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ON LIBRATION POINT ORBITS

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H/P 2009/04/29-13:24:24UT

OPS-G FORUMMartin Hechler, GFAESOC 2009/4/3

56 slides

Libration 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Orbits at L2

Missions Going there

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Libration Points in Sun-Earth

- Libration Points:
- 5 Lagrange Points
- L1 and L2 of interest for space missions

Lagrange 1736-1813

- 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Orbits at L2

y

Satellite in L2

- Does notwork 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)

x

Libration 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

- free transfer possible, stable manifold
- Lissajous orbits:
- small amplitude possible
- large insertion ∆V

- can be free of eclipse for 6 years
- condition on initial state

- small amplitude possible

- not periodic

- ‘quasi periodic’: z-frequency = x-y-frequency

Libration 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

Libration 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

- Sun and Earth nearly aligned as seen from spacecraft
- Drawbacks:
- 1.5 x 106 km for communication
- However development of deep space communications
technology (X-band, K-band) ameliorates

this disadvantage

- However development of deep space communications
- 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”)

- 1.5 x 106 km for communication

Libration 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

View from Earth

Herschel

JWST

LISA Pathfinder (L1)

XEUS

EUCLID

PLATO

SPICA

Planck

GAIA

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

ESA Missions to L2 (and L1)

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Herschel + Planck

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

GAIA

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

ESA Missions to L2 in Planning Phase

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Numerical Orbit Construction

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Basics: Linear Theory of Orbits at L2 (1)

Circular restricted problem

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Basics: Linear Theory of Orbits at L2 (2)

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

Ax= 1/c2 Ay

Fast variables

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Basics: Linear Theory of Orbits at L2 (3)

Libration 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

Libration 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

x-y rotating = in ecliptic

Sun-Earth on x-axis

To solar system

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Outward Escape on Unstable Manifold

1 m/s

10 cm/s

10 m/s

1 revolution ≈ 180 days

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Inward Escape on Unstable Manifold

-10 cm/s

-1 m/s

-10 m/s

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Stable Manifold of HERSCHEL Orbit

“Stable manifold” = surface-structure in space, which flows into orbit

Herschel orbit

Transfer

Perigee of stable manifold

Of Herschel Orbit

e-λt

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Stable Manifold of PLANCK Orbit

Planck orbit

Transfer

Jump onto stable manifold

e-λt

Perigee of stable manifold

of Planck Orbit

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Stable Manifold and Weak Stability Boundary

Backward

Forward

- 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

- 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

fast

Example bisection

slow

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Example Bisection in Perigee Velocity

Integrate until > 2000000 km or < 800000 km

Bisection in velocity

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

To Earth

To Sun

2 revolutions

“Non-escape” if >450 days

Stop box

1 mm/s ≈ 1% of radiation pressure effect over 10 days

Computer word length

Integrator accuracy

mm/s ≈ orbit determination accuracy ≈ dynamic noise

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Numerical Construction of Orbits to/at L2

- 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

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

- Weak Stability Boundary Orbit Construction Method
- also works for perturbed gravity field

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Orbit with 10 x Radiation Pressure

Shift towards sun

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

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)

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Orbit Maintenance

later

Transfer Optimisation and

Launch Windows

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Result: Optimum Planck Transfer

Herschel “free” transfer

Planck ≠ Herschel from day 2

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Transfer Optimisation (Planck/GAIA)

Herschel/Planck: (i, ω, Vp, Tp~Ω) all fixed

GAIA: ω and Ω free

Solved by forward/backward shooting

Cost functional (Σ║ΔV║=min)

Departure variables

launch (i, Ω, ω, Vp, Tp)

Arrival variables

Lissajous (Ay,Az,Фz,Ty=0)

With prescribed properties:

(α < 15o and no eclipse)

Tp

Day 2

Tm

Fast Transfer:

Ti – Tp < 50 days

Ti

Matching constraint (ΔX=0)

Number of manoeuvres depends on case

Libration 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”

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Launch Windows Constraints:

- 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)

- 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

Libration 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

4/29

J2 corrected

- Osculating at Planck S/C separation
- J2 must be on in integration

Remaining degree of freedom: Ω

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Herschel/Planck Launch Window (as of 2007)

- 140 launch orbit

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Herschel/Planck Launch Window (as of 2008)

- 60 launch orbit

Libration 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

Libration 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

Libration 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

Libration 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

Libration 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

L2 Transfer with Lunar Gravity Assist

- Currently discussed again for Baikonur back-up

Libration 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Recovery from Escape due to Noise

Extreme example

- 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

20 m/s

30 days

65 m/s

Cost increases with delay

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Navigation Process

Real World

Random

number

Execution error

Noise

Manoeuvre

Execution

XRW

Dynamics

XRW

Errors

Measurements

∆V

z

Ground System

State extended

By noise model

parameters

Orbit Determination

Manoeuvre

Optimisation

Measurements Model

Estimation

XEST

Objective

= no escape

Dynamics Model

with noise model

XEST

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Navigation Mission Analysis

Real World Simulation

Noise

Random numbers

Execution error

Random numbers

Manoeuvre

Execution

xsim

Dynamics Model

xsim

Errors

Random Numbers

Measurements Model

Monte-Carlo

|∆V| sampling

∆V

z

Ground System Representation

OD + Covariance analysis

Manoeuvre

Optimisation

Measurements Model

Estimation

Filter

x, C

Objective

= no escape

Dynamics Model

with noise model

x, C

Knowledge

Covariance

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Navigation: Example Noise Modelling (GAIA)

Estimated random variables

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Navigation for Libration Point Orbits

- Orbit Determination as for any other mission
- Kalman filter for covariance mission analysis
- Expansion of state vector e.g. by radiation pressure model parameters

- Kalman filter for covariance mission analysis
- 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

- Retargeting to non-escape orbit (by bisection along escape direction)

- During transfer as for interplanetary navigation

Libration 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

Orbit Determination Accuracy (Doppler + Range)

Manoeuvre execution errors

Zero declination

Requirement

(from one ground station)

Plane of Sky measurement (POS)

Libration 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

Libration 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

Libration Point Orbits - M. Hechler - ESOC 2009/4/3

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