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OPS Forum Libration Orbits 03.04.2009

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ESA's Herschel and Planck missions will fly on orbits around the libration point L2 in the Sun-Earth system. Future astronomy missions such as GAIA and JWST will also use this type of orbit. This …

ESA's Herschel and Planck missions will fly on orbits around the libration point L2 in the Sun-Earth system. Future astronomy missions such as GAIA and JWST will also use this type of orbit. This forum will provide a comprehensive introduction to the basic mechanics of libration point orbits.


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    • 1. 1Libration Point Orbits - M. Hechler - ESOC 2009/4/3 ON LIBRATION POINT ORBITS 56 slides OPS-G FORUM Martin Hechler, GFA ESOC 2009/4/3 H/P 2009/04/29-13:24:24UT
    • 2. 2Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Contents • Lagrange points in Sun-Earth system and orbits around them • ESA missions at L2 and L1 (Sun-Earth): Why there ? In which orbits ? • From basics of linear theory to numerical orbit construction • Transfers to L2 or L1 : Stable manifold and weak stability boundary • The freely reachable orbits (Herschel, JWST) • Transfer optimisation to Lissajous orbits (Planck, GAIA) • Launch windows (Herschel/Planck, GAIA) • Lunar flybys, transfers with apogee raising sequence (LISA pathfinder) • Navigation: orbit determination and orbit correction manoeuvres
    • 3. 3Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Libration Points Orbits at L2 Missions Going there
    • 4. 4Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Libration Points in Sun-Earth Satellite in L2: • Centrifugal force (R=1.01 AU) balances central force (Sun + Earth)  1 year orbit period at 1.01 AU with Sun + Earth attracting  Satellite remains in L2 • However: Theory of Lagrange only valid if Earth moves on circle and Earth+Moon in one point  But orbits around L2 exist Lagrange 1736-1813Libration Points: • 5 Lagrange Points • L1 and L2 of interest for space missions
    • 5. 5Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Satellite in L2 • Does not work in exact problem • Would also be in Earth half-shadow • And difficult to reach (much propellant) Satellites in orbits around L2 • With certain initial conditions a satellite will remain near L2 also in exact problem  called Orbits around L2  Different Types of Orbits classified by their motion in y-z (z=out of ecliptic) Orbits at L2 x y
    • 6. 6Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Orbit Families at Libration Points • ‘Strict’ Halo orbits:  ‘quasi periodic’: z-frequency = x-y-frequency  large amplitudes (AY ≥ 600000 km)  loss of one degree of freedom in initial state  in general no free transfer  free of eclipse by definition • Quasi Halo orbits:  not periodic  free transfer possible, stable manifold “touches” launch conditions  can be free of eclipse for long time • Lissajous orbits:  small amplitude possible  large insertion ∆V  can be free of eclipse for 6 years  condition on initial state
    • 7. 7Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Criteria for Selection of Orbit Class at L2 • No essential advantage of ‘strict’ Halos  Orbit selection on mission requirements alone • Typical selection criteria:  no eclipse  limit on sun-spacecraft-Earth angle e.g. for communication system design  ∆V budget  Two families of orbits of most interest  Minimum transfer ∆V Quasi-Halos  Lissajous orbits without eclipse for 6 years
    • 8. 8Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Why do Astronomy Missions go to L2 ? • Advantages for Astronomy Missions:  Sun and Earth nearly aligned as seen from spacecraft  stable thermal environment with sun + Earth IR shielding  only one direction excluded form viewing (moving 360o per year)  possibly medium gain antenna in sun pointing  Low high energy radiation environment • Drawbacks:  1.5 x 106 km for communication  However development of deep space communications technology (X-band, K-band) ameliorates this disadvantage  Long transfer duration  Fast transfer in about 30 days with +10 m/s  Instable orbits  frequent manoeuvres, escape in case of problems and at end of mission (L2 region “self-cleaning”)
