ops forum 10.12.2010 on the brink of gravity - mission analysis for space astronomy

Libration Point Mission AnalysisLandgraf, Renk, De Pascale, Paulino

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On the Brink of Gravity - Mission Analysis for Space Astronomy

Dr. rer. nat. Markus Landgraf, Dr.-Ing. Florian Renk, Paolo de Pascale, and Tatiana Paulino

• Before I start, let me quickly introduce our little libration point mission analysis team that is mainly working on Gaia, LISA Pathfinder, and Cosmic Vision missions:• Dr.-Ing. Florian Renk: Gaia, LISA Pathfinder, Euclid, Plato, Space Time Explorer• Paolo de Pascale: Gaia• Tatiana Paulino: Euclid, Gaia, Space Time Explorer• Bram de Vogeleer (not presenting): LISA Pathfinder, Euclid• Marcel Düring (not presenting): LISA Pathfinder• myself: Gaia, LISA Pathfinder, Euclid, PLATO, IXO• If we are to explore not only our solar system, but the cosmos in more general, we have to rely on telescopes to collect photons• This is most efficiently done outside the Earthʼs atmosphere and thus follows the need for installation and operation of space telescopes• Globally speaking Europe leads astronomy research, due to our history, but also due to the missions currently operated in ESOC: Herschel/Planck, XMM-Newton, Integral• This has been recognized by the Association Aeronautique et Astronautique de France (3AF), which awarded the award for outstanding space endeavors to the Herschel/Planck mission• The first generation of space telescopes, represented by Hubble, was about exploiting the improved seeing• The second generation, which is operational today, covered the full range of the electro-magnetic spectrum from microwaves (COBE) to gamma rays (Compton)• The third generation, now in the study phase (in Europe: Cosmic Vision) will see instruments dedicated to more specific tasks: mapping the extra-galactic sky in near-IR or staring at hundreds of thousands of stars for years in the optical, ...• All of those missions share a set of basic requirements:• maximum accessible sky area (no bright objects distributed over the sky)• high thermal stability• low radiation exposure due to sensitive instrumentation• ideally no eclipses of the Sun by the Earth or the Moon• Fixed Sun-spacecraft-Earth geometry


Overview• The best spot for space astronomy

• Getting there

• What if anything goes wrong?

• Manoeuvre history of Herschel/Planck

• The way of the future

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The Best Spot for Space Astronomy

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[TP presenting]


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• The requirements of the survey-kind of space telescopes are nearly demonstrated by Planck• Planck spins to image (in the sub-mm waveband) the sky background using a telescope line of

sight perpendicular to the spinning axis• The only possible configuration for solar arrays and the antenna is thus with a bore-sight along this

axis• Consequently must the Sun and the Earth be somewhere along this axis, ideally on the same side• The ideal place for Planck is thus a permanent station on the night side of the Earth• But we cannot just park a spacecraft at a spot in space, or can we?


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• Another example is the Euclid mission, a proposal to Cosmic Vision• Its goal is to map the sky outside the plane of the Milky Way using a step-and-stare strategy• But, a (hopefully small) thermal shield must be pointed to the Sun and the Earth such that the detectors and optics

are not unnecessarily heated• A strategy must be found, in which the exposures can be performed such that the Sun and Earth remain behind the

shield• Again it would be nice to stay at 00:00 local time far away from the Earth• We know that for a satellite orbiting the Earth there is no such parking spot• Only if we have highly elliptical orbits we experience a period of quasi-stationarity at the apogee (like for XMM-

Newton and Integral)• If we increase the apogee distance further, we get to the edge of the Earthʼs gravitational sphere of influence• So it is logical to consider the three-body problem of the Sun, Earth, and the spacecraft and to look for “parking

spots”


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[ML presenting]• In the search for a “parking spot” we have a look at the potential energy, because local extrema always mean local stability• In a frame that is rotating with the Earth around the Sun (thus, with the Sun at the centre the Earth is always on the positive x-axis) there are three forces: Gravity from the

Sun, gravity from the Earth, and the centrifugal force of the rotation (the Coriolis force is not considered for stationary objects)• Here we find nice spots to “park” labelled L1 to 5, which are the Lagrange Points, first discovered by Leonard Euler in 1750, but named after Joseph Louis Lagrange, who

worked intensively on expanding Newtonian Mechanics beyond the two-body problem.• Lagrange described the restricted circular three-body problem in 1772.• We cannot make right away a statement about their stability as we are not in an inertial frame, in which the acceleration is merely the gradient of the potential energy• Indeed L1 to L3 are stable in two directions and unstable in one (mathematically speaking there are four Eigenvalues of the characteristic polynomial: positive real, negative

real, positive imaginary and negative imaginary)• L4 and L5 are stable in all three directions (for any relevant celestial system)• L2 certainly fulfils our requirement of having the Sun and the Earth in a fixed geometry on one side of the spacecraft• However, exactly at L2 the Sun never shines• We will have to swing about that point, in first order we create a three-dimensional pendulum


