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Instabilities in Bounded and Composite Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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Unbounded, Complete Polytropes

Free-Energy Function and Its Derivatives

The free-energy function that is relevant to a discussion of the structure and stability of unbounded configurations having polytropic index, ~n, has the form,

~\mathcal{G}(x)

~=


-ax^{-1} +b x^{-3/n} + \mathcal{G}_0
\, ,

where ~x \equiv R/R_\mathrm{SWS} identifies the radius of the configuration and \mathcal{G}_0 is an arbitrary constant. If the coefficients, ~a, b, and ~c, are held constant while varying the configuration's size, we see that,

~\frac{d\mathcal{G}}{dx}

~=


x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} \biggr]
\, ,

and,

~\frac{d^2\mathcal{G}}{dx^2}

~=


x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} \biggr]
\, .

In terms of the system's mass and its structural form factors, ~\mathfrak{f}_M, ~\mathfrak{f}_A, and ~\mathfrak{f}_W, the two relevant coefficients are,

~a

~=

~\frac{3}{5}\biggl(\frac{n+1}{n}\biggr) \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^2
\mathfrak{f}_W  \, ,

~b

~=

~n \biggl( \frac{3}{4\pi}\biggr)^{1/n} \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\mathfrak{f}_M} \biggr]^{(n+1)/n}
\mathfrak{f}_A \, .

Equilibrium Radius

A configuration's equilibrium radius, ~x_\mathrm{eq}, can be determined in one of two ways:

Extrema in the Free Energy

Equilibria are identified by extrema in the free-energy function. Setting d\mathcal{G}/dx = 0, we find,

~x_\mathrm{eq}

~=


\biggl(\frac{an}{3b} \biggr)^{n/(n-3)}

~\Rightarrow ~~~~~ R_\mathrm{eq}^{n-3}

~=

\frac{R_\mathrm{SWS}^{n-3}}{M_\mathrm{SWS}^{n-1}} \biggl( \frac{4\pi}{3\cdot 5^n}\biggr) \biggl(\frac{n+1}{n}\biggr)^n 
\biggl[ \frac{\mathfrak{f}_W^n \mathfrak{f}_M^{1-n}}{\mathfrak{f}_A^n}   \biggr] M^{n-1}

 

~=

\biggl( \frac{4\pi}{3\cdot 5^n}\biggr)  
\biggl[ \frac{\mathfrak{f}_W^n \mathfrak{f}_M^{1-n}}{\mathfrak{f}_A^n}  \biggr]  G^n K^{-n} M^{n-1} \, .

If one assumes that the equilibrium system has no internal structure — that is, that the interior density and temperature are uniform throughout — then ~\mathfrak{f}_M = \mathfrak{f}_A = \mathfrak{f}_W = 1 and this derived expression gives a good estimate of the equilibrium radius, given any choice of the pair of parameters, ~M and ~K.


Detailed Force Balance

Alternatively, a solution of the,

Lane-Emden Equation

~\frac{1}{\xi^2} \frac{d}{d\xi}\biggl( \xi^2 \frac{d\Theta_H}{d\xi} \biggr) = - \Theta_H^n

gives information regarding the detailed interior structure of the equilibrium system via knowledge of the properties of the Lane-Emden function, ~\Theta_H(\xi), as well as an exact expression for the equilibrium radius, namely,

~R_\mathrm{eq}^{n-3}
= \frac{3^n}{(n+1)^n} \biggl(\frac{4\pi}{3}\biggr) G^n K^{-n} M^{n-1} \cdot \frac{\mathfrak{f}_M^{1-n}}{\xi_1^{2n}} \, ,

where,

~\mathfrak{f}_M \equiv \frac{\bar\rho}{\rho_c} = \biggl(- \frac{3\Theta_H^'}{\xi} \biggr)_{\xi_1} \, .

Together

Now, once the Lane-Emden function, ~\Theta_H, is known from a detailed force-balance model, the other two structural form factors also can be straightforwardly determined. For unbounded, complete polytropes, the relevant expressions are,

~\mathfrak{f}_W

~=

~\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta_H^'}{\xi} \biggr]^2_{\xi_1} \, ,

~\mathfrak{f}_A

~=

~\frac{3(n+1) }{(5-n)} ~\biggl[ \Theta_H^' \biggr]^2_{\xi_1} \, .

