Proximity to Dynamical Instability

The most critical aspect of dynamical systems is their stability, and, therefore, a substantial amount of research has investigated the dynamical stability of planetary n i i i |i i i i | i i i i | ii i i | i i i i |

.nn.lhnn m

.nn.lhnn m

Fig. 7.7. The distribution of proximities to an apsidal separatrix (defined as e = 0). For systems with e < 0.01 eccentricity oscillations of two orders of magnitude are present in the system. Nearly half of all adjacent pairs are on orbits such that they are near a separatrix. If the distribution of eccentricities were uniform and random, only a few per cent of systems should interact such that e < 0.01 (Barnes & Greenberg, 2006a).

systems, e.g. (Gozdziewski & Maciejewski, 2001; Gozdziewski et al., 2001; Kiseleva-Eggleton et al., 2002; Barnes & Quinn, 2004; Barnes & Greenberg, 2006b, 2007b). Many known systems appear to lie near Lagrange instability (the type of stability in which at least one planet is ejected from the system within several million years, see Sect. 7.2.3).

Investigations into stability have found that Lagrange unstable regions exist within the 1 standard deviation error ellipses for systems with MMRs, such as 55 Cnc and HD 82943, as well as those without, such as HD 12661 and 47 UMa as shown in Fig. 7.8. This figure shows the results of numerical simulations within observationally permitted parameter space for these four systems (see (Barnes & Quinn, 2004) for more details). The parameter space was sampled as a Gaussian with a peak at the best fit values (at the time of the simulations) with a standard deviation equal to the published error. Therefore the centers of each panel are more highly sampled than the edges. The shading indicates the fraction of initial conditions, in a certain range of orbital element space, that give Lagrange stable behavior (no ejections or exchanges) after — 106 years: White regions contained only stable configurations, black only unstable, and darkening shades of gray correspond to decreasing fractions of simulations which predict stability. In this figure Pc/Pb is the ratio of the orbital periods.

The contours represent values of p, the proximity to the Hill stability boundary (Barnes & Greenberg, 2006b, 2007b). For non-resonant systems, the limit of Lagrange stability corresponds to values of p slightly greater than 1. But in the presence of a resonance, Lagrange stable orbits can be found at p values as small as 0.75. Lagrange stability boundaries are qualitatively different for resonant and non-resonant pairs. The former have a "stability peninsula" located at the resonance, while the latter have a large contiguous "stability plateau".

The apparent correspondence between Lagrange stability and values of p is evidence that the expression for p (Eq. [1] in (Barnes & Greenberg, 2006b)), is a valid representation of the limits of dynamical stability in systems of two planets. In Fig. 7.9, we plot the current distribution of p values for two planet systems. Most systems lie near p =1, and resonant systems tend to have p < 1 (the resonance protects the system from instability) (Barnes & Greenberg, 2007b).

Figs. 7.8 - 7.9 suggest that many planetary systems are dynamically full (no additional companions can survive in between the observed planets), and leads to the Packed Planetary Systems (PPS) hypothesis (Barnes & Quinn, 2004; Barnes & Raymond, 2004; Raymond & Barnes, 2005; Raymond, Barnes & Kaib, 2006): All pairs of planets lie close to dynamical instability. The observation that so many systems (using minimum masses) lie near Lagrange instability, despite incompleteness issues (see Sect. 7.1.2), has led to investigations to identify regions of Lagrange stability between the few pairs that are more separated (Menou & Tabachnik, 2003; Barnes & Raymond, 2004; Dvorak et al., 2003; Funk et al., 2004; Raymond & Barnes, 2005; Raymond, Barnes & Kaib, 2006; Rivera & Haghighipour, 2007). This research has shown that the gaps between HD 74156 b-c, HD 38529 b-c, and 55 Cnc c-d are large enough to support additional Saturn-mass planets. The detection of planets

55 Cnc HO 82943

55 Cnc HO 82943

Fig. 7.8. Lagrange stability boundary in relation to the Hill stability boundary for some exoplanetary systems (see text for a discussion of the simulations summarized in these figures). In these plots white regions represent bins in which all configurations were stable, black bins contained no stable configurations, darker shades of gray correspond to regions in which the fraction of stable simulations were smaller (see Barnes & Quinn 2004 for more details). The curves represent contour lines of ¡3. Contour lines follow the shape of the Lagrange stability boundary, except in resonance, where there the Lagrange stability region is larger. Top left: Stability of the 55 Cnc system depends on the parameters of the 3:1 resonant pair; the eccentricity of the larger planet, and the ratio of the periods. When Pc/Pe = 3, the Lagrange stability boundary is located at 3 ~ 1-03. Top right: HD 82943's stability depends on the eccentricity of the larger planet and the ratio of the planets' periods. The Lagrange stable region shown exists wholly in a region that would be considered unstable from Hill stability theory. Bottom left: The stability of 47 UMa depends on the eccentricities of the two planets. The Lagrange stability boundary corresponds to 3 ~ 1-015. Bottom right: The stability of the HD 12661 system depends on the eccentricities of the two outer planets. The Lagrange stability boundary lies near the 3 =1-1 contour.

