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Astrodynamics
(AERO0024)
6. Interplanetary Trajectories
Gaëtan Kerschen
Space Structures &
Systems Lab (S3L)
Course Outline
THEMATIC UNIT 1: ORBITAL DYNAMICS
Lecture 02: The Two-Body Problem
Lecture 03: The Orbit in Space and Time
Lecture 04: Non-Keplerian Motion
THEMATIC UNIT 2: ORBIT CONTROL
Lecture 05: Orbital Maneuvers
Lecture 06: Interplanetary Trajectories
2
Motivation
3
6. Interplanetary Trajectories
6.1 Patched conic method
6.2 Lambert’s problem
6.3 Gravity assist
4
6. Interplanetary Trajectories
6.1 Patched conic method
6.1.1 Problem statement
6.1.2 Sphere of influence
6.1.3 Planetary departure
6.1.4 Hohmann transfer
6.1.5 Planetary arrival
6.1.6 Sensitivity analysis and launch windows
6.1.7 Examples
5
?
Hint #1: design the Earth-Mars transfer using known concepts
Hint #2: division into simpler problems
Hint #3: patched conic method
6
What Transfer Orbit ? Constraints ?
Dep.
“∆
∆V1”
Arr.
Arr.
Dep.
“∆
∆V2”
Dep.
Arr.
Motion in the
heliocentric
reference frame
7
Planetary Departure ? Constraints ?
v∞
vEarth / Sun
∆V1
?
β
Motion in the
planetary
reference frame
8
Planetary Arrival ? Similar Reasoning
Transfer
ellipse
SOI
Arrival
hyperbola
SOI
Departure
hyperbola
9
Patched Conic Method
Three conics to patch:
1. Outbound hyperbola (departure)
2. The Hohmann transfer ellipse (interplanetary travel)
3. Inbound hyperbola (arrival)
6.1.1 Problem statement
10
Patched Conic Method
Approximate method that analyzes a mission as a
sequence of 2-body problems, with one body always being
the spacecraft.
If the spacecraft is close enough to one celestial body, the
gravitational forces due to other planets can be neglected.
The region inside of which the approximation is valid is
called the sphere of influence (SOI) of the celestial body. If
the spacecraft is not inside the SOI of a planet, it is
considered to be in orbit around the sun.
6.1.1 Problem statement
11
Patched Conic Method
Very useful for preliminary mission design (delta-v
requirements and flight times).
But actual mission design and execution employ the most
accurate possible numerical integration techniques.
6.1.1 Problem statement
12
Sphere of Influence (SOI) ?
Let’s assume that a spacecraft is within the Earth’s SOI if
the gravitational force due to Earth is larger than the
gravitational force due to the sun.
GmE msat GmS msat
>
2
2
rE , sat
rS , sat
rE , sat < 2.5 × 105 km
The moon lies outside the SOI and is in orbit about
the sun like an asteroid !
6.1.2 Sphere of influence
13
Sphere of Influence (SOI)
Third body:
sun or planet
Spacecraft
m2
mj
d
ρ
r
m1
O
Central body:
sun or planet
6.1.2 Sphere of influence
µ
 d ρ 
&&
r + 3 r = −Gm j  3 + 3 
r
ρ 
d
Disturbing
function (L04)
14
If the Spacecraft Orbits the Planet
&&
rpv +
G ( m p + mv )
rpv3
 rsv rsp 
rpv = −Gms  3 + 3 
r

r
sp 
 sv
p:planet
v: vehicle
s:sun
&&
rpv − A p = Ps
Primary
gravitational
acceleration due
to the planet
6.1.2 Sphere of influence
Perturbation
acceleration due
to the sun
15
If the Spacecraft Orbits the Sun
 rpv rsp 
G ( ms + mv )
&&
rsv +
rsv = −Gm p  3 + 3 
3
r

