SSC/Structure/Polytropes

Polytropic Spheres

Lane (1870)

Part I:  Isolated Polytropes

Part II:  Known Analytic Solutions

Part III:  Numerical Solutions

Isolated Polytropes

Isolated Polytropes

Here we will supplement the simplified set of principal governing equations with a polytropic equation of state, as defined in our overview of supplemental relations for time-independent problems. Specifically, we will assume that $\rho$ is related to $H$ through the relation,

$\rho =[\frac{H}{(n+1)K_{\mathrm{n}}}{]}^n$

It will be useful to note as well that, for any polytropic gas, the three key state variables are always related to one another through the simple expression,

$(n+1)P=H\rho$ .

In his effort to model the Sun's interior, 📚 J. H. Lane (1870, The American Journal of Science and Arts, Vol. 50, pp. 57 - 74) was the first to couch an examination of stellar structure in the context of, what is now usually referred to as, "polytropic structures." He examined the structural properties of spherically symmetric models having, effectively, indexes of $n=\frac{3}{2}$ and $n=\frac{5}{2}$. In an accompanying chapter, we review this early groundbreaking work highlighting the quantitative care with which Lane carried out his analysis. A much more expansive study of polytropic (and isothermal) structures was subsequently published by 📚 Emden, R. 1907, Gaskugeln (Leipzig), who was aware of Lane's work — see, for example, the footnote on p. 462 of his book. It is largely Emden's notation — especially as employed by [C67] — that is adopted in current discussions of polytropic structures, including our discussion which follows.

Governing Relations

Lane-Emden Equation

Adopting solution technique #2, we need to solve the following second-order ODE relating the two unknown functions, $\rho$ and $H$:

$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dH}{dr}\right)=-4\pi G\rho$ .

It is customary to replace $H$ and $\rho$ in this equation by a dimensionless polytropic enthalpy, ${\mathrm{\Theta }}_H$, such that,

${\mathrm{\Theta }}_H\equiv \frac{H}{H_c}=(\frac{\rho }{{\rho }_c}{)}^{1/n},$

where the mathematical relationship between $H/H_c$ and $\rho /{\rho }_c$ comes from the adopted barotropic (polytropic) relation identified above. To accomplish this, we replace $H$ with $H_c{\mathrm{\Theta }}_H$ on the left-hand-side of the governing differential equation and we replace $\rho$ with ${\rho }_c{\mathrm{\Theta }}_H^n$ on the right-hand-side, then gather the constant coefficients together on the left. The resulting ODE is,

$\left[\frac{1}{4\pi G}\right(\frac{H_c}{{\rho }_c}\left)\right]\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d{\mathrm{\Theta }}_H}{dr}\right)=-{\mathrm{\Theta }}_H^n$ .

The term inside the square brackets on the left-hand-side has dimensions of length-squared, so it is also customary to define a dimensionless radius,

$\xi \equiv \frac{r}{a_{\mathrm{n}}},$

where,

$a_{\mathrm{n}}\equiv \left[\frac{1}{4\pi G}\right(\frac{H_c}{{\rho }_c}){]}^{1/2}=[\frac{(n+1)K_n}{4\pi G}\cdot {\rho }_c^{(1-n)/n}{]}^{1/2},$

in which case our governing ODE becomes what is referred to in the astronomical literature as the,

Lane-Emden Equation

$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d{\mathrm{\Theta }}_H}{d\xi }\right)=-{\mathrm{\Theta }}_H^n$

§IV.2 of [C67], p. 88, Eq. (11)

Our task is to solve this ODE to determine the behavior of the function ${\mathrm{\Theta }}_H(\xi )$ — and, from it in turn, determine the radial distribution of various dimensional physical variables — for various values of the polytropic index, $n$. In particular, from time to time we will find it useful to realize that the mass interior to $r$ is given by the expression,

$M_r$	$=$	$\underset{0}{\overset{r}{\int }}4\pi \rho r^{2}dr=4\pi {\rho }_c{a}_n^3\underset{0}{\overset{\xi }{\int }}{\mathrm{\Theta }}_H^n{\xi }^{2}d\xi$
	$=$	$-4\pi {\rho }_c{a}_n^3\underset{0}{\overset{\xi }{\int }}\left[\frac{1}{{\xi }^2}\frac{d}{d\xi }\right({\xi }^2\frac{d{\mathrm{\Theta }}_H}{d\xi }\left)\right]{\xi }^{2}d\xi$
	$=$	$-4\pi {\rho }_c{a}_n^3\left({\xi }^2\frac{d{\mathrm{\Theta }}_H}{d\xi }\right).$
§IV.5.b of [C67], p. 97, Eq. (67)

