User:Shmigheghi

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1 This is lame
2 What I Do
3 Who I Do It With
4 Where I Do It

[edit] This is lame

$88632 = \frac {2 \pi r_s^{3/2}}{42828^{1/2}}\$
$T_s = 88632 \,$
$r_s = 20425.987 \mbox{ km } \,$
$v_s = \sqrt { \frac {k}{r_s}\ } = \sqrt { \frac {42828}{20425.987}\ } = 1.44801 \mbox{ km/s }$
$l_s = r_s v_s = 29577.0886 \dfrac{\mbox{km}^2}{\mbox{s}} \,$

$H = \dfrac{\mbox{m}^2 \cdot \mbox{km}}{\mbox{C}^2}$
$889258.49 = \frac {\ell_1^2/42828}{1 + e_1 \cos(2.809)}\$
$582775.22 = \frac {\ell_1^2/42828}{1 + e_1 \cos(2.759)}\$
$\sqrt { 582775.22 \cdot 42828 (1 + e_1 \cos(2.759)) }$
$e_1 = \frac {(\frac {889258.49}{582775.22}\ - 1)}{\cos(2.759) - (\frac {889258.49}{582775.22}\ \cos(2.809))}\ = 1.022 > 1$
$\ell_1 = 35988.1428 \dfrac{\mbox{km}^2}{\mbox{s}}$
$\ell_1 = \sqrt {k r_p (1 + e_1)}$
$\frac {k}{r_s}\$
$r_p = \frac {35988.1428^2}{42828(2.022)}\ = 14955.8095 \mbox{ km }$
$v_p = \sqrt { \frac {42828(2.022)}{41955.8095}\ } = 2.4063 \mbox{ km/s }$
$M_h = -F + e \sinh(F) = \frac {k^2 (e^2 - 1)^{3/2} t}{\ell^3}\$
$\tanh(F/2) = \sqrt { \frac {1.022 - 1}{1.022 + 1}\ } \tan( \frac {2.759}{2}\ ) = 0.53861$
$F = 2( \mathrm{arctanh} (0.53861)) = 1.2044 \,$
$t = 938221.1669 \mbox{ s} \,$
$M_h = 0.008492 \,$
$0 = e \sinh(F) - F - M_h \,$
$F_{n+1} = F_n - \frac {e\sinh(F_n) - F_n - M_h}{e\cosh(F_n) - 1}\$
$F_1 = 0.2 \,$
$F_2 = 0.2 - \frac {1.022\sinh(0.2) - 0.2 - 0.008492}{1.022\cosh(0.2) - 1}\ = 0.264143$
$F_3 = 0.25603217 \,$
$F_4 = 0.25587253 \,$
$\tanh( \frac {0.255872}{2}\ ) = \sqrt { \frac {1.022 - 1}{1.022 + 1}\ } \tan(\phi/2)$
$\phi = 1.76824 \mbox{ rad} \,$
$r = \frac {\ell^2/k}{1 + e\cos(\phi)}\ = 37823.45092 \mbox{ km} \,$
$e_2 = \frac {r_s - r_p}{r_s + r_p}\ = 0.154604 \,$
$\ell_2 = \sqrt { \frac {2kr_s r_p}{r_s + r_p}\ } = 27194.7733 \dfrac{\mbox{ km}^2}{\mbox{s}}$
$\Delta v_1 = \sqrt { \frac {k}{r_p}\ } ( \sqrt { \frac {2r_s}{r_s + r_p}\ } - \sqrt {1 + e_1} ) = -0.587957 \mbox{ km/s}$
$T_2 = \frac {2\pi}{ \sqrt {k} } \ ( \frac {r_s + r_p}{2} \ )^{3/2} = \frac {2\pi}{ \sqrt {42828} } ( \frac {20425.987 + 14955.8095}{2} \ )^{3/2} \$
$T_2 = 71439.893 \mbox{ seconds} \,$
$\frac {T_2}{2} = 35719.95 \mbox{ seconds}$
$\Delta v_2 = \sqrt { \frac {k}{r_s} \ } (1 - \sqrt { \frac {2r_p}{r_s + r_p} \ } ) = \sqrt { \frac {42828}{20425.987} \ } ( 1 - \sqrt { \frac {2(14955.8095)}{20425.987 + 14955.8095} \ } )$
$\Delta v_2 = 0.11663 \mbox{ km/s} \,$
$T(\epsilon) \approx \frac {2\pi r_s^{3/2} }{ \sqrt {k} } \ + \frac {3\pi r_s^{3/2} \epsilon^2}{ \sqrt {k} } \$
$T(\epsilon) - T \leq 300$
$\frac {2\pi}{ \sqrt {42828} } \ ( \frac {20425.987}{1 - \epsilon^2} \ ) ^{3/2} - 88632 \leq 300$
$\epsilon \leq 0.047436$
$r_0 = r_s \,$
$r\prime (\epsilon) = -r_s (1+\epsilon \cos(\phi))^{-2} \cos(\phi)$
$r\prime (\epsilon) = \frac {-r_s \cos(\phi) }{(1 + \epsilon \cos(\phi))^2} \$
$r\prime (0) = -r_s \cos(\phi))$
