F ( u ) = 0 {\displaystyle \mathbf {F} (\mathbf {u} )=0}
[ y − x 1 ] = [ 1 0 b 0 1 − a 0 0 1 ] [ y ′ − x ′ 1 ] {\displaystyle \left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}1&0&b\\0&1&-a\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]}
[ y − x 1 ] = [ c − s 0 s c 0 0 0 1 ] [ y ′ − x ′ 1 ] {\displaystyle \left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}c&-s&0\\s&c&0\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]}
[ E ] = [ c − s b s c − a 0 0 1 ] {\displaystyle [E]=\left[{\begin{array}{ccc}c&-s&b\\s&c&-a\\0&0&1\end{array}}\right]}
J = [ y 1 y 2 y 3 − x 1 − x 2 − x 3 1 1 1 ] {\displaystyle J=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]}
J − 1 = 1 d e t J [ a 23 s ^ 23 T a 31 s ^ 31 T a 12 s ^ 12 T ] {\displaystyle J^{-1}={\frac {1}{detJ}}\left[{\begin{array}{c}a_{23}{\hat {s}}_{23}^{T}\\a_{31}{\hat {s}}_{31}^{T}\\a_{12}{\hat {s}}_{12}^{T}\end{array}}\right]}
d e t J = a 23 s ^ 23 T S ^ 1 = a 31 s ^ 31 T S ^ 2 = a 12 s ^ 12 T S ^ 3 {\displaystyle detJ=a_{23}{\hat {s}}_{23}^{T}{\hat {S}}_{1}=a_{31}{\hat {s}}_{31}^{T}{\hat {S}}_{2}=a_{12}{\hat {s}}_{12}^{T}{\hat {S}}_{3}}
A = [ y 1 0 y 3 − x 1 1 − x 3 1 0 1 ] {\displaystyle A=\left[{\begin{array}{ccc}y_{1}&0&y_{3}\\-x_{1}&1&-x_{3}\\1&0&1\end{array}}\right]}
B = [ y 1 y 2 y 3 − x 1 − x 2 − x 3 1 1 1 ] {\displaystyle B=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]}
C = [ 0 − 1 0 1 0 1 0 0 0 ] {\displaystyle C=\left[{\begin{array}{ccc}0&-1&0\\1&0&1\\0&0&0\end{array}}\right]}
D = [ 1 0 cos γ 0 − x 2 sin γ 0 1 0 ] {\displaystyle D=\left[{\begin{array}{ccc}1&0&\cos {\gamma }\\0&-x_{2}&\sin {\gamma }\\0&1&0\end{array}}\right]}
T ^ = ω [ y Q − x Q 1 ] = [ v O x v O y ω ] {\displaystyle {\hat {T}}=\omega \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}v_{Ox}\\v_{Oy}\\\omega \end{array}}\right]}
δ D ^ = δ ϕ [ y Q − x Q 1 ] = [ δ r O x δ r O y δ ϕ ] {\displaystyle \delta {\hat {D}}=\delta \phi \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}\delta r_{Ox}\\\delta r_{Oy}\\\delta \phi \end{array}}\right]}
T ^ = J γ ^ {\displaystyle {\hat {\mathbf {T} }}=\mathbf {J} {\hat {\mathbf {\gamma } }}}
δ D ^ = J δ θ ^ {\displaystyle \delta {\hat {\mathbf {D} }}=\mathbf {J} \delta {\hat {\mathbf {\theta } }}}
Δ t = i n i t i a l _ m a s s m e ˙ ( 1 − e − Δ v i s p ⋅ g 0 ) {\displaystyle \Delta t={\frac {initial\_mass}{\dot {m_{e}}}}\left(1-e^{\frac {-\Delta v}{isp\cdot g_{0}}}\right)}
C ( z ) = { 1 − c o s z z ( z > 0 ) c o s h − z − 1 − z ( z < 0 ) 1 2 ( z = 0 ) {\displaystyle C(z)={\begin{cases}{\frac {1-cos{\sqrt {z}}}{z}}&(z>0)\\{\frac {cosh{\sqrt {-z}}-1}{-z}}&(z<0)\\{\frac {1}{2}}&(z=0)\end{cases}}} S ( z ) = { z − s i n z ( z ) 3 ( z > 0 ) s i n h − z − − z ( − z ) 3 ( z < 0 ) 1 6 ( z = 0 ) {\displaystyle S(z)={\begin{cases}{\frac {{\sqrt {z}}-sin{\sqrt {z}}}{({\sqrt {z}})^{3}}}&(z>0)\\{\frac {sinh{\sqrt {-z}}-{\sqrt {-z}}}{({\sqrt {-z}})^{3}}}&(z<0)\\{\frac {1}{6}}&(z=0)\end{cases}}}
P S O M = P S O E − E x s i n ( P S O E ) + E y c o s ( P S O E ) {\displaystyle PSO_{M}=PSO_{E}-E_{x}sin(PSO_{E})+E_{y}cos(PSO_{E})}
