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Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
The Orbit Design Process
Establish orbit types
Determine orbit related requirements
Assess launch options
Create ΔV budget
Perform orbit design trades
Determine orbit related requirements
Temperature gradient
Absolute temperature
Straylight reduction
Max eclipse time
Communication requirements ( volume, timeliness )
Sun-Spacecraft-Earth angle ( >5deg for communication )
Scan strategy
Attitude disturbance reduction
Radiation (total dose over mission time)
…
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Equation of motion:
Energy:
Orbit angular momentum:
Two-Body System
3· 0r r
r
m- =
2
2
v
r
mx = -
h r v= ´
Conic Sections
Equation of a conic section (trajectory equation):
Slicing the cone with a plane:e>1 Hyperbola0<e<1 Ellipsee=0 Circlee=1 Parabola
1 cos( )
pr
e n=
+
Ellipse
Focus F and F‘
Semimajor axis a
Semiminor axis b
Semiparameter p
Eccentricity e
True anomaly n
Apoapsis rmax = ra
Earth = Apogee; Sun = Aphelion; Moon = Aposelen
Periapsis rmin = rp
Earth = Perigee; Sun = Perihelion; Moon = Periselen
Ellipse
2PF PF a¢ + =
2 2 2 2( ) 1b ae a b a e+ = = -
(1 )1 cos(0) 1
(1 )1 cos( ) 1
p
a
p pr a e
e ep p
r a ee ep
= - = =+ ⋅ +
+ = ==+ ⋅ -
2 2(1 ) /p a e b a= - =
2a pr r a+ =
a p
a p
r re
r r
-=
+
Hyperbola
1 cos( )
1cos( ) lim
10
1
r
pe
rp
er e
e
e
n
n¥ ¥
+ =
æ ö÷ç= - ÷ç ÷ç ÷è ø
= -
= -
n¥
1arccos( )
en¥ = -
Two-Body Motion
Orbital period:
Orbital energy:
Vis-Viva equation:
Eccentric anomaly E:
Mean anomaly M:
3
·2·a
P pm
=
2 2 1v
r amæ ö÷ç= - ÷ç ÷ç ÷è ø
2
2 2
v
a r
m mx = - = -
3sin( ) ( )pM E e E t t
a
m= - = -
cos( )cos( )
1 cos( )
eE
e
nn
+=
+
Orbital Elements
a semimajor axis
e eccentricity
I inclination
Ω ascending node
ω argument of periapsis
ν true anomaly
0 0
0 0 0 0
0 0
( ) ( )
( ) ( ) ( ) ( )
( ) ( )
x x
y y
z z
r t v t
r t r t v t v t
r t v t
æ ö æ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç= =÷ ÷ç ç÷ ÷ç ç÷ ÷÷ ÷ç ç÷ ÷ç çè ø è ø
Ecsape Velocity from a Circular Orbit
Circlevr
m=
2Parabolav
r
m⋅=
2CiParabol ca r lev v v
r r
m m⋅D = = --
( )2 1D = ⋅ -vr
m
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Hohmann Transfer
1D = -p p Kv v v
2D = -a K av v v
D = D + DHohmann p av v v
1 2
2 2
+ += =p ar r r r
a
( )1 1
2
1
21 1
1 1
2Hohmann
kv
r k r
k
k
r
r
k
m mæ ö æ öæ ö⋅ ÷ ÷ç ç ÷ç÷ ÷ç çD = - + - ÷ç÷ ÷ç ÷ç ç÷ ÷÷÷÷ çç è ø+ +è øè
=
ø
:
Solar System Parameters
Planet e i[°] r[AU] vF[km/s] v[km/s] Period[yr]
Mercury 0.205 7.0 0.39 4.3 47.9 0.241
Venus 0.007 3.4 0.72 10.4 35.0 0.615
Earth 0.017 0.0 1.00 11.2 29.8 1.000
