| 29 | | Thoughts on binary sims |
| 30 | | |
| 31 | | given the velocity as a function of position and time |
| 32 | | |
| 33 | | [[latex($\vec{v}(\vec{x})=(v_w-R_p \Omega \sin{q}) [\cos{\eta}, \sin{\eta}] + v_w \sin{\gamma}$)]] |
| 34 | | |
| 35 | | where |
| 36 | | |
| 37 | | Start with the assumption that |
| 38 | | |
| 39 | | [[latex($t_0=t$)]] |
| 40 | | |
| 41 | | [[latex($|v|=v_w$)]] |
| 42 | | |
| 43 | | DO |
| 44 | | [[latex($d = |\vec{x}-\vec{X}_p(t_r)|$)]] |
| 45 | | |
| 46 | | [[latex($t_r=t-\frac{d}{v_w}$)]] |
| 47 | | |
| 48 | | [[latex($\sin {q} =\frac{|x|}{d}\sin{(\Omega t_r+\alpha)}$)]] |
| 49 | | |
| 50 | | [[latex($|v|=v_w-R_p \Omega \sin{q}$)]] |
| 51 | | END DO |
| 52 | | |
| 53 | | [[latex($\sin {\eta} = \frac{R_p}{d}\sin{(\Omega t_r+\alpha)}$)]] |
| 54 | | |
| 55 | | If we switch to a rotating frame that rotates counter to the orbit so the angular speed is [[latex($\Omega$)]], then |
| 56 | | |
| 57 | | [[latex($\vec{v}_r=\vec{v}-\Omega \times \vec{x}$)]] |
| 58 | | |
| 59 | | and |
| 60 | | |
| 61 | | [[latex($\alpha=\alpha_0+\Omega t$)]] |
| 62 | | |
| 63 | | so that |
| 64 | | |
| 65 | | [[latex($\Omega t_r+\alpha = \alpha_0 - \Omega \frac{d}{v_w}$)]] |
| 66 | | |