rsv
r
pv
sp


p:planet
v: vehicle
s:sun
&&
rsv − A s = Pp
Primary
gravitational
acceleration due
to the sun
6.1.2 Sphere of influence
Perturbation
acceleration due
to the planet
16
SOI: Correct Definition due to Laplace
It is the surface along which the magnitudes of the
acceleration satisfy:
Pp
Ps
=
As Ap
Measure of the planet’s
influence on the orbit of the
vehicle relative to the sun
Measure of the deviation of
the vehicle’s orbit from the
Keplerian orbit arising from
the planet acting by itself
2
5
rSOI
6.1.2 Sphere of influence
 mp 
≈
 rsp
 ms 
17
SOI: Correct Definition due to Laplace
If
Pp
Ps
the spacecraft is inside the SOI of the planet.
>
As Ap
The previous (incorrect) definition was
6.1.2 Sphere of influence
Ap
As
>1
18
SOI Radii
Planet
SOI Radius (km)
SOI radius
(body radii)
Mercury
1.13x105
45
Venus
6.17x105
100
OK ! Earth
9.24x105
145
Mars
5.74x105
170
Jupiter
4.83x107
677
Neptune
8.67x107
3886
6.1.2 Sphere of influence
19
Validity of the Patched Conic Method
The Earth’s SOI is 145 Earth radii.
This is extremely large compared to the size of the Earth:
The velocity relative to the planet on an escape hyperbola is
considered to be the hyperbolic excess velocity vector.
vSOI ≈ v∞
This is extremely small with respect to 1AU:
During the elliptic transfer, the spacecraft is considered to be
under the influence of the Sun’s gravity only. In other words, it
follows an unperturbed Keplerian orbit around the Sun.
6.1.2 Sphere of influence
20
Outbound Hyperbola
The spacecraft necessarily escapes using a hyperbolic
trajectory relative to the planet.
Hyperbolic excess
speed
Lecture 02:
v =v +v
2
2
∞
2
esc
When this velocity vector is added to the
planet’s heliocentric velocity, the result is
the spacecraft’s heliocentric velocity on the
interplanetary elliptic transfer orbit at the
SOI in the solar system.
≈v
2
SOI
+v
2
esc
Is vSOI the velocity on
the transfer orbit ?
6.1.3 Planetary departure
21
Magnitude of VSOI
The velocity vD of the spacecraft relative to the sun is
imposed by the Hohmann transfer (i.e., velocity on the
transfer orbit).
H. Curtis, Orbital Mechanics for
Engineering Students, Elsevier.
6.1.3 Planetary departure
22
Magnitude of VSOI
By subtracting the known value of the velocity v1 of the
planet relative to the sun, one obtains the hyperbolic
excess speed on the Earth escape hyperbola.
vSOI = vD − v1 =
Imposed
Known
6.1.3 Planetary departure
µ sun 

2 R2
− 1 ≈ v∞

R1  ( R1 + R2 ) 
Lecture05
23
Direction of VSOI
What should be the direction of vSOI ?
For a Hohmann
transfer, it should
be parallel to v1.
6.1.3 Planetary departure
24
Parking Orbit
A spacecraft is ordinary launched into an interplanetary
trajectory from a circular parking orbit. Its radius equals the
periapse radius rp of the departure hyperbola.
H. Curtis, Orbital Mechanics for
Engineering Students, Elsevier.
6.1.3 Planetary departure
25
∆V Magnitude and Location
2
Lecture 02:
h
rp =
µ (1 + e)
e = 1+
2
p ∞
rv
µ
known
known
v∞ =
µ
a
h2
1
a=
µ e2 − 1
h=
µ e2 − 1
v∞
h
µ
∆v = v p − vc = −
rp
rp
6.1.3 Planetary departure
h = rp
2µ
v +
rp
2
∞
1
β = cos
e
−1
26
Planetary Departure: Graphically
Departure to outer or inner planet ?
H. Curtis, Orbital Mechanics for Engineering Students, Elsevier.
6.1.3 Planetary departure
27
Circular, Coplanar Orbits for Most Planets
Planet
Inclination of the orbit
to the ecliptic plane
Eccentricity
Mercury
7.00º
0.206
Venus
3.39º
0.007
Earth
0.00º
0.017
Mars
1.85º
0.094
Jupiter
1.30º
0.049
Saturn
2.48º
0.056
Uranus
0.77º
0.046
Neptune
1.77º
0.011
Pluto
17.16º
0.244
6.1.4 Hohmann transfer
28
Governing Equations
vD − v1 =
v2 − v A =
µ sun 

