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

Timestamp:

06/29/15 13:14:09 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/UniformCollapse

@@ -14 +14 @@
 - which means that the mass contained within each concentric sphere remains a constant over the collapse
 
-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.
+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. We will look at these equations next.
 
-Velocity:
+'''Velocity equation'''
 
 [[latex($\frac{dr}{dt} = -\sqrt{\frac{8\pi}{3} G \rho_0 r_0^2(\frac{r_0}{r}-1)}$)]]
 
 - describes the outer velocity of a sphere that contains mass M_r as a function of r = r(t)
-- at radius r = r_0, dr dt = 0 by construction
-- if r were decreasing linearly over time, then plotting v over time linearly would show that v decreases as the square root. However, a look at r vs t shows that r does not decrease linearly, so v does not strictly go at the negative square root. 
+- at radius [[latex($r = r_0$)]], [[latex($\frac{dr} {dt} = 0$)]] by construction
+- if r were decreasing linearly over time, then plotting v over time on a linear scale would show that v decreases as the square root. However, a look at r vs. t shows that r does not decrease linearly, so v does not strictly go at the negative square root. 
+
+