The density exponent in $ϵ={ϵ}_0{\rho }^a{T}^b$ reflects the number of reacting pairs per unit mass; for ordinary two-body hydrogen burning, $a\simeq 1$. The temperature exponent is the local logarithmic sensitivity of the thermally averaged nuclear cross-section. Near $2\times 10^7\mathrm{K}$,are standard approximations.

Spherical regularity makes central scalar profiles even in $r$. Since density and temperature decrease outward in a normal stellar core, they have expansionswith $\alpha ,\beta >0$. The high power $T^b$ then suppresses burning rapidly away from the centre, especially for the CNO cycle.

The luminosity equation isExpanding and integrating givesThe luminosity generated by a thin shell scales locally as $r^2\exp [-((a+1)\beta +b\alpha )r^2]$. Its maximum therefore haswhen the temperature term dominates. Comparable central structures then give

Near the centre, mass conservation gives $m(r)=4\pi {\rho }_c{r}^3/3+O(r^5)$. Substitution into hydrostatic equilibrium and integration yieldsFor the perfect-gas law $P=\rho \mathcal{R}T/\mu$,Taking the two-body value $a=1$ in the earlier expression for $\gamma$ and eliminating $\alpha$ gives

A stellar polytrope of index $n$ obeysCombining hydrostatic equilibrium, $dP/dr=-Gm\rho /r^2$, with mass conservation, $dm/dr=4\pi r^2\rho$, gives the Lane-Emden equationRegularity and normalization at the stellar centre requireIntegrating the Lane-Emden equation from the centre then gives the enclosed mass

For $n=5$, direct substitution into the equation findsso $\beta =1/3$ and $\gamma =-1/2$. This profile has no finite first zero and hence has infinite radius, but its total mass converges:Its mean density over the full, infinite configuration is consequently zero.

For a perfect gas, $T\propto P/\rho \propto {\rho }^{1/5}$, so $T/T_c=\theta$. The luminosity from CNO cycle burning with $ϵ={ϵ}_0\rho T^{11}$ is thereforewhereThus $A$ is of order unity. The model is physically poor because the $n=5$ polytrope has infinite radius and zero mean density, while strongly temperature-sensitive CNO burning changes the thermal gradient and commonly creates convection. Real cores also have evolving composition, non-polytropic opacity and energy transport, and boundaries supplied by the surrounding star.