Planets in Mean-Motion Resonances and the System Around HD45364

TL;DR

This paper reviews the dynamics of planets in mean-motion resonances, focusing on the HD 45365 system, and discusses how such resonances inform us about planetary formation and evolution.

Contribution

It provides a detailed analysis of resonant planetary dynamics and applies these methods specifically to the HD 45365 system, enhancing understanding of resonant orbital evolution.

Findings

Resonant planets exhibit unique orbital behaviors.

Mean-motion resonances can reveal planetary formation history.

Application to HD 45365 demonstrates these dynamics in practice.

Abstract

In some planetary systems, the orbital periods of two of its members present a commensurability, usually known by mean-motion resonance. These resonances greatly enhance the mutual gravitational influence of the planets. As a consequence, these systems present uncommon behaviors, and their motions need to be studied with specific methods. Some features are unique and allow us a better understanding and characterization of these systems. Moreover, mean-motion resonances are a result of an early migration of the orbits in an accretion disk, so it is possible to derive constraints on their formation. Here we review the dynamics of a pair of resonant planets and explain how their orbits evolve in time. We apply our results to the HD 45365 planetary system.

Tables

Table 1. Table 1: Extra-solar planets confirmed to be trapped in a mean-motion resonance.

Star	Planets	Resonance	Reference
GJ 876	c,b	2$:$1	Lee and Peale (2002)
GJ 876	b,e	2$:$1	Rivera et al. (2010)
HD 73526	b,c	2$:$1	Tinney et al. (2006)
HD 82943	c,b	2$:$1	Lee et al. (2006)
HD 128311	b,c	2$:$1	Vogt et al. (2005)
HD 45364	b,c	3$:$2	Correia et al. (2009)
HD 204313	b,d	3$:$2	Robertson et al. (2012)
HD 5319	b,c	4$:$3	Giguere et al. (2015)
HD 60532	b,c	3$:$1	Laskar and Correia (2009)
HD 33844	b,c	5$:$3	Wittenmyer et al. (2016)
HD 202206	b,c	5$:$1	Correia et al. (2005)

Table 2. Table 2: Quasi-periodic decomposition of the resonant angle σ 1 = 2 ​ λ 1 − 3 ​ λ 2 + ϖ subscript 𝜎 1 2 subscript 𝜆 1 3 subscript 𝜆 2 italic-ϖ \sigma_{1}=2\lambda_{1}-3\lambda_{2}+\varpi for an integration over 10 5 superscript 10 5 10^{5}  yr of the orbital solution of the HD 45364 planetary system (Correia et al. 2009 ) .

Combination					${\nu }_i$	$A_i$	${\varphi }_i$
$n_b$	$n_c$	$g_1$	$g_2$	$l_{\sigma }$	(deg/yr)	(deg)	(deg)
\svhline 0	0	0	0	1	19.8207	68.444	-144.426
0	0	-1	1	0	0.8698	13.400	136.931
0	0	1	-1	1	18.9509	8.606	168.643
0	0	-1	1	1	20.6905	8.094	82.505
0	0	-2	2	0	1.7396	2.165	-176.138
0	0	-2	2	1	21.5603	0.622	-50.564
0	0	0	0	3	59.4621	0.540	-73.279

Equations

$$\frac{P_2}{P_1}\approx \frac{p+q}{p},$$

$$\hat{H}=\hat{K}+{\hat{H}}_1,$$

$$\hat{K}=-\sum _{i=1}^2\frac{{\mu }_i^2{\beta }_i^3}{2{\hat{\Lambda }}_i^2},$$

$${\hat{H}}_1=-\mathcal{G}\frac{m_1{m}_2}{{\Delta }_{12}}+\frac{{\tilde{\mathbf{r}}}_1\cdot {\tilde{\mathbf{r}}}_2}{m_0},$$

$$(p+q)n_2-p{n}_1\approx 0,$$

$$\hat{G}={\hat{G}}_1+{\hat{G}}_2={\hat{\Lambda }}_1\sqrt{1-e_1^2}+{\hat{\Lambda }}_2\sqrt{1-e_2^2}.$$

