Laser Doppler Interferometry Mission for determination of ...emits.sso.esa.int/emits-doc/5125-RD12-LDI-Executive-Summary.pdf · alcatel alenia space italia s.p.a. doc. no : sd-rp-ai-0447

Laser Doppler Interferometry Mission for determination of the

Earth’s Gravity Field

Executive Summary

ESTEC Contract No: 18456/04/NL/CP

SD-RP-AI-0447 December 2005

Alcatel Alenia Space Italia S.p.A.

DOC. No : SD-RP-AI-0447 ISSUE : 01 DATE : 19/DEC/05 PAGE : 2 of 60 CLASS. : DC

DOCUMENT CHANGE RECORD ISSUE DATE REASON FOR CHANGE AFFECTED PARAGRAPHS 01 19/DEC/05 First Issue



TABLE OF CONTENTS

1. INTRODUCTION ..............................................................................................6

2. APPLICABLE AND REFERENCE DOCUMENTS.............................................7

2.1 APPLICABLE DOCUMENTS .........................................................................7 2.2 REFERENCE DOCUMENTS .........................................................................7

3. GEOPHYSICAL PHENOMENA AND THEIR INFLUENCE ON EARTH’S GRAVITY FIELD .....................................................................................................8 3.1 GEOPHYSICAL PHENOMENA INVOLVING MASS REDISTRIBUTION

REVIEW........................................................................................................8 3.2 GEOPOTENTIAL MODIFICATION PRODUCED BY SUMATRA

EARTHQUAKE............................................................................................11

4. GRAVITY FIELD MEASUREMENT.................................................................13 4.1 ANALYSIS OF GRAVITY FIELD MEASUREMENT BY SATELLITE-

SATELLITE TRACKING ..............................................................................13 4.2 GRAVITY FIELD MEASUREMENT NUMERICAL SIMULATION...................18 4.3 SUMATRA EARTHQUAKE DETECTABILITY BY GRAVIMETRIC

MISSIONS ..................................................................................................20

5. SSI MISSION REFERENCE PARAMETERS AND PERFORMANCE REQUIREMENTS..................................................................................................23 5.1 MISSION PARAMETERS ............................................................................23 5.2 SATELLITE ELEMENTS CONCURRING TO THE SCIENTIFIC

MEASUREMENT ........................................................................................25 5.3 PERFORMANCE REQUIREMENTS............................................................26

5.3.1 Satellite-Satellite Distance Variation Measurement Error Tree ...............26 5.3.2 Non-gravitational Relative Acceleration Measurement Error Tree...........27

6. METROLOGY SYSTEM FOR THE SSI MISSION ...........................................29

6.1 LASER METROLOGY SYSTEM FUNCTIONAL AND PERFORMANCE REQUIREMENTS........................................................................................29

6.1.1 Functional Requirements......................................................................29 Main.....................................................................................................29 6.1.2 Performance Requirements ..................................................................29

6.2 RELATIVE DISTANCE VARIATION MEASUREMENT SYSTEM ..................32 6.3 ORIENTATION RELATIVE TO THE LASER BEAM MEASUREMENT

SYSTEM.....................................................................................................34 6.4 LATERAL DISPLACEMENT MEASUREMENT SYSTEM..............................35 6.5 BEAM STEERING MECHANISM.................................................................36 6.6 LASER SOURCE AND STABILISATION SYSTEM.......................................38 6.7 OPTICAL LINK BUDGET.............................................................................40

7. CONTROL SYSTEM FOR THE SSI MISSION.................................................41



7.1 CONTROL SYSTEM REQUIREMENTS SUMMARY ....................................41 7.2 FUNCTIONS AND ORGANISATION OF THE CONTROL SYSTEM .............43 7.3 LINEAR MOTION CONTROL ......................................................................43 7.4 ANGULAR MOTION CONTROL ..................................................................45 7.5 CONTROL SYSTEM PERFORMANCES .....................................................46

8. PRELIMINARY SATELLITE CONFIGURATION .............................................48

9. FIRST PROOF-OF-CONCEPT FOR THE METROLOGY SYSTEM .................55 9.1 ANGULAR SENSOR TEST .........................................................................55 9.2 PERFORMANCE IMPROVEMENT BY AMPLITUDE MODULATION ............56 9.3 LATERAL DISPLACEMENT METROLOGY CONCEPT VERIFICATION.......58

10. ACRONYMS AND ABBREVIATIONS.............................................................60



DISTRIBUTION LIST

Name Q.ty Name Q.ty Bruno LEONE (ESTEC) 20 Barraco Ignazio (AAS-I) 1 Giuseppe Finocchiaro (AAS-I) 1 Gianclaudio Cassisa (AAS-I) 1 Stefano CESARE (AAS-I) 3 Gianfranco Sechi (AAS-I) 1 Luciana BONINO (AAS-I) 1 Marco PISANI (IMGC) 1 Andrea Milani (UniPi) 1 Roberto Sabadini (UniMi) 1 Federica Migliaccio (PoliMi) 1



1. INTRODUCTION This executive summary synthesises the output of the activities performed under the “Laser Doppler Interferometry Mission for Determination of the Earth’s Gravity Field” contract and relative CCN, by Alcatel Alenia Space Italia and its subcontractor/consultants. A detailed description of the study outcomes is provided in the Final Report. The objectives of the activity performed under this contract are established in the SOW ([AD 1]). They are:

1. To define a suitable SSI mission to study atmospheric, oceanographic and hydrological phenomena and Earth interior processes, and to derive the characteristics and performances of its laser metrology system,

2. To establish an appropriate design for the SSI laser metrology system,

3. To define an R&D programme leading to the realisation of the laser metrology system of a SSI mission. The term Satellite-Satellite Interferometry (SSI) defines the Earth gravity field measurement technique based on the tracking of the distance variation between a satellite pair in LEO by means of a laser metrology system (Figure 1-1). The denomination SSI Mission will be used throughout the document equivalently to Laser Doppler Interferometry Mission. From the inter-satellite distance variation it is possible to get information about the gravitational accelerations acting on the satellites and then on the geopotential. The technique is similar to that adopted on the GRACE mission (the so called Low-Low Satellite-Satellite Tracking), but here the use of a laser interferometer rather than a radio-frequency ranging system allows in principle to increase significantly the precision in the retrieval of the gravity field, provided that the distance measurement resolution is matched by the measurement resolution of the non-gravitational accelerations acting on the satellites. The objective of the SSI mission is fact the monitoring of the modifications of Earth gravity field map at high resolution (up to harmonic degree 180-240) over long periods of time of at least 5 to 10 years (ref. [AD 1]). This means substantially to achieve the typical GOCE resolution in the geopotential variation determination over a time period typical of the CHAMP and GRACE missions. This study has been led by Alcatel Alenia Space Italia (AAS-I) as Prime Contractor with the fundamental contribution of the subcontractor Institute of Metrology “G. Colonnetti” (laser metrology and proof of concept test) and of the consultants University of Milano (geophysical phenomena review and analysis), Polytechnic of Milano (gravity field measurement analysis), University of Pisa (gravity field measurement numerical simulation and SSI mission measurement error model). The study team was composed by the following key personnel: - Stefano Cesare (study manager), Luciana Bonino, Fulvio Bresciani, Giuseppe Catastini, Simona De Sanctis,

Massimo Dumontel, Gianfranco Sechi (AAS-I) - Marco Pisani (Institute of Metrology “G. Colonnetti”) - Roberto Sabadini, Anna Maria Marotta, Giorgio Dalla Via (University of Milano) - Alberta Albertella, Federica Migliaccio, Mirko Reguzzoni, Fernando Sansò (Polytechnic of Milano) - Andrea Milani, Alessandro Rossi, Daniela Villani (University of Pisa)

From the Agency side, the study has been managed by Mr. Bruno Leone from the Opto-electronics Section.

Earth

satellite 2 satellite 1

g2 g1

∆d

Figure 1.3-1: Principle of the SSI technique: the inter-satellite distance variation ∆d produced by the gravity

accelerations g1, g2 is measured by a laser metrology system.



