PX282 - D1 - stellar interiors

consider a spherically-symmetric star of a radius, $r$ , and a mass, $M_{r}$ , which has a spherical shell of a thickness, $d r$ , and a mass, $d m$ , with a density, $ρ (r)$
$L_{r}$ is the luminosity coming from inside the shell
from hydrostatic equilibrium, the stellar structure equations:

\begin{matrix} (1) & hydrostatic equilibrium: & \frac{d P}{d r} = \frac{- G M_{r} ρ}{r^{2}} \\ mass conservation: & d m = 4 π r^{2} d r ρ \\ (2) & \frac{d M_{r}}{d r} = 4 π r^{2} ρ \end{matrix}

with equation $(1)$ , the pressure and the temperature at the core can be estimated
assuming $\frac{d P}{d r} = c o n s t a n t = - \frac{P_{c o r e}}{R_{*}} :$

P_{c o r e} = - R_{*} \frac{d P}{d r} = R_{*} \frac{G M_{r} ρ}{R_{*}^{2}}

average $ρ = \frac{M}{V} = \frac{3 M_{*}}{4 π R_{*}^{3}} :$

P_{c o r e} = \frac{3 M_{*} G M_{r}}{4 π R_{*}^{4}}

since $M_{r} \approx M_{*} :$

P_{c o r e} \approx \frac{3 G}{4 π} \frac{M_{*}^{2}}{R_{*}^{4}}

this gives: $P_{⊙, c o r e} \approx 2.7 \times 10^{14} N m^{- 2}$
the real values is $\sim 10^{16}$ due to increased $ρ$ near the core
assuming the core is an ideal gas, $P = n k T : T_{c o r e} \approx 1.3 \times 10^{7} K$