Evolutionary characteristics of the orbits of outer Saturnian, Uranian, and Neptunian satellites

ISSN 0038-0946, Solar System Research, 2008, Vol. 42, No. 6, pp. 488–504. © Pleiades Publishing, Inc., 2008.Original Russian Text © M.A. Vashkov’yak, N.M. Teslenko, 2008, published in Astronomicheskii Vestnik, 2008, Vol. 42, No. 6, pp. 523–539.

488

This work is a natural continuation of our study ofthe orbital evolution of the outer (distant) satellitesorbiting the giant planets (Vashkov’yak and Teslenko,2008). It supplements the above paper devoted to theouter satellites of Jupiter with the results of our calcu-lations of the orbital evolution of Saturnian, Uranian,and Neptunian satellites. The notation used here, thestructure of the tables, figures and the applied methodsof investigation are completely identical to thosedescribed in our previous paper (Vashkov’yak andTeslenko, 2008). Therefore, avoiding the repetition ofcommon sections as much as possible, we implicitlyassume that the reader is familiar with the above paper.

The initial orbital elements of the outer Saturnian,Uranian, and Neptunian satellites (see Table 1)retrieved from the server of the Sternberg AstronomicalInstitute (Moscow State University) organized byN.V. Emel’yanov(http://Infm1.sai.msu.su/neb/nss/index/htm) served asthe input data for our calculations. These elements havealready been improved based on all available observa-tions.

1. In contrast to the family of outer Jovian satellites,the ranges of the semimajor axes in the system of Sat-urn overlap for the groups of prograde and retrogradeouter satellite orbits (see Table 2). The maximumeccentricities are about 0.7–0.8, while the maximumangles between the orbital and ecliptical planes slightlyexceed

50°

.The system of Uranus currently numbers only nine

outer satellites, with only one of them having a pro-grade motion. The semimajor axis of its orbit lieswithin the range of semimajor axes for the retrogradeorbits (see Table 3). The maximum eccentricities are

approximately the same (0.70–0.85), while the maxi-mum angles between the orbital and ecliptical planesare about 65

°

and 40

°

, respectively, for the progradeand retrograde orbits. It is interesting to recall that, until1997, Uranus was the only giant planet without observ-able outer satellites.

Neptune has a record small number of discoveredouter satellites. At the same time, however, the maxi-mum eccentricity of one of the evolving orbits and thelargest mean planetocentric distance of one of the Nep-tunian satellites hold records among the satellite sys-tems (see Table 4).

The decrease in the number of discovered distantsatellites from Jupiter to Neptune is probably attribut-able to the increasing observational difficulties as thedistance to the planet of observation increases.

2. Tables 5–7 give the extreme orbital parameters ofthe outer Saturnian, Uranian, and Neptunian satellites,respectively. In addition to the satellite designations(column 1), these tables contain the minimum and max-imum semimajor axes (columns 2 and 3), eccentricities(columns 4 and 5), and ecliptical inclinations (columns 6and 7). The characteristics of the temporal variations inthe arguments of the pericenters and the longitudes ofnodes are listed in columns 8 and 9, respectively; thecirculation periods and mean motions are given on theleft and right, respectively. In the first rows of all col-umns, the values obtained numerically are highlightedin boldface. For comparison, the second rows of col-umns 2–7 list the corresponding values obtained usingthe numerical–analytical method. The correspondingvalues of

v

G

and

v

H

from Beaugé and Nesvorny (2007)are given in columns 8 and 9 under the mean motions

v

ω

and

v

Ω

. A comparison of the calculated mean

Evolutionary Characteristics of the Orbits of Outer Saturnian, Uranian, and Neptunian Satellites

M. A. Vashkov’yak and N. M. Teslenko

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russia

Received April 1, 2008

Abstract

—We present the results of our systematic study of the long-period orbital evolution of all of the outerSaturnian, Uranian, and Neptunian satellites known to date. The plots of the orbital elements against time givea clear idea of the pattern of the orbital evolution of each satellite. The tabular data allow us to estimate the basicparameters of the evolving orbits, including the ranges of variation in the semimajor axes, eccentricities, andecliptical inclinations as well as the variation periods and mean motions of the arguments of pericenters and thelongitudes of the nodes. We compare the results obtained by numerically integrating the rigorous equations ofthe perturbed motion of the satellites with the analytical and numerical–analytical results. The satellite orbitswith a librational pattern of variation in the arguments of pericenters are set apart.

PACS

numbers:

95.10.Ce, 91.10.Sp, 96.30.N-, 96.30.Qk, 96.30.Td

DOI:

10.1134/S0038094608060038

SOLAR SYSTEM RESEARCH

Vol. 42

No. 6

2008

EVOLUTIONARY CHARACTERISTICS OF THE ORBITS OF OUTER SATURNIAN 489

Tab

le 1

.

Ini

tial o

rbita

l ele

men

ts o

f th

e ou

ter

Satu

rnia

n, U

rani

an, a

nd N

eptu

nian

sat

ellit

es

Sat

ellit

eJD

(

t

0

), d

ays

n

, deg

. day

–1

a

, km

ei

, deg

M

0

, deg

ω

, deg

Ω

, deg

S9

2416

791.

5 0.

6586

3278

5 12

8922

23.3

96

0.15

5951

55

174.

8371

71

22.1

8228

5 30

0.53

1554

22

8.66

0499

S19

2451

763.

5 0.

2759

5748

0 23

0247

68.4

55

0.39

0267

88

172.

9400

70

285.

2994

96

40.9

6084

1 20

5.01

3061

S20

2451

763.

5 0.

