Resonance Properties of Extra-Solar Two-Planet Systems

c Pleiades Publishing, Ltd., 2010. ISSN 1063-7729, Astronomy Reports, 2010, Vol. 54, No. 6, pp. 550–561.  c E.D. Kuznetsov, 2010, published in Astronomicheski˘ı Zhurnal, 2010, Vol. 87, No. 6, pp. 605–616. Original Russian Text 

Resonance Properties of Extra-Solar Two-Planet Systems E. D. Kuznetsov M. Gorky Ural State University, Yekaterinburg, Russia Received August 19, 2009; in final form, November 26, 2009

Abstract—The properties of the distribution of resonance zones in the two-planet problem are analyzed as a function of the semi-major axes and masses of the planets on cosmogonic time scales. A solution to equations averaged using the Hori–DePrit method with accuracy to second order in the small parameter of the problem is used, and the translation from the averaged to the osculating elements taken into account. Conditions for the overlap of resonance zones are obtained. It is shown that motion of pairs of planets in the extrasolar planetary systems HD 73526, 47 UMa, HD 181433 (c–d), GJ 581 (b–c), and HD 155358 can occur in regions of resonance overlap. The orbital evolution of the two-planet system 47 UMa is analyzed in the absence of resonance, in the vicinity of the 1 : 2, 3 : 7, 2 : 5, 3 : 8, 1 : 3, and 1 : 7 resonances when the initial data correspond to a region of wide resonance, and in regions of overlap of wide-resonance zones. The possibility of chaotic motions of the planets in the 47 UMa system is demonstrated. DOI: 10.1134/S1063772910060089

1. INTRODUCTION Studies of resonance properties of planetary systems are aimed at obtaining answers to questions such as the following. What are the conditions for the onset of resonances? Where are resonance zones located? How does orbital evolution proceed in the vicinity of resonance zones? We restrict the current study to resonances of mean motions due to the presence of commensurate frequencies of the orbital motions of planets. We consider the resonance properties of two-planet systems in which the more massive planet is located in the inner orbit. The dimensions of resonance zones and conditions for resonance overlap are determined for such systems. A detailed analysis of the properties of the orbital evolution of the 47 UMa system is presented. Our analysis of the orbital evolution makes use of the results of [1–5]. The method applied in these studies is the following. The position of a planet is described in Jacobian coordinates [6], and a system close to Keplerian osculating elements is used. All the positional elements are small and dimensionless:  a= 0 0  (a − a )/a , e, I = sin(I/2). The angular elements are the longitudes: α = l + g + Ω, β = g + Ω, γ = Ω. Here, a, a0 , e, I, l, g, and Ω are the semi-major axis and its mean value, the eccentricity, the inclination to a fixed plane that is close to the constant Laplace plane, the mean anomaly, the argument of periastron, and the longitude of the ascending node. The perturbing Hamiltonian is represented by a Poisson series (a Fourier series in all the angular

variables with coefficients in the form of a Taylor series in all positional variables). Expansions with symbolic parameters are used [3–5]; i.e., symbolic expressions are preserved for all the orbital elements and the mass, which makes it possible to work with arbitrary planetary systems. Hori–Deprit averaging is applied [7], and the averaged equations of motion are integrated numerically. The change-of-variable functions corresponding to the translation between the osculating and averaged elements as a function of the latter are computed analytically. Change-of-variable functions are useful when analyzing the resonance properties of planetary systems. First, these functions are used to estimate the widths of resonance zones. Second, they make it possible to translate between the averaged and osculating elements, which is necessary if we wish to correctly describe the orbital evolution in the vicinity of resonances. 2. RESONANCE ZONES AND REGIONS OF RESONANCE OVERLAP IN TWO-PLANET SYSTEMS Our investigation of the resonance properties of two-planet systems will be based on the approach proposed in [8–10]. We write the resonance condition using the system of notation of [3–5, 10]: (1) n1 ω1 + n4 ω2 = 0. Here, n1 and n4 are the indices for the mean longitudes of the first and second planet in the expansion of the Hamiltonian in a Poisson series, ωs = κs a−3/2 s 550

(s = 1, 2)

RESONANCE PROPERTIES OF EXTRA-SOLAR TWO-PLANET SYSTEMS

is the mean motion of the planets, and κs are gravitational parameters defined by the formulas

semi-major axis a2 and k is the multi-index of the positional-variable vector

κ12 = Gm0 (1 + μm1 ), 1 + μm1 + μm2 , κ22 = Gm0 1 + μm1 where G is the gravitational constant, m0 the mass of the central star, and μm1 and μm2 the masses of the two-planetary. Three parameters are introduced for the masses of the two-planetary, which preserves some freedom in selecting their values. For example, if we set m1 = 1, μ will then be the ratio of the mass of the first planet to the mass of the star. Relation (1) is rarely satisfied exactly. We introduce the resonance value of the semi-major axis of the second planet’s orbit for which (1) is satisfied:  2/3   n4 1 + μm1 + μm2 1/3 res a2 = a1 . (2) n1 (1 + μm1 )2 A solution based on expanding the Hamiltonian and the generating function of the Hori–Deprit transform in a Poisson series is not applicable, or is only applicable to a limited extent, for semi-major axes in the interval res a2 ∈ [ares 2 − Δa, a2 + Δa]. Here, Δa is the width of the resonance zone, which depends on the perturbation amplitude. The method of averaging over short-period perturbations involves the change-of-variable function describing the relationship between the averaged and osculating elements. We can determine Δa from various conditions. We will use the concept of narrow- and wide-resonance zones. A rigorous definition requiring several pages of text can be found in [8, 9]. However, the essence of this concept is not complicated. If the small parameter of the problem is sufficiently small, the zones associated with various resonances do not overlap. The motion is conventionally taken to be periodic outside a resonance zone, at least in a first approximation in μ. When determining a narrow zone, only one resonance harmonic for which condition (1) is satisfied is taken into account. When determining a wide zone, the influence of additional non-resonance, short-period terms must be included. According to [8, 9], the width of a narrowresonance zone is given by  Akn xk  (3) Δa = μb, b = . 1.5κ 2 a−5/2 nω=0

