Project Topics
Get Complete Project Material File(s) Now! »
Application of the adiabatic theory to resonant capture
The simplest model of a resonance with resonant angle and canonically conjugated action makes use of an integrable Hamiltonian. The \ rst model for resonance » is the pendulum, which as we saw in Subsect. 2.1.6.3 is a local approximation for the dynamics. In the problems of Celestial Mechanics, a global integrable approximation for resonance usually takes the form of an Andoyer Hamiltonian, cfr. Equation (2.56), which I rewrite here for readability:
where , , c are real numbers, with 6= 0 and c 6= 0. The rst two -independent addends are usually the larger contribution (which comes e.g. from the Keplerian Hamiltonian in a Celestial Mechanics resonance problem); it is expanded in power series in as is usually (at least initially) small. Notice that the last addend satis es the d’Alembert rules in the canonical pair ( ; ) (this means that the singularity at the origin is only apparent and can be removed passing in canonical cartesian coordinates); this last term is usually smaller than the rst (it comes e.g. from the perturbing Hamiltonian in Celestial Mechanics), which we can express by writing c = C where j j is a small parameter (e.g. the planet-to-star mass ratio).
As we saw in Subsect. 2.1.6.3, the system (2.125) is integrable, and we can draw phase diagrams such as the ones shown in Figures 2.3 and 2.4. There, we also recalled that resonances are not necessarily synonymous with libration and they are strictly linked to the presence of the separatrices, which enclose the resonant region(s) of the phase space. The emergence of the separatrices depends on the speci c values of the parameters , and c, and the fact that these coe cients are constants is true only in the conservative case. Whenever there are dissipative processes acting on the system (for example in the case of mean motion resonances it might be migration in protoplanetary disc, migration due to tides, …) the coe cients will change, therefore the phase diagrams will change and the resulting dynamics will re ect these e ects.
The general idea is the following. We consider a system which is initially close to an equilibrium point at (or near) the origin, so that 0, and whose dynamics is governed by (2.125) for some k. Over the evolution of , for k = 1 the dynamics around the equilibrium point is purely adiabatic, while for k > 1 the adiabatic principle breaks. As in Subsect. 2.1.6.3, we consider the cases k = 1 and k = 2 only, since they are enough to show when the adiabatic principle can be used and when it fails.
Consider the case k = 1 rst, so the di erent phase diagrams for HAnd;1 are the ones shown in Figure 2.3 (also reproduced in Figures 2.8 and 2.9 for readability, see below). Although the resonant region does not coincide with the libration region, at the resonant equilibrium point (the stable equilibrium point inside the resonant region).
Figure 2.8: Depiction over the evolution of of the dynamics described in the text, in the case k = 1 and with _ < 0, so resonant capture is ensured by the adiabatic principle. The phase portraits are the same as in Figure 2.3, and the reference orbit is now marked with a thick black line. The system is initially close to the origin and with small amplitude of libration around the centre (panel (a)); it follows adiabatically the equilibrium point until the separatrix appears (panel (b)), so the orbit naturally ends up inside the resonant region (panel (c)). This situation corresponds to the black arrows in panel (b) of Figure 2.7.