5281

6740

3 14

9362

75.7

48

0.46

3977

62

47.1

7272

0 60

.362

565

237.

4502

81

353.

1217

65

S21

2451

810.

5 0.

3949

3794

8 18

1302

64.5

80

0.63

7772

09

34.8

3617

6 35

5.40

6137

28

5.67

4581

94

.438

292

S22

2451

810.

5 0.

7959

7923

4 11

3628

92.3

90

0.36

2546

33

49.2

6960

0 23

9.18

7490

69

.968

736

151.

6581

93

S23

2451

810.

5 0.

3620

7043

2 19

2115

03.1

14

0.09

6547

35

174.

6855

96

55.0

8242

5 61

.665

920

253.

4178

15

S24

2451

763.

5 0.

7989

7980

3 11

3344

25.6

47

0.16

2538

75

48.6

2495

0 34

6.22

2921

89

.937

853

352.

4615

06

S25

2451

810.

5 0.

3854

3535

9 18

4270

44.4

29

0.25

9954

00

169.

5131

09

206.

6774

42

300.

3459

62

79.8

9417

0

S26

2451

857.

5 0.

4690

9452

1 16

1652

76.7

98

0.48

5397

17

37.3

5235

2 16

6.74

2989

60

.641

003

110.

9636

98

S27

2451

818.

5 0.

4941

9006

2 15

6132

76.3

63

0.20

6543

97

148.

5370

39

245.

8196

54

208.

1757

89

284.

8711

51

S28

2451

810.

5 0.

4209

4561

9 17

3755

83.1

65

0.61

8404

22

34.1

5207

2 59

.306

333

280.

3012

20

145.

8832

95

S29

2451

810.

5 0.

3978

7953

9 18

0407

94.0

17

0.37

4787

23

48.5

2592

5 30

4.15

7484

63

.944

888

63.7

8227

9

S30

2451

810.

5 0.

3252

9572

3 20

6334

16.7

69

0.52

4777

31

174.

9250

93

76.3

7160

3 91

.813

056

246.

8699

90

S31

2452

675.

5 0.

3640

6579

4 19

1412

42.8

90

0.32

4753

66

135.

8074

50

158.

5863

04

177.

8758

94

182.

6649

32

S36

2453

351.

5 0.

3236

9555

9 20

7013

60.6

67

0.20

8128

17

167.

4342

84

236.

2353

11

265.

1548

08

194.

4063

91

S37

2451

811.

5 0.

4299

3858

5 17

1324

35.4

41

0.36

3761

16

42.2

2119

2 26

9.17

8985

9.

9048

12

196.

8444

33

S38

2453

351.

5 0.

3679

0944

8 19

0076

93.7

01

0.15

3205

23

157.

2950

14

223.

1555

51

154.

0391

06

213.

7791

66

S39

2451

810.

5 0.

3358

5589

2 20

1986

06.3

05

0.64

0584

58

146.

0516

09

330.

6454

93

87.0

4214

9 28

4.07

3678

S40

2453

351.

5 0.

3361

5235

1 20

1867

28.8

61

0.21

9434

16

157.

6754

30

179.

3577

17

336.

7191

04

145.

5366

61

S41

2453

352.

5 0.

2968

4387

5 21

9316

50.8

97

0.12

1341

99

163.

1271

31

272.

3717

47

148.

3559

26

237.

5513

01

S42

2453

351.

5 0.

2550

9237

4 24

2637

76.8

03

0.16

5842

83

168.

1389

68

271.

0471

81

351.

8181

27

270.

6194

96

S43

2453

351.

5 0.

3577

3628

9 19

3663

62.3

64

0.31

2858

56

162.

4442

49

89.8

9785

9 15

.806

984

323.

2351

22

S44

2453

351.

5 0.

3857

7653

0 18

4161

78.5

70

0.36

2692

79

153.

5700

97

274.

2089

16

267.

0355

64

46.5

9688

4

S45

2453

351.

5 0.

3008

4363

6 2

1736

827.

852

0.34

7636

20

149.

2447

50

106.

7586

99

164.

9605

29

290.

4414

01

S46

2453

740.

5 0.

3105

4360

921

2817

98.2

24

0.46

5503

84

151.

1014

03

80.6

2963

8 19

1.74

9100

22

0.14

5635

S47

2453

351.

5 0.

4173

2288

517

4759

95.2

64 0

.380

9335

2 15

5.76

7107

58

.878

350

198.

9142

49

297.

0557

95

S48

2453

351.

5 0.

2894

4570

8 22

3037

88.7

39

0.36

4541

28

167.

0517

64

278.

7188

94

330.

8431

37

257.

2129

76

490


Vol. 42

No. 6

2008

VASHKOV’YAK, TESLENKO

Tab

le 1

.

(C

ontd

.)

Sat

ellit

eJD

(

t

0

), d

ays

n

, deg

. day

–1

a

, km

ei

, deg

M

0

, deg

ω

, deg

Ω

, deg

S/20

04 S

7 24

5335

1.5

0.32

8339

708

2050

5692

.799

0.

5510

6481

16

5.58

4692

27

5.35

8489

10

0.72

3773

34

6.29

7482

S / 2

004

S12

2453

351.

5 0.

3474

6742

3 19

7460

69.3

03

0.40

0760

42

163.

9856

20

268.

1539

58

97.9

1078

4 31

3.74

0077

S / 2

004

S13

2453

351.

5 0.

4013

2532

7 17

9373

79.9

03

0.25

6978

32

167.

3491

88

14.3

3533

9 5.

5219

72

221.

3433

59

S / 2

004

S17

2453

352.

5 0.

3620

4517

5 19

2123

96.5

68

0.22

0699

82

166.

