Analysing the dynamics of the Kepler-90 planetary system

ABSTRACT

Kepler-90 system has a set of eight planets in a hierarchical structure. In this work, we used frequency analysis to study several Kepler-90 analogues to analyse in detail how the values of the eccentricity of each planet can alter the stability of the system. The system was formed by the star and eight planets for three different intervals of eccentricity (⁠|$e$|⁠). Our results show that for the first and second intervals, all the systems are stable. However, no set of Kepler-90 systems with large values of |$e$| survived up to |$10^5$| orbits of Kepler-90h. In the low-eccentricity interval, planets Kepler-90b and Kepler-90c were found to be in a 5:4 mean motion resonance, and the pair Kepler-90g and Kepler-90h in a 3:2 mean motion resonance, although these two pairs are in resonance in different Kepler-90 analogues. In the medium-eccentricity interval, only the pair Kepler-90g and Kepler-90h is in resonance. Kepler-90b and Kepler-90c are in quasi-resonance 5:4; the critical angle circulates and librates intermittently in different periods of time. The results statistically indicate that these resonances directly affect the most of them. The influence covers a wide range of possibilities: (i) a strong resonant mean motion coupling added to a coupling of the longitude of the pericentres; (ii) just a single resonant argument librating; (iii) intermittent behaviour, interleaving libration, and circulation of the resonant arguments; and (iv) the quasi-resonant influence due to the near commensurability.

1 INTRODUCTION

Among more than 4000 planetary systems already discovered, one is of particular interest because it has eight planets around a star, resembling our own Solar system. Thanks to Kepler mission, Cabrera et al. (2014) first reported the discovery of the transiting Kepler-90 planets Kepler-90b, Kepler-90c, Kepler-90e, and Kepler-90f in orbit around the star KIC 11442793. The previous discovery of planets Kepler-90d, Kepler-90g, and Kepler-90h (Borucki et al. 2010) brings the total number of seven planets located between 0.074 and 1.01 au. Cabrera et al. (2014) revised the planets’ orbital parameters and analysed the dynamics presented in this interesting system. Their results showed that the outermost planet, Kepler-90h, is located at 1.01 au. This system is hierarchical, the smaller planets are closer, while the larger ones are further away from the star. The two innermost planets (Kepler-90b and Kepler-90c) have sizes close to Earth’s radii, and Kepler-90d, Kepler-90e, and Kepler-90f are super-Earths with sizes between 2 and 3 Earth radii. Kepler-90g and Kepler-90h are gas giant planets located in the external region. Through numerical simulations including the larger five planets (the gravitational effects of the smaller and further planets, Kepler-90c and Kepler-90d, were neglected), Cabrera et al. (2014) verified that the system is stable if the eccentricity of all five planets is smaller than 0.001. Shallue & Vanderburg (2018) identified Kepler-90i, the eighth planet around Kepler-90 system, through a deep learning method. Their results showed that Kepler-90i is located between the orbits of Kepler-90c and Kepler-90d.

The masses (in Earth mass unit), radii (in Earth radius unit), semimajor axes (⁠|$a_\mathrm{ p}$|⁠), and inclinations (⁠|$i_\mathrm{ p}$|⁠) for all eight planets are shown in Table 1.

Contreras & Boley (2018) analysed the stability in planetary systems with tightly packed inner planets (planets at a very close distance from the star) through numerical simulations and secular theory. This study was applied to verify how the analogues of Kepler-11 and Kepler-90 systems behave under the effects of a perturbing Jupiter-like planet at different locations, exterior to the outermost planet of each system. They estimated the masses of Kepler-90 planets from a mass–radius relation described in Wright et al. (2011); their values are presented in Table 1. The results regarding the Kepler-90 planetary system over 10 Myr, without the hypothetical Jupiter-like planet, showed that about 20 per cent of the numerical simulations turned out to be unstable, even when the initial eccentricity of all Kepler-90 planets was assumed to be zero.

