1. Some problems in the optimization of the LISA orbits Guangyu Li 1 ， Zhaohua Yi 1,2 ， Yan Xia 1 Gerhard Heinzel 3 Oliver Jennrich 4 1 、 Purple Mountain Observatory,CAS, Nanjing 2 、 Nanjing University 3 、 Max Planck Institute for Gravitational Physics 4 、 European SpaceResearch and Technology Center Third International ASTROD Symposium

2. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits LISA ( Laser Interferometer Space Antenna ) Introduction LISA is an ESA-NASA mission. It’s primary scientific goal is to detect gravitational waves with wavelength from 0.1 mHz to 1 Hz. Three spacecrafts will be launched around 2015, and reach it’s orbits around the Sun after one year. Observation will continue 5-10 years. LISA Pathfinder will be launch in 2009 to test the technology.

3. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the LISA Constellation Upper left: The LISA constellation Upper right: Orbit around the sun Lower right: Animation

4. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Requirements on the stability of the LISA constellation D= 5 ×10 6 km, Δ D= ±5 ×10 4 km α =60 degree, Δα= ± 1.5degree Δ v r = ± 15m/s Trailing angle of the constellation center δ=20 degree, Δδ small enough Figure of merit: Q(a i,e i,ω i Ω i ν i ) = w d Δ D 2 ＋ w a Δα 2 ＋ w v Δ v r 2 + w l Δδ 2 Aim of the constellation orbit optimization: Find a set of parameters a i,e i,ω i Ω i ν i (i=1,2,3) to minimize Q.

5. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits 3 levels to study LISA constellation orbits Motion of the constellation center----plane co-orbit restricted problem. Motion of a single spacecraft relative to the constellation center ---- general co-orbit restricted problem. breathing motion of the constellation arm

6. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Effect of Earth and Jupiter on the Variation of the armlength

7. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Plan for the orbit optimization research In the frame of plane co-orbit restricted problem, analyze the motion of the constellation center, obtain analytical formulae. In the frame of general co-orbit restricted problem, analyze the motion of a single spacecraft, obtain analytical formulae. Analyze the conditions to form the constellation Analyze the breathing motion of the constellation arms Analyze the effects of other solar system bodies to the constellation motion.

8. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Plan for the orbit optimization research Select parameters for numerical optimization, using the analytical results. Consider the effects of all solar-system bodies and post-Newtonian effects. Analyze the effect and weight of every initial parameter. Optimize the initial parameters 。

9. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center Plane co-orbit restricted 3-body problem Take the sun S as origin ， the direction to the Earth-Moon barycenter E as x-axis (co-rotating non-inertial coordinate system). The equation of motion of the constellation center C is ： Normalized to length unit 1AU, time unit 1 day and mass unit 1 solar mass. n=0.0172021251 rad/day is the mean angular velocity of the Earth-moon barycenter around the sun (in an inertial system), i.e. the angular velocity of the co-rotating coordinate system. 0.00000304

10. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center Equation of motion in polar coordinates （ r,θ ）

11. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center - first order analytical solution Because the mass parameterμis small ， we can obtain the following first-order analytical solution ： Takeμ=0 (no perturbation by Earth)

12. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center - first order analytical solution We get the co-orbit solution: as special solution and the general solution: with h, A and B as integration constants

13. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center - first order analytical solution From here find approximation by iterative substitution ：

14. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center - first order analytical solution with the parameters:

15. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the constellation center - first order analytical solution Comparison with high-precision numerical solution ：

16. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Conclusion: The precision of the first-order approximate analytical solution satisfies the requirements of the space-craft orbit design Motion of the constellation center -first order analytical solution Comparison with high-precision numerical solution ：

17. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Several conclusions concerning the motion of the LISA constellation center k describes the orbit instability. Near the earth, k is large and the orbits are not stable. If degree ， C is at the Lagrange point L5 ， (classical result)

18. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits For r 0 =1AU, in our parameter range of interest [-60,-10] ， the angleθand the heliocentric distance Δr are nearly monotonic functions of time and thus reach their maximum at the end of the mission lifetime (t = 3700 days) 。 The following diagrams show the relation between and θ 0 and the maximum values: Several conclusions concerning the motion of the LISA constellation center

19. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Several conclusions concerning the motion of the LISA constellation center If θ 0 =-20 degrees ， we get θ max = -7.85 degrees ， Δr = 0.36x10 6 km 。 With time ， the distance to the sun increases ， the average angular velocity decreases ， and thus the earth- trailing angle increases 。 If we choose r 0 slightly smaller than 1.0AU ， it will gradually increase to more than 1.0AU ， so we can decrease the variation of θ 。 We can use this formula to get an initial parameter r 0 for the numerical optimisation 。 The following diagram shows an optimization result where the variation of theta is below 2.6 degrees. during 80 of the mission lifetime, it is less than 0.5 degrees. The variation of Delta r is larger than before, about 0.8x10 6 km.

20. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the Constellation Preliminary discussion Equation of motion of spacecraft P i ：

21. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the Constellation Preliminary discussion Moving the coordinate system origin from the sun to the constellation center C yields ：

22. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the Constellation Preliminary discussion Equation of motion of spacecraft P i ：

23. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the Constellation Preliminary discussion Equation of motion of the constellation arms ：

24. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Motion of the Constellation Preliminary discussion In the first-order approximation we get the following differential equation ：

25. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Next steps Study the elliptic restricted co-orbit 3-body problem Study stability and stability region of the solution Find higher-order analytical solutions Apply to study motion of various small bodies Apply to orbit design of other spacecraft

26. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Preliminary results one of the optimised orbits J2000 bary-heliocentric coordinates Initial state: epoch: JD2457387.75 (2016 Jan. 0.25 ) r (AU) (AU/day) S/C 1 0.24319326082354764 0.90270876558251567 0.37772248037472733 -0.016566080056811327 0.0039404117401137233 0.0014652709946148183 S/C 2 0.24702209212800164 0.87579482214883408 0.39609082970493692 -0.016816242428327844 0.0040130016578734298 0.0016349634284733734 S/C 3 0.27263517900240519 0.88889572154315666 0.37953184457686123 -0.016566603729254115 0.0040094698237708492 0.0020088078816390222

27. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Range of parameter variations in 3700 days (numerically integrated orbit) Armlength (10 4 km) relative velocity (m/s) Angle (degree) Target 495.0~505.0-15.0~15.058.5~61.5 S/C 1 487.7 ~ 497.5-10.0 ~ 11. 459.0~ 61.0 S/C 2 488.1~ 498.2-12. 5 ~ 9.559. 3 ~ 60.7 S/C 3 486.6~ 498.1-14.6 ~ 11. 359.0 ~ 61.0

28. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits armlengths variation

29. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits variation of relative velocities along arms (m/s)

30. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits variation of angles

31. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits trail-back angle of constellation center behind earth

32. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Another optimized orbit J2000 bary-heliocentric coordinates Initial state: epoch JD2457388.5 (2016 Jan. 1.0 ) r (AU) (AU/day) S/C 1 0.24332544827637306 0.90323495941223553 0.37794109569487494 -0.016561207425831212 0.0039393487640362271 0.0014648487709437339 S/C 2 0.24723777766725949 0.87629715712772249 0.39631902010035003 -0.016810817655221109 0.004012777068478429 0.0016349129525411396 S/C 3 0.2728475755292819 0.88939504547586623 0.37974276977013616 -0.016561633944598177 0.0040088568182004751 0.0020085272687749892

33. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Range of parameter variations in 3700 days (numerically integrated orbit) Armlength (10 4 km) relative velocity (m/s) Angle (degree) Target 495.0~505.0-15.0~15.058.5~61.5 S/C 1 487.6 ~ 497. 4-9.0 ~ 9.359.2 ~ 60.7 S/C 2 488.5 ~ 497. 5-8.5 ~ 9.759.2 ~ 60.7 S/C 3 488. 6 ~ 497.3-7.4 ~ 10.659.3 ~ 60.9

34. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits armlengths

35. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits relative velocities (m/s)

36. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits angles

37. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits trail-back angle of constellation center behind earth

38. July 14-16, 2006 Beijing Li Guangyu: Some problems in the optimization of LISA orbits Welcome to visit PMO! gyl@pmo.ac.cngyl@pmo.ac.cn THANK YOU !