Difference between revisions of "User:Tohline/Appendix/Ramblings/NumericallyDeterminedEigenvectors"

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(Begin new "Ramblings" chapter)
 
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{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Setup==
We'll begin with the linear-adiabatic wave equations that describe oscillations of the core and envelope, separately.  We also will immediately restrict our investigation to configurations for which,
We'll begin with the linear-adiabatic wave equations that describe oscillations of the core and envelope, separately.  We also will immediately restrict our investigation to configurations for which,
<div align="center">
<div align="center">
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(1 - \eta^2)\frac{d^2x}{d\eta^2} +   
(1 - \eta^2)\frac{d^2x}{d\eta^2} +   
( 4 - 6\eta^2 )  \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
( 4 - 6\eta^2 )  \frac{1}{\eta} \cdot \frac{dx}{d\eta}  
+ \mathfrak{F}_\mathrm{core} x \, .
+ \mathfrak{F}_\mathrm{core} x \, ,
</math>
</math>
   </td>
   </td>
Line 53: Line 54:
   <td align="right">
   <td align="right">
<math>~0</math>
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 + \frac{(g^2-\mathcal{B}) \xi}{\mathcal{A}} - \mathcal{D} \xi^3\biggr] \frac{d^2x}{d\xi^2}
+ \biggl\{ 3  + \frac{4(g^2-\mathcal{B}) \xi}{\mathcal{A}} - 6\mathcal{D} \xi^3 \biggr\}
\frac{1}{\xi} \cdot \frac{dx}{d\xi}
+ \biggl[
\mathcal{D} \biggl(\frac{\rho_c}{\rho_e}\biggr)  \biggl( \mathfrak{F}_\mathrm{env} +  2\alpha_e  -2\alpha_e\frac{\rho_e}{\rho_c} \biggr)\xi^3 -\alpha_e 
\biggr]\frac{x}{\xi^2}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 90: Line 72:
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl(1 - \frac{\rho_e}{\rho_c} \biggr)  \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1  + 2\biggl(\frac{\rho_e}{\rho_c}\biggr)  - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{\mathcal{A}}\biggl( \frac{\rho_e}{\rho_c}\biggr)^2  = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr]
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e}  - 2\alpha_e
\, .
</math>
  </td>
</tr>
</table>
</div>
In a [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|separate chapter on astrophysical interesting ''equilibrium structures'']], we have derived analytical expressions that define the equilibrium properties of bipolytropic configurations having <math>~(n_c, n_e) = (0, 0)</math>, that is, bipolytropes in which both the core and the envelope are uniform in density, but the densities in the two regions are different from one another.  Letting <math>~R</math> be the radius and <math>~M_\mathrm{tot}</math> be the total mass of the bipolytrope, these configurations are fully defined once any two of the following three key parameters have been specified:  The envelope-to-core density ratio, <math>~\rho_e/\rho_c</math>; the radial location of the envelope/core interface, <math>~q \equiv r_i/R</math>; and, the fractional mass that is contained within the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>.  These three parameters are related to one another via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{q^3}{\nu} \biggl( \frac{1-\nu}{1-q^3} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Equilibrium configurations can be constructed that have a wide range of parameter values; specifically,
<div align="center">
<math>~0 \le q \le 1 \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~0 \le \nu \le 1 \, ;</math>
&nbsp; &nbsp; &nbsp; &nbsp;
and,
&nbsp; &nbsp; &nbsp; &nbsp;
<math>~0 \le \frac{\rho_e}{\rho_c} \le 1 \, .</math>
</div>
(We recognize from buoyancy arguments that any configuration in which the envelope density is larger than the core density will be Rayleigh-Taylor unstable, so we restrict our astrophysical discussion to structures for which <math>~\rho_e < \rho_c</math>.)
By employing the [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|linear stability analysis techniques described in an accompanying chapter]], we should, in principle, be able to identify a wide range of eigenvectors &#8212; that is, radial eigenfunctions and accompanying eigenfrequencies &#8212; that are associated with adiabatic radial oscillation modes in any one of these equilibrium, bipolytropic configurations.  Using numerical techniques, [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy &amp; Fiedler (1985)], for example, have carried out such an analysis of bipolytropic structures having <math>~(n_c, n_e) = (1,5)</math>.  A ''pair'' of  [[User:Tohline/SSC/Perturbations#2ndOrderODE|linear adiabatic wave equations (LAWEs)]] must be solved &#8212; one tuned to accommodate the properties of the core and another tuned to accommodate the properties of the envelope &#8212;  then the pair of eigenfunctions must be matched smoothly at the radial location of the interface; the identified core- and envelope-eigenfrequencies must simultaneously match. 
