PX282 - E3 - main sequence

image: nelson bridwell

H-burning stars in PP chains
overtime, He builds up in the core
the average mass per particle in the core goes from $μ_{H +} = \frac{1}{2}$ to $μ_{H e +} = \frac{4}{3}$
the pressure:

p = n k_{B} T = \frac{ρ}{μ m_{H}} k_{B} T

therefore, as $μ$ increases, the pressure decreases, and the core gets compressed
this leads to the temperature increasing to balance it out, hence the rate of nuclear reaction increases
the nuclear timescale:

τ_{N} = \frac{E_{N}}{L} \approx 7 \times 10^{9} y e a r s (sun)

scaling law for low mass stars

the luminosity from energy transfer:

L \propto \frac{r^{2}}{κ ρ} T^{3} \frac{d T}{d r} \propto \frac{r^{2} T^{6.5}}{ρ^{2}} \frac{d T}{d r}

- where, $κ \propto ρ T^{- 3.5}$ from kramer's laws

therefore:

\frac{d T}{d r} = \frac{T}{R}

where, $ρ \sim \frac{M}{R^{3}}$ and $r \sim R :$

\begin{matrix} (1) & L \propto \frac{R^{7} T^{7.5}}{M^{2}} \end{matrix}

taking the hydrostatic equilibrium equation:

\frac{d P}{d r} = - \frac{G M_{r}}{r^{2}} ρ

\frac{P}{R} \propto \frac{M^{2}}{R^{5}}, P \propto \frac{M^{2}}{R^{4}}

for an ideal gas:

P = \frac{ρ}{μ m_{H}} k_{B} T \propto \frac{M^{2}}{R^{4}} \propto \frac{M}{μ R^{3}} T

⟹ T \propto \frac{M μ}{R}

for a constant $T :$

R \propto M μ

back to equation $(1) :$

L \propto μ^{7} M^{5}

so, if $μ$ increases, the luminosity also increases in main sequence

τ_{N} = D \times \frac{m_{c o r e} c^{2}}{L} \propto \frac{M}{L} \propto M^{- 4}

higher mass stars have shorter main-sequence lifetime