$$\Gamma =\frac{p+q}{p}{\hat{\Lambda }}_1+{\hat{\Lambda }}_2.$$

$${\sigma }_i=-\frac{p}{q}{\lambda }_1+\frac{p+q}{q}{\lambda }_2-{\varpi }_i,i=1,2,$$

$$I_i={\Lambda }_i-G_i=\frac{1}{2}{\Lambda }_i{\xi }_i^2\propto e_i^2,i=1,2,$$

$${\xi }_i=\sqrt{2\left(1-\sqrt{1-e_i^2}\right)}\approx e_i.$$

$$x_i={\sqrt{I}}_i{\mathbf{e}}^{\mathbf{i}{\sigma }_i}.$$

$$H=K(I_i)+\tilde{S}(I_i,\Delta \varpi )+\tilde{R}(I_i,{\sigma }_i),$$

$${\Lambda }_1={\Lambda }_{1,0}-\frac{p}{q}\left(D-\delta \right),{\Lambda }_2={\Lambda }_{2,0}+\frac{p+q}{q}\left(D-\delta \right),$$

$$D=I_1+I_2={\Lambda }_1-G_1+{\Lambda }_2-G_2=\sum _{i=1}^2{x}_i{\bar{x}}_i,$$

$$\delta ={\Lambda }_{1,0}-G_1+{\Lambda }_{2,0}-G_2,$$

$$H=K(D)+{\tilde{S}}_q(I_i,\Delta \varpi )+\sum _{k=0}^q{R}_k(x_1^k{x}_2^{q-k}+{\bar{x}}_1^k{\bar{x}}_2^{q-k}),$$

$$\frac{d{x}_i}{dt}=\mathbf{i}\frac{\partial H}{\partial {\bar{x}}_i}.$$

$$H=K(D)+R_1(x_1+{\bar{x}}_1)+R_0(x_2+{\bar{x}}_2),$$

$$R_1=R\cos \varphi ,R_0=R\sin \varphi ,$$

$$u_1=|u_1|{\mathbf{e}}^{\mathbf{i}\theta }=x_1\cos \varphi +x_2\sin \varphi ,u_2=-x_1\sin \varphi +x_2\cos \varphi ,$$

$$H=K(D)+R(u_1+{\bar{u}}_1),\mathrm{with}D=\sum _{i=1}^2{u}_i{\bar{u}}_i.$$

$$(p+q){\dot{\lambda }}_2-p{\dot{\lambda }}_1-q{\dot{\varpi }}_i=0,i=1,2,$$

$$(p+q)n_2-p{n}_1-q{g}_i=0.$$

$$\dot{\delta }{|}_d$$

$$⟨\dot{A}⟩=\frac{1}{2R{\delta }^{q/2}}\left(⟨ϵ\dot{ϵ}{|}_d⟩-\frac{q}{4\delta }⟨{ϵ}^2\dot{\delta }{|}_d⟩\right),$$

$$ϵ=D-\delta ,\mathrm{and}\dot{ϵ}{|}_d=-\frac{q}{p}\frac{{\Lambda }_1{\Lambda }_2}{2}{\frac{\dot{\alpha }}{\alpha }|}_d.$$

$${\frac{{\dot{\hat{G}}}_i}{{\hat{G}}_i}|}_d=-\frac{1}{T_{m,i}},\mathrm{and}{\frac{{\dot{e}}_i}{e_i}|}_d=-\frac{1}{T_{e,i}},$$

$${\frac{{\dot{\xi }}_i}{{\xi }_i}|}_d\approx {\frac{{\dot{e}}_i}{e_i}|}_d=-\frac{1}{T_{e,i}},$$

$${\frac{\dot{\alpha }}{\alpha }|}_d=\frac{2}{T_m}+\frac{4}{{\Lambda }_1{\Lambda }_2}\left(\frac{{\Lambda }_1{\sin }^2\varphi }{T_{e,2}}-\frac{{\Lambda }_2{\cos }^2\varphi }{T_{e,1}}\right)D,$$

$$\dot{\delta }{|}_d=\frac{1}{T_M}-\frac{D}{T_E},\mathrm{and}⟨\dot{\delta }{|}_d⟩=\frac{1}{T_M}-\frac{\delta }{T_E},$$

