Analytical Methods

The basis for analytical methods lies in consideration of the "disturbing function", the difference between the gravitational potential of a planet due to a star, and that due to a star and one or more additional planets. We will focus on the Fourier series expansion of the disturbing function. Analytic methods can describe two phenomena often seen in planetary systems: resonant and secular evolution. Resonant and secular theory assume certain terms in the series will average to zero, and can therefore be ignored when modeling orbits. Secular theory ignores all terms that depend on the mean motion, n (the orbital frequency if the planet were on a circular orbit), and, in many cases, all terms that are of order 3 or higher, i.e. e3 « 0. Resonant theory adds terms that do depend on mean motions, but only those related to the resonance in question. For a detailed description of the disturbing function, secular theory and resonance theory, consult Murray & Dermott (1999).

Secular Theory

To visualize secular theory, imagine the distribution of the planets' masses over a long timescale. The distribution would be that of a ring of matter. Secular theory predicts how the shapes of these rings will change with time, and hence how the orbit changes with time. In this second order approximation, motion out of the plane is decoupled from motion in the plane.

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Fig. 7.2. The secular oscillations of the planets in orbit about HIP 14810 (Wright et al., 2007). Note that the oscillations of both e's (inner is black, outer red) and Aw have the same period. Also note that when the e's are at their extrema, Aw is at its equilibrium value, 180° in this case.

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Fig. 7.2. The secular oscillations of the planets in orbit about HIP 14810 (Wright et al., 2007). Note that the oscillations of both e's (inner is black, outer red) and Aw have the same period. Also note that when the e's are at their extrema, Aw is at its equilibrium value, 180° in this case.

In nearly all planetary system cases, secular theory predicts the e's and Aw's (the difference between two w's) oscillate. The movement of w is often called precession. An example of a secular interaction is shown in Fig. 7.2. Secular theory assumes a is constant, and therefore conservation of angular momentum requires that as one planet's eccentricity drops, the other rises (orbital angular momentum is proportional to e). If Aw oscillates about 0, the system is experiencing aligned libration. If Aw librates about n, then the system is undergoing anti-aligned li-bration. If Aw oscillates through 2n then the "apsides" (the points of closest and furthest approach to the origin of an orbit) are circulating. The type of oscillation depends on initial conditions. Exoplanet examples of these types of behavior are shown in Fig. 7.3. The motion of Aw is analogous to that of a swinging pendulum. When the pendulum swings back and forth, the oscillation is libration. If the pendulum swings all the way around, the oscillation is circulation. Note there is a clear boundary between these types of motion: the swing that brings the pendulum up to a perfectly vertical position. This boundary between qualitatively different types of oscillation is known as a "separatrix".

Recently it has been noted that many systems lie near an "apsidal separatrix" (Ford et al., 2005; Barnes & Greenberg, 2006a,c) (the point on the orbit that is closest to the origin is known as the "apse"). The simplest apsidal separatrix is the boundary between libration (either aligned or anti-aligned) and circulation. For

Fig. 7.3. Examples of apsidal oscillations in exoplanetary systems The y-axis is the difference between the longitudes of periastron, Aw. Top: HD 12661 undergoes apsidal libration in an anti-aligned mode. When exactly anti-aligned (Aw = n) the direction of periastron of one planet is the same as apoastron of the other. Middle: HD 37124 c-d undergoes apsidal libration in an aligned mode. When aligned (Aw = 0) the ellipses are oriented such that the two periastra point in the same direction. Bottom: The planets in HD 168443 undergo apsidal circulation since Aw rotates through 360°.

Fig. 7.3. Examples of apsidal oscillations in exoplanetary systems The y-axis is the difference between the longitudes of periastron, Aw. Top: HD 12661 undergoes apsidal libration in an anti-aligned mode. When exactly anti-aligned (Aw = n) the direction of periastron of one planet is the same as apoastron of the other. Middle: HD 37124 c-d undergoes apsidal libration in an aligned mode. When aligned (Aw = 0) the ellipses are oriented such that the two periastra point in the same direction. Bottom: The planets in HD 168443 undergo apsidal circulation since Aw rotates through 360°.

systems of just two planets, the apsidal separatrix can only separate circulation and libration. This type of separatrix is a "libration-circulation separatrix". An example of the libration-circulation separatrix is shown in the left panels of Fig. 7.4.

In systems of more than two planets, things get more complicated. In addition to the libration-circulation separatrix, the system may interact with different numbers of rotations of Aw through 360° during one eccentricity oscillation. The boundary between interactions with different numbers of circulations in one eccentricity cycle is called a "circulation-mode separatrix", and an example is shown in the right panels of Fig. 7.4.

