Herschel, Planck and GAIA Orbit Design Martin Hechler, Jordi Cobos Mission Analysis Section, ESA/ESOC, D-64293 Darmstadt, Germany TERMA, Flight Dynamics Division, ESA/ESOC, Darmstadt, Germany 7th International Conference on Libration Point Orbits and Application Parador d Aiguablava, Girona, Spain 10-14 June 2002
Contents ESA missions using Orbits at L 2 and L 1 Numerical Orbit Construction Method Use of linearised equations in circular restricted 3-body problem Escape and non-escape directions Numerical construction of transfer orbits Numerical construction of Non-escape orbits at L 2 Amplitudes reduction maneuvers Eclipse avoidance maneuvers Applications to ESA Projects Herschel/Planck launch window (L 2 Transfer from ARIANE launch) Navigation and orbit maintenance Conclusions M. Hechler - ESOC/TOS-GMA 1
Orbits at L 2 in Sun Earth System for Astronomy Missions Y Orbit 1.5 106 km L 1 X L 2 Earth/Moon Advantages for Astronomy Missions: L 4 L 5 Sun and Earth nearly aligned from spacecraft 60 o 60 o SUN stable thermal environment only one specific direction excluded for viewing (moving 360 per year) medium gain antenna in sun pointing possible Drawbacks: 1.5 10 6 km for communications L3 long transfer duration ( 90 days) instability of orbits frequent manoeuvres M. Hechler - ESOC/TOS-GMA 2
ESA Missions in Libration Point Orbits (Sun-Earth System) Project Launch Objective Orbit, Remarks SOHO 1995 Sun observations Halo at L 1. Still in operations Smart 2 2006 Technology demonstration Cluster of S/C near L 1, drag free, drift for LISA and Darwin away Herschel 2007 Far infrared astronomy Large amplitude Lissajous at L 2. Stable manifold transfer from ARIANE launch Planck 2007 Cosmic background 15 Lissajous at L 2. Double launch with Herschel on ARIANE Eddington 2008 Star seismology Lissajous at L 2, Hersche/Planck spacecraft bus reuse SGST 2009+ Second Generation Space Lissajous at L 2, with NASA Telescope GAIA 2010+ Astrometry 15 Lissajous at L 2, orbit control with FEEPS M. Hechler - ESOC/TOS-GMA 3
Halos Versus Lissajous Orbits Halo orbits: quasi-periodic : z-frequency = x-y-frequency large amplitudes (A y 600000 km) free of eclipse by definition (at L 2 ) Lissajous orbits: z-frequency x-y-frequency small amplitudes possible, e.g. S yz = A 2 y + A 2 z 200000 km or free transfer possible for ARIANE or SOYUZ (but then large amplitudes) free of eclipse up to 6 years, small V for another 6 years Z (KM) 400000 200000 0-200000 -400000 y-z-rotating Earth shadow (13000 km) -400000-200000 0 200000 400000 Y (KM) No essential advantage of Halos for the candidate space missions M. Hechler - ESOC/TOS-GMA 4
Mission Analysis Tasks for a Libration Point Missions Mission Analysis Tasks: Orbit selection for given spacecraft tasks (amplitudes, phasing, eclipses) Construction of the transfer orbit (for given launcher), amplitude reduction manoeuvre Strategies for orbit maintenance and eclipse avoidance Launch window Propellant budget Definition of conditions for spacecraft design (e.g. thruster mounting) Mathematics and Software: Understanding of dynamical properties (theory) Selection of methods (e.g. starting from linear theory) Necessary extension of theory (e.g. amplitude reduction, eclipse avoidance) Numerical implementation (with exact dynamics) M. Hechler - ESOC/TOS-GMA 5