Full text

1 Introduction

In addition to the solar system, about 600 multi-planetary systems are known, that is, systems that contain at least two planets. Among these systems, at least eleven pairs of planets have been identified to have resonant orbits, or, more precisely, to be trapped in a mean-motion resonance (see Table 1). Many other pairs are listed as possible resonances, but our present knowledge of their orbital elements is not sufficient to confirm the resonant behaviour.

A mean-motion resonance occurs when two planets exert a regular, periodic gravitational influence on each other. The physics principle behind is similar in concept to a driven harmonic oscillator. As for the oscillator, a planet has a natural orbital frequency. Another planet doing the “pushing” will act in periodic repetition and has a cumulative effect on the motion of the considered planet. In mean-motion resonances, the orbital periods are related by a ratio of two small integers:

[TABLE]

where $P_{1}$ and $P_{2}$ are the orbital periods of the inner and outer planet, respectively, and $p$ and $q$ are integers. The $q$ value also designates the order of the resonance, and the resonance is called a $(p+q)$:$p$ mean-motion resonance. Mean-motion resonances greatly enhance the mutual gravitational interactions between the planets. In many cases, the result is an unstable interaction, in which the planets exchange momentum and shift orbits until the resonance no longer exists. For instance, the Kirkwood gaps in the spatial distribution of the asteroids correspond to orbital periods that are integer fractions of Jupiter’s period. However, under some circumstances, mean-motion resonances can be self-correcting, so that the system remains stable with the planets in resonance. This is the case for all the planetary systems observed in mean-motion resonances (Table 1).

In the solar system, we observe a 3:2 mean-motion resonance between Neptune and Pluto, which means that Pluto completes two orbits in the time it takes Neptune to complete three. Another example of a 3:2 resonance is present in the HD 45364 planetary system, but involving two giant planets (Correia et al. 2009). In Fig. 1, we show the evolution of the HD 45364 system over $10^{5}$ yr in the rotating frame of the inner and of the outer planet. Due to the 3:2 mean-motion resonance trapping, the relative positions of the two planets are repeated, preventing close encounters and the consequent destruction of the system. The paths of the planets in the rotating frame illustrate the relationship between the resonance and the frequency of conjunctions with the internal or external planet: the orbital configuration of the system is repeated every 3 orbits of the inner planet and every 2 orbits of the outer planet. In this particular frame, we are also able to see the libration of each planet around its equilibrium position.

The most commonly observed mean-motion resonance is the 2:1, followed by the 3:2, but many other configurations are also possible (Table 1). Sometimes more than two planets can be involved in a mean-motion resonance. In the solar system this is the case of the Jupiter’s moons Io, Europa and Ganymede, which are involved in a 4:2:1 resonance, also known as Laplace resonance. For exoplanets, a similar example is known for the GJ 876 planets $b$, $c$ and $e$ (Rivera et al. 2010).

2 Conservative dynamics

We refer to the star as body 0, to the inner planet as body 1, and to the outer planet as body 2. Denoting the masses of the three bodies $m_{i}$, we introduce for both planets $\mu_{i}=\mathcal{G}(m_{0}+m_{i})$ and $\beta_{i}=m_{0}m_{i}/(m_{0}+m_{i})$, where $\mathcal{G}$ is the gravitational constant. We let $\mathbf{r}_{i}$ be the position vectors of the planets with respect to the star and $\tilde{\mathbf{r}}_{i}$ the canonically conjugated momenta (in astrocentric coordinates, see Laskar and Robutel 1995). As usual in the literature, semi-major axes are noted $a_{i}$, eccentricities $e_{i}$, mean longitudes $\lambda_{i}$, and longitudes of periastron $\varpi_{i}$. For simplicity, we only consider the planar case. The Hamiltonian of the three-body problem reads

[TABLE]

where $\hat{K}$ is the Keplerian part (star-planets interactions) and $\hat{H}_{1}$ is the perturbative part (planet-planet interactions). The Keplerian part is given by

[TABLE]

where $\hat{\Lambda}_{i}=\beta_{i}\sqrt{\mu_{i}a_{i}}$ is the circular angular momentum of planet $i$.

