Low Energy Transfer Trajectories

Paolo Teofilatto

Scuola di Ingegneria Aerospaziale

Università di Roma “La Sapienza”

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

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

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

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

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

The minimum is at boundary of R

12

31 2 3( ) ( ) ( )J J J

( ) minJ R J

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

Transfer in Keplerian field n.2X

xp

F

I

= d d X p pV X x

, X X p p -

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

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

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

Variation of Inclination

Moon assisted Earth orbital transfers: GTO

Lunar assisted GTO

Lunar assisted GTO with reduced apogee

Optimal lunar assisted GTO are in the unstable Earth-Moon region

Earth-Moon Zero Velocity Curves

WSB: a Low Energy Transferto the Moon

(up to 20% more of the final payload mass)

The Sun gravity-gradient effect

Zero Velocity curves during WSB transfer

Zero Velocity curves during WSB transfer

Zero Velocity curves during WSB transfer

Zero Velocity curves during WSB transfer

Zero Velocity curves during WSB transfer

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

VariationV

X Y

Ztransf = er 20 / time a t :44 dayV cm s

6 perturbed trajectories

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

SavingV

0var 35 iation of inclination i

40 m/snom aposeleniumV

5 m/s (reduction of periselenium)WSBV

35 /SAVEDV m s

Different separation times

Different separation times

?

Different final parameters

There are two families of trajectories having the “same” periselenium distance

Different separation times

Different separation times

?

Different final parameters

There are two families of trajectories having the “same” periselenium distance

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

The two Keplerian ellipses

rv

v

A

B

equal energy curve

Case A

apogee

1P

0P

rv 0

Case B

Lambert

Moon orbit

0r

1r

0v

0 1 06 Kmr e

1 384400 Kmr

The problem in the restricted 3bp

M

E M t

y

x

0P

1P

minr1

X

Y xy

03

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

Case 00 0

The Earth_Moon Jacobi “constant”

Jacobi constant of the exterior Lagrangian point L2

Hadjidemetriou curves for 00.012 and 0

Argument of periselenium

Effect of planetary eccentricity on ballistic capture in the solar system

Jupiter Comets

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

Satellite Eccentricity at capture

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