PHYS 4110 – Dynamics of Space Vehicles
Chapter 9: Interplanetary Trajectories
Earth, Moon, Mars, and Beyond
Dr. Jinjun Shan, Professor of Space Engineering
Department of Earth and Space Science and Engineering
Room 255, Petrie Science and Engineering Building
Tel: 416-736 2100 ext. 33854
Email: [email protected]
Homepage: http://www.yorku.ca/jjshan
Outline
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Interplanetary Hohmann Transfers
Rendezvous Opportunities
Sphere of Influence
Method of Patched Conics
Planetary Departure
Planetary Rendezvous
Planetary Flyby
Prof. Jinjun Shan
Interplanetary Trajectories - 2
Interplanetary Hohmann Transfers
Prof. Jinjun Shan
Interplanetary Trajectories - 3
Interplanetary Hohmann Transfers
Prof. Jinjun Shan
Interplanetary Trajectories - 4
Example - 1
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Calculate the total delta-v required by a mission from the
earth to Mars and the flight time.
Prof. Jinjun Shan
Interplanetary Trajectories - 5
Interplanetary Hohmann Transfers
Hohmann Transfers from the earth to
other planets in the solar system
Earth
Mercury
Venus
Mars
Jupiter
Saturn
Uranus
Neptune
delta-V
17.14
5.202
5.593
14.44
15.735
15.939
15.707
105
146
259
998
(2.73)
2,222
(6.09)
5,862
(16.1)
11,171
(30.6)
Km/s
Flight time
days
Prof. Jinjun Shan
Interplanetary Trajectories - 6
Rendezvous Opportunities - 1
θ1 = θ10 + n1t
θ 2 = θ 20 + n 2 t
φ = θ 2 − θ1 = φ 0 + ( n 2 − n1 ) t
Tsyn
€
Prof. Jinjun Shan
2π
T1T2
=
=
n2 − n1 T1 − T2
Interplanetary Trajectories - 7
Example - 2
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Calculate the synodic periods of Mars and Venus relative to
the earth.
Prof. Jinjun Shan
Interplanetary Trajectories - 8
Rendezvous Opportunities - 2
Synodic periods in the Solar System, relative to Earth:
Sid. P. (a) Syn. P. (a) Syn. P. (d)
Mercury
0.241
0.317
115.9
Venus
0.615
1.599
583.9
Earth
1
—
—
Moon
0.0748 0.0809
29.5306
Mars
1.881
2.135
777.9
Jupiter
11.87
1.092
398.9
Saturn
29.45
1.035
378.1
Uranus
84.07
1.012
369.7
Neptune 164.9
1.006
367.5
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Interplanetary Trajectories - 9
Rendezvous Opportunities - 3
Prof. Jinjun Shan
Interplanetary Trajectories - 10
Example - 3
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Calculate initial and final phase for a earth-Mars mission.
Calculate the minimum waiting time for initiating a return trip
from Mars to the earth.
Prof. Jinjun Shan
Interplanetary Trajectories - 11
Sphere of Influence
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Interplanetary Trajectories - 12
Example - 4
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Calculate the radius of the earth’s SOI.
Calculate the radius of the moon’s SOI (for earth-moon
system).
Prof. Jinjun Shan
Interplanetary Trajectories - 13
Method of Patched Conics
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Interplanetary Trajectories - 14
Planetary Departure - 1
ΔvD = v∞ =
⎞
2 R2
− 1⎟⎟
⎜
R1 ⎝ R1 + R2
⎠
µsun ⎛⎜
h2 1
rp =
µ1 1 + e
e = 1+
rp v∞
v∞ =
µ
h
e2 − 1
2
h = rp v∞
µ1
2
v p = v∞ +
2 µ1
rp
vc =
2
2 µ1
+
rp
µ1
rp
Δv = v p − v c
⎛
⎜
1
⎜
β = cos−1 ⎜
2
rp v∞
⎜ 1 +
⎜
µ1
⎝
Prof. Jinjun Shan
⎞
⎟
⎟
⎟
⎟
⎟
⎠
Interplanetary Trajectories - 15
Planetary Departure - 2
Prof. Jinjun Shan
Interplanetary Trajectories - 16
Example - 5
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A spacecraft is launched on a mission to Mars starting from 300
km circular parking orbit. Calculate (a) the delta-v required; (b)
the location of perigee of the departure hyperbola; (c) the amount
of propellant required as a percentage of the spacecraft mass
before the delta-v burn, assuming a specific impulse of 300
seconds. [Example 8.4]
R1 = 1.496 × 108 km
R2 = 2.279 × 108 km
11
3
µSun = 1.327 × 10 km /s
2
µEarth = 3.986 × 105 km 3/s2
Prof. Jinjun Shan
Interplanetary Trajectories - 17
Planetary Rendezvous - 1
v ∞ = v 2 − v A (v )
v ∞ = v A (v ) − v 2
% (
−1 1
δ = 2sin ' *
& e)
v ∞2 +
vp =
h2
Δ=
µ2
vp
)
hyp
Δv = v p
rp =
)
hyp
€
2 µ2
+
rp
− vp
2 µ2 1 − e
v ∞ 2 1+ e
Δv min = v ∞
Prof. Jinjun Shan
= rp 1+
e 2 −1
2
µ2
2 µ2
rp
1
= v∞
€
e = 1+
rp v ∞ 2
)
vp
)
cap
2 µ2
rp v ∞ 2
µ2 (1+ e)
=
rp
cap
ra =
1−e
2
2 µ2
v ∞2
Δ = rp
2
1−e
Interplanetary Trajectories - 18
Planetary Rendezvous - 2
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Interplanetary Trajectories - 19
Planetary Rendezvous - 3
2 µ2
Δ = rp 1+
rp v ∞ 2
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Interplanetary Trajectories - 20
Example - 6
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After a Hohmann transfer from the earth, calculate the minimum
delta-v required to place a spacecraft in Mars orbit with a period
of seven hours. Also calculate the periapse radius, the aiming
radius and the angle between periapse and Mars’ velocity vector.
