Changes between Version 2 and Version 3 of u/erica/UniformCollapse

Timestamp:

06/29/15 12:57:22 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/UniformCollapse

@@ -6 +6 @@
 
 
-Integrating this equation under the boundary condition, the initial velocity [[latex($\frac{dr}{dt}=0$)]] at the initial radius of the sphere [[latex($r=r_0$)]], leads to an equation that describes the radius of the outer sphere over time (i.e. over the course of collapse), and the velocity at this radius over time.
+This acceleration is due to all of the mass ''inside'' of the sphere, i.e. the mass contained within a radius r, [[latex($M_r$)]]. Since this equation governs the local acceleration of the gas at any radius, we can consider how the acceleration of any shell within the sphere behaves over time, once we know r(t) of course. Initially (t=0), however, we do know the radius, so we can check the behavior of this equation at t=0 for various shells within the sphere. 
 
+Consider first the outer most radius. A sphere with radius [[latex($r=r_0$)]] has within it mass that goes like [[latex($M_{r0} \propto r_0^3$)]]. Therefore, we see that initially, the acceleration of the gas is proportional to [[latex($r_0$)]]. The density of the sphere is uniform, and this means repeating this procedure for any smaller radii shows that the acceleration of interior shells decreases as r decreases.  
+
+
+  Integrating this equation under the boundary condition, the initial velocity [[latex($\frac{dr}{dt}=0$)]] at the initial radius of the sphere [[latex($r=r_0$)]], leads to an equation that describes the radius of the outer sphere over time (i.e. over the course of collapse), and the velocity at this radius over time.
+