After identifying the precise form of the LAWEs that apply to the case of <math>~(n_c, n_e) = (0,0)</math> bipolytropes, we discovered that, for a restricted range of key parameters, the pair of equations can both be solved ''analytically''.
==Two Separate LAWEs==
In an [[User:Tohline/SSC/Perturbations#2ndOrderODE|accompanying discussion]], we derived the so-called,
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />
{{User:Tohline/Math/EQ_RadialPulsation01}}
</div>
<!--
<div align="center" id="2ndOrderODE">
<font color="#770000">'''Adiabatic Wave Equation'''</font><br />
<math>
\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{r_0} - \biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl(\frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr)\biggl[\omega^2 + (4 - 3\gamma_\mathrm{g})\frac{g_0}{r_0} \biggr]  x = 0 \, ,
</math>
</div>
-->
For both regions of the bipolytrope, we define the dimensionless (Lagrangian) radial coordinate,
<div align="center">
<math>~\xi  \equiv \frac{r_0}{r_i} \, .</math>
</div>
So, the interface is, by definition, located at <math>~\xi = 1</math>; and, the surface is necessarily at <math>~\xi = q^{-1}</math>.  As the material in the bipolytrope's core (envelope) is compressed/de-compressed during a radial oscillation, we will assume that heating/cooling occurs in a manner prescribed by an adiabat of index <math>~\gamma_c ~(\gamma_e)</math>; in general, <math>~\gamma_e \ne \gamma_c</math>.  For convenience, we will also adopt the frequently used shorthand "alpha" notation,
<div align="center">
<math>~\alpha_c \equiv 3 - \frac{4}{\gamma_c} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\alpha_e \equiv 3 - \frac{4}{\gamma_e} \, .</math>
</div>
===The Core's LAWE===
After adopting, for convenience, the function notation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~g^2</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
1  + \biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_c} \biggr) \biggl( 1-q \biggr) +
\frac{\rho_e}{\rho_c} \biggl(\frac{1}{q^2} - 1\biggr) \biggr]  \, ,
</math>
  </td>
</tr>
</table>
</div>
we [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Match_Prasad-like_Envelope_Eigenvector_to_the_Core_Eigenvector|have deduced]] that, for the core, the LAWE may be written in the form,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
(1 - \eta^2)\frac{d^2x}{d\eta^2} + 
( 4 - 6\eta^2 )  \frac{1}{\eta} \cdot \frac{dx}{d\eta}
+ \mathfrak{F}_\mathrm{core} x \, .
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>~\eta \equiv \frac{\xi}{g} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c\, .</math>
</div>
Not surprisingly, this is identical in form to the eigenvalue problem that was first presented &#8212; and solved analytically &#8212; by [[User:Tohline/SSC/UniformDensity#Setup_as_Presented_by_Sterne_.281937.29|Sterne (1937)]] in connection with his examination of radial oscillations in ''isolated'' uniform-density spheres.  As is demonstrated below, for the core of our zero-zero bipolytrope, we can in principle adopt any one of the [[User:Tohline/SSC/UniformDensity#Sterne.27s_General_Solution|polynomial eigenfunctions and corresponding eigenfrequencies]] derived by Sterne. 
===The Envelope's LAWE===
Subsequently, we also [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#More_General_Solution|have deduced]] that, for the envelope, the governing LAWE becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 + \frac{(g^2-\mathcal{B}) \xi}{\mathcal{A}} - \mathcal{D} \xi^3\biggr] \frac{d^2x}{d\xi^2}
+ \biggl\{ 3  + \frac{4(g^2-\mathcal{B}) \xi}{\mathcal{A}} - 6\mathcal{D} \xi^3 \biggr\}
\frac{1}{\xi} \cdot \frac{dx}{d\xi}
+ \biggl[
\mathcal{D} \biggl(\frac{\rho_c}{\rho_e}\biggr)  \biggl( \mathfrak{F}_\mathrm{env} +  2\alpha_e  -2\alpha_e\frac{\rho_e}{\rho_c} \biggr)\xi^3 -\alpha_e 
\biggr]\frac{x}{\xi^2} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~2\biggl(\frac{\rho_e}{\rho_c}\biggr)  \biggl(1 - \frac{\rho_e}{\rho_c} \biggr)  \, ;
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathcal{B}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~1  + 2\biggl(\frac{\rho_e}{\rho_c}\biggr)  - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2 
\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\mathcal{D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{1}{\mathcal{A}}\biggl( \frac{\rho_e}{\rho_c}\biggr)^2  = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr]
\, ,
</math>
  </td>
</tr>