As discussed in Cabrera et al. (2014), the ratio between the periods of Kepler-90b and Kepler-90c suggests that the pair could be in a 5:4 mean motion resonance (MMR), while Kepler-90d, Kepler-90e, and Kepler-90f are near a 4:2:1 Laplace resonance. However, as Contreras & Boley (2018) pointed out, it is necessary to analyse the temporal evolution of the resonant angles to confirm if the planets are indeed in resonance. They found that the resonant angles for the two pairs librate: Kepler-90b and Kepler-90c are in 5:4 MMR, and Kepler-90g and Kepler-90h are in 3:2 MMR. Both angles librate with large amplitudes around 0° and |$180^{\circ }$|⁠, respectively. Their analysis regarding the three-body resonance shows that the resonant angle circulates during all the numerical simulations.

In an attempt to add new information to the dynamical analysis carried out by previous authors, we study the stability of the Kepler-90 planetary system for a range of different values of eccentricity. The stability of the system is analysed through the frequency analysis technique (Laskar 1990, 1993). For each range of eccentricity, a set of 100 Kepler-90 planetary systems analogues was numerically integrated. We also search for MMRs between the planets. Our analysis revealed that several analogues of the pairs Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h can be in 5:4 and 3:2 MMR, respectively. By computing the temporal variation of the critical angle, corresponding to each MMR (5:4 and 3:2 MMRs), different regimes could be identified, which will be described in the following sections.

It is important to point out the work by Liang, Robnik & Seljak (2021), which through an analysis of transiting time variations of both planets, Kepler-90g and Kepler-90h, obtained different masses for the two giant planets. Their best-fitting masses for Kepler-90g and Kepler-90h are 15 and 203 |$\mathrm{ M}_{{\oplus }}$|⁠, respectively. However, in this work, we will adopt the orbital and physical parameters derived from Contreras & Boley (2018) and Shallue & Vanderburg (2018) (Table 1).

This paper is divided as follows. In Section 2, we use frequency analysis to study the stability of a sample of Kepler-90 system analogues within a range of initial values of eccentricities. In Section 3, we analyse the different behaviours of the critical angle of the two pairs of planets (Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h). In Section 4, we discuss our results.

2 STABILITY OF KEPLER-90 ANALOGUES

In this section, we explore a sample of Kepler-90 system analogues to evaluate their stability for different values of the initial eccentricities of the planets; we carried out this exploration using the frequency analysis technique.

2.1 Frequency analysis overview

Since originally introduced by Laskar (1990), frequency analysis has been successfully used to characterize the long-term dynamics of many multidimensional systems with n-degrees of freedom. The frequency analysis method is based on Fourier techniques for which computationally inexpensive, short-term numerical integrations are sufficient to analyse the overall dynamics of large regions of the accessible phase space of the system (see e.g. Laskar 1993; Nesvornỳ & Morbidelli 1998; Robutel & Laskar 2001; Muñoz-Gutiérrez et al. 2022; Alves-Carmo, Vaillant & Correia 2023).

In this work, we follow a standard approach developed in our previous studies (e.g. Muñoz-Gutiérrez & Giuliatti Winter 2017; Gaslac Gallardo et al. 2020), to estimate the stability of the eight Kepler-90 planets through frequency analysis. A summary of our considerations is provided below, but the reader is referred to the above works for more complete descriptions of the procedure.

We used the frequency modified Fourier transform (FMFT) algorithm, which was developed by Šidlichovský & Nesvorný (1996), to calculate the main frequency, |$\nu _0$|⁠, of a dynamical variable defined as |$z^{\prime }(t)=a(t) \, \mathrm{ e}^{\mathrm{ i}\lambda (t)}$|⁠, where |$a$| and |$\lambda$| are the semimajor axis and mean longitude of each test planet. Both variables are dependent on time and extracted as time series from a short numerical integration. The FMFT algorithm returns an approximate decomposition of |$z^{\prime }(t)$| of the form

The first term in equation (1) contains the main parameters of the decomposition provided by the frequency analysis technique. In the case of a simple Keplerian orbit, the following conditions would be fulfilled: |$N=0$|⁠, |$|\alpha _0| = a_\mathrm{ p}$|⁠, and |$\nu _0 = n_\mathrm{ p}$|⁠, where |$a_\mathrm{ p}$| and |$n_\mathrm{ p}$| are the semimajor axis and mean motion of the planet, respectively.