2 R2
− 1


R1  ( R1 + R2 ) 
µ sun 
2 R1 
1 −
 v

R2 
( R1 + R2 )  2
vD
v1
vA
Signs ?
v2 - vA, vD - v1 >0 for transfer to an outer planet
v2 - vA, vD - v1 <0 for transfer to an inner planet
6.1.4 Hohmann transfer
29
Schematically
Transfer to outer planet
Transfer to inner planet
6.1.4 Hohmann transfer
30
Arrival at an Outer Planet
For an outer planet, the spacecraft’s heliocentric
approach velocity vA is smaller in magnitude than that of
the planet v2.
v2 + v∞ = v A
v2 > v A
and v∞ have
opposite signs.
v2
6.1.5 Planetary arrival
31
Arrival at an Outer Planet
The spacecraft
crosses the
forward portion
of the SOI
6.1.5 Planetary arrival
32
Enter into an Elliptic Orbit
If the intent is to go into orbit around the planet, then ∆
must be chosen so that the ∆v burn at periapse will
occur at the correct altitude above the planet.
2µ
∆ = rp 1 + 2
rp v∞
∆v = v p ,hyp − v p ,capture
6.1.5 Planetary arrival
h
µ (1 + e)
2µ
µ (1 + e)
2
= −
= v∞ +
−
rp
rp
rp
rp
33
Planetary Flyby
Otherwise, the specacraft will simply continue past
periapse on a flyby trajectory exiting the SOI with the
same relative speed v∞ it entered but with the velocity
vector rotated through the turn angle δ.
e = 1+
6.1.5 Planetary arrival
rp v∞2
µ
1
δ = 2sin
e
−1
34
Sensitivity Analysis: Departure
The maneuver occurs well within the SOI, which is just a
point on the scale of the solar system.
One may therefore ask what effects small errors in position
and velocity (rp and vp) at the maneuver point have on the
trajectory (target radius R2 of the heliocentric Hohmann
transfer ellipse).
2µ1

v∞ +

rp δ v p
δ rp
δ R2
µ
2
1

=
+
2
R1vD  vD v∞ rp rp
R2
vD
vp
1−
2µ sun 
6.1.6 Sensitivity analysis and launch windows






35
Sensitivity Analysis: Earth-Mars, 300km Orbit
µ sun = 1.327 ×10 km / s , µ1 = 398600 km / s
11
3
2
3
2
R1 = 149.6 × 106 km, R2 = 227.9 ×106 km, rp = 6678 km
vD = 32.73 km / s, v∞ = 2.943 km / s
δ R2
R2
= 3.127
δ rp
rp
+ 6.708
δ vp
vp
A 0.01% variation in the burnout speed vp changes the
target radius by 0.067% or 153000 km.
A 0.01% variation in burnout radius rp (670 m !) produces
an error over 70000 km.
6.1.6 Sensitivity analysis and launch windows
36
Sensitivity Analysis: Launch Errors
Ariane V
Trajectory correction maneuvers are clearly mandatory.
6.1.6 Sensitivity analysis and launch windows
37
Sensitivity Analysis: Arrival
The heliocentric velocity of Mars in its orbit is roughly
24km/s.
If an orbit injection were planned to occur at a 500 km
periapsis height, a spacecraft arriving even 10s late at
Mars would likely enter the atmosphere.
6.1.6 Sensitivity analysis and launch windows
38
Cassini-Huygens
Existence of Launch Windows
Phasing maneuvers are not practical due to the large
periods of the heliocentric orbits.
The planet should arrive at the apse line of the transfer
ellipse at the same time the spacecraft does.
6.1.6 Sensitivity analysis and launch windows
40
Rendez-vous Opportunities
θ1 = θ10 + n1t
θ 2 = θ 20 + n2t
φ = θ 2 − θ1
φ0 − 2π = φ0 + ( n2 − n1 ) Tsyn
φ = φ0 + ( n2 − n1 ) t
Tsyn
2π
=
n1 − n2
Tsyn
T1T2
=
T1 − T2
Synodic period
6.1.6 Sensitivity analysis and launch windows
41
Transfer Time
Lecture 02
π
t12 =
µ sun
 R1 + R2 


