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


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


Libration Points

Orbits at L2

Missions Going there


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


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


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


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


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”)


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


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


Herschel + Planck


GAIA


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


From Linear Theory to

Numerical Orbit Construction


Basics: Linear Theory of Orbits at L2 (1)

Circular restricted problem



Lis-ELEVEC AND Lis-VECELE (for t=0) :

Fast variables

Ax= 1/c2 Ay





Exact problem inherits properties from linear problem Use ∆V-direction of linear problem in numerical method


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


Outward Escape on Unstable Manifold

10 m/s

1 m/s

10 cm/s

1 revolution ≈ 180 days


Inward Escape on Unstable Manifold

-10 m/s

-1 m/s -10 cm/s


Central Manifold

No escape

-10 cm/s

+10 cm/s


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


Stable Manifold of PLANCK Orbit

e-λt

Perigee of stable manifoldof Planck Orbit

Transfer

Planck orbit

Jump onto stable manifold


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


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


• 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


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


Orbit with 10 x Radiation Pressure

Shift towards sun


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)


PLANCK Orbit (Lissajous)

2.5 years propagation


Transfer Optimisation and

Launch Windows

Navigation and

Orbit Maintenance

later


Result: Optimum Planck Transfer

Herschel “free” transfer

Planck ≠ Herschel from day 2


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


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”


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


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


Herschel/Planck Launch Window (as of 2007)

• 140 launch orbit


Herschel/Planck Launch Window (as of 2008)

• 60 launch orbit


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


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


GAIA Fast Transfer to L2 and 6 Years Orbit

• Cycle eclipse to eclipse > 6 years choice of initial z-phase


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


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


L2 Transfer with Lunar Gravity Assist

• Currently discussed again for Baikonur back-up


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


Navigation and

Orbit Maintenance


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


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


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


Navigation: Example Noise Modelling (GAIA)

Estimated random variables


• 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


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


Orbit Determination Accuracy (Doppler + Range)

(from one ground station)

Plane of Sky measurement (POS)

Manoeuvre execution errors

Zero declination

Requirement


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


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

Technology

OPS Forum Libration Orbits 03.04.2009