Plugging the appropriate ratio of these two functions, namely,

~\biggl[\frac{\mathfrak{f}_W}{\mathfrak{f}_A} \biggr]^n

~=

~\biggl( \frac{3\cdot 5}{n+1} \biggr)^n \biggl[ \frac{1}{\xi} \biggr]^{2n}_{\xi_1} \, ,

into the expression for the equilibrium radius obtained from the free-energy analysis gives precisely the same answer as was obtained from the detailed force-balance analysis. Using either method of determination we conclude, therefore, that,

~\frac{K^n R_\mathrm{eq}^{n-3}}{G^n M^{n-1}}
= \frac{4\pi}{(n+1)^n} \cdot \xi_1^{-(n+1)} (-\Theta_H^')^{1-n}_{\xi_1} \, .

Stability

The procedure that has been used to obtain a detailed force-balanced model of unbounded polytropes cannot readily be extended to provide a stability analysis of such systems. However, the free-energy analysis can be readily extended. If the first derivative of the free-energy function is zero — that is, if you have identified an equilibrium configuration — and the second derivative of the free-energy function for that configuration is positive, then the equilibrium system is dynamically stable. If, however, the second derivative is negative, then the equilibrium system is dynamically unstable.

Using the expression for the second derivative of the free-energy function derived above, we deduce that equilibrium configurations are dynamically unstable when,

~\biggl[\frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} - 2a \biggr]_{x_\mathrm{eq}}

~<


~0

~\Rightarrow ~~~~~ x_\mathrm{eq}^{(n-3)/n}

~<


~\frac{2an^2}{3(3+n)b}

~\Rightarrow ~~~~~ \frac{an}{3b}

~<


~\frac{2an^2}{3(3+n)b}

~\Rightarrow ~~~~~ n

~>


~3 \, .


Bounded (Pressure-Truncated) Polytropes

Throughout this section we will rely on the following definitions of two normalization constants:

M_\mathrm{SWS} \equiv 
\biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,


R_\mathrm{SWS} \equiv \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, .

Free-Energy Function and Its Derivatives

The free-energy function that is relevant to a discussion of the structure and stability of a pressure-truncated configuration having polytropic index, ~n, has the form,

~\mathcal{G}(x)

~=


-ax^{-1} +b x^{-3/n} + c x^3 + \mathcal{G}_0
\, ,

where ~x \equiv R/R_\mathrm{SWS} identifies the radius of the configuration and \mathcal{G}_0 is an arbitrary constant. If the coefficients, ~a, b, and ~c, are held constant while varying the configuration's size, we see that,

~\frac{d\mathcal{G}}{dx}

~=


ax^{-2} - \frac{3b}{n}\cdot x^{-(3+n)/n} + 3c x^2

 

~=


x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} + 3c x^4 \biggr]
\, ,

and,

~\frac{d^2\mathcal{G}}{dx^2}

~=


x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} + 6c x^4 \biggr]
\, .

In terms of the system's mass and its structural form factors, ~\tilde\mathfrak{f}_M, ~\tilde\mathfrak{f}_A, and ~\tilde\mathfrak{f}_W, the three relevant coefficients are,

~a

~=

~\frac{3}{5}\biggl(\frac{n+1}{n}\biggr) \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2
\tilde\mathfrak{f}_W  \, ,

~b

~=

~n \biggl( \frac{3}{4\pi}\biggr)^{1/n} \biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}
\tilde\mathfrak{f}_A \, ,

~c

~=

~\frac{4\pi}{3} \, .

Equilibrium Radius

A configuration's equilibrium radius, ~x_\mathrm{eq}, can be determined in one of two ways:

Extrema in the Free Energy

Equilibria are identified by extrema in the free-energy function, that is, by setting d\mathcal{G}/dx = 0. Hence, ~x_\mathrm{eq} is given by the root(s) of the polynomial expression that is often referred to as the,

Scalar Virial Theorem

~x^{(n-3)/n}_\mathrm{eq}

~=

\frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] \, .

