Changes between Version 18 and Version 19 of u/erica/JeansInstability

Timestamp:

06/28/13 22:46:23 (13 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/JeansInstability

@@ -1 +1 @@
 [[PageOutline]]
 
+= Linearization of the Fluid Equations =
+
+The system of equations to describe a fluid with pressure and self-gravity are the following:
+
+[[latex($\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{v} )  = 0 ~(continuity)$)]]
+
+
+[[latex($\frac{\partial \vec{v}}{\partial t} + \vec{v}(\nabla \cdot \vec{v}) = - \frac{\nabla P}{\rho} - \nabla \phi ~(Euler)$)]]
+
+[[latex($\nabla ^ 2 \phi = 4 \pi G \rho ~(Poisson)$)]]
+
+A equilibrium solution to this non-linear system of PDE's is easily given by:
+
+[[latex($\rho = \rho_0 = ~constant $)]]
+
+[[latex($v  = 0 $)]]
+
+[[latex($P = P(\rho) ~(constant ~entropy ~S)$)]]
+
+However, note that this is the source of the infamous 'Jeans Swindle', as plugging this equilibrium solution into the Euler equation implies that '''''phi is a constant''''' ([[latex($\phi = \phi_0$)]]), in contradiction with Poisson's equation. Ignoring this though, we can do a standard linear analysis on these equations to check for stability of small perturbations. As usual, a linear analysis 1) transforms the original non-linear system into a solvable linear system, and 2) provides valuable insight into the basic physics within the fluid. 
+
+Inducing a small linear perturbation to the equilibrium solution gives the following:
+
+[[latex($\rho = \rho_0 + \rho_1$)]]
+
+[[latex($P = P_0 + P_1$)]]
+
+[[latex($v = v_1$)]]
+
+[[latex($\phi = \phi_0 $)]]
+
+Plugging these solutions into the above system of equations leads to the Jeans Instability as described as follows.
+
 = The Jeans Instability =
+
+
+
 
 Linear perturbative analyis on the self-gravitating fluid equations under the assumption of infinite, constant ([[latex($v_0$)]]), static background density ([[latex($\rho_0$)]]) results in the following 2nd order PDE in the density perturbation ([[latex($\rho_1$)]]):