= Solution
A thin spherical shell has outward <pressure force> $-4\pi r^2(dP/dr)dr$ and inward <gravitational force> $GM(r)\,4\pi r^2\rho(r)dr/r^2$. <Hydrostatic equilibrium> requires
$$
\boxed{\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2},\qquad \frac{dM}{dr}=4\pi r^2\rho(r).}
$$
A uniform-<density> <solar model> gives $M(r)=M_\odot(r/R_\odot)^3$, $\bar\rho=3M_\odot/(4\pi R_\odot^3)$, and after <integration> from $R_\odot$ to zero,
$$
P_c=\frac{3GM_\odot^2}{8\pi R_\odot^4}=1.34\times10^{14}\,\mathrm{Pa},\qquad
T_c=\frac{P_c\mu m_p}{\bar\rho k_B}=6.9\times10^6\,\mathrm K\quad(\mu=0.6).
$$
Thus the appropriate orders of magnitude are $10^{14}$–$10^{16}\,\mathrm{Pa}$ and $10^7\,\mathrm K$. The uniform-<density> approximation underestimates the actual <central pressure> because the Sun is centrally concentrated. <Radiation pressure> is neglected in applying the <ideal gas law>.
Back to article page