9303

15

146.

1133

66

177.

1081

69

19.9

2042

4

S / 2

006

S1

2453

739.

5 0.

3755

9816

2 18

7474

01.4

66

0.14

0705

11

154.

0994

00

217.

7237

65

140.

5763

92

340.

7256

59

S / 2

006

S4

2453

740.

5 0.

3862

9125

7 18

3998

15.4

08

0.35

8163

39

172.

6645

66

79.8

6430

5 14

3.60

7470

34

3.55

4570

S / 2

006

S5

2453

352.

5 0.

2858

4891

7 22

4904

95.2

08

0.17

1643

05

165.

7099

48

131.

8500

32

15.1

3033

5 34

2.27

7989

S / 2

006

S6

2453

740.

5 0.

3605

8161

0 19

2643

48.9

64

0.18

5668

25

163.

1035

25

259.

4265

12

237.

0663

91

19.7

8983

9

S / 2

007

S1

2453

740.

5 0.

4045

7723

5 17

8411

32.7

42

0.12

9324

41

50.7

4267

7 30

3.38

6683

61

.618

966

92.3

5745

9

S / 2

007

S2

2454

118.

5 0.

4521

2625

0 16

5672

42.8

09

0.21

5800

76

176.

6777

88

145.

6967

14

63.2

1204

8 11

3.06

3082

S / 2

007

S3

2454

118.

5 0.

3698

5141

6 18

9410

99.8

99

0.16

5742

62

177.

0214

99

172.

3247

07

280.

9944

02

96.1

0760

4

U16

24

4585

2.5

0.62

0765

698

7169

336.

902

0.08

1225

45

139.

4698

98

88.4

8732

6 34

0.60

0180

17

3.93

9166

U17

24

4585

2.5

0.27

8871

736

1222

2530

.504

0.

4530

4274

15

3.76

1496

11

3.74

0111

14

.616

222

255.

1721

29

U18

24

5137

7.5

0.18

0417

268

1633

9749

.556

0.

3224

0284

14

6.29

4739

21

8.54

6870

17

0.53

8555

32

0.09

5815

U19

24

5137

7.5

0.16

2443

200

1752

3859

.865

0.

5519

0696

14

6.73

6281

14

5.32

4953

3.

9370

69

249.

7426

50

U20

24

5137

7.5

0.53

1070

815

7955

408.

884

0.14

3934

17

141.

5401

72

166.

6431

33

28.5

6686

1 18

9.34

4003

U21

24

5213

4.5

0.48

0642

795

8502

549.

563

0.22

2253

00

166.

1858

09

105.

7314

74

161.

6010

82

198.

7793

77

U22

24

5213

4.5

1.34

7701

587

4275

973.

156

0.13

7491

94

147.

5131

91

96.5

3322

4 12

3.43

5305

103

.147

657

U23

24

5213

4.5

0.21

4656

001

1455

2427

.379

0.

7956

1391

50

.893

940

91.5

4441

9 77

.267

773

19.9

0533

5

U24

24

5213

4.5

0.12

8978

296

2043

7120

.510

0.

4434

9088

16

7.13

9432

96

.815

121

166.

7597

63

225.

3742

44

N9

2451

377.

5 0.

1914

5506

4 16

5886

57.0

20

0.25

8998

34

111.

8468

79

196.

6600

75

157.

6202

17

217.

4791

18

N10

24

5211

4.5

0.03

9129

229

4781

0315

.229

0.

3096

4279

12

3.79

3240

20

6.15

3091

12

7.51

7540

32

0.01

0959

N11

24

5211

4.5

0.12

4134

104

2214

4288

.126

0.

1397

9487

53

.085

681

85.6

8623

4 61

.171

031

62.1

6397

8

N12

24

5211

4.5

0.11

4063

356

2342

9249

.102

0.

3783

9627

37

.859

106

220.

5892

54

132.

4039

5252

.790

973

N13

24

5250

0.5

0.03

6121

744

5042

8570

.970

0.61

7665

10

141.

6470

52

252

.978

015

74.5

4475

040

.931

081


Vol. 42

No. 6

2008


motions of the arguments of the pericenters and nodeswith the results of the above paper reveals differencesthat do not exceed one-tenth of a degree per year. Onlyfor the orbits of satellites S42 and N10 is this differencelarger by a factor of one-and-a-half and two, respec-tively. The discrepancies between the results can beexplained by both the different initial elements adoptedin the calculations and by the different time scales uponwhich the calculations were performed. We failed tofind any comparative data in the literature for the

ω

-libration orbits marked in the tables by asterisks and forthe 12 orbits of Saturnian satellites given in the secondhalf of Table 5.

3. In Figs. 1–52, four orbital elements of the outerSaturnian, Uranian, and Neptunian satellites are plottedagainst time. The time

t

in years is along the horizontalaxis and the following parameters are along the verticalaxis: 1, eccentricity

e

; 2, ecliptical inclination

i

indegrees; 3, argument of pericenter

ω

in degrees; and 4,longitude of ascending node

Ω

in degrees. In these fig-ures, the current time

t

is measured from the initial time

t

0

and the corresponding Julian dates JD (

t

0

) are listed inTable 1. In Figs. 3, 5, 7, 12, 36, 37, 38 (the orbits of sat-ellites S20, S22, S24, S29, S/2007 S1, S/2007 S2, andS/2007 S3) and Figs. 39–52 (the orbits of the Uranianand Neptunian satellites), the solid, dotted, and dashedlines correspond to the numerical integration of the rig-orous equations of motion, the numerical–analyticalmethod, and the analytical solution of the doubly aver-aged Hill problem, respectively. The remaining figuresshow the time dependences of the orbital elementsobtained only by the numerical and numerical–analyti-cal methods. The analytical method in the model ofdouble averaging yields mainly good qualitativeresults. Quantitatively, there are differences that stem

from the fact that the Hill approximation is limited andthat averaging the perturbing functions introduces sim-plifications. Therefore, the characteristics of the evolv-ing orbits for several outer Saturnian, Uranian, andNeptunian satellites that were previously obtained ana-lytically (Vashkov’yak, 1999; 2001a; 2001b; 2003)should be considered only as approximate means.Since, in addition, these were calculated for the prelim-inary initial elements of the satellite orbits taken fromMPEC, the more realistic values from Tables 5–7should be used in the subsequent analysis.