    • 9. 9Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Used Orbit Classes for Missions at L1 and L2 Lissajous orbits: • Earth aspect <15o  Survey Missions scanning in sun pointing spin Quasi Halo orbits: • “Free” transfer  Observatories Herschel JWST LISA Pathfinder (L1) XEUS EUCLID PLATO SPICA Planck GAIA View from Earth
    • 10. 10Libration Point Orbits - M. Hechler - ESOC 2009/4/3 ESA Missions to L2 (and L1) Mission Launch Orbit Objective + Spacecraft Herschel Ariane 2009/4/29 Quasi Halo L2 Far infrared photometry and spectroscopy 3.5 m Ø primary mirror Helium cryostat, launch mass 3415 kg Sun shade on side (viewing ┴ sun direction ± ) Planck Ariane 2009/4/29 with Herschel Lissajous L2  Map anisotropies of microwave background  1911 kg spacecraft, refrigerator pumps, <1 K  spinning at 1 rpm, sun pointing  optical axis prescribes small-circles over sky GAIA Soyuz/Fregat 2011/2012 (French Guyana) Lissajous L2  Astrometry to micro-arc-sec for 109 stars  spinning at 1 rev per 6 hours  spin axis coning at 45o from sun in 63 days  Two telescopes ┴ spin axis, 106.5o apart LPF VEGA 2011 (Kourou) Quasi Halo L1  Verification of drag free controller for LISA  LISA (2018): gravity wave detector (3 S/C)  462 kg S/C (sun pointing) + prop. module  1910 kg launch mass
    • 11. 11Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Herschel + Planck
    • 12. 12Libration Point Orbits - M. Hechler - ESOC 2009/4/3 GAIA
    • 13. 13Libration Point Orbits - M. Hechler - ESOC 2009/4/3 ESA Missions to L2 in Planning Phase Mission Launch Orbit Objective + Spacecraft PLATO Soyuz from Guiana 2017/2018 Quasi Halo L2  Precision photometry of 105 of nearby milky way stars  Observation of transitions of planetary companions  Observations of stellar oscillations  Sun-shield, Passive cooling  Array of up to 30 individual telescopes EUCLID Soyuz from Guiana 2017/2018 Quasi Halo L2  Optical and Near-Infrared Imaging and Spectroscopy  Mapping of weak gravitational lensing  Coverage of galactic sky above 30 deg latitude  3-axis stabilised, sun-shield and HGA, 1.2 m mirror SPICA Soyuz from Guiana 2017/2018 Quasi Halo L2  Infrared astronomy IXO (XEUS) Ariane 5 ECA 2018 Quasi Halo L2  X-ray imaging and spectroscopy of the hot universe  Physics of black holes and hot accretion discs  long (35m) focal length by formation flying of mirror and detector spacecraft
    • 14. 14Libration Point Orbits - M. Hechler - ESOC 2009/4/3 From Linear Theory to Numerical Orbit Construction
    • 15. 15Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Basics: Linear Theory of Orbits at L2 (1) Circular restricted problem
    • 16. 16Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Basics: Linear Theory of Orbits at L2 (2) Lis-ELEVEC AND Lis-VECELE (for t=0) : Fast variables Ax= 1/c2 Ay
    • 17. 17Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Basics: Linear Theory of Orbits at L2 (3)
    • 18. 18Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Basics: Linear Theory of Orbits at L2 (4)  Exact problem inherits properties from linear problem  Use ∆V-direction of linear problem in numerical method
    • 19. 19Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Exact Problem: Unstable Manifold Orbits at L2 are unstable  escape for small deviation  generates unstable manifold e+λt To solar system x-y rotating = in ecliptic Sun-Earth on x-axis
    • 20. 20Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Outward Escape on Unstable Manifold 10 m/s 1 m/s 10 cm/s 1 revolution ≈ 180 days
    • 21. 21Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Inward Escape on Unstable Manifold -10 m/s -1 m/s -10 cm/s
    • 22. 22Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Central Manifold No escape -10 cm/s +10 cm/s