Review: Linearised Dynamics of Orbits Around L 2 Differential equations for relative motion in frame rotating with Earth around sun ẍ 2 ẏ (1 + 2K) x = 0 ÿ + 2 ẋ (1 K) y = 0 z + K z = 0 Complete solution of linearised problem (x-y motion and z-motion are uncoupled) x = A 1 e λ xyt + A 2 e λ xyt + A x cos (ω xy t + φ xy ) y = A 1 c 1 e λ xyt A 2 c 1 e λ xyt A x c 2 sin (ω xy t + φ xy ) z = A z cos (ω z t + φ z ) Choice of initial conditions such that A 1 = A 2 = 0 Lissajous Orbits x = A x cos(ω xy t) y = A y sin(ω xy t), with A y = c 2 A x z = A z cos(ω z t + φ z ). c 1, c 2, ω xy, λ xy and ω z are constants depending on K, K depends on the masses only, K=4.001012 for L 1, K=3.940522 for L 2 c 2 =3.187 for L 2 REMARK: Formulation for L 1 identical except for K and ref. system transformation M. Hechler - ESOC/TOS-GMA 6
Escape and Non-Escape Directions (in Linear Theory) Integration constants A 1, A 2 linear functions of initial conditions: A 1 A 2 = c 2 ω xy 2d 1 c 2 ω xy 2d 1 ω xy 2d 2 c 2 1 2d 2 2d 1 ω xy c 2 1 2d 2 2d 2 2d 1 x 0 y 0 ẋ 0 + ẋ 0 ẏ 0 + ẏ 0 with d 1 = c 1 λ xy + c 2 ω xy d 2 = c 1 ω xy c 2 λ xy X-Y part of state vector (x 0, y 0, ẋ 0, ẏ 0 ) T on Lissajous orbit A 1 = 0 Then velocity increment V = ( ẋ 0, ẏ 0 ) T with ( c 2 d 2, 1 d 1 ) ẋ 0 ẏ 0 = 0 will not lead to an escape from the family of orbits around L 2. M. Hechler - ESOC/TOS-GMA 7
Escape and Non-Escape Directions (2) Define (in the x-y-plane) escape direction: V component along u excites unstable motion ±u T = ± ( c 2 d 2, 1 d 1 ) non-escape direction orthogonal to u ±s T = ± ( 1 d 1, c 2 d 2 ) The escape line is +28.6 from the x-axis (=sun to Earth axis) The non-escape line is -61.4 from the x-axis Then In the linear problem these directions do not depend on the point in orbit (homogeneous) Velocity increment components along ±u control the stability Velocity increment components in the plane spanned by s and the z-direction will only change amplitude or phase of a non-escape orbit Remark: This holds for the linearised problem M. Hechler - ESOC/TOS-GMA 8
Escape and Non-Escape Directions (3) x u s u s y u s L 2 u s Remark: Property used for thruster mounting on Herschel and Planck M. Hechler - ESOC/TOS-GMA 9
Numerical Construction of Transfers and Orbits to L 2 (Step 1: Scan) Scan over perigee velocity to find location of fuzzy boundary (nearly parabolic) 7 DV 747.0000000000 TFIN 46.694 RFIN 2571489.5 IP -1 DVTOT 747.000 from +x 90.0 8 DV 744.0000000000 TFIN 53.521 RFIN 2573648.1 IP -1 DVTOT 744.000 from +x 90.0 9 DV 741.0000000000 TFIN 64.336 RFIN 2525932.4 IP -1 DVTOT 741.000 from +x 90.0 10 DV 738.0000000000 TFIN 107.157 RFIN 2546203.3 IP -1 DVTOT 738.000 from +x 90.0 11 DV 735.0000000000 TFIN 77.880 RFIN 90498.8 IP 1 DVTOT 735.000 from +x 90.0 12 DV 732.0000000000 TFIN 55.806 RFIN 27342.1 IP 1 DVTOT 732.000 from +x 90.0 13 DV 729.0000000000 TFIN 45.162 RFIN 12583.2 IP 1 DVTOT 729.000 from +x 90.0 14 DV 726.0000000000 TFIN 38.207 RFIN 9398.2 IP 1 DVTOT 726.000 from +x 90.0 15 DV 723.0000000000 TFIN 33.219 RFIN 7589.2 IP 1 DVTOT 723.000 from +x 90.0 1e+06 x-y-rotating 500000 Y (KM) 0-500000 -1e+06-2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) REMARK: The orbit construction idea comes from the work of Edward Belbruno M. Hechler - ESOC/TOS-GMA 10