The small perturbation parameter

δ

=

n

1

/

n

, where

n

1

and

n

are the mean motions of the planet and the satel-lite, respectively, is very important for the analyticaland numerical–analytical methods. The developed ver-sion of the numerical–analytical method allows itsproper application only for

δ

which does not exceedabout 0.1. In the system of Saturn, the outer satellitesS19, S28, S29, S41, S42, S45, S46, S48, and S/2006 S5do not satisfy this condition. Therefore, a discrepancybetween the numerical and numerical–analyticalresults, particularly in the dependences

ω

(

t

)

, is clearlyseen in the corresponding figures.

For the orbits of the outer Saturnian satellites S20,S29, and S/2007 S1, analysis of the doubly averagedHill problem suggests that the variations in the argu-ments of the pericenters are librational in pattern. How-ever, our calculations performed by the more accuratenumerical and numerical–analytical methods showthat, in fact, the arguments of the pericenters for theseorbits, “on average,” vary circulationally (see Figs. 3,12, and 36). This qualitative discrepancy between theresults stems from the fact that the phase points in theargument of the pericenter–eccentricity plane are closeto the separatrix of the integrable doubly averaged Hill

Table 2.

Characteristics of the family of evolving orbits for the outer Saturnian satellites

Types of orbits Number of satellites

a

min

,

million km

a

max

,

million km

e

min

e

max

i

min

,

deg

i

max

,

deg

Prograde orbits

9 11.3 18.8 0.02 0.71 28 55

Retrograde orbits

29 12.8 26.0 0.08 0.77 128 179

Table 3.

Characteristics of the family of evolving orbits for the outer Uranian satellites


a

min

,

million km

a

max

,

million km

e

min

e

max

i

min

,

deg

i

max

,

degPrograde orbits

1 14.5 14.7 0.44 0.85 47 66

Retrograde orbits

8 4.3 20.8 0.07 0.70 139 172

Table 4.

Characteristics of the family of evolving orbits for the outer Neptunian satellites


a

min

, million km

a

max

, million km

e

min

e

max

i

min

, deg

i

max

, deg

Prograde orbits 2 22.0 23.7 0.06 0.63 30 55Retrograde orbits 3 16.6 52.2 0.07 0.90 110 147

492


Vol. 42

No. 6

2008


Table 5.

Extreme semimajor axes, eccentricities, inclinations, variation periods, and precession rates of the arguments of thepericenters (L is the libration of