Plugging the definitions of the three free-energy coefficients into this virial expression gives the mass-radius relationship for pressure-truncated, polytropic equilibrium configurations, namely,

~\biggl( \frac{3}{4\pi} \biggr)^{1/n} \tilde\mathfrak{f}_A \biggl[
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n}

~=

\frac{1}{5} \cdot \tilde\mathfrak{f}_W 
\biggl( \frac{n+1}{n}\biggr) \biggl[\biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{2} 
+ \frac{4\pi}{3} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \, .

If one assumes that the equilibrium system has no internal structure — that is, that the interior density and temperature are uniform throughout — then ~\mathfrak{f}_M = \mathfrak{f}_A = \mathfrak{f}_W = 1 and this derived expression provides a good, approximate mass-radius relationship for configurations having a given ~K that are truncated by a surrounding, massless medium exerting a pressure, ~P_e, on the system.

Detailed Force Balance

From the detailed force-balance analysis presented in Appendix B of Steven W. Stahler (1983), we see that the mass, ~M, associated with the equilibrium radius, ~R_\mathrm{eq}, of bounded (pressure-truncated) polytropic spheres is given through the following pair of parametric relations:


~\frac{M}{M_\mathrm{SWS}}

~=~


\biggl( \frac{n^3}{4\pi} \biggr)^{1/2}  \tilde\Theta_H^{(n-3)/2} \tilde\xi^2 (-\tilde\Theta_H^') \, ,


~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }

~=~


\biggl( \frac{n}{4\pi} \biggr)^{1/2}  \tilde\xi \tilde\Theta_H^{(n-1)/2}  \, .

Together

As was realized in the case of unbounded polytropes, once the Lane-Emden function, ~\tilde\Theta_H(\xi), and its radial derivative, ~\tilde\Theta_H^'(\xi), are known from a detailed force-balance analysis, all three structural form factors can be straightforwardly determined. For bounded (pressure-truncated) polytropes, the relevant expressions are,

~\tilde\mathfrak{f}_M

~=

~ - \frac{3\tilde\Theta_H^'}{\tilde\xi}  \, ,

\tilde\mathfrak{f}_W

~=

\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\Theta_H^{n+1} + 3 (\tilde\Theta_H^')^2 - \tilde\mathfrak{f}_M \tilde\Theta_H \biggr] \, ,

~
\tilde\mathfrak{f}_A

~=

~\frac{1}{(5-n)} \biggl\{ 6\tilde\Theta_H^{n+1} +  (n+1)
\biggl[3 (\tilde\Theta_H^')^2 - \tilde\mathfrak{f}_M \tilde\Theta_H \biggr] \biggr\} \, .

We have noticed, as well, that the relationship between ~\tilde\mathfrak{f}_A and ~\tilde\mathfrak{f}_W is relatively simple, namely,

~
\tilde\mathfrak{f}_A

~=

~
\tilde\Theta_H^{n+1} +\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2\cdot \tilde\mathfrak{f}_W \, .

As it turns out, it is sufficient to use this relationship — rather than the separate, explicit definitions of ~\tilde\mathfrak{f}_A and ~\tilde\mathfrak{f}_W — in order to obtain identical expressions for the equilibrium radius from both the detailed force-balance and free-energy analyses. By way of demonstration, let's plug this simpler expression for ~\tilde\mathfrak{f}_A into the virial equilibrium relation.

~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4

~=


\biggl( \frac{3}{4\pi} \biggr)^{(n+1)/n} 
\biggl[ \tilde\Theta_H^{n+1} +\frac{(n+1)}{3\cdot 5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W  \biggr] \biggl[
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
- ~\frac{3}{5 \cdot 4\pi} \cdot \tilde\mathfrak{f}_W 
\biggl( \frac{n+1}{n}\biggr) \biggl[\biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{2}

 

~=

 
\biggl[ \tilde\Theta_H^{n+1} +\frac{(n+1)}{3\cdot 5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W  \biggr] \mathfrak{M}^{(n+1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
- ~\frac{4\pi}{3\cdot 5} \cdot \tilde\mathfrak{f}_W \biggl( \frac{n+1}{n}\biggr) \mathfrak{M}^{2}

 