Numerical Construction of Transfers and Orbits to L 2 (Step 2: Bisection) Bisection in V s along perigee velocity 1. forward integration for e.g. 450 days and stop if orbit escapes from Earth system (e.g. 2 10 6 km ) or orbit comes close to Earth (e.g. 0.5 10 6 km ) 2. change initial velocity and repeat 1 (bisection depending on stop) 3. if stop conditions not reached Non-escape orbit at L 2 1 DV 738.0000000000 TFIN 107.157 RFIN 2546203.3 IP -1 DVTOT 738.000 from +x 90.0 2 DV 736.5000000000 TFIN 115.362 RFIN 451451.3 IP 1 DVTOT 736.500 from +x 90.0 3 DV 737.2500000000 TFIN 136.113 RFIN 2502975.5 IP -1 DVTOT 737.250 from +x 90.0 4 DV 736.8750000000 TFIN 262.092 RFIN 2543153.7 IP -1 DVTOT 736.875 from +x 90.0 5 DV 736.6875000000 TFIN 137.911 RFIN 649127.3 IP 1 DVTOT 736.688 from +x 90.0 6 DV 736.7812500000 TFIN 168.178 RFIN 530412.6 IP 1 DVTOT 736.781 from +x 90.0 7 DV 736.8281250000 TFIN 183.741 RFIN 255171.1 IP 1 DVTOT 736.828 from +x 90.0 8 DV 736.8515625000 TFIN 197.399 RFIN 75861.5 IP 1 DVTOT 736.852 from +x 90.0 9 DV 736.8632812500 TFIN 213.156 RFIN 11826.7 IP 1 DVTOT 736.863 from +x 90.0 10 DV 736.8691406250 TFIN 232.520 RFIN 14927.4 IP 1 DVTOT 736.869 from +x 90.0 11 DV 736.8720703125 TFIN 258.844 RFIN 85512.6 IP 1 DVTOT 736.872 from +x 90.0 12 DV 736.8735351562 TFIN 334.092 RFIN 2535923.7 IP -1 DVTOT 736.874 from +x 90.0 13 DV 736.8728027344 TFIN 276.132 RFIN 193892.9 IP 1 DVTOT 736.873 from +x 90.0 14 DV 736.8731689453 TFIN 295.617 RFIN 431594.3 IP 1 DVTOT 736.873 from +x 90.0 15 DV 736.8733520508 TFIN 339.052 RFIN 620920.9 IP 1 DVTOT 736.873 from +x 90.0 16 DV 736.8734436035 TFIN 372.092 RFIN 2563350.2 IP -1 DVTOT 736.873 from +x 90.0 17 DV 736.8733978271 TFIN 361.540 RFIN 286026.0 IP 1 DVTOT 736.873 from +x 90.0 18 DV 736.8734207153 TFIN 385.041 RFIN 29649.1 IP 1 DVTOT 736.873 from +x 90.0 19 DV 736.8734321594 TFIN 412.092 RFIN 2500385.8 IP -1 DVTOT 736.873 from +x 90.0 20 DV 736.8734264374 TFIN 407.022 RFIN 16355.0 IP 1 DVTOT 736.873 from +x 90.0 21 DV 736.8734292984 TFIN 453.494 RFIN 194775.5 IP 1 DVTOT 736.873 from +x 90.0 M. Hechler - ESOC/TOS-GMA 11
Numerical Construction of Transfers and Orbits to L 2 (Step 2: Bisection) 1e+06 x-y-rotating 500000 Y (KM) 0-500000 -1e+06-2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) M. Hechler - ESOC/TOS-GMA 12
Numerical Construction of Orbits at L 2 ( V along Escape Direction) Same type of bisection, but with V along escape direction u of linear theory Initial guess either from transfer orbit or from a state vector for given amplitudes and phase calculated by any analytic theory Forward integration to next crossing of x-z-plane and repeat of correction method z-component uncontrolled 1 DV 0.0000000000 TFIN 341.391 RFIN 194782.7 IP 1 DVTOT 0.000 from +x 0.0 2 DV 2.5000000000 TFIN 148.000 RFIN 2503830.5 IP -1 DVTOT 2.500 from +x 151.4 3 DV 1.2500000000 TFIN 174.000 RFIN 2509335.5 IP -1 DVTOT 1.250 from +x 151.4 4 DV 0.6250000000 TFIN 196.000 RFIN 2535459.8 IP -1 DVTOT 0.625 from +x 151.4 5 DV 0.3125000000 TFIN 212.000 RFIN 2582775.8 IP -1 DVTOT 0.312 from +x 151.4 6 DV 0.1562500000 TFIN 224.000 RFIN 2584026.5 IP -1 DVTOT 0.156 from +x 151.4 7 DV 0.0781250000 TFIN 236.000 