ρ = ρ 0 e − h ρ ( Z − Z 0 ) {\displaystyle \rho =\rho _{0}e^{-h_{\rho }(Z-Z_{0})}}
A t t = t 0 { X = 0 Y = 0 Z = z 0 X ˙ = 3 2 ω z 0 Y ˙ = 0 Z ˙ = 0 {\displaystyle At\ t=t0{\begin{cases}X=0\\Y=0\\Z=z_{0}\\{\dot {X}}={\frac {3}{2}}\omega z_{0}\\{\dot {Y}}=0\\{\dot {Z}}=0\\\end{cases}}}
M ( Δ t ) = [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 3 2 μ a − 5 2 Δ t 0 0 0 0 0 ] C O V ( t + Δ t ) = M ⋅ C O V ( t ) ⋅ M t {\displaystyle {\begin{matrix}M(\Delta t)={\begin{bmatrix}0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\0&0&0&0&0&0\\-{\frac {3}{2}}{\sqrt {\mu }}a^{\frac {-5}{2}}\Delta t&0&0&0&0&0\\\end{bmatrix}}\\\\COV(t+\Delta t)=M\cdot COV(t)\cdot M^{t}\end{matrix}}}
C O R = R R ∗ (Cholesky decomposition) X p × n ∼ N ( 0 , 1 ) d r a w n V a l u e s = R ⋅ X ⋅ s d + m e a n V a l u e s {\displaystyle {\begin{array}{lcl}COR=RR^{*}\ {\mbox{(Cholesky decomposition)}}\\{\underset {p\times n}{X}}\sim N(0,1)\\drawnValues=R\cdot X\cdot sd+meanValues\\\end{array}}}
Δ m = m ( 1 − e − Δ v g 0 i s p ) {\displaystyle \Delta m=m(1-e^{\frac {-\Delta v}{g_{0}isp}})}
F = g 0 i s p Δ m Δ t {\displaystyle F=g_{0}isp{\frac {\Delta m}{\Delta t}}}
s m r a = t s i d − 2 π ( c j d − i n t ( c j d ) ) + π (1) {\displaystyle smra=tsid-2\pi (cjd-int(cjd))+\pi \qquad {\mbox{(1)}}}
l o n = r a − t s i d (2) {\displaystyle lon=ra-tsid\qquad {\mbox{(2)}}}
m h a = r a − s m r a (3) {\displaystyle mha=ra-smra\qquad {\mbox{(3)}}}
t h a = r a − s m r a (4) {\displaystyle tha=ra-smra\qquad {\mbox{(4)}}}
h l = 12 π ( h + π ) m o d u l o 24 (5) {\displaystyle hl={\frac {12}{\pi }}(h+\pi )\quad modulo\quad 24\qquad {\mbox{(5)}}}
d ( Ω ) d t = − 3 n c o s ( i ) R 2 J 2 2 a 2 ( 1 − e 2 ) 2 (1) {\displaystyle {\frac {d(\Omega )}{dt}}={\frac {-3ncos(i)R^{2}J2}{2a^{2}(1-e^{2})^{2}}}\qquad {\mbox{(1)}}}
e x = e c o s ( ω ) e y = e s i n ( ω ) {\displaystyle e_{x}=ecos(\omega )\qquad e_{y}=esin(\omega )}
e x = e c o s ( ω + Ω ) e y = e s i n ( ω + Ω ) {\displaystyle e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )}
i x = 2 s i n ( i 2 ) c o s ( Ω ) i y = 2 s i n ( i 2 ) s i n ( Ω ) {\displaystyle i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )}
keplerian parameters: ( a , e , i , ω , Ω , M ) a : semi major axis e : excentricity i : inclination ω : periapsis argument Ω : right ascension of ascending node M : mean anomaly {\displaystyle {\begin{array}{lcl}{\mbox{keplerian parameters:}}\ (a,e,i,\omega ,\Omega ,M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ i:\ {\mbox{inclination}}\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}}
circular adapted parameters: ( a , e x , e y , i , Ω , ω + M ) a : semi major axis e : excentricity e x = e c o s ( ω ) e y = e s i n ( ω ) i : inclination ω : periapsis argument Ω : right ascension of ascending node M : mean anomaly {\displaystyle {\begin{array}{lcl}{\mbox{circular adapted parameters:}}\ (a,e_{x},e_{y},i,\Omega ,\omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega )\qquad e_{y}=esin(\omega )\\\ i:\ {\mbox{inclination}}\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}}
