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QUASI PERIODIC ORBITS IN THE VICINITY OF THE SUN-EARTH L2 POINT AND THEIR IMPLEMENTATION IN "SPECTR-RG" & "MILLIMETRON" MISSIONS
I VA N I L I N , A N D R E W T U C H I N
K E L D Y S H I N S T I T U T E O F A P P L I E D M AT H E M AT I C S RUSSIAN ACADEMY OF SCIENCES

24th International Symposium on Space Flight Dynamics Laurel, Maryland, 5-9 May, 2014

APPLICATION OF QUASI PERIODIC ORBITS NEAR LIBRATION POINTS
ESA space telescope "Gaia"
launched on 19.12.13, direct transfer to a Lissajous orbit in the vicinity of the Sun-Earth L2 point

Roscosmos space telescope "Spectr-RG"
launch: scheduled at the end of 2015 direct transfer to a Lissajous orbit in the vicinity of the Sun-Earth L2 point Scientific mission: X-rays and Gamma range high precision sky survey, black holes, neutron stars, supernova explosions and galaxy cores study.

YZ

Roscosmos space telescope "Millimetron"
launch: scheduled in 2018 direct transfer to a big radius quasi-halo orbit in the vicinity of the Sun-Earth L2 point Scientific mission: space observation in millimeter, sub-millimeter and infrared ranges. The 12m space telescope will operate at cryogenic temperatures near 4K providing unique sensibility.

YZ

2

PERIODIC AND QUASI-PERIODIC MOTIONS AROUND L2 POINT

3

THREE BODY PROBLEM APPROXIMATION
1. Solution of the system of the linearized equations, describing circular restricted TBP

1 A t cos 1t 1 t C t et ,

Amin A(t ) Amax
t

2 k2 A t sin1t 1 t k1C t e
3 B t cos 2t 2 t ,

Bmin B(t ) Bmax

C (t ) C

max

2. Ricahrdson 3d order analytical approximation of periodic motion about the collinear points, obtained with the help of Lindstedt-Poincaree technique applied to Legendre polinomial expansion of the classical CRTBP equations of motion
2 2 3 x a21 Ax a22 Az2 Ax cos 1 (a23 Ax a24 Az2 )cos 2 1 (a31 Ax a32 Ax Az2 )cos 3 2 3 y kAx sin 1 (b21 Ax b22 Az2 )sin2 1 (b31 Ax b32 Ax Az2 )sin3 1 1 1

2 z n Az cos 1 nd21 Ax Az (cos 2 1 3) n (d32 Az Ax d31 Az3 )cos 3

n 2 n, n 1,3

Az 0

Ax 0

Ax

min

Q

4

QUASI PERIODIC SOLUTION IN RTBP TRANSITION FROM CIRCULAR RTBP TO ELLIPTIC RTBP
CRTBP

ERTBP

· Quasi-periodic orbit approximation:
Richardson model

· Libration points in the RTBP are homographic,
which means they keep their relative position when transfer to the ERTBP is performed

· State vector X (t) , lying on the obtained
quasi periodic solution
x y z
dx dx dt df dt df

· Transfer from dimensionless true anomaly f ,
describing evolution of the elliptic system, to the dimensionless time t of the TBP is performed
tg E tg 1 2 1 f 2 e e
M E e sinE tdimensionless M

p 1 e cos f

dt p3 2 df 1 e cos f

2

· A periodic halo orbit approximation is built with
the help of Richardson model

x

p x(e sin f ) (1 e cos f )

32

p3 2 y y(e sin f ) (1 e cos f ) p3 2 z z(e sin f ) (1 e cos f )

· TBP state vector ( x, y, z, x, y, z) is converted to
Nehvill dimensionless variables

( , , , , , )

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TRANSFER TRAJECTORY DESIGN - THE ISOLINE METHOD
The transfer trajectory to the selected quasiperiodic orbit is searched within the invariant manifold of the restricted three body problem with help of the isoline method. This method provides connection between periodic orbit dots and geocentric transfer trajectory parameters the isoloines of transfer trajectory pericentre height function depending on periodic orbit parameters are built. This provides one-impulse transfer from LEO to the quasi-periodic orbit.

x (x , y , z, x , y , z)

17 rL 24

( , , , , , )

rL

Earth
LEO parameters:

L
r , r , i, , ,
Periodic orbit parameters:

2

A, B, C, D, 1,

2

1-impulse flight trajectories are separated out with the condition:

r r *
r (1 ,2 ) r *

With the fixed A, B and C = 0 the isoline is built in the 1, 2 plane:

6

SPACECRAFT TRAJECTORY CALCULATION ALGORITHM STRUCTURE
Periodic orbit parameters A, B, LEO parameters r , r

Data, inclination selection

Quasi-periodic solution obtained in ERTBP

Isoline building algorithm

Transfer trajectory initial approximation building algorithm
Edge conditions: correction of B value, C = 0; max t in the vicinity of periodic solution

Selection of trajectories with min V

Shadow zones and radio visibility zones calculation, solution analysis

Stationkeeping impulses calculation

Transfer trajectory exact calculation

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STATIONKEEPING IMPULSES CALCULATION
The correction impulse vector is calculated according to the condition of the maximum time of the spacecraft staying in the L2 point vicinity of the stated radius after the correction has been implemented. The maximum time is searched for with the help of the gradient method.

