Changes between Version 15 and Version 16 of u/erica/UniformCollapse

Timestamp:

06/29/15 13:59:53 (10 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/UniformCollapse

@@ -18 +18 @@
 '''Position equation'''
 
-Start by describing this so to exaplain the numerical solution but,
+Integrating the above equation twice with a change of variables yields the equation of motion:
 
+[[latex($\xi + \frac{1}{2} \sin(2\xi) = kt$)]]
+
+where
+
+[[latex($k=\sqrt{\frac{8 \pi}{3} G \rho_0}$)]]
+
+and
+
+[[latex($\frac{r}{r_0}=\cos^2(\xi)$)]]
+
+Now, that equation of motion is a real hassle to deal with, so numerical solution is necessary. This, we can use mathematica for to find the roots of this equation in terms of [[latex($\xi,t$)]]. Once we have [[latex($\xi,t$)]] pairs, we can take the [[latex($\cos^2(\xi)*r_0$)]] to get a list of values for r for a list of discrete t.
+
+[[latex($\boxed{r(t) = r_0 \cos^2(\xi(t))}$)]]
 
 '''Velocity equation'''
 
-[[latex($\frac{dr}{dt} = -\sqrt{\frac{8\pi}{3} G \rho_0 r_0^2(\frac{r_0}{r}-1)}$)]]
+[[latex($\boxed{\frac{dr}{dt} = -\sqrt{\frac{8\pi}{3} G \rho_0 r_0^2(\frac{r_0}{r(t)}-1)}}$)]]
 
-- describes the outer velocity of a sphere that contains mass M_r as a function of r = r(t)
+- describes the outer velocity of a sphere that contains mass M_r as a function of r(t), where r(t) is a list of discrete radii as described above
 - 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.