For an interaction to lie near a separatrix, the amplitude of eccentricity oscillations are generally two orders of magnitude or more. Since 0 < e < 1 for bound planets, this means that at least one planet in near-separatrix interactions (both libration-circulation and circulation-mode) periodically is on a nearly circular orbit. The proximity to the separatrix can be parameterized by e, which is approximately equal to the minimum e divided by the average e over a secular cycle (Barnes & Greenberg, 2006c). When e = 0 the pair is on an apsidal separatrix, and one eccentricity periodically reaches zero.

Fig. 7.4. Examples of near-separatrix motion in planetary systems. Left Panels: A libration-circulation separatrix. Two possible evolutions of v And c and d (the middle and outer planet of the system) assuming different estimates of the current orbits. The black points are the system from (Butler et al., 2006), the red from (Ford et al., 2005). Although the best-fit orbits in these two cases are very similar, they result in qualitatively different types of evolution of Aw. The older data predict aligned libration, whereas the updated data predict circulation (top). Note that the evolution of ec is similar in both cases, and periodically reach near-zero values (bottom). Right Panels: A circulation-mode separatrix. HD 69830 c and d evolve near the circulation-mode separatrix. The black data are from (Lovis et al., 2006), and the red data are for a fictitious system in which the inner planet's, b's, eccentricity was changed from 0.1 to 0.15. In the first 104 years, the actual Aw undergoes 1 complete rotation through 360°, but in the fictitious system, Aw undergoes 2 complete circulations (top). We again see that the middle planet's eccentricity periodically drops to near-zero values in both cases (bottom).

Fig. 7.4. Examples of near-separatrix motion in planetary systems. Left Panels: A libration-circulation separatrix. Two possible evolutions of v And c and d (the middle and outer planet of the system) assuming different estimates of the current orbits. The black points are the system from (Butler et al., 2006), the red from (Ford et al., 2005). Although the best-fit orbits in these two cases are very similar, they result in qualitatively different types of evolution of Aw. The older data predict aligned libration, whereas the updated data predict circulation (top). Note that the evolution of ec is similar in both cases, and periodically reach near-zero values (bottom). Right Panels: A circulation-mode separatrix. HD 69830 c and d evolve near the circulation-mode separatrix. The black data are from (Lovis et al., 2006), and the red data are for a fictitious system in which the inner planet's, b's, eccentricity was changed from 0.1 to 0.15. In the first 104 years, the actual Aw undergoes 1 complete rotation through 360°, but in the fictitious system, Aw undergoes 2 complete circulations (top). We again see that the middle planet's eccentricity periodically drops to near-zero values in both cases (bottom).

Although secular theory provides a method for identifying components of the dynamics of planetary systems, it must be used with caution on extra-solar planetary systems. Their proximity to the apsidal separatrix, can obscure the true motion of the system (Barnes & Greenberg, 2006a). The inclusion of additional terms may be more useful in these cases (Lee & Peale, 2003; Michtchenko & Malhotra, 2004; Libert & Henrard, 2005; Veras & Armitage, 2007), however W-body methods may be the most practical method for determining the secular behavior of exoplanets.

Finally, it is necessary to digress and discuss the term "secular resonance". Currently two definitions exist in the literature for this term, which often leads to confusion. One definition states that Aw is librating. The other definition is that the ratio of two (or more) precessional frequencies is close to a ratio of two small integers. The latter definition is preferable as it is closer to the true meaning of a resonance, a

Fig. 7.5. Example of a secular resonance in the v And system without general relativity. General relativity suppresses the resonance by inducing a high frequency apsidal precession in v And b. Top: The eccentricity evolution of v And b from just planet c. Middle: b's evolution from just planet d. Bottom: b's evolution from both c and d. The amplitude of eccentricity oscillation is nearly an order of magnitude larger than from either perturber alone, a direct result of the secular resonance.

Fig. 7.5. Example of a secular resonance in the v And system without general relativity. General relativity suppresses the resonance by inducing a high frequency apsidal precession in v And b. Top: The eccentricity evolution of v And b from just planet c. Middle: b's evolution from just planet d. Bottom: b's evolution from both c and d. The amplitude of eccentricity oscillation is nearly an order of magnitude larger than from either perturber alone, a direct result of the secular resonance.

commensurability of frequencies, so we will use this definition. The former should be referred to as apsidal libration. The difference is illustrated in Figs. 7.3 and 7.5 with exoplanet examples. In Fig. 7.3 the librational behavior of HD 12661 (anti-aligned) and HD 37124 c-d (aligned) are shown (top and middle panels). Fig. 7.5 shows a secular resonance in the u And system, with orbital elements from (Ford et al., 2005). The top two panels show the eccentricity evolution of u And b due to only planet c (top) and only d (middle), while the bottom panel is b's evolution affected by both planets. The secular resonance pushes b's eccentricity to values much larger than that of either planet alone. Note that for this example we have neglected the effects of general relativity, which overwhelms the secular resonance in this system. For more on libration and secular resonances consult (Barnes & Greenberg, 2006a), who also show that a secular resonance is impossible in a two-planet system.