2 2

Here, Akn are the coefficients of the Poisson series representing the change-of-variables function for the ASTRONOMY REPORTS

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as − a0s , x3s−1 = es , a0s Is = sin , s = 1, 2. 2

x3s−2 = x3s

The size of a resonance zone for a wide resonance is

 Δa = μC + 2 μb,

C=

 Bkn xk .

(4)

nω=0

Here, Bkn are the coefficients of an echeloned Poisson series representing the change-of-variables function for the semi-major axis a2 . Non-resonance terms are taken into account as a sum of amplitudes. Let us determine the resonance values of the semimajor axes and estimate the sizes of the resonance zones Δa for two-planet systems in which the more massive planet is located closer to the central star: m1 > m2 , a1 < a2 . In the computations using (3) and (4), we used the change-of-variable function for the semi-major axis of the second planet [4]. Tables 1 and 2 give the resonance values (2) for the semi-major axis of the second planet’s orbit, ares 2 /a1 , in units of a1 ; the corresponding values of the indices n1 , n4 ; the sizes (3), (4) of the resonance zones Δa; res and the minimum ares min and maximum amax semimajor axes corresponding to the boundaries of the narrow- and wide-resonance zones. The empty cells in Tables 1 and 2 correspond to resonances for which it was not possible to reliably estimate the size Δa and boundary positions of the wide-resonance zones res ares min and amax due to the effect of resonance overlap. Results are presented for two values of the small parameter, μ = 1/300 (Table 1) and μ = 1/100 (Table 2). Depending on the selected definition of the small parameter, m1 + m2 m1 or μ = , μ= m0 m0 we obtain for the masses of the planets 10 10 m1 = mJ or m1 + m2 = mJ 3 3 1 , for μ = 300 m1 = 10mJ or m1 + m2 = 10mJ 1 , for μ = 100 where mJ is the mass of Jupiter. The main method used to search for extrasolar planets, the radial-velocity method, does not enable determination of the mass m of the planet, and only

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KUZNETSOV

Table 1. Resonance values for the semi-major axes and sizes of these resonance zones for μ = 1/300 n1

n4

ares 2 /a1

Narrow resonance

Broad resonance

Δa

ares min

ares max

Δa

ares min

ares max

8

−11

1.235

0.130

1.105

1.365

5

−7

1.250

0.179

1.071

1.430

7

−10

1.267

0.107

1.160

1.375

9

−13

1.277

0.080

1.196

1.357

2

−3

1.309

0.265

1.044

1.574

9

−14

1.341

0.037

1.304

1.379

0.252

1.090

1.593

7

−11

1.350

0.051

1.299

1.402

0.255

1.095

1.605

5

−8

1.367

0.068

1.299

1.434

0.257

1.110

1.623

8

−13

1.381

0.028

1.353

1.409

0.172

1.209

1.553

3

−5

1.404

0.117

1.287

1.522

0.309

1.095

1.714

7

−12

1.431

0.022

1.409

1.453

0.117

1.314

1.548

4

−7

1.451

0.053

1.398

1.503

0.162

1.289

1.613

5

−9

1.478

0.030

1.449

1.508

0.108

1.370

1.586

6

−11

1.496

0.016

1.480

1.513

0.077

1.419

1.574

7

−13

1.509

0.009

1.500

1.519

0.061

1.448

1.571

1

−2

1.586

0.107

1.479

1.693

0.232

1.354

1.818

5

−11

1.690

0.005

1.685

1.695

0.026

1.664

1.716

4

−9

1.715

0.008

1.707

1.724

0.031

1.685

1.746

3

−7

1.758

0.015

1.743

1.772

0.040

1.717

1.798

2

−5

1.840

0.025

1.815

1.865

0.058

1.782

1.898

3

−8

1.921

0.005

1.916

1.927

0.017

1.904

1.938

1

−3

2.078

0.035

2.043

2.113

0.073

2.005

2.151

2

−7

2.303

0.003

2.300

2.306

0.009

2.294

2.312

1

−4

2.517

0.013

2.504

2.531

0.028

2.489

2.546

1

−5

2.921

0.005

2.916

2.926

0.011

2.911

2.932

1

−6

3.299

0.002

3.297

3.300

0.004

3.295

3.303

1

−7

3.656

0.001

3.655

3.656

0.002

3.654

3.657

provides an estimate of the product m sin i, where i is the inclination of the orbit to the plane of the sky, i.e., the angle between the line of sight toward the center of mass of the system and the area vector [11]. The inclinations i are not known for the extrasolar planetary systems considered below. We will assume that the masses of the planets satisfy the condition 10 mJ  m1  10mJ 3

