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Comparing Stability Analyses of Zero-Zero Bipolytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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In separate chapters we have discussed the following interrelated aspects of Bipolytropes that have ~(n_c,n_e) = (0,0):

Building on these separate discussions, here we examine what might be learned from a comparison of the two traditional approaches to stability analysis, namely:  (1) solutions of the LAWE, and (2) a free-energy analysis.

Key Attributes of Equilibrium Configurations

Physical Properties

This adopted parameter notation pays tribute to the notation that was introduced by Chandrasekhar and his collaborators in the early 1940s in papers associated with the discovery of the Schönberg-Chandrasekhar mass limit.
Aside from specifying its radius, ~R, and total mass, ~M_\mathrm{tot}, there are three particularly interesting dimensionless parameters that characterize the internal structure of a bipolytrope having ~(n_c,n_e) = (0,0). They are, the radial location of the core/envelope interface,

~q \equiv \frac{r_i}{R} \, ;

the ratio of the density of the envelope material to the density of the core, ~0 \le \rho_e/\rho_c \le 1; and the fraction of the total mass that is contained in the core,

~\nu \equiv \frac{M_\mathrm{core}}{M_\mathrm{tot}} \, .

Identifying a unique bipolytropic configuration requires the specification of two of these three dimensionless parameters; the third parameter is, then, necessarily determined via what we will refer to as the,

Primary Algebraic Constraint

~\frac{\rho_e}{\rho_c}

=

~\frac{q^3(1-\nu)}{\nu(1-q^3)} \, .

It is also relatively straightforward to appreciate that, in dimensional units, the value of the central density is,

~\rho_c

~=

~\frac{3M_\mathrm{tot}}{4\pi G R^3} \cdot \frac{\nu}{q^3} \, .

Our study of equilibrium configurations has shown that once, for example, the pair of parameters, ~q and ~\rho_e/\rho_c, has been specified, other properties of the associated equilibrium configuration can be succinctly expressed in terms of the function,

~g^2

~\equiv


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] \, .

For example, the central pressure is given by the expression,

~P_c

  ~= 

\biggl( \frac{3}{2^3\pi} \biggr) \frac{\nu^2 g^2}{q^4} \biggl[ \frac{GM_\mathrm{tot}^2}{R^4} \biggr] \, .

Sequences

Figure 1:
Equilibrium Sequences of Constant ~\rho_e/\rho_c

Constant Density Sequences

Across the two-dimensional, ~(q,\nu) parameter space that is defined by the full range of physically viable values of ~q and ~\nu, namely,

~0 \le q \le 1 \, ,      and       ~0 \le \nu \le 1 \, ,


an equilibrium model sequence can be defined by, for example, specifying that all models along the sequence have the same density jump at the interface. Drawing on the above primary algebraic constraint, each choice of ~\rho_e/\rho_c will generate a sequence governed by the function,

~\nu

=

~\biggl[\frac{(1-q^3)}{q^3} \biggl( \frac{\rho_e}{\rho_c} \biggr) + 1\biggr]^{-1}

 

=

~\frac{q^3}{q^3 + (1-q^3)(\rho_e/\rho_c)} \, .

Figure 1 displays several such equilibrium sequences across the ~(q,\nu) plane — see also a related figure associated with our free-energy determination of stability. The curves show how ~\nu varies with ~q along sequences for which the specified density ratio is ~\tfrac{1}{2} (blue), ~\tfrac{1}{4} (green), and ~\tfrac{1}{10} (maroon). We have employed a free-energy analysis (see summary, below) to examine whether a transition from stable to unstable configurations is encountered while traversing — that is, while evolving along — such sequences.

Figure 2:
Analytic Eigenvector Constraint

Analytic Eigenvector Sequence

In a separate search for eigenvectors that simultaneously satisfy the linear adiabatic wave equation (LAWE) for the core and the LAWE for the envelope (see summary, below), we discovered that eigenvectors for some radial modes of oscillation can be specified fully analytically along a sequence of equilibrium models that is defined by what we will refer to as the,

Analytic Eigenvector Constraint

~g^2

~\equiv

~1  + 2\biggl(\frac{\rho_e}{\rho_c}\biggr)  - 3\biggl(\frac{\rho_e}{\rho_c}\biggr)^2  
\, .