Mars 0.094 1.9 1.52 5.0 24.1 1.88
Jupiter 0.049 1.3 5.20 59.5 13.1 11.9
Saturn 0.057 2.5 9.58 35.5 9.7 29.4
Uranus 0.046 0.8 19.20 21.3 6.8 83.7
Neptune 0.011 1.8 30.05 23.5 5.4 163.7
Pluto 0.244 17.2 39.24 1.1 4.7 248.0
Hohmann Transfer, Δv Magnitude in the Solar System
0 5 10 15 20 25 30 35 400
2
4
6
8
10
12
14
16
Zielorbit [AU]
delta
V [k
m/s
]
Hohmann Transfer von der Erde
Mars
dvdvp
dva
Hohmann Transfer
0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
5
10
15
20
25
Zielorbit [AU]
delta
V [k
m/s
]
Hohmann Transfer von der Erde
Mars
Venus
Merkur dvdvp
dva
Hohmann Transfer, Time-of-Flight
3
2
P at p
mD = =
Time-of-Flight:
( )1 23
1.52 1 1
2 2
+
Å
æ ö+ ÷ç= = ⋅ ÷ç ÷ç ÷è ø
r r
kPp
m
0.2 0.4 0.6 0.8 1 1.2 1.4 1.60.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Zielorbit [AU]
Tran
sfer
zeit
[yr]
Hohmann Transfer von der Erde
Mars
VenusMerkur
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
40
45
50
Zielorbit [AU]
Tran
sfer
zeit
[yr]
Hohmann Transfer von der Erde
Phase Angle γ
( )1 23
2( )
+
D = D =
r r
Mars Mars Marsn t nn pm
( )1 23
2
+
D =
r r
Mars Marsnn pm
( )1 23
2
32
+
D =
r r
Marsr
mn p
m
1.51 1
2
æ ö+ ÷ç ÷çD = ⋅ ÷ç ÷ç ÷÷çè ø
kMarsn p
1.51 11
2
æ öæ ö ÷ç + ÷÷çç ÷÷çç= - D = ⋅ - ÷÷çç ÷÷çç ÷÷÷çç ÷è ø ÷çè ø
kMarsg p n p
1 0
1 0
k
k
gg
> >< <
outbound, target planet must be in front of Earthinbound, target planet must be behind Earth
0 5 10 15 20 25 30 35 40-300
-250
-200
-150
-100
-50
0
50
100
150
Zielorbit [AU]
Pha
senw
inke
l [d
eg]
Hohmann Transfer von der Erde
Phase Angle γ
1.51 11
2
æ öæ ö ÷ç + ÷÷çç ÷÷çç= - D = ⋅ - ÷÷çç ÷÷çç ÷÷÷çç ÷è ø ÷çè ø
kMarsg p n p
Synodic Period
2n
P
p=
Mean angular velocity:
( )1 1.51
2 1
1Syn
k
Pn
pt = = ⋅
D -
2 1n n nD = -Relative angular velocity between two planets:
( )1.51
1
1Syn
k
Pt Å= ⋅-
Synodic period:
Synodic Period
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Zielorbit [AU]
Syn
odis
che
Per
iode
[J
ahre
]
Hohmann Transfer von der Erde
( )1.51
1
1Å= ⋅
-Syn
k
Pt
Δv for a change in Inclination of an Elliptical Orbit
1 2 cos= = ⋅ v v v f
cos sin2 2
æ öD D ÷ç= ⋅ ⋅ ÷ç ÷ç ÷è øv i
v f
2 cos sin2
æ öD ÷çD = ⋅ ⋅ ⋅ ÷ç ÷ç ÷è øi
v v f
Inklination [deg]
[deg
]
dv
[km/s] für r=7000 km
0 20 40 60 80 100 120 140 160 180
5
10
15
20
25
30
35
40
45
0
1
2
3
4
5
Node Change
( )2 212
2 1 cos ( ) sin ( ) cos( )D = ⋅ ⋅ - - ⋅ DWv v i i sin( )
sin hd
r adt h i
w n+W= ⋅
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Lambert Problem
1 2 2 1, , ,r r t t We know :
1 2r r
We are looking for the ellipse or hyperbola which connects und
In the real world the orbits of the planets are neither coplanar nor circular.