In a more complex and more realistic system, subsequent terms appear, i.e. |$N\gt 0$|⁠, and in this case, ||$\alpha _0$|| and |$\nu _0$| will not remain constant. However, for a regular or quasi-periodic orbit, ||$\alpha _0$|| and |$\nu _0$| will remain close to |$a_\mathrm{ p}$| and |$n_\mathrm{ p}$| at all times, meaning that |$\alpha _0 \gg \alpha _j$|⁠. On the other hand, for unstable orbits, these variables will change quickly during the numerical integration and the |$N\gt 0$| terms may be comparable in magnitude to the zero-order term.

To estimate the stability of the orbits, we define a diffusion parameter for each planet, |$D$|⁠, as follows (Correia et al. 2005; Muñoz-Gutiérrez & Giuliatti Winter 2017):

where |$\nu _{01}$| and |$\nu _{02}$| correspond to the mean frequencies of the orbits of the given planet in the integration intervals [|$0, \, T$|] and [|$T, \, 2T$|], respectively, where |$T$| is half the total integration time.

A small value of the diffusion parameter indicates that the mean frequencies in both intervals remain almost constant, indicating a stable trajectory. A large value of the diffusion parameter indicates that the mean frequencies show large variations from one integration interval to the next, indicating an unstable or even chaotic orbit of a planet.

2.2 Stability through frequency analysis: simulations and analysis of results

We generated 100 Kepler-90 analogues for each interval of the eccentricity, giving a total of 300 Kepler-90 analogues. Since the system has eight planets, a total of 2400 initial conditions were analysed for a period of |$10^4 \, T_{\rm h}$| (where |$T_{\rm h}$| is the orbital period of Kepler-90h, |$T_{\rm h} = 331.6 \, {\rm d}$|⁠). Three different intervals of eccentricity were studied, the first interval ranges random values from 0 to |$10^{-3}$|⁠, the second interval from |$10^{-3}$| to |$10^{-2}$|⁠, and the third interval from |$10^{-2}$| to |$10^{-1}$|⁠. The argument of the pericentre (⁠|$\omega$|⁠), the longitude of the ascending node (⁠|$\Omega$|⁠), and the mean anomaly (⁠|$M$|⁠) of each planet were randomly chosen from 0° to |$360^\circ$|⁠. The initial values of the semimajor axis (⁠|$a$|⁠), inclination (⁠|$i_{\rm p}$|⁠), mass, and radius for each planet were extracted from Table 1. The mass and radius of the star are assumed to be |$1.2 \, {\rm M}_{\odot }$| and |$1.2 \, {\rm R}_{\odot }$|⁠, respectively, where |${\rm M}_{\odot }$| and |${\rm R}_{\odot }$| are the mass and radius of the Sun, respectively.

To carry out the simulations, we used the Bulirsch–Stoer integrator of the integration package mercury (Chambers 1999). Each nominal system, formed by the star and the eight planets, is identified by its initial conditions. A nominal system will be removed from the numerical simulation set if there is an intersection between the orbits of the planets, a collision occurs between them, or if one planet has an orbital distance larger than 8 au. After the numerical simulations were completed, we applied the FMFT method to calculate the main frequencies of each planet in both consecutive time intervals. With both values, we obtain the diffusion parameter of the corresponding orbit in each nominal system according to equation (2).

