Space Propulsion - IAP/TU Wien

Space Propulsion
Gravitational trajectories
Assumption: conservative 1/r 2 force field
F
= −
dU
dr
mμ
= −
2
r
V
2
2
μ
− = ε = const.
r
conservation of total energy; ε is specific total
energy;
V
2
⎛
= ⎜
⎝
dr
dt
2
⎞
⎟
⎠
⎛
+ ⎜r
⎝
2
dθ
⎞
⎟
dt ⎠
magnitude 2 of velocity in polar coordinates (r, θ)
μ
− +
r
1
2
⎛
⎜
⎝
dr
dt
2
⎞
⎟
⎠
+
2
1 ⎛ dθ
⎞
⎜r
⎟
2 ⎝ dt ⎠
= ε
2
2 2
h ⎛ dr ⎞ h 2μ
differential equ. of
⎜ ⎟ + − = ε
4
2
r ⎝ dθ
⎠ r r
trajectory
dr
dt
dr dθ
dr ⎛ h
= ⎜
dθ
dt dθ
⎝ r
=
2
⎞
⎟
⎠
dθ
V sinφ
= =
dt r
h
2
r
dθ
=
h / r
2
2μ
h
ε + −
r r
2
2
dr
subst.
r=1/u
dθ
= −
h
ε + 2μ.
u − h
2
u
2
du
θ + C
= −h
∫
du
2
ε + 2μ.
u − h u
2
general solution