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GNSS-R concept GNSS-R concept extended by a fine orbit tuningextended by a fine orbit tuning

Jaroslav KLOKOČNÍKJaroslav KLOKOČNÍK a a, Aleš BEZDĚK , Aleš BEZDĚK aa, Jan KOSTELECKÝ, Jan KOSTELECKÝ b,cb,c

a a Astronomical Institute, Academy of Sciences of the Czech Republic, Astronomical Institute, Academy of Sciences of the Czech Republic, CZ-251 65 Ondřejov, Czech Republic, Tel .:+420323620158,CZ-251 65 Ondřejov, Czech Republic, Tel .:+420323620158,

jklokocn@asu.cas.czjklokocn@asu.cas.cz, , bezdek@asu.cas.cz,bezdek@asu.cas.cz,b b CEDR – Research Institute for Geodesy, Topography and Cartography, CEDR – Research Institute for Geodesy, Topography and Cartography,

CZ-250 66 Zdiby, Czech RepublicCZ-250 66 Zdiby, Czech Republic, , c c Department of Advanced Geodesy, Czech Technical University, Department of Advanced Geodesy, Czech Technical University,

CZ-166 29 Praha 6, Thákurova 7,Czech RepublicCZ-166 29 Praha 6, Thákurova 7,Czech Republic , , kost@fsv.cvut.czkost@fsv.cvut.cz

22ndnd CNES CCT workshop on passive reflectometry using radiocom space signals, CNES CCT workshop on passive reflectometry using radiocom space signals, Space Reflecto2011, Calais, France 27 and 28 Oct 2011Space Reflecto2011, Calais, France 27 and 28 Oct 2011

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OutlineOutline

DefinitionsOrbit tuning for altimetry mission ERS 1, DGFI 1989GRACE and GOCE Inspiration for GNSS-RExamples for GNSS-R, second satellite Instead of ConclusionReferences

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Definitions

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Orbital resonancesOrbital resonances

Orbital resonance β/α takes place, if: Groundtracks are exactly the same after

β nodal revolutions and α nodal days Equivalent names: repeat orbit, resonant orbit

Density of ground tracks at equator

D = o/β,

where the circumference

o = 40 075 km (for the Earth)

Number of single-satellite crossover points (among the ascending and descending tracks)

where u = 1 for I < 90 and

u = –1 for I > 90 deg

(see, e.g., Farless 1986 or Kim 1997).

1 uN x

5

mean motion vs semi-major axes, 1mean motion vs semi-major axes, 1stst order analytical theory order analytical theory

Recalling the exact resonance β:α; the phase in LPE at the exact resonance becomes . The relevant satellite mean motion n:

(1)

Now we have to put the LPEs for the orbital elements Ω, ω, and M0 into (1); we get:

where the minus sign is valid for the normal rotation and the plus sign for the retrograde rotation of the planet. Finally we arrive at

(2)

where for and for , the former case for the normal rotation, the latter for the retrograde rotation. Note that

0MSn

,,cos2

3

2

3cos61

22

2

a

RzIIzJSn

0

1coscos4

2

31 2

2

2 IIa

RJSn

202 5 CJ 0S 1 0S 1

Klokočník, J., Kostelecký, J., and Gooding, R. H., “On fine orbit selection for particular geodetic and oceanographic missions involving passage through resonances”, Journal of

Geodesy, Vol. 77, No. 1, 2003, pp. 30–40. doi: 10.1007/s00190-002-0276-3

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Historical notes - altimetry and GRACE

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History: orbit tuning for altimetry missions: oceanography vs space geodesy

8

D = 932 km

9

D = 80 km

10

future

11

Motivation to study resonance regimes for freely decaying GRACE:

dramatic changes in the accuracy of monthly solutions for variations of

the geopotential near low-order resonances

(here the case of 61/4 in Autumn 2004):

Courtesy of S. Bettadpur (2004, 2006, priv. commun.) Note logarithmic scales on the y-axes.

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GOCE

(pre)-selected orbits with high-order resonances

for measuring phases with the onboard gradiometer,

orbit keeping by means of the ion motor to ± 5m

13

14

D=2505 km

D=41 km

15

GNSS-RGNSS-R

small demonstration satellite piggybacked to another satellite to sun-synchronous, retrograde, dawn-dusk (SSO) orbit (6 am and 6 pm are times of equator crossings)

at height 500–800 km.

