Lane-Emden variables for a stellar polytrope ($r=\alpha \xi ,\rho ={\rho }_c{\theta }^n$)

For a spherical Newtonian stellar polytrope with $P=K{\rho }^{1+1/n}$, constant $K>0$ and polytropic index $n>0$, defineHere ${\rho }_c$ 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 equationThe 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 $M=4\pi {\rho }_c{\alpha }^3[-{\xi }_1^2{\theta }^{\prime }({\xi }_1)]$.

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.