10 mJ  m1 + m2  10mJ . 3 In this case, the inclinations i of the planetary orbits to the plane of the sky should be large. Considering the limiting cases corresponding to μ = 1/300 and 1/100, we will take the resonance conditions for real systems to lie within these boundaries. It follows from Tables 1 and 2 that, on average, the size Δa of a resonance zone decreases with growth in or

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Table 2. Resonance values of the semi-major axes and sizes of these resonance zones for μ = 1/100 n1

n4

ares 2 /a1

Narrow resonance

Broad resonance

Δa

ares min

ares max

Δa

ares min

ares max

8

−11

1.233

0.220

1.013

1.452

7

−10

1.265

0.182

1.083

1.446

9

−13

1.274

0.135

1.139

1.409

9

−14

1.338

0.063

1.276

1.401

7

−11

1.347

0.087

1.261

1.434

5

−8

1.364

0.115

1.249

1.479

8

−13

1.378

0.047

1.331

1.425

3

−5

1.401

0.199

1.203

1.600

7

−12

1.428

0.037

1.391

1.464

0.288

1.140

1.716

4

−7

1.448

0.090

1.358

1.537

0.346

1.102

1.794

5

−9

1.475

0.050

1.425

1.526

0.244

1.231

1.719

6

−11

1.493

0.028

1.466

1.521

0.186

1.307

1.680

7

−13

1.506

0.016

1.490

1.522

0.157

1.349

1.663

1

−2

1.583

0.184

1.399

1.766

0.420

1.162

2.003

5

−11

1.686

0.008

1.678

1.694

0.065

1.621

1.752

4

−9

1.712

0.014

1.698

1.726

0.070

1.642

1.782

3

−7

1.754

0.025

1.729

1.779

0.083

1.670

1.837

2

−5

1.836

0.043

1.793

1.879

0.109

1.727

1.945

3

−8

1.917

0.009

1.908

1.926

0.037

1.880

1.954

1

−3

2.074

0.060

2.013

2.134

0.131

1.943

2.204

2

−7

2.298

0.006

2.292

2.304

0.018

2.280

2.316

1

−4

2.512

0.023

2.489

2.535

0.050

2.462

2.563

1

−5

2.915

0.008

2.907

2.923

0.019

2.896

2.934

1

−6

3.292

0.003

3.289

3.295

0.008

3.284

3.300

1

−7

3.648

0.001

3.647

3.649

0.004

3.644

3.652

the resonance value of the semi-major axis ares 2 /a1 . As a rule, the widths of neighboring resonance zones decrease with growth of the order of the resonance, ν = |n1 | + |n4 |. Broad-resonance zones are larger than narrow-resonance zones in size by factors of two to seven. Overlapping of the resonance zones is observed for small values of a2 /a1 . In the case of a narrow resonance, this occurs at a2 /a1  1.693 for μ = 1/300 (possible overlapping of the 8 : 11, 5 : 7, 7 : 10, 9 : 13, 2 : 3, 9 : 14, 7 : 11, 5 : 8, 8 : 13, 3 : 5, 7 : 12, 4 : 7, 5 : 9, 6 : 11, 7 : 13, 1 : 2, 5 : 11 resonance zones) and at a2 /a1  1.766 for μ = 1/100 (8 : 11, 7 : 10, 9 : 13, ASTRONOMY REPORTS

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9 : 14, 7 : 11, 5 : 8, 8 : 13, 3 : 5, 7 : 12, 4 : 7, 5 : 9, 6 : 11, 7 : 13, 1 : 2, 5 : 11, 4 : 9, 3 : 7). In the case of a wide resonance, this occurs at a2 /a1  1.818 for μ = 1/300 (9 : 14, 7 : 11, 5 : 8, 8 : 13,3 : 5, 7 : 12, 4 : 7, 5 : 9, 6 : 11, 7 : 13, 1 : 2, 5 : 11, 4 : 9, 3 : 7, 2 : 5) and at a2 /a1  2.003 for μ = 1/100 (7 : 12, 4 : 7, 5 : 9, 6 : 11, 7 : 13, 1 : 2, 5 : 11, 4 : 9, 3 : 7, 2 : 5, 3 : 8, 1 : 3). The data of Tables 1 and 2 can be used to determine which resonance zones overlap for a specified value of a2 /a1 for μ = 1/300 and 1/100. Interaction of the resonances in the region of resonance overlap leads to chaotization of the motions [12]. This must be taken