ASIDE: In an accompanying discussion of pressure-truncated polytropes, we adopt the following length normalization: $R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $\left[\right(\frac{G}{K_n}{)}^n{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^{n-1}{]}^{1/(n-3)}.$ Let's see how the traditional Lane-Emden length scale, $a_n$, relates. $a_n^2$ $=$ $\left[\frac{(n+1)K_n}{4\pi G}\right]\left[\frac{{\rho }_c}{\overline{\rho }}{]}^{(1-n)/n}\right[\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi (a_n{\xi }_1{)}^3}{]}^{(1-n)/n}$   $=$ $[\frac{(n+1)K_n}{3G}\cdot M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{(1-n)/n}]{\mathfrak{𝔣}}_M^{(n-1)/n}{\xi }_1^{3(n-1)/n}(\frac{3}{4\pi }{)}^{1/n}{a}_n^{3(n-1)/n}$ $\Rightarrow a_n^{(n-3)/n}$ $=$ $\left[\frac{3}{(n+1)}\right(\frac{G}{K_n}\left)M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{(n-1)/n}\right]\left[{\mathfrak{𝔣}}_M{\xi }_1^3{]}^{(1-n)/n}\right(\frac{4\pi }{3}{)}^{1/n}$ $\Rightarrow \frac{a_n}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$ $=$ $\left[\frac{3}{(n+1)}{]}^{n/(n-3)}\right[{\mathfrak{𝔣}}_M{\xi }_1^3{]}^{(1-n)/(n-3)}(\frac{4\pi }{3}{)}^{1/(n-3)}.$ where, we have made use of the relation drawn from our accompanying discussion of structural form factors — see, also, here — ${\mathfrak{𝔣}}_M=[-\frac{3{\theta }^{\text{'}}}{\xi }{]}_{{\xi }_1},$ denotes the equilibrium ratio of the mean-to-central density. We conclude, therefore, that, in terms of $R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$, the equilibrium radius of an isolated polytrope is, $[\frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{(n-3)}$ $=$ $[\frac{a_n{\xi }_1}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}^{(n-3)}=[\frac{3}{(n+1)}{]}^n\left[{\mathfrak{𝔣}}_M{\xi }_1^3{]}^{(1-n)}\right(\frac{4\pi }{3}){\xi }_1^{(n-3)}$   $=$ $\frac{4\pi }{(n+1{)}^n}[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}^{(1-n)}{\xi }_1^{(n-3)}$ $\Rightarrow \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$ $=$ $\left[\frac{4\pi }{(n+1{)}^n}{]}^{1/(n-3)}\right[(-{\theta }^{\text{'}}){\xi }^2{]}_{{\xi }_1}^{(1-n)/(n-3)}{\xi }_1.$ This matches the expression presented in an accompanying summary supporting a PowerPoint presentation.

Boundary Conditions

Given that it is a $2^{\mathrm{n}\mathrm{d}}$-order ODE, a solution of the Lane-Emden equation will require specification of two boundary conditions. Based on our definition of the variable ${\mathrm{\Theta }}_H$, one obvious boundary condition is to demand that ${\mathrm{\Theta }}_H=1$ at the center $(\xi =0)$ of the configuration. In astrophysically interesting structures, we also expect the first derivative of many physical variables to go smoothly to zero at the center of the configuration — see, for example, the radial behavior that was derived for $P$, $H$, and $\mathrm{\Phi }$ in a uniform-density sphere. Hence, we will seek solutions to the Lane-Emden equation where $d{\mathrm{\Theta }}_H/d\xi =0$ at $\xi =0$ as well.

Known Analytic Solutions

Go here.

Numerical Solutions

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Related Discussions

• Lane-Emden equation

• Polytrope

• A. T. Service (1977, ApJ 211, 908) — Concise Approximation Formulae for the Lane-Emden Functions

• Mathematical Tables: Emden functions being solutions of emden's equation together with certain associated functions. II, Volume 2

• Power-Law density distribution