ω

) and the longitudes of the nodes for the orbits of distant Saturnian satellites

1 2 3 4 5 6 7 8 9

Saturnian satellite

a

min

, million km

a

max

,million km

e

min

e

max

i

min

,deg

i

max

, deg

T

ω

, thousand

years

vω, vG, deg. yr–1

TΩ, thousand

years

vΩ, vH, deg. yr–1

S9 12.85 12.99 0.139 0.186 172.7 177.8 0.4588 0.7847 0.7179 0.5015

12.85 12.99 0.141 0.186 172.7 177.8 0.7748 0.4937

S19 22.38 23.86 0.233 0.440 169.5 175.8 0.2148 1.6760 0.2733 1.3185

22.35 23.73 0.245 0.435 169.4 176.3 1.6844 1.2986

S20 14.84 15.15 0.113 0.644 38.42 54.54 0.5779 0.6230 0.5573 –0.6460

14.84 15.17 0.115 0.642 38.52 54.59 0.6869 –0.6897

S21 17.87 18.82 0.363 0.704 28.57 47.59 0.1913 1.8814 0.2727 –1.3200

17.92 18.73 0.363 0.686 29.10 47.26 1.8778 –1.2952

S22* 11.30 11.39 0.119 0.562 41.01 53.84 0.5690 – 0.8659 –0.4157

11.30 11.40 0.117 0..565 41.06 53.83 (L) – –

S23 19.12 19.75 0.076 0.160 173.5 178.7 0.2793 1.2888 0.3783 0.9517

19.11 19.74 0.077 0.156 173.5 178.7 1.2299 0.8888

S24* 11.27 11.37 0.142 0.549 40.61 53.60 0.5700 – 0.8476 –0.4247

11.28 11.37 0.144 0.548 40.57 53.38 (L) – –

S25 18.32 18.91 0.154 0.273 163.9 170.5 0.3193 1.1275 0.4299 0.8374

18.31 18.88 0.161 0.274 164.0 170.2 1.1523 0.8519

S26 16.11 16.62 0.329 0.634 28.47 45.39 0.2513 1.4326 0.3631 –0.9914

16.10 16.62 0.323 0.630 28.95 45.43 1.4460 –0.9850

S27 15.42 15.71 0.186 0.372 147.9 156.7 0.4727 0.7617 0.5415 0.6648

15.42 15.71 0.185 0.366 147.9 156.8 0.7518 0.6641

S28 17.17 17.90 0.318 0.640 28.83 45.61 0.2221 1.6212 0.3268 –1.1018

17.16 17.85 0.322 0.632 29.27 45.84 1.6382 –1.1053

S29 17.58 18.25 0.092 0.550 41.42 54.24 0.4420 0.8145 0.4637 –0.7763

17.60 18.22 0.072 0.552 41.40 54.10 0.8347 –0.8168

S30 19.86 20.88 0.368 0.566 172.1 178.4 0.2277 1.5808 0.3073 1.1714

19.89 20.86 0.374 0.560 172.0 178.8 1.6230 1.1727

S31 18.97 19.68 0.238 0.666 133.6 152.7 0.3393 1.0609 0.3391 1.0615

18.97 19.66 0.239 0.660 133.8 151.9 1.1144 1.0864

S36 20.29 21.18 0.179 0.337 162.6 170.2 0.2731 1.3183 0.3488 1.0321

20.29 21.17 0.178 0.326 162.7 170.2 1.2379 0.9371

S37 16.75 17.40 0.311 0.638 30.08 45.90 0.2376 1.5155 0.3407 –1.0565

16.77 17.38 0.315 0.632 29.94 45.68 1.5081 –1.0289

S38 18.96 19.58 0.091 0.215 155.4 162.4 0.3565 1.0098 0.4241 0.8488

18.95 19.57 0.091 0.209 155.5 162.2 0.9958 0.8310

S39 19.95 20.88 0.271 0.768 128.3 152.5 0.2832 1.2714 0.2977 1.2094

19.93 20.77 0.278 0.762 128.8 152.0 1.3299 1.2880

SOLAR SYSTEM RESEARCH Vol. 42 No. 6 2008


Table 5. (Contd.)

1 2 3 4 5 6 7 8 9

Saturnian satellite

amin, million

km

amax,million

kmemin emax

imin,deg

imax, deg

Tω, thousand

years


TΩ, thousand

years

Saturnian satellite

S40 19.91 20.73 0.153 0.349 151.2 159.4 0.3342 1.0772 0.3734 0.9642

19.91 20.70 0.153 0.338 151.2 160.2 1.1044 0.9606

S41 21.83 22.92 0.078 0.209 161.2 168.2 0.2769 1.3002 0.3346 1.0758

21.82 22.88 0.077 0.199 161.3 168.2 1.3353 1.1043

S42 24.19 25.92 0.122 0.313 166.5 173.6 0.2212 1.6273 0.2643 1.3619

24.22 25.94 0.120 0.285 166.6 173.5 1.4838 1.669

S43 19.32 20.10 0.281 0.466 161.4 169.2 0.2644 1.3614 0.3425 1.0510

19.33 20.11 0.284 0.460 160.9 169.3 1.3806 1.0507

S44 18.04 18.65 0.204 0.477 143.1 154.7 0.3811 0.9446 0.4138 0.8700

18.06 18.63 0.211 0.475 143.2 154.7 – –

S45 21.50 22.84 0.309 0.639 143.9 161.4 0.2281 1.5782 0.2687 1.3398

21.49 22.75 0.318 0.637 144.2 161.5 – –

S46 20.72 21.82 0.315 0.601 148.8 163.0 0.2406 1.4960 0.2913 1.2357

20.71 21.77 0.326 0.586 148.9 163.3 – –

S47 17.42 17.97 0.362 0.570 153.0 165.4 0.2875 1.2521 0.3703 0.9722

17.41 17.96 0.368 0.567 153.1 165.6 – –

S48 22.18 23.64 0.331 0.566 163.6 173.4 0.1991 1.8084 0.2530 1.4231

22.20 23.59 0.332 0.552 163.8 173.5 – –

S/2004 S7 20.45 21.63 0.407 0.650 156.9 170.2 0.2097 1.7170 0.2710 1.3286

20.44 21.54 0.410 0.644 157.2 170.0 1.6468 1.2138

S/2004 S12 19.41 20.23 0.243 0.419 160.0 168.8 0.2746 1.3110 0.3543 1.0162

19.40 20.18 0.245 0.417 160.1 168.8 1.3125 0.9974

S/2004 S13 17.81 18.37 0.191 0.313 165.6 171.5 0.3053 1.1793 0.4141 0.8694

17.81 18.35 0.193 0.310 165.8 171.4 1.1981 0.8710

S/2004 S17 19.09 19.73 0.129 0.239 164.4 171.0 0.3054 1.1790 0.4053 0.8881

19.08 19.75 0.130 0.238 164.4 171.0 1.1794 0.8715

S/2006 S1 18.43 18.97 0.090 0.214 152.1 159.5 0.4010 0.8978 0.4516 0.7971

18.42 18.96 0.089 0.210 152.0 159.4 – –

S/2006 S4 18.09 18.71 0.253 0.385 171.2 177.2 0.2867 1.2555 0.3989 0.9025

18.09 18.69 0.254 0.383 171.3 177.4 – –

S/2006 S5 22.33 23.57 0.120 0.268 164.3 171.2 0.2506 1.4363 0.3061 1.1759

22.34 23.58 0.121 0.260 164.5 170.8 – –

S/2006 S6 18.98 19.66 0.156 0.292 159.4 166.9 0.3166 1.1370 0.4054 0.8879

18.98 19.65 0.153 0.288 159.7 166.7 – –

S/2007 S1 17.51 18.07 0.017 0.391 41.67 51.16 0.8513 0.4229 0.5814 –0.6192

17.48 18.06 0.034 0.438 41.40 51.17 – –

S/2007 S2 16.50 16.87 0.142 0.222 171.2 176.7 0.3771 0.9547 0.5706 0.6309

16.50 16.87 0.146 0.221 171.2 176.8 – –

S/2007 S3 18.60 19.21 0.138 0.239 171.5 177.2 0.3052 1.1794 0.4270 0.8432

18.59 19.18 0.138 0.236 171.6 177.2 – –

494



Table 6. Extreme semimajor axes, eccentricities, inclinations, variation periods, and precession rates of the arguments of thepericenters (L is the libration of ω) and the longitudes of nodes for the orbits of distant Uranian satellites

