Changes between Version 5 and Version 6 of u/erica/JeansInstability

Timestamp:

06/21/13 11:21:03 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/JeansInstability

@@ -17 +17 @@
 When 
 
+[[latex($\omega^2 > 0$)]] 
+
+or 
+
+[[latex($4 \pi G \rho_0 < C_s^2 k^2 $)]]  
+
+[[latex($\omega$)]] is real, and the density perturbation has the form:
+
+[[latex($\rho_1 = e^{i(\vec{k} \cdot \vec{x})} e^{-i(^+_- \omega t)} $)]]
+
+which is oscillatory. Thus the density perturbation propagates through the medium as a sound wave, i.e. pressure gradients are strong enough to support the medium against self-gravitational collapse. On the other hand, when
+
 [[latex($\omega^2 < 0$)]] 
 
@@ -33 +45 @@
 [[latex($\rho_1 = \cos(kx)e^{^+ _-\omega t}$)]] 
 
-Thus, the perturbation takes on a form that is oscillatory in x and exponential in time. What we'd like to do now is determine, for which frequencies is the perturbation ''unstable'' (exponentially growing), and for what frequencies is the perturbation ''stable'' (exponentially decaying). In order to determine this we can rewrite [[latex($\omega$)]] into a more intuitive form.
+Thus, the perturbation takes on a form that is oscillatory in x and exponential in time. To get the critical length scale (aka Jeans length) for the oscillations, we write the oscillatory condition
+
+[[latex($C_s^2 k^2 > 4 \pi G \rho_0$)]]
+
+Substituting [[latex($2 \pi / \lambda $)]] for k and solving for [[latex($\lambda$)]] we have
+
+[[latex($ \lambda < \frac{2 \pi C_s}{(4 \pi G \rho_0)^{1/2}} $)]]
+
+That is, for wavelengths smaller than [[latex($2 \pi C_s / (4 \pi G \rho_0)$)]], the perturbation is stable to collapse. You can think of this wavelength as length scales over which the average [[latex($\rho = \rho_0$)]]. The critical length is known as the Jeans length and is given by:
+
+[[latex($\lambda_J = Cs(\frac{\pi}{G \rho_0})^{1/2}$)]]
+
+
+
+
+What we'd like to do now is determine, for which frequencies is the perturbation ''unstable'' (exponentially growing), and for what frequencies is the perturbation ''stable'' (exponentially decaying). In order to determine this we can rewrite [[latex($\omega$)]] into a more intuitive form.