Changes between Version 25 and Version 26 of GravoTurbulence

Timestamp:

11/30/11 12:31:06 (13 years ago)

Author:

Jonathan

Comment:

—

GravoTurbulence

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+== Clumplets ==
+
+An alternative is to put create a heirarch  of clumps of uniform density each with a radius and mean density that puts them in approximate virial equilibrium...
+
+|| [[latex($r_p$)]] || radius of parent clump ||
+|| [[latex($\rho_p$)]] || nominal density of parent clump (not including contributions from children) ||
+|| [[latex($\overline{\rho_p}$)]] || mean density of parent clump ||
+|| [[latex($r_c$)]] || radius of child clump ||
+|| [[latex($\rho_c$)]] || nominal radius of child clump (not including contributions from its children) ||
+|| [[latex($\overline{\rho_c}$)]] || mean density of child clump ||
+|| [[latex($f$)]]  || volume filling fraction
+|| [[latex($\chi$)]] || nominal density contrast ||
+|| [[latex($n$)]] || number of child clumps ||
+|| [[latex($\xi_c$)]] || ratio of child nominal density to child mean density ||
+
+These 10 quantities are related by the following 6 equations.
+
+[[latex($n \left(\frac{r_c}{r_p} \right)^{3} = f$)]]  - from geometry
+
+[[latex($\frac{\overline{\rho_c}}{ \overline{\rho_p} }=\left( \frac{r_p}{r_c} \right)^2 $)]]  - from jeans length scaling
+
+[[latex($\overline{\rho_p}=f \overline{\rho_c} + (1-f)\rho_p$)]]   - volume weighted mean density
+
+[[latex($\frac{\rho_p}{\rho_c} = \chi$)]]  - definition
+
+[[latex($r_p=c_s\sqrt{\frac{\pi}{G\overline{\rho_p}}}$)]]  - jeans criterion
+
+[[latex($\frac{\rho_c}{\overline{\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}+(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$)]]
+
+[[latex($f=n\left(\frac{r_c}{r_p}\right)^3=n \left(A f + B \right) ^{3/2}$)]] 
+
+which gives a cubic for [[latex($f$)]]
+
+[[latex($f^2=n^2(Af+B)^3=n^2(Af^3+3A^2f^2B+3AfB^2+B^3)$)]] 
+
+or
+
+[[latex($n^2Af^3+(3n^2A^2B-1)f^2+3n^2AB^2f+n^2B^3$)]]
+
+Consider the trivial case of [[latex($n=1$)]] child clump with the same density [[latex($\chi=1$)]] 
+
+We then have [[latex($B=1$)]] and [[latex($A=0$)]] and the cubic becomes:
+
+[[latex($-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$)]]
+
+
+If we have multiple levels 
+
+ - Highest level of clump has no children so mean density is nominal density.
 
 == Velocity Perturbations ==