$r\prime \prime (\epsilon) = 2r_s \cos^2(\phi) (1 + \epsilon \cos(\phi))^{-3}$
$r\prime \prime (\epsilon) = 2r_s \cos^2(\phi)$
$r(\epsilon) \approx r_s - r_s\epsilon \cos (\phi) + r_s \epsilon^2 \cos^2 (\phi)$
$\frac {2\pi(r_s)^{3/2}}{ \sqrt {k} } \ + \frac {3\pi(r_s)^{3/2} \epsilon^2}{ \sqrt {k} } \ \leq 88932$
$\frac {3\pi(r_s)^{3/2} \epsilon^2}{ \sqrt {k} } \ \leq 300$
$\epsilon \leq 0.0475$
$\phi = 2 \arctan ( \sqrt { \frac {1 + \epsilon}{1 - \epsilon } \ } \tan(\psi/2) )$
$\phi (0) = \psi \,$
$\phi \prime (\epsilon) = 2[ \frac {\tan(\psi/2) \frac {1}{2} \ ( \frac {1 + \epsilon}{1 - \epsilon} \ )^{-1/2} ( \frac {1 - \epsilon - (1+ \epsilon)(-1)}{(1 - \epsilon)^2} \ ) }{1 + ( \frac {1 + \epsilon}{1 - \epsilon} \ ) \tan^2(\psi /2)} \ ]$
$\phi \prime \prime (\epsilon) = \frac {2 \tan(\psi/2)}{(1-\epsilon)^{3/2}(1+\epsilon)^{1/2}+(1-\epsilon)^{1/2}(1+\epsilon)^{3/2} \tan^2(\psi/2)} \$
$\phi \prime \prime (0) = (2 \sin (\psi /2)\cos(\psi /2))(\cos^2(\psi/2)+\sin^2(\psi/2)) = \sin \psi$
$\phi = \psi + \epsilon \sin(\psi) + \frac {\epsilon^2}{2} \ \sin(\psi) + 0(\epsilon^3)$
$F_0 (t) = \frac {k^2 t}{\ell_s ^3} \ = \psi$
$F_1 (t) = \sin(\psi) = \sin( \frac {k^2 t}{\ell_s ^3} \ )$
$F_2 (t) = \frac {\sin(\psi)}{2} \ = \sin( \frac {k^2 t}{\ell_s ^3} \ )/2$
$\phi (t,\epsilon) = \frac {k^2 t}{\ell_s ^3} \ + \epsilon \sin( \frac {k^2 t}{\ell_s ^3} \ ) + \frac {\epsilon^2}{2} \ \sin ( \frac {k^2 t}{\ell_s ^3} \ )$
$T(\epsilon) = \frac {2\pi r_s^{3/2}}{ \sqrt {k} } \ (1 + \frac {3}{2} \ \epsilon^2 )$
$M = \frac {2\pi t}{T(\epsilon) } \ = \frac {2\pi t}{ \frac {2\pi r_s^{3/2}}{ \sqrt {k} } \ (1 + \frac {3}{2} \ \epsilon^2 ) } \ = \frac {tk^2}{\ell^3} \ (1 + \frac {3}{2} \ \epsilon^2 )^{-1}$
$\mbox {let} \frac {k^2}{\ell^3} \ = a$
$\psi = M + \epsilon \sin (M) + \frac {\epsilon^2}{2} \ \sin(2M)$
$\psi = t a (1 - \frac {3}{2} \ \epsilon^2 ) + \epsilon \sin (at (1 - \frac {3}{2} \ \epsilon^2 )) + \frac {\epsilon^2}{2} \ \sin (2at(1 - \frac {3}{2} \ \epsilon^2 ))$
$\psi \prime (\epsilon) = -3ta \epsilon - 3at\epsilon^2 \cos (at(1 - \frac {3}{2} \ \epsilon^2 )) + \sin(at(1 - \frac {3}{2} \ \epsilon^2 )) -3\epsilon^3 at \cos (2at(1 - \frac {3}{2} \ \epsilon^2 )) + \epsilon \sin (at(1 - \frac {3}{2} \ \epsilon^2 ))$
$\psi \prime \prime (\epsilon) = -3ta + \sin (2at)$
$\psi (\epsilon) = \psi (0) + \epsilon \psi \prime (0) + \frac {\epsilon^2}{2} \ \psi \prime \prime (0)$
$\psi (\epsilon) = \frac {tk^2}{\ell^3} \ + \epsilon \sin ( \frac {tk^2}{\ell^3} \ ) + \epsilon^2 ( \frac {-3}{2} \ \frac {tk^2}{\ell^3} \ + \frac {1}{2} \ \sin (\frac {2tk^2}{\ell^3} \ )$
$\phi (t) = \psi + \epsilon \sin (\psi) + \frac {\epsilon^2}{2} \ \sin (\psi)$
$f_0 (t) = \psi = \frac {tk^2}{\ell^3} \$
$\phi (t) = \frac {tk^2}{\ell^3} \ + \epsilon \sin ( \frac {tk^2}{\ell^3} \ ) + \frac {\epsilon^2}{2} \ \sin ( \frac {tk^2}{\ell^3} \ )$
$\epsilon^2 \,$