SSC/Structure/BiPolytropes/Analytic15

BiPolytrope with n_c = 1 and n_e = 5

Murphy (1983) Analytic (nc, ne) = (1, 5)

Comment by J. E. Tohline on 12 April 2015: I became aware of the published discussions of this system by Murphy (1983) and Murphy & Fiedler (1985b) in March of 2015 after searching the internet for previous analyses of radial oscillations in polytropes and, then, reading through Horedt's (2004) §2.8.1 discussion of composite polytropes.

Here we construct a system of bipolytropic configurations in which the core has an

$n_c=1$

polytropic index and the envelope has an

$n_e=5$

polytropic index. As in the case of our separately discussed, "mirror image" bipolytropic configurations having

$(n_c,n_e)=(5,1)$

, this system is particularly interesting because the entire structure can be described by closed-form, analytic expressions. Bipolytropes of this type were first constructed by J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175), and attributes of their physical structure were further discussed by J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219); additional, closely related references are given below. In the discussion that follows, we will be heavily referencing Murphy's (1983) work.

 

Part I:   Steps 2 thru 7

Part II:  Analytic Solution of Interface Relation

III:  Modeling

Steps 2 & 3

Based on the discussion presented elsewhere of the structure of an isolated $n=1$ polytrope, the core of this bipolytrope will have the following properties:

$\theta (\xi )=\frac{\sin \xi }{\xi }\Rightarrow {\theta }_i=\frac{\sin {\xi }_i}{{\xi }_i};$

$\frac{d\theta }{d\xi }=[\frac{\cos \xi }{\xi }-\frac{\sin \xi }{{\xi }^2}]\Rightarrow (\frac{d\theta }{d\xi }{)}_i=[\frac{\cos {\xi }_i}{{\xi }_i}-\frac{\sin {\xi }_i}{{\xi }_i^2}].$

The first zero of the function $\theta (\xi )$ and, hence, the surface of the corresponding isolated $n=1$ polytrope is located at ${\xi }_s=\pi$. Hence, the interface between the core and the envelope can be positioned anywhere within the range, $0<{\xi }_i<\pi$.

Step 4: Throughout the core (0 ≤ ξ ≤ ξ_i)

Specify: $K_c$ and ${\rho }_0\Rightarrow$
$\rho$	$=$	${\rho }_0{\theta }^{n_c}$	$=$	${\rho }_0\left(\frac{\sin \xi }{\xi }\right)$
$P$	$=$	$K_c{\rho }_0^{1+1/n_c}{\theta }^{n_c+1}$	$=$	$K_c{\rho }_0^2(\frac{\sin \xi }{\xi }{)}^2$
$r$	$=$	$[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$	$=$	$[\frac{K_c}{2\pi G}{]}^{1/2}\xi$
$M_r$	$=$	$4\pi \left[\frac{(n_c+1)K_c}{4\pi G}{]}^{3/2}{\rho }_0^{(3-n_c)/(2n_c)}\right(-{\xi }^2\frac{d\theta }{d\xi })$	$=$	$4\pi \left[\frac{K_c}{2\pi G}{]}^{3/2}{\rho }_0\right[\sin \xi -\xi \cos \xi ]$

Step 5: Interface Conditions

			Setting $n_c=1$ and $n_e=5\Rightarrow$
$\frac{{\rho }_e}{{\rho }_0}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{n_c}{\varphi }_i^{-n_e}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i{\varphi }_i^{-5}$
$\left(\frac{K_e}{K_c}\right)$	$=$	${\rho }_0^{1/n_c-1/n_e}(\frac{{\mu }_e}{{\mu }_c}{)}^{-(1+1/n_e)}{\theta }_i^{1-n_c/n_e}$	$=$	$\left[{\rho }_0^4\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-6}{\theta }_i^4{]}^{1/5}$
$\frac{{\eta }_i}{{\xi }_i}$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^{(n_c-1)/2}{\varphi }_i^{(1-n_e)/2}$	$=$	$\left(\frac{1}{3}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\varphi }_i^{-2}$
$(\frac{d\varphi }{d\eta }{)}_i$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}{\theta }_i^{-(n_c+1)/2}{\varphi }_i^{(n_e+1)/2}\right(\frac{d\theta }{d\xi }{)}_i$	$=$	$\left(\frac{1}{3}{)}^{1/2}{\theta }_i^{-1}\right(\frac{d\theta }{d\xi }{)}_i{\varphi }_i^3$