When combined with the above primary algebraic constraint, this is equivalent to demanding that,

~\nu

~=

~\tfrac{1}{3}(1+2q^3) \, ,

and, simultaneously,

~\frac{\rho_e}{\rho_c}

~=

~\frac{2q^3}{1+2q^3} \, .

The behavior of these two functions is displayed in Figure 2; the variation of ~\nu with ~q is traced by the dark blue squares while the variation of ~\rho_e/\rho_c with ~q is marked by the small, circular black dots.

Radial Oscillation Frequencies

In a separate chapter, we have summarized some of the quantitative characteristics of five radial oscillation modes that we have determined analytically for bipolytropes that have ~(n_c, n_e) = (0,0). Table 1 details some of these characteristics; among them is the dimensionless oscillation frequency,

~\sigma_c^2 \equiv \frac{3\omega^2}{2\pi \gamma_c G \rho_c} \, .


Table 1
Quantum Numbers ~q ~\nu ~\gamma_c ~\gamma_e ~\sigma_c^2
~\ell ~j
2 1 0.794385 0.668 2.254 1.194 16.45
2 2 0.768375 0.636 1.046 1.209 34.37
3 1 0.396061 0.375 1.023 1.344 12.17
3 2 0.594040 0.473 1.025 1.056 34.20
3 3 0.645515 0.513 1.325 1.840 65.97

Our free-energy analysis of these bipolytropic configurations has shown that each model's characteristic radial oscillation frequency is given by the expression,

~\sigma_\mathfrak{G}^2 \equiv \frac{3\omega_\mathfrak{G}^2}{2\pi \gamma_c G\rho_c}

~=

~
\frac{3q^2 \nu }{5\gamma_c}\biggl[ 
2( 3\gamma_e - 4)  f 
+ 3(\gamma_e - \gamma_c)(3 - 5 g^2)  \biggr] \, ,

where,

~f

~\equiv~

1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl(\frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] \, .

Here we will restrict our discussion to models that obey the analytic eigenvector constraint, in which case,

~f

~=~

1 + \frac{5}{2} \biggl( \frac{2q^3}{1+2q^3} \biggr) \biggl(\frac{1-q^2}{q^2} \biggr)
+ \biggl( \frac{2q^3}{1+2q^3} \biggr)^2 \biggl[ \biggl(\frac{1-q^5}{q^5} \biggr) - \frac{5}{2}\biggl(\frac{1-q^2}{q^2} \biggr) \biggr]

 

~=~

1 + 5 \biggl[ \frac{q(1-q^2)}{1+2q^3} \biggr] 
+ \biggl[ \frac{2q}{(1+2q^3)^2} \biggr] \biggl[ 2 (1-q^5 ) - 5q^3 (1-q^2 ) \biggr]

 

~=~

1 + 5 \biggl[ \frac{q(1-q^2)}{1+2q^3} \biggr] 
+ \biggl[ \frac{2q}{(1+2q^3)^2} \biggr] \biggl[ 2 - 5q^3 +3q^5  \biggr]

 

~=~

\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) + 5 [ q(1-q^2)(1+2q^3) ] 
+ 2q  ( 2 - 5q^3 +3q^5 ) 
\biggr\}

 

~=~

\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) 
+ q (5 - 5q^2 + 10q^3 - 10q^5  +  4 - 10q^3 + 6q^5 ) 
\biggr\}

 

~=~

\frac{1}{(1+2q^3)^2} \biggl\{ (1 + 4q^3 + 4q^6) 
+ (9q - 5q^3  - 4q^6  ) 
\biggr\}

 

~=~

\frac{1}{(1+2q^3)^2} \biggl[ 1 + 9q - q^3   \biggr] \, ,

and,

~(3-5g^2)

~=

~3 - 5\biggl[1 + 2\biggl( \frac{2q^3}{1+2q^3} \biggr) - 3\biggl( \frac{2q^3}{1+2q^3} \biggr)^2 \biggr]