The perturbative part can be decomposed in direct and indirect interactions,

[TABLE]

with $\Delta_{12}=||\mathbf{r}_{1}-\mathbf{r}_{2}||$. This perturbation can be expressed as a function of elliptical orbital elements by expanding it in Fourier series of the mean longitudes $\lambda_{i}$, and longitudes of the pericenter $\varpi_{i}$ (e.g., Laskar and Robutel 1995). For a given mean-motion resonance $(p+q)$:$p$, the corresponding combination of the mean longitudes undergoes slow variations (Eq. (1))

[TABLE]

where $n_{i}=\dot{\lambda}_{i}\approx 2\pi/P_{i}$, while other (non-resonant) combinations of these angles circulate rapidly. The long-term evolution of the orbits is thus accurately described by the averaged Hamiltonian over the non-resonant combinations of the mean longitudes. By performing this averaging and the classical angular momentum reduction, one obtains a two degrees of freedom problem (e.g., Delisle et al. 2012) and two constants of motion. The first constant of motion is the total angular momentum,

[TABLE]

The second, which comes from the averaging, is a combination of the circular angular momenta (or semi-major axes, e.g., Michtchenko and Ferraz-Mello 2001),

[TABLE]

As shown in Delisle et al. (2012), the constant $\Gamma$ can be used as a scaling factor and does not influence the dynamics of the system except by changing the scales of the problem (in space, energy, and time). The elimination of $\Gamma$ is achieved by performing the following change of coordinates: $\Lambda_{i}=\hat{\Lambda}_{i}/\Gamma$, $G_{i}=\hat{G}_{i}/\Gamma$, $H=\Gamma^{2}\hat{H}$, and $t=\hat{t}/\Gamma^{3}$, while angle coordinates are unchanged. Using these new coordinates, the dynamics of the system depends on only one parameter, $G=\hat{G}/\Gamma=G_{1}+G_{2}$, the renormalized angular momentum. The remaining two degrees of freedom can be represented by both resonant angles,

[TABLE]

and both angular momentum deficits (Laskar 2000), which are canonically conjugated to the resonant angles,

[TABLE]

with

[TABLE]

We can also introduce rectangular coordinates,

[TABLE]

It should be noted that for small eccentricities, $|x_{i}|\propto e_{i}$. The averaged Hamiltonian takes the form

[TABLE]

where $\tilde{S}$ is the secular part of the Hamiltonian depending on the difference of longitudes of periastron ($\Delta\varpi=\sigma_{2}-\sigma_{1}$), but not on mean longitudes of the planets, and $\tilde{R}$ is the resonant part. These two parts can be expanded as power series of eccentricities, or, more precisely, of $x_{i}$ (Laskar and Robutel 1995; Delisle et al. 2012).

The Keplerian part can be expressed as a function of the momenta $I_{i}$ by substituting the expressions of $\Lambda_{i}$ in Equation (3),

[TABLE]

where $D$ is the total angular momentum deficit (Laskar 2000),

[TABLE]

$\delta$ is the total angular momentum deficit at exact commensurability (which is a constant of motion),

[TABLE]

and $\Lambda_{i,0}$ denotes the value of $\Lambda_{i}$ at exact commensurability.