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into account when analyzing the dynamical evolution of systems located in a region of resonance overlap. A comparison of Tables 1 and 2 shows that, as the value of the small parameter μ increases, the resonances values of the semi-major axis ares 2 /a1 decrease, as well as the distance between them, leading to a growth in the regions of resonance overlap. Note that, if the more massive planet is located further from the central star than the less massive planet, the pattern is reversed: the resonance values ares 2 /a1 and the distance between them grow with increasing μ. We used data from regularly updated catalogs of extrasolar planets [13–15] to selected eight twoplanet systems located in the vicinity of resonance regions and satisfying the conditions m1 > m2 , a1 < a2 . Tables 3 and 4 present data on regions of resonance overlap and resonance zones in which the two-planet systems could be located for μ = 1/300 and 1/100. Column (1) gives the required information about the system: the name of the star (if the system has multiple planets, the pair of planets considered is indicated in parentheses); range of semimajor orbital axes for the second planet in units of a1 , a2 /a1 ; and references for data on the semi-major axes. Columns (2) and (3) present the ratios of the mean motions of the planets ω2 /ω1 corresponding to resonance relations in the cases of narrow and wide resonances for the indicated range of a2 /a1 . The interacting resonances with intersecting zones that form a region of resonance overlap are indicated in parentheses. With μ = 1/300, the two systems HD 73526 and 47 UMa could be located in regions of overlap of narrow resonances (Table 3). The important role of capture in resonance was noted in [16], as a result of which the migration of orbits is hindered and the system is stabilized in the vicinity of the 1 : 2 resonance. However, the range of possible semi-major orbital axes presented in [16] does not enable unambiguous identification of the set of resonances determining the dynamical evolution of the HD 73526 system. One region of resonance overlap and six individual resonances are located in the range of semi-major orbital axes for the 47 UMa system. A description and analysis of the resonance properties of 47 UMa will be presented in Sections 3 and 4. The HD 155358 system may be located in one of four resonances. The nominal semi-major axes given in [20] yield the non-resonance value a2 /a1 = 1.949, which lies between the 3 : 8 and 1 : 3 resonances. Since the semi-major axes for HD 181433 (c– d) [18] and GJ 581 (b–c) [19] are given without errors, the nominal values of a2 /a1 are presented for these

planets in Table 3. There are no lower-order resonances (ν  16) for HD 181433 (c–d), whereas the GJ 581 system (b–c) is located in the 4 : 9 resonance. The situation is simpler for values a2 /a1 > 2. The 1 : 3 resonance is possible in the 55 Cnc system (b– c), and the 1 : 4 resonance may be present in the HD 108874 and HD 102272 systems. These conclusions are in agreement with the results of [22–24]. A consideration of the conditions for the twoplanet systems to fall in regions of wide resonance (Tables 1, 3) shows that the systems end up in regions of resonance overlap when a2 /a1 < 2. Note that only regions of resonance overlap that do not intersect regions corresponding to narrow resonances are presented for the HD 73526 and 47 UMa systems in Table 3. The conditions for the HD 181433 (c–d), GJ 581 (b–c), and HD 155358 systems change when we consider the wide-resonance zones—all these systems could be in regions of resonance overlap. This must be taken into account when analyzing their orbital evolution. Overlapping of zones of wide resonances could lead to a slow (possibly very slow, compared to the conditions in the regions of overlap of zones of narrow resonances) drift in the parameters of the system and a gradual chaotization of the orbital evolution on cosmogonic time scales. The resonance properties of the 55 Cnc (b–c), HD 108874, and HD 102272 systems were preserved in the transition from narrow- to wide-resonance zones. Increasing the small parameter to μ = 1/100 did not change the resonance properties of 55 Cnc (b–c), HD 108874, and HD 102272, but led to qualitative and quantitative variations in the properties of the remaining systems (Table 4). For HD 73526 and 47 UMa, the set of regions of resonance overlap and individual resonances changed, for both the narrow- and wide-resonance zones. The characteristics of HD 181433 (c–d) and GJ 581 (b–c) correspond to regions of overlap of narrow resonances, while the broad-resonance overlap region contains a larger number of resonance zones. The HD 155358 system could also be in a region of overlapping narrow resonances, although, as for the case μ = 1/300, the nominal value a2 /a1 = 1.949 lies between the 3 : 8 and 1 : 3 resonances. This analysis enables us to draw the following conclusion. If the planetary masses in the considered extrasolar two-planet systems are such that the corresponding value of the small parameter is μ = 1/300−1/100, and the semi-major axes for the planetary orbits satisfy 1 < a2 /a1 < 2, the orbital evolution of these systems may have a chaotic character due to resonance-zone overlap. The data in Tables 1 and 2 can be used to precisely establish the upper boundary ASTRONOMY REPORTS Vol. 54 No. 6

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Table 3. Overlap of resonances and individual resonances for two-planet systems for μ = 1/300 ω2 /ω1

System Narrow resonance

Broad resonance

(2 : 3, 3 : 5, 4 : 7, 5 : 9, 1 : 2), (2 : 3, 3 : 5, 5 : 9, 1 : 2), (2 : 3, 3 : 5, 1 : 2), (2 : 3, 1 : 2), 1 : 2, (1 : 2, 5 : 11), 5 : 11, 4 : 9, 3 : 7, 2 : 5

(3 : 5, 1 : 2, 5 : 11, 4 : 9), (1 : 2, 5 : 11, 4 : 9), (1 : 2, 4 : 9), (1 : 2, 4 : 9, 3 : 7), (1 : 2, 3 : 7), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5), 2 : 5

1 : 2, (1 : 2, 5 : 11), 5 : 11, 4 : 9, 3 : 7, 2 : 5, 3 : 8

(3 : 5, 1 : 2, 5 : 11, 4 : 9), (1 : 2, 5 : 11, 4 : 9), (1 : 2, 4 : 9), (1 : 2, 4 : 9, 3 : 7), (1 : 2, 3 : 7), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5), 2 : 5, 3 : 8, 1 : 3