2 1t t t- = DIf we specify the time-of-flight( ), only one soultion exists.
Earth-Mars Transfer Lambert
Launch Date: 61344 ModJDate 31 Oct 2026Transfer Time: 309 daysArrival Date: 61654 ModJDate 6 Sep 2027
Earth-Venus Transfer Lambert
Launch Date: 60676 ModJDate 1 Jan 2025Transfer Time: 127 daysArrival Date: 60803 ModJDate 8 May 2025
Earth-Mercury Transfer Lambert
Launch Date: 60806 ModJDate 11 May 2025Transfer Time: 100 daysArrival Date: 60907 ModJDate 20 Aug 2025
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
„Hyperbola Elements“
Hyperbola elements are useful for describing an orbit when leaving or approaching a planet.
6 elements are needed:
rp
C3 energy = (v∞)2
Right ascension α and declination δ of theoutgoing asymptote
Velocity azimuth at periapsis
True anomaly
Example Mars-Transfer in 2003
Solve Lambert problem:
Launch Epoch [ModJDate]
Tran
sfer
Tim
e [D
ays]
Cost Function = dv1 + dv2 [km/s]
5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675 5.6675
5.265 5.27 5.275 5.28 5.285 5.29 5.295 5.3
x 104
100
150
200
250
300
350
400
5.8
6
6.2
6.4
6.6
6.8
7
7.2
7.4
7.6
7.8
2
2
1
2
2.90
0.55
0.28
10.73
5.46
2.97
3 8.786
kms
kms
kms
dv v
Mag
C v
ad
¥
¥
¥
¥
æ ö÷ç ÷ç ÷ç ÷ç= = - ÷ç ÷ç ÷÷ç- ÷çè ø
= - = -
=
= =
In spherical coordinates:
Example Mars-Transfer in 2003
2
1 1.1452pr ve
m¥
Å
⋅= + =
1acos 150.83
en¥
æ ö÷ç= - ÷ = ç ÷ç ÷è ø
1 2
sin sin sin( )
sin sinasin 180 asin
sin sin
i
i i
d w nd d
w n w n
¥
¥ ¥
= +æ ö æ ö÷ ÷ç ç= ÷ - = - ÷ -ç ç÷ ÷ç ç÷ ÷è ø è ø
64.8
200ParkingOrbit
ParkingOrbit
i
h km
= =
Start from Baikanour:
1
2
203
35
ww
= =
2 solutions:
0 50 100 150 200 250 300 350-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1sin(i)*sin( + ) ; i=64.8
[deg]
sin()
0 50 100 150 200 250 300 350-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1sin(i)*sin( + ) ; i=3
[deg]
sin()
Example Mars-Transfer in 2003
( ) ( )tan cos( )sin ; cos
tan cosi
d w na a
d¥ ¥
¥ ¥¥
+-W = -W =
For the ascending node we can derive:
tan cos( )atan ,
tan cosi
d w na
d¥ ¥
¥¥
Wæ ö+ ÷ç ÷W = - ç ÷ç ÷çè ø
goes from 0..360°, quadrant check must be performed:
1
2
351.84
346.66
W = W =
w·
· W
·
is controlled via the time spent in the parking orbit
is controlled via the daily launch time
the launch date comes out of the solution to the Lambert problem
Approaching the Target Planet, B-Plane
vS
v¥
¥
=
Definition B-Plane:
zT S e= ´
ze can be the North Pole or e.g. the ecliptic
R
R S T= ´
forms a right-handed orthogonal system:
Approaching the Target Planet, B-Plane
cos cos cos i
qq d¥ =
We can derive a relation for the B-Plane angle :
.i d¥³Out of this we can see that the inclination is constrained:
0 50 100 150 200 250 300 3500
20
40
60
80
100
120
140
160
180
[deg]
