Changes between Version 9 and Version 10 of u/johannjc

Timestamp:

03/30/12 22:48:48 (14 years ago)

Author:

Jonathan

Comment:

—

u/johannjc

@@ -43 +43 @@
 where 
 
-[[latex($\hat{x}_0 = \frac{\vec{x}_0-\vec{X_p}}{|\vec{x}_0-\vec{X_p}|}$)]]
+[[latex($\hat{x}_0 = \frac{\vec{x}_0-\vec{X_p}}{S}$)]] and [[latex($|\vec{x}_0-\vec{X}_p| = S$)]]
 
-And then the velocity is constant along characteristics given by [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 t$)]] 
+And then the velocity is constant along characteristics given by [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)$)]] 
 
-Given [[latex($\vec{x}$)]] and [[latex($t$)]] we wish to find [[latex($\vec{x}_0$)]] and [[latex($\vec{v}_0$)]] so that [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 t$)]] 
+Given [[latex($\vec{x}$)]] and [[latex($t$)]] we wish to find [[latex($\vec{x}_0$)]] and [[latex($t_0$)]] (and [[latex($\vec{v}_0=v$)]]) so that [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)$)]] 
 
 substituting the expression for [[latex($\vec{v}_0$)]] we have
 
-[[latex($\vec{x}-\vec{x}_0 = (\vec{v_p}+\hat{x}_0 v_w) t$)]] 
+[[latex($\vec{x}-\vec{x}_0(t_0) = (\vec{v_p}(t_0)+\hat{x}_0(t_0) v_w) (t-t_0)$)]] 
 
+or 
+
+[[latex($\vec{x}-\vec{x}_0(t_0) = (\vec{v_p}(t_0)+\frac{\vec{x}_0(t_0)-\vec{X_p}(t_0)}{|\vec{x}_0(t_0)-\vec{X_p}(t_0)|} v_w) (t-t_0)$)]] 
+
+which gives two non-linear equations which along with the constraint that [[latex($|\vec{x}_0-\vec{X}_p| = S$)]] can be solved for [[latex($\vec{x_0}$)]] and [[latex($t_0$)]].
+
+We can simplify things if we assume that [[latex($v_w >> R_p\Omega$)]]
+
+[[latex($\vec{x}-\vec{x}_0(t_0) = \hat{x}_0 v_w (t-t_0)$)]] 
+
+and that [[latex($|x| >> |x_0|$)]] since we can then solve for [[latex($t-t_0=|x|/v_w$)]].  
+
+Then we can calculate [[latex($\vec{v_p}(t_0)= R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0]$)]] where [[latex($\phi=-\Omega |x|/v_w$)]]
+
+
+Furthermore, [[latex($\vec{x}-\vec{v}_p(t_0)(t-t_0)=\vec{x}_0(t_0)+\hat{x}_0v_w(t-t_0)$)]]
+
+
+And since [[latex($\vec{x_0}=\vec{X}_p+S\hat{x}_0$)]] 
+
+
+we have 
+
+[[latex($\vec{x}-\vec{v}_p(t_0)(t-t_0)-\vec{X}_p(t_0)=S\hat{x}_0+\hat{x}_0v_w(t-t_0)$)]]
+
+and [[latex($\hat{x}_0 = \frac{\vec{x}-\vec{v}_p(t_0)(t-t_0)-\vec{X}_p(t_0)}{S+v_w(t-t_0)}$)]]
+
+
+so that [[latex($\vec{v}=\vec{v}_0=\vec{v_p}+\hat{x}_0 v_w=R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0] +\frac{\vec{x}-\vec{v}_p(t_0)(t-t_0)-\vec{X}_p(t_0)}{S+v_w(t-t_0)}v_w$)]]
+
+[[latex($\vec{v}=R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0] \left ( \frac{1}{1+\frac{|x|}{S}} \right) + \hat{x} v_w \left(\frac{1}{1+\frac{S}{|x|}} \right ) - \frac{\vec{X}_p}{|x|} v_w \left ( \frac{1}{1+\frac{S}{|x|}}\right )$)]]
+
+or
+
+[[latex($\vec{v} \approx \hat{x}v_w-\frac{R_p}{|x|} v_w [\cos{(\Omega t+\phi)}, \sin{(\Omega t +\phi)}, 0] + \frac{S}{|x|} R_p\Omega [-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0]$)]]
+
+and if we express [[latex($\vec{x}=|x|[[\cos(\alpha), \sin(\alpha), 0]$)]] then [[latex($j=\vec{x} \times \vec{v}=-R_p v_w \sin(\beta)+R_p S\Omega \cos(\beta)$)]] where [[latex($\beta=\alpha - \Omega t - \phi$)]] and in particular - if we consider the orbit of the secondary we have [[latex($\alpha=\Omega t + \pi$)]] so that [[latex($j=-R_p v_w \sin(\pi-\phi)$)]]