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• If you superimpose pendulum motion (oscillation, representing the imaginary Eigenvalues) in two directions with two different frequencies, you get Lissajous figures (described first by Nathaniel Bowditch in 1815, but more completely by Jules Antoine Lissajous in 1857)

• Conveniently displayed on an oscilloscope it can be seen how the difference in frequency in x- and y-direction creates a continuous shift in the phase

• For equal frequencies ellipses are created, the semi-major axis ratio depends on the phase difference


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• At L2 the oscillation frequency is given by the curvature of the local potential, which is different in the plane of the Earthʼs motion (we all it x-y-plane or simply the Ecliptic plane) than in the out-of plane direction(we call it ʻzʼ).

• The oscillation in the plane is coupled, thus there can be no phase drift• Thus we will get a closed curve in the x-y plane and a Lissajous figure in the x-z and y-z planes• The Lissajous behaviour leads to crossing of the Sun-Earth line at some point, i.e. eclipses• If we go further away from the point of equilibrium, the curvature of the potential changes, which

leads to deviations from the harmonic oscillator• In first order the frequencies of the oscillations now depend on the chosen amplitude. • Indeed a ratio of in- and out-of-plane amplitude can be found for which both frequencies match


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• Closed curves - solutions are found: the Halo orbit• Only the ratio of in- and out-of plane amplitude is fixed, so there are big and small ones• Also there are two mirror-symmetric classes: pro and retrograde orbits (not shown)


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• For now we only looked at the curves created by the stable motion (imaginary Eigenvalues)• The unstable motion is an exponential approach (or departure) from (or to) the stable solution• Depending on the final (initial) phase of the arrival (departure) there are many curves, together they

for the stable (unstable) manifold, a two-dimensional object in 3-D space (which is actually a projection of the 4-D hyperplane in 6-D phase space)

• Here the view in the plane is shown


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• View “from behind”, i.e. the y-z- plane or the view from the Earth (at 0,0)• It can be seen that parts of the stable manifold lie close to the Earth• When on the stable manifold the spacecraft drifts towards the libration orbit without any

manoeuvres• The launcher simply has to inject into the stable manifold• For Lissajous orbits (here a halo is shown) the stable manifold is as far away as 100,000km, so a

transfer with additional insertion manoeuvre is needed


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• From the effective potential of the three-body problem it appears that any displacement or impulse in the Sun- or anti-Sun direction would lead to an escape

• Due to the additional complication of the Coriolis force in the rotating frame, the ʻescape directionʼ for displacements and impulses are different

• Operationally the escape direction for impulses is relevant, it is in-plane, 28.5deg off the Sun-direction

• The escape direction does not depend on where the spacecraft is along the orbit (in the linear approximation)


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• In addition to Lissajous and Halo orbits there are quasi-halo orbits, which have large amplitudes to approximate the halo condition, but that condition is not prescribed in order to minimise the insertion delta-v

• Indeed the stable manifold of large amplitude obits intersects with near- Earth space, so that theoretically a quasi-halo orbit can be reached directly by injection from the launcher without any further deterministic manoeuvres

• Such solutions are exploited for mass-critical missions like LISA Pathfinder• Shown here are three orbits around L1 for LISA Pathfinder following a launch by Rockot• The animation shows the real scale of the Sun-Earth three-body problem• The co-linear points are 1.5 million km from earth, one hundredth of the Sun-Earth distance


Getting There

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• Here a typical direct ascent like by Ariane 5 from Kourou can be seen• After the first vertical ascent the gravity turn starts building up orbital velocity• During the fully optimised flight phase of the upper stage, lower altitudes are reached as the most

efficient increase of orbital energy is achieved at high speeds• After the injection and separation, the spacecraft rises steeply over the indian ocean (altitude not to

scale with Earthʼs surface)• Thus, the naturally the acquisition of signal occurs over the ground-stations in Australia• The spent stages end up in the Atlantic ocean• target apogee radius (Herschel/Planck, Ariane 5 direct ascent) 1,223,699km


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• As an example we show the Herschel/Planck Launch Window• Libration orbits around the co-linear Lagrange points are reached when the apse line of the transfer orbit (orbit after injection

as represented by the state vector) is oriented (roughly)along the Sun-Earth line• The ideal (to reach the smallest possible free quasi-Halo) targeting angle (in-plane angle between apse line and line towards

