Widdekind Posted January 25, 2012 Share Posted January 25, 2012 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 mH, 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 ? Link to comment Share on other sites More sharing options...
homie12 Posted February 4, 2012 Share Posted February 4, 2012 I thought neutral gases in a vacuum do not coelesce? Link to comment Share on other sites More sharing options...
swansont Posted February 4, 2012 Share Posted February 4, 2012 I thought neutral gases in a vacuum do not coelesce? Not chemically, but they would gravitationally. Link to comment Share on other sites More sharing options...
ajb Posted February 4, 2012 Share Posted February 4, 2012 Virial Theorem, I am getting flashbacks to my undergraduate lectures in astrophysics. Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now