PX158 - I5b - temperature at the centre of stars

using the ideal gas equation:

p V = n R_{g} T

where, $n = N / N_{A}$ , $N =$ number of particles

using $R_{g} / N_{A} = k_{B} :$

\begin{aligned} p V & = N k_{B} T \\ p & = \frac{N}{V} k_{B} T \\ = \frac{ρ}{μ} k_{B} T \end{aligned}

where, $μ =$ mass per particle, $ρ = \frac{N μ}{V}$

from pressure at the centre of stars, $p = ρ \frac{G M}{R}$ :

k_{B} T = \frac{G M μ}{R}

$k_{B} T \sim$ thermal energy
$\frac{G M_{s} μ}{R_{s}} \sim$ gravitational energy

the mass at the centre of a star is mostly protons and electrons, ie: $μ \to p^{+} + e^{-}$ , and $m_{p} + m_{e} \approx m_{p} :$

μ \approx \frac{m_{p}}{2}

eg: what is the central temperature of the sun? $T_{c ⊙} \approx 12 \times 10^{6} K$
- computer model gives, $T_{c ⊙} = 15 \times 10^{6} K$