RFIN 2585050.7 IP -1 DVTOT 0.078 from +x 151.4 8 DV 0.0390625000 TFIN 248.000 RFIN 2537365.8 IP -1 DVTOT 0.039 from +x 151.4 9 DV 0.0195312500 TFIN 262.000 RFIN 2519552.8 IP -1 DVTOT 0.020 from +x 151.4 10 DV 0.0097656250 TFIN 280.000 RFIN 2559225.2 IP -1 DVTOT 0.010 from +x 151.4 11 DV 0.0048828125 TFIN 298.000 RFIN 2514475.0 IP -1 DVTOT 0.005 from +x 151.4 12 DV 0.0024414062 TFIN 324.000 RFIN 2518329.7 IP -1 DVTOT 0.002 from +x 151.4 13 DV 0.0012207031 TFIN 366.000 RFIN 2505636.3 IP -1 DVTOT 0.001 from +x 151.4 14 DV 0.0006103516 TFIN 405.238 RFIN 616851.1 IP 1 DVTOT 0.001 from +x 151.4 15 DV 0.0009155273 TFIN 388.000 RFIN 2515481.6 IP -1 DVTOT 0.001 from +x 151.4 16 DV 0.0007629395 TFIN 410.000 RFIN 2515283.6 IP -1 DVTOT 0.001 from +x 151.4 17 DV 0.0006866455 TFIN 452.379 RFIN 44090.8 IP 1 DVTOT 0.001 from +x 151.4 M. Hechler - ESOC/TOS-GMA 13
Numerical Construction of Orbits at L 2 (Herschel 2007/2/15 Launch) 1e+06 x-y-rotating manoeuvers 1e+06 y-z-rotating Earth shadow (13000 km) manoeuvers 500000 500000 Y (KM) 0 Y (KM) 0-500000 -500000-1e+06-1e+06-2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) -500000 0 500000 Z (KM) Z (KM) 500000 0-500000 -2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) x-z-rotating manoeuvers Herschel Orbit: Transfer construction method leads to selection of Lissajous orbit (A y, A z, φ z ) around L 2 such that its stable manifold touches best ARIANE launch conditions M. Hechler - ESOC/TOS-GMA 14
Amplitude Reduction Manoeuvres (along Non-Escape Direction) Manoeuvres along non-escape direction will create transition onto stable manifold of another orbit (A 2 0)) Optimum time 4.6 days before y = 0 V function of amplitude (size) change (Jordi Cobos, MAS WP 398, 1997) Start value from linear theory: V = A y 3.648001 10 7 s 1 Similar formula for z-component + criteria how to combine Numerical correction (bisection) along escape direction 1 DV 0.0000000000 TFIN 89.409 RFIN 121548.0 IP 1 DVTOT 170.994 from +x 117.6... 7 DV 21.9531250000 TFIN 288.000 RFIN 2516405.7 IP -1 DVTOT 172.397 from +x 124.9 8 DV 21.9140625000 TFIN 241.967 RFIN 210446.5 IP 1 DVTOT 172.392 from +x 124.9 9 DV 21.9335937500 TFIN 260.296 RFIN 112360.6 IP 1 DVTOT 172.395 from +x 124.9 10 DV 21.9433593750 TFIN 283.628 RFIN 72628.2 IP 1 DVTOT 172.396 from +x 124.9 11 DV 21.9482421875 TFIN 358.160 RFIN 335736.2 IP 1 DVTOT 172.397 from +x 124.9 12 DV 21.9506835938 TFIN 308.000 RFIN 2542317.6 IP -1 DVTOT 172.397 from +x 124.9 13 DV 21.9494628906 TFIN 330.000 RFIN 2553409.8 IP -1 DVTOT 172.397 from +x 124.9 14 DV 21.9488525391 TFIN 354.000 RFIN 2537415.6 IP -1 DVTOT 172.397 from +x 124.9 15 DV 21.9485473633 TFIN 388.000 RFIN 2527299.6 IP -1 DVTOT 172.397 from +x 124.9 16 DV 21.9483947754 TFIN 391.641 RFIN 328911.6 IP 1 DVTOT 172.397 from +x 124.9 17 DV 21.9484710693 TFIN 430.000 RFIN 2556265.1 IP -1 DVTOT 172.397 from +x 124.9 18 DV 21.9484329224 TFIN 414.843 RFIN 217892.5 IP 1 DVTOT 172.397 from +x 124.9 19 DV 21.9484519958 TFIN 446.901 RFIN 102305.4 IP 1 DVTOT 172.397 from +x 124.9 20 DV 21.9484615326 TFIN 450.000 RFIN 2241871.7 IP 1 DVTOT 172.397 from +x 124.9 M. Hechler - ESOC/TOS-GMA 15
