ON LIBRATION POINT ORBITS
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ON LIBRATION POINT ORBITS . H/P 2009/04/29-13:24:24UT. OPS-G FORUM Martin Hechler, GFA ESOC 2009/4/3. 56 slides. Contents. Lagrange points in Sun-Earth system and orbits around them ESA missions at L 2 and L 1 (Sun-Earth): Why there ? In which orbits ?

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Libration point orbits m hechler esoc 2009

ON LIBRATION POINT ORBITS

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

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


Libration point orbits m hechler esoc 2009

Libration Points

Orbits at L2

Missions Going there

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


Libration points in sun earth

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 l 2

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

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

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


Criteria for selection of orbit class at l 2

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 l 2

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”)

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


Used orbit classes for missions at l 1 and l 2

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 l 2 and l 1

ESA Missions to L2 (and L1)

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


Herschel planck

Herschel + Planck

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


Libration point orbits m hechler esoc 2009

GAIA

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


Esa missions to l 2 in planning phase

ESA Missions to L2 in Planning Phase

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


Libration point orbits m hechler esoc 2009

From Linear Theory to

Numerical Orbit Construction

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


Basics linear theory of orbits at l 2 1

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 l 2 2

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 l 2 3

Basics: Linear Theory of Orbits at L2 (3)

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


Basics linear theory of orbits at l 2 4

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


Libration point orbits m hechler esoc 2009

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


Libration point orbits m hechler esoc 2009

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


Libration point orbits m hechler esoc 2009

Inward Escape on Unstable Manifold

-10 cm/s

-1 m/s

-10 m/s

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


Libration point orbits m hechler esoc 2009

Central Manifold

No escape

+10 cm/s

-10 cm/s

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


Libration point orbits m hechler esoc 2009

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


Libration point orbits m hechler esoc 2009

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

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

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 l 2

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

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

Orbit with 10 x Radiation Pressure

Shift towards sun

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


Libration point orbits m hechler esoc 2009

HERSCHEL Orbit (Halo)

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


Libration point orbits m hechler esoc 2009

PLANCK Orbit (Lissajous)

2.5 years propagation

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


Libration point orbits m hechler esoc 2009

Navigation and

Orbit Maintenance

later

Transfer Optimisation and

Launch Windows

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


Result optimum planck transfer

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

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

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

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

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


    Herschel planck launch window v p selection

    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

    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

    Herschel/Planck Launch Window (as of 2008)

    • 60 launch orbit

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


    Herschel planck launch window now

    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

    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 l 2 and 6 years orbit

    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

    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

    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


    L 2 transfer with lunar gravity assist

    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

    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


    Libration point orbits m hechler esoc 2009

    Navigation and

    Orbit Maintenance

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


    Recovery from escape due to noise

    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

    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

    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

    Navigation: Example Noise Modelling (GAIA)

    Estimated random variables

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


    Navigation for libration point orbits

    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

    • 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

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


    Correction manoeuvres in 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

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


    Orbit determination accuracy doppler range

    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

    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

    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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