Linear-density stellar model ($\rho (r)={\rho }_c(1-r/R)$)

An imposed spherical profile $\rho ={\rho }_c(1-r/R)$ gives $M=\pi {\rho }_c{R}^3/3$ in hydrostatic equilibrium. For an ideal gas with zero surface pressure, $P=P_c(1-x{)}^2(1+2x-9x^2/5)$ and $T=T_c(1-x)(1+2x-9x^2/5)$, where $x=r/R$, $P_c=5G{M}^2/(4\pi R^4)$ and $T_c=5GM/(12\mathcal{R}R)$. The density has a central cusp and temperature initially rises outward, so positive outward radiative diffusion with regular positive nuclear heating cannot satisfy this imposed profile near the centre.