2. APPLICABLE AND REFERENCE DOCUMENTS

2.1 APPLICABLE DOCUMENTS [AD 1] Laser Doppler Interferometry Mission For Determination of the Earth’s Gravity Field. Statement of

Work, TOS-MME/2003/489, Issue 1, Sept. 2003

2.2 REFERENCE DOCUMENTS [RD 1] Laser Doppler Interferometry Mission: Final Report, SD-RP-AI-0445, December 2005



3. GEOPHYSICAL PHENOMENA AND THEIR INFLUENCE ON EARTH’S GRAVITY FIELD

3.1 GEOPHYSICAL PHENOMENA INVOLVING MASS REDISTRIBUTION REVIEW In order to pursue a sustainable development it is necessary to model the geophysical processes characterizing the Solid Earth (post glacial rebound, tectonics subduction), the Atmosphere (global and small-scale circulation), the Cryosphere (ice mass unbalance, mountain glaciers melting) and the Hydrosphere (sea level changes, ocean dynamics). These geophysical processes involve mass redistribution. Thus their evolution can be monitored through the variations they induce in the Earth geopotential and in the related quantities: geoid height (the geoid altitude measured relatively to the reference Earth ellipsoid) and gravity anomaly (the difference between the actual gravity acceleration in a given location and that referred to the reference Earth ellipsoid in the same location). For understanding the long-term climatic variations, it is of particular importance the identification of the trends produced by these phenomena on the gravity field parameters. A summary of all the geophysical phenomena reviewed during the study provided in Table 3.1-1, together with:

• required measurement accuracy of geoid undulation or gravity anomaly needed to detect the phenomenon;

• the degree ℓ of the geopotential spherical harmonic corresponding to the typical spatial resolution of the phenomenon (ℓ = 20000 km/spatial resolution km);

• the typical temporal period of the phenomenon;

• the latitude range over which the phenomenon takes place. The only geophysical phenomenon, among those analyzed, for which the corresponding variation of the gravity field (in terms of gravity anomaly) is not detectable by the SSI mission, is the tectonics subduction because the time scale of this phenomenon is millions of years (apart from the catastrophic events represented by the earthquakes). Nonetheless the SSI mission can provide a very detailed static map of the gravity anomaly produced by the tectonics subduction, useful for a better knowledge and modelling of this phenomenon. The detectability by the SSI mission of the effect produced by earthquakes on the Earth geopotential is addressed in sections 3.2, 4.3 considering the example of the catastrophic event occurred in Sumatra at the end of the year 2004. Figure 3.1-1 shows effect of two of the smaller secular geophysical phenomena on the geoid height variation per harmonic degree: in practice this is the smaller signal produced by the various geophysical phenomena to be detected by the SSI mission. The blue line is the time derivative of the zonal components of the geoid per degree, in mm/yr, up to degree 200, due to the uniform ice loss over Greenland, equivalent to global sea level rise of 0.4 mm/yr. This simulation is of importance to separate the present-day contribution from Post Glacial Rebound (PGR). It is likely that ice loss in Antarctica and Greenland is not uniform. The green curve in Figure 3.1-1 shows the time derivative of the geoid per degree, in mm/yr, assuming that ice loss is occurring only in the southernmost part of Greenland, over a disk centered at the colatitude of 24 degree and longitude of 318 degree (equivalent to a mountain glacier melting). The small ice disk can be considered also representative of the hydrological cycle of the smallest basins, since both grounded ice and water load are treated in the same fashion by our modelling. The only difference is that melting in Greenland stands for the secular term, thus remaining constant for years, while the numbers in Figure 3.1-1 have a cycle of six months, once referred to the water load of hydrological basins.



Required measurement accuracy Geophysical Phenomena

Geoid height Gravity anomaly

Max. harmonic degree (spatial

resolution)

Typical time period

Extension in latitude

Post Glacial Rebound

1 mm/year (cumulative error)

Summed from ℓ =2 to 200

Secular Northern hemisphere: +40° ÷ + 85° Southern hemisphere: -90° ÷ -60° Antarctica: -90° ÷ -60°

Ice mass unbalance in Antarctica and Greenland

0.05÷ 0.7 mm/year (single harmonic degree variance)

ℓ = 200 (100 km)

Secular

Greenland: +60° ÷ +83.5°

Pacific fire belt:

-70° ÷ +70°

Solid Earth

Tectonics - Subduction

25 ÷ 100 mgal 2 ÷ 9⋅10-6 mgal/yr

(30 ÷ 100 km) 10 Myr

Calabrian and Aegean: +30° ÷ +40°

Alps: +44° ÷ +48°

Mountain Glaciers 0.02÷ 0.07 mm/yr (single harmonic degree variance)

ℓ = 200 (100 km)

Annual to secular

Patagonian Ice fields:

-56° ÷ -44° World wide:

-56° ÷ 85° Hydrology: global hydrological cycle and land water storage

0.02÷0.07 mm/yr (single harmonic degree variance, Po plain basin)

ℓ = 200 (100 km)

Months

Regional scale (Po plain): +44° ÷ +46°

Sea Level Changes 1 mm/yr (cumulative error)

Summed from ℓ =2 to 200

Secular -60° ÷ 90°

Basin scale mass changes

1 cm ℓ = 4 to 20 Several years

Narrow scale components of global ocean circulation

5 ÷ 10 mm ℓ = 400 to 2000 Annual

Mean flow – eddy flow interaction

5 ÷ 10 mm ℓ = 200 to 2000 Annual

Ocean Dynamics

Bottom currents

0.1 ÷ 1 mm

(cumulative errors)

ℓ = 100 to 2000 Months to decades

-60° ÷ 90°

Global circulation

0.5 mm/yr ℓ = 4 (5000 km) Annual to secular

All latitudes Atmospheric Dynamics

Small scale circulation (2.5° × 2.5° data sampling)

0.5 mm/yr (upper bound)

ℓ = 70 (300 km) Annual

Table 3.1-1: Gravity signal of the entire range of geophysical process



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180 200

Geopotential harmonic degree

Geo

id h

eigh

t var

iatio

n [m

m/y

ear]

Mountain Glaciers

Ice Mass Unbalance

Figure 3.1-1: Geoid height variation generated by two of the smaller secular geophysical phenomena as a function of the harmonic degree



3.2 GEOPOTENTIAL MODIFICATION PRODUCED BY SUMATRA EARTHQUAKE The 2004 Sumatra earthquake (parameter given in Table 3.2-1) displaced a large amount of material in the radial direction with respect to the Earth's centre (Figure 3.2-1). The shallow, almost instantaneous occurrence of uplift and subsidence, with an offset of about 6 m triggered the tsunami which flooded the coasts throughout the Indian Ocean. This earthquake, like other similar events, is thus expected to have permanently changed the gravity in a broad region surrounding its causative fault.

Ms 4.05 x 1022 Nm Strike 320° Dip 11°

Fault length 1000 km Fault width 90 km

Homogeneous slip over the fault 15 m Epicenter coordinates 3.1°N - 95.7°E

Table 3.2-1: Sumatra Earthquake parameters

Figure 3.2-1: Vertical coseismic displacement in metres in the geographical reference frame caused by Sumatra earthquake. Upwarping of the crust is in red, downwarping in blue. The inset shows the displacement along the section perpendicular to the fault.



The upward and downward displacement of the order of metres of a large portion of the ocean bottom in the Sumatra trench region produces a gravity signal, represented in Figure 3.1-12 by the geoid anomaly, that translates the perturbed gravity field into the displacement of the hydrostatic component of the sea-surface after the earthquake, thereby excluding the effects of winds, currents and tides. The pattern of the geoid anomaly is smoother than the vertical displacement due to the slow decay with distance of the gravitational signal from the displaced crustal material. A positive geoid anomaly of 12 mm was induced by the uplifted lithosphere, while the subsided region caused a negative geoid anomaly of -6 mm; the westernmost part of Sumatra is overprinted by a geoid anomaly of about -2 mm. The possibility of detecting this Sumatra earthquake generated geoid anomaly by the currently operative (GRACE), planned (GOCE) and designed (SSI) gravity missions is addressed in section 4.3, where the effect of this anomaly on the quantities measured by these space missions is evaluated.

Figure 3.1-11: Geoid anomaly in millimetres produced by the Sumatra earthquake. Positive geoid anomalies are given in red and negative ones in blue.



4. GRAVITY FIELD MEASUREMENT

4.1 ANALYSIS OF GRAVITY FIELD MEASUREMENT BY SATELLITE-SATELLITE TRACKING A semianalytic model has been developed in order to express the geopotential measurement error as function of the main parameters of the SSI mission, in which the information about the Earth gravity are obtained from the tracking of the distance variation between two satellites flying on the same orbital path at low altitude. In the development of this model, the gravitational potential is split into two parts: a model potential V and the residual part U (anomalous potential) which is what we want to retrieve. The final expression obtained for the variance of the estimation error of the anomalous potential Fourier coefficients is:

( )( ) ( )

)ˆ()ˆ(

)910( )28(4

)1()12(k)ˆ(

22

2422422222

222

2

2

2222

22222

,,

kk

k

AB

SJJkkJkkM

JSMk

J

Jk

JkAkakI

σσ

ω

εωσ νν

=

++−

++⋅−+−−

−=lll

from which the errors on the determination of the geoid undulations (N) and to the gravity anomalies (∆g) can be obtained as follows

∑

∑

−

=∆

=

l

l

l

l

l 222

22

222

)1( )ˆ(

)ˆ(

σσ

σσ

RGMg

RN

where

∑=

=l

ll

0

22 )ˆ(2m

mCσσ (error degree variance), and

∑=

−

=l

ll

l

02

22,

2

)ˆ(

)(1)ˆ(

p k

pmm

pA

IFC

σ

σ .