The secular part contains terms of degree two and more in eccentricities while the resonant part contains terms of degree $q$ and more. Thus, the simplest model of the resonance should take into account at least those terms of order $q$ in eccentricities in the perturbative part,

[TABLE]

where $\tilde{S}_{q}$ is the secular part truncated at degree $q$, and $R_{k}$ are constant coefficients (see Delisle et al. 2012). The equations of motion are given by

[TABLE]

This problem is much simpler than the initial four degrees of freedom problem. However, in general it is still non-integrable since it presents two degrees of freedom. For a first-order mean-motion resonance (such as the 2:1 or 3:2 resonances) the simplest Hamiltonian reads

[TABLE]

where there are no secular terms since they only appear at degree two. It is well known that the Hamiltonian (18) is integrable (see Sessin and Ferraz-Mello 1984; Henrard et al. 1986; Wisdom 1986). Introducing $R$ and $\phi$ such that

[TABLE]

and the new coordinates

[TABLE]

the Hamiltonian (18) reads

[TABLE]

In these coordinates, the Hamiltonian does not depend on the angle associated with $u_{2}$. It depends only on the action $u_{2}\bar{u}_{2}$, which is thus a new constant of motion of the system. We are then left with only one degree of freedom ($u_{1}$, $\bar{u}_{1}$) and the problem is integrable. However, if one introduces second-order terms in the Hamiltonian of a first-order resonance, this simplification no longer occurs. It does not occur either, in the case of higher order mean-motion resonances, even at the minimal degree of development of the Hamiltonian. It is nevertheless possible to make some additional hypothesis in order to obtain an integrable system that still captures most of the characteristics of the resonant motion (see Delisle et al. 2014).

In Fig. 2 we show the energy levels of the Hamiltonian (21) on the plane defined by $u_{2}=0$, in the case of the HD 45364 planetary system (3:2 mean-motion resonance, $\phi=-31.45^{\circ}$), for increasing values of $\delta$ (see Eq. (15)). We distinguish three areas of interest: two zones of circulation (internal circulation for low eccentricities and external circulation for high eccentricities), and a libration zone (banana-shaped level curves) separated from both circulation areas by two separatrices. Fixed points of the system, corresponding to $\dot{u}_{i}=0$, are marked with green and blue dots (for stable ones), and a red cross (for the unstable one). The green dot corresponds to the libration center. In Fig. 3 we show the positions of these three fixed points as a function of the parameter $\delta$. When $\delta$ increases, the libration area moves to higher eccentricities (i.e. higher values of $|u_{1}|$, Fig. 3) and the internal circulation area takes more space (Fig. 2). On the contrary, for smaller values of $\delta$, the internal circulation and the libration areas tend to shrink and eventually completely disappear, leaving only the external circulation area (Fig. 2).

3 Occurrence of mean-motion resonances

When multi-planetary systems are found, one may ask whether a pair of its members is in resonance or not. A straightforward test is to check if the period ratio can be given by the ratio of two integers (Eq. (1)). However, this simple test can be misleading because there is always a rational number close to any real number. Moreover, the resonant ratio does not depend solely on the orbital periods. Indeed, form expression (8), the exact resonance occurs for

[TABLE]

where $\dot{\varpi}_{i}$ corresponds to the precession rate of the apsidal line of planet $i$. A frequently used test is then to check if the resonant angle $\sigma_{i}$ (Eq. (8)) is in libration. Nevertheless, for small eccentricities this libration can be artificial (Delisle et al. 2012): the center of the circulation zone is shifted from zero (Fig. 2), and the system describes a small circle around the equilibrium point. Thus, the angles $\sigma_{i}$ appear to librate because their origin is taken at zero (Fig. 2).

The ultimate test to confirm the presence of a mean-motion resonance is to perform a frequency analysis (Laskar 1988, 1990) of the orbital solution and check if the mean-motions $n_{i}$ and the eigenmodes of $\varpi_{i}$, that we denote $g_{i}$, are indeed resonant:

[TABLE]

In addition, unless the libration amplitude of the resonant angle $\sigma_{i}$ is fully damped, we should be able to detect a true libration frequency associated with this motion (see Fig. 1). In Table 3, we provide a quasi-periodic decomposition of the resonant angle $\sigma_{1}$ in terms of decreasing amplitude for the 3:2 resonant HD 45364 planetary system (Correia et al. 2009). All the quasi-periodic terms are easily identified as integer combinations of the fundamental frequencies. The fact that we are able to express all the main frequencies of $\sigma_{1}$ in terms of exact combinations of the fundamental frequencies is a signature of a very regular resonant motion.