PX282 - D5 - energy sources and timescales

timescale = \frac{quantity}{rate of change}

gravitational timescale

$Ω$ is a dynamical time scale:

τ_{d y n} = \frac{R}{v_{e s c}} = \sqrt{\frac{R^{3}}{2 G M}}

where, $v_{e s c} = \sqrt{\frac{2 G M}{R}}$ is the escape velocity

for the sun: $τ_{d y n} \approx 1000 s$ , which is the time for the sun to collapse if all other forces are removed
it relevant for supernovae

thermal timescale

from the virial theorem:

U = - \frac{Ω}{2} \approx \frac{G M^{2}}{2 R}

τ_{K H} = \frac{U}{L} = \frac{G M^{2}}{2 R L}

where, $K H$ stands for kelvin-holmholtz

for the sun: $τ_{K H} = 1.5 \times 10^{7} y r$
it is known that stars are more than $10^{9}$ years old
hence, thermal energy cannot be the key energy source
$τ_{K H}$ is relevant for stars collapsing during formation

nuclear timescale

τ_{N} = \frac{E_{N}}{L}

the nuclear energy:

E_{N} = η X m_{c o r e} c^{2}

where,
$η =$ efficiency,
$X =$ H mass fraction,
$m_{c o r e} =$ mass available for fusion,

eg: $η \approx 0.007$ , $X \approx 0.7$ , $m_{c o r e} \approx 0.1 M :$

∴ τ_{N} \approx 7 \times 10^{9} years

this shows that nuclear energy drives the main-sequence lifetime