Changes between Version 15 and Version 16 of u/johannjc

Timestamp:

04/13/12 13:13:16 (13 years ago)

Author:

Jonathan

Comment:

—

u/johannjc

@@ -28 +28 @@
 == Random Ruminations ==
 
+
+So setting up the initial solution can be accomplished by the following:
+
+First start with the assumption that the retarted time is the current time
+
+[[latex($t_r=t$)]]
+
+
+DO
+
+  Calculate the displacement vector from the primary at the retarded time
+
+  [[latex($\vec{d} = \vec{x}-\vec{X}_p(t_r)$)]]
+
+  Calculate the wind normal so that 
+
+  [[latex($(v_w \hat{n} + \vec{V}_p(t_r)) \times \vec{d} = 0$)]]
+ 
+  [[latex($\hat{n} \times \vec{d} = \frac{-1}{v_w}\vec{V}_p(t_r) \times \vec{d}$)]]
+
+  It is possible to solve [[latex($\hat{a} \times \vec{b} = \vec{c}$)]] for the unit vector [[latex($\hat{a}$)]] 
+
+  [[latex($\hat{a}=\frac{\vec{b} \times \vec{c} + \sqrt{\vec{b} \cdot \vec{b} - \frac{(\vec{b} \times \vec{c}) \cdot (\vec{b} \times \vec{c})}{\vec{b} \cdot \vec{b}}} \vec{b}}{\vec{b} \cdot \vec{b}}$)]]
+
+
+  Once we have an estimate for [[latex($\hat{n}$)]] we can improve the estimate by modifying the trajectory to account for the gravity from the secondary as follows:
+
+   Calculate the wind velocity from the primary
+ 
+   [[latex($\vec{v}(t_r, x_p(t_r))=|v_w \hat{n}+\vec{V}_p(t_r)|$)]]
+
+   Now solve for the trajectory from the primary that leaves at [[latex($\vec{X_p}(t_r)$)]] at velocity [[latex($\vec{v}(t_r, x_p(t_r))$)]] taking into account the force from the secondary.  
+
+   Determine the path's distance of closest approach to [[latex($\vec{x}$)]] and call that [[latex($\vec{x}'(t')$)]]
+  
+   Estimate the change in initial velocity needed [[latex($\vec{v'}(t_r,x_p(t_r) = \vec{v'}(t_r,x_p(t_r))+\frac{x-x'(t')}{t'-t_r}$)]]
+
+
+
+
+
+
+  Update the retarded time using the new distance and wind speed
+
+  [[latex($t_r=t-\frac{d}{v}$)]]
+
+END DO
+
+The only problem occurs when there are multiple solutions for the retarded time...
+
+This will occur once we reach distances of order [[latex($v_w/\Omega$)]]
+
+If we switch to a rotating frame that rotates counter to the orbit so the angular speed is [[latex($\Omega$)]], then 
+
+[[latex($\vec{v}_r=\vec{v}-\Omega \times \vec{x}$)]]
+
+and
+
+[[latex($\alpha=\alpha_0+\Omega t$)]]
+
+so that 
+
+[[latex($\Omega t_r+\alpha = \alpha_0 - \Omega \frac{d}{v_w}$)]]
+
+
+
+
+= Threading =
  What is the maximum combined work we can give each processor
 
@@ -39 +107 @@
 
 
-