88632 = 2 π r s 3 / 2 42828 1 / 2 {\displaystyle 88632={\frac {2\pi r_{s}^{3/2}}{42828^{1/2}}}\ } T s = 88632 {\displaystyle T_{s}=88632\,} r s = 20425.987 km {\displaystyle r_{s}=20425.987{\mbox{ km }}\,} v s = k r s = 42828 20425.987 = 1.44801 km/s {\displaystyle 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 km 2 s {\displaystyle l_{s}=r_{s}v_{s}=29577.0886{\dfrac {{\mbox{km}}^{2}}{\mbox{s}}}\,}
H = m 2 ⋅ km C 2 {\displaystyle H={\dfrac {{\mbox{m}}^{2}\cdot {\mbox{km}}}{{\mbox{C}}^{2}}}} 889258.49 = ℓ 1 2 / 42828 1 + e 1 cos ( 2.809 ) {\displaystyle 889258.49={\frac {\ell _{1}^{2}/42828}{1+e_{1}\cos(2.809)}}\ } 582775.22 = ℓ 1 2 / 42828 1 + e 1 cos ( 2.759 ) {\displaystyle 582775.22={\frac {\ell _{1}^{2}/42828}{1+e_{1}\cos(2.759)}}\ } 582775.22 ⋅ 42828 ( 1 + e 1 cos ( 2.759 ) ) {\displaystyle {\sqrt {582775.22\cdot 42828(1+e_{1}\cos(2.759))}}} e 1 = ( 889258.49 582775.22 − 1 ) cos ( 2.759 ) − ( 889258.49 582775.22 cos ( 2.809 ) ) = 1.022 > 1 {\displaystyle e_{1}={\frac {({\frac {889258.49}{582775.22}}\ -1)}{\cos(2.759)-({\frac {889258.49}{582775.22}}\ \cos(2.809))}}\ =1.022>1} ℓ 1 = 35988.1428 km 2 s {\displaystyle \ell _{1}=35988.1428{\dfrac {{\mbox{km}}^{2}}{\mbox{s}}}} ℓ 1 = k r p ( 1 + e 1 ) {\displaystyle \ell _{1}={\sqrt {kr_{p}(1+e_{1})}}} k r s {\displaystyle {\frac {k}{r_{s}}}\ } r p = 35988.1428 2 42828 ( 2.022 ) = 14955.8095 km {\displaystyle r_{p}={\frac {35988.1428^{2}}{42828(2.022)}}\ =14955.8095{\mbox{ km }}} v p = 42828 ( 2.022 ) 41955.8095 = 2.4063 km/s {\displaystyle v_{p}={\sqrt {{\frac {42828(2.022)}{41955.8095}}\ }}=2.4063{\mbox{ km/s }}} M h = − F + e sinh ( F ) = k 2 ( e 2 − 1 ) 3 / 2 t ℓ 3 {\displaystyle M_{h}=-F+e\sinh(F)={\frac {k^{2}(e^{2}-1)^{3/2}t}{\ell ^{3}}}\ } tanh ( F / 2 ) = 1.022 − 1 1.022 + 1 tan ( 2.759 2 ) = 0.53861 {\displaystyle \tanh(F/2)={\sqrt {{\frac {1.022-1}{1.022+1}}\ }}\tan({\frac {2.759}{2}}\ )=0.53861} F = 2 ( a r c t a n h ( 0.53861 ) ) = 1.2044 {\displaystyle F=2(\mathrm {arctanh} (0.53861))=1.2044\,} t = 938221.1669 s {\displaystyle t=938221.1669{\mbox{ s}}\,} M h = 0.008492 {\displaystyle M_{h}=0.008492\,} 0 = e sinh ( F ) − F − M h {\displaystyle 0=e\sinh(F)-F-M_{h}\,} F n + 1 = F n − e sinh ( F n ) − F n − M h e cosh ( F n ) − 1 {\displaystyle F_{n+1}=F_{n}-{\frac {e\sinh(F_{n})-F_{n}-M_{h}}{e\cosh(F_{n})-1}}\ } F 1 = 0.2 {\displaystyle F_{1}=0.2\,} F 2 = 0.2 − 1.022 sinh ( 0.2 ) − 0.2 − 0.008492 1.022 cosh ( 0.2 ) − 1 = 0.264143 {\displaystyle 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 {\displaystyle F_{3}=0.25603217\,} F 4 = 0.25587253 {\displaystyle F_{4}=0.25587253\,} tanh ( 0.255872 2 ) = 1.022 − 1 1.022 + 1 tan ( ϕ / 2 ) {\displaystyle \tanh({\frac {0.255872}{2}}\ )={\sqrt {{\frac {1.022-1}{1.022+1}}\ }}\tan(\phi /2)} ϕ = 1.76824 rad {\displaystyle \phi =1.76824{\mbox{ rad}}\,} r = ℓ 2 / k 1 + e cos ( ϕ ) = 37823.45092 km {\displaystyle r={\frac {\ell ^{2}/k}{1+e\cos(\phi )}}\ =37823.45092{\mbox{ km}}\,} e 2 = r s − r p r s + r p = 0.154604 {\displaystyle e_{2}={\frac {r_{s}-r_{p}}{r_{s}+r_{p}}}\ =0.154604\,} ℓ 2 = 2 k r s r p r s + r p = 27194.7733 km 2 s {\displaystyle \ell _{2}={\sqrt {{\frac {2kr_{s}r_{p}}{r_{s}+r_{p}}}\ }}=27194.7733{\dfrac {{\mbox{ km}}^{2}}{\mbox{s}}}} Δ v 1 = k r p ( 2 r s r s + r p − 1 + e 1 ) = − 0.587957 km/s {\displaystyle \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 = 2 π k ( r s + r p 2 ) 3 / 2 = 2 π 42828 ( 20425.987 + 14955.8095 2 ) 3 / 2 {\displaystyle 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 seconds {\displaystyle T_{2}=71439.893{\mbox{ seconds}}\,} T 2 2 = 35719.95 seconds {\displaystyle {\frac {T_{2}}{2}}=35719.95{\mbox{ seconds}}} Δ v 2 = k r s ( 1 − 2 r p r s + r p ) = 42828 20425.987 ( 1 − 2 ( 14955.8095 ) 20425.987 + 14955.8095 ) {\displaystyle \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}}\ }})} Δ v 2 = 0.11663 km/s {\displaystyle \Delta v_{2}=0.11663{\mbox{ km/s}}\,} T ( ϵ ) ≈ 2 π r s 3 / 2 k + 3 π r s 3 / 2 ϵ 2 k {\displaystyle T(\epsilon )\approx {\frac {2\pi r_{s}^{3/2}}{\sqrt {k}}}\ +{\frac {3\pi r_{s}^{3/2}\epsilon ^{2}}{\sqrt {k}}}\ } T ( ϵ ) − T ≤ 300 {\displaystyle T(\epsilon )-T\leq 300} 2 π 42828 ( 20425.987 1 − ϵ 2 ) 3 / 2 − 88632 ≤ 300 {\displaystyle {\frac {2\pi }{\sqrt {42828}}}\ ({\frac {20425.987}{1-\epsilon ^{2}}}\ )^{3/2}-88632\leq 300} ϵ ≤ 0.047436 {\displaystyle \epsilon \leq 0.047436} r 0 = r s {\displaystyle r_{0}=r_{s}\,} r ′ ( ϵ ) = − r s ( 1 + ϵ cos ( ϕ ) ) − 2 cos ( ϕ ) {\displaystyle r\prime (\epsilon )=-r_{s}(1+\epsilon \cos(\phi ))^{-2}\cos(\phi )} r ′ ( ϵ ) = − r s cos ( ϕ ) ( 1 + ϵ cos ( ϕ ) ) 2 {\displaystyle r\prime (\epsilon )={\frac {-r_{s}\cos(\phi )}{(1+\epsilon \cos(\phi ))^{2}}}\ } r ′ ( 0 ) = − r s cos ( ϕ ) ) {\displaystyle r\prime (0)=-r_{s}\cos(\phi ))} r ′ ′ ( ϵ ) = 2 r s cos 2 ( ϕ ) ( 1 + ϵ cos ( ϕ ) ) − 3 {\displaystyle r\prime \prime (\epsilon )=2r_{s}\cos ^{2}(\phi )(1+\epsilon \cos(\phi ))^{-3}} r ′ ′ ( ϵ ) = 2 r s cos 2 ( ϕ ) {\displaystyle r\prime \prime (\epsilon )=2r_{s}\cos ^{2}(\phi )} r ( ϵ ) ≈ r s − r s ϵ cos ( ϕ ) + r s ϵ 2 cos 2 ( ϕ ) {\displaystyle r(\epsilon )\approx r_{s}-r_{s}\epsilon \cos(\phi )+r_{s}\epsilon ^{2}\cos ^{2}(\phi )} 2 π ( r s ) 3 / 2 k + 3 π ( r s ) 3 / 2 ϵ 2 k ≤ 88932 {\displaystyle {\frac {2\pi (r_{s})^{3/2}}{\sqrt {k}}}\ +{\frac {3\pi (r_{s})^{3/2}\epsilon ^{2}}{\sqrt {k}}}\ \leq 88932} 3 π ( r s ) 3 / 2 ϵ 2 k ≤ 300 {\displaystyle {\frac {3\pi (r_{s})^{3/2}\epsilon ^{2}}{\sqrt {k}}}\ \leq 300} ϵ ≤ 0.0475 {\displaystyle \epsilon \leq 0.0475} ϕ = 2 arctan ( 1 + ϵ 1 − ϵ tan ( ψ / 2 ) ) {\displaystyle \phi =2\arctan({\sqrt {{\frac {1+\epsilon }{1-\epsilon }}\ }}\tan(\psi /2))} ϕ ( 0 ) = ψ {\displaystyle \phi (0)=\psi \,} ϕ ′ ( ϵ ) = 2 [ tan ( ψ / 2 ) 1 2 ( 1 + ϵ 1 − ϵ ) − 1 / 2 ( 1 − ϵ − ( 1 + ϵ ) ( − 1 ) ( 1 − ϵ ) 2 ) 1 + ( 1 + ϵ 1 − ϵ ) tan 2 ( ψ / 2 ) ] {\displaystyle \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)}}\ ]} ϕ ′ ′ ( ϵ ) = 2 tan ( ψ / 2 ) ( 1 − ϵ ) 3 / 2 ( 1 + ϵ ) 1 / 2 + ( 1 − ϵ ) 1 / 2 ( 1 + ϵ ) 3 / 2 tan 2 ( ψ / 2 ) {\displaystyle \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)}}\ } ϕ ′ ′ ( 0 ) = ( 2 sin ( ψ / 2 ) cos ( ψ / 2 ) ) ( cos 2 ( ψ / 2 ) + sin 2 ( ψ / 2 ) ) = sin ψ {\displaystyle \phi \prime \prime (0)=(2\sin(\psi /2)\cos(\psi /2))(\cos ^{2}(\psi /2)+\sin ^{2}(\psi /2))=\sin \psi } ϕ = ψ + ϵ sin ( ψ ) + ϵ 2 2 sin ( ψ ) + 0 ( ϵ 3 ) {\displaystyle \phi =\psi +\epsilon \sin(\psi )+{\frac {\epsilon ^{2}}{2}}\ \sin(\psi )+0(\epsilon ^{3})} F 0 ( t ) = k 2 t ℓ s 3 = ψ {\displaystyle F_{0}(t)={\frac {k^{2}t}{\ell _{s}^{3}}}\ =\psi } F 1 ( t ) = sin ( ψ ) = sin ( k 2 t ℓ s 3 ) {\displaystyle F_{1}(t)=\sin(\psi )=\sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )} F 2 ( t ) = sin ( ψ ) 2 = sin ( k 2 t ℓ s 3 ) / 2 {\displaystyle F_{2}(t)={\frac {\sin(\psi )}{2}}\ =\sin({\frac {k^{2}t}{\ell _{s}^{3}}}\ )/2} ϕ ( t , ϵ ) = k 2 t ℓ s 3 + ϵ sin ( k 2 t ℓ s 3 ) + ϵ 2 2 sin ( k 2 t ℓ s 