 2 
3/ 2
φ0 = π − n2t12
6.1.6 Sensitivity analysis and launch windows
42
Earth-Mars Example
Tsyn
365.26 × 687.99
=
= 777.9 days
365.26 − 687.99
It takes 2.13 years for a given configuration of Mars
relative to the Earth to occur again.
t12 = 2.2362 ×10 s = 258.8 days
7
φ0 = 44
o
The total time for a manned Mars mission is
258.8 + 453.8 + 258.8 = 971.4 days = 2.66 years
6.1.6 Sensitivity analysis and launch windows
43
Earth-Jupiter Example: Hohmann
Galileo’s original mission was
designed to use a direct
Hohmann transfer, but
following the loss of
Challenger Galileo's intended
Centaur booster rocket was
no longer allowed to fly on
Shuttles. Using a lesspowerful solid booster rocket
instead, Galileo used gravity
assists instead.
6.1.7 Examples
44
Earth-Jupiter Example: Hohmann
Velocity when leaving Earth's SOI:
vD − v1 = v =
E
∞
µ sun 

2 R2
− 1 = 8.792km/s

R1  ( R1 + R2 ) 
Velocity relative to Jupiter at Jupiter's SOI:
v2 − vA = v =
J
∞
µ sun 
2 R1 
1 −
 = 5.643km/s

