User:Shmigheghi

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)}}\$
${\sqrt {582775.22\cdot 42828(1+e_{1}\cos(2.759))}}$
$e_{1}={\frac {({\frac {889258.49}{582775.22}}\ -1)}{\cos(2.759)-({\frac {889258.49}{582775.22}}\ \cos(2.809))}}\ =1.022>1$
$\ell _{1}=35988.1428{\dfrac {{\mbox{km}}^{2}}{\mbox{s}}}$
$\ell _{1}={\sqrt {kr_{p}(1+e_{1})}}$
${\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)}}\$
$\phi \prime \prime (0)=(2\sin(\psi /2)\cos(\psi /2))(\cos ^{2}(\psi /2)+\sin ^{2}(\psi /2))=\sin \psi$
$\phi =\psi +\epsilon \sin(\psi )+{\frac {\epsilon ^{2}}{2}}\ \sin(\psi )+0(\epsilon ^{3})$
$F_{0}(t)={\frac {k^{2}t}{\ell _{s}^{3}}}\ =\psi$
$F_{1}(t)=\sin(\psi )=\sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )$
$F_{2}(t)={\frac {\sin(\psi )}{2}}\ =\sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )/2$
$\phi (t,\epsilon )={\frac {k^{2}t}{\ell _{s}^{3}}}\ +\epsilon \sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )+{\frac {\epsilon ^{2}}{2}}\ \sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )$
$T(\epsilon )={\frac {2\pi r_{s}^{3/2}}{\sqrt {k}}}\ (1+{\frac {3}{2}}\ \epsilon ^{2})$
$M={\frac {2\pi t}{T(\epsilon )}}\ ={\frac {2\pi t}{{\frac {2\pi r_{s}^{3/2}}{\sqrt {k}}}\ (1+{\frac {3}{2}}\ \epsilon ^{2})}}\ ={\frac {tk^{2}}{\ell ^{3}}}\ (1+{\frac {3}{2}}\ \epsilon ^{2})^{-1}$
${\mbox{let}}{\frac {k^{2}}{\ell ^{3}}}\ =a$
$\psi =M+\epsilon \sin(M)+{\frac {\epsilon ^{2}}{2}}\ \sin(2M)$
$\psi =ta(1-{\frac {3}{2}}\ \epsilon ^{2})+\epsilon \sin(at(1-{\frac {3}{2}}\ \epsilon ^{2}))+{\frac {\epsilon ^{2}}{2}}\ \sin(2at(1-{\frac {3}{2}}\ \epsilon ^{2}))$
$\psi \prime (\epsilon )=-3ta\epsilon -3at\epsilon ^{2}\cos(at(1-{\frac {3}{2}}\ \epsilon ^{2}))+\sin(at(1-{\frac {3}{2}}\ \epsilon ^{2}))-3\epsilon ^{3}at\cos(2at(1-{\frac {3}{2}}\ \epsilon ^{2}))+\epsilon \sin(at(1-{\frac {3}{2}}\ \epsilon ^{2}))$
$\psi \prime \prime (\epsilon )=-3ta+\sin(2at)$
$\psi (\epsilon )=\psi (0)+\epsilon \psi \prime (0)+{\frac {\epsilon ^{2}}{2}}\ \psi \prime \prime (0)$
$\psi (\epsilon )={\frac {tk^{2}}{\ell ^{3}}}\ +\epsilon \sin({\frac {tk^{2}}{\ell ^{3}}}\ )+\epsilon ^{2}({\frac {-3}{2}}\ {\frac {tk^{2}}{\ell ^{3}}}\ +{\frac {1}{2}}\ \sin({\frac {2tk^{2}}{\ell ^{3}}}\ )$
$\phi (t)=\psi +\epsilon \sin(\psi )+{\frac {\epsilon ^{2}}{2}}\ \sin(\psi )$
$f_{0}(t)=\psi ={\frac {tk^{2}}{\ell ^{3}}}\$
$\phi (t)={\frac {tk^{2}}{\ell ^{3}}}\ +\epsilon \sin({\frac {tk^{2}}{\ell ^{3}}}\ )+{\frac {\epsilon ^{2}}{2}}\ \sin({\frac {tk^{2}}{\ell ^{3}}}\ )$
$\epsilon ^{2}\,$