Changes between Version 9 and Version 10 of u/erica/JeansInstability

Timestamp:

06/21/13 12:12:14 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/JeansInstability

@@ -76 +76 @@
 Taking the positive root and plugging into [[latex($\rho_1$)]], we see
 
-[[latex($\rho_1 \propto e^{-i i [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]t}$)]]
+[[latex($\rho_1 \propto e^{-i i [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}$)]]
 
 which gives
 
-[[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]t}$)]]
+[[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}$)]]
 
 
 Thus we have found the condition on [[latex($\omega$)]] which gives unstable perturbation modes. Now, the growth rate of these modes is given by 
 
-[[latex($\Gamma = 4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})$)]] 
+[[latex($\Gamma = [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}$)]] 
 
 and so the characteristic timescale for exponential growth, that is the time in which the perturbation increases by a factor of e, is given by
@@ -91 +91 @@
 [[latex($\tau = \frac{1}{\Gamma}$)]]
 
-= High lambda limit, i.e. pressure-less collapse, i.e. uniform collapse = 
+= High lambda limit, i.e. pressure-less collapse, or 'uniform collapse' = 
 
- 
+From 
+
+[[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]^{1/2}t}$)]] 
+
+it is easy to see that in the limit 
+
+[[latex($\lambda \rightarrow \infty$)]]
+
+for finite [[latex($\lambda_J$)]] (i.e.  pressure gradients are null/gravity dominates), the characteristic collapse time becomes
+
+[[latex($\tau = [\frac{1}{4 \pi G \rho0}]^{1/2} ~ \frac{1}{2} t_ff$)]]
+
+where [[latex($t_ff$)]] is the freefall time for a spherical region of uniform gas. 
+
+= Writing a module to test self-gravity using the Jeans Instability = 
+
+I go through the procedure of setting up the problem on this page here.
+