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Some problems in the optimization of the LISA orbits

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

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Some problems in the optimization of the LISA orbits

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  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. 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. Li Guangyu: Some problems in the optimization of LISA orbits

  3. Motion of the LISA Constellation • Upper left: The LISA constellation • Upper right: Orbit around the sun • Lower right: Animation Li Guangyu: Some problems in the optimization of LISA orbits

  4. Requirements on the stability of the LISA constellation D= 5 ×106 km, Δ D= ±5 ×104 km α =60 degree, Δα= ± 1.5degree Δ vr= ± 15m/s Trailing angle of the constellation center δ=20 degree, Δδ small enough Figure of merit: Q(ai,ei,ωiΩiνi) = wdΔ D2＋ wa Δα2 ＋ wv Δ vr2+ wlΔδ2 Aim of the constellation orbit optimization: Find a set of parameters ai,ei,ωiΩiνi (i=1,2,3) to minimize Q . Li Guangyu: Some problems in the optimization of LISA orbits

  5. 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 Li Guangyu: Some problems in the optimization of LISA orbits

  6. Effect of Earth and Jupiter on the Variation of the armlength Li Guangyu: Some problems in the optimization of LISA orbits

  7. 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. Li Guangyu: Some problems in the optimization of LISA orbits

  8. 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。 Li Guangyu: Some problems in the optimization of LISA orbits

  9. Motion of the constellationcenterPlane 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： 0.00000304 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. Li Guangyu: Some problems in the optimization of LISA orbits

  10. Motion of the constellation centerEquation of motion in polar coordinates（r,θ） Li Guangyu: Some problems in the optimization of LISA orbits

  11. 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) Li Guangyu: Some problems in the optimization of LISA orbits

  12. 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 Li Guangyu: Some problems in the optimization of LISA orbits

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

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

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

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

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

  18. Several conclusions concerning the motion of the LISA constellation center • For r0=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 θ0 and the maximum values: Li Guangyu: Some problems in the optimization of LISA orbits

  19. Several conclusions concerning the motion of the LISA constellation center • Ifθ0=-20 degrees，we getθmax= -7.85 degrees， Δr = 0.36x106km。With time，the distance to the sun increases，the average angular velocity decreases，and thus the earth-trailing angle increases。If we choose r0 slightly smaller than 1.0AU，it will gradually increase to more than 1.0AU，so we can decrease the variation ofθ。We can use this formula to get an initial parameter r0 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 0.8x106km. Li Guangyu: Some problems in the optimization of LISA orbits

  20. Motion of the ConstellationPreliminary discussion Equation of motion of spacecraft Pi： Li Guangyu: Some problems in the optimization of LISA orbits

  21. Motion of the ConstellationPreliminary discussion Moving the coordinate system origin from the sun to the constellation center C yields： Li Guangyu: Some problems in the optimization of LISA orbits

  22. Motion of the ConstellationPreliminary discussion Equation of motion of spacecraft Pi： Li Guangyu: Some problems in the optimization of LISA orbits

  23. Motion of the ConstellationPreliminary discussion Equation of motion of the constellation arms： Li Guangyu: Some problems in the optimization of LISA orbits

  24. Motion of the ConstellationPreliminary discussion In the first-order approximation we get the following differential equation： Li Guangyu: Some problems in the optimization of LISA orbits

  25. 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 Li Guangyu: Some problems in the optimization of LISA orbits

  26. Preliminary resultsone of the optimised orbits • J2000 bary-heliocentric coordinates • Initial state: epoch: JD2457387.75 (2016 Jan. 0.25 ) Li Guangyu: Some problems in the optimization of LISA orbits

  27. Range of parameter variationsin 3700 days (numerically integrated orbit) Li Guangyu: Some problems in the optimization of LISA orbits

  28. armlengths variation Li Guangyu: Some problems in the optimization of LISA orbits

  29. variation of relative velocities along arms (m/s) Li Guangyu: Some problems in the optimization of LISA orbits

  30. variation of angles Li Guangyu: Some problems in the optimization of LISA orbits

  31. trail-back angle of constellation center behind earth Li Guangyu: Some problems in the optimization of LISA orbits

  32. Another optimized orbit • J2000 bary-heliocentric coordinates • Initial state: epoch JD2457388.5 (2016 Jan. 1.0 ) Li Guangyu: Some problems in the optimization of LISA orbits

  33. Range of parameter variationsin 3700 days (numerically integrated orbit) Li Guangyu: Some problems in the optimization of LISA orbits

  34. armlengths Li Guangyu: Some problems in the optimization of LISA orbits

  35. relative velocities (m/s) Li Guangyu: Some problems in the optimization of LISA orbits

  36. angles Li Guangyu: Some problems in the optimization of LISA orbits

  37. trail-back angle of constellation center behind earth Li Guangyu: Some problems in the optimization of LISA orbits

  38. Welcome to visit PMO! gyl@pmo.ac.cn THANK YOU ! Li Guangyu: Some problems in the optimization of LISA orbits

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