= Solution
A thin spherical shell of thickness $dr$ has <mass> $4\pi r^2\rho\,dr$. Balancing its inward gravitational <force> with the <pressure> difference across its faces gives
$$
\boxed{\frac{dP}{dr}=-\frac{GM_r\rho}{r^2},\qquad
\frac{dM_r}{dr}=4\pi r^2\rho.}
$$
For a transparent estimate take constant <density> $\rho=3M_\odot/(4\pi R_\odot^3)$ and negligible surface <pressure>. Then $M_r=M_\odot r^3/R_\odot^3$, so integration gives
$$
\boxed{P_c=\frac{3GM_\odot^2}{8\pi R_\odot^4}\simeq1.3\times10^{14}\,\mathrm{Pa}.}
$$
Combining with the <ideal gas law> and $\mu\simeq0.6$ yields $T_c\simeq\mu m_HGM_\odot/(2k_BR_\odot)\simeq7\times10^6$ K. The less specific virial scaling gives $T_c\sim\mu m_HGM_\odot/(k_BR_\odot)\sim10^7$ K.
The real Sun is strongly centrally concentrated: a uniform-density model substantially underestimates <central pressure> and is not a detailed solar model. Its value is a justified scale estimate, while more realistic structure produces <central temperatures> of order $1.5\times10^7$ K and <central pressures> of order $10^{16}$ Pa. <Hydrostatic equilibrium> alone does not uniquely determine both profiles without an equation of state and a <density>/thermal model.
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