The above expressions allows to quickly compute with a good approximation the achievable error on the geopotential coefficients, the geoid height and the gravity anomaly as function of the satellite orbit altitude and inclination, the mission duration, the inter-satellite distance and its measurement error and of the error in the measurement of the non-gravitational accelerations affecting the two satellites. Figure 4.1-1 provides the error degree variances (for l = 200) as a function of the inter-satellite distance. The error decreases up to about L0 = 10 km and then becomes stable, suggesting the choice of such a value for the reference inter-satellite distance. For a fixed inter-satellite distance of 10 km, a mission-length of one year (corresponding to about eight repeat periods) the error degree variances on the geopotential harmonics have been computed for several measurement errors of the inter-satellite distance and of the non-gravitational accelerations. This parametric analysis enables to find the spectral densities of these measurement errors suitable to fulfil the objectives of the SSI mission: they are shown in Figure 4.1-2. The resulting error degree variance and the cumulative errors in terms of geoid undulations and gravity anomalies are shown in Figure 4.1-3a,b,c respectively. The error degree variance is compared with three different signal degree variances: the upper curve represents the gravity field degree variances obtained using Kaula’s rule; the two lower curves represent 10% and 1% of Kaula’s degree variances and can be associated to the gravity field variations to be monitored. The computation has been carried out for two altitudes: 325 km and 350 km. Of course, the lower the satellites fly, the lower the error is, especially at high degrees. In any case, the global error at degree 200, for instance in terms of geoid undulations, is of the order of few millimeters, which should fulfil the expected geophysical requirements.



The cumulative error of the geoid undulations as a function of the latitude, has been also computed and shown in Figure 4.1-4. This representation is useful to study the spatial distribution of the error; in this figure we can note, for instance, the effect of the orbit inclination that, being sun-synchronous, leaves the polar caps not covered by the satellite tracks: in fact the error increases in proximity of the poles. Concerning the phenomena with secular variations, as a first approximation we can estimate the trend of these phenomena by least squares interpolation of 1-year solutions (linear regression). The estimation error is computed on the basis of the a-priori error variance of the 1-year solutions. Figure 4.1-5 represents the cumulative errors of this trend estimation procedure, both for a 7-year mission at an altitude of h = 350 km and for a 6-year mission at an altitude of h = 325 km. In this figure these errors are compared to the signal generated by two of the smaller secular geophysical phenomena (section 3.1). Although the mission length is shorter, the solution at an altitude of 325 km is still better: only for harmonics higher than ℓ = 160, the melting of the smallest ice disk cannot be captured.

Figure 4.1-1: Error degree variances as a function of the inter-satellite distance



1 10 4 0.001 0.01 0.1 11 10 13

1 10 12

1 10 11

1 10 10

1 10 9

1 10 8

1 10 7

1 10 6

relative acceleration measurement error SDrelative distance measurement error SD expressed in accel.relative acceleration total measurement error SD

frequency [Hz]

acce

l. sp

ectra

l den

sity

[m/s

2/sq

rt(H

z)]

0.001 0.1

Figure 4.1-2: Spectral densities of the measurement error of the satellite-satellite non-gravitational relative

acceleration and of the inter-satellite distance (converted in acceleration) suitable to fulfil the SSI mission objectives. The resulting overall spectral density (RSS of the two) represents the measurement error of the satellite-satellite relative acceleration produced by the gravity field.



Figure 4.1-3: Performance for a mission length of one year; the inter-satellite distance is fixed at 10 km; two different satellite altitudes are considered; the inter-satellite distance and of the non-gravitational accelerations measurement errors have the spectral densities shown in Figure 4.1-2.



Figure 4.1-4: Cumulative error of the geoid undulation as a function of the latitude, up to degree 200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180 200Geopotential harmonic degree

Geo

id h

eigh

t va

riat

ion

[mm

/yea

r]

Geoid variation measurement errorMountain GlaciersIce Mass Unbalance

h = 350 km, 7 years measurement phase

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140 160 180 200

Geopotential harmonic degree

Geo

id h

eigh

t var

iatio

n [m

m/y

ear]

Geoid variation measurement errorMountain Glaciers

Ice Mass Unbalance

h = 325 km, 6 years measurement phase

Figure 4.1-5: Geoid height variation measurement error per harmonic degree (red curve) compared to the signal

generated by two of the smaller secular geophysical phenomena (blue and green curves). Geoid variation measurement error computed for h = 350 km over a 7 years measurement phase (left) and for h = 325 km over a 6 years measurement phase (right).



4.2 GRAVITY FIELD MEASUREMENT NUMERICAL SIMULATION The semianalytic simulation, described in Section 4.1, had the purpose of defining the minimum measurement error requirements compatible with the science goals. Once these have been selected, a new simulation cycle was started to check that the scientific goals can be achieved with such measurement errors, taking into account a number of effect not accounted for in the semianalytic simulation. Some of these effects were predictable, on the basis of existing analytical theories; however, the main purpose of a full numerical simulation is to look for the unexpected. The starting point of the numerical simulation is an implementation of the measurement error model for the inter-satellite distance and the non-gravitational accelerations defined in section 4.1. In addition to the random measurement noise represented by the power spectrum of Figure 4.1-2, a systematic component of the error has been introduced, represented by 4 pure sinusoidal waves at frequencies corresponding to 1, 2, 3 and 4 cycles per orbital period of the satellites; these are meant to represent mostly thermal effects, but also perturbations driven by the orbit and drag-free control, acting at frequencies below the lower end of the Measurement Band (1 mHz). An orbit was also generated, assuming non gravitational perturbations partly compensated by the orbit and drag-free control. Starting from these data, the exact signal from the interferometer was simulated, expressed by the equivalent gravity gradient measurements (there is an approximation involved in this conversion, but the error is negligible given the short baseline) and the simulated noise was added. The main difference between semianalytic and numerical simulation, which was expected, was due to the polar gap effect. Although also the semianalytic simulations predict a degradation of the solution over the unobserved polar caps, most of this effect cannot be described by the current semianalytic theories. This is due to the fact that the geopotential spherical harmonics appearing in the principal components are very strongly correlated, up to 99.99%, in the full numerical solution, while the correlation is neglected in the semianalytic theory. It can be shown that the unobservable gravity anomalies concentrated on the polar caps are mostly formed with spherical harmonic of high degree ℓ but low order m. Thus this is a large effect, but it is well understood and not worrying: the propagation of the errors resulting from the unobserved polar gaps to the geoid has negligible values outside of the polar gaps themselves! Another feature highlighted by the numerical simulation is a pronounced signature of the error on the geopotential spherical harmonics coefficients with low ℓ - m. This is due to another weakness of the solution, due to the almost polar orbit. Also in this case the effect is more pronounced in the numerical simulation than in the semianalytic one, which means correlations play a significant role. However, the overall effect is just one order of magnitude, since the Sun-synchronous orbit is not too close to a polar one. The conclusion is that the true error from the numerical simulation is a reliable estimate, consistent with the semianalytic results as much as it can be, taking into account that the latter does not account for correlations. The effect of the sectorial weakness does exist, although it is not dramatic. The effect of the polar gaps is, on the contrary, huge in that the harmonics with very low order m can have error to signal ratio >1; however, when this error is represented on the surface of the Earth, it is concentrated on the polar caps not overflown by the satellite. The final result of the gravity field measurement numerical simulation is represented as a function of the degree ℓ only, computing the RMS over the different orders, in Figure 4.2-1. The representation of the signal and noise as a function of the degree ℓ only, although it is the traditional way to describe the performance of a gravimetry mission, is not completely appropriate in this case, and has to be composed with great care. Indeed, Figure 4.2-1 shows as many as 7 lines: starting from the top on the left, Kaula's rule (green), the “ground truth” field EGM96 used in the simulation (red), the true error for the low m harmonics strongly affected by the polar gaps (light blue), the true error summed only for m >15 (black), the formal RMS for low m (violet), the formal RMS for high m (dark blue), the uncertainty estimated by the semianalytic method (red). It is clear that because of the polar gap the low order m coefficients can be 100% wrong already around degree ℓ = 80. On the contrary, for large m, thus on the surface outside the polar caps, the signal to noise ratio at ℓ = 80 is about 1000, and is > 7 up to degree ℓ = 200. The semianalytic estimate appears optimistic for low degree ℓ, but it becomes more and more realistic as the degree ℓ increases. This effect is not easy to understand from this figure, but is clear: in fact the two opposite effects of the sectorial weakness and of the high frequency increase of the



noise balance around ℓ = 200, giving the (accidental) coincidence of the formal numerical and semianalytic estimate at that degree. In conclusion, all the simulations confirm that the assumed measurement error model, even taking into account the systematic effects from the gradiometer calibration problems, is compatible with a good determination of the spherical harmonics coefficients up to degree and order 200, with the only exception of the coefficients of low m, exception expressing the very obvious fact that there are no measurements over the polar caps not overflown by the satellite.

