Kepler轨道基本公式
Kepler 轨道基本公式：B121 4：00——6：00 1 运动微分方程 =-r
7 J2摄动效果
3πJ 2cos i
∆Ωrev =-
(a /R E ) 2(1-e 2) 2
μr 3
, ∆ωrev
r
3πJ 2(5cos2i -1) =- 2(a /R E ) 2(1-e 2) 2
8 hohmann转移初始相位角
⎡3/2⎤
2 三常量
H
=d (r ⨯v )=d (r ⨯r
)H =r ⨯v =const （
d t d t =r
⨯r +r ⨯r =r ⨯-μ
）
r 3
r =0
r
r =-μ
r 3(r r )E =v 22-μ
r
=const （r r =1d 2d t (r r )=1d 1d 22d t (v v )=2d t (v )）
-μμd ⎛μr 3(r
r )=-r
(rr )=d t ⎫3⎝r ⎪⎭H ⨯d v μ
d t =-r 3⎡⎣
(r ⨯v )⨯r ⎤⎦L =v ⨯H -μr
r
=const （=-μr 3
⎡⎣r 2v -(v r )r ⎤⎦
）
=-μ⎡r
2
⎢⎣
r v -d r d t r ⎤⎥⎦=-μd ⎛r ⎫d t ⎝r ⎪⎭L 2=μ2+2EH 2
3 轨道方程
L r =(v ⨯H ) r -μ
r
r r =(r ⨯v ) H -μr =H 2-μr
p =H
2
L
μ
, e =
μ
, p =a (1-e 2)
r =
p 1+e cos θ
E =-
μ
2a
L =v ⨯H -μr
r
e t
t =
L μ
4 速度分量
v u =
+e cos θ), v r =
sin θ, tan γ=
e sin θ
1+e cos θ
5时间历程
1+e cos θ=1-e 2θE
1-e cos E , tan 2=2M =E -e sin E , M =
t -t p ), T =26轨道摄动
a =-μ
d a 2a 2d E d E
2E ⇒d t =μd t , d t
=v f =v r f r +v u f u +v h f h
d a 2
⇒d t =(e sin θ)f r +(1+e cos θ)f u ⎤⎦θ⎛r 1+r 2⎫H =π⎢⎢1- ⎣⎝2r ⎪⎥
2⎭⎥⎦v ep =
v ea =
9 C-W
x -2Ωz =f x y +Ω2y =f y
z -3Ω2z +2Ωx
=f z x 4x
02z
0t =(
ω
-6z 0)sin(ωt ) -
ω
cos(ωt ) +(6ωz 2z
00-3x
0) t +(x 0+ω
)
y t ) +
y
0t =y 0cos(ωω
sin(ωt )
z 2x
0t ) +
z
02x
t =(
ω
-3z 0)cos(ωωsin(ωt ) +(4z 00-
ω
)
10 常用计算常数
μ=GM =3.986005⨯1014m 3/s2
恒星日 86164s 太阳日 86400s
地球半径 6378.14km 1年 365.2422天
11 影响球
R =(
m L m ) 2/5
D EL E
12 常用的导数关系
2
d M =n d t =⎛ r ⎫
r ⎫⎝a ⎪⎭d E =a ⎪⎭d f
d E =n ⎛ a ⎫⎝r ⎪⎭d t =⎛ a ⎫
⎝r ⎪⎭d M =r ⎫a ⎪⎭
d f
2
2
d f =a ⎫r ⎪⎭d t =a ⎫r ⎪⎭d M =a ⎫
r ⎪⎭
d E
2
d t =11⎛r ⎫r ⎫n d M =n ⎝a ⎪⎭d E =a ⎪⎭
d f