i [de
g]
=0°
=30°
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Launcher Performance
VEGA: S-ClassSSO@800 ~= 1300 kg
Soyuz: M-ClassSSO@800 ~= 4400 kg
Ariane 5: L-ClassSSO@800 ~= 10000 kg
Ariane 5ME: XL-ClassSSO@800 ~= 30000 kg
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Gravity Assist 2D Case
2
1cos
1
)(
peri
Planet
er v
e
n
m
¥
¥
= -
⋅= +
12 arcsin
ea
æ ö÷ç= ⋅ ÷ç ÷ç ÷è ø
Deflection angle:
( )v va¥- ¥+= ⋅Rot
12 ¥D = ⋅Satv v
e
Gravity Assist 2D Case
In front of the planet => Heliocentric velocity decrease
Behind the planet => Heliocentric velocity increase
Gravity Assist 3D Case
( ) ( ( ))pv r vq a¥- ¥+= ⋅ ⋅Rot Rot
Sat Planetv v v¥- -= -
Sat Planetv v v+ ¥+= +
You can choose θ and rp for free(without spending any deltaV), because the asymptote can bemoved around the B-plane.
Gravity Assist, Tisserand‘s Graph
( )22 1 cos( )SatSat Sat
Sat
v
r aT e i
a rÅ
Å
¥
= + ⋅ - =
=
constant for sequential fly-by's at a planet
constant (it's just a rotation of the velocity vector relative to the planet )
Can be used to find possible GA sequences
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
rp [AU]
r a [AU
]
E 3E 4E 5E 6V 3V 4V 5V 6
0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.780.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
1.25
rp [AU]
r a [AU
]
E 3E 4E 5E 6V 3V 4V 5V 6
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Low thrustEngine performance Fregat Upper stage
0 50 100 150 200 2501000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
Time [s]
mt [k
g]
Isp = 320s m1 = 2000kg Thrust = 14000 N
0 50 100 150 200 2500
0.5
1
1.5
2
2.5
Time [s]
dv [k
m/s
]
Isp = 320s m1 = 2000kg Thrust = 14000 N
Low thrustEngine performance ion engine
0 1 2 3 4 5 6 71000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
Time [year]
mt [k
g]
Isp = 2000s m1 = 2000kg Thrust = 0.1 N
0 1 2 3 4 5 6 70
2
4
6
8
10
12
14
Time [year]
dv [k
m/s
]
Isp = 2000s m1 = 2000kg Thrust = 0.1 N
Ion thrusters survey
ESA (Qinetiq T6):ISP 4200 sFmax 0.2 NP 5-6 kWMissions:
• Bepi Colombo
ESA (PPS 1350 SNECMA):ISP 1660 sFmax 0.09 NP 1.5 kWMissions:
• SMART
Credit: ESA
Credit: ESA
Ion thrusters survey
NASA(NSTAR):ISP 3100 sFmax 0.09 NP 2.6 kWMissions:
• Deep Space 1• Dawn
Credit: NASA/JPL
Credit: NASA/JPL-Caltech
Credit: NASA/JPL
Ion thrusters survey
JAXA(NEC):ISP 3200 sFmax 0.008 NP 0.35 kWMissions:
• Hayabusa
Lambert Solver for Low ThrustShape-Based Approach
Exponential sinuoid used by Petropoulos:
Inverse polynominal used by Wall and Conway:
Example Mars Transfer
tof = 700 days
m0 = 2100 kg (Soyus)
v_inf = 1.0 km/s
m_final = 1765kg
Isp = 4000 s
F_max = 0.2 N
ESA cornerstone mission to MercuryLaunch: 9 July 2016
MPO: Mercury Planetary Orbiter (480 x 1500 km)
