Changes between Version 26 and Version 27 of GravoTurbulence


Ignore:
Timestamp:
11/30/11 17:34:46 (13 years ago)
Author:
Jonathan
Comment:

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  • GravoTurbulence

    v26 v27  
    362362[[latex($\overline{\rho_p}=f \overline{\rho_c} + (1-f)\rho_p$)]]   - volume weighted mean density
    363363
    364 [[latex($\frac{\rho_p}{\rho_c} = \chi$)]]  - definition
     364[[latex($\frac{\rho_c}{\rho_p} = \chi$)]]  - definition
    365365
    366366[[latex($r_p=c_s\sqrt{\frac{\pi}{G\overline{\rho_p}}}$)]]  - jeans criterion
    … …  
    371371If we are given [[latex($r_p$)]], [[latex($n$)]], [[latex($\chi$)]] and [[latex($\xi_c$)]] we can solve for the other quantities...
    372372
    373 [[latex($r_c=\sqrt{\frac{f\overline{\rho_c}+(1-f)\chi \rho_c}{\overline{\rho_c}}}r_p = \sqrt{f+(1-f)\chi \xi_c}r_p = \sqrt{A f+B}r_p$)]] where [[latex($B=\chi\xi_c \mbox{ and } A=1-B$)]]
     373[[latex($r_c=\sqrt{\frac{f\overline{\rho_c}+\frac{1-f}{\chi} \rho_c}{\overline{\rho_c}}}r_p = \sqrt{f+\frac{1-f}{\chi} \xi_c}r_p = \sqrt{A f+B}r_p$)]] where [[latex($B=\frac{\xi_c}{\chi} \mbox{ and } A=1-B$)]]
    374374
    375375[[latex($f=n\left(\frac{r_c}{r_p}\right)^3=n \left(A f + B \right) ^{3/2}$)]]
    … …  
    377377which gives a cubic for [[latex($f$)]]
    378378
    379 [[latex($f^2=n^2(Af+B)^3=n^2(Af^3+3A^2f^2B+3AfB^2+B^3)$)]]
     379[[latex($f^2=n^2(Af+B)^3=n^2(A^3f^3+3A^2f^2B+3AfB^2+B^3)$)]]
    380380
    381381or
    382382
    383 [[latex($n^2Af^3+(3n^2A^2B-1)f^2+3n^2AB^2f+n^2B^3$)]]
     383[[latex($n^2A^3f^3+(3n^2A^2B-1)f^2+3n^2AB^2f+n^2B^3$)]]
    384384
    385385Consider the trivial case of [[latex($n=1$)]] child clump with the same density [[latex($\chi=1$)]]
    … …  
    392392
    393393[[latex($-f^3+5f^2-12f+8=0$)]]
     394
     395Surprisingly this cubic has no real positive roots!  To see why consider the radius ratio of the isolated parent and child clump.  From the Jeans relation, the child clump radius should be .7071  and it would occupy 35% of the volume.  This would then give a mean parent density of 2*.35+.65 = 1.35 that will require the parent radius be shrunk.  This however will only increase the mean density so the solution is stuck between shrinking the parent cloud to smaller and smaller radii is too high.  To lower the mean parent density we can increase the radius of the parentTo increase the mean parent density we need to If we include this additional contribution to the parent clump, the parent clump radius would shrink and the child clump would occupy even more then 35% of the volume.  On the other hand since the ratio of
    394396
    395397