Ballistic Support for "SPECTR-RG" Spacecraft Flight to L2 Point of Sun-Earth System

1. The ballistic support of the “SPECTR-RG” spacecraft flight to the L2 point of the Sun-Earth system I.S. Ilin, G.S. Zaslavskiy, S.M. Lavrenov, V.V. Sazonov, V.A. Stepaniants, A.G. Tuchin, D.A. Tuchin, V.S. Yaroshevskiy Keldysh Institute if Applied Mathematics RAS 2012

2. 100 400 0 1000 1000 -150 -500 1600 1600 0 -800 -1000 -400 -400 The quasi-periodicorbits in the vicinityof the L2point of the Sun-Earth system

3. Missions to the L2 point of the Sun-Earth system • Two Russian missions are to be sent to the vicinity of the L2 point during the next few years: • The «Spectr-RG» spacecraft, flying to the L2 point of the Sun-Earth system and staying at the halo orbitin it’s vicinity.NPO S.A. Lavochkina, 2015. • The «Millimetron» spacecraft, flying to the L2 point of the Sun-Earth system and staying at the halo orbit in it’s vicinity. The spacecraft has to go out far from the ecliptics plane.NPO S.A. Lavochkina, 2018. • The examples of theL2 point missions that have already been implemented: • NASA spacecraft «WMAP», (2001 – 2009) • ESA spacecraft «Planck» + space observatory «Hershel» (2009) • ESA space observatory – spacecraft «Gaia» should go to the vicinity of the L2 point of the Sun-Earth system in 2013

4. The «Spectr-RG» mission • The «Spectr-RG» mission presupposes the flight to the vicinity of the Sun-Earth system L2 point and the halo orbit motion in the L2 point vicinity during the 7 years period. • The halo orbit in the vicinity of the Sun-Earth system L2 point is opportune because of the possibility of reaching it with a single-impulse flight with no correction at it’s end. • To keep the spacecraft in the halo orbit the stationkeeping is needed. Total stationkeeping costs for the 7 years period must not overcome 200 m/sec.

5. ξ3 ξ2 L2 ξ1 x3 x2 Earth x1 The direction to the Sun The isoline of the pericentre height function building method. • The isoline method for the approximate description of the Earth – L2 trajectories was suggested in M.L. Lidov’s papers. It was applied for the direct single-impulse flights without any Lunar swing by maneuver. • The spacecraft motion is described in the rotating reference frames: in the geocentric reference frame and in the reference frame with the beginning in the L2 libration point The average values ofA(t)и B(t) are chosen at the halo orbit designing stage. The average value of C(t) must be close to 0.

6. The linearized equations of the spacecraft motion in the quasi-periodicorbit in the rotating reference frame

7. The integration constants µ1, µ–theSun and the Earth gravitational constants rL1, rL– the distances fromthe L2point to the Sun and the Earth; a1– the astronomical unit; n1– the averageanglespeedof the Earth orbital motion.

8. The isoline building algorithm • The search of the pericentre height function according to the following algorithm: • The spacecraft state vector is calculated in the inertial reference frame, obtained by the fixation of the rotating reference frame axes at a fixed moment of time according to the parameters: А, B, and . • The obtained state vector is converted into the non-rotating geocentric ecliptic reference frame. • The geocentric orbit elements are counted with the help of the obtained state vector with the pericentre height among them. • The first isoline dot search • The extension of the isoline to the next dot

9. The first isoline dot search The scanning is performed within the intervalfrom 0 to 360° for φ1 and within the intervalfrom –180° to 180° for φ2 with the step of 45º for φ2 and 1º for φ1 . The φ1value satisfyingthe following condition is looked for: With the help of the bisection method theφ1mvalue satisfyingthe following conditionis searched: The pair ofφ1m, φ2values found is the isoline beginning point.

10. φ2 φ1b, φ2b φ1i+1, φ2i+1 φ1i, φ2i φ1i-1, φ2i-1 φ1 The extension of the isoline from the current point

11. The examples of the obtained isolines The isolines within the 27.01.14 launch windowwith the Moon swing by maneuver The isolines within the 18.12.14 launch windowwith the Moon swing by maneuver and 1 lap at the LEO The isolines without the Moon swing by maneuver φ2 φ2 φ2 φ1 φ1 φ1 from 0.18 to 0.2. = 0.1

12. The structure of the nominal transfer trajectory calculation algorithm • The isolines built are the income data for the flight trajectory initial kinematics' parameters calculation algorithm – the initial approximation of the transfer to the halo orbit. • The initial approximation built is used for the exact calculationof the flight from the Earth orbit with the fixed height to the given halo orbit. The kinematics' parameters vectoris counted more precisely according to the edge conditions. • The velocity impulses, needed for the stationkeeping of the spacecraft in the given area around L2 point are counted. • The shadow zones and radiovisibility zones for the locating stations, situated on Russian territory are counted for the whole spacecraft lifetime.

