= Lane-Emden variables for a stellar polytrope
{c}
{title2=$r=\alpha\xi,\quad\rho=\rho_c\theta^n$}
For a spherical Newtonian <stellar polytrope> with $P=K\rho^{1+1/n}$, constant $K>0$ and <polytropic index> $n>0$, define
$$
\rho(r)=\rho_c\theta(\xi)^n,\qquad r=\alpha\xi,\qquad
\alpha^2=\frac{(n+1)K}{4\pi G}\rho_c^{(1-n)/n}.
$$
Here $\rho_c$ is the central <mass density>, $\alpha$ is a length scale, and $\xi,\theta$ are <dimensionless>. Substitution into the <stellar hydrostatic equation> and spherical <mass conservation> removes the dimensional constants and gives the <Lane-Emden equation>
$$
\frac1{\xi^2}\frac d{d\xi}\left(\xi^2\theta'\right)=-\theta^n,
\qquad \theta(0)=1,\quad\theta'(0)=0.
$$
The central conditions normalize the <mass density> and impose spherical regularity. When the regular solution has a finite first zero $\xi_1$, the zero-pressure surface is $R=\alpha\xi_1$ and the <Lane-Emden mass formula> gives $M=4\pi\rho_c\alpha^3[-\xi_1^2\theta'(\xi_1)]$.
The first-zero construction is a condition on the solution, not an automatic finite boundary for every <polytropic index>. For example, a <polytrope of index one> has $\theta=\sin\xi/\xi$ and $\xi_1=\pi$, whereas a <polytrope of index five> extends to infinite <radius> while retaining finite total <mass>. External <pressure> can instead define a truncated boundary before a zero is reached.
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