circular equatorial adapted parameters: ( a , e x , e y , i x , i y , ω + Ω + M ) a : semi major axis e : excentricity e x = e c o s ( ω + Ω ) e y = e s i n ( ω + Ω ) i : inclination i x = 2 s i n ( i 2 ) c o s ( Ω ) i y = 2 s i n ( i 2 ) s i n ( Ω ) ω : periapsis argument Ω : right ascension of ascending node M : mean anomaly {\displaystyle {\begin{array}{lcl}{\mbox{circular equatorial adapted parameters:}}\ (a,e_{x},e_{y},i_{x},i_{y},\omega +\Omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )\\\ i:\ {\mbox{inclination}}\\\ i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}}
keplerian parameters: ( a , e , i , ω , Ω , M ) circular equatorial adapted parameters: ( a , e x , e y , i x , i y , ω + Ω + M ) a : semi major axis e : excentricity e x = e c o s ( ω + Ω ) e y = e s i n ( ω + Ω ) i : inclination i x = 2 s i n ( i 2 ) c o s ( Ω ) i y = 2 s i n ( i 2 ) s i n ( Ω ) ω : periapsis argument Ω : right ascension of ascending node M : mean anomaly {\displaystyle {\begin{array}{lcl}{\mbox{keplerian parameters:}}\ (a,e,i,\omega ,\Omega ,M)\\{\mbox{circular equatorial adapted parameters:}}\ (a,e_{x},e_{y},i_{x},i_{y},\omega +\Omega +M)\\\ a:\ {\mbox{semi major axis}}\\\ e:\ {\mbox{excentricity}}\\\ e_{x}=ecos(\omega +\Omega )\qquad e_{y}=esin(\omega +\Omega )\\\ i:\ {\mbox{inclination}}\\\ i_{x}=2sin({\frac {i}{2}})cos(\Omega )\qquad i_{y}=2sin({\frac {i}{2}})sin(\Omega )\\\ \omega :\ {\mbox{periapsis argument}}\\\ \Omega :\ {\mbox{right ascension of ascending node}}\\\ M:\ {\mbox{mean anomaly}}\\\end{array}}}
M v → R = v → S {\displaystyle M{\vec {v}}_{R}={\vec {v}}_{S}}
d r p o m = − 3 n ⋅ j 2 ⋅ e r 2 ( 1 − 5 c o s ( i ) c o s ( i ) ) 4 ( 1 − e 2 ) 2 a 2 d r g o m = − 3 n ⋅ j 2 ⋅ e r 2 c o s ( i ) 2 ( 1 − e 2 ) 2 a 2 d r M = n + 3 n ⋅ j 2 ⋅ e r 2 ( 3 c o s ( i ) c o s ( i ) − 1 ) 4 ( 1 − e 2 ) 1.5 a 2 d p o m ( 1 , : ) = d ( d r p o m ) d ( a ) = − 3.5 ⋅ d r p o m a d g o m ( 1 , : ) = d ( d r g o m ) d ( a ) = − 3.5 ⋅ d r g o m a d M ( 1 , : ) = d ( d r M ) d ( a ) = 2 n − 3.5 ⋅ d r M a d p o m ( 2 , : ) = d ( d r p o m ) d ( e ) = 4 e ⋅ d r p o m 1 − e 2 d g o m ( 2 , : ) = d ( d r g o m ) d ( e ) = 4 e ⋅ d r g o m 1 − e 2 d M ( 2 , : ) = d ( d r M ) d ( e ) = 3 e ( d r M − n ) 1 − e 2 d p o m ( 3 , : ) = d ( d r p o m ) d ( i ) = 5 K s i n ( 2 i ) d g o m ( 3 , : ) = d ( d r g o m ) d ( i ) = − 2 K s i n ( i ) d M ( 3 , : ) = d ( d r M ) d ( i ) = 3 K s i n ( 2 i ) ( 1 − e 2 ) 0.5 where K = − 3 n ⋅ j 2 ⋅ e r 2 4 ( 1 − e 2 ) 2 a 2 {\displaystyle {\begin{array}{lcl}\ drpom={\frac {-3n\cdot j2\cdot er^{2}(1-5cos(i)cos(i))}{4(1-e^{2})^{2}a^{2}}}\\\ \\\ drgom={\frac {-3n\cdot j2\cdot er^{2}cos(i)}{2(1-e^{2})^{2}a^{2}}}\\\ \\\ drM=n+{\frac {3n\cdot j2\cdot er^{2}(3cos(i)cos(i)-1)}{4(1-e^{2})^{1.5}a^{2}}}\\\ \\\ dpom(1,:)={\frac {d(drpom)}{d(a)}}={\frac {-3.5\cdot drpom}{a}}\\\ \\\ dgom(1,:)={\frac {d(drgom)}{d(a)}}={\frac {-3.5\cdot drgom}{a}}\\\ \\\ dM(1,:)={\frac {d(drM)}{d(a)}}={\frac {2n-3.5\cdot drM}{a}}\\\ \\\ dpom(2,:)={\frac {d(drpom)}{d(e)}}={\frac {4e\cdot drpom}{1-e^{2}}}\\\ \\\ dgom(2,:)={\frac {d(drgom)}{d(e)}}={\frac {4e\cdot drgom}{1-e^{2}}}\\\ \\\ dM(2,:)={\frac {d(drM)}{d(e)}}={\frac {3e(drM-n)}{1-e^{2}}}\\\ \\\ dpom(3,:)={\frac {d(drpom)}{d(i)}}=5Ksin(2i)\\\ \\\ dgom(3,:)={\frac {d(drgom)}{d(i)}}=-2Ksin(i)\\\ \\\ dM(3,:)={\frac {d(drM)}{d(i)}}=3Ksin(2i)(1-e^{2})^{0.5}\\\ \\\ {\mbox{where }}K={\frac {-3n\cdot j2\cdot er^{2}}{4(1-e^{2})^{2}a^{2}}}\\\end{array}}}
g t = T k , n ⋅ 2 π T s i − d g o m where T k , n is the keplerian period or the nodal period {\displaystyle {\begin{array}{lcl}\ gt=T_{k,n}\cdot {\frac {2\pi }{T_{si}-dgom}}\\\ \\\ {\mbox{where }}T_{k,n}{\mbox{ is the keplerian period or the nodal period}}\\\end{array}}}
![{\displaystyle \left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}1&0&b\\0&1&-a\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28a68d4eec21bc6bf6925ec04cac265f09013699)
![{\displaystyle \left[{\begin{array}{c}y\\-x\\1\end{array}}\right]=\left[{\begin{array}{ccc}c&-s&0\\s&c&0\\0&0&1\end{array}}\right]\left[{\begin{array}{c}y'\\-x'\\1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ef1563a1a5c3b5262a27aea24501067698d3255)
![{\displaystyle [E]=\left[{\begin{array}{ccc}c&-s&b\\s&c&-a\\0&0&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b978d53cad8bd4c9a2a0aa40eb2482a3d3002359)
![{\displaystyle J=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35951f50c33b1d5c4ced0d5b90c28d7357b954e2)
![{\displaystyle J^{-1}={\frac {1}{detJ}}\left[{\begin{array}{c}a_{23}{\hat {s}}_{23}^{T}\\a_{31}{\hat {s}}_{31}^{T}\\a_{12}{\hat {s}}_{12}^{T}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6050c4ffecbe35df30c4568477ea6563b533e01d)
![{\displaystyle A=\left[{\begin{array}{ccc}y_{1}&0&y_{3}\\-x_{1}&1&-x_{3}\\1&0&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/815453b0f08372acd49b37603745ff118e5ca5a0)
![{\displaystyle B=\left[{\begin{array}{ccc}y_{1}&y_{2}&y_{3}\\-x_{1}&-x_{2}&-x_{3}\\1&1&1\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90a024d72cd5e36c646d45768edb157dee25333f)
![{\displaystyle C=\left[{\begin{array}{ccc}0&-1&0\\1&0&1\\0&0&0\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e51f4e6f8d3e7a118df99052343e4d802ee5f41)
![{\displaystyle D=\left[{\begin{array}{ccc}1&0&\cos {\gamma }\\0&-x_{2}&\sin {\gamma }\\0&1&0\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde3b401267582760a7a84f5a84f64afdd3d92c5)
![{\displaystyle {\hat {T}}=\omega \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}v_{Ox}\\v_{Oy}\\\omega \end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d251b13396fa0aea271896c99f6ed00c3a0f3254)
![{\displaystyle \delta {\hat {D}}=\delta \phi \left[{\begin{array}{c}y_{Q}\\-x_{Q}\\1\end{array}}\right]=\left[{\begin{array}{c}\delta r_{Ox}\\\delta r_{Oy}\\\delta \phi \end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/859010c87785ad2f5a37b688b63f204e425d1d25)