~=

 
\biggl[ \tilde\Theta_H^{n} \mathfrak{M}\biggr]^{(n+1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
+ ~\biggl[ \mathfrak{M}^{(n+1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
- ~\frac{4\pi}{n} \biggl( \frac{\mathfrak{M}}{\tilde\xi}\biggr)^{2} \biggr] \frac{(n+1)}{3\cdot 5} \cdot \tilde\xi^2 \tilde\mathfrak{f}_W \, ,

where, we have temporarily adopted the shorthand notation,

~\mathfrak{M}

~\equiv

~\frac{3}{4\pi} 
\biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{1}{\tilde\mathfrak{f}_M} = \biggl( \frac{M}{M_\mathrm{SWS}} \biggr) \frac{\tilde\xi}{4\pi (- \tilde\Theta_H^')} \, .

Regrouping terms from this last virial expression, we also can write,

~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl\{1 -
\biggl[ \tilde\Theta_H^{n} \mathfrak{M} 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{-3} \biggr]^{(n+1)/n} 
\biggr\}

~=

 
~\mathfrak{M}^2 \biggl\{ \frac{n\tilde\xi^2}{4\pi} \cdot \mathfrak{M}^{(1-n)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
- ~1 \biggr\} \frac{4\pi(n+1)}{3\cdot 5n} \cdot \tilde\mathfrak{f}_W \, .

Now, this relation will be satisfied without having to plug in the fully explicit definition of ~\tilde\mathfrak{f}_W if we demand that the expressions inside of the curly braces on both sides of the equation are zero, independently. This demand implies that (from the LHS),

~\tilde\Theta_H^{n} \mathfrak{M} 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{-3}

~=

 
~1 \,  ,

and, independently (from the RHS),

~
\frac{n\tilde\xi^2}{4\pi} \cdot \mathfrak{M}^{(1-n)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n}

~=

 
~1 \, .

This pair of conditions, in turn, imply that,

~
\biggl[ \tilde\Theta_H^{n} 
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{-3}  \biggr]^{(n-1)/n}  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n}

~=

 
~\frac{4\pi}{n\tilde\xi^2}

~\Rightarrow ~~~~~
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{2}

~=

 
~\frac{n}{4\pi} \cdot \tilde\xi^{2} \tilde\Theta_H^{n-1} \, ;

and,

~\mathfrak{M}

~=

 
~\biggl(\frac{n}{4\pi} \cdot \tilde\xi^{2} \tilde\Theta_H^{n-1}\biggr)^{3/2} \tilde\Theta_H^{-n} = 
\biggl(\frac{n}{4\pi}\biggr)^{3/2} \cdot \tilde\xi^{3} \tilde\Theta_H^{(n-3)/2}

~\Rightarrow~~~~~\biggl( \frac{M}{M_\mathrm{SWS}} \biggr)

~=

 
~
\biggl(\frac{n^3}{4\pi}\biggr)^{1/2} (- \tilde\xi^{2}\tilde\Theta_H^')\tilde\Theta_H^{(n-3)/2} \, ,

which match exactly the pair of parametric equations relating ~M to ~R_\mathrm{eq} that Stahler derived via his detailed force-balance analysis. We note as well that, after eliminating the parameter, ~P_e — which appears in the definition of both normalizations, ~M_\mathrm{SWS} and ~R_\mathrm{SWS} — using either method of determination we have,


~\frac{K^n R_\mathrm{eq}^{n-3}}{G^n M^{n-1}}
= \frac{4\pi}{(n+1)^n} \cdot \tilde\xi^{-(n+1)} (-\tilde\Theta_H^')^{1-n} \, .

Stability

As before, if the first derivative of the free-energy function is zero — that is, if you have identified an equilibrium configuration — and the second derivative of the free-energy function for that configuration is positive, then the equilibrium system is dynamically stable. If, however, the second derivative is negative, then the equilibrium system is dynamically unstable.