the condition = 0 must hold, so libration is a necessary condition for resonance at the equilibrium point. Since initially is vanishingly small and the last term in (2.125) is small compared to the rst two, we calculate which shows that measures the initial distance from the resonance when the orbit is at (or close to) the origin. Therefore, for this orbit to cross and be captured in the resonance, must cross the value 0. Let us assume to x ideas that changes from positive to negative, and write (t) as a function of some \time » t (Figure 2.7 panel (a)). Initially 0, so and therefore is initially positive and approaching 0. Then at one moment in \time », ’ (t ) = 0: for there to be a capture, we need to keep 0 even for t > t when actually (t) < 0. Looking at equation (2.126), ’ + , the only way for this to happen is if > 0, so that the now negative term is cancelled by the positive term ; in order for this to work though, must keep increasing as keeps becoming more negative, and capture was successful. If the change in is much slower compared to the timescale of the evolution of the variables ( ; ), this process is perfectly explained by adiabatic theory. If goes from positive to negative and > 0, we are in the same situation studied in Subsect. 2.1.6.3 and we read the change in the phase diagrams in Figure 2.3 from left to right; the same sequence is reproduced in Figure 2.8 for readability, where we also mark the reference orbit as a thick curve. Assuming that we have a small amplitude of libration around the stable equilibrium point in panel (a), as decreases slowly we adiabatically follow the equilibrium as it shifts away from the origin, maintaining the same area enclosed by the orbit. Then, a bifurcation generates the separatrices (panels (b) and (c)) which enclose the centre equilibrium point followed by the orbit, which is the true condition for resonance. Therefore, for k = 1 when decreases slowly enough and is positive, at small amplitude of libration around the equilibrium point resonant capture is guaranteed by adiabatic theory. Still in the case of decreasing , we saw at the beginning of Subsect. 2.1.6.3 that having instead < 0 is equivalent to changing the sign of the time variable, and now the evolution is obtained reading Figure 2.3 right to left; this sequence is now reproduced in Figure 2.9 inverting the order of the panels (so Figure 2.9 can be read left to right like Figure 2.8), where again the reference orbit is marked as a thick curve. Initially we are still at small , so now the evolution starts from the stable centre near the origin outside of the banana-shaped resonant region (panel (a) of Figure 2.9). As changes, the orbit is invested by the already present separatrix as it closes in towards the origin, and the adiabatic principle does not hold anymore (cfr. the last subsection). After the orbit crosses the separatrix, it typically goes into the region of growing area, that is into the the outer circulation zone in this case (panel (b)), and the orbit now encloses a bigger area than it did before, equal to the area enclosed by the separatrix at the moment of the crossing, and is then maintained adiabatically (Figure 2.9, panel (c)).
Consider now k = 2 so we refer to Figure 2.4, and again the case of going from positive to negative values to x ideas. If > 0 we can again read the panels of Figure 2.3 left-to-right/top-to-bottom to understand the amplitude is then maintained adiabatically (panel (c), thick grey line). This situation corresponds to the grey arrows in panel (b) of Figure 2.7.
dynamics. We start again at vanishing , that is at the centre equilibrium point at the origin in panel (a). As becomes smaller, there is a bifurcation (panel (b)) and a separatrix emerges from the origin itself. In this case, the adiabatic principle is not applicable anymore since the orbit crosses the separatrix. The system may enter one of the resonant lobes in panels (c), (d); then, after another bifurcation occurs (panel (e)), another stable centre appears at the origin enclosed by two new separatrices (panel (f)): if the orbit is still close enough to the separatrix, the adiabatic principle is again not applicable18 and the system may follow one of the resonant equilibria in the two resonant lobes, or end up in the inner circulation region. Therefore, for k = 2 when decreases slowly enough and is positive, even at small amplitude of libration around the equilibrium point at the origin, resonant capture is a probabilistic phenomenon (the probability to fall into a region of expanding area or another is given by the ratio of the derivatives of the areas). In the case < 0 (and still going from positive to negative) we again read the panels and the \time » in reverse order, starting as always from the equilibrium point at vanishing , that is from the centre at the origin in panel (f). Now, as the value of changes the librating orbit is invested by the separatrices which are collapsing towards the origin (panel (e)), gets excited and no capture is possible.
In both cases (k = 1 and k = 2), when (t) changes from negative to positive, the condition on the sign of is ipped, as one can see by changing the sign to the time variable. Therefore, capture is only possible when _ < 0.