    • 23. 23Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Stable Manifold of HERSCHEL Orbit “Stable manifold” = surface-structure in space, which flows into orbit e-λtPerigee of stable manifold Of Herschel Orbit Transfer Herschel orbit
    • 24. 24Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Stable Manifold of PLANCK Orbit e-λt Perigee of stable manifold of Planck Orbit Transfer Planck orbit Jump onto stable manifold
    • 25. 25Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Stable Manifold and Weak Stability Boundary Weak Stability Boundary: • Perigee from launch conditions (i, Ω, ω) • Scan and bisection in perigee velocity (Vp)  One non escape solution (free transfer)  Transfers to Quasi-Halos Example bisection Forward Stable Manifold: • Backward integration from orbit at L1/2 • “Jump onto stable manifold”  Two local minima in ΔV (fast/slow)  Used for transfers to “constrained” orbits slow fast Backward
    • 26. 26Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Example Bisection in Perigee Velocity mm/s ≈ orbit determination accuracy ≈ dynamic noise Integrator accuracy Computer word length To Sun To Earth bisec dv(m/s) days rstop(km) 0 3.013107155600253 103.5 855325.3 0 4.013107155600253 119.7 2319036.2 0 3.513107155600253 140.6 2044164.8 1 3.263107155600253 137.6 1094563.3 2 3.388107155600254 160.1 2070930.6 3 3.325607155600254 213.0 827084.2 4 3.356857155600254 174.8 2080976.5 5 3.341232155600254 193.6 2052903.9 6 3.333419655600254 293.1 2324254.3 7 3.329513405600254 230.4 801319.6 8 3.331466530600254 248.6 828429.7 9 3.332443093100254 272.2 825422.6 10 3.332931374350254 321.7 2019561.9 11 3.332687233725253 291.1 859153.3 12 3.332809304037754 346.9 1158669.1 13 3.332870339194003 338.0 2084535.5 14 3.332839821615878 364.8 2092294.5 15 3.332824562826816 384.0 870165.7 16 3.332832192221348 450.0 2280789.3 1 mm/s ≈ 1% of radiation pressure effect over 10 days Bisection in velocity Integrate until > 2000000 km or < 800000 km “Non-escape” if >450 days 2 revolutions Stop box
    • 27. 27Libration Point Orbits - M. Hechler - ESOC 2009/4/3 • Same procedure from any point in orbit • Initial guess of state  from forward integration of transfer  or from analytic theory • Correction of velocity by scan + bisection along escape direction u • Integrate e.g. 1/2 revolution and repeat forward process  “Mathematical” ∆V’s ≈ 1 mm/s per revolution (far below navigation ∆V)  No gradients, no terminal conditions (only non-escape)  Orbit construction based on Weak Stability Boundary method Numerical Construction of Orbits to/at L2
    • 28. 28Libration Point Orbits - M. Hechler - ESOC 2009/4/3  Weak Stability Boundary Orbit Construction Method also works for perturbed gravity field Non-gravitational Accelerations • No difference for orbit generation method if other deterministic perturbations are included in dynamics:  Radiation pressure (may be lifting – GAIA)  attitude manoeuvre effects, if predictable  wheel off-loading (may be used to correct orbit – Herschel)  large known manoeuvres
    • 29. 29Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Orbit with 10 x Radiation Pressure Shift towards sun
    • 30. 30Libration Point Orbits - M. Hechler - ESOC 2009/4/3 HERSCHEL Orbit (Halo) 4 years propagation Launch 2009/4/29 – 13:24:24 Remark: For the nominal launch time the Herschel orbit is nearly a Halo (by chance)
    • 31. 31Libration Point Orbits - M. Hechler - ESOC 2009/4/3 PLANCK Orbit (Lissajous) 2.5 years propagation
    • 32. 32Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Transfer Optimisation and Launch Windows Navigation and Orbit Maintenance later
    • 33. 33Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Result: Optimum Planck Transfer Herschel “free” transfer Planck ≠ Herschel from day 2