HD 181433 (c–d), a2 /a1 = 1.705 [18]

There are no resonances below ν  16

(3 : 5, 1 : 2, 5 : 11, 4 : 9)

GJ 581 (b–c), a2 /a1 = 1.707 [19]

4:9

(3 : 5, 1 : 2, 5 : 11, 4 : 9)

HD 155358, a2 /a1 = 1.764−2.146 [20]

3 : 7, 2 : 5, 3 : 8, 1 : 3

(1 : 2, 3 : 7), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5), 2 : 5, 3 : 8, 1 : 3

55 Cnc (b–c), a2 /a1 = 2.085−2.086 [21]

1:3

1:3

HD 108874, a2 /a1 = 2.471−2.655 [22]

1:4

1:4

HD 102272, a2 /a1 = 2.472−2.643 [23]

1:4

1:4

HD 73526, a2 /a1 = 1.366−1.853 [16]

47 UMa, a2 /a1 = 1.578−2.005 [17]

for the value of a2 /a1 that provides resonance overlap, depending on the small parameter μ and the form of the considered resonance zone. 3. THE EXTRASOLAR PLANETARY SYSTEM 47 UMa The available observational information about the planetary system of 47 UMa (HD 95128, HR 4277, HIP 53721, GJ 407) is contradictory. The first planet, 47 UMa b, was discovered in 1996 [25], and its existence was confirmed by many subsequent observations [22, 26–29]. The second planet, 47 UMa c, was discovered in 2002 [26]; however, analysis of subsequent observations did not yield unambiguous results. On the one hand, several solutions describing the orbit of the planet 47 UMa c were constructed in [28]. On the other hand, the existence of 47 UMa c was not confirmed in [22, 29]. Several systems of orbital elements for the planets have been constructed. The first set of elements for the orbits of both planets was determined in [26, 27]. The catalog of orbital elements of extrasolar planets [17] gives a set of parameters that is close to the former. The entire available set of radial-velocity ASTRONOMY REPORTS

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observations for 47 UMa is analyzed in [28], yielding a solution that differs appreciably from previous solutions. Refined solutions for the orbit of 47 UMa b are given in the studies [22, 29], which did not confirm the existence of 47 UMa c. The characteristics of the 47 UMa planetary system are presented in Table 5. Here, T0 is the Julian date of the pericenter passage, m the mass of the planet in Jupiter masses mJ , and m0 the mass of the star in solar masses M . The argument g and longitude β of the pericenter coincide, since we took the longitudes of the ascending nodes of the planetary orbits to be zero in our subsequent computations. The uncertainties in the parameters in units of the last digit are given in parentheses. The absence of firm evidence for the existence of the second planet in the 47 UMa system does not reduce interest in the dynamical properties of this system. The range of possible semi-major orbital axes for 47 UMa c includes regions of resonance-zone overlap and a number of individual resonances. Small variations in the semi-major axes of the planetary orbits can lead to substantial variations of their orbital evolution and the resonance and stochastic properties of their motion.

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Table 4. Overlap of resonances and individual resonances for two-planet systems for μ = 1/100 ω2 /ω1

System Narrow resonance

Broad resonance

(5 : 8, 3 : 5, 4 : 7, 1 : 2), (5 : 8, 3 : 5, 4 : 7, 5 : 9, 1 : 2), (3 : 5, 4 : 7, 5 : 9, 1 : 2), (3 : 5, 4 : 7, 1 : 2), (3 : 5, 1 : 2), 1 : 2, (1 : 2, 5 : 11), (1 : 2, 4 : 9), (1 : 2, 3 : 7), 3 : 7, 2 : 5

(4 : 7, 1 : 2, 4 : 9, 3 : 7, 2 : 5), (4 : 7, 1 : 2, 3 : 7, 2 : 5), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5)

(3 : 5, 1 : 2), 1 : 2, (1 : 2, 5 : 11), (1 : 2, 4 : 9), (1 : 2, 3 : 7), 3 : 7, 2 : 5, 3 : 8, 1 : 3

(4 : 7, 1 : 2, 4 : 9, 3 : 7, 2 : 5), (4 : 7, 1 : 2, 3 : 7, 2 : 5), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5), (1 : 2, 2 : 5, 3 : 8), (1 : 2, 2 : 5, 3 : 8, 1 : 3), (1 : 2, 3 : 8, 1 : 3), (1 : 2, 1 : 3), 1 : 3

HD 181433 (c–d), a2 /a1 = 1.705 [18]

(1 : 2, 4 : 9)

(4 : 7, 5 : 9, 1 : 2, 4 : 9, 3 : 7)

GJ 581 (b–c), a2 /a1 = 1.707 [19]

(1 : 2, 4 : 9)

(4 : 7, 5 : 9, 1 : 2, 4 : 9, 3 : 7)

(1 : 2, 3 : 7), 3 : 7, 2 : 5, 3 : 8, 1 : 3

(4 : 7, 1 : 2, 4 : 9, 3 : 7, 2 : 5), (4 : 7, 1 : 2, 3 : 7, 2 : 5), (1 : 2, 3 : 7, 2 : 5), (1 : 2, 2 : 5), (1 : 2, 2 : 5, 3 : 8), (1 : 2, 2 : 5, 3 : 8, 1 : 3), (1 : 2, 3 : 8, 1 : 3), (1 : 2, 1 : 3), 1 : 3

HD 73526, a2 /a1 = 1.366−1.853 [16]

47 UMa, a2 /a1 = 1.578−2.005 [17]

HD 155358, a2 /a1 = 1.764−2.146 [20]

Table 5. Parameters of the 47 UMa planetary system no.