Figure 4.2-1: Signal and various estimates of the error as a function of degree ℓ (see text for explanation of the different lines).



4.3 SUMATRA EARTHQUAKE DETECTABILITY BY GRAVIMETRIC MISSIONS The possibility of detecting the geoid changes due to the Sumatra earthquake, or to similar geophysical phenomena, by exploiting the information coming from gravity space missions (GRACE, GOCE, SSI) has been investigated. The procedure consisted in propagating the variation of the geoid undulation at ground level due to the Sumatra earthquake (section 3.2) to the quantities measured by the on board GOCE gradiometer, and to the distance variation between the two satellites of the GRACE mission and of the SSI mission. The spherical harmonic coefficients reproducing the geoid modified by the Sumatra earthquake were used to propagate the effects of the Sumatra earthquake at satellite orbit:

• GOCE: altitude = 250 km, inclination = 96.5° • GRACE: altitude = 494 km, inclination = 89.0°, inter-satellite distance = 220 km • SSI mission: altitude = 325 km, inclination = 96.78°, inter-satellite distance = 10 km

A series of ascending and descending orbital arcs were simulated over an area larger than the area of interest (-9° ≤ ϕ ≤ 15°; 80° ≤ λ ≤ 110°). Figure 4.3-1 shows the radial component of the gravity gradient tensor (the quantity measured by the GOCE gradiometer) produced along the selected orbital tracks by the gravity field modified by the Sumatra earthquake. The blue line is the spectral density of the gravity gradient components measurement error of GOCE gradiometer, according to its currently specified performance requirements. The spectral density of the GRACE inter-satellite acceleration produced by the gravity field modified by the Sumatra earthquake is shown in Figure 4.3-2, together with the combined noise spectral density (SD) of the K-band ranging (KBR) system and of the accelerometer. The spectral density of the SSI inter-satellite acceleration produced by the gravity field modified by the Sumatra earthquake is shown in Figure 4.3-3, together with the combined noise SD of the interferometer (inter-satellite distance measurement error) and of the accelerometers (non-gravitational acceleration measurement error). In addition, recalling that each frequency component of the discrete power spectrum is distributed as a 2

2χ , assuming a significance level α = 1%, it is possible to derive a “threshold” SD, above which the signal SD is very likely larger than the noise SD. This threshold is shown in both Figure 4.3-2 and Figure 4.3-3 (upper green line). As a conclusion regarding the study of the effects of the Sumatra 2004 earthquake, the following points can be stated: • It is apparent that the signal produced by the geoid anomaly caused by the Sumatra earthquake is significantly

below the GOCE measurement error. So its effect cannot be detected by the GOCE instrument, at least using a time-wise approach.

• The effects of Sumatra earthquake on the gravity field can be detected both by the GRACE and the SSI

mission (the performance of the SSI satellite-to-satellite tracking mission seems to be quite better), at least on the basis of the nominal noise SD.



10-4 10-3 10-2 10-1 10010-8

10-6

10-4

10-2

100

102

frequency [Hz]

Trr PSD [mE/Hz1/2]

Figure 4.3-1: SD’s of the Trr data (track by track), as compared with the nominal noise SD of the gradiometer

(blue line). Ascending arcs are represented in red, while descending arcs in purple

10-4 10-3 10-2 10-1 10010-14

10-12

10-10

10-8

10-6

10-4

acceleration variation PSD [m/s2/Hz1/2]

frequency [Hz] Fig. 4.3-2: SD’s of the acceleration variation (track by track), as compared with the nominal noise SD of the

interferometer (blue line) and of the accelerometers (cyan line). Combined noise SD in green solid line and the corresponding threshold SD for a significance level of 1% (upper green line). Ascending arcs are represented in red, while descending arcs in purple



10-4 10-3 10-2 10-1 10010-16

10-14

10-12

10-10

10-8

10-6acceleration variation PSD [m/s2/Hz1/2]

frequency [Hz] Figure 4.3-3: SD’s of the acceleration variation (track by track), as compared with the nominal noise SD of the

interferometer (blue line) and of the accelerometers (cyan line). Combined noise SD in green solid line and the corresponding threshold SD for a significance level of 1% (upper green line). Ascending arcs are represented in red, while descending arcs in purple



5. SSI MISSION REFERENCE PARAMETERS AND PERFORMANCE REQUIREMENTS

5.1 MISSION PARAMETERS A summary of the SSI Mission main parameters and of their reference values is provided in the following Table 5.1-1. Mission parameter Reference value

Orbit altitude (h), eccentricity (e) h = 325 km

Eccentricity (e) e = 0 (circular orbit, nominally)

Orbit inclination (i) i = 96.78° (sun-synchronous orbit)

Longitude of the ascending node (Ω) Ω = RA ± 90° (dusk-dawn/dawn-dusk orbit)

In-flight measurement phase duration

6 years

Number of satellites 2

Satellite arrangement Satellites moving along the same orbital path

Inter-satellite distance (d) d = 10 km Upper limit of the measurement error spectral density of the non-gravitational relative accelerations between the two satellites ( )(~δ 2,1 fD ) Note: this spectral density it the same shown in Figure 4.1-2 (relative acceleration measurement error SD).

1 10 4 0.001 0.01 0.1 11 10 12

1 10 11

1 10 10

1 10 9

1 10 8

1 10 7

frequency [Hz]

rel.

acc.

mea

s. er

ror S

D [m

/s2/

sqrt(

Hz)

]

.1 10 3 10 1

Upper limit of the measurement error spectral density of the satellite-satellite relative distance ( )(~δ fd ) Note: this spectral density it the same shown in Figure 4.1-2 (relative distance measurement error SD), but expressed in acceleration.

1 10 4 0.001 0.01 0.1 11 10 9

1 10 8

1 10 7

1 10 6

frequency [Hz]

rel.d

ista

nce

mea

s. e

rror

SD

[m/s

qrt(H

z)] 10 3 10 1

Table 5.1-1: Reference parameters of the Satellite-Satellite Interferometry Mission



The scientific mission objective, i.e. the high resolution measurement of the Earth gravity field for the investigation and monitoring of various geophysical phenomena, calls for the lowest possible altitude. In fact, the anomalous gravitation potential decreases with the altitude h more and more rapidly as the harmonic degree increases. On the other hand, the lower the altitude the large the propellant consumption for the aerodynamic drag compensation. Therefore, for a given amount of propellant available on-board (we assumed 40 kg of Xenon for feeding the ion thrusters, like in GOCE), the orbit altitude h and the measurement phase duration T are linked by a univocal relationship. The combination h = 325 km, T = 6 years has been selected because it provides a better scientific return (see section 4.1). The sun-synchronous orbit (Figure 5.5-1), although doesn’t allow to cover all the latitudes (it leaves outside the polar caps), has been preferred to the polar orbit because: 1) On a polar orbit the measurements are insensitive to the spherical harmonics depending only upon longitude,

the so-called sectorials (with l = m). This implies a poor sensitivity to the geophysical phenomena which have a development along the parallels (e.g. oceanic currents).

2) With a polar orbit the sun crosses the orbit plane twice per year and so blinds the laser interferometer sensors

and upsets the thermal environment on the optical bench. 3) On a sun-synchronous (dawn-dusk or dusk-dawn) orbit the angle between the sun and the orbit plane is nearly

constant along the year and always far from the laser beam direction. Also the power generation and management and the thermal control of a satellite in a sun-synchronous orbit is greatly simplified.

A circular orbit allows to measure the Earth gravity field with the same resolution at all the latitudes, avoids the coupling of the Earth thermal radiation and the atmospheric drag with the orbit eccentricity, and minimizes the variations of the aerodynamic drag along the orbit. Two is the minimum number of satellites allowing the implementation of the SSI mission. A larger number of satellites implies a significant mission complication and is unnecessary to achieve the mission objectives.

