User:Shmigheghi

This is lame

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

$H=\frac{{\text{m}}^2\cdot \text{km}}{{\text{C}}^2}$
$889258.49=\frac{{\ell }_1^2/42828}{1+e_1\cos (2.809)}$
$582775.22=\frac{{\ell }_1^2/42828}{1+e_1\cos (2.759)}$
$\sqrt{582775.22\cdot 42828(1+e_1\cos (2.759))}$
$e_1=\frac{(\frac{889258.49}{582775.22}-1)}{\cos (2.759)-(\frac{889258.49}{582775.22}\cos (2.809))}=1.022>1$
${\ell }_1=35988.1428\frac{{\text{km}}^2}{\text{s}}$
${\ell }_1=\sqrt{k{r}_p(1+e_1)}$
$\frac{k}{r_s}$
$r_p=\frac{35988.1428^2}{42828(2.022)}=14955.8095\text{ km }$
$v_p=\sqrt{\frac{42828(2.022)}{41955.8095}}=2.4063\text{ km/s }$
$M_h=-F+e\sinh (F)=\frac{k^2(e^2-1{)}^{3/2}t}{{\ell }^3}$
$\tanh (F/2)=\sqrt{\frac{1.022-1}{1.022+1}}\tan (\frac{2.759}{2})=0.53861$
$F=2(\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(0.53861))=1.2044$
$t=938221.1669\text{ s}$
$M_h=0.008492$
$0=e\sinh (F)-F-M_h$
$F_{n+1}=F_n-\frac{e\sinh (F_n)-F_n-M_h}{e\cosh (F_n)-1}$
$F_1=0.2$
$F_2=0.2-\frac{1.022\sinh (0.2)-0.2-0.008492}{1.022\cosh (0.2)-1}=0.264143$
$F_3=0.25603217$
$F_4=0.25587253$
$\tanh (\frac{0.255872}{2})=\sqrt{\frac{1.022-1}{1.022+1}}\tan (\varphi /2)$
$\varphi =1.76824\text{ rad}$
$r=\frac{{\ell }^2/k}{1+e\cos (\varphi )}=37823.45092\text{ km}$
$e_2=\frac{r_s-r_p}{r_s+r_p}=0.154604$
${\ell }_2=\sqrt{\frac{2k{r}_s{r}_p}{r_s+r_p}}=27194.7733\frac{{\text{ km}}^2}{\text{s}}$
$\mathrm{\Delta }{v}_1=\sqrt{\frac{k}{r_p}}(\sqrt{\frac{2r_s}{r_s+r_p}}-\sqrt{1+e_1})=-0.587957\text{ km/s}$
$T_2=\frac{2\pi }{\sqrt{k}}(\frac{r_s+r_p}{2}{)}^{3/2}=\frac{2\pi }{\sqrt{42828}}(\frac{20425.987+14955.8095}{2}{)}^{3/2}$
$T_2=71439.893\text{ seconds}$
$\frac{T_2}{2}=35719.95\text{ seconds}$
$\mathrm{\Delta }{v}_2=\sqrt{\frac{k}{r_s}}(1-\sqrt{\frac{2r_p}{r_s+r_p}})=\sqrt{\frac{42828}{20425.987}}(1-\sqrt{\frac{2(14955.8095)}{20425.987+14955.8095}})$
$\mathrm{\Delta }{v}_2=0.11663\text{ km/s}$
$T(ϵ)\approx \frac{2\pi r_s^{3/2}}{\sqrt{k}}+\frac{3\pi r_s^{3/2}{ϵ}^2}{\sqrt{k}}$
$T(ϵ)-T\le 300$
$\frac{2\pi }{\sqrt{42828}}(\frac{20425.987}{1-{ϵ}^2}{)}^{3/2}-88632\le 300$
$ϵ\le 0.047436$
$r_0=r_s$
$r\prime (ϵ)=-r_s(1+ϵ\cos (\varphi ){)}^{-2}\cos (\varphi )$
$r\prime (ϵ)=\frac{-r_s\cos (\varphi )}{(1+ϵ\cos (\varphi ){)}^2}$
$r\prime (0)=-r_s\cos (\varphi ))$