Planet

1

47 UMa b

e

g, deg

T0 , JD

2.09

0.061(14)

171.8(15.2)

2450356.0(33.6)

2.54

47 UMa c

3.73

0.005(115)

127.0(55.8)

2451363.5(495.3)

0.76

47 UMa b

2.09

0.061(14)

171.8(15.2)

2453622.9(33.6)

2.54

47 UMa c

3.73

0.005(115)

127.0(55.8)

2451363.5(495.3)

0.76

47 UMa b

2.13(12)

0.061(14)

172(15)

2450356(34)

2.63(23)

47 UMa c

3.79(24)

0.00(12)

127(56)

2451360(500)

0.79(13)

47 UMa b

2.11(4)

0.049(14)

111(22)

2450173(65)

2.60(13)

47 UMa c

7.73(58)

0.005

127

2452134(146)

1.34(22)

5

47 UMa b

2.11

0.097(39)

300(20)

2452915(64)

6

47 UMa b

2.100(22)

0.012(23)

147(117)

2449222(347)

2

3

4

a, AU

Detailed studies of the stability of the 47 UMa two-planet system based on set of parameters no. 1 (Table 5) were carried out in [30], where it was shown that the orbital evolution of the system becomes stochastic in the case of motions in the region of the 2 : 5, 3 : 7, 4 : 9, and 5 : 11 resonances. The

m sin i, mJ

m0 , M 

Source

1.03

[26]

1.03

[27]

1.08

[17]

1.07

[28]

2.76(10)

1.03

[29]

2.45(10)

1.063(29)

[22]

properties of the system obtained by varying the initial epoch and mean anomalies of the planets within the ranges of their observational uncertainties are described. The stability of the system is analyzed as a function of the initial eccentricity of the orbit of the outer planet and the inclination of the orbital plane ASTRONOMY REPORTS Vol. 54 No. 6

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RESONANCE PROPERTIES OF EXTRA-SOLAR TWO-PLANET SYSTEMS

of the planets to the plane of the sky (the case of co-planar orbits is considered). The complex, nonlinear dependence of the dynamical and resonance properties of the system on the initial conditions is demonstrated. Let us consider the orbital evolution and resonance properties of the 47 UMa two-planet system for parameter sets nos. 3 and 4 (Table 5) by varying the semi-major axes of the planetary orbits within their uncertainties. This makes it possible to investigate the evolution of the system not only within the zones of individual resonances, but also in the vicinity of regions of resonance overlap. Determining the conditions corresponding to stability of the orbital evolution makes it possible to constrain the range of possible orbital elements in searches for new planets in the 47 UMa system. 4. ORBITAL EVOLUTION OF THE 47 UMa TWO-PLANET SYSTEM We base our analysis of the dynamical evolution of the 47 UMa two-planet system on parameter sets no. 3 and no. 4 (Table 5), which represent two appreciably different possible structures for this system. We used the solution for the planetary three-body problem in averaged elements constructed in [1–5]. It is known that the averaged elements provide a smoother (and in this sense “prettier” [10]) picture of the evolution than do the osculating elements. In the applied solution, the change-of-variable functions make it possible to translate between the mean and osculating elements, which provides substantially more information about the results obtained. We used the masses and orbital elements corresponding to parameter sets no. 3 and no. 4 (Table 5) as the input data. We used the nominal (“mean”) values for all parameters except the semi-major axes a1 , a2 and the eccentricity e2 in set no. 3. The semimajor orbital axes were varied over the entire range of possible values. The eccentricity e2 was taken to be 0.005. Table 6 shows the structure of the resonance zones for the range of possible values of the semi-major orbital axes for the 47 UMa planetary system for μ = 1/300. The first three columns give the intervals of a2 /a1 and the frequency ratios ω2 /ω1 corresponding to resonances. The structure of the broadresonance zones is indicated only for regions in which there are no low-order narrow resonances. The three last columns present for the selected wide-resonance zones and some non-resonance regions the initial values of the semi-major axes a1 and a2 used to model the evolution of the two-planet system and the model number. ASTRONOMY REPORTS