Using the expression for the second derivative of the free-energy function derived above, we deduce that equilibrium configurations are dynamically unstable when,

~\biggl[\frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} - 2a  + 6cx^4\biggr]_{x_\mathrm{eq}}

~<


~0

~\Rightarrow ~~~~~ x_\mathrm{eq}^{(n-3)/n}

~<


~\frac{n^2}{3(3+n)b}\biggl[ 2a - 6cx^4_\mathrm{eq} \biggr]

~\Rightarrow ~~~~~ \frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr]

~<


~\frac{n^2}{3(3+n)b}\biggl[ 2a - 6cx^4_\mathrm{eq} \biggr]

~\Rightarrow ~~~~~ (3+n) [a + 3c x^4_\mathrm{eq} ]

~<


~n[ 2a - 6cx^4_\mathrm{eq} ]

~\Rightarrow ~~~~~ [3c(3+n) + 6nc] x^4_\mathrm{eq}

~<


~[2n  - (3+n)]a

~\Rightarrow ~~~~~ x^4_\mathrm{eq}

~<


~\frac{(n-3)a}{9c(n+1)} \, .

Plugging in the expressions for the free-energy coefficients, ~a and ~c, this in turn implies that an equilibrium system will be dynamically unstable if,

~\biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^4

~<


~\frac{(n-3)}{20\pi n} 
\biggl[ \biggl( \frac{M}{M_\mathrm{SWS}}\biggr) \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^2
\tilde\mathfrak{f}_W

~\Rightarrow ~~~~~ \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^4 \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^{-2}

~<


~\biggl[ \frac{(n-3)}{12\pi n (5-n)} \biggr] \biggl[\frac{1}{\tilde\xi(-\tilde\Theta_H^')^2} \biggr]
\biggl[\tilde\xi \tilde\Theta_H^{n+1} + 3\tilde\xi (\tilde\Theta_H^')^2 - 3(-\tilde\Theta_H^')\tilde\Theta_H \biggr]

~\Rightarrow ~~~~~ \frac{P_e R_\mathrm{eq}^4}{GM^2}

~<


~\biggl[ \frac{(n-3)}{12\pi (5-n)(n+1)} \biggr] \biggl[\frac{1}{\tilde\xi(-\tilde\Theta_H^')^2} \biggr]
\biggl[\tilde\xi \tilde\Theta_H^{n+1} + 3\tilde\xi (\tilde\Theta_H^')^2 - 3(-\tilde\Theta_H^')\tilde\Theta_H \biggr] \, .


Composite Polytropes (Bipolytropes)

Throughout this section we will rely on the following definitions of three normalization constants:

~R_\mathrm{norm}

~=

~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)}  \, ,

~P_\mathrm{norm}

~=

~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)}  \, ,

~E_\mathrm{norm}

~=

~ 
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, .

Free-Energy Function and Its Derivatives

The free-energy function that is relevant to a discussion of the structure and stability of a composite polytropic configurations of index, ~n_c, for the core and, ~n_e, for the envelope, has the form,

~\mathcal{G}(x)

~=


-(a_c + a_e) x^{-1} +b_c x^{-3/n_c} +b_e x^{-3/n_e} + \mathcal{G}_0
\, ,

where ~x \equiv R/R_\mathrm{norm} identifies the radius of the configuration and \mathcal{G}_0 is an arbitrary constant. If the coefficients, ~a_c, a_e, b_c, and ~b_e, are held constant while varying the configuration's size, we see that,

~\frac{d\mathcal{G}}{dx}

~=


(a_c + a_e)x^{-2} - \frac{3b_c}{n_c}\cdot x^{-(3+n_c)/n_c} - \frac{3b_e}{n_e}\cdot x^{-(3+n_e)/n_e}

 

~=


x^{-2} \biggl[ (a_c + a_e) - \frac{3b_c}{n_c}\cdot x^{(n_c-3)/n_c}  - \frac{3b_e}{n_e}\cdot x^{(n_e-3)/n_e}  \biggr]
\, ,

and,

~\frac{d^2\mathcal{G}}{dx^2}

~=


x^{-3} \biggl[ -2(a_c + a_e) + \frac{3(3+n_c)b_c}{n_c^2}\cdot x^{(n_c-3)/n_c} + \frac{3(3+n_e)b_e}{n_e^2}\cdot x^{(n_e-3)/n_e} \biggr]
\, .

