Changes between Version 9 and Version 10 of u/erica/UniformCollapse

Timestamp:

06/29/15 13:31:33 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/UniformCollapse

@@ -16 +16 @@
 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.
 
+'''Position equation'''
+
+Start by describing this so to exaplain the numerical solution but,
+
+
 '''Velocity equation'''
 
@@ -26 +31 @@
 
 [[Image(velocityplot.png, 40%)]]
-[[br]]'''''Velocity of outer radius over time.''''' '''This plot shows that for initial concentric spheres of the same density, by varying outer radius, spheres with larger radii have steeper velocity profiles over time. Note this velocity is the velocity out the outer edge of the sphere. The free-fall time here, as can be seen where all of the curves begin to asymptote, is about [[latex($t_{ff} \approx .24$)]]. The smallest sphere plotted has an initial radius of [[latex($\frac{1}{1000}r_0$)]] (this is the green line), whereas the velocity of the outer most sphere within initial radius [[latex($r_0$)]] is the blue line.
+[[br]]'''''Velocity of outer radius over time.''''' '''This plot shows that for initial concentric spheres of the same density, by varying outer radius, spheres with larger radii have steeper velocity profiles over time. Note this velocity is the velocity out the outer edge of the sphere. The free-fall time here, as can be seen where all of the curves begin to asymptote, is about [[latex($t_{ff} \approx .24$)]]. The smallest sphere plotted has an initial radius of [[latex($\frac{1}{1000}r_0$)]] (this is the green line), whereas the velocity of the outer most sphere within initial radius [[latex($r_0$)]] is the blue line. Note that this shows that all shells continue to accelerate over time, that the furthest away shells have the highest initial acceleration, all consistent with the discussion above.
+
+Now, instead of seeing this plot in terms of time, we can