Download - OPS Forum Libration Orbits 03.04.2009
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
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
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Libration Points
Orbits at L2
Missions Going there
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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
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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
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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
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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
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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”)
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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
HerschelJWSTLISA Pathfinder (L1)XEUSEUCLIDPLATOSPICA
PlanckGAIA
View from Earth
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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
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Herschel + Planck
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GAIA
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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
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From Linear Theory to
Numerical Orbit Construction
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Basics: Linear Theory of Orbits at L2 (1)
Circular restricted problem
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Basics: Linear Theory of Orbits at L2 (2)
Lis-ELEVEC AND Lis-VECELE (for t=0) :
Fast variables
Ax= 1/c2 Ay
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Basics: Linear Theory of Orbits at L2 (3)
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Basics: Linear Theory of Orbits at L2 (4)
Exact problem inherits properties from linear problem Use ∆V-direction of linear problem in numerical method
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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
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Outward Escape on Unstable Manifold
10 m/s
1 m/s
10 cm/s
1 revolution ≈ 180 days
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Inward Escape on Unstable Manifold
-10 m/s
-1 m/s -10 cm/s
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Central Manifold
No escape
-10 cm/s
+10 cm/s
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Stable Manifold of HERSCHEL Orbit
“Stable manifold” = surface-structure in space, which flows into orbit
e-λtPerigee of stable manifoldOf Herschel Orbit
Transfer
Herschel orbit
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Stable Manifold of PLANCK Orbit
e-λt
Perigee of stable manifoldof Planck Orbit
Transfer
Planck orbit
Jump onto stable manifold
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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
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Example Bisection in Perigee Velocity
mm/s ≈ orbit determination accuracy ≈ dynamic noise
Integrator accuracy
Computer word length
To SunTo 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
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• 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
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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
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Orbit with 10 x Radiation Pressure
Shift towards sun
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HERSCHEL Orbit (Halo)
4 years propagationLaunch 2009/4/29 – 13:24:24
Remark: For the nominal launch time the Herschel orbit is nearly a Halo (by chance)
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PLANCK Orbit (Lissajous)
2.5 years propagation
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Transfer Optimisation and
Launch Windows
Navigation and
Orbit Maintenance
later
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Result: Optimum Planck Transfer
Herschel “free” transfer
Planck ≠ Herschel from day 2
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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 fixedGAIA: ω and Ω free
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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”
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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
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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 corrected4/29
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Herschel/Planck Launch Window (as of 2007)
• 140 launch orbit
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Herschel/Planck Launch Window (as of 2008)
• 60 launch orbit
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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
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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
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GAIA Fast Transfer to L2 and 6 Years Orbit
• Cycle eclipse to eclipse > 6 years choice of initial z-phase
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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
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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
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L2 Transfer with Lunar Gravity Assist
• Currently discussed again for Baikonur back-up
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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
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Navigation and
Orbit Maintenance
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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
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Navigation Process
Dynamics
Measurements
XRW
Noise
Errors
XEST
z
Estimation
Orbit Determination
ManoeuvreOptimisation
Objective= no escape
Ground System
XEST
ManoeuvreExecution
Execution error
XRW
∆V
Real World
Measurements Model
Dynamics Modelwith noise model
State extendedBy noise modelparameters
Randomnumber
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Navigation Mission Analysis
Dynamics Model
Measurements Model
xsim
NoiseRandom numbers
ErrorsRandom Numbers
x, C
z
EstimationFilter
OD + Covariance analysis
ManoeuvreOptimisation
Objective= no escape
Ground System Representation
x, C
ManoeuvreExecution
Execution errorRandom numbers
xsim
∆V
Real World Simulation
Measurements Model
Dynamics Modelwith noise model
|∆V| sampling
KnowledgeCovariance
Monte-Carlo
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Navigation: Example Noise Modelling (GAIA)
Estimated random variables
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• 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
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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
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Orbit Determination Accuracy (Doppler + Range)
(from one ground station)
Plane of Sky measurement (POS)
Manoeuvre execution errors
Zero declination
Requirement
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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
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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