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557

When 1.578  a2 /a1  1.695, the 47 UMa twoplanet system is in a region of narrow resonances, and an overlap of two resonances is observed when 1.685  a2 /a1  1.693. A region of wide resonances on which individual narrow resonances are superposed is present for a2 /a1 = 1.898. When a2 /a1  1.865, the system is either in a region of broadresonance overlap or in a narrow-resonance zone. An analogous analysis of the positions of resonance zones in the 47 UMa system carried out for μ = 1/100 yielded the following results. The region in which narrow-resonance zones can continually be traced corresponds to 1.578  a2 /a1  1.779. Overlapping of two resonance zones is observed for 1.578  a2 /a1  1.600 (3 : 5, 1 : 2), 1.678  a2 /a1  1.694 (1 : 2, 5 : 11), 1.698  a2 /a1  1.726 (1 : 2, 4 : 9), and 1.729  a2 /a1  1.766 (1 : 2, 3 : 7). When 1.779  a2 /a1  2.003, the system is either in a zone of broad-resonance overlap or in a narrowresonance zone. We expect that the orbital evolution of the 47 UMa system will acquire a chaotic character when the planets move in regions of resonance overlap and in resonance zones. We carried out our analysis of the orbital evolution of the 47 UMa two-planet system with the aim of searching for signs of chaotization of the planets’ motions. The averaged equations of motion for the twoplanetary problem obtained with accuracy to second order in the small parameter of the problem [1–5] were numerically integrated using the method of Everhart [31], which has 15th order and implements automated step choice. The integration interval was 106 yrs. The small parameter was taken to be μ = 1/300. When specifying the initial data based on parameter set no. 3 (Table 5), we selected the following masses for the star and planets for μ = 1/300 (in this case, the orbital inclination is i = 44.2◦ ): m0 = 1.08 M , m1 = 3.77 mJ , m2 = 1.14 mJ . We have for set no. 4 m0 = 1.07 M , m1 = 3.74 mJ , m2 = 1.93 mJ . The analytical solution of [1–5] is not applicable in regions of narrow resonance, but can be used to investigate the orbital evolution in regions of wide resonance and in non-resonance zones. We described the stochastic properties of the planetary motions based on an analysis of the integrated autocorrelation function (IACF) A [32]. The IACF A is defined as the mean of the squares of a sequence of product-moment autocorrelation function:

N −k ¯i )(ri+k − r¯i+k ) i=0 (ri − r Ak =  1/2 ,

N −k

N −k 2 2 (r − r ¯ ) (r − r ¯ ) i i i+k i+k i=0 i=0

558

KUZNETSOV

Table 6. Overlap of resonances and individual resonances for the 47 UMa two-planet system for μ = 1/300 Initial values

ω2 /ω1

a2 /a1 Narrow resonance

Broad resonance

No. of

a1 , AU

a2 , AU

models

(3 : 5, 1 : 2, 5 : 11, 4 : 9)

2.13

3.622

1

(1 : 2, 4 : 9, 3 : 7)

2.13

3.692

2

1.772−1.782

(1 : 2, 3 : 7)

2.13

3.785

3

1.782−1.798

(1 : 2, 3 : 7, 2 : 5)

2.13

3.813

4

1.798−1.815

(1 : 2, 2 : 5)

2.13

3.848

5

2:5

2.13

4.008

6

2.01

3.821

7

2.01

3.839

8

2.01

3.963

9

2.01

4.030

10

2.11

7.364

11

2.11

7.711

12

Parameter set no. 3 (Table 5) 1.578−1.685

1:2

1.685−1.693

(1 : 2, 5 : 11)

1.693−1.695

5 : 11

1.695−1.707 1.707−1.724

4:9

1.724−1.743 1.743−1.772

1.815−1.865

3:7

2:5

1.865−1.898

No resonances with ν  16

1.898−1.904

3:8

1.904−1.916 1.916−1.927

3:8

1.927−1.938

3:8 No resonances with ν  16

1.938−2.004

1:3

2.005−2.005 Parameter set no. 4 (Table 5)

No resonances with ν  16

3.326−3.655

1:7

3.654−3.655 3.655−3.656

1:7

3.656−3.658

1:7 No resonances with ν  16

3.658−4.015

where r¯s = (s + 1)−1 si=0 ri is the mean value of an s-element subset of a uniform time series (t0 is the initial time and h the constant time step), r = {ri : ri = r(f (t0 + ih)), 0  i  N }. We computed A using a representation in the form of a sum,

 K  1 (Jk − Jk−1 )A2Jk . 1+ A(JK ) = JK k=1

Here, Jk = jK − jK−k specifies the shift of the argument of the IACF in a sequence of exponentially

distributed points, j0 = 0,

j1 = 1, . . . , jk = jk−1 + 2[(k−1)/B] , k = 0, 1, . . . , K, jK  N,

where [z] is the integer part of z and B is the baseline number. The IACF A asymptotically approaches unity for constant time series. For a uniform time series representing a periodic sine function, A = 0.5. For other periodic and quasi-periodic time series, A approaches a finite value close to 0.5. For chaotic trajectories, A asymptotically approaches zero. ASTRONOMY REPORTS Vol. 54 No. 6

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RESONANCE PROPERTIES OF EXTRA-SOLAR TWO-PLANET SYSTEMS

559

Table 7. Values of the IACF A and ranges of variation of the semi-major axes and eccentricities of the 47 UMa planetary system for μ = 1/300 A

No. of models

Mean elements

Osculating elements Osculating elements

a1 , а.u.

e1

a2 , а.u.

