PX282 - E1a - star formation

image magnus vilhelm

stellar evolution begins with star formation, which happens from the gravitational collapse of clouds of material
from the virial theorem:

2 U + Ω = 0

if $2 U > | Ω |$ , the cloud expands, and if $2 U < | Ω |$ , the cloud collapses

Ω_{t o t} = - \int_{0}^{R} \frac{G M_{r} (r)}{r} ρ 4 π r^{2} d r

for a constant $ρ$ ,

M_{r} = M {(\frac{r}{R})}^{3}

where, $M_{r}$ is the mass within the radius, $r$ , and $M$ is the total mass if the cloud with radius, $R$ , and density:
$$\rho = \frac{M}{\frac{4}{3}\pi R^{3}}$$

Ω_{t o t} = - \frac{4 π G M ρ}{R^{3}} \int_{0}^{R} r^{4} d r

Ω = - \frac{3}{5} \frac{G M^{2}}{R}

since $U = \frac{3}{2} k_{B} T$ per particle:

U_{t o t} = \frac{3}{2} N k_{B} T

- where, $N = \frac{M}{μ m_{H}}$ is the number of particles

for collapse:

3 N k_{B} T < \frac{3}{5} \frac{G M^{2}}{R} ⟹ M > \frac{5 R k_{B} T}{μ m_{H} G}

removing $R :$

M_{J} > {(\frac{5 k_{B} T}{μ m_{H} G})}^{\frac{3}{2}} {(\frac{3}{4 π ρ})}^{\frac{1}{2}}

- where, $M_{J}$ is called the jeans mass

a typical molecular cloud has $T = 20 K$ , $R \approx 2 p c$ , $μ = 2$ , and therefore, $M >\sim 100 M_{⊙}$