The takeaway is the following. Resonant capture can occur only if _ < 0. If the orbit is initially near = 0, for k = 1 the capture in resonance is ensured by the adiabatic principle and for k = 2 (or larger) is probabilistic. The capture into resonance is followed by a monotonic increase of over time. Instead, when _ > 0, an orbit with initially 0 (whatever the k value) necessarily jumps across the resonance, with a consequent impulsive increase of . The path taken by the evolution when capture is successful is shown in Figure 2.7 panel (b) in bold black arrows (for the case _ < 0 to x ideas): initially follows the evolution of as long as 0, and when vanishes there is a capture in a resonance lobe which keeps = 0 and forces to increase as the resonant equilibrium gets farther and farther away from the origin. In grey arrows we show the case when there is no capture, so does not lock at 0 and gets a non-adiabatic kick due to the interaction with the separatrix; after the disappearance of the separatrix, the orbit maintains adiabatically its excited amplitude.
As a nal remark, we should keep in mind that the Andoyer Hamiltonians (2.125) (or even the simpler pen-dulum model) represent only an integrable approximation to the real dynamics. In the real system (which is often non-integrable) the separatrices are replaced by chaotic bands. In this case, when an orbit meets the chaotic region around the separatrix, the probability to jump into a region or another is no longer simply pro-portional to the ratio of the derivatives of said areas. It depends also on the mixing properties of the chaotic zone [Henrard and Henrard(1991), Henrard and Morbidelli(1993)].
18 This is typically the case when the second bifurcation occurs for a value of very close to that corresponding to the rst bifurcation, as in the case of the 3:1 mean motion resonance and the secondary resonance studied in Chapter 5.
Two planets { The structure of resonant pairs and capture into mean motion resonance
We begin in this chapter our investigation of planetary dynamics in mean motion resonance, focusing especially on Super-Earths/Mini-Neptunes. Recall from the Introduction, Subsections 1.2.2 and 1.2.3, that such planets are expected to form within the lifetime of their protoplanetary disc and interact gravitationally with it. A common outcome of planet-disc interactions is inward migration, which is halted at the inner edge of the disc. When multiple planets are embedded in the same disc, their slow convergent migration usually leads to the formation of a chain of mutual mean motion resonances. Due to this renewed interest in resonant captures, we revisit in this chapter the problem of capture in rst order resonances of two equal-mass coplanar planets in convergent migration, using a semi-analytical approach and numerical simulations. In Section 3.1 we compute analytically the locus of equilibrium points of rst-order resonances, where both the resonant and secular oscillations of the planetary orbits have a null amplitude. Our calculations are developed for unexpanded Hamiltonians, which allows to follow the dynamics up to arbitrarily large eccentricities (e.g. [Beaug et al.(2006)], [Michtchenko et al.(2006)]). We compare the results with those obtained with rst and second order expansions of the Hamiltonian in the eccentricity, showing qualitative and quantitative disagreements. The quantitative accuracy of our results is validated with simulations in which planets are forced to migrate towards each-other, without any eccentricity damping. These simulations have to follow the loci of the equilibrium points, and show perfect agreement with the unexpanded model. Moreover, we calculate the two frequencies of libration around the equilibrium points, therefore obtaining a complete understanding of the system; we again check the validity of the analytical calculations against numerical simulations in which the amplitudes of resonant and secular librations are slightly excited and the frequencies of oscillation of the semi major axis and the eccentricity are measured. In Section 3.2 we introduce the eccentricity damping exerted by the disc onto the planets. This leads to a nal equilibrium con guration where convergent migration stops. The analytic calculation of the equilibrium eccentricities and semi major axes ratio is presented in Subsection 3.2.2. We check against numerical simulations the validity of these analytical predictions, showing excellent agreement. The content of this chapter has been published in [Pichierri et al.(2018)].