Alternative: In our introductory description of how to build a bipolytropic structure, we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [C67] found it useful to employ, instead, the ratio of the $3^{\mathrm{r}\mathrm{d}}$ to $4^{\mathrm{t}\mathrm{h}}$ expressions, which in the present case produces,

$\frac{{\eta }_i{\varphi }_i^5}{(d\varphi /d\eta {)}_i}=\frac{{\xi }_i{\theta }_i}{(d\theta /d\xi {)}_i}\left(\frac{{\mu }_e}{{\mu }_c}\right),$

and the product of the $3^{\mathrm{r}\mathrm{d}}$ and $4^{\mathrm{t}\mathrm{h}}$ expressions, which in the present case generates,

$\frac{3{\eta }_i(d\varphi /d\eta {)}_i}{{\varphi }_i}=\frac{{\xi }_i(d\theta /d\xi {)}_i}{{\theta }_i}\left(\frac{{\mu }_e}{{\mu }_c}\right).$

In what follows we will sometimes refer to the first of these two expressions as Chandrasekhar's "U-constraint" and we will sometimes refer to the second as Chandrasekhar's "V-constraint." As is explained in an accompanying discussion, Murphy (1983) followed Chandrasekhar's lead and extracted fitting conditions from this last pair of expressions. In seeking the most compact analytic solution, we have found it advantageous to invoke our standard $3^{\mathrm{r}\mathrm{d}}$ fitting expression in tandem with the Chandrasekhar's V-constraint.

Step 6: Envelope Solution

Comment by J. E. Tohline on 20 April 2015: There is a type-setting error in this function expression as published in the upper left-hand column of the second page of the article by Murphy (1983); the sine function in the denominator should be sine-squared, as presented here.

Following the work of Murphy (1983) and of Murphy & Fiedler (1985a), we will adopt for the envelope's structure the F-Type solution of the

$n=5$

Lane-Emden function discovered by S. Srivastava (1968, ApJ, 136, 680) and described in an accompanying discussion, namely,

$\varphi$	$=$	$\frac{B_0^{-1}\sin [\ln (A_0\eta {)}^{1/2})]}{{\eta }^{1/2}\{3-2{\sin }^2[\ln (A_0\eta {)}^{1/2}]{\}}^{1/2}}$
	$=$	$\frac{B_0^{-1}\sin \mathrm{\Delta }}{{\eta }^{1/2}(3-2{\sin }^2\mathrm{\Delta }{)}^{1/2}},$

Note that our homology factor and scaling coefficient serve virtually the same roles as the homology factor, A, and scaling coefficient, B, used by Murphy (1983), but they are not mathematically identical so we have added a subscript "0" to highlight the distinctions.

where

$A_0$

is a "homology factor" and

$B_0$

is an overall scaling coefficient — the values of both will be determined presently from the interface conditions — and we have introduced the notation,

$\mathrm{\Delta }\equiv \ln (A_0\eta {)}^{1/2}=\frac{1}{2}(\ln A_0+\ln \eta ).$

The first derivative of Srivastava's function is,

$\frac{d\varphi }{d\eta }$

$=$

$\frac{B_0^{-1}[3\cos \mathrm{\Delta }-3\sin \mathrm{\Delta }+2{\sin }^{}\mathrm{\Delta }]}{2{\eta }^{3/2}(3-2{\sin }^2\mathrm{\Delta }{)}^{3/2}}.$

As has been explained in the context of our more general discussion of Srivastava's function, if we ignore, for the moment, the additional "$m\pi$" phase shift that can be attached to a determination of the angle, $\mathrm{\Delta }$, the physically viable interval for the dimensionless radial coordinate is, $e^{2\pi }\ge A_0\eta \ge {\eta }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\equiv e^{2{\tan }^{-1}(1+2^{1/3})}.$

For this bipolytropic configuration, it is worth emphasizing how the dimensionless radial coordinate of the $n_e=5$ envelope, $\eta$, is related to the dimensionless radial coordinate of the $n_c=1$ core, $\xi$. Referring to the general setup procedure for constructing any bipolytropic configuration that has been presented in tabular form in a separate discussion, it is clear that in order for the radial coordinate, $r$, to carry a consistent meaning throughout the model, we must have,

