Herschel, Planck and GAIA Orbit Design. Martin Hechler, Jordi Cobos

Transcription

1 Herschel, Planck and GAIA Orbit Design Martin Hechler, Jordi Cobos Mission Analysis Section, ESA/ESOC, D Darmstadt, Germany TERMA, Flight Dynamics Division, ESA/ESOC, Darmstadt, Germany 7th International Conference on Libration Point Orbits and Application Parador d Aiguablava, Girona, Spain June 2002

2 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

3 Orbits at L 2 in Sun Earth System for Astronomy Missions Y Orbit 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: km for communications L3 long transfer duration ( 90 days) instability of orbits frequent manoeuvres M. Hechler - ESOC/TOS-GMA 2

4 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 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 Second Generation Space Lissajous at L 2, with NASA Telescope GAIA Astrometry 15 Lissajous at L 2, orbit control with FEEPS M. Hechler - ESOC/TOS-GMA 3

5 Halos Versus Lissajous Orbits Halo orbits: quasi-periodic : z-frequency = x-y-frequency large amplitudes (A y 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 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) y-z-rotating Earth shadow (13000 km) Y (KM) No essential advantage of Halos for the candidate space missions M. Hechler - ESOC/TOS-GMA 4

6 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

7 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= for L 1, K= 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

8 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

9 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 from the x-axis (=sun to Earth axis) The non-escape line is 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

10 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

11 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 TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x e+06 x-y-rotating Y (KM) e+06-2e e+06-1e e+06 X (KM) REMARK: The orbit construction idea comes from the work of Edward Belbruno M. Hechler - ESOC/TOS-GMA 10

12 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 km ) or orbit comes close to Earth (e.g 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 TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x 90.0 M. Hechler - ESOC/TOS-GMA 11

13 Numerical Construction of Transfers and Orbits to L 2 (Step 2: Bisection) 1e+06 x-y-rotating Y (KM) e+06-2e e+06-1e e+06 X (KM) M. Hechler - ESOC/TOS-GMA 12

14 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 TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x M. Hechler - ESOC/TOS-GMA 13

15 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 Y (KM) 0 Y (KM) e+06-1e+06-2e e+06-1e e+06 X (KM) Z (KM) Z (KM) e e+06-1e e+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

16 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 s 1 Similar formula for z-component + criteria how to combine Numerical correction (bisection) along escape direction 1 DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP -1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x DV TFIN RFIN IP 1 DVTOT from +x M. Hechler - ESOC/TOS-GMA 15

17 Planck Orbit (15 Maximum Earth-Spacecraft-Sun Angle) 1e+06 x-y-rotating manoeuvers 1e+06 y-z-rotating Earth shadow (13000 km) manoeuvers Y (KM) 0 Y (KM) e+06-1e+06-2e e+06-1e e+06 X (KM) Z (KM) x-z-rotating manoeuvers Planck Orbit: Z (KM) Double launch on ARIANE 5 with Herschel Separation at launch Same transfer orbit -2e e+06-1e e+06 X (KM) Amplitude reduction M. Hechler - ESOC/TOS-GMA 16

18 Eclipse Avoidance Manoeuvres (in xy-plane and/or in z) Z (KM) without ecl. dv Earth shadow (13000 km) manoeuvers after ecl. dv 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 c 2 ω z = 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 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

19 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

20 Herschel/Planck Launch Window (ARIANE Performance) Remarks: A5E/SV A5E/CA delayed ign. immediate inclination i argument of perigee ω ascending node Ω K (relative Kourou at launch) perigee altitude h p km km time from lift off to S/C sep min 24.9 min true anomaly at injection f inj mass in orbit 5310 kg 6267 kg coast duration min 0 coast arc apogee altitude km - impact longitude of H impact latitude of H 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

21 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

22 Herschel/Planck Launch Window (Size of Herschel Orbit) M. Hechler - ESOC/TOS-GMA 21

23 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

24 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

25 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

26 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

27 References [1] R. Farquhar, Halo Orbits and Lunar Swingby Missions of the 1990 s, Acta Astronautica, Vol 24, 1991, pp [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 /97/D/IM(SC), 1999 M. Hechler - ESOC/TOS-GMA 26