Later the operational satellite again on SSO orbit

but at height 1300–1500 km

both orbits sun-synchronous, I = 98–101 deg,

revisit time about 3 days for the former and about 2 days for the latter mission.

Swath is 900 km or 1500 km, respectively, spatial resolution not worse than 10×100 kmxkm

16

0 5 10 15 20 25 30 35 40 45 50 55 60450

500

550

600

650

700

750

800

850

15:1

29:2

43:3

44:3

46:3

57:4

59:4

61:4

71:5

72:5

73:5

74:5

76:5

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

17

0 5 10 15 20 25 30 35 40 45 50 55 60650

655

660

665

670

675

680

44:3

103:7

117:8

147:10

161:11

191:13

205:14

235:16

249:17

250:17

278:19

279:19

293:20

323:22

337:23

338:23

353:24

366:25

367:25

381:26

395:27

397:27

411:28

425:29

426:29

439:30

454:31

455:31

456:31

469:32

485:33

499:34

512:35

513:35

514:35

527:36

529:36

542:37

543:37

544:37

557:38

559:38

571:39

587:40

600:41

601:41

602:41

603:41

617:42

629:43

630:43

631:43

632:43

645:44

647:44

659:45

661:45

662:45

673:46

675:46

688:47

689:47

690:47

691:47

703:48

717:49

718:49

719:49

720:49

733:50

746:51

749:51

761:52

763:52

765:52

776:53

777:53

778:53

779:53

791:54

793:54

806:55

807:55

808:55

809:55

821:56

823:56

835:57

838:57

849:58

851:58

853:58

863:59

864:59

865:59

866:59

867:59

868:59

881:60

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

18

5 10 15 20 25 30 35 40 45 50 55 60750

755

760

765

770

775

780

43:3

72:5

115:8

158:11

187:13

201:14

244:17

259:18

273:19

287:20

302:21

330:23

331:23

358:25

359:25

373:26

388:27

389:27

401:28

403:28

416:29

417:29

431:30

444:31

445:31

446:31

459:32

461:32

475:33

487:34

489:34

502:35

503:35

517:36

530:37

531:37

532:37

533:37

545:38

547:38

560:39

573:40

588:41

589:41

590:41

605:42

616:43

617:43

618:43

619:43

631:44

633:44

646:45

647:45

659:46

661:46

673:47

674:47

675:47

676:47

677:47

689:48

691:48

702:49

703:49

704:49

705:49

706:49

717:50

719:50

733:51

734:51

745:52

747:52

749:52

759:53

760:53

761:53

762:53

763:53

775:54

788:55

789:55

791:55

803:56

818:57

820:57

821:57

831:58

833:58

835:58

845:59

846:59

847:59

848:59

849:59

850:59

863:60

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

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0 5 10 15 20 25 30 35 40 45 50 55 601250

1300

1350

1400

1450

1500

1550

13:1

25:2

37:3

38:3

51:4

62:5

63:5

64:5

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

20

0 5 10 15 20 25 30 35 40 45 50 55 601440

1445

1450

1455

1460

1465

1470

25:2

138:11

163:13

188:15

213:17

238:19

262:21

263:21

287:23

288:23

301:24

312:25

313:25

337:27

338:27

351:28

362:29

363:29

387:31

388:31

401:32

412:33

413:33

437:35

438:35

439:35

451:36

462:37

463:37

464:37

487:39

488:39

501:40

512:41

513:41

514:41

537:43

538:43

539:43

549:44

551:44

562:45

563:45

577:46

587:47

588:47

589:47

599:48

601:48

612:49

613:49

614:49

627:50

637:51

638:51

649:52

651:52

662:53

663:53

664:53

665:53

677:54

687:55

688:55

689:55

699:56

701:56

712:57

713:57

715:57

727:58

737:59

738:59

739:59

740:59

749:60

751:60

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

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Example of possible orbit tuning for GNSS-R satelliteExample of possible orbit tuning for GNSS-R satellite

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

780

782

784

786

788

790

792

794

796

798

800

100

(790.8)

143

(786.1)

157

(795.1)

186

(783.5)

214

(797.2)

229

(781.9)

243

(788.0)

257

(793.5)

271

(798.3)

272

(780.8)315

(780.0)