1 2 3 4 5 6 7 8 9

Uranian satellite

amin, million km

amax,million km emin emax

imin,deg

imax, deg

Tω, thousand

years


TΩ, thousand

years


U16 7.16 7.17 0.071 0.317 138.9 143.2 9.000 0.0400 6.500 0.05547.16 7.17 0.072 0.314 139.1 144.1 0.0396 0.0567

U17 12.14 12.25 0.446 0.597 151.8 162.4 1.390 0.2590 1.750 0.205712.14 12.25 0.445 0.597 151.7 162.4 0.2618 0.1890

U18 16.07 16.39 0.316 0.581 143.3 155.6 1.140 0.3158 1.350 0.266616.07 16.39 0.317 0.576 143.1 155.6 0.3231 0.2768

U19 17.31 17.80 0.459 0.702 145.7 161.6 0.820 0.4390 0.960 0.375017.32 17.80 0.458 0.700 145.9 161.6 0.4474 0.3440

U20 7.94 7.96 0.124 0.339 141.1 146.5 5.480 0.0657 5.200 0.06927.95 7.96 0.123 0.340 141.1 146.4 0.0656 0.0689

U21 8.49 8.52 0.202 0.238 165.8 168.3 2.630 0.1369 4.300 0.08378.49 8.51 0.202 0.239 165.8 168.2 0.1403 0.0858

U22 4.27 4.28 0.091 0.188 145.5 148.0 12.350 0.0291 14.350 0.02514.28 4.28 0.093 0.189 145.6 147.8 0.0296 0.0256

U23 * 14.50 14.71 0.440 0.855 47.5 66.4 0.520 – 0.980 –0.3673 14.51 14.71 0.445 0.856 47.4 66.3 (L) – –

U24 20.01 20.75 0.316 0.487 166.5 171.6 0.750 0.4800 0.990 0.363620.01 20.75 0.325 0.479 166.5 171.6 0.4899 0.3565

Table 7. Extreme semimajor axes, eccentricities, inclinations, variation periods, and precession rates of the arguments of thepericenters (L is the libration of ω) and the longitudes of the nodes for the orbits of distant Neptunian satellites

1 2 3 4 5 6 7 8 9

Neptunian satellite

amin, million km

amax,million km emin emax

imin,deg

imax, deg

Tω, thousand

years


TΩ, thousand

years


N9 16.56 16.63 0.190 0.904 110.9 144.2 5.500 0.0655 5.400 0.0667

16.56 16.63 0.193 0.903 110.9 144.2 0.0650 0.0660

N10 45.52 49.52 0.074 0.879 117.2 147.2 1.530 0.2353 1.440 0.2500

45.63 49.37 0.073 0.868 117.2 147.2 0.4258 0.3985

N11* 22.06 22.30 0.064 0.628 39.6 55.1 3.120 – 4.160 –0.0865

22.06 22.30 0.044 0.632 39.6 55.2 (L) – –

N12 23.38 23.72 0.286 0.555 30.3 43.4 2.260 0.1593 3.250 –0.1108

23.39 23.71 0.284 0.554 30.3 43.3 0.1596 –0.1111

N13* 47.38 52.22 0.149 0.872 117.9 145.2 0.600 – 1.090 0.3303

47.66 52.08 0.150 0.866 118.0 144.9 (L) – –

problem. Allowance for the small perturbations revealsthe actual circular pattern of evolution of the argumentsof the pericenters for these orbits and the necessity ofimproving the simplest model of evolution for thedomains of orbital parameters e, i, and ω in close(together with δ) proximity to the separatrix.

Since the maximum value of δ in the system of outerUranian satellites is about 0.09 (for satellite U24,Fig. 47), the agreement between the results of the abovetwo methods for the remaining satellites is good. Forthe orbit of satellite U22 (Fig. 45) with the smallestsemimajor axis, the discrepancy in the dependences i(t)



for t > 11000 years probably stems from the fact that itis improper to use the polynomials for the time thatdescribe the evolution of the planetary orbits in theapplied algorithm of the numerical method.

In the system of Neptune, two outer satellites, N10and N13, have orbits with large semimajor axes reach-ing about 49 and 52 million km (Table 7), while the cor-responding maximum values of δ can be 0.15 and 0.17.Therefore, the difference between the numerical andnumerical–analytical results is clearly seen in Fig. 49(N10). In contrast, for this satellite, the analyticalmethod even yields a qualitatively different (libra-tional) pattern of variation in the argument of the peri-center with time than do the numerical and numerical–analytical methods (Fig. 49, fragment 3). This orbit hassuch elements that the phase point in the (ω, e) plane isvery close to the separatrix of the integrable doublyaveraged Hill problem. Similar qualitative changeshave already been pointed out for the orbits of the Sat-urnian satellites S20, S29, and S/2007 S1. In contrast tothese orbits and the orbit of N10, the analysis of theorbital evolution of the Neptunian satellite N13 in termsof the doubly averaged Hill problem yields a circulationof the argument of the pericenter. The more accuratemethods suggest that it librates (Fig. 52).

