low energy transfer trajectories
DESCRIPTION
Low Energy Transfer Trajectories. Paolo Teofilatto Scuola di Ingegneria Aerospaziale Università di Roma “La Sapienza”. Summary. Impulsive transfers in Keplerian field Earth orbits transfers in non Keplerian field Weak Stability Boundary lunar transfers - PowerPoint PPT PresentationTRANSCRIPT
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Low Energy Transfer Trajectories
Paolo Teofilatto
Scuola di Ingegneria Aerospaziale
Università di Roma “La Sapienza”
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Summary
1. Impulsive transfers in Keplerian field
2. Earth orbits transfers in non Keplerian field
3. Weak Stability Boundary lunar transfers
4. Low energy lunar constellation deployment
5. Eccentricity effect in interplanetary transfers
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1) Impulsive transfers in Keplerian Field
• Lawden “primer vector” theory• Cicala-Miele optimization theory via Green’s
theorem• Hazelrigg definitive contribution in the 2D case• Other important contributes: Ting, Edelbaum,
Rider, Eckel, Marec, Marchal , ....
T. Edelbaum: “How many impulses ?”
Astronautics and Aeronautics, vol.5, 1967
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Lawden COAXIAL RULE
• If the initial or terminal orbit of a transfer is circular then all the other transfer orbits must be coaxial to the point of entrance or exit on the circular orbit.
• Optimal time-open, angle-open, transfers between optimally oriented orbits: coaxial transfer orbits
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Cicala-Miele application of the Green’s Theorem
Space state: two dimensional bounded region R
Cost Function: min ( )J
1 1 2 2( ) J dx dx
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Green’s Theorem
2 1 2 1 2 1
2 1( ) ( )J J J d
2
2 1If <0 on R then ( ) ( ) 0J J J
R R
1
21
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The minimum is at boundary of R
12
31 2 3( ) ( ) ( )J J J
( ) minJ R J
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Transfer in Keplerian field
xa=apogee distance , X=1/xa
xp=perigee distanceX
xpparabolas
increasingapogee I
F
circles
X*xp=1
= d d X p pV X x
3/ 2
1
2 (1 )X pp
xXx
3/ 2
1
2 (1 )ppp
X
Xxx
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Transfer in Keplerian field n.2X
xp
F
I
= d d X p pV X x
, X X p p -
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Transfers with X geq XF
X
I
F
xp
XF
a2
a1 b1
b2
1 1 2
1
Ia b b F
V
2 2
2
Ia b F
V
1 2 2 1 1
2 1
a a b b a
= < 0 , ( =d )V V V
Two impulses are better than fourHohmann strategy is optimal if the constraint is imposed
FX X
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Transfers with X leq XF
X
I
F
xp
XF
Three impulses are better than six
Biparabolic transfer is optimal if the constraint is imposed
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Local Analisys
Hohmann vs Bielliptic : local analysis
Biparabolic is better than Hohmann if
Any bielliptic is better than Hohmann if / 11.8F IR R
/ 15.4F IR R
INCLINATION VARIATION
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Variation of Inclination
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Moon assisted Earth orbital transfers: GTO
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Lunar assisted GTO
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Lunar assisted GTO with reduced apogee
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Optimal lunar assisted GTO are in the unstable Earth-Moon region
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Earth-Moon Zero Velocity Curves
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WSB: a Low Energy Transferto the Moon
(up to 20% more of the final payload mass)
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The Sun gravity-gradient effect
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Zero Velocity curves during WSB transfer
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Zero Velocity curves during WSB transfer
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Zero Velocity curves during WSB transfer
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Zero Velocity curves during WSB transfer
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Zero Velocity curves during WSB transfer
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Weak Stability Boundary Trajectories for the deployment of
lunar spacecraft constellations• Take advantage of the weak stability dynamics in order
to deploy a constellation of lunar spacecraft with a small • Consider a nominal WSB trajectory with periselenium
distance • Consider a cluster of small impulses (10:20 cm/s) from a
certain point of the nominal trajectory.• Select those impulses such that the injected spacecraft
have a periselenium distance “close” to • Since small variations in initial conditions imply large
variations in the final conditions (‘instability’) we may expect rather different lunar orbit parameters with respect to the nominal ones (constellation deployment)
200pr Km
pr
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VariationV
X Y
Ztransf = er 20 / time a t :44 dayV cm s
6 perturbed trajectories
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Final parametersOnly one of the 6 burns leads the spacecraft to a periselenium
“close” to rp
• Nominal parameters (at Moon):
•Perturbated parameters (at Moon):
55.36 d37172 , 0.9373 , , 143.41 deg , 77.42 g d ge ea Km e i
transfer time : 91.92 days , periselenium distance : 236 Km
89.99 d32900 , 0.9441 , , 0.034 deg , 120.00e deg ga Km e i
transfer time : 90.85 days , periselenium distance : 596 Km
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SavingV
0var 35 iation of inclination i
40 m/snom aposeleniumV
5 m/s (reduction of periselenium)WSBV
35 /SAVEDV m s
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Different separation times
Different separation times
?
Different final parameters
There are two families of trajectories having the “same” periselenium distance
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Different separation times
Different separation times
?
Different final parameters
There are two families of trajectories having the “same” periselenium distance
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Keplerian Case
• Given find the velocities in to reach 0 1 and r r
0P 1P
r 0 1 0 1
1v a( , , ) + b( , , ) v
vr r r r
0v
0 r
1 r
0P
1P
0v
v
vr
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The two Keplerian ellipses
rv
v
A
B
equal energy curve
Case A
apogee
1P
0P
rv 0
Case B
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Lambert
Moon orbit
0r
1r
0v
0 1 06 Kmr e
1 384400 Kmr
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The problem in the restricted 3bp
M
E M t
y
x
0P
1P
minr1
X
Y xy
03
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Hadjidemetriou work
22 2 2 2 2min 0 0 0 0 min 0 0 0 0
0
1 2[ 2 ( ) 2] 2 [ 2 2 ( ) ] 0
4r v v r r r r v r r
r
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Case 00 0
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The Earth_Moon Jacobi “constant”
Jacobi constant of the exterior Lagrangian point L2
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Hadjidemetriou curves for 00.012 and 0
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Argument of periselenium
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Effect of planetary eccentricity on ballistic capture in the solar system
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Jupiter Comets
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Capture Condition
2 2
2 22
2
2 22
42 2 cos( )
1
42 cos( ) 0
1
pc c
p p p m
p
p p m
xe c e c Bc ix x x e
xc Bc i B
x x e
( ) 0ic LC C
0
21 cos 1 cos
iLc
m c m
B Ie e
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Satellite Eccentricity at capture
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Conclusion• Global results are at disposal for optimal
(low energy) orbit transfers in the Keplerian field
• Lower energy transfer orbits can be obtained by a third body (e.g. Moon) gravitational help
• The effect of a four body (e.g. Sun) is important in low energy lunar transfer
• Planet eccentricity has a role in planetary ballistic capture