Planck Orbit (15 Maximum Earth-Spacecraft-Sun Angle) 1e+06 x-y-rotating manoeuvers 1e+06 y-z-rotating Earth shadow (13000 km) manoeuvers 500000 500000 Y (KM) 0 Y (KM) 0-500000 -500000-1e+06-1e+06-2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) -500000 0 500000 Z (KM) x-z-rotating manoeuvers Planck Orbit: Z (KM) 500000 0 Double launch on ARIANE 5 with Herschel Separation at launch -500000 Same transfer orbit -2e+06-1.5e+06-1e+06-500000 0 500000 1e+06 X (KM) Amplitude reduction M. Hechler - ESOC/TOS-GMA 16
Eclipse Avoidance Manoeuvres (in xy-plane and/or in z) Z (KM) 200000 100000 0-100000 -200000 without ecl. dv Earth shadow (13000 km) manoeuvers after ecl. dv -100000 100000 Y (KM) 2007/3/21 launch of Planck (z-reversal, 10 orbit ) Calculated using linear theory with numerical correction along escape direction If amplitude ratio A z A y 5.8397258 c 2 ω z = 0.923 then manoeuvre along z, else in xy-plane z-manoeuvre turns around z-velocity at point with maximum y estimate for size of V v 2 ωz 2 + 5.8397258 c 2 2 r 0 S yz S 2 yz r 2 0 For the Planck orbit with a maximum sunspacecraft-earth angle of 15 a limit of 15 m/s has been estimated M. Hechler - ESOC/TOS-GMA 17
Herschel/Planck Launch Window (Principle) For a launch with ARIANE for maximum performance the argument of pericentre is fixed this comes from the fact that the current upper stage is very small for the overall launcher performance, so the lower composite puts the upper stage with the spacecraft into an eccentric orbit. The right ascension of the ascending node is a function of launch hour (as usual) For each launch time (date and hour) the perigee altitude and the direction of the perigee velocity is prescribed Modulus of perigee velocity to a non-escape orbit then by bisection All other parameters (e.g. size of amplitude reduction manoeuvre) result from this Tables in launch date and launch time Level line figures with condition shading M. Hechler - ESOC/TOS-GMA 18
Herschel/Planck Launch Window (ARIANE Performance) Remarks: A5E/SV A5E/CA delayed ign. immediate inclination i 6.1 12.73 argument of perigee ω 110.1 188.8 ascending node Ω K 239.1 209.84 (relative Kourou at launch) perigee altitude h p 235.1 km 114.3 km time from lift off to S/C sep. 133.2 min 24.9 min true anomaly at injection f inj 36.3 47.83 mass in orbit 5310 kg 6267 kg coast duration 106.9 min 0 coast arc apogee altitude 3661.8 km - impact longitude of H155 89.9 impact latitude of H155 5.55 A5E/SV with delayed ignition of versatile upper stage (mission baseline) A5E/CA with sub-optimum ascent to shift line of apsides by 15 M. Hechler - ESOC/TOS-GMA 19
Herschel/Planck Launch Window (ARIANE 5 ESV with Delayed Ignition) Launch window for 15 maximum sun aspect angle and 180 m/s total V on Planck (deterministic only) M. Hechler - ESOC/TOS-GMA 20
Herschel/Planck Launch Window (Size of Herschel Orbit) M. Hechler - ESOC/TOS-GMA 21
Herschel/Planck Launch Window (ARIANE 5 ECA Direct Injection) Launch window for 15 maximum sun aspect angle and 325 m/s total V on Planck (deterministic only), backup scenario M. Hechler - ESOC/TOS-GMA 22
