a central density, <math>~\rho_c</math>, and constituent fluid sound speed, <math>~c_s</math>, have been specified, the internal structure of an equilibrium, isothermal sphere can be completely described in terms of the function, <math>~\psi(\xi) \equiv \ln(\rho_c/\rho)</math>, which is a solution of the,
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<font color="maroon"><b>Isothermal Lane-Emden Equation</b></font> <p></p>
{{ Math/EQ_SSLaneEmden02 }}
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subject to the boundary conditions, <math>~ \psi = 0</math> and <math>~d\psi/d\xi = 0</math> at <math>~\xi = 0</math>.  Using numerical integration techniques, the function, <math>~\psi(\xi)</math>, and its first derivative have been evaluated at discrete locations throughout the isothermal sphere and published by [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden (1907)]] and by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar &amp; Wares (1949)].  In an [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|accompanying discussion]], we describe the steps that we have used to independently integrate this 2<sup>nd</sup>-order ODE in order to construct numerical models of equilibrium configurations for the stability analyses described herein. 

In isolation, the isothermal sphere extends to infinity.  But configurations of finite extent can be constructed by truncating the function, <math>~\psi</math>, at some radius, <math>~0 < \tilde\xi < \infty</math> &#8212; such that the surface density is finite and set by the value of <math>~\tilde\psi \equiv \psi(\tilde\xi)</math> &#8212; and embedding the configuration in a hot, tenuous medium that exerts an "external" pressure, <math>~P_e = c_s^2 \rho_c e^{-\tilde\psi}</math>, uniformly across the surface of the &#8212; now, truncated &#8212; sphere. The internal structure of such a "pressure-truncated" isothermal sphere is completely describable in terms of the same function, <math>~\psi(\xi)</math>, that describes the structure of the isolated isothermal sphere, except that beyond <math>~\tilde\xi</math> the function becomes physically irrelevant.  

A ''sequence'' of equilibrium, pressure-truncated isothermal spheres is readily defined by varying the value of <math>~\tilde\xi</math>.  Figure 1 displays the behavior of such an equilibrium sequence, as viewed from three different astrophysical perspectives (in all cases, <math>~c_s</math> is held fixed while <math>~\tilde\xi</math> is varied monotonically along the sequence): &nbsp; ''Left panel'' &#8212; A diagram showing how the truncated configuration's equilibrium volume varies with the externally applied pressure, if the configuration's mass is held fixed.  ''Center panel'' &#8212; A diagram showing how the truncated configuration's mass varies with the equilibrium radius, if the external pressure is held fixed.  ''Right panel'' &#8212; A diagram that shows how the configuration's mass varies with central density, if the external pressure is held fixed.

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<b>Figure 1:</b> &nbsp; Equilibrium Sequences of Pressure-Truncated Isothermal Spheres<br />(viewed from three different astrophysical perspectives)
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  <td align="center"><font color="black" size="+2">&#x25CF;</font></td><td align="center"><math>~\xi_e</math></td>
  <td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
  <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
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  <td align="center" colspan="1"><font color="yellow" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">4.05</td>
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[[File:IsothermalTrunc4.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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[[File:IsothermalTrunc12.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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[[File:IsothermalTrunc11.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">6.45</td>
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  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">9.00</td>
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  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">67.00</td>
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  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">98.50</td>
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  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">735.00</td>
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  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="right" colspan="1">1060.00</td>
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  <td align="center" colspan="2">&nbsp;</td>
  <td align="center" colspan="1"><math>~\biggl(\frac{2^8}{3^4\cdot 5^3}\biggr) \biggl[\xi^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2 e^{-\psi}\biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl[ \biggl(\frac{2^2}{3 \cdot 5}\biggr) \xi  \biggl(\frac{d\psi}{d\xi}\biggr)\biggr]^{-3}_\tilde\xi</math><br />
See also Fig. 1 of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] </td>
  <td align="center" colspan="1"><math>~\biggl[ \frac{\xi^2}{\sqrt{4\pi}}  \biggl(\frac{d\psi}{d\xi}\biggr) e^{-\psi/2}\biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>~\frac{\tilde\xi}{\sqrt{4\pi}} ~e^{-\tilde\psi/2}</math><br />See also Fig. 12b of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]</td>
  <td align="center" colspan="1"><math>~\biggl[ \frac{\xi^2}{\sqrt{4\pi}}  \biggl(\frac{d\psi}{d\xi}\biggr) e^{-\psi/2} \biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>~e^{\tilde\psi}</math><p></p>&nbsp;<br />See also Fig. 12a of [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]</td>
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<sup>&dagger;</sup>This is the classic P-V diagram that shows up in most discussions of [[SSC/Structure/BonnorEbert#Fig1|Bonnor-Ebert spheres]].<br />
<sup>&Dagger;</sup>In a similar diagram in which the radius (rather than external pressure) is held fixed, the purple (rather than green) markers identify mass extrema.
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In Figure 1, the small, yellow circular marker identifies the configuration along each equilibrium sequence for which <math>~\tilde\xi = 4.05</math>.  In the center panel it marks the approximate location of a "turning point" along the mass-radius equilibrium sequence; specifically, it identifies the configuration along the sequence that exhibits the maximum equilibrium radius.  This yellow marker does not appear to be associated with any particularly special feature of the equilibrium sequence displayed in the left figure panel or the one displayed in the panel on the right.  Moving along each of the displayed sequences, past the yellow marker, one encounters a small, green circular marker that identifies the configuration for which <math>~\tilde\xi = 6.45</math>.  In the left panel, it marks the approximate location of another "turning point"; specifically, the turning point that identifies the external pressure above which no equilibrium configurations exist.  This will henceforth be referred to as the <math>~P_e</math>-max turning point.  It is clear that, in both the center panel and the right panel, this green marker is also associated with a turning point &#8212; the so-called ''Bonnor-Ebert limiting mass''. 

<span id="BonnorCondition">As</span> we have [[SSC/Structure/BonnorEbert#Limiting_Pressure_and_Maximum_Mass|reviewed in a separate chapter]], Bonnor was the first to point out, in the context of the <math>~P-V</math> diagram (left panel of Figure 1), that the precise location of the <math>~P_e</math>-max turning point is given by the configuration for which,
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<math>~\biggl[e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2\biggr]_{\tilde\xi} = 2 \, .</math>
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We (and others before us) have determined numerically that this condition is satisfied for <math>~\tilde\xi \approx 6.451053</math>.

Moving still further along each Figure 1 equilibrium sequence, one encounters in succession:&nbsp; a purple marker <math>~(\tilde\xi = 9)</math>;  a green marker <math>~(\tilde\xi = 67)</math>; a purple marker <math>~(\tilde\xi = 98.5)</math>; a green marker <math>~(\tilde\xi = 735)</math>; and, finally,  a purple marker <math>~(\tilde\xi = 1060)</math>.  The two additional green markers were specifically put in place in the left panel of Figure 1 to identify the approximate locations of additional local extrema in the external pressure; and the three purple markers were put in place to identify the approximate locations of local extrema in the configuration volume. We see from the center and right panels that the configurations identified by the green markers are also associated with additional local extrema in the mass, but the purple markers do not appear to be associated with any particularly special feature of these two equilibrium sequences.