<tr>
<tr>
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</div>
</div>


<span id="KeyConstraint">We have been unable</span> to demonstrate that this governing equation can be solved analytically for ''arbitrary'' pairs of the key model parameters, <math>~q</math> and <math>~\rho_e/\rho_c</math>.  But, if we choose parameter value pairs that satisfy the constraint,
==Initial Focus==
<div align="center">
<math>~g^2 = \mathcal{B} </math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~g = \frac{1}{1+2q^3} \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; &nbsp;
<math>~q^3 = \mathcal{D} =  \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] </math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3} \, ,</math>
</div>
 
 
<div align="center">
<table border="1" cellpadding="8" align="center" width="70%">
<tr><td align="left">
<font color="red">'''WRONG!'''</font>  &nbsp; The expression that relates <math>~g^2</math> to <math>~q^3</math> should read,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~g^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1+8q^3}{(1+2q^3)^2} = \frac{1+8\mathcal{D} }{(1+2\mathcal{D})^2}</math>
  </td>
</tr>
</table>
</div>
</td></tr>
</table>
</div>
 
 
then the LAWE that is relevant to the envelope simplifies.  Specifically, it takes the form,
 
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
( 1  - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + ( 3  - 6q^3 \xi^3 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi}
+
\biggl[ q^3  \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e 
\biggr]\frac{x}{\xi^2}
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x}{\xi^2}\biggl\{
( 1  - q^3 \xi^3 )
\biggl[ \frac{d}{d\ln\xi} \biggl( \frac{d\ln x}{d\ln \xi} \biggr) - \biggl(  1 -  \frac{d\ln x}{d\ln \xi} \biggr)\cdot \frac{d\ln x}{d\ln \xi}\biggr]
+ ( 3  - 6q^3 \xi^3 )  \frac{d\ln x}{d\ln \xi}
+ \biggl[ q^3  \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e 
\biggr] \biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>
Shortly after deriving this last expression, we realized that one possible solution is a simple power-law eigenfunction of the form,
<div align="center">
<math>~x=a_0 \xi^{c_0} \, ,</math>
</div>
where the (constant) exponent is one of the roots of the quadratic equation,
<div align="center">
<math>~c_0^2 + 2c_0 - \alpha_e = 0 \, ,</math>
&nbsp; &nbsp; &nbsp; &nbsp; <math>~\Rightarrow</math> &nbsp; &nbsp; &nbsp; &nbsp;
<math>~c_0 = -1 \pm \sqrt{1+\alpha_e} \, .</math>
</div>
This power-law eigenfunction must be paired with the associated, dimensionless eigenfrequency parameter,
<div align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_0(c_0+5) = 3c_0 + \alpha_e</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e}  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 3(c_0 + \alpha_e) = 3[\alpha_e -1 \pm \sqrt{1+\alpha_e}] \, .</math>
  </td>
</tr>
</table>
</div>


Next, [[User:Tohline/SSC/Stability/BiPolytrope0_0Details#Eureka_Regarding_Prasad.27s_1948_Paper|we noticed]] the strong similarities between the mathematical properties of this eigenvalue problem and the one that was studied by [http://adsabs.harvard.edu/abs/1948MNRAS.108..414P C. Prasad (1948, MNRAS, 108, 414-416)] in connection with, what we now recognize to be, a closely related problem.  Drawing heavily from Prasad's analysis, we discovered an infinite number of eigenfunctions (each, a truncated polynomial expression) and associated eigenfrequencies that satisfy this governing envelope LAWE.  The eigenvectors associated with the lowest few modes are tabulated, below.




{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Revision as of 00:24, 14 January 2017

Numerically Determined Eigenvectors of a Zero-Zero Bipolytrope

Here we build on the analytic foundation summarized in an accompanying chapter and attempt to numerically construct a variety of eigenvectors that describe radial oscillations of bipolytropes for which, <math>~(n_c, n_e) = (0,0)</math>.

Whitworth's (1981) Isothermal Free-Energy Surface
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Setup

We'll begin with the linear-adiabatic wave equations that describe oscillations of the core and envelope, separately. We also will immediately restrict our investigation to configurations for which,

<math>~g^2 = \mathcal{B} </math>         <math>~\Rightarrow</math>         <math>~g^2 = \frac{1+8q^3}{ (1+2q^3)^2 } \, ,</math>         and,         <math>~q^3 = \mathcal{D} = \biggl[ \frac{\rho_e/\rho_c}{2(1-\rho_e/\rho_c)} \biggr] </math>         <math>~\Rightarrow</math>         <math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3} \, .</math>

For the core we have,

<math>~0</math>

<math>~=</math>

<math>~ (1 - \eta^2)\frac{d^2x}{d\eta^2} + ( 4 - 6\eta^2 ) \frac{1}{\eta} \cdot \frac{dx}{d\eta} + \mathfrak{F}_\mathrm{core} x \, , </math>

where,

<math>~\eta \equiv \frac{\xi}{g} \, ,</math>         and         <math>~\mathfrak{F}_\mathrm{core} \equiv \frac{3\omega_\mathrm{core}^2}{2\pi G\gamma_c \rho_c} - 2\alpha_c\, .</math>

And, for the envelope we have,

<math>~0</math>

<math>~=</math>

<math>~ ( 1 - q^3 \xi^3 ) \frac{d^2x}{d\xi^2} + ( 3 - 6q^3 \xi^3 ) \frac{1}{\xi} \cdot \frac{dx}{d\xi} + \biggl[ q^3 \mathfrak{F}_\mathrm{env} \xi^3 -\alpha_e \biggr]\frac{x}{\xi^2} \, , </math>

where,

<math>~\mathfrak{F}_\mathrm{env}</math>

<math>~\equiv</math>

<math>~\frac{3\omega^2_\mathrm{env}}{2\pi G \gamma_e \rho_e} - 2\alpha_e \, . </math>

Initial Focus

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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