$r=[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$	$=$	$[\frac{(n_e+1)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(1-n_e)/(2n_e)}\eta$
$\Rightarrow \left(\frac{K_c}{3K_e}\right){\rho }_e^{4/5}$	$=$	$(\frac{\eta }{\xi }{)}^2.$

Referring back to the already established interface conditions, above, to relate ${\rho }_e$ to ${\rho }_0$, and to re-express the ratio, $K_e/K_c$, we therefore have,

$(\frac{\eta }{\xi }{)}^2$	$=$	$\frac{1}{3}\left[{\rho }_0^4\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-6}{\theta }_i^4{]}^{-1/5}\left[{\rho }_0\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i{\varphi }_i^{-5}{]}^{4/5}$
$\Rightarrow \frac{\eta }{\xi }$	$=$	$\left(\frac{1}{3}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\varphi }_i^{-2}.$

While this result is not a surprise because the right-hand-side is the same expression that was presented, above, as the interface condition for the ratio, ${\eta }_i/{\xi }_i$, it is nevertheless useful because it shows that the same relation works throughout the system — not just at the interface — and it clearly defines how we can swap back and forth between the two dimensionless radial coordinates when examining the structure and characteristics of this composite bipolytropic structure.

First Constraint

Calling upon Chandrasekhar's V-constraint, as just defined above — see also our accompanying discussion for elaboration on Murphy's (1983) "V_5F" and "V_1E" function notations — one fitting condition at the interface is,

$-\frac{2{\xi }_i}{3{\theta }_i}\left(\frac{d\theta }{d\xi }{)}_i\right(\frac{{\mu }_e}{{\mu }_c})$	$=$	$2[\frac{\eta (-d\varphi /d\eta )}{\varphi }{]}_i$
	$=$	$\frac{[3\sin {\mathrm{\Delta }}_i-2{\sin }^{}{\mathrm{\Delta }}_i-3\cos {\mathrm{\Delta }}_i]}{\sin {\mathrm{\Delta }}_i(3-2{\sin }^{}{\mathrm{\Delta }}_i)}$
	$=$	$\frac{3-2{\sin }^2{\mathrm{\Delta }}_i-3\cot {\mathrm{\Delta }}_i}{(3-2{\sin }^2{\mathrm{\Delta }}_i)}.$

The left-hand side of this expression is inherently positive over the physically viable radial coordinate range, $0\ge {\xi }_i\ge \pi$ and its value is known once the radial coordinate of the edge of the core has been specified. So, defining the interface parameter,

${\kappa }_i\equiv -\frac{2{\theta }_i^{\text{'}}{\xi }_i}{3{\theta }_i}\left(\frac{{\mu }_e}{{\mu }_c}\right),$

we will recast the first constraint into, what will henceforth be referred to as, the

Key Nonlinear Interface Relation ${\kappa }_i$ $=$ $\frac{3-2{\sin }^2{\mathrm{\Delta }}_i-3\cot {\mathrm{\Delta }}_i}{(3-2{\sin }^2{\mathrm{\Delta }}_i)}.$

In a separate subsection of this chapter, below, we present a closed-form analytic solution, ${\mathrm{\Delta }}_i({\kappa }_i)$, to this nonlinear equation.

Second Constraint

Obtained from Third Interface Condition

Our 3^rd interface condition, as detailed above, states that,

$\frac{{\eta }_i}{{\xi }_i}$

$=$

$3^{-1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\varphi }_i^{-2}.$

If we now choose to normalize the interface amplitude such that, ${\varphi }_i=1$, then this condition establishes two relations: First, from the 3^rd interface condition alone,

${\eta }_i$

$=$

$3^{-1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\xi }_i;$

and, second, from the definition of Srivastava's function, $\varphi$, we deduce that the overall scaling parameter is,

$B_0^2$	$=$	$\frac{{\sin }^2{\mathrm{\Delta }}_i}{{\eta }_i(3-2{\sin }^2{\mathrm{\Delta }}_i)}$
	$=$	$\frac{\sqrt{3}}{{\xi }_i}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{-1}.$

Notice that, after the solution, ${\mathrm{\Delta }}_i({\kappa }_i)$, of the key nonlinear interface relation has been determined, the first of these two relations also permits us to write,