328

(799.1)

329

(784.6)

343

(788.8)

357

(792.7)

371

(796.3)

385

(799.6)

386

(787.3)

414

(794.1)

415

(782.6)

443

(789.3)

457

(792.3)

472

(785.1)

485

(797.8)

501

(781.3)

515

(784.2)

528

(796.0)

529

(787.0)

543

(789.6)

557

(792.0)

571

(794.4)

585

(796.6)

586

(788.5)

587

(780.4)

599

(798.8)

601

(782.9)

614

(793.0)

615

(785.3)

629

(787.6)

643

(789.8)

644

(782.4)

657

(791.8)

671

(793.9)

672

(786.8)

685

(795.8)

699

(797.6)

701

(784.0)

713

(799.4)

728

(794.5)

730

(781.5)

743

(789.9)

756

(798.0)

757

(791.7)

758

(785.4)

773

(781.1)

786

(789.1)

787

(783.0)

799

(796.8)

801

(784.9)

814

(792.5)

815

(786.6)

827

(799.8)

829

(788.4)

842

(795.7)

843

(790.0)

844

(784.4)

857

(791.6)

859

(780.5)

870

(798.6)

871

(793.2)

872

(787.7)

873

(782.2)

885

(794.7)

887

(783.9)

899

(796.1)

901

(785.5)

902

(780.3)

913

(797.5)

915

(787.1)

927

(798.9)

928

(793.7)

929

(788.6)

943

(790.1)

956

(796.5)

957

(791.5)

958

(786.6)

959

(781.6)

971

(792.9)

973

(783.1)

985

(794.3)

986

(789.4)

999

(795.6)

1013

(796.8)

1014

(792.2)

1015

(787.5)

1016

(782.8)

1027

(798.1)

1041

(799.3)

1042

(794.7)

1043

(790.2)

1044

(785.6)

1045

(781.1)

1057

(791.5)

1059

(782.5)

1072

(788.3)

1073

(783.8)

1085

(793.9)

1087

(785.2)

1098

(799.5)

1101

(786.5)

1102

(782.2)

1115

(787.8)

1127

(797.4)

1128

(793.2)

1129

(789.0)

1130

(784.8)

1131

(780.6)

1141

(798.6)

1143

(790.2)

1156

(795.5)

1157

(791.4)

1159

(783.2)

1171

(792.6)

1173

(784.4)

1184

(797.7)

1185

(793.7)

1186

(789.7)

1187

(785.7)

1188

(781.7)

1199

(794.8)

1201

(786.9)

1212

(799.8)

1213

(795.9)

1214

(791.9)

1216

(784.1)

1217

(780.2)

1227

(796.9)

1229

(789.2)

1231

(781.4)

1241

(797.9)

1243

(790.3)

1244

(786.4)

1255

(798.9)

1257

(791.4)

1259

(783.8)

1269

(799.9)

1270

(796.2)

1271

(792.4)

1272

(788.7)

1273

(784.9)

1274

(781.2)

1298

(798.1)

1299

(794.5)

1301

(787.2)

1313

(795.5)

1315

(788.2)

1317

(781.0)

1327

(796.4)

1328

(792.9)

1330

(785.7)

1331

(782.1)

1341

(797.4)

1343

(790.3)

1345

(783.2)

1356

(794.8)

1357

(791.3)

1358

(787.8)

1359

(784.3)

1369

(799.2)

1373

(785.4)

1384

(796.7)

1385

(793.3)

1386

(789.8)

1387

(786.4)

1388

(783.0)

1399

(794.2)

1401

(787.4)

1403

(780.6)

1412

(798.5)

1414

(791.8)

1415

(788.4)

1417

(781.7)

1427

(796.0)

1429

(789.4)

1431

(782.7)

Orbital resonances of a satellite of Earth (i=98.5 deg)

repeat period (nodal days)

altit

ude

(SM

A –

637

8.1

km)

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243/17 872/61

h=788.0 km, D = 165 km h=787.7 km, D = 46 km

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Table 1. Density of the ground tracks and number of the cross-over points for different repeat orbits

D = o/β,

u = 1 for I < 90 and u = –1 for I > 90 deg

1 uN x

24

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The Fine Orbit Tuning for (the second) ESA’s GNSS-R satellite The Fine Orbit Tuning for (the second) ESA’s GNSS-R satellite