4. The orbits of satellites S22, S24, U23, and N11with librational variations in the argument of pericenterω are under the Lidov–Kozai resonance conditions

Ò1 = (1 – Â2)cos2i ≤ 3/5, Ò2 = e2(2/5 – sin2isin2ω) < 0 andwe see from the corresponding Figs. 5, 7, 46, and 50that the periodic solar perturbations (with a periodapproximately equal to half the orbital period of theplanet) are small compared to the long-period ones. Inthe upper fragment of Fig. 53, the curves correspondingto the bifurcation value of the constant Ò1 = 3/5 areshown in the plane of initial parameters (i0, e0). In itslower fragment, the Ò2 = 0 separatrixes are shown in theplane of initial parameters (i0, ω0). In both fragments,different symbols mark the orbits of ω librators, onlyone of which (N13) is retrograde.

A significant variation in the eccentricity and an out-of-phase variation in the sine of the inclination calledthe Lidov–Kozai mechanism are possible during theevolution of the outer satellite orbits. In the system ofSaturn, this mechanism shows up particularly clearly inthe orbital evolution of satellites S20 and S22, respec-tively, with librational and circulational patterns ofvariation in the arguments of pericenters—the ampli-tudes of the eccentricity variations are very large andcan reach 0.45.

This paper completes the compilation of the so-called “evolutionary atlas” of orbits for all of the outersatellites of the giant planets known to date. However,there is every reason to believe that the technicalimprovement of observational instruments will allow

0.200.160.12180175170400200

0400200

0 100 200 300 400 500 600 700 800 9001000

-1-

-2-

-3-

-4-

Fig. 1. Orbital elements of satellite S9 versus time; t (years)is along the horizontal axis and the following parameters arealong the vertical axis: 1, e; 2, i (deg.); 3. ω (deg.);4, Ω (deg.); the solid and dotted lines represent the numeri-cal and numerical–analytical methods, respectively.

0.5

0.30.2180175

165400200

0400200

0 100 200 300

-1-

-2-

-3-

-4-

0.4

170

150 25050

Fig. 2. Same as Fig. 1 for S19.

0.80.4

6050

30400200

0400200

0 200 400 600

-1-

-2-

-3-

-4-

40

300 500100

0

Fig. 3. Same as Fig. 1 for S20 (the dashed line represents theanalytical method).

496



0.80.6

5545

25400200

0400200

0

-1-

-2-

-3-

-4-

35

0.2

100 200 300150 25050

0.4


0.60.4

0605040

120100

60400200

0 100 200 300 400 500 600 700 800 9001000

-1-

-2-

-3-

-4-

0.2

80


0.200.15

180175

400200

0400200

0

-1-

-2-

-3-

-4-

170

0.05

100 200 300150 25050

0.10

350 400 450 500


0.60.4

06050

30120100

60400200

0 100 200 300 400 500 600 700 800 9001000

-1-

-2-

-3-

-4-

0.2

80

40


0.300.25

175170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.15

100 200 300150 25050

0.20

350 400 450 500

165


0.70.5

3530

400200

0400200

0

-1-

-2-

-3-

-4-

25

0.3

100 200 300150 25050 350 400 450 500




0.40.3

160155

145400200

0400200

0 200 400 600

-1-

-2-

-3-

-4-

150

300 500100

0.10.2


0.70.5

4535

400200

0400200

0

-1-

-2-

-3-

-4-

25

0.3

100 200 300150 25050 350400450500


0.60.5

180175

400200

0400200

0

-1-

-2-

-3-

-4-

170

0.3

100 200 300150 25050 350 400 450 500

0.4


0.80.6

160150

400200

0400200

0

-1-

-2-

-3-

-4-

130

0.2

100 200 300150 25050 350 400 450 500

0.4

140


0.6

0.2

5550

400200

0400200

0

-1-

-2-

-3-

-4-

40

0

100 200 300150 25050 350 400 450 500

45

0.4


0.350.25

175

165

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.15

100 200 300150 25050 350 400 450 500

170


498



0.70.5

5040

400200

0400200

0

-1-

-2-

-3-

-4-

30

0.3

100 200 300150 25050 350 400 450 500


0.250.15

165160

400200

0400200

0

-1-

-2-

-3-

-4-

155

0.05

100 200 300150 25050 350 400 450 500


0.80.6

160

400200

0400200

0

-1-

-2-

-3-

-4-

120

0.2

100 200 300150 25050 350 400 450 500

0.4

140


0.40.3

160155

400200

0400200

0

-1-

-2-

-3-

-4-

150

0.1

100 200 300150 25050 350 400 450 500

0.2


0.250.15

170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.05

100 200 300150 25050 350 400 450 500

165


0.350.20

175

400200

0400200

0

-1-

-2-

-3-

-4-

165

0.05

100 200 300150 25050 350 400 450 500

170




0.5

0.3

170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.2

100 200 300150 25050 350 400 450 500

165

0.4


0.5

0.3

155

400200

0400200

0

-1-

-2-

-3-

-4-

140

0.2

100 200 300150 25050 350 400 450 500

145

0.4

150


0.7

170

400200

0400200

0

-1-

-2-

-3-

-4-

140

0.3

100 200 300150 25050 350 400 450 500

150

0.5

160


0.60.5

165155

400200

0400200

0

-1-

-2-

-3-

-4-

145

0.3

100 200 300150 25050 350 400 450 500

0.4


0.60.5

170160

400200

0400200

0

-1-

-2-

-3-

-4-

150

0.3

100 200 300150 25050 350 400 450 500

0.4


0.60.5

175170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.3

100 200 300150 25050

0.4

165


500



0.70.6

170

160

400200

0400200

0

-1-

-2-

-3-

-4-

155

0.4

100 200 300150 25050 350 400 450 500

0.5

165

Fig. 28. Same as Fig. 1 for S/2004 S7.