$A_0{\eta }_i$	$=$	$e^{2{\mathrm{\Delta }}_i}$
$\Rightarrow A_0$	$=$	$3^{1/2}(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\xi }_i^{-1}{e}^{2{\mathrm{\Delta }}_i}.$

Throughout the envelope, therefore, the angle,

$\mathrm{\Delta }$

$=$

$\ln (A_0\eta {)}^{1/2}=\frac{1}{2}\ln [\xi \cdot {\xi }_i^{-1}{e}^{2{\mathrm{\Delta }}_i}]={\mathrm{\Delta }}_i+\ln (\frac{\xi }{{\xi }_i}{)}^{1/2}.$

Obtained from Chandrasekhar's U-constraint

We shall now demonstrate that the same expression for the scaling coefficient, $B_0$, can alternatively be obtained from Chandrasekhar's U-constraint, without assuming that ${\varphi }_i=1$, after taking into account the result that already has been obtained from the V-constraint. As described above, the U-constraint is an alternative interface condition that may be written as,

$\frac{{\xi }_i{\theta }_i}{(-d\theta /d\xi {)}_i}\left(\frac{{\mu }_e}{{\mu }_c}\right)$

$=$

$\frac{{\eta }_i{\varphi }_i^5}{(-d\varphi /d\eta {)}_i},$

which, in the particular case being examined here, becomes — again, see our accompanying discussion for elaboration on Murphy's (1983) "U_5F" and "U_1E" function notations —

$\frac{2{\xi }_i^2}{3{\kappa }_i}(\frac{{\mu }_e}{{\mu }_c}{)}^2$	$=$	$(U_{\mathrm{5}\mathrm{F}}{)}_i$
	$=$	$\frac{2B_0^{-4}{\sin }^4{\mathrm{\Delta }}_i}{(3-2{\sin }^2{\mathrm{\Delta }}_i)(3-2{\sin }^2{\mathrm{\Delta }}_i-3\cot {\mathrm{\Delta }}_i)}.$

Now, from our discussion, above, of the first constraint, we know that,

$(3-2{\sin }^2{\mathrm{\Delta }}_i-3\cot {\mathrm{\Delta }}_i)$

$=$

$(3-2{\sin }^2{\mathrm{\Delta }}_i){\kappa }_i.$

Hence, Chandrasekhar's U-constraint becomes,

$\frac{2{\xi }_i^2}{3{\kappa }_i}(\frac{{\mu }_e}{{\mu }_c}{)}^2$	$=$	$\frac{2B_0^{-4}{\sin }^4{\mathrm{\Delta }}_i}{(3-2{\sin }^2{\mathrm{\Delta }}_i{)}^2{\kappa }_i}$
$\Rightarrow B_0^4$	$=$	$\frac{3{\sin }^4{\mathrm{\Delta }}_i}{{\xi }_i^2(3-2{\sin }^2{\mathrm{\Delta }}_i{)}^2}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}$
$\Rightarrow B_0^2$	$=$	$\frac{\sqrt{3}}{{\xi }_i}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{-1},$

which, as predicted, is identical to what we learned from the third interface condition, alone.

Comment on Murphy's Scalings

Murphy's (1983) derivations also include an homology factor, $A$, and an overall scaling factor, $B$, but they are calculated differently from our $A_0$ and $B_0$. In the righthand column of the third page of his paper, Murphy states that,

$A=\frac{{\xi }_J}{{\zeta }_J},$

which, when translated into our notation $({\zeta }_J\to {\xi }_i$ and ${\xi }_J\to A_0{\eta }_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}})$ gives,

$A=\frac{A_0{\eta }_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}}{{\xi }_i}.$

Now, in our derivation, ${\eta }_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$ is synonymous with the location of the envelope interface, ${\eta }_i$, as expressed in terms of the dimensionless radial coordinate associated with Srivastava's Lane-Emden function, so we can equally well state that,

$A=\frac{A_0{\eta }_i}{{\xi }_i}.$

Recalling that ${\varphi }_i=1$, we know from the interface conditions detailed above that,

$\frac{{\eta }_i}{{\xi }_i}=\frac{1}{3^{1/2}}\left(\frac{{\mu }_e}{{\mu }_c}\right).$

Hence Murphy's homology factor, $A$, is related to our homology factor, $A_0$ via the expression,

$A=\frac{A_0}{3^{1/2}}\left(\frac{{\mu }_e}{{\mu }_c}\right).$

It is usually the value of this quantity, rather than simply our derived value of $A_0$, that is tabulated below — both here and here — as we make quantitative comparisons between the characteristics of our derived models and those published by Murphy (1983) and by Murphy & Fiedler (1985a).