$r\prime \prime (ϵ)=2r_s{\cos }^2(\varphi )(1+ϵ\cos (\varphi ){)}^{-3}$
$r\prime \prime (ϵ)=2r_s{\cos }^2(\varphi )$
$r(ϵ)\approx r_s-r_sϵ\cos (\varphi )+r_s{ϵ}^2{\cos }^{}(\varphi )$
$\frac{2\pi (r_s{)}^{3/2}}{\sqrt{k}}+\frac{3\pi (r_s{)}^{3/2}{ϵ}^2}{\sqrt{k}}\le 88932$
$\frac{3\pi (r_s{)}^{3/2}{ϵ}^2}{\sqrt{k}}\le 300$
$ϵ\le 0.0475$
$\varphi =2\arctan (\sqrt{\frac{1+ϵ}{1-ϵ}}\tan (\psi /2))$
$\varphi (0)=\psi$
$\varphi \prime (ϵ)=2[\frac{\tan (\psi /2)\frac{1}{2}(\frac{1+ϵ}{1-ϵ}{)}^{-1/2}(\frac{1-ϵ-(1+ϵ)(-1)}{(1-ϵ{)}^2})}{1+(\frac{1+ϵ}{1-ϵ}){\tan }^2(\psi /2)}]$
$\varphi \prime \prime (ϵ)=\frac{2\tan (\psi /2)}{(1-ϵ{)}^{3/2}(1+ϵ{)}^{1/2}+(1-ϵ{)}^{1/2}(1+ϵ{)}^{3/2}{\tan }^2(\psi /2)}$
$\varphi \prime \prime (0)=(2\sin (\psi /2)\cos (\psi /2))({\cos }^{}(\psi /2)+{\sin }^{}(\psi /2))=\sin \psi$
$\varphi =\psi +ϵ\sin (\psi )+\frac{{ϵ}^2}{2}\sin (\psi )+0({ϵ}^3)$
$F_0(t)=\frac{k^{2}t}{{\ell }_s^3}=\psi$
$F_1(t)=\sin (\psi )=\sin (\frac{k^{2}t}{{\ell }_s^3})$
$F_2(t)=\frac{\sin (\psi )}{2}=\sin (\frac{k^{2}t}{{\ell }_s^3})/2$
$\varphi (t,ϵ)=\frac{k^{2}t}{{\ell }_s^3}+ϵ\sin (\frac{k^{2}t}{{\ell }_s^3})+\frac{{ϵ}^2}{2}\sin (\frac{k^{2}t}{{\ell }_s^3})$
$T(ϵ)=\frac{2\pi r_s^{3/2}}{\sqrt{k}}(1+\frac{3}{2}{ϵ}^2)$
$M=\frac{2\pi t}{T(ϵ)}=\frac{2\pi t}{\frac{2\pi r_s^{3/2}}{\sqrt{k}}(1+\frac{3}{2}{ϵ}^2)}=\frac{t{k}^2}{{\ell }^3}(1+\frac{3}{2}{ϵ}^2{)}^{-1}$
$\text{let}\frac{k^2}{{\ell }^3}=a$
$\psi =M+ϵ\sin (M)+\frac{{ϵ}^2}{2}\sin (2M)$
$\psi =ta(1-\frac{3}{2}{ϵ}^2)+ϵ\sin (at(1-\frac{3}{2}{ϵ}^2))+\frac{{ϵ}^2}{2}\sin (2at(1-\frac{3}{2}{ϵ}^2))$
$\psi \prime (ϵ)=-3taϵ-3at{ϵ}^2\cos (at(1-\frac{3}{2}{ϵ}^2))+\sin (at(1-\frac{3}{2}{ϵ}^2))-3{ϵ}^{3}at\cos (2at(1-\frac{3}{2}{ϵ}^2))+ϵ\sin (at(1-\frac{3}{2}{ϵ}^2))$
$\psi \prime \prime (ϵ)=-3ta+\sin (2at)$
$\psi (ϵ)=\psi (0)+ϵ\psi \prime (0)+\frac{{ϵ}^2}{2}\psi \prime \prime (0)$
$\psi (ϵ)=\frac{t{k}^2}{{\ell }^3}+ϵ\sin (\frac{t{k}^2}{{\ell }^3})+{ϵ}^2(\frac{-3}{2}\frac{t{k}^2}{{\ell }^3}+\frac{1}{2}\sin (\frac{2t{k}^2}{{\ell }^3})$
$\varphi (t)=\psi +ϵ\sin (\psi )+\frac{{ϵ}^2}{2}\sin (\psi )$
$f_0(t)=\psi =\frac{t{k}^2}{{\ell }^3}$
$\varphi (t)=\frac{t{k}^2}{{\ell }^3}+ϵ\sin (\frac{t{k}^2}{{\ell }^3})+\frac{{ϵ}^2}{2}\sin (\frac{t{k}^2}{{\ell }^3})$
${ϵ}^2$

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