 

~=

~3 - \frac{5}{(1+2q^3)^2 }\biggl[(1+2q^3)^2 + 4q^3(1+2q^3) - 12q^6 \biggr]

 

~=

~\frac{1}{(1+2q^3)^2 }\biggl[-2(1+2q^3)^2 - 20q^3(1+2q^3) + 60q^6 \biggr]

 

~=

~\frac{1}{(1+2q^3)^2 }\biggl[-2-8q^3  - 8q^6  -20q^3 - 40q^6 + 60q^6 \biggr]

 

~=

~- \frac{2(1 + 14q^3  - 6q^6 ) }{(1+2q^3)^2 } \, .

Hence,

~\sigma_\mathfrak{G}^2

~=

~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
2\biggl( \frac{3\gamma_e}{\gamma_c} - \frac{4}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  \biggr)
-6 \biggl( \frac{\gamma_e}{\gamma_c} - 1 \biggr)\biggl( 1 + 14q^3  - 6q^6 \biggr)  \biggr]

 

~=

~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
\biggl( - \frac{8}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  \biggr)
+\biggl( \frac{6\gamma_e}{\gamma_c} \biggr)  \biggl( 1 + 9q - q^3  
-1 - 14q^3  + 6q^6 \biggr)  
+ 6 \biggl( 1 + 14q^3  - 6q^6 \biggr)  
\biggr]

 

~=

~
\frac{3q^2 \nu }{5(1+2q^3)^2}\biggl[ 
6 ( 1 + 14q^3  - 6q^6 ) - \frac{8}{\gamma_c}  \biggl( 1 + 9q - q^3  \biggr)
+6q\biggl( \frac{\gamma_e}{\gamma_c} \biggr)  \biggl( 9  - 15q^2  + 6q^5 \biggr)  
\biggr]

 

~=

~
\frac{q^2 }{5(1+2q^3)}\biggl[ 
6 ( 1 + 14q^3  - 6q^6 ) - \frac{8}{\gamma_c}  \biggl( 1 + 9q - q^3  \biggr)
+6q\biggl( \frac{\gamma_e}{\gamma_c} \biggr)  \biggl( 9  - 15q^2  + 6q^5 \biggr)  
\biggr] \, ,

where, making this last step, we have replaced the leading factor of ~\nu with its (above) expression in terms of ~q.

Analysis

If we hold ~q and ~\gamma_e fixed, at what value of ~\gamma_c does the Free-energy frequency go to zero? The answer is as follows.

~0

~=

~ 
6 ( 1 + 14q^3  - 6q^6 ) - \frac{8}{[\gamma_c]_\mathrm{crit} }  \biggl( 1 + 9q - q^3  \biggr)
+6q\biggl( \frac{\gamma_e}{[\gamma_c]_\mathrm{crit} } \biggr)  \biggl( 9  - 15q^2  + 6q^5 \biggr)

~\Rightarrow~~~ 6 ( 1 + 14q^3  - 6q^6 ) [\gamma_c]_\mathrm{crit}

~=

~ 
8 ( 1 + 9q - q^3 ) - 6q\gamma_e  ( 9  - 15q^2  + 6q^5 )

~\Rightarrow~~~  [\gamma_c]_\mathrm{crit}

~=

~ 
\frac{4 ( 1 + 9q - q^3 ) - 3q\gamma_e  ( 9  - 15q^2  + 6q^5 )  }{ ( 1 + 14q^3  - 6q^6 )}

Table 2
from Table 1 Global Stability
Quantum Numbers ~q ~\nu ~\gamma_c ~\gamma_e ~\sigma_c^2 ? ~[\gamma_c]_\mathrm{crit}
~\ell ~j
2 1 0.794385 0.668 2.254 1.194 16.45 S 1.358
2 2 0.768375 0.636 1.046 1.209 34.37 U 1.361
3 1 0.396061 0.375 1.023 1.344 12.17 U 1.318
3 2 0.594040 0.473 1.025 1.056 34.20 U 1.520
3 3 0.645515 0.513 1.325 1.840 65.97 S 1.075

Related Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

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