    • 34. 34Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Transfer Optimisation (Planck/GAIA) Solved by forward/backward shooting Departure variables launch (i, Ω, ω, Vp, Tp) Arrival variables Lissajous (Ay,Az,Фz,Ty=0) With prescribed properties: (α < 15o and no eclipse) Matching constraint (ΔX=0) Cost functional (Σ║ΔV║=min) Fast Transfer: Ti – Tp < 50 days Ti Tp Day 2 Tm Number of manoeuvres depends on case Herschel/Planck: (i, ω, Vp, Tp~Ω) all fixed GAIA: ω and Ω free
    • 35. 35Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Planck Manoeuvre Model • All manoeuvres are done in sun pointing mode • Decomposition modelled in optimisation  ΔV of each thruster + phase angle (5 variables)  ║ΔV║ = sum of ΔV’s of thrusters ~ propellant  Optimisation in general converges to “pure manoeuvres”
    • 36. 36Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Launch Windows Definition of Launch Window: • Dates (seasonal) and hours (daily) for which a launch is possible • “Best” launcher target conditions (possibly as function of day and hour) Constraints: • Propellant on spacecraft required to reach a given orbit (type) • Geometric conditions: – eclipses – sun aspect angles Typical method: • Calculate orbits for scan in launch times • shade areas for which one of the conditions is not satisfied
    • 37. 37Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Herschel/Planck Launch Window: Vp Selection • Double launch on ARIANE 5: rp, i, ω for maximum mass • Fixed launch conditions in Earth fixed frame at lift-off  only one flight program on launcher (cost saving)  Vp to be fixed • Both spacecraft correct perigee velocity Vp (~ ra)  Fast transfer vp about 2 m/s below vp of stable manifold Launch conditions of Ariane: Vp = Vesc - 30.32 m/s  Ra = 1 200 000 km Osculating at Planck S/C separation  J2 must be on in integration Remaining degree of freedom: Ω J2 corrected 4/29
    • 38. 38Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Herschel/Planck Launch Window (as of 2007) • 140 launch orbit
    • 39. 39Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Herschel/Planck Launch Window (as of 2008) • 60 launch orbit
    • 40. 40Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Herschel/Planck Launch Window (now) • 60 launch orbit • Both S/C tanks full • Change of launcher axis at fairing separation  15 min lost 4/29 Launch window on 2009/4/29: 13:24:24 – 14:06:24 UT
    • 41. 41Libration Point Orbits - M. Hechler - ESOC 2009/4/3 GAIA Seasonal Launch Window • Soyuz from French Guyana to circular parking orbit at 15o inclination • 2nd Fregat burn at any time in circular orbit to reach L2 transfer (free ω)  Two degrees of freedom (Ω, ω)  optimum transfer near ecliptic plane 165 m/s One optimum launch time per day
    • 42. 42Libration Point Orbits - M. Hechler - ESOC 2009/4/3 GAIA Fast Transfer to L2 and 6 Years Orbit • Cycle eclipse to eclipse > 6 years  choice of initial z-phase
    • 43. 43Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Transfer with Lunar Gravity Assist • Perigee of stable manifold of small amplitude Lissajous orbits above 50000 km • Intersects Moon orbit at two points → two solutions from Moon to given orbit Cross section  Launch orbit near lunar orbit plane
    • 44. 44Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Lunar Flyby Opportunities + Phasing • One opportunity per month • Earth to moon 2-3 days  Navigation difficult • Necessity of phasing orbits – Launch to orbit below moon (200000 km) – Sequence of apogee raising manoeuvres – Also perigee raising manoeuvres necessary against perturbations – Launcher dispersion correction with manoeuvres at perigee
    • 45. 45Libration Point Orbits - M. Hechler - ESOC 2009/4/3 L2 Transfer with Lunar Gravity Assist • Currently discussed again for Baikonur back-up
    • 46. 46Libration Point Orbits - M. Hechler - ESOC 2009/4/3 LISA Pathfinder Apogee Raising Sequence • Large liquid propulsion stage (440 N, 321 s) connected to S/C • ~15 manoeuvres at perigee to raise apogee from 900 km to 1.3 x 106 *km (to L1) • Gravity loss limited to 1.5% (in total ΔV) Example case above for old Rockot launch scenario
    • 47. 47Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Navigation and Orbit Maintenance
