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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 requiresA 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,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.

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 requiresA 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,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.

If radiation dominates the central pressure, equate $P_c\sim G{M}_{\ast }^2/R_{\ast }^4$ to $a{T}_c^4/3$. This givesAt roughly fixed burning temperature, the corresponding mass-radius scaling is $R_{\ast }\propto M_{\ast }^{1/2}$, rather than the gas-pressure estimate $R_{\ast }\propto M_{\ast }$. Radiation pressure alone does not determine a unique radius at a specified mass: a thermal or nuclear-burning condition is still needed to specify $T_c$. The relation is a homology estimate, not a universal exact law for all massive stars.

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 givesFor 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 givesCombining with the ideal gas law and $\mu \simeq 0.6$ yields $T_c\simeq \mu m_{H}G{M}_{\odot }/(2k_B{R}_{\odot })\simeq 7\times 10^6$ K. The less specific virial scaling gives $T_c\sim \mu m_{H}G{M}_{\odot }/(k_B{R}_{\odot })\sim 10^7$ 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 $1.5\times 10^7$ 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.