Changes between Version 11 and Version 12 of u/johannjc


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Timestamp:
04/01/12 18:57:51 (13 years ago)
Author:
Jonathan
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  • u/johannjc

    v11 v12  
    2929Thoughts on binary sims
    3030
    31 [[Image(binary.png, width=400)]]
     31given the velocity as a function of position and time
    3232
    33 First we have position of primary
    34 
    35 [[latex($\vec{X_p}=R_p [\cos{\Omega t}, \sin{\Omega} t, 0]$)]]
    36 
    37 and velocity of primary as
    38 
    39 [[latex($\vec{v_p}=R_p \Omega [-\sin{\Omega t}, \cos{\Omega t}, 0]$)]]
    40 
    41 The velocity [[latex($v_0$)]] at the surface of the primary [[latex($x_0$)]] can be written as
    42 
    43 [[latex($\vec{v}_0=\vec{v_p}+\hat{x}_0 v_w$)]]
     33[[latex($\vec{v}(\vec{x})=(v_w-R_p \Omega \sin{q}) [\cos{\eta}, \sin{\eta}] + v_w \sin{\gamma}$)]]
    4434
    4535where
    4636
    47 [[latex($\hat{x}_0 = \frac{\vec{x}_0-\vec{X_p}}{S}$)]] and [[latex($|\vec{x}_0-\vec{X}_p| = S$)]]
     37Start with the assumption that
    4838
    49 And then the velocity is constant along characteristics given by [[latex($\vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)$)]]
     39[[latex($t_0=t$)]]
    5040
    51 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($\vec{x}-\vec{x}_0 = \vec{v}_0 (t-t_0)$)]]
     41DO
     42  [[latex($d = |\vec{x}-\vec{X}_p(t_r)|$)]]
    5243
    53 substituting the expression for [[latex($\vec{v}_0$)]] we have
     44  [[latex($t_r=t-\frac{d}{v_w}$)]]
     45END DO
    5446
    55 [[latex($\vec{x}-\vec{x}_0(t_0) = (\vec{v_p}(t_0)+\hat{x}_0(t_0) v_w) (t-t_0)$)]]
     47[[latex($\sin {q} =\frac{|x|}{d}\sin{(\Omega t_r+\alpha)}$)]]
    5648
    57 or
     49[[latex($\sin {\eta} = \frac{R_p}{d}\sin{(\Omega t_r+\alpha)}$)]]
    5850
    59 [[latex($\vec{x}-\vec{x}_0(t_0) = (\vec{v_p}(t_0)+\frac{\vec{x}_0(t_0)-\vec{X_p}(t_0)}{|\vec{x}_0(t_0)-\vec{X_p}(t_0)|} v_w) (t-t_0)$)]]
     51If we switch to a rotating frame that rotates counter to the orbit so the angular speed is [[latex($\Omega$)]], then
    6052
    61 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($t_0$)]].
     53[[latex($\vec{v}_r=\vec{v}-\Omega \times \vec{x}$)]]
     54
     55and
     56
     57[[latex($\alpha=\alpha_0+\Omega t$)]]
     58
     59so that
     60
     61[[latex($\Omega t_r+\alpha = \alpha_0 - \Omega \frac{d}{v_w}$)]]
    6262
    6363
    64 
    65 
    66 
    67 
    68 
    69 
    70 
    71 
    72 
    73 
    74 
    75 
    76 
    77 
    78 == Approximations ==
    79 
    80 
    81 We can simplify things if we assume that [[latex($v_w >> R_p\Omega$)]]
    82 
    83 [[latex($\vec{x}-\vec{x}_0(t_0) = \hat{x}_0 v_w (t-t_0)$)]]
    84 
    85 and that [[latex($|x| >> |x_0|$)]] since we can then solve for [[latex($t-t_0=|x|/v_w$)]]. 
    86 
    87 Then we can calculate [[latex($\vec{v_p}(t_0)= R_p\Omega[-\sin{(\Omega t+\phi)}, \cos{(\Omega t +\phi)}, 0]$)]] where [[latex($\phi=-\Omega |x|/v_w$)]]
    88 
    89 and [[latex($\vec{X_p(t_0)}=R_p [\cos{(\Omega t + \phi)}, \sin{(\Omega t + \phi)}, 0]$)]]
    90 
    91 
    92 
    93 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($X_p(t_0)$)]]
    94 
    95 
    96 We then have that [[latex($\vec{j}=\vec{x}\times\vec{v} = |x|(v_w+R\Omega\sin{\gamma})\sin{\gamma}$)]]
    97 
    98 and [[latex($\gamma =
    99 
    100 
    101 
    102 
    103 
    104 
    105 
    106 The wind that is arriving at [[latex($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}$)]]
    107 where [[latex($\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)$)]].
    108 
    109 If we consider the specific angular momentum [[latex($\vec{j}=\vec{x} \times \vec{v}$)]] we can identify three components:
    110 
    111 [[latex($\vec{j}=\vec{x} \times \vec{v}_p(t_0)+v_w \vec{x} \times \hat{n}$)]]
    112 
    113 
    114 now [[latex($\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]$)]]
    115 
    116 
    117 
    118 
    119 
    120 
    121 Furthermore, [[latex($\vec{x}-\vec{v}_p(t_0)(t-t_0)=\vec{x}_0(t_0)+\hat{x}_0v_w(t-t_0)$)]]
    122 
    123 
    124 And since [[latex($\vec{x_0}=\vec{X}_p+S\hat{x}_0$)]]
    125 
    126 
    127 we have
    128 
    129 [[latex($\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)$)]]
    130 
    131 and [[latex($\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)}$)]]
    132 
    133 
    134 so that [[latex($\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$)]]
    135 
    136 [[latex($\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 )$)]]
    137 
    138 or
    139 
    140 [[latex($\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]$)]]
    141 
    142 and if we express [[latex($\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)$)]]