1 V max Vi q Fc 2 Fc

T

Vmax - the biggest possible value of the impulse;

2 2 R A ,B rL B 1 k2

q - step decrease controlling coefficient.
2 A

toutL2 tinL2 FC 1 t1 T T B t BrL t1

2

C t

2

dt

8

THE ISOLINE METHOD FOR THE MOON SWING BY TRANSFERS
Advantages: Moon swing by maneuver allows to obtain more compact orbit Disadvantages: Trajectories become time-dependent and mission robustness decreases as the maneuver execution errors may cost a lot of V to correct
1 x 10
9

X-Y

1

x 10

9

X-Y

0.8

50 40

60 70

0.8

0.6

30 80

0.6

0.4

20

0.4

7 60 0 50

80 90

40

0.2

90

10 4 1

0.2

280 00 0 13 460 20 10 4 1

0

460 280 100

0

-0.2

-0.2

-0.4

-0.4

150

-0.6
150

-0.6

-0.8

-0.8

-1 -4

-2

0

2

4

6

8

10

12

14

16 x 10
8

-1 -4

-2

0

2

4

6

8

10

12

14

16 x 10
8

Quasi-periodic orbit obtained without the Moon swing by maneuver Ay = 0.8 million km

Quasi-periodic orbit obtained with the help of Moon swing by maneuver Ay = 0.5 million km

9

The XY plane view of the rotating reference frame, mln. km.

QUASI PERIODIC ORBITS FOR "SPECTR-RG" & "MILLIMETRON" MISSIONS

10

Total V costs are less than 15 m/s for 7 years period.

QUASI PERIODIC ORBITS FOR "SPECTR-RG" & "MILLIMETRON" MISSIONS
XY, XZ, YZ PROJECTIONS ON THE ROTATING REFERENCE FRAME

1000

1000

1000

-1000

-200

-600

-200 -600

-1000

Dimension: thousands of km

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EVOLUTION OF DIMENSIONLESS PARAMETERS DESCRIBING THE QUASI-PERIODIC ORBIT GEOMETRY
A
B
A RL

B RL RL

B

A

C

12

t , days

RESEARCH RESULTS
A new method of quasi periodic orbits construction, generalizing LindstedtPoincaree-Richardson technique for the ERTBP case has been developed and programmed. M.L. Lidov's isoline building method providing one-impulse transfers from LEO to a quasi-periodic orbit in the vicinity of a collinear libration point has been extend on gravity assist trajectory class. An algorithm calculating stationkeeping impulses for the quasi periodic orbit maintainance has been developed and programmed. It provides stationkeeping V costs are within 15 m/sec. strategies for spacecraft lifetime over 7 years, total Nominal trajectories for Spectr-RG and Millimetron missions have been obtained by performing the calculation described above in the full Solar system ballistic model. All the restrictions such as Earth and Moon shadow avoidance conditions and constant radio visibility from the Northern hemisphere have been met.

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LITERATURE LIST
· P.Gurfil, D.Meltzer "Stationkeeping on Unstable Orbits: Generalization to the Elliptic Restricted Three-Body Problem", The Journal of astronautical Sciences, Vol.54, 1, January-March 2006, pp. 29-51 · GСmez, G; Llibre, J; MartМnez, R and SimС, C (2001). Dynamics and Mission Design near Libration Points - Volume 1. Fundamentals: The Case of Collinear Libration Points. World Scientific, Singapore. · Howell, K C; Barden, B T; Wilson, R S and Lo, M W (1998). Trajectory Design Using a Dynamical Systems Approach with Application to GENESIS. Advances in the Astronautical Sciences 97: 1665-1684. · Wiggins, S (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York. · Meyer, K R and Hall, G R (1992). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Applied Mathematical Sciences, vol. 90. Springer-Verlag, New York. · Richardson D.L. Analytic construction of periodic orbits about the collinear points. Celestial Mechanics, vol. 22, p. 241 253. Oct 1980. · I.S. Ilin, G.S. Zaslavskiy, S.M. Lavrenov, V.V. Sazonov, V.A. Stepanyants, A.G. Tuchin, D.A. Tuchin, V.S. Yaroshevsky. Halo orbits in the vicinity of the Sun Earth system L2 point and the transfer to such orbits. Cosmic Research, vol. 3, 2014. · I.S. Ilin, V.V. Sazonov, A.G. Tuchin. Construction of the local orbits near the libration point of the Sun Earth system. KIAM RAS preprint, 2012. · I.S. Ilin, V.V. Sazonov, A.G. Tuchin. Construction of the flights from low Earth orbits to local orbits near the libration point in the Sun Earth system. KIAM RAS preprint, 2012. · M.L. Lidov, V.A. Lyakhova, N.M. Teslenko. One impulse flight to the quasi periodic orbit in the vicinity of the Sun Earth system L2 point and some related common problems. // Cosmic. research. 1987. . XXV. 2. P. 163185. · M.L. Lidov, V.A. Lyakhova, N.M. Teslenko. The trajectories of the Earth Moon halo orbit in the vicinity of the Sun Earth system L2 point flight // Cosmic. research. 1992. . 30. 4. P. 435454.

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