Lagrange point)is 15deg• Many constraints close the launch window, however, e.g. when going to L2 eclipses during transfer are possible and have to

be avoided by choosing the launch date and time• Other constraints are: • limited delta-v for insertion into the Lissajous orbit for Planck• solar aspect angle at separation from the launcher• sun aspect angle at fairing separation• eclipses during the in-orbit phase for Herschel• H3 is s/c separation• „two manoeuvres to final orbit“ means that delta-v for two manoeuvres is smaller than a single one


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• A very different ascent can be seen for Soyuz from Baikonur• Here the upper stage first injects the nose module (upper stage and payload) into a circular parking

orbit between 180 and 190km• On this high inclination parking orbit the nose module coasts for up to one revolution• A second ignition of the Fregat injects the spacecraft into its transfer towards the libration point• In this scenario the orientation of the apse line (basically the target direction) can be chosen with

much more flexibility (within the plane of the parking orbit, though)• For different launch dates different orientations of the transfer will be required as the libration points

lie in the Sun (L1) or ant-Sun (L2) direction, which moves along the ecliptic plane by about 1deg per day.

• This, the ground-station, over which initial acquisition occurs changes with the launch date, depending on where the second burn of the Fregat occurs


What if Anything Goes Wrong?

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[FR presenting for PDP]


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• The two critical phases of a conventional mission to a Lagrange point are the launch and the TCM1

• If the TCM1 is delayed the manoeuvre can be performed later (with some limits) with extra delta-v• If the launcher underperforms, however, the spacecraft remains in Earth orbit• Recovery strategies call for an attempt to raise the apogee at the next perigee (left figure)• Problem: during the one orbit the Sun has moved (~1deg per day), or, in our rotating frame our

apse line (to negative angles)• Therefore the departure will no longer go to the L2 direction, but some tens of degrees off• However, large-amplitude libration orbits can still be reached • The limit of the off-targeting is somewhere around 30deg•


• For smaller orbits the Moon can have an extreme effect on the trajectory, so that the inclination can be changed, the perigee raised, etc.

• The natural solution is a lunar fly-by as the fly-by angle (angle between apse line and anti-Sun direction) is around -50deg (corresponds to 1 to 2 month wait between nominal transfer and lunar fly-by)


Contingency Rules for Herschel

• if apogee >950,000km, perform TCM1 as normal, preferably on day 1 (only if Δv<100m/s)

• below 950,000km and above 600,000km stay on eccentric orbit for one revolution, perform perigee raising (if needed) at apogee, and perform new perigee manoeuvre

• below 600,000km more complex recovery scenarios might be needed (lunar gravity assist, extended wait), recommendation: establish resonant orbit with the Moon

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[FR presenting]


Herschel/Planck Manoeuvre HistoryHerschel (CReMA Budget:77.5m/s)Herschel (CReMA Budget:77.5m/s) Planck (CReMA Budget: 293.5 m/s)Planck (CReMA Budget: 293.5 m/s)

2009-05-14T13:12:02 launch 2009-05-14T13:12:02 launch

2009-05-15T15:28:21 OCM LEOP: 9.0m/s 2009-05-15T21:26:17 OCM LEOP: 14.4m/s

2009-05-18T18:13:03 OCM touch-up: 1.0m/s 2009-06-06T04:30:23 OCM-z: 153.6m/s

2009-06-10T13:31:43 trans. OCM 1: 0.7m/s 2009-06-17T18:52:58 OCM-z touch: 12.6m/s

2009-06-24T13:04:59 trans. OCM 2: 0.2m/s 2009-07-02T18:48:16 OCM ins.: 59.9m/s

2009-07-17T12:31:50 s-k 1: 0.4m/s 2009-07-15T11:04:43 OCM ins. touch: 0.5m/s

2009-08-14T11:29:06 s-k 2: 0.2m/s 2009-08-14T12:31:06 s-k 1: 0.11m/s

2009-09-11T21:33:14 s-k 3: 0.1m/s 2009-09-11T21:01:45 s-k 2: 0.04m/s

2009-11-13T18:34:27 s-k 4:0.1 m/s 2009-12-04T17:50:40 s-k 3:0.05m/s

2009-12-18T15:10:10 s-k 5:0.1m/s 2010-01-15T21:59:38 s-k 4:0.06m/s

2010-01-22T16:18:04 s-k 6:0.1m/s 2010-02-26T23:03:35 s-k 5:0.04m/s

2010-03-04T17:59:53 s-k 7:0.3m/s 2010-04-26T17:56:56 s-k 6:0.03m/s

total spent 12.2m/s total spent 242.2m/s

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Use is well below budget for two reasons:1.Accurate launch2.Launch date required perigee velocity close to the injection value (one flight program for Herschel

and Planck covering a full launch period)3.Normally stochastic manoeuvres are budgeted conservatively, i.e. the budget will be above the

actual expenditure for missions like Herschel with a large component of stochastic manoeuvres


The way of the Future

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A libration orbit around L2 can achieve: • no eclipses during transfer and in the operational phase• thermal stability• Earth-spacecraft distance below 1.8 million km for efficient communications (compared to e.g.

