Possible explanation for planets formations ?

[Widdekind]

Fig. 21.16, on page 893, of Carroll & Ostlie's Intro. to Mod. Astrophys. [1st ed.], shows that (1) for stars more massive than [math]2 M_{\odot}[/math] (A5), Specific Angular Momentum (L/M) increases as M^2/3; and, that (2) for stars less massive than said same value, the star's S.A.M. increases as M⁵ (with the deficit probably borne by planets, according to the caption). We seek an order-of-magnitude calculation, which can indicate some sort of suggestion, as to the origin of this [math]2 M_{\odot}[/math] threshold.

 

 

 

S.A.M. for massive stars

 

Please ponder an idealized Giant Molecular Cloud core, of constant density, from which a spherical fragment begins to collapse. The initial Angular Momentum of said spherical cloud at onset of collapse is:

 

[math]L = \frac{2}{5} M R^{2} \omega_{g} = \frac{2}{5} M \left(\frac{M}{\frac{4 \pi \rho}{3}}\right)^{2/3} \omega_{g}[/math]

 

And so:

 

[math]\frac{L}{M} = \frac{2}{5} \left(\frac{M}{\frac{4 \pi \rho}{3}}\right)^{2/3} \omega_{g} \propto M^{2/3}[/math]

 

Thus, it is completely consistent, w/ Carroll & Ostlie (ibid.), to consider the collapse of cloud cores as occurring in a constant density, and w/ a constant initial angular velocity (being [math]\omega_{g}[/math], the local galactic angular velocity, of rotation around the galactic center). More massive stars have higher S.A.M. merely b/c they "reach out farther" across the (quasi-)constant density cloud core, accumulating amounts of matter (dM) having higher & higher S.A.M. (dL ~ dM x R²).

 

Note that, at the threshold mass ([math]2 M_{\odot}[/math]), the S.A.M. is 10^17.3 cm² s^-1, from which we can calculate the (effective) cloud core density, as:

 

[math]\rho = \left( \frac{2 \omega_{g}}{5 \frac{L}{M} } \right)^{3/2} M \frac{3}{4 \pi} \approx 7.3 \times 10^{-17} g \; cm^{-3}[/math]

 

Reassuringly, this value favorably compares w/ Carroll & Ostlie's figures for dense cloud cores (~10^-16).

 

But, below the threshold mass value of [math]2 M_{\odot}[/math], more and more of that S.A.M. is missing from the central star (& seemingly deposited into a disk of protoplanets). Why would this be?

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Merged post follows:

Consecutive posts merged

This author's investigation of Extra-solar Planetary Disks:

 

http://www.scienceforums.net/forum/showthread.php?t=47222

[granpa]

Specific Angular Momentum? is that anything like angular velocity?

 

no. more like angular velocity times r^2

[Widdekind]

From their figure 21.16, Specific Angular Momentum is the ratio of Angular Momentum to Mass, L / M. Since (ignoring coefficients) L = I w, and I = M R², SAM has the units of R² w (as you said).