Changes between Version 28 and Version 29 of GravoTurbulence

Timestamp:

12/06/11 16:34:33 (13 years ago)

Author:

Jonathan

Comment:

—

GravoTurbulence

@@ -352 +352 @@
 || [[latex($\chi$)]] || nominal density contrast ||
 || [[latex($n$)]] || number of child clumps ||
-|| [[latex($\xi_c$)]] || ratio of child nominal density to child mean density ||
+|| [[latex($\xi_c$)]] || ratio of child mean density to child nominal density ||
 
 These 10 quantities are related by the following 6 equations.
@@ -366 +366 @@
 [[latex($r_p=c_s\sqrt{\frac{\pi}{G\overline{\rho_p}}}$)]]  - jeans criterion
 
-[[latex($\frac{\rho_c}{\overline{\rho_c}}=\xi_c$)]]
+[[latex($\frac{\overline{\rho_c}}{\rho_c}=\xi_c$)]]
 
 
 If we are given [[latex($r_p$)]], [[latex($n$)]], [[latex($\chi$)]] and [[latex($\xi_c$)]] we can solve for the other quantities...
 
-[[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($B=\frac{\xi_c}{\chi} \mbox{ and } A=1-B$)]]
+[[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($B=\frac{1}{\chi\xi_c} \mbox{ and } A=1-B$)]]
 
 [[latex($f=n\left(\frac{r_c}{r_p}\right)^3=n \left(A f + B \right) ^{3/2}$)]] 
@@ -389 +389 @@
 [[latex($-f^2+1=0$)]] and we recover correctly that [[latex($f=1$)]]
 
-What if we now have [[latex($n=1$)]] clump with a density contrast [[latex($\chi=2$)]] ?  [[latex($B=2$)]] and [[latex($A=-1$)]] and  the cubic becomes:
-
-[[latex($-f^3+5f^2-12f+8=0$)]]
-
-Surprisingly 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 over density of 2*.35+.65 = 1.35 that will require the parent radius be shrunk.  This however will also increase the mean density at a rate faster then making the clump smaller can compensate for.  At higher density contrasts and lower filling fractions, this obstacle can be overcome and a solution exists.
-
-
-If we have multiple levels we can start on the highest level 'L' where [[latex($\xi_L=1$)]] and solve for the filling fraction for level 'L-1'.  This can then be used to calculate [[latex($\overline{\rho_{L-1}}$)]] and [[latex($\xi_{L-1}$)]] which then allows us to solve for the filling fraction for the next higher level and so on.  Finally once we have [[latex($\xi_0$)]] and [[latex($r_0$)]] we can calculate [[latex($\rho_0$)]] which can then be used to calculate [[latex($\rho_1,\rho_2,...,\rho_L$)]]
-
- - Highest level of clump has no children so mean density is nominal density.
+What if we now have [[latex($n=1$)]] clump with a density contrast [[latex($\chi=2$)]] ?  [[latex($B=1/2$)]] and [[latex($A=1/2$)]] and  the cubic becomes:
+
+[[latex($f^3-5f^2+3f+1=0$)]]
+
+Surprisingly this cubic has no real positive roots between 0 and 1!  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 over density of 2*.35+.65 = 1.35 that will require the parent radius be shrunk.  This however will also increase the mean density at a rate faster then making the clump smaller can compensate for.  At higher density contrasts and lower filling fractions, this obstacle can be overcome and a solution exists.
+
+
+If we have multiple levels we can start on the highest level 'L' where [[latex($\xi_L=1$)]] and solve for the filling fraction for level 'L-1'.  This can then be used to calculate [[latex($\xi_{L-1} = f_{L-1}\chi\xi_L+(1-f_{L-1})$)]] which then allows us to solve for the filling fraction for the next higher level and so on.  Finally once we have [[latex($\xi_0$)]] and [[latex($r_0$)]] we can calculate [[latex($r_i=r_{i-1} f_{i-1}^{1/3}$)]] and [[latex($\rho_i=\frac{\mbox{JeansDensity}(r_i)}{\xi_i}$)]] 
+
+
+in 2D there are only a few modifications
+
+[[latex($f=n\left(\frac{r_c}{r_p}\right)^2=n \left(A f + B \right) ^{2/2} = n(A f + B)$)]] 
+
+and we have [[latex($f=\frac{nB}{1-nA}=\frac{nB}{1-n(1-B)}=\frac{nB}{1-n+B}$)]]
+
+Note that [[latex($nB >= B >= 1-n+B$)]] for [[latex($n >= 1 \mbox{ and } B >> 0$)]] so [[latex($f >= 1$)]] which does not allow for any solutions of interest.
 
 == Velocity Perturbations ==