Spitzer, in the extended mission >100 million km)• limited variation of Sun-spacecraft Earth angle (depending on amplitude)


•Large scale exploration mission scenarios are considered, which exploit the use of not only the Earth-Sun libration points (SEL), but also the Earth-Moon libration points (MEL)•Here a scenario with multiple launches to an assembly station at EML2 is shown•EML2 has the advantages of constant Earth-Moon geometry (permanent comms link) and

Moon synchrony (surface accessibility, infinite launch window)•From this hub, lunar exploration as well as servicing of space telescopes at SEL2 can be

performed


to Earth

•The libration points of the Earth-Moon system are roughly located 60,000 km away from the moon on the Earth-Moon line.•Due to the bounded rotation of the Moon the relative geometry of the lunar surface does

not change with respect to the libration points•The EML 2 point behind the moon can be reached cheaper than the EML 1 point between

the Earth and the Moon due to the existence of lunar fly-by trajectories. In addition a free transfer (no insertion manoeuvre at EML arrival) exist to the EML 2 point via the Weak Stability Boundary region of the Earth.•This might provide advantages for the assembly of large space infrastructure, since it can

happen in close vicinity to Earth without mass penalties.•Due to the similar energy level of the Sun-Earth and the Earth Moon system connections

between both regions exist (next slide)


•Plot of the unstable manifold departing the EML 2 point in the rotating frame of the Earth. •A bifurcation can be clearly seen. •The transfer is free and does not cost any Δv. •Monthly transfer opportunities exist as well as “exotic” options that can be applied in

order to influence the size of the SEL orbit. •A transfer back from the SEL region towards the EML regions is also possible, however, it

is much more constrained due to the stringent departure conditions on the SEL together with the required phasing of the moon.


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[ML presenting]• The lunar fly-by can be used to recover a contingency, or as the baseline• Without going into details, more elaborate mission designs are considered• Main objectives:

- maximise mass to final orbit- avoid critical manoeuvres (like TCM1)

• Here a lunar fly-by scenario for Gaia after launch from Baikonur (back-up) is shown• Soyuz would launch Gaia into a highly elliptical orbit with apogee at 200,000km (available launch performance 2,235 instead of 2,096kg)• A phasing strategy is used and small manoeuvres are performed with the on-board CPS to reach the Moon• Because with a lunar fly-by it is possible to directly jump onto the stable manifold of a Lissajous orbit, no (or very little) insertion delta-v is needed.• The insertion delta-v (current assignment 165 m/s) can be spent for apogee raising• The Earth-orbiting and fly-by phase add to the time spent in transfer: 203d for the fly-by vs. 35d for the baseline fast transfer• The final total delta-v is even smaller (196 vs 255m/s) than for the baseline mission from Kourou (launch mass 2,160kg)• The mission does not contain any critical manoeuvres


• L2 offers more science opportunities than are currently exploited• Besides the benign radiation and thermal environment there is the very flat gravity environment• This allows spacecraft formations to be installed with minimum Δv requirement• Forced motion is practically “manoeuvre free” below baselines of thousands of kilometres• Longer baselines could follow natural motion, i.e. libration orbits on different phase angles• The example shows three spacecraft on a halo orbit separated by 120° in phase angle• This could be used for VLBI, Planet hunting, or gravity wave detection


• Formation flying missions pose new challenges to mission analysis:• Here a formation rendezvous is shown• The target quasi-halo orbit (green) is populated with a partial formation• A new spacecraft is to be added by using the black transfer• In order to achieve a finite launch window the transfer on the stable manifold of one libration orbit

must be connected to the stable manifold of the target orbit• Here a two manoeuvre strategy is shown that is analoguous to the Lambert problem of the two-

body system• Future application of libration orbits will require research in methods, mathematics, and software.


Радиоастрон Миллиметрон

LISA Darwin

•Formations at the libration points allow extreme baselines•Currently the following science projects require such a design:-Darwin/Terrestrial Planet Finder (Nulling Interferometry, ESA/NASA)-LISA (Michaelson Interferometer, Gravity Waves, ESA/NASA)-sub-millimetre imaging like Radioastron and Millimetron (VLBI, Roskosmos)•Those missions promise astonishing science:-Detection and imaging of extra-solar planets-Detection of life on extra-solar planets-Detection of gravity waves: “gravity astronomy”

ops forum 10.12.2010 on the brink of gravity - mission analysis for space astronomy

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