[Example 8.5]
Prof. Jinjun Shan
Interplanetary Trajectories - 21
Example - 7
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A spacecraft is launched on a mission to Mars starting from ISS
in a 300 km circular parking orbit, and is required to be placed in
Mars orbit with a period of 7 hours. Calculate (a) the total delta-v
required; (b) the location of perigee of the departure hyperbola;
(c) the periapse radius and the aiming radius of arrival hyperbola,
the angle between periapse and Mars’ velocity vector; (d) if the
spacecraft mass in circular parking orbit is 5,000 kg, what is the
mass of spacecraft in Mars orbit, assuming a specific impulse of
300 seconds; (e) the total delta-v required to land on Mars from a
250 km circular orbit in Mars’ equatorial plane.
µ Sun = 1.327 × 1011 km3 /s 2
µ Earth = 3.986 × 105 km3 /s 2
µ Mars = 4.2828 × 104 km3 /s 2
Prof. Jinjun Shan
rEarth = 6378 km
R1 = REarth = 1.496 × 108 km
rMars = 3396 km
R2 = RMars = 2.279 × 108 km
Interplanetary Trajectories - 22
Planetary Flyby - 1
A spacecraft which enters a planet’s sphere
of influence and does not impact the planet
or go into orbit around it will continue in its
hyperbolic trajectory through periapse and
exit the sphere of influence.
! (v) ! !
v1 = v + v∞1
! (v) ! !
v2 = v + v∞ 2
!
! (v) ! (v)
!
Δv ( v ) = v2 − v1 = Δv∞
Prof. Jinjun Shan
Interplanetary Trajectories - 23
Planetary Flyby - 2
! (v) ! !
! (v) ! !
!
! (v) ! (v)
!
v1 = v + v∞1
v2 = v + v∞ 2
Δv ( v ) = v2 − v1 = Δv∞
! (v)
(v)
(v)
v1 = v1 V uˆ V + v1 S uˆ S = v⊥1uˆ V − vr1uˆ S
!
(v)
(v)
v∞1 = [v∞1 ]V uˆ V + [v∞1 ]S uˆ S = v1 cos α1 − v uˆ V + v1 sin α1uˆ S
[ ]
2
[ ]
h = rp v∞ +
2µ
rp
(
e = 1+
)
rp v∞
2
µ
(v)
[
]
v
v
sin α1
−1
−1
∞1 S
1
φ1 = tan
= tan
φ2 = φ1 + δ
(v)
[v∞1 ]V
v1 cos α1 − v
!
v∞ 2 = [v∞1 ]V uˆ V + [v∞1 ]S uˆ S = v∞ cos φ2uˆ V + v∞ sin φ2uˆ S
! (v) ! !
(v)
(v)
v2 = v + v∞ 2 = [v2 ]V uˆ V + [v2 ]S uˆ S = (v + v∞ cos φ2 )uˆ V + v∞ sin φ2uˆ S
h2 = Rv⊥ 2
Prof. Jinjun Shan
R=
h2
2
µ sun
1
1 + e2 cosθ 2
vr 2 =
µ sun
h2
e2 sin θ 2
Interplanetary Trajectories - 24
Gravity Assist Maneuvers - 1
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Interplanetary Trajectories - 25
Gravity Assist Maneuvers - 2
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Interplanetary Trajectories - 26