R2 
( R1 + R2 ) 
Transfer time: 2.732 years
6.1.7 Examples
45
Earth-Jupiter Example: Departure
Velocity on a circular parking orbit (300km):
vc =
µE
RE + h
= 7.726km/s
2µ
∆v = v +
− 7.726 km/s = 6.298 km/s
rp
2
∞
e = 1+
rp v∞2
µ
6.1.7 Examples
= 2.295
46
Earth-Jupiter Example: Arrival
Final orbit is circular with radius=6R J
2µ
µ (1 + e)
∆v = v +
−
= 24.95 − 17.18 = 7.77km/s
rp
rp
2
∞
e=1.108
6.1.7 Examples
47
Hohmann Transfer: Other Planets
Planet
v∞ departure
(km/s)
Transfer time
(days)
Mercury
7.5
105
Venus
2.5
146
Mars
2.9
259
Jupiter
8.8
998
Saturn
10.3
2222
Pluto
11.8
16482
Assumption of circular, co-planar orbits and
tangential burns
6.1.7 Examples
48
Venus Express: A Hohmann-Like Transfer
C3 = 7.8 km2/s2
Time: 154 days
Real data
Why ?
C3 = 6.25 km2/s2
Hohmann
Time: 146 days
6.1.7 Examples
49
6.1.7 Examples
6. Interplanetary Trajectories
6.2 Lambert’s problem
51
Motivation
Section 6.1 discussed Hohmann interplanetary transfers,
which are optimal with respect to fuel consumption.
Why should we consider nontangential burns (i.e., nonHohmann transfer) ?
L05
6.2 Lambert’s problem
52
Non-Hohmann Trajectories
Solution using Lambert’s
theorem (Lecture 05):
If two position vectors and the
time of flight are known, then the
orbit can be fully determined.
6.2 Lambert’s problem
53
Venus Express Example
6.2 Lambert’s problem
54
Porkchop Plot: Visual Design Tool
C3 contours
C3 contours
Arrival date
Departure date
In porkchop plots, orbits are considered to be non-coplanar and elliptic.
6.2 Lambert’s problem
55
6.2 Lambert’s problem
Type II transfer for
cargo: the
spacecraft travels
more than a 180°true
anomaly
Type I transfer for
piloted: the
spacecraft travels less
than a 180°true
anomaly
6.2 Lambert’s problem
6.2 Lambert’s problem
6. Interplanetary Trajectories
6.3 Gravity assist
6.3.1 Basic principle
6.3.2 Real-life examples
61
∆V Budget: Earth Departure
SOYUZ
Planet
C3
(km2/s2)
Mercury
[56.25]
Venus
6.25
Mars
8.41
Jupiter
77.44
Saturn
106.09
Pluto
[139.24]
Assumption of circular,
co-planar orbits and
tangential burns
62
∆V Budget: Arrival at the Planet
A spacecraft traveling to an inner planet is accelerated
by the Sun's gravity to a speed notably greater than the
orbital speed of that destination planet.
If the spacecraft is to be inserted into orbit about that
inner planet, then there must be a mechanism to slow
the spacecraft.
Likewise, a spacecraft traveling to an outer planet is
decelerated by the Sun's gravity to a speed far less than
the orbital speed of that outer planet. Thus there must
be a mechanism to accelerate the spacecraft.
6.3.1 Basic principle
63
Prohibitive ∆V Budget ? Use Gravity Assist
Also known as planetary flyby trajectory, slingshot
maneuver and swingby trajectory.
Useful in interplanetary missions to obtain a velocity
change without expending propellant.
This free velocity change is provided by the gravitational
field of the flyby planet and can be used to lower the ∆v
cost of a mission.
6.3.1 Basic principle
64
What Do We Gain ?
Spacecraft
outbound velocity
SOI
Spacecraft
inbound velocity
Vout = Vin
6.3.1 Basic principle
65
Gravity Assist in the Heliocentric Frame
Resultant Vout
SOI
Resultant Vin
Planet’s sun
relative velocity
∆v = v ∞ ,out − v ∞ ,in
6.3.1 Basic principle
66
A Gravity Assist Looks Like an Elastic Collision
Inertial frame
Frame attached to
the train
Frame attached to
the train
Inertial frame
6.3.1 Basic principle
67
Leading-Side Planetary Flyby
A leading-side flyby results in a
decrease in the spacecraft’s
heliocentric speed (e.g., Mariner
10 and Messenger).
6.3.1 Basic principle
68
Trailing-Side Planetary Flyby
A trailing-side flyby results in an
increase in the spacecraft’s
heliocentric speed (e.g.,
Voyager and Cassini-Huygens).
6.3.1 Basic principle
69
What Are the Limitations ?
Launch windows may be rare (e.g., Voyager).
Presence of an atmosphere (the closer the spacecraft can
get, the more boost it gets).
Encounter different planets with different (possibly harsh)
environments.
What about flight time ?
6.3.1 Basic principle
70
V
V
E
J
G
A
See Lecture 1
71
Messenger
6.3.2 Real-life examples
72
6.3.2 Real-life examples
Hohmann Transfer vs. Gravity Assist
Gravity assist
Planet
C3
(km2/s2)
Transfer time
(days)
Real
mission
C3
Transfer time
(km2/s2)
(days)
Mercury
[56.25]
105
Messenger
16.4
2400
Saturn
106.09
2222
Cassini
Huygens
16.6
2500
Remark: the comparison between the transfer times is difficult,
because it depends on the target orbit. The transfer time for gravity
assist mission is the time elapsed between departure at the Earth
and first arrival at the planet.
6.3.2 Real-life examples
74
New Horizons: Mission to Pluto
6.3.2 Real-life examples
75
Direct Transfer Was Also Envisioned
6.3.2 Real-life examples
76
Launch Window Combining GA and Direct
6.3.2 Real-life examples
77
6.3.2 Real-life examples
6. Interplanetary Trajectories
6.1 PATCHED CONIC METHOD
6.1.1 Problem statement
6.1.2 Sphere of influence
6.1.3 Planetary departure
6.1.4 Hohmann transfer
6.1.5 Planetary arrival
6.1.6 Sensitivity analysis and launch windows
6.1.7 Examples
6.2 LAMBERT’S PROBLEM
6.3 GRAVITY ASSIST
6.3.1 Basic principle
6.3.2 Real-life examples
79
Even More Complex Trajectories…
80
Astrodynamics
(AERO0024)
6. Interplanetary Trajectories
Gaëtan Kerschen
Space Structures &
Systems Lab (S3L)