Navigation and Orbit Maintenance (1) Navigation = orbit determination + stochastic orbit correction manoeuvres Modelling of all error sources in tracking and dynamics Covariance analysis or Monte-Carlo analysis Derivation of statistics for orbit correction manoeuvres 99-percentile for propellant loading During transfer: Removal of launcher dispersion within 2 days Monte-Carlo analysis with correction manoeuvre along velocity Same type of bisection to create non escape orbit as above Result: 99-percentile of 55 m/s for ARIANE 5 ESV case for first correction Touch-ups and Lissajous orbit injection manoeuvre correction: 3 m/s M. Hechler - ESOC/TOS-GMA 23
Navigation and Orbit Maintenance (2) In Lissajous orbit: Simulations for Doppler and range tracking Several orbit maintenance strategies tested: 1. Classical interplanetary navigation with a shifting target position. 2. Linear Quadratic Control. 3. Removal of the velocity component along the escape direction. 4. Re-computation of future periodic trajectory at each manoeuvre time. For all strategies 1 m/s per year seems to be sufficient to maintain orbit near L 2. Strategy along escape direction preferred clear a priori knowledge on manoeuvre directions (homogeneous) thruster mounting and calculations for manoeuvre decomposition based on this. M. Hechler - ESOC/TOS-GMA 24
Conclusions Lissajous orbits around L 2 selected for ESA missions (free transfer, small size) Herschel/Planck, GAIA, Eddington, Smart 2 Linear theory provides good initial guesses for construction of these orbits: Numerical corrections along escape direction to generate non-escape orbits Size reduction manoeuvres along non-escape direction (numerically refined) Eclipse avoidance manoeuvres 15 m/s for another 6 year without eclipse Transfer optimisation: Herschel injected by ARIANE onto stable manifold of large size Lissajous orbit Planck injected to small amplitude orbit from this transfer by size reduction V 1 m/s per year sufficient for orbit maintenance rather independent of strategy Other related studies: GAIA (concept before 2001): transfer from midday GTO via L 1 (bifurcations) GAIA (now): direct transfer from SOYUZ launch or use of lunar gravity assists. GAIA orbit maintenance using FEEPS (very low thrust electric propulsion) Smart 2: around L 1 for ARIANE double launch, slow escape without orbit control M. Hechler - ESOC/TOS-GMA 25
References [1] R. Farquhar, Halo Orbits and Lunar Swingby Missions of the 1990 s, Acta Astronautica, Vol 24, 1991, pp. 227-234 [2] G. Gómez, A. Jorba, J. Masdemont, C. Simó, Study Refinement of Semi-Analytical Halo Orbit Theory, Final Report ESOC Contract 8625/89/D/MD(SC), Barcelona, April 1991 [3] E. Belbruno, G.B. Amata, Low Energy Transfer to Mars and the Moon Using Fuzzy Boundary Theory, Alenia Torino, SD-RP-AI-0202, July 1996 [4] M. Hechler, GAIA/FIRST Mission Analysis: ARIANE and the Orbits around L 2, MAS WP 393, ESOC February 1997 [5] M. Hechler, J. Cobos, FIRST Mission Analysis: Transfers to Small Lissajous Orbits around L 2, MAS WP 398, ESOC July 1997 [6] M. Hechler, J. Cobos, FIRST/PLANCK and GAIA Mission Analysis: Launch Windows with Eclipse Avoidance Manoeuvres, MAS WP 402, ESOC December 1997 [7] J. Cobos, M. Hechler, FIRST/PLANCK Mission Analysis: Transfer to Lissajous Orbit Using the Stable Manifold, MAS WP 412, ESOC December 1998 [8] M. Belló Mora, F. Blesa Moreno, Study on Navigation for Earth Libration Points, Final Report ESA Contract No. 12571/97/D/IM(SC), 1999 M. Hechler - ESOC/TOS-GMA 26