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. The quasi-periodic orbits in the vicinity of the L 2 point of the Sun-Earth system -400 1600 1000 -1000 -150 100 0 0 1600 -400 -800 1000 -500 400

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 orbit in 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 the L2 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. 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 direction to the Sun x1x1 Earth ξ1ξ1 L2L2 ξ3ξ3 ξ2ξ2 x3x3 x2x2 The average values of A(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-periodic orbit in the rotating reference frame

7. The integration constants µ 1, µ – the Sun and the Earth gravitational constants a 1 – the astronomical unit; r L1, r L – the distances from the L 2 point to the Sun and the Earth; n 1 – the average angle speed of the Earth orbital motion.

8. The isoline building algorithm The search of the pericentre height function according to the following algorithm: 1.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. 2.The obtained state vector is converted into the non-rotating geocentric ecliptic reference frame. 3.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 interval from 0 to 360° for φ1 and within the interval from –180° to 180° for φ2 with the step of 45º for φ2 and 1º for φ 1. The φ 1 value satisfying the following condition is looked for: With the help of the bisection method the φ 1m value satisfying the following condition is searched : The pair of φ 1m, φ 2 values found is the isoline beginning point.

10. The extension of the isoline from the current point φ 1i, φ 2i φ 1i+1, φ 2i+1 φ 1i-1, φ 2i-1 φ 1b, φ 2b φ1φ1 φ2φ2

11. The examples of the obtained isolines The isolines within the 27.01.14 launch window with the Moon swing by maneuver from 0.18 to 0.2. = 0.1 φ2φ2 φ2φ2 φ2φ2 φ1φ1 φ1φ1 φ1φ1 The isolines within the 18.12.14 launch window with the Moon swing by maneuver and 1 lap at the LEO The isolines without the Moon swing by maneuver

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 calculation of the flight from the Earth orbit with the fixed height to the given halo orbit. The kinematics' parameters vector is 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. The initial approximation calculation. The transition from the transfer trajectory to the halo-orbit Earth L2L2 A, B, C, D, φ 1, φ 2 r π, r α, i, Ω, ω, τ LEO parameters: Halo-orbit parameters: The condition to select the one impulse transfer trajectories: With the fixed A, B и C = 0 an isoline is build in the φ1, φ2 plane:

14. The stages of the nominal trajectory calculation 1.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 of the parameters B and C = 0. 2.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 L 2 area of the following radius:

15. The calculation of the stationkeeping impulses, keeping the spacecraft in the halo orbit in the L 2 area,, - the partial derivatives of the F C 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 sphere till the entrance of the L 2 point vicinity. For calculations of the pericentre height corresponding to the given halo orbit the trajectory is divided into 3 parts: It is opportune to use a Moon swing by maneuver for the halo orbit transfer trajectories, as it allows to find the orbits coming closer to the L2 point. 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, YZ plane 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 500 Dimension: thousands of km -2001500 200 -2001500 500 200 -500 The total characteristic velocity costs for the stationkeeping are about 30 m/sec for the 7 years period.

21. 500 -2001500 200 -2001500 500 200 -500 The total characteristic velocity costs for the stationkeeping are about 30 m/sec for the 7 years period. The XY, XZ, YZ plane 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

22. The halo-orbit, calculated for the «Millimetron» project. The XY, XZ, YZ plane views in the rotating reference frame 1100 -11001500 900 -700 900 -11001500 Dimension: thousands of km The total characteristic velocity costs for the stationkeeping are about 14 m/sec for the 7 years period.

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

24. The transfer to the L 2 vicinity with the help of the Moon swing by maneuver The dates of the transition to the L 2 vicinity for 2014 year Month θAθA Launch dateThe duration of the launch window, hours January 0.142014012836 0.152014012872 February 0.142014022740 0.152014022648 March 0.122014032946 April 0.122014042724 May 0.122014052920 0.132014052928 0.142014052936 0.152014052952 June 0.122014062522 0.132014062533.5 0.142014062540.5 0.152014062560 July 0.152014072541 August 0.142014082414.5 0.152014082357.5 September 0.122014092212.5 October 0.12201410236 November 0.122014112123.5 December 0.122014121822

25. 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. The contingencies for the «Specter-RG» spacecraft orbit

26. 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. The results of the research