In the lefthand column of the fourth page of his paper, Murphy (1983) defines the coefficient $B$ in such a way that the value of the envelope function, ${\varphi }_{5F}$, equals the value of the core function, ${\theta }_{1E}$, at the interface. Specifically, he sets,

$B$	$=$	$\left[\frac{{\zeta }_J}{\sin {\zeta }_J}\right]\left[\frac{A^{1/2}\sin (\ln \sqrt{A{\zeta }_J})}{(A{\zeta }_J{)}^{1/2}\{2+\cos [\ln (A{\zeta }_J)]{\}}^{1/2}}\right]$
	$=$	$\left[\frac{{\zeta }_J}{\sin {\zeta }_J}\right]\left[\frac{\sin (\ln \sqrt{A{\zeta }_J})}{{\zeta }_J^{1/2}\{3-2{\sin }^2(\ln \sqrt{A{\zeta }_J}){\}}^{1/2}}\right].$

Switching to our terminology, that is, setting,

$\ln \sqrt{A{\zeta }_J}\to {\mathrm{\Delta }}_i$     and, as before,      ${\zeta }_J\to {\xi }_i,$

gives,

$B$	$=$	$\left[\frac{{\xi }_i}{\sin {\xi }_i}\right]\left[\frac{\sin {\mathrm{\Delta }}_i}{{\xi }_i^{1/2}(3-2{\sin }^2{\mathrm{\Delta }}_i{)}^{1/2}}\right]$
	$=$	${\theta }_i^{-1}\left[{\xi }_i^{-1}\right(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{-1}{]}^{1/2}.$

Hence, in terms of the definition of our scaling coefficient, $B_0$, derived above, we have,

$B$

$=$

$\frac{B_0}{3^{1/4}}(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i^{-1}.$

As we make quantitative comparisons between the characteristics of our derived models and those published by Murphy (1983) and by Murphy & Fiedler (1985a), below, we usually will tabulate the value of this quantity, rather than simply our derived value of $B_0$.

Step 7: Identifying the Surface

Because Shrivastava's function — and, along with it, the envelope's density — drops to zero when,

$\mathrm{\Delta }={\mathrm{\Delta }}_s\equiv \pi ,$

we know that the radius, ${\xi }_s$, of the bipolytropic configuration is given by the expression,

$\pi$	$=$	${\mathrm{\Delta }}_i+\ln (\frac{{\xi }_s}{{\xi }_i}{)}^{1/2}$
$\Rightarrow \ln (\frac{{\xi }_s}{{\xi }_i}{)}^{1/2}$	$=$	$\pi -{\mathrm{\Delta }}_i$
$\Rightarrow {\xi }_s$	$=$	${\xi }_i{e}^{2(\pi -{\mathrm{\Delta }}_i)}.$

In terms of the natural radial coordinate of the envelope, this is,

${\eta }_s(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}$

$=$

$3^{-1/2}{\xi }_i{e}^{2(\pi -{\mathrm{\Delta }}_i)}.$

Key References

• S. Srivastava (1968, ApJ, 136, 680) A New Solution of the Lane-Emden Equation of Index n = 5
• H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115): Remark on the Polytrope of Index 5 — the result of this work by Buchdahl has been highlighted inside our discussion of bipolytropes with $(n_c,n_e)=(5,1)$.
• J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37): A Finite Radius Solution for the Polytrope Index 5
• J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41): On the F-Type and M-Type Solutions of the Lane-Emden Equation
• J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205): Physical Characteristics of a Polytrope Index 5 with Finite Radius
• J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376): A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation
• J. O. Murphy (1983, Australian Journal of Physics, 36, 453): Structure of a Sequence of Two-Zone Polytropic Stellar Models with Indices 0 and 1
• J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175): Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices n = 1 and n = 5
• J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219): Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models
• J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222): Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models

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