Changes between Version 8 and Version 9 of u/johannjc

Timestamp:

03/30/12 19:26:59 (13 years ago)

Author:

Jonathan

Comment:

—

u/johannjc

@@ -27 +27 @@
 
 
+Thoughts on binary sims
+
+First we have position of primary 
+
+[[latex($\vec{X_p}=R_p [\cos{\Omega t}, \sin{\Omega} t, 0]$)]]
+
+and velocity of primary as
+
+[[latex($\vec{v_p}=R_p \Omega [-\sin{\Omega t}, \cos{\Omega t}, 0]$)]]
+
+The velocity [[latex($v_0$)]] at the surface of the primary [[latex($x_0$)]] can be written as 
+
+[[latex($\vec{v}_0=\vec{v_p}+\hat{x}_0 v_w$)]]
+
+where 
+
+[[latex($\hat{x}_0 = \frac{\vec{x}_0-\vec{X_p}}{|\vec{x}_0-\vec{X_p}|}$)]]
+
+And then the velocity is constant along characteristics given by [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 t$)]] 
+
+Given [[latex($\vec{x}$)]] and [[latex($t$)]] we wish to find [[latex($\vec{x}_0$)]] and [[latex($\vec{v}_0$)]] so that [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 t$)]] 
+
+substituting the expression for [[latex($\vec{v}_0$)]] we have
+
+[[latex($\vec{x}-\vec{x}_0 = (\vec{v_p}+\hat{x}_0 v_w) t$)]] 
+