Step 1 standard orbit selection (already done by ESA)

SSO orbits 500–800 km and 1300–1500 km,

I = 98–101 deg,

Step 2 suggestion for possible fine orbit tuning (done now & here)

Step 3 the definitive, more specific orbit selection (by ESA),

to be aware of possibility of the fine orbit tuning

Step 4 definitive fine orbit tuning for the final orbit selection

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Case of GOCE we are ready to repeat something like that for any satellite

27Orbits with no syubcycles, orbits with subcycles, the basic difference…

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GOCE groundtrack pattern animation for a hypothetical free fall

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actual61-day orbit

planned 61-day orbit

20/21-day subcycles

41-day subcycle

62-day orbit

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Temporal evolution of an orbit Temporal evolution of an orbit – – withwith//without a subcyclewithout a subcycle

Repeat orbit with no subcycles→ gradually filling up two large equatorial gaps

Repeat orbit with a subcycle→ groundtracks laid down in two (or more) almost homogeneous grids

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AnalytiAnalytical vs. numerical modellingcal vs. numerical modelling

So far, graphs based on simple theory with only the zonal term J2 (flattening of Earth)

What happens when all other orbital perturbations (geopotential, lunisolar, tides, radiation, …) are added?

Peaks in histograms widened Repeat character is keptEarth coverage graphs almost the same

(0.02° ↔ 2 km)

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Orbits near the actual Orbits near the actual 61-day orbit of GOCE61-day orbit of GOCE

Equator with groundtracks after 65 days Different mean altitudes

Repeat orbits: 61-day (selected for MOP1) 41-day subcycle

62-day orbit compared with 61-dayhas more regular groundtrack grid is only by 200 m higher

62

62

41

41

61

61

6162

4141

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Proposal for tProposal for the he 145-day orbit with 62 or 83145-day orbit with 62 or 83 day subcyclesday subcycles

Planned 2327:145 repeat orbit node spacing ≈ 17.2 km

Nearest repeat orbits 62-day, lower by 30 m 83-day, higher by 23 m

Ion thruster performance ±50 m

145-day repeat is a good choice → node spacing at least

40.3 km for 62-day repeat30.1 km for 83-day repeat

Resonant orbits ordered by height

D R h (km)

1653 103 255,062

995 62 255,105

2327 145 255,135

1332 83 255,158

1669 104 255,19

2006 125 255,211

2343 146 255,226

83145

62

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Instead of ConclusionInstead of Conclusion

Analyses done for GRACE and GOCE

can be repeated for GNSS-R satellite(s).

Number of measuring points (nadir and off-nadir) for a bistatic altimetry

with the fine orbit tuning

may increase by 1-2 orders,

in comparison with

the number of monostatic (nadir) altimetry (sub-satellite) points

in a “general” orbit (ignoring any fine orbit tuning).

Additional costs for the inclusion of the fine orbit tuning and orbit keeping

are theoretically zero.

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Thank you for your attentionThank you for your attention

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More information: jklokocn@asu.cas.cz www.asu.cas.cz/~jklokocn, ~bezdek

References

Klokocnik et al. 2003: On Fine Orbit Selection for Particular Geodetic and Oceanographic Missions Involving Passage Through Resonances,J. GEOD., Vol. 77, No. 1, 30–40, doi:10.1007/s00190-002-0276-3

Wagner et al. 2006: Degradation of Geopotential Recovery from Short Repeat-Cycle Orbits: Application to GRACE Monthly Fields, J. GEOD.,Vol. 80, No. 2, 94–103, doi:10.1007/s00190-006-0036

Klokocnik et al. 2008: Variations in the Accuracy of Gravity Recovery due to Ground Track Variability: GRACE, CHAMP, and GOCE, J. GEOD, Vol. 82, No. 12, 917–927, doi:10.1007/s00190-008-0222-0

Bezděk et al. 2009: Simulation of Free Fall and Resonances in the GOCE Mission, J. GEODYN., Vol. 48, No. 1, 47–53, doi:10.1016/j.jog.2009.01.007

Klokocnik et al. 2010: Orbit Tuning of Planetary Orbiters for Accuracy Gain in Gravity-Field Mapping, JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS, Vol. 33, No. 3, May–June, doi: 10.2514/1.46223.