3 ) {\displaystyle \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 ( ϵ ) = 2 π r s 3 / 2 k ( 1 + 3 2 ϵ 2 ) {\displaystyle T(\epsilon )={\frac {2\pi r_{s}^{3/2}}{\sqrt {k}}}\ (1+{\frac {3}{2}}\ \epsilon ^{2})} M = 2 π t T ( ϵ ) = 2 π t 2 π r s 3 / 2 k ( 1 + 3 2 ϵ 2 ) = t k 2 ℓ 3 ( 1 + 3 2 ϵ 2 ) − 1 {\displaystyle 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}} let k 2 ℓ 3 = a {\displaystyle {\mbox{let}}{\frac {k^{2}}{\ell ^{3}}}\ =a} ψ = M + ϵ sin ( M ) + ϵ 2 2 sin ( 2 M ) {\displaystyle \psi =M+\epsilon \sin(M)+{\frac {\epsilon ^{2}}{2}}\ \sin(2M)} ψ = t a ( 1 − 3 2 ϵ 2 ) + ϵ sin ( a t ( 1 − 3 2 ϵ 2 ) ) + ϵ 2 2 sin ( 2 a t ( 1 − 3 2 ϵ 2 ) ) {\displaystyle \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}))} ψ ′ ( ϵ ) = − 3 t a ϵ − 3 a t ϵ 2 cos ( a t ( 1 − 3 2 ϵ 2 ) ) + sin ( a t ( 1 − 3 2 ϵ 2 ) ) − 3 ϵ 3 a t cos ( 2 a t ( 1 − 3 2 ϵ 2 ) ) + ϵ sin ( a t ( 1 − 3 2 ϵ 2 ) ) {\displaystyle \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}))} ψ ′ ′ ( ϵ ) = − 3 t a + sin ( 2 a t ) {\displaystyle \psi \prime \prime (\epsilon )=-3ta+\sin(2at)} ψ ( ϵ ) = ψ ( 0 ) + ϵ ψ ′ ( 0 ) + ϵ 2 2 ψ ′ ′ ( 0 ) {\displaystyle \psi (\epsilon )=\psi (0)+\epsilon \psi \prime (0)+{\frac {\epsilon ^{2}}{2}}\ \psi \prime \prime (0)} ψ ( ϵ ) = t k 2 ℓ 3 + ϵ sin ( t k 2 ℓ 3 ) + ϵ 2 ( − 3 2 t k 2 ℓ 3 + 1 2 sin ( 2 t k 2 ℓ 3 ) {\displaystyle \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}}}\ )} ϕ ( t ) = ψ + ϵ sin ( ψ ) + ϵ 2 2 sin ( ψ ) {\displaystyle \phi (t)=\psi +\epsilon \sin(\psi )+{\frac {\epsilon ^{2}}{2}}\ \sin(\psi )} f 0 ( t ) = ψ = t k 2 ℓ 3 {\displaystyle f_{0}(t)=\psi ={\frac {tk^{2}}{\ell ^{3}}}\ } ϕ ( t ) = t k 2 ℓ 3 + ϵ sin ( t k 2 ℓ 3 ) + ϵ 2 2 sin ( t k 2 ℓ 3 ) {\displaystyle \phi (t)={\frac {tk^{2}}{\ell ^{3}}}\ +\epsilon \sin({\frac {tk^{2}}{\ell ^{3}}}\ )+{\frac {\epsilon ^{2}}{2}}\ \sin({\frac {tk^{2}}{\ell ^{3}}}\ )} ϵ 2 {\displaystyle \epsilon ^{2}\,}
I am a wiki mini-janitor. I clean up the messes YOU make.
Trypa <- still awesome Gugilymugily Grevlek DarkSerge
Final Fantasy VI for the ongoing drama. Something Awful because it sure is something awful har har har.
![{\displaystyle \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)}}\ ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12001b795a205ef4b27166c6ac5dbb457418fa1)
