From the Virial Theorem,

 

[math]K = -\frac{1}{2} U[/math]

 

[math]\frac{M k_B T}{\mu} \approx \frac{3}{10} \frac{G M^2}{R}[/math]

 

[math]\therefore R \; T = \frac{3}{10} \frac{G \mu}{k_B} M[/math]

Now, the Luminosity, radiated away as heat:

 

[math]L = 4 \pi \sigma R^2 T^4 = \frac{4 \pi \sigma}{R^2} \left( R \; T \right)^4[/math]

is balanced by the release, of GPE:

 

[math]L = -\frac{dU}{dt} = \frac{3}{5} \frac{G M^2}{R^2}\dot{R}[/math]

Er go,

 

[math]\frac{4 \pi \sigma}{R^2} \left( R \; T \right)^4 = \frac{3}{5} \frac{G M^2}{R^2}\dot{R}[/math]

 

[math]4 \pi \sigma \left( \frac{3}{10} \frac{G \mu}{k_B} M \right)^4 = \frac{3}{5} G M^2 \dot{R}[/math]

 

[math]\frac{2 \pi \sigma}{G} \left( \frac{3}{10} \right)^3 \left( \frac{G \mu}{k_B} \right)^4 M^2 = \dot{R}[/math]

Assuming primordial gas composition (X = 3/4, Y = 1/4), so that the average particle mass is ~0.6 m_H, w.h.t.:

 

[math]\dot{R} \approx 100 km/s \times \left( \frac{M}{M_{\odot}} \right)^2[/math]

 

[math]\approx \frac{1}{3} 10^{-3} c \times \left( \frac{M}{M_{\odot}} \right)^2[/math]

If so, then the "implosion speed" of collapse [math]\dot{R} \rightarrow c[/math] near [math]M \rightarrow 50 M_{\odot}[/math]. Are such speeds plausible ?

 

Such massive proto-stars collapse, on the MS, in ~10⁴yrs:

 

And, Molecular Cloud 'cores' are typically <1 lyr across ; however, cloud collapse occurs isothermally (Sterzik 2003, Sterzik 2003). Perhaps isothermal collapse accounts for the slower observed collapse speeds ?

I thought neutral gases in a vacuum do not coelesce?

I thought neutral gases in a vacuum do not coelesce?

 

Not chemically, but they would gravitationally.

Virial Theorem, I am getting flashbacks to my undergraduate lectures in astrophysics.