13. Earth L2 LEO parameters: Halo-orbit parameters: rπ, rα,i, Ω, ω, τ A, B, C, D, φ1, φ2 The initial approximation calculation. The transition from the transfertrajectory to the halo-orbit The condition to select the one impulse transfer trajectories: With the fixed A, Bи C=0 an isoline is build in the φ1, φ2plane:

14. The stages of the nominal trajectory calculation • The velocity vector of the hyperbolic transfer trajectory, obtained from the initial approximation is counted more precisely according to the edge conditions which are the given values ofthe parameters B andC = 0. • The velocity vector, obtained at the stage 1 is counted more precisely according to the condition of the maximum time of the halo-orbit staying in the L2 area of the following radius:

15. The calculation of the stationkeeping impulses, keeping the spacecraft in the halo orbit in theL2 area , , - the partial derivatives of theFC function with respect to the components of the velocity vector - the biggest possible value of the impulse; - the coefficient, controlling the step decrease.

16. The isoline method for the Moon swing by transfers • the flight from Earth to the entrance into the Moon incidence sphere, • the flight inside the Moon’s incidence sphere, • the flight after leaving the Moon’s incidence spheretill the entrance of the L2 point vicinity. It is opportune to use a Moon swing by maneuverfor the halo orbit transfer trajectories, as it allows to find the orbits coming closer to the L2 point. For calculations of the pericentre height corresponding to the given halo orbit the trajectory is divided into 3 parts: For searching the pericentre height these parts of trajectory are passed backwards. The function of the pericentre height also depends on time in case of the Moon swing by maneuver being applied.

17. The transfer trajectory without the Moon swing by maneuver The XY plane view, the rotating reference frame, mln. km.

18. The transfer trajectory with the Moon swing by maneuver The XY plane view, the rotating reference frame, mln. km.

19. The transfer trajectory with the Moon swing by maneuver and the preliminary lap at the LEO The XY plane view, the rotating reference frame, mln. km.

20. The XY, XZ, YZplane views of the halo-orbit in the rotating reference frame. The transfer to the halo-orbit is performed with the help of the Moon swing by maneuver Dimension: thousands of km 200 200 500 -200 1500 -200 1500 -500 500 The total characteristic velocity costsfor the stationkeepingareabout 30 m/secfor the 7 years period.

21. The XY, XZ, YZplane views of the halo-orbit in the rotating reference frame. The transfer to the halo-orbit is performed with the help of the Moon swing by maneuver. There was 1 preliminary lap at the LEO. Dimension: thousands of km 200 200 500 -200 1500 -200 1500 -500 500 The total characteristic velocity costsfor the stationkeepingareabout 30 m/secfor the 7 years period.

22. The halo-orbit, calculated for the «Millimetron» project. TheXY, XZ, YZplane views in the rotating reference frame Dimension: thousands of km 900 1100 900 -1100 1500 -1100 1500 -700 1500 The total characteristic velocity costsfor the stationkeepingareabout 14m/secfor the 7 years period.

23. The evolution of the orbit parameters , and t, days

24. The transfer to the L2 vicinity with the help of the Moon swing by maneuverThe dates of the transition to the L2 vicinity for 2014 year

25. The contingencies for the «Specter-RG» spacecraft orbit • To provide the needed level of solar cell panels luminance and radiovisibilty conditions for the Russian tracking stations, the following circumstances were taken into account at the orbit design stage: • If the spacecraft comes too close to the ecliptic plane, the penumbra area entrance is possible; • If the spacecraft goes too far from the ecliptic plane, long periods of no radiovisibility are highly probable.

26. The results of the research • The ballistic problem of obtaining halo orbits with the given geometric dimensions in the ecliptic plane and in the plane orthogonal to it has been solved. • A new method of transfer trajectories building for the flight from LEO to the family of halo orbits in the vicinity of the Sun-Earth system L2 point is developed. These trajectories need no impulse for the transfer from the flight trajectory to the halo-orbit. • The stationkeeping velocity costs are evaluated. • The primary evaluations of the orbit parameters determination and the forecast accuracy have been obtained.