Changes between Version 3 and Version 4 of u/erica/UniformCollapse

Timestamp:

06/29/15 13:06:36 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/UniformCollapse

@@ -8 +8 @@
 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.  
+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. In other words, the furthest shells have the highest acceleration. They speed up from an initial velocity of 0 the fastest. This has to be the case, as 
 
+- all shells reach the origin at the same time (so outer shells have furthest to go in the same amount of time, i.e. they must travel faster)
+- shells do not cross over the collapse 
+- which means that the mass contained within each concentric sphere remains a constant over the collapse
 
-  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 last point is useful for solving integrating the equation. Since the mass is constant in any shell (i.e. shell by the way here is used to mean concentric sphere), we can replace [[latex($M_r$)]] in the equation by [[latex($M_r = \frac{4}{3} \pi r_0^3 \rho_0$)]]. 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.