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Journal Club for March 22
Thoughts on binary sims
First we have position of primary
\vec{X_p}=R_p [\cos{\Omega t}, \sin{\Omega} t, 0]
and velocity of primary as
\vec{v_p}=R_p \Omega [-\sin{\Omega t}, \cos{\Omega t}, 0]
The velocity v_0 at the surface of the primary x_0 can be written as
\vec{v}_0=\vec{v_p}+\hat{x}_0 v_w
where
\hat{x}_0 = \frac{\vec{x}_0-\vec{X_p}}{S} and |\vec{x}_0-\vec{X}_p| = S
And then the velocity is constant along characteristics given by \vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)
Given \vec{x} and t we wish to find \vec{x}_0 and t_0 (and \vec{v}_0=v) so that \vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)
substituting the expression for \vec{v}_0 we have
\vec{x}-\vec{x}_0(t_0) = (\vec{v_p}(t_0)+\hat{x}_0(t_0) v_w) (t-t_0)
or
\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 |\vec{x}_0-\vec{X}_p| = S can be solved for \vec{x_0} and t_0.
We can simplify things if we assume that v_w >> R_p\Omega
\vec{x}-\vec{x}_0(t_0) = \hat{x}_0 v_w (t-t_0)
and that |x| >> |x_0| since we can then solve for t-t_0=|x|/v_w.
Then we can calculate \vec{v_p}(t_0)= R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0] where \phi=-\Omega |x|/v_w
Furthermore, \vec{x}-\vec{v}_p(t_0)(t-t_0)=\vec{x}_0(t_0)+\hat{x}_0v_w(t-t_0)
And since \vec{x_0}=\vec{X}_p+S\hat{x}_0
we have
\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 \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 \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
\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
\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 \vec{x}=|x|[[\cos(\alpha), \sin(\alpha), 0] then j=\vec{x} \times \vec{v}=-R_p v_w \sin(\beta)+R_p S\Omega \cos(\beta) where \beta=\alpha - \Omega t - \phi and in particular - if we consider the orbit of the secondary we have \alpha=\Omega t + \pi so that j=-R_p v_w \sin(\pi-\phi)