Resonant Interactions

Two bodies may be in a mean motion resonance (MMR) when the ratio of their periods is close to a ratio of small integers. When this occurs, the planets periodically line up at the same points in their orbits, which introduces a repetitive force that cannot be assumed to average to zero over long timescales. Resonant effects can be comparable to secular effects, depending on masses and orbits. Eight systems have two planets that are in a resonance.

Resonances can stabilize a system by preventing close approaches that might eject a planet. Stable resonances tend to prevent conjunction from occurring near the minimum distance between two orbits. Consider the 3:2 case of Neptune and Pluto: Although the orbits cross, the resonance is such that conjunction can never occur at this danger zone.

Resonances are often described in terms of the planets' mean longitudes A. The mean longitude is similar to the true longitude, but it measures the position of a planet assuming its angular velocity is constant (only true for a circular orbit). When resonances occur, the mean longitudes and angles of periastron evolve in certain, regular ways. Resonances also force circular orbits to become non-circular. If conjunction occurs at periastron of the inner planet, then at the following conjunction the apsides will not be perfectly aligned because the change in e will change the apsidal frequency. This non-alignment will introduce a net torque on the orbit that tends to pull the orbits back toward alignment. In this way, resonances maintain themselves, but the alignment will oscillate about an equilibrium position.

From the qualitative description above, it is clear that a resonance occurs if certain combinations of angles librate about fixed values. If we denote the outer planet with a prime, then the resonant dynamics are important if the "resonant argument",

varies slowly relative to the orbital motion. Note the integers jk obey

in all terms of the disturbing function. For any pair of planets, integers can be identified that solve Eqs. (7.1 - 7.2), but the resonance will only be effective if its order is low enough. The order of a resonance is defined as the difference between |3i| and 1321. If the order is — 4 or less and the larger number (31) is small (<5) then the resonance is at least as important as secular effects in an exoplanet systems. High order resonances are present and important in the Solar System, including resonances between three planets (Murray & Holman, 1999), but their role is unknown in exoplanets because the observational errors are too large for the effects of these interactions to be unambiguously determined.

In exoplanet systems, some resonances show simple behavior: 1 or more resonant arguments are always librating, see top panel of Fig. 7.6. But some peculiar examples of resonances have been uncovered. The planets around HD 108874, for example, evolve with one resonant argument always librating, but the other arguments alternate between libration and circulation (Barnes & Greenberg, 2006c), as shown in the bottom panel of Fig. 7.6. Note that this system has an e value of 0.2, suggesting it lies far from the apsidal separatrix. However, the resonance alters the apsidal motion, and therefore e is not always a valid description of near-separatrix motion in the case of mean motion resonances.

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Fig. 7.6. Examples of librating resonance arguments in exoplanet systems. The y-axis, 0, represents the resonant arguments. Top: The planets GJ 876 c and b are in a 2:1 resonance, and both possible combinations of resonant angles librate. The short period means the oscillation should be observable. Bottom: The planets in HD 108874 are in a 4:1 resonance and 4 possible resonant angles exist. In this system one angle, 0 = 4A' — A — w — 2w', (black curve) is always librating, but 0 = 4A' — A — 3w' (red curve) alternates between libration and circulation on 105 year timescales.

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Fig. 7.6. Examples of librating resonance arguments in exoplanet systems. The y-axis, 0, represents the resonant arguments. Top: The planets GJ 876 c and b are in a 2:1 resonance, and both possible combinations of resonant angles librate. The short period means the oscillation should be observable. Bottom: The planets in HD 108874 are in a 4:1 resonance and 4 possible resonant angles exist. In this system one angle, 0 = 4A' — A — w — 2w', (black curve) is always librating, but 0 = 4A' — A — 3w' (red curve) alternates between libration and circulation on 105 year timescales.

If $ librates for multiple combinations of j's, then the system is in an "apsidal corotation resonance" (Ferraz-Mello et al., 2005; Michtchenko, Beauge & Ferraz-Mello, 2006), and Aw will also librate. For more on the physics of resonances, consult (Peale, 1976; Greenberg, 1977; Beauge et al., 2003), or (Murray & Dermott, 1999, Chap. 8).

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