ZJ2000

XJ2000

YJ2000

sun synchronous orbit

line of nodes

sun

γ constant

i > 90°

S2S1



Figure 5.1-1: Sun-synchronous orbit geometry

5.2 SATELLITE ELEMENTS CONCURRING TO THE SCIENTIFIC MEASUREMENT In the SSI mission the information about the Earth gravity field is derived from the variation of the distance d between the COMs of the two satellites produced by the gravity acceleration (let’s call this variation ∆dG). The instrument utilized to measure this distance variation is the laser interferometer. But on a low earth orbit the two satellites are subject to the aerodynamic drag which, together with other non-gravitational forces, contributes to the variation of the inter-satellite distance (let’s call ∆dD this variation induced by the non-gravitational accelerations). The measure of the laser interferometer gives therefore an inter-satellite distance variation (∆d) which includes both the gravitational and the non-gravitational contribution:

∆d = ∆dG + ∆dD In order to separate ∆dG from ∆dD, the latter must be measured with an independent instrument. Such instrument is an accelerometer, installed on each satellite, nominally centered in the COM. Since the laser interferometer has to measure the distance between the satellite COMs, its reference end points (materialized by hollow corner cubes, see section 6) must be embedded in the accelerometer proof mass. Although the SSI mission is based on the use of accelerometers instead of inertial sensors, a drag-free control (and thus specific drag-free actuators) is anyway necessary:

1) to reduce the non-gravitational acceleration maxim level within the dynamic range of the sensor; 2) to reduce the non-gravitational acceleration fluctuations so to minimize the their measurement errors

which are proportional to the non-gravitational acceleration themselves (through the accelerometer scale factor).

The non-gravitational accelerations measurement process requires also a fine control of the angular accelerations and of the angular rates. The sensors in charge of providing the information for feeding this control are star trackers and the accelerometer themselves, utilized as angular acceleration sensors. The attitude control in instead driven by the laser beam pointing between the satellites. For this purpose a Beam Steering Mechanism (BSM) has been introduces, to avoid imposing too stringent requirements on the attitude control of the whole satellite. The BSM on the satellite 1 is driven substantially by measurements of the satellite 2 offsets relative to the laser beam provided by a specific lateral displacement metrology (this implies the realization of an inter-satellite link in order to close this control loop. The non-gravitational accelerations measurement process requires the precise knowledge of the accelerometer sensitive axis orientation relative to the line joining the COMs of the two satellites. The inter-satellite distance measurement process requires the precise knowledge of the satellite orientation relative to the laser beams. Both these pieces of information are provided by a specific angle metrology. Finally, navigation receivers (either GPS or GALILEO) placed on each satellite provides (together with the laser metrology) the information utilized for controlling the satellite orbit and relative position, for keeping the satellites in an Earth-pointing attitude and for reconstructing (a posteriori) the inertial orientation of the vector joining the two satellites. The main measurement and control elements of the SSI mission are schematically represented in Figure 5.2-1.



Earth

satellite 2 satellite 1g2 g1

GPS/Greceiver

startracker

Navigationsatellites(GPS/Galileo)

drag and attitude controlactuators

accelerometer

angle and lateraldisplacement metrology interferometer

BSMnon-gravitational

forces d

Figure 5.2-1: Main elements of the measurement and control systems of the SSI mission 5.3 PERFORMANCE REQUIREMENTS The top-level performance requirements of the SSI mission concerns the measurement of the satellite-satellite distance variation and of the non-gravitational relative acceleration between the satellites, and are provided in Table 5.5-1 (expressed as upper limit of the measurement error spectral density as function of the frequency f).

5.3.1 Satellite-Satellite Distance Variation Measurement Error Tree The geometry of the inter-satellite distance measurement by the laser metrology is shown in Figure 5.3.1-1. The quantity to be measured is the variation of the distance d between the COMs of the two satellites. The errors affecting this distance variation measurement depend on:

• The intrinsic measurement error of the laser interferometer, which measured the distance variation along the path A-B-C followed by the laser beam (A and B are the retro-reflectors at the extremity of the interferometer, B is the point in which the beam is bent by the BSM).

• The stability of the laser metrology reference points A, B and the COMs of the two satellites.

• The coupling of the displacement of the points A, B, C from the COM with the satellite rotations relative to the laser beam travelling between the two satellites.

• The coupling of the variations of the pointing of the laser beam emitted by the satellite 1 with the deviation of the laser wavefront from a perfect spherical shape at satellite 2.

The error tree for the satellite-satellite distance variation measurement and the apportionment of the top-level requirements among the various contributors is provided in the chart of Figure 5.3.1-2. This apportionment defines in turn the performance requirements for the various elements concurring to the error terms:

- The laser interferometer - The metrology for measuring the satellite orientation relative to the laser beam - The separation and stability between the laser beam reflection point A and bending point B on satellite 1 - The Beam Steering Mechanism which must keep the laser beam precisely and tightly pointed to satellite 2 - The satellite COM position and stability relative to the laser beam reflection points - The satellite attitude control relative to the laser beam



S1S2XL

ZL

XS2

ZS2 ZS1

OS1OS2

XS1

AB

C

d

r1

r2

d1d2 θ1

θ1'

θ2'θ2

Figure 6.2.3-1: Geometry of the satellite-satellite COMs distance measurement by the laser metrology

Figure 5.3.1-2: Top-level requirement on the satellite-satellite distance measurement error spectral density

( )(~δ fd ) and its apportionment among the main error contributors. The errors are added RSS. 5.3.2 Non-gravitational Relative Acceleration Measurement Error Tree The scenario of the non-gravitational acceleration measurement by the accelerometers installed on the satellites S1 and S2 is shown in Figure 5.3.2-1, where D1 and D2 denotes the drag acceleration (due to aerodynamic forces and radiation pressure) on the satellites S1 and S2. The errors affecting the measurement of the non-gravitational relative accelerations between the two satellites depend on:

• The intrinsic measurement noise of the accelerometers.

• The coupling of the accelerometer scale factor and sensitive axes misalignment with the in-track and lateral components of the drag.

• The coupling of the satellite angular dynamics (angular rates and angular accelerations) with the displacement between the accelerometer proof mass and the satellite COM.

The error tree for the non-gravitational relative acceleration measurement between the two satellites and the apportionment of the top-level requirements among the various contributors is provided in the chart of Figure 5.3.2-2. This apportionment defines in turn the requirements for the various elements concurring to the error terms:

Satellite-Satellite COM distance measurement error

)(~δ fd =

Laser interferometer measurement

error )(~δ fL =

0.5⋅ )(~δ fd

Laser metrology reference points-

S/C COM stability

)(~δ 0α fd = 0.283⋅ )(~δ fd

Reference points-COM lateral displacement coupling with laser beam orientation

)(~δ 1α fd =

0.617⋅ )(~δ fd

1 104

0.001 0.01 0 .1 11 10

9

1 108

1 107

1 10 6

frequency [Hz]

rel.d

ista

nce

mea

s. e

rror

SD

[m/s

qrt(

Hz)

] 103

101

Reference points-COM longitudinal displacement coupling with laser beam

orientation )(~δ 2α fd =

0.53⋅ )(~δ fd

Laser beam pointing

coupling with wavefront error

)(~δ fdϕ = 0.164⋅ )(~δ fd



- The accelerometer (considered also as sensor for the measurement of the satellite angular accelerations about the COM)

- The control system of the drag linear accelerations, the angular accelerations and the angular rates - The metrology for measuring the accelerometer main measurement axis orientation in the Laser Beam

Reference Frame (which is the same utilised for measuring the satellite orientation relative to the laser beam; the same requirements apply too)

- The satellite COM position and stability relative to the accelerometer proof mass center In this mission, contrarily than on GOCE, the air drag accelerations measured by the accelerometers on the two satellites cannot be considered correlated. This implies a more stringent requirement on the knowledge of the accelerometer absolute scale factor, for a special calibration method shall be devised.

flight direction

local vertical

S1

S2XL

ZL

D1D2

A2

A1

X2

Z2

X1

Z1

OS1 θ1θ2q2

q1

O1OS2

O2

Figure 5.3.2-1: Measurement scenario of the non-gravitational accelerations on S1, S2 COMs

Figure 5.3.2-2: Top-level requirement on the non-gravitational relative acceleration measurement error spectral density ( )(~δ 2,1 fD ) and its apportionment among the main error contributors. The errors are added RSS. LBRF = Laser Beam Reference Frame, with X-axis passing through the COMs of the two satellites, Z-axis in the radial direction.