The four relevant coefficients are,

~a_c

~\equiv


\biggl( \frac{\nu}{q^3} \biggr) \int_0^{q} 3\biggl[\frac{M_r(y)}{M_\mathrm{tot}} \biggr]_\mathrm{core}  \biggl[ \frac{\rho(y)}{\bar\rho} \biggr]_\mathrm{core}  y dy 
\, ,

~a_e

~\equiv


\biggl( \frac{1-\nu}{1-q^3} \biggr) \int_{q}^{1} 3\biggl[\frac{M_r(y)}{M_\mathrm{tot}} \biggr]_\mathrm{env}  \biggl[ \frac{\rho(y)}{\bar\rho} \biggr]_\mathrm{env}  y dy \, ,

~b_c

~\equiv


\frac{4\pi n_c}{3} \biggl[ \frac{P_{ic} x^{3(n_c+1)/n_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_0^q  3\biggl[\frac{1 - p_c(y)}{1-p_c(q)} \biggr]  y^2 dy 
= 
\frac{4\pi n_c}{3} \biggl[ \frac{P_{ic} x^{3(n_c+1)/n_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} q^3 s_\mathrm{core} 
\, ,

~b_e

~\equiv


\frac{4\pi n_e}{3} \biggl[ \frac{P_{ie} x^{3(n_e+1)/n_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \int_q^1  3\biggl[1 - p_e(y) \biggr]  y^2 dy 
=
\frac{4\pi n_e}{3} \biggl[ \frac{P_{ie} x^{3(n_e+1)/n_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} (1-q^3)s_\mathrm{env}
\, .


Specifically for analytically definable composite polytropes having ~(n_c, n_e) = (5, 1),

~a_c

~\equiv


\frac{3}{5}
\biggl[\biggl(\frac{\nu}{q^3} \biggr)^2 \biggl( 1 + \ell_i^2 \biggr)^{3}  \biggr]_\mathrm{eq}
\biggl(\frac{5}{2^4}\biggr) \biggl( \frac{q}{\ell_i}\biggr)^{5} 
\biggl[ \ell_i \biggl(\ell_i^4 - \frac{8}{3} \ell_i^2 -1 \biggr) (1 + \ell_i^2)^{-3}  + \tan^{-1}(\ell_i) \biggr]  
\, ,

~a_e

~\equiv


\biggl( \frac{\mu_e}{\mu_c} \biggr)^2  \biggl[ \frac{3^2 A^2 }{b_\eta^4 } \biggr] \biggl( \frac{\nu^2 \theta_i^4}{q^6} \biggr)
\frac{1}{8b_\eta} \biggl[6 b_\eta y  - 3\sin[2(b_\eta y-B)] - 4b_\eta y \sin^2(b_\eta y - B)  \biggr]_q^1

 

~=


\biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
\frac{3A^2 }{8b_\eta^3} \biggl[2 b_\eta y  - \sin[2(b_\eta y-B)] - \frac{4}{3}\cdot b_\eta y \sin^2(b_\eta y - B)  \biggr]_q^1  \, ,

~q^3 s_\mathrm{core}

~\equiv


~\frac{3}{2^3}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3
\biggl[ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \biggr]
\, ,

~(1-q^3)s_\mathrm{env}

~\equiv


~\biggl( \frac{P_{ic}}{P_{ie}}  \biggr) \frac{3A^2}{4b_\eta^3} \biggl[ 2b_\eta y -\sin[2(b_\eta y - B)] \biggr]_q^1   \ ,

where,

~\ell

~\equiv~

~\frac{\xi}{\sqrt{3}} \, ,

~A

~=~

~\frac{\eta_i}{\sin(\eta_i - B)} \, ,

~(\eta_s - B)

~=~

~\pi       …      

~[\mathrm{hence}, ~\sin^2(\eta_s - B) = 0 ]

~\eta_i

~=~

~3\biggl( \frac{\mu_e}{\mu_c} \biggr) \ell_i ( 1 + \ell_i^2 )^{-1}\, ,

~b_\eta

~=~

~\eta_s = \frac{\eta_i}{q} \, .