e2

Parameter set no. 3 (Table 5), m0 = 1.08 M , m1 = 3.77 mJ , m2 = 1.14 mJ 1

0.493

0.289

2.128−2.136

0.033−0.063

3.620−3.624

0.004−0.021

2

0.492

0.360

2.128−2.135

0.035−0.063

3.689−3.693

0.004−0.022

3

0.491

0.412

2.129−2.134

0.036−0.062

3.784−3.786

0.004−0.022

4

0.488

0.418

2.129−2.134

0.037−0.062

3.811−3.814

0.004−0.022

5

0.492

0.432

2.129−2.133

0.037−0.062

3.846−3.849

0.004−0.022

6

0.489

0.456

2.129−2.132

0.039−0.062

4.007−4.009

0.004−0.021

7

0.486

0.459

2.009−2.012

0.040−0.062

3.820−3.822

0.004−0.021

8

0.490

0.464

2.009−2.012

0.040−0.062

3.838−3.839

0.004−0.021

9

0.484

0.466

2.009−2.012

0.040−0.062

3.961−3.963

0.004−0.021

10

0.486

0.468

2.009−2.011

0.040−0.063

4.029−4.030

0.003−0.021

Parameter set no. 4 (Table 5), m0 = 1.07 M , m1 = 3.74 mJ , m2 = 1.93 mJ 11

0.496

0.494

2.110

0.051−0.061

7.363−7.364

0.004−0.019

12

0.474

0.471

2.110

0.052−0.061

7.711−7.712

0.004−0.018

The baseline number for which the IACF could be reliably estimated with an integration interval of 106 yrs was determined to be B = 400, based on a series of numerical experiments. It was established that a more informative definition of A is based on a time series of the orbital eccentricity for the inner planet e1 . The eccentricity e1 experiences the maximum (relative) perturbations among the positional elements for the planetary orbits in the 47 UMa system. Table 7 presents estimates of the IACF A and the ranges of variation of a and e for the 47 UMa system for μ = 1/300. The model numbers in Tables 6 and 7 coincide. We computed values of A for the mean and osculating elements (Table 7). The solution for the averaged elements does not show any signs of chaotic motion within wide-resonance zones (models 6, 8, 10, 12) or in regions of broad-resonance overlap (models 1–5). In all the models considered, the solution for the averaged elements is quasi-periodic. The solution for the osculating elements obtained using the change-of-variable functions reveals stochastic properties in regions of broad-resonance overlap. The values A = 0.289−0.432 for models 1– 5 are a consequence of chaotization of the motion in regions of broad-resonance overlap. ASTRONOMY REPORTS

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In non-resonance regions (models 7, 9, 11) and in wide-resonance zones (models 6, 8, 10, 12), the motion is quasi-periodic, and no manifestation of chaotic properties of the motion is detected over the considered interval of 106 yrs. As we can see from Table 7, the values of the IACF A computed using the osculating elements approach 0.5 with growth in a2 /a1 , indicating that the motion is quasi-periodic. When analyzing the stochastic properties of the motion, we assumed that the manifestation of chaotic properties becomes important when A < 0.45. The positional elements of the planetary orbits were varied within small limits. The inclination of the planetary orbits was initially taken to be 0.001◦ , and oscillates with an amplitude of several milliarcseconds. The range of variation of the osculating semimajor axes and eccentricities are given in Table 7. The variation of the semi-major axes did not exceed 0.01 AU. The eccentricities oscillated within 0.03. The amplitude of the oscillations of the orbital elements of the inner, more massive, planet is larger, since its orbital eccentricity appreciably exceeds that for the outer planet. Signs of chaotic properties of the motion are observed in regions of resonance-zone overlap. Chaotization of the trajectories grows with decrease in the

560

KUZNETSOV

relative value of the semi-major axis of the second planet a2 /a1 , which corresponds to a decrease in the distance between the planets’ orbits. Our conclusions on the resonance properties of extrasolar planetary systems are in agreement with the results of other studies. It was shown in [33] based on the results of numerical simulations of the motions of test particles that the 47 UMa could be in a sharp resonance, a weak resonance, or a nonresonance state. A number of studies have investigated the evolution of the 47 UMa planetary system in the case of resonances of the mean motions: 5 : 11, 4 : 9 [30], 3 : 7 [27, 30], 3 : 8 [34], 2 : 5 [35, 36]. It was concluded in [37] that there is no evidence for loworder resonances of the mean motions in the 47 UMa system, but this system is located in a secular resonance associated with motion of the lines of apsides. The results of our analysis of the resonance properties of the 47 UMa two-planet system in the vicinity of weak resonances leads us to conclude that, for small values of the ratio a2 /a1 , the presence of the system in a region of resonance overlap could lead to chaotization of the orbital evolution. 5. CONCLUSION The method for describing the resonance properties of planetary systems we have presented is simple and universal. Our estimates of the resonance values of the semi-major axes and widths of resonance zones in relative units for characteristic values of the small parameter of the problem make it easy to classify and describe the resonance properties of planetary systems. All our results were obtained in the framework of the averaged two-planetary problem [1–5]. The analytical solution we have used is not applicable in the case of resonance, but, with the use of changeof-variables functions, we were able to estimate the widths of resonance zones and investigate the orbital evolution near the boundaries of narrow-resonance zones. The conditions for the onset of resonances and resonance overlap that we have found will be useful for studies of new extrasolar planetary systems. ACKNOWLEDGMENTS The author thanks K.V. Kholshevnikov for useful discussions and valuable comments. This work was partially supported by the Program of State Support for Leading Scientific Schools of the Russian Federation (NSh-1323.2008.2) and the Analytical Departmental Targeted Program of the Federal Agency on Education of the Ministry of Education and Science of the Russian Federation “The Development of the Scientific Potential of Higher Education (2009– 2010)” (projects 2.2.3.1/1842, 2.1.1/504).

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Translated by D. Gabuzda