Non-gravitational relative acceleration measurement error

)(~δ 2,1 fD =

Accelerometer noise, scale factor,

misalignment in LBRF (satellite 1)

)(~δ21

)(~δ 2,1A1 fDfD =

Accelerometer noise, scale factor,

misalignment in LBRF (satellite 2)

)(~δ21 )(~δ 2,1A2 fDfD =

Satellite 1 dynamics, gravity gradient and accelerometer-COM

offset

)(~δ21

)(~δ 2,1S1 fDfD =

1 10 4 0.001 0.01 0.1 11 10 12

1 10 11

1 10 10

1 10 9

1 10 8

1 10 7

frequency [Hz]

rel. a

cc. m

eas.

erro

r SD

[m/s2

/sqrt(

Hz)

]

.1 10 3 10 1

Satellite 2 dynamics, gravity gradient and accelerometer-COM

offset

)(~δ21

)(~δ 2,1S2 fDfD =



6. METROLOGY SYSTEM FOR THE SSI MISSION 6.1 LASER METROLOGY SYSTEM FUNCTIONAL AND PERFORMANCE REQUIREMENTS

6.1.1 Functional Requirements For scientific purpose of determining the variation of the relative distance between the COMs of the satellites S1, S2, the laser metrology system shall accomplish the two following functions:

1. to measure the variation of the relative distance δL = δ(d1 + d2) between the laser beam reflection points A and C (i.e. the vertices of the corner cube retro-reflectors embedded respectively in the proof mass of the accelerometer A1, located on S1, and of the accelerometer A2, located on S2);

2. to measure the rotation angles θ′1, ψ′1, θ′2, ψ′2 expressing the orientation of the laser beam in the Satellite

Reference Frame (SRF) of the satellites S1, S2 (θ′i = rotation around the Y axis, ψ′i = rotation around the Z axis).

In addition, the laser metrology system shall be able:

3. to measure the lateral displacement of the reflection point C on the satellite 2 relative to the axis of the laser beam emitted by the satellite S1, in the plane orthogonal to the incoming laser beam axis.

The last function is for driving the BSM on S1 so to keep the laser beam precisely and tightly pointed to the reflection point C on S2. The function 2 can be also used to supply information for the satellite relative position and attitude control, if necessary.

6.1.2 Main Performance Requirements The laser interferometer shall provide the measurement of the distance variation between the points A and C with an error spectral density in the measurement bandwidth (MBW) of the SSI mission (a spectral band between 1 mH and 100 mHz in which the gravity field measurement performance is optimised):

<⋅⋅

≥⋅

≤ −

−

119

19

for105.2

for105.2)(~δ

LL

L

fff

fff

fL

m/Hz1/2

(fL1 = 0.01 Hz) Error standard deviation in the MBW = 1.16 nm 1σ 1 10 4 0.001 0.01 0.1 1

1 10 9

1 10 8

1 10 7

1 10 6

frequency [Hz]

inter

fero

mete

r erro

r SD

[m/sq

rt(H

z)]

10 3 10 1



The laser metrology system shall provide the measurement of the rotation angles (θ′1, ψ′1), (θ′2, ψ′2) and of their variation (δθ′1, δψ′1, δθ′2, δψ′2) about the Y, Z axes of the S1, S2 SRF relative to the laser beam with: A maximum error 1θ′ , 1ψ′ , 2θ′ , 2ψ′ , 1θδ ′ , 1ψδ ′ , 2θδ ′ , 2ψδ ′ ≤ 1⋅10-5 rad An error spectral density in the MBW

<⋅⋅

≥⋅

≤′′′′

′′′′−

−

Hzrad

for105.1

for105.1

)(ψ~δ ),(θ

~δ),(ψ

~δ ),(θ

~δ

)(ψ~

),(θ~

),(ψ~ ),(θ

~

117

17

2211

2211

LL

L

fff

fff

ffff

ffff

Error standard deviation in the MBW = 0.014” 1σ

1 10 4 0.001 0.01 0.1 11 10 7

1 10 6

1 10 5

1 10 4

frequency [Hz]

pitc

h/ya

w a

ngle

err

or S

D [r

ad/sq

rt(H

z)] 10 3 10 1

During the scientific measurements, the BSM shall keep the laser beam nominally aligned to the segment joining the beam bending point B on S1 to the retro-reflection point C on S2 with:

A maximum pointing error ∆θ′1, ∆ψ′1 ≤ 1⋅10-5 rad A pointing stability spectral density in the MBW Hz

radfor108

for108)(ψ~),(θ~

118

18

11

<⋅⋅

≥⋅

≤′∆′∆ −

−

LL

L

fff

fff

ff

Pointing stability standard deviation in MBW = 0.008” 1σ

1 10 4 0.001 0.01 0.1 11 10 8

1 10 7

1 10 6

1 10 5

frequency [Hz]

Poin

ting

stab

ility

SD

[rad

/sqr

t(Hz)

]

10 3 10 1



This requirement is derived from the measurement error generated by coupling of the laser beam pointing stability with the deviation of the laser wavefront from a perfect spherical shape centred on the transmitting satellite 1. It has been derived together with the specification for the amplitude of curvature error in the far-field laser wavefront: w = λ/10, being λ the laser wavelength. The fulfilment of the above pointing and pointing stability requirements implies in turn the fulfilment of the following requirements for the measurement the lateral displacement of the reflection point C on the satellite 2 relative to the axis of the laser beam emitted by the satellite S1, in the plane orthogonal to the incoming laser beam axis: • Maximum lateral position measurement error

Y∆ , Z∆ < 0.1 m • Spectral density of the lateral position

measurement error

<⋅⋅

≥⋅≤∆∆ −

−

114

14

ZY for108

for108)(~),(~

LL

L

fff

fff

ff

m/Hz1/2 Error standard deviation in the MBW = 0.37 mm 1σ 1 10 4 0.001 0.01 0.1 1

1 10 4

0.001

0.01

0.1

frequency [Hz]

Late

ral p

ositi

on e

rror

SD

[m/s

qrt(H

z)] 10 3 10 1

A maximum pointing error of 1⋅10-5 rad of the laser beam implies also a far-field divergence angle sufficiently large to intercept the retro-reflection point on S2 in presence of such an error, without loosing much power. In the present design we consider a laser beam outgoing from S1 with a full-cone far-field divergence angle θb = 2⋅10-4 rad at 1/e2 relative intensity (= 1.2⋅10-4 rad, at 50% relative intensity), as shown in Figure 6.1.2-1.

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.50

0.5

1intensity profile at d = 10 km

distance from the laser beam axis [m]

norm

aliz

ed ra

dial

inte

nsity

of t

he la

ser 0.6 1

Figure 6.1.2-1: Far-field divergence angle of the laser beam θb (above) and normalized laser beam intensity at S2

(d = 10 km) as function of the distance from the beam axis (below)

S1S2θs θb

d

Chapters 6.2 to 6.7 contain proprietary information and are therefore not displayed.



7. CONTROL SYSTEM FOR THE SSI MISSION

7.1 CONTROL SYSTEM REQUIREMENTS SUMMARY In Table 7.1-1, a summary of the requirements applicable to the satellite attitude control and drag control and to the relative position control is provided. In general the reference frames in which the requirements are expressed have the X-axis aligned to the line joining the COMs of the two satellites, the Z-axis in the radial direction and the Y-axis in the out-of-orbit-plane direction. Satellite attitude control Requirement 1 Rotation angles about the Y, Z axes (pitch, yaw) θi, ψi ≤ ±1.0° (i = 1, 2 = satellite index) 2 Rotation angle about the X axis (roll) ϕi ≤ ±3.0° 3 Spectral density of rotation angles about the Y, Z axes

(fL1 = 0.01 Hz)

<⋅⋅

≥⋅≤ −

−

115

15

for105.2

for105.2

)(ψ~)(θ~

LL

L

i

i

fff

fff

ff

rad/Hz1/2

4 Spectral density of rotation angle about the X axis (fL1 = 0.01 Hz)

<⋅⋅

≥⋅≤ −

−

114

14

for105.2

for105.2)(~

LL

L

i fff

fff

fϕ

rad/Hz1/2 Satellite angular rates control about COM Requirement 5 Angular rates about the X, Z axes ωiX ≤ 1⋅10-4, ωZ ≤ 1⋅10-4 rad/s 6 Angular rate about the Y axis ωiY = 1.15⋅10-3 (satellite spin rate for Earth pointing) 7 Spectral density of angular rates about the X, Z axes

(fa1 = 0.001 Hz, fa2 = 0.01 Hz)

>⋅

<⋅

≤≥⋅

≤

−

−

−

2

2

2

5

1

215

215

Z

X

for103

for103

andfor103

)(ω~)(ω~

aa

aa

aa

i

i

ffff

fff

f

ffff

ff

rad/s/Hz1/2 (see Figure 8.1-1) 8 Spectral density of angular rate about the Y axis

(fa1 = 0.001 Hz, fa2 = 0.01 Hz)

>⋅

<⋅

≤≥⋅

≤

−

−

−

2

2

2

6

1

216

216

Y

for102

for102

andfor102

)(ω~

aa

aa

aa

i

ffff

ffff

ffff

f

rad/s/Hz1/2 Satellite angular accelerations control about COM Requirement 9 Angular accelerations about the Y, Z axes