Fig. 7.9. The distribution of proximities to the Hill stability boundary, ¡3 = 1. All systems with 3 < 1 are in mean motion resonance, implying that the resonance is protecting the system from instability. Most non-resonant systems have configuration with 3 < 2, suggesting these planetary systems are packed (Barnes & Greenberg, 2007b); companions between those that are known would disrupt the system.

Fig. 7.9. The distribution of proximities to the Hill stability boundary, ¡3 = 1. All systems with 3 < 1 are in mean motion resonance, implying that the resonance is protecting the system from instability. Most non-resonant systems have configuration with 3 < 2, suggesting these planetary systems are packed (Barnes & Greenberg, 2007b); companions between those that are known would disrupt the system.

in these locations would support the packed nature of planetary systems, and would be an exciting achievement for the nascent field of exoplanet dynamics.

In order to verify or disprove the PPS hypothesis, a quantitative definition of "close" is required. As stated in Sect. 2.3 there exists no quantitative definition of Lagrange stability for any number of planets, and no definition for Hill stability for systems with more than 2 planets. With limited data available, it appears that when 3 < 1.5 - 2 a system is packed (Barnes & Greenberg, 2007b). Future work should reveal how robust this limit is, as well as identify packing limit for systems of more than 2 planets.

The PPS hypothesis received a major boost with the discovery of a Saturn-mass planet in the HD 74156 system between planets b and c (Bean et al., 2007). The revised system is unstable, but the new planet lies close (less than one standard deviation) to the most stable orbit identified by numerical simulations (Raymond & Barnes, 2005). This detection offers strong support for the PPS hypothesis, as it verifies a prediction. At this point, one such detection does not confirm the PPS hypothesis, but it is worth noting that the first 6 multiple planet systems detected now show evidence of dynamical packing.

Table 7.1. Summary of Dynamical Properties of Multiple Planet Systems

System

Pair

MMR

AM

e

ß

Class

47 UMa

b-c

-

Ca

0

1.025

S

55 Cnc

e-b

-

C

0.067

-

T

b-c

3:1

C

0.11

-

R

c-d

-

C

0.158

-

S

GJ 876

d-c

-

Ca

0

-

T

c-b

2:1

A

0.34

-

R

Gl 581

b-c

-

C

0.15

-

T

c-d

-

C

0.20

-

T

HD 12661

b-c

-

C

0.003

1.199

S

HD 37124

b-c

-

C

0.009

-

S

c-d

-

A

0.096

-

S

HD 38529

b-c

-

C

0.44

2.070

S

HD 69830

b-c

-

C

0.095

-

T

c-d

-

C

0.04

-

S

HD 74156b

b-c

-

C

0.36

1.542

S

HD 73526

b-c

2:1

AA

0.006

0.982

R

HD 82943

b-c

2:1

C

0.004

0.946

R

HD 128311

b-c

2:1

C

0.091

0.968

R

HD 108874

b-c

4:1

C/AAc

0.2

1.107

R

HD 155358

b-c

-

AA

0.21

1.043

S

HD 168443

b-c

-

C

0.22

1.939

S

HD 169830

b-c

-

C

0.33

1.280

S

HD 190360

c-b

-

C

0.38

1.701

T

HD 202206

b-c

5:1

C

0.096

0.883

R

HD 217107

b-c

-

C

0.46

7.191

T

HIP 14810

b-c

-

AA

0.05

1.202

T

SS

J-S

-

C

0.19

-

S

S-U

-

C

0.006

-

S

U-N

-

C

0.004

-

S

ß Ara

c-d

-

C

0.002

-

T

d-b

2:1

C

0.003

-

R

b-e

-

C

0.13

-

S

v And

b-c

-

C

1.8 x 10-4

-

T

c-d

-

C

2.8 x 10~4

-

S

a The current eccentricity of one planet is 0, placing the pair on an apsidal separatrix. b These values do not incorporate the new planet (Bean et al., 2007) c This pair alternates between circulation and anti-aligned libration.

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