Changes between Version 10 and Version 11 of u/johannjc

Timestamp:

03/31/12 13:18:28 (13 years ago)

Author:

Jonathan

Comment:

—

u/johannjc

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 Thoughts on binary sims
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+[[Image(binary.png, width=400)]]
 
 First we have position of primary 
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 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($t_0$)]].
 
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+== Approximations ==
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 We can simplify things if we assume that [[latex($v_w >> R_p\Omega$)]]
 
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 Then we can calculate [[latex($\vec{v_p}(t_0)= R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0]$)]] where [[latex($\phi=-\Omega |x|/v_w$)]]
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+and [[latex($\vec{X_p(t_0)}=R_p [\cos{(\Omega t + \phi)}, \sin{(\Omega t + \phi)}, 0]$)]]
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+The wind arriving at [[latex($x(t)$)]] left the surface of the primary when it was at [[latex($X_p(t_0)$)]] travelling at a speed somewhere between [[latex($v_w-R\Omega$)]] and [[latex($v_w+R\Omega$)]].  We can make the assumption that the wind speed goes like [[latex($v_w+R\Omega\sin(\gamma)$)]] where [[latex($\gamma=\alpha-\Omega t_0$)]] where [[latex($\alpha$)]] is the angle towards [[latex($x$)]] and [[latex($\gamma$)]] is the angle between [[latex($\vec{x}(t)$)]] and [[latex($X_p(t_0)$)]]
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+We then have that [[latex($\vec{j}=\vec{x}\times\vec{v} = |x|(v_w+R\Omega\sin{\gamma})\sin{\gamma}$)]]
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+and [[latex($\gamma = 
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+The wind that is arriving at [[latex($x(t)$)]] left from the surface of the primary when it was at a position [[latex($X_p(t_0)$)]] travelling at a velocity of [[latex($v_p(t_0) + v_w \hat{n}$)]] 
+where [[latex($\hat{n}=\frac{x(t)-X'(t)}{|x(t)-X'(t)|}$)]] where [[latex($X'(t)=X_p(t_0)+v_p(t_0)(t-t_0)$)]].
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+If we consider the specific angular momentum [[latex($\vec{j}=\vec{x} \times \vec{v}$)]] we can identify three components:
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+[[latex($\vec{j}=\vec{x} \times \vec{v}_p(t_0)+v_w \vec{x} \times \hat{n}$)]]
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+now [[latex($\vec{n} = |x|[\cos{\alpha}, \sin{\alpha}, 0]-R_p[\cos{\Omega t_0}, \sin{\Omega t_0}, 0] - R_p \Omega (t-t_0) [-\sin{\Omega t_0}, \cos{\Omega t_0}, 0]$)]]
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