    • 48. 48Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Recovery from Escape due to Noise 20 m/s 30 days Extreme example Cost increases with delay 65 m/s • Unknown random accelerations or events perturb orbit • S/C will move away on unstable manifold (in or out)  Orbit Maintenance necessary  back on stable manifold of another non-escape orbit
    • 49. 49Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Navigation Process Dynamics Measurements XRW Noise Errors XEST z Estimation Orbit Determination Manoeuvre Optimisation Objective = no escape Ground System XEST Manoeuvre Execution Execution error XRW ∆V Real World Measurements Model Dynamics Model with noise model State extended By noise model parameters Random number
    • 50. 50Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Navigation Mission Analysis Dynamics Model Measurements Model xsim Noise Random numbers Errors Random Numbers x, C z Estimation Filter OD + Covariance analysis Manoeuvre Optimisation Objective = no escape Ground System Representation x, C Manoeuvre Execution Execution error Random numbers xsim ∆V Real World Simulation Measurements Model Dynamics Model with noise model |∆V| sampling Knowledge Covariance Monte-Carlo
    • 51. 51Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Navigation: Example Noise Modelling (GAIA) Estimated random variables
    • 52. 52Libration Point Orbits - M. Hechler - ESOC 2009/4/3 • Orbit Determination as for any other mission  Kalman filter for covariance mission analysis  Expansion of state vector e.g. by radiation pressure model parameters • Orbit correction manoeuvres (stochastic part)  During transfer as for interplanetary navigation  Retarget to position at insertion + remove escape component at insertion  Apply linear algorithm + |∆V| statistics by sampling  Alternatively by Monte-Carlo analysis re-optimising transfer  In orbit at Libration Point  Retargeting to non-escape orbit (by bisection along escape direction) requires 50% of propellant compared to methods using a reference orbit  Monte-Carlo simulations used for |∆V| accumulated over 1 year in orbit Navigation for Libration Point Orbits
    • 53. 53Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Correction Manoeuvres in Libration Point Orbits • Use estimated state from Monte-Carlo analysis • Calculate ∆V to remove escape component  with bisection method along escape direction as for reference orbit generation • For usual noise assumptions  5 cm/s to 20 cm/s every 30 days  reference orbit generation (mm/s every 180 days) in-bedded in maintenance Independent of orbit type
    • 54. 54Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Orbit Determination Accuracy (Doppler + Range) (from one ground station)  Plane of Sky measurement (POS) Manoeuvre execution errors Zero declination Requirement
    • 55. 55Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Orbit Determination Accuracy with POS • 1 day batch processing of tracking date, with estimation of manoeuvre errors • Required velocity accuracy = 2.5 mm/s (for aberration) only reached with Plane of Sky (POS) measurements
    • 56. 56Libration Point Orbits - M. Hechler - ESOC 2009/4/3 Summary on Libration Point Orbits • Orbit generation at Libration points:  Orbit in general not prescribed, only properties for space missions prescribed  Bisection method along unstable direction generates non escape orbits  Orbit generation method works also for perturbed field • Transfers to Libration point orbit:  Stable manifold free transfers to Quasi-Halos (Herschel, JWST)  Optimum transfers to Lissajous orbits  Forward/backward shooting (Planck, GAIA)  Sequence of Highly Eccentric Orbits HEO  manoeuvres at perigee (LPF)  Lunar flyby improves mass budget  Option for GAIA from Baikonur • Orbit determination and maintenance:  Orbit determination requires noise models with parameter estimation  Manoeuvres with same method as orbit generation (removal of unstable component) Herschel/Planck, but also GAIA, LPF, JWST on the way