Yωi& ≤ 1⋅10-6, Zωi& ≤ 1⋅10-6 rad/s2



10 Spectral density of the angular accelerations about the Y, Z axes

>⋅

<⋅

≤≥⋅

≤

−

−

−

2

2

2

8

1

218

218

Z

Y

for101

for101

andfor101

)(ω~)(ω~

aa

aa

aa

i

i

ffff

fff

f

ffff

ff

&

&

rad/s2/Hz1/2 Satellite non-gravitational accelerations control about

COM Requirement

11 Drag accelerations about the X, Y, Z axes DiX ≤ 2⋅10-6, DiY ≤ 2⋅10-6, DiZ ≤ 1⋅10-6 m/s2 12 Spectral density of drag accelerations about the X axis

(fa1 = 0.001 Hz, fa2 = 0.01 Hz)

>⋅

<⋅

≤≥⋅

≤

−

−

−

2

2

2

8

1

218

218

X,

for101

for101

andfor101

)(~

aa

aa

aa

i

ffff

fff

f

ffff

fD

m/s2/Hz1/2 13 Spectral density of drag accelerations about the Y axis

(fa1 = 0.001 Hz, fa2 = 0.01 Hz)

≤

>⋅

<⋅

≤≥⋅

−

−

−

2

2

2

7

1

217

217

Y

for101

for101

andfor101

)(~

aa

aa

aa

i

ffff

fff

f

ffff

fD

m/s2/Hz1/2 14 Spectral density of drag accelerations about the Z axis

(fa1 = 0.001 Hz, fa2 = 0.01 Hz)

≤

>⋅

<⋅

≤≥⋅

−

−

−

2

2

2

8

1

218

218

Z

for101

for101

andfor101

)(~

aa

aa

aa

i

ffff

ffff

ffff

fD

m/s2/Hz1/2 Satellite-satellite relative position control Requirement 15 S2-S1 displacement along X axis ∆dX = ±500 m (about the nominal distance d = 104

m) 16 S2 displacement along Y, Z axes ∆dY, ∆dZ = ±50 m Table 7.1-1: Attitude, drag, and relative position control requirements for the SSI mission



7.2 FUNCTIONS AND ORGANISATION OF THE CONTROL SYSTEM The control system of the SSI mission is in charge of the following tasks: 1. To compensate the non-gravitational accelerations acting on the satellite COM motion (Drag-Free Control -

DFC). 2. To control the relative position of satellite COMs (Relative Position Control - RPC) so as to keep the

formation against gravitational perturbations and residual drag accelerations. The RPC shall not apply its action inside the MBW, otherwise it would affect the scientific measurements.

3. To permit the control of the spacecrafts orbits (Orbit Control, OC). This control may be done autonomously

or by ground support. As for the PRP, the OC shall not apply its action inside the MBW, otherwise it would affect the scientific measurements.

4. To maintain each satellite aligned with the LBRF (Relative Attitude Control - RAC), whose X-axis represents

the nominal satellite-to-satellite Line Of Sight (LOS).

5. To limit angular rates and accelerations about COM of each satellite (Angular Rate and Acceleration Control - ARAC), in order to reduce the impacts of satellite motion on the accelerometer measurements.

6. To maintain the laser beam orientation (Beam Steering Control - BSC), which has to be nominally aligned to

the segment joining the beam bending point on satellite 1 (S1) to the retro-reflector on satellite 2 (S2). The SSI control system is composed by the following control functions: − Spacecrafts Linear motion control that has in charge the relative position control (formation keeping) and the

drag-free control for each spacecraft. It has in charge the relative position control and drag-free controls. It may integrate also a loose satellite formation/orbit control loop.

− Spacecrafts Angular motion control that has in charge the angular acceleration, angular rate and relative

attitude control. It has in charge the relative attitude control and the angular rate and angular acceleration control.

− Beam steering control that has in charge the pointing of the laser beam emitted by satellite 1 to the satellite 2.

7.3 LINEAR MOTION CONTROL The key idea for the linear motion control is to have two local DFC acting independently on each satellite. The preliminary control architecture regarding the relative satellite COM dynamics is shown in Figure 7.3-1. In the block diagram, the hierarchical decomposition of DFC and RPC is highlighted: − red lines denote the two wide band DFC, which are local acceleration controls on each of the two satellite

COM dynamics. iu denotes the force command to be realized on satellite Si (ion thrusters have been considers, see below). ia~ denotes the non-gravitational acceleration measurement provided by the accelerometer on the satellite Si. ia is the reference acceleration to be tracked by DFC. The latter would be nominally zero, but it may be imposed by the RPC and OC. Finally, iiai aae ~−= denotes the input to DFC, which is the difference between desired and measured linear acceleration.

− green lines denote the narrow band RPC, which act on the relative satellite COM dynamics. In Figure