MMO: Mercury Magnetospheric Orbiter (JAXA)(600 x 11800 km)
Bepi Colombo (Credits: Rüdiger Jehn, ESOC)
Low thrust trajectory optimization
Specific impulse 4022 sMaximum thrust 0.290 NSEP availability 90%Fuel consumption for navigation 5%
4.1 tons
pull
Bepi Colombo
Optimization ConstraintsConstraint ValueEarth flyby altitude > 300 kmVenus 1 flyby altitude < 1500 kmVenus 2 flyby altitude > 300 kmMercury flyby altitudes > 200 kmLast Mercury flyby altitude > 300 kmDuration of coast arc before flybys > 30 days
Duration of coast arc after flybys > 7 days
Duration of coast arc after launch > 90 days
Duration of coast arc before MOI > 60 days
Solar aspect angle (outside 0.8 AU) 77° < SAA < 112.5°
Solar aspect angle (inside 0.8 AU) 77° < SAA < 94.5°
Bepi Colombo
Bepi Colombo
Departure Launch Date 9 July 2016Escape velocity 3.475 km/sEscape declination -3.8°Initial mass 4100 kg
Cruise ∆V SEP 4.254 km/sTotal impulse 16.6 MNsCruise time 7.5 years
Arrival Date 1 Jan 2024Mercury true anomaly
67.8°
Velocity at periherm
3.793 km/s
Ωarr 67.7°ωarr (South pole) -2°
Content
Mission Analysis in the Context of System Engineering
Keplerian Elements
Hohmann Transfer / Inclination Change
Lambert Problem
Arriving and Leaving a planet
Launcher Performance
Gravity Assist
Low Thrust
Special Orbit Types
Sun Synchronous Orbit (SSO)
A SSO maintains the same orientation with respect to the sun all year round. This natural phenomen is due to the irregular shape of the earth.
Imagined by Jean-Pierre Penot (CNES) and Bernard Nicolas, illustrated by Bernard Nicolas
Sun Synchronous Orbit (SSO)
22
23( ) sin
2(3 1)Å Å- ⋅
⋅= -
Potential:R
U r Jr
mj
2
24(sin2 sin( ))
3
2h ia u
RJ
r
mÅ Å= -⋅
⋅⋅
⋅
32
sin( )
sin
W» ⋅ h
udr a
du h i
72
9.96 cosÅ
Å
æ ö÷ç ÷çDW = - ⋅÷ç ÷÷ç +è øday
Ri
h R
72
0.9856 9.96 cosÅ
Å
æ ö÷ç ÷ç = - ⋅÷ç ÷÷ç +è ø
Ri
h R
72
acos 0.0989 Å
Å
æ ö÷ç æ ö ÷ç + ÷÷çç ÷÷çç= - ÷÷çç ÷÷÷çç ÷è ø ÷ç ÷çè ø
SSOSSO
R hi
R
0 1000 2000 3000 4000 5000 600090
100
110
120
130
140
150
160
170
180
Height (km)
SS
O In
clin
atio
n (d
eg)
Geostationary Orbit
3
2 23h56min
42162km
= =
=
Orbital Period:
rP
r
pm
1 Jan 26 Feb 12 Apr 29 Aug 14 Oct 31 Dec22
22.5
23
23.5
24
Time
Sun
light
[hr]
Sunlight GEO
Highly elliptical orbit
Elliptical orbit:Different regions can be studied
Two satellites:Time derivations can be determined
Literatur
Interplanetary Mission Analysis and Design, Stephen KembleISBN 3-540-29913-0
Fundamentals of Astrodynamics and Applications, David ValladoISBN 978-1881883142
Space Mission Engineering: The new SMAD, James R. WertzISBN 978-1-881-883-15-9
Web
http://nssdc.gsfc.nasa.gov/planetary/planetfact.html
http://naif.jpl.nasa.gov/naif/spiceconcept.html
http://gmat.gsfc.nasa.gov/