Structure of rst-order mean motion resonances
Consider two planets of masses m1 and m2 orbiting the same star of mass M in a canonical heliocentric refer-ence frame ([Poincare(1892)], see also Subsect. 2.2.2), assuming coplanar orbits for simplicity. We introduce the Hamiltonian for the planar three-body problem H = Hkepl + Hpert, in modi ed Delaunay variables ( i; i; i; i) (cfr. Equations (2.95) to (2.98)); the subscripts i = 1; 2 indicate the inner and outer planet respectively. We then impose a rst order mean motion resonance between the two planets, that is we assume that the two mean motions G(M + m1)=a13 G(M + m2)=a23 satisfy the resonance condition kn2 (k 1)n1 0, where n1 = and n2 = is a positive integer, k 2. In order to consider the resonant interactions only, we eliminate perturbatively the non-resonant angles: to rst order in the planetary mass, this corresponds to averaging the Hamiltonian over
the fast angles (cfr. Subsect. 2.3.1). In fact, since the Keplerian part Hkepl does not depend on the angles, only the perturbation Hamiltonian Hpert needs averaging. We note that we need to integrate Hpert e.g. with respect to the angle 1 over the interval [0; 2k ], corresponding to k revolutions of the inner planet around the star (which in turn by the resonance condition is equivalent to (k 1) revolutions of the outer planet), in order to fully recover the periodicity of the Hamiltonian. This leads to a new averaged perturbing Hamiltonian which we denote with Hres.
Table of contents :
1 Introduction
1.1 A brief history of Planetary Science
1.1.1 The discovery of exoplanetary systems
1.1.2 The exoplanets sample
1.1.3 The Solar System in perspective and implications of the exoplanets sample
1.2 Planetary formation
1.2.1 Protoplanetary discs
1.2.2 Building the planets: Overview of accretion processes
1.2.3 Shaping the planets’ orbits during the disc phase: Planetary type-I migration and eccentricity damping
1.3 This thesis in context
2 Hamiltonian mechanics and the planetary problem
2.1 Hamiltonian systems
2.1.1 Link with Lagrangian formalism
2.1.2 Dynamical variables
2.1.3 Canonical transformations
2.1.4 Integrable dynamics and action-angle variables
2.1.5 Equilibrium points and linear stability
2.1.6 Basic examples of Hamiltonian systems
2.2 Planetary systems in Hamiltonian mechanics
2.2.1 The two-body problem
2.2.2 The planetary problem
2.3 Elements of Hamiltonian perturbation theory
2.3.1 First order perturbation theory
2.3.2 An introduction to adiabatic theory
2.3.3 Application of the adiabatic theory to resonant capture
3 Two planets { The structure of resonant pairs and capture into mean motion resonance
3.1 Structure of rst-order mean motion resonances
3.1.1 First and higher order expansions of the Hamiltonian in the eccentricities
3.1.2 Equilibrium points of the averaged Hamiltonian
3.1.3 Frequencies in the limit of small amplitude of libration
3.2 Capture into resonance by type-I migration
3.2.1 Convergent inward migration in disc and resonant capture
3.2.2 Planet-disc interactions and evolution in mean motion resonance
4 Three-planet systems and the near-resonant population
4.1 The near-resonant population
4.1.1 Methods and physical setup
4.2 Analytical model for three resonant planets
4.2.1 Resonant equilibrium points
4.2.2 Resonant repulsion for three-planets systems
4.3 A scenario for dissipative evolution of three-planet systems
4.3.1 Choice of systems
4.3.2 Analytical maps
4.3.3 Numerical simulations
4.4 Results
4.4.1 Probabilistic measure of a resonant conguration in Kepler-305, YZ Cet and Kepler-
4.4.2 The 5:4 { 4:3 resonant chain on Kepler-60 and other near-resonant systems with k >
4.5 Conclusions
5 The onset of instability in resonant chains
5.1 Introduction
5.2 2 Planets
5.3 3 Planets
5.3.1 Numerical stability maps for N = 3 and k = 3
5.3.2 Numerical and analytical investigation of the phenomenon
5.3.3 Rescaled Hamiltonian and new set of canonical variables
5.3.4 Purely resonant dynamics
5.3.5 The synodic contribution
5.3.6 Dependence on k
5.4 N Planets
6 Extreme secular excitation of eccentricity inside mean motion resonance
6.1 Small bodies driven into star-grazing orbits by planetary perturbations
6.2 Planetary Hamiltonian
6.3 Studying the averaged Hamiltonian
6.4 Eect of short-range forces
6.5 Results
6.6 Conclusions
7 Conclusions
7.1 Future perspectives