12 ggg aaa −=∆ is the relative acceleration due to gravity field perturbations, 12~~~ rrd −= and 12

~~~ vvv −=d



denote relative COM position and velocity measurements, to be obtained by the Satellite Navigation System. Finally, Ou denotes the relative acceleration command requested by the RPC. Ou should be converted into local acceleration reference trajectories 1a and 2a by means of a suitable command allocation strategy.

The linear motion control makes use of the following sensors:

• accelerometers, for measuring the non-gravitational linear accelerations;

• GPS or GALILEO Satellite Navigation System to provide the inertial position of both satellites. A suitable data link between the satellites has to be provided for the exchange of the GPS/GALILEO measurements, but at low frequency (not critical). The linear motion control makes use of ion thrusters (IT) as actuators. Four ITs arranged on the back of the satellite in a pyramid configuration allow to compensate the drag accelerations and to adjust the satellite relative position along the three axes (a failure reduces the controllability to 2 axes only). Alternative solutions considered but discarded for their complexity and the large on-board resources demand were: two ITs for in-track drag and orbit control (with redundancy) plus FEEP or cold-gas thrusters for the drag control along the other axes.

FORMATION FLIGHT

DYNAMICS

S1

+

Accelerometer

Drag-Free Control

ITA dynamics +

1D

ITA,1F 1a

1a~

S2

+

Accelerometer

ITA dynamics +

2D

2a

2a~

+−

Relative orbit kinematics

and dynamics

a∆

Relative Position Control

dvd,

Command allocation /

equalisation

1u

2u

Ou

Earth gravity

field

+

+

ga∆

Coupling with the laser beam pointing control (BSC)

2a 1a

1a

−1ae

2ae

+

−GPS / GALILEO

Inertial Navigation System

ITA,2F

Drag-Free Control

dvd ~,~

m

m

Figure 7.3-1: Linear motion control – Architecture block diagram



7.4 ANGULAR MOTION CONTROL Angular motion control has the following objectives:

• to achieve and maintain the optical link between the satellites, so as to allow the satellite-to-satellite interferometer measurements.

• to limit attitude, angular rate and angular accelerations of each of the two spacecrafts below defined requirements, so as to reduce their effects on laser interferometer and accelerometer measurement errors.

The architecture of the angular motion control system follows from some important considerations: 1. the alignment of the laser beam to maintain the satellite-to-satellite optical path can be achieved by means of

two independent degrees of freedom: (i) the spacecraft attitude with respect to a Local Orbital Reference Frame (LORF), and (ii) the laser beam orientation with respect to the Satellite Reference Frame (SRF).

2. Beam Steering Control (BSC) will be realized by means of the Beam Steering Mechanism, which, being

composed of light moving parts, it is accredited to have high-frequency dynamics. Therefore, it is possible to relax the pointing requirements to the RAC (which shall be compatible with “low-cost” attitude control actuation systems like magnetic torque rods or inertia wheels), while leaving to the BSC the responsibility to recover high-frequency misalignment residuals. The proposed solution for the angular motion control has been based on two cooperative and parallel different control loops:

− a 3-axis attitude, angular rate and acceleration controls on each satellite. The RAC control bandwidth must be sufficient to reject angular acceleration disturbances below target requirements. Each satellite attitude control is independent and then no inter-satellite link is needed;

− a wide band Beam Steering Control (BSC), to be in charge of the primary objective and to ensure laser beam pointing requirements. Information from attitude control and linear motion are not needed. Since the measurement and actuator are on different spacecrafts, a real-time inter-satellite link is needed.

The preliminary control architecture regarding the angular motion is depicted in Figure 7.4-1. In the block diagram:

− red lines denote the two local and de-coupled Attitude Controls, each realized on each spacecraft. Since the target of both attitude controls is to align each satellite with respect to the same reference frame (i.e. the LBRF), the ensemble of local attitude controls is the Relative Attitude Control (RAC). In the block diagram,

iq is the inertial attitude of the satellite Si, iq~ is the attitude measure, iq is the reference attitude (i.e. LBRF inertial attitude), iω is the inertial angular rate of the satellite Si and iω is the reference angular rate. Each of the two attitude controls realizes (via the selected actuator) the torque iuT to reject disturbances idT and to achieve the pointing performance requirement.

− blue lines denote the Beam Steering Control loop, which is driven by laser beam misalignment/displacement measurements. The output of the BSC is the command to BSM BSMu , which in turn can be thought as a set-

point for BSM prisms orientation. The pointing [ ]Tb 111 ψθϕ ′′′=θ of the laser beam with respect to S1 SRF must compensate for attitude error 1qe .

The sensors used by attitude control system are:

− Star Trackers (STR), for inertial attitude measurement;

− Accelerometer, for angular acceleration measurement;

− Inertial Navigation System (GPS/GALILEO receiver), for inertial position and velocity measurements. They are employed by RAC to reconstruct on-board the inertial orientation of the LORF or the local LBRF.



Laser metrology misalignment/displacement measurements are employed by the BSC to ensure the primary objective of the angular motion control. As already introduced, the actuator for BSC will be the BSM. To recover high-frequency jitter left by the RAC, the BSM shall have an extended frequency dynamics, at least over 0.2-0.3 Hz. The identified reference actuators for the attitude control are reaction wheels (RW) and magnetic torquers (MTR). A set of four RWs reaction wheels is utilised for the attitude, angular rate and angular acceleration control. One of the RWs is used to provide a momentum bias, so to keep far from zero the angular velocity of the momentum wheels during the operations (this to minimise the generation of low-frequency disturbances. A set of three MTRs is used to download the angular momentum accumulated by the RWs and to support the RWs for the control about the out-of-plane axis of the satellite (long which the MTRs provides a full controllability).

INTERFEROMETER

S1

+

Accelerometer and star-tracker

Attitude dynamics

+

11,qω&

1uT

11~,

~qω&

Attitude Control

1dT

S2

+

Displacement/ Misalignment

Sensor Unit Attitude

dynamics +

22 ,qω&

2uT

2,2~~ qω&

2q

Attitude Control

2dT

1~r 1

~v

2~r 2

~v

1q

2q

d

Beam Steering

Control

Accelerometer and star-tracker

pee ~,~θ

Beam Steering

Mechanism

BSMu

bθ

1q

Figure 7.4-1: Angular motion control – preliminary architecture block diagram

7.5 CONTROL SYSTEM PERFORMANCES The control system performances have been assessed by means of numerical simulations based on the single spacecraft, 6 Degrees of Freedom (orbit + attitude dynamics) End to End simulator developed for GOCE mission, suitably adapted to the SSI mission. The results of the SSI control system performance simulations are provided for the most demanding parameters: linear acceleration along the X-axis (Figure 7.5-1), angular accelerations about the Y and Z axes (Figure 7.5-2,3). The drag-free control simulation results show that the performance requirements can be met. The required specifications for the sensors (accelerometers) and actuators (ion thrusters) on which this control is based can be met by the state-of-the art technology.



The simulations of the angular motion control based on RW and MTR show that the performance requirements can be met, if the specified performances for the RW and MTR are also fulfilled. The open points are related to the low frequency noise that can be generated by the RW, for which very few data are currently available (specific characterisation tests are needed).

Figure 7.5-1: Unilateral spectral density and time series of linear acceleration (X-axis) (blue line: requirement)

Figure 7.5-2: Unilateral spectral density and time series of angular acceleration (Y-axis) (blue line: requirement)

Figure 7.5-3: Unilateral spectral density and time series of angular acceleration (Z-axis) (blue line: requirement)



8. PRELIMINARY SATELLITE CONFIGURATION

The two satellites for the SSI mission are designed to fit the Eurokot fairing in a back-to-back arrangement (like GRACE), as shown in Figure 8-1. Each satellite is long, narrow, and symmetric and has no appendages to minimize the perturbation forces and torques produced by the air drag and by the radiation pressure, when flying in the nominal attitude (longitudinal axis aligned to the XL axis of the LBRF. The body-mounted solar panels occupy one half of the satellite, on the side which is always illuminated in the sun-synchronous orbit. On the side which is opposite to the Sun and to the Earth, there are the star trackers for the attitude measurement. On the Zenith-pointing edge of the satellite there is the antenna for of the navigation system (GPS/GALILEO). A similar antenna is accommodated on the opposite edge (Nadir-pointing) for making the configuration symmetric (a similar solution is adopted in GOCE). Four ion thrusters, in a pyramidal shape, “push” the satellite from the side opposite to the orbital velocity (Figure 8-2).

Figure 8-1: SSI satellites under the Eurokot fairing



Figure 8-2: Satellite external configuration

Figure 8-3: Satellite internal configuration

Solar panels GPS antenna

Ion thrusters Ion thrusters

Star trackers

Tank

Interferometer Optical Bench

Baffle

Thermal enclosure

Baffle

Tank

Ion thrusters

Star trackers

GPS antenna

GPS receiver

Laser Optical Bench

0.814 m

3.824 m

1.45 m

3.468 m



In the middle of the satellite there the interferometer optical bench supporting the accelerometer, the optics and the detectors of the laser metrology (Figures 8-3, 8-4). The optical bench is wrapped by a thermally controlled enclosure. The center of the proof mass is located in correspondence of the satellite COM. The propellant (Xenon) supplying the ion thrusters is split in two tanks located at the sides of the optical bench. The mid points of the tanks lie on the plane normal to the satellite longitudinal axis and containing the COM, so the propellant depletion does not change at first order the COM position. The ion thruster (2 operative + 2 redundant) are tilted to apply force components along the in-track direction (main component) and cross-track direction (for side drag compensation). The ion thruster force application line passes in proximity of the COM; a small offset could be applied if useful for supporting the yaw attitude control. From the extremities of the optical bench, two conical baffles long till the top and bottom panels of the satellite screens the path of the laser beam outgoing from the satellite 1 and incoming to satellite 2 (and viceversa after the back reflection). Intermediate floors arranged along the satellite longitudinal axis, supports the internal equipment (star trackers, navigation system receivers, reaction wheels, magnetic torquers, avionics,… ). On the internal floor placed just in front of the interferometer optical bench (on the side of the telescope) there is the shutter which will be closed during the launch and in case of contingency for protecting the optical bench detectors from direct sun illumination. On the same shutter, the retro-reflector for the calibration of the Beam Steering Mechanism is installed (see Figure 8-5). The internal floor placed below the interferometer optical bench (on the side of the ion thrusters) is the laser optical bench hosting the laser source and its driving electronics, the optical resonator for the laser frequency stabilisation, the analogue and digital electronics of the laser metrology, the front end and the digital electronics of the accelerometer. The configuration of the laser optical bench is shown in Figure 8-6. The in-orbit nominal orientation of a satellite is shown in Figure 8-7. The nominal mutual orientation of the two satellites in the measurement phase is shown in Figure 8-8. The relative position of the two satellites in orbit (and their role) can be interchanged. For instance, in case of failure of the metrology laser source on the satellite 1 (the following satellite) it can be moved in place of the satellite 2 (the leading satellite) and the latter can take the role of the first one (master satellite) in the relative distance measurement.

Pages 51 - 53 contain proprietary information and are therefore not displayed.



Figure 8-7: Nominal orientation of one satellite in orbit

Figure 8-8: Nominal mutual orientation of two satellites in the measurement phase

satellite 2

satellite 1 satellite 2

satellite 1

Chapter 9 contains proprietary information and is therefore not displayed.



10. ACRONYMS AND ABBREVIATIONS

AM Amplitude Modulator AOM Acousto-Optic Modulator ARAC Angular Rate and Acceleration Control BSC Beam Steering Control BSM Beam Steering Mechanism CCN Contract Change Notice CHAMP CHAllenging Minisatellite Payload COM Centre of Mass DC Direct Current DFC Drag Free Control E Eötvös (acceleration gradient measurement unit; 1 E = 10-9 m/s2/m) ESA European Space Agency FEEP Field Emission Electric Propulsion FS Frequency Shifter Gal acceleration measurement unit; 1 Gal = 10-2 m/s2 GOCE Gravity field and steady-state Ocean Circulation Explorer GPS Global Positioning System GRACE Gravity Recovery And Climate Experiment IMGC Istituto di Metrologia “G. Colonnetti” IT Ion Thruster LBRF Laser Beam Reference Frame LEO Low Earth Orbit LISA Laser Interferometer Space Antenna LORF Local Orbital Reference Frame MBW Measurement Bandwidth of the SSI mission (preliminarily set between 1 mHz and 100 mHz) MTR Magnetic Torquers NEP Noise Equivalent Power OC Orbit Control pbs Polarizing Beam Splitter PSD Position Sensing Detector RAC Relative Attitude Control RF Radio Frequency RPC Relative Position Control RSS Root Sum Square RW Reaction Wheel S1 (S2) Satellite 1 (Satellite 2) SD Spectral Density SOW Statement of Work SRF Satellite Reference Frame SSI Satellite-Satellite Interferometry STR Star Trackers w.r.t. with respect to

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