0.5

0.3

170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.2

100 200 300150 25050 350 400 450 500

165

0.4


0.35

175

400200

0400200

0

-1-

-2-

-3-

-4-

165

0.15

100 200 300150 25050 350 400 450 500

170

0.25


0.25

175

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.10

100 200 300150 25050 350 400 450 500

170

0.200.15

165


0.25

160

400200

0400200

-1-

-2-

-3-

-4-

150

0.05

155

0.15


0.400.35

180175

400200

0400200

0

-1-

-2-

-3-

-4-

170

0.25

100 200 300150 25050

0.30

350 400 450 500


0 100 200 300150 25050 350 400 450 500



0.30.2

175170

400200

0400200

0

-1-

-2-

-3-

-4-

160

0.1

100 200 300150 25050 350 400 450 500

165

0.300.25

170165

400200

0400200

-1-

-2-

-3-

-4-

155

0.150.20

160



0.60.4

0554535

400200

0400200

0 100 200 300 400 500 600 700 800 9001000

-1-

-2-

-3-

-4-

0.2


0.40.2

180175

400200

0400200

0

-1-

-2-

-3-

-4-

170

0

100 200 300150 25050 350 400 450 500


0.250.20

180

400200

0400200

0

-1-

-2-

-3-

-4-

170

0.10

100 200 300150 25050

0.15

350 400 450 500

175


0.40.2

0145140135400200

0400200

0 100 200 300 400 500 600 700 800 9001000

-1-

-2-

-3-

-4-

Fig. 39. Same as Fig. 3 for U16.

0 100 200 300150 25050 350 400 450 500

502



0.80.60.4170160150400200

0400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-


0.60.5

0.3160150140400200

0400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-

0.4


0.80.60.4180160140400200

0400200

0 500 1000 1500

-1-

-2-

-3-

-4-


0.40.2

0150145140400200

0400200

0 10002000

30004000

9000

-1-

-2-

-3-

-4-

80007000

60005000


0.25

0.20170168166

400200

0400200

0 10002000

30004000

9000

-1-

-2-

-3-

-4-

80007000

60005000


0.200.15

0.05148.0146.5145.0

400200

0400200

0 5000 15000

-1-

-2-

-3-

-4-

10000

0.10




1.00.5

0806040

15010050

400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-


0.450.40

175170165400200

0400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-

0.35


1.00.6

0150130110400200

0400200

0 10002000

30004000

9000

-1-

-2-

-3-

-4-

80007000

60005000

0.2

Fig. 48. Same as Fig. 3 for N9.

1.00.5

150130110400200

0400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-

0


0.80.4

0554535

1208040

400200

0 1000 2000 3000 4000

-1-

-2-

-3-

-4-

60005000


0.60.40.2453530

400200

0400200

0 1000 2000 3000 4000

-1-

-2-

-3-

-4-

60005000

40


504



quite a few distant satellites to be discovered in theneighborhood of these planets.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundationfor Basic Research (project nos. 07-02-92169-NTsNI_a and 07-02-91229-YaF) and the scientificschool grant no. NSh-1123.2008.1.

REFERENCES

Vashkov’yak, M.A., Evolution of the Orbits of Distant Satel-lites of Uranus, Pis’ma Astron. Zh., 1999, vol. 25, no. 7,pp. 554–560 [Astron. Lett. (Engl. Transl.), vol. 25, no. 7,pp. 476–481].

Vashkov’yak, M.A., Orbital Evolution of Uranus’s New Outer Satellites, Pis’ma Astron. Zh., 2001a, vol. 27,no. 6, pp. 470–475 [Astron. Lett. (Engl. Transl.), vol. 27,no. 6, pp. 404–409].

Vashkov’yak, M.A., Orbital Evolution of Saturn’s New Outer Satellites and Their Classification, Pis’ma Astron.Zh., 2001b, vol. 27, no. 7, pp. 533–542 [Astron. Lett.(Engl. Transl.), vol. 27, no. 7, pp. 455–463].

Vashkov’yak, M.A., Orbital Evolution of New Distant Nep-tunian Satellites and omega-Librators in the SatelliteSystems of Saturn and Jupiter, Pis’ma Astron. Zh., 2003,vol. 29, no. 10, pp. 782–793 [Astron. Lett. (Engl.Transl.), vol. 29, no. 10, pp. 695–703].

Vashkov’yak, M.A. and Teslenko, N.M., Evolusion Charac-teristics of Jupiter’s Outer Satellites, Astron. Vestn.,2008, vol. 42, no. 4, pp. 301–316 [Sol. Syst. Res. (Engl.Transl.), vol. 42, no. 4, pp. 281–295].

Beaugé, C. and Nesvorny, D., Proper Elements and Secular Resonances for Irregular Satellites, Astron. J., 2007,vol. 133, pp. 2537–2558.

http://lnfm1.sai.msu.su/neb/nss/index.htm.

0.80.4

150130110400200

0400200

0 500 1000 1500 2000 2500 3000

-1-

-2-

-3-

-4-

0


0.900.750.600.450.300.15

0180

135

90

45

0 30 60 90 120 150 180

e

c1 > 3/5 c1 > 3/5c1 < 3/5

c1 = 3/5S24S22U23N11N13

i

c2 > 0

c2 > 0

c2 < 0

c2 = 0S24S22U23N11N13

ω

Fig. 53. Locations of the ω-libration orbits for the outer Sat-urnian, Uranian, and Neptunian satellites in the planes ofinitial parameters (i0, e0) and (i0, ω0).

Documents

Evolutionary characteristics of the orbits of outer Saturnian, Uranian, and Neptunian satellites