Let's try putting these terms all together in the context of the virial theorem.

a_c - \frac{3b_c}{n_c}\cdot x^{(n_c-3)/n_c}

~=


\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 \biggl\{ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2  
\biggl[ - \frac{8}{3} \ell_i^3  + ( 1 + \ell_i^2 )^{3}\cdot \mathfrak{L}_1  \biggr]  
- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq}  \biggl[  (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1 \biggr]\biggr\}

 

~=


\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 \biggl\{ 
- \frac{2^3 \nu^2 \ell_i}{3 q^4 }   +\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2    
- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1
\biggr\}

 

~=


- \frac{ \nu^2 }{2 q \ell_i^2}   +\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2    
- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1 \, ,

a_e - \frac{3b_e}{n_e}\cdot x^{(n_e-3)/n_e}

~=


\frac{3A^2 }{8b_\eta^3}  \biggl\{ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
\biggl[\mathfrak{L}_2- \frac{4}{3}\cdot b_\eta y \sin^2(b_\eta y - B)  \biggr]_q^1 
- 8\pi \cdot x^4 \biggl[ \frac{P_{ic} }{P_\mathrm{norm}} \biggr]_\mathrm{eq} \biggl[ \mathfrak{L}_2\biggr]_q^1 \biggr\}

 

~=


\frac{3A^2 }{8b_\eta^3}  \biggl\{ 
- \frac{2^2 \nu^2 b_\eta }{3 q^4 \ell_i^2}  \biggl[ y \sin^2(b_\eta y - B)  \biggr]_q^1 
+\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \biggl[ \mathfrak{L}_2\biggr]_q^1 
\biggr\}

 

~=


- \frac{\nu^2 }{ 2q^4 \ell_i^2}  \cdot \frac{A^2 }{b_\eta^2}   \biggl[ y \sin^2(b_\eta y - B)  \biggr]_q^1 
+\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \frac{3A^2 }{8b_\eta^3}  \biggl[ \mathfrak{L}_2\biggr]_q^1

 

~=


+ \frac{\nu^2 }{ 2q \ell_i^2}  +\biggl[ \biggl( \frac{\nu }{q^2 \ell_i} \biggr)^2
- 8\pi \cdot x^4 \biggl( \frac{P_{ic} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}\biggr] \frac{3A^2 }{8b_\eta^3}  \biggl[ \mathfrak{L}_2\biggr]_q^1 \, ,

where the last step has been made by appreciating that,

~\frac{A^2 }{b_\eta^2}   \biggl[ y \sin^2(b_\eta y - B)  \biggr]_q^1

~=


~\biggl(\frac{q}{\eta_i}\biggr)^2\biggl[ \frac{\eta_i}{\sin(\eta_i-B)} \biggr]^2
\biggl[ \sin^2(b_\eta - B) - q \sin^2(b_\eta q - B)  \biggr]

 

~=


~\biggl(\frac{q}{\eta_i}\biggr)^2\biggl[ \frac{\eta_i}{\sin(\eta_i-B)} \biggr]^2
\biggl[ \cancelto{0}{\sin^2(\eta_s - B)} - q \sin^2(\eta_i - B)  \biggr]

 

~=


~-q^3 \, ,

and we have adopted the shorthand notation,

~\mathfrak{L}_1

~\equiv

~ \ell_i (\ell_i^4 - 1 )(1+\ell_i^2)^{-3}   + \tan^{-1}(\ell_i) \, ,

~\mathfrak{L}_2

~\equiv

~2b_\eta y -\sin[2(b_\eta y - B)] \, .

Adding these two expressions together and setting the result equal to zero (as prescribed by the virial theorem) gives,

~0

~=

~
\biggl[ \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2    
- 8\pi \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \biggl( \frac{P_{i} }{P_\mathrm{norm}} \biggr)_\mathrm{eq} \biggr] \biggl\{
\frac{3}{2^4}\biggl( \frac{q}{\ell_i} \biggr)^3 (1 + \ell_i^2)^3 \cdot \mathfrak{L}_1 
+\frac{3A^2 }{8b_\eta^3}  \biggl[ \mathfrak{L}_2\biggr]_q^1 
\biggr\}

~\Rightarrow ~~~~  \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \biggl( \frac{P_{i} }{P_\mathrm{norm}} \biggr)_\mathrm{eq}

~=

~
\frac{1}{8\pi} \biggl(\frac{\nu}{q^2 \ell_i} \biggr)^2

~\Rightarrow ~~~~  \frac{R_\mathrm{eq}^4 P_{i} }{GM_\mathrm{tot}^2}

~=

~
\frac{3}{8\pi} \biggl(\frac{\nu}{q^2 \xi_i} \biggr)^2 \, .

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