We mapped the initial eccentricity versus |$\log D$| for three intervals of |$e$| classified as low (⁠|$0 \lt e \lt 10^{-3}$|⁠), medium (⁠|$10^{-3} \lt e \lt 10^{-2}$|⁠), and high (⁠|$ 10^{-2} \lt e \lt 10^{-1}$|⁠) intervals. Figs 1(a)–(i) show nine panels. The left column presents the low values of |$e$|⁠, the middle column presents the medium values of |$e$| (medium interval), and the right column presents the high values of |$e$| (high interval). The planets are represented by different colours and shapes. In the top row, the blue, red, and green squares indicate Kepler-90b, Kepler-90c, and Kepler-90i planets; in the middle row, the blue, red, and green triangles indicate Kepler-90d, Kepler-90e, and Kepler-90f planets; finally, in the bottom row, the green and red circles indicate Kepler-90g and Kepler-90h, respectively.

These figures show the values of |$\log D$| only for those planets that survived during the total numerical integration. As can be seen, all surviving planets have |$\log D \lt {-}5$|⁠.

In the first interval, the cluster of planets has |$-13 \le \log D \le -7$| (Figs 1a, d, and g). In this case, 100 per cent of systems are stable for initially low-eccentricity values between 0 and |$10^{-3}$|⁠. The orbit of the planets presented an average |$\log D$| of −9.9, −11.9, −10.7, −9.6, −8.7, −8.7, −9.8, and −11.1 for the eight planets in increasing distance from the star, respectively.

Similarly, we observed that in the medium-eccentricity interval, the planets have |$-13\le \log D\le -5.5$| (Figs 1b, e, and h). As in the low-eccentricity interval, 100 per cent of systems are stable. In this case, the average value of |$\log D$| is equal to −9.9, −11.9, −10.7, −9.6, −8.7, −8.7, −9.8, and −11.0 from Kepler-90b to Kepler-90h, respectively.

In contrast, for the high-eccentricity interval, initial eccentricities ranging from |$10^{-2}$| to |$10^{-1}$|⁠, only per cent of systems survived during the whole integration time. In this case, their values of |$\log D$| range from |$-10$| to |$-5$| (Figs 1c, f, and i). In 65 per cent of the systems, collisions were detected between the planets, while in 11 per cent, there were intersections between their orbits, and in 18 per cent of the systems, a planet was ejected. In most systems, Kepler-90f and Kepler-90e were ejected, the closest planets to the exterior gas giant planets.

We also verified the variation of the eccentricities of each planet in each interval. The value of |$\Delta e = e_{\rm max}-e_{\rm min}$|⁠, where |$e_{\rm max}$| and |$e_{\rm min}$| are the maximum and minimum values of eccentricity, respectively, was extracted from the time evolution of the eccentricity of each planet. Fig. 2 shows the minimum (⁠|$\Delta {e}_{\rm min}$|⁠), the maximum (⁠|$\Delta {e}_{\rm max}$|⁠), and the average |$\Delta {\overline{e}}$| values of the eccentricity that the planets reached in all nominal systems, in low (Fig. 2a) and medium (Fig. 2b) eccentricity intervals. In the low-eccentricity interval, Kepler-90g assumed a larger variation of |$e$|⁠, |${\Delta {e}}_{\rm min} = 0.018$|⁠, |${\Delta {e}}_{\rm max}= 0.028$|⁠, and |$\Delta {\overline{e}}= 0.024$| compared to other planets. The eccentricities of Kepler-90i, Kepler-90d, Kepler-90e, and Kepler-90f vary due to the effects of the massive planets, Kepler-90g and Kepler-90h, while the most massive planet, Kepler-90h, experiences only a small variation in |$e$|⁠. The same did not happen to Kepler-90b and Kepler-90c, since they are further away from the massive planets, they present a small variation in |$e$|⁠.

Similar to the results found in the low-eccentricity interval, in the medium-eccentricity interval, Kepler-90b, Kepler-90c, and Kepler-90h have a small variation of |$e$|⁠. This occurs because Kepler-90b and Kepler-90c are further from Kepler-90h, while the outer planet is the most massive of the system. Larger variations in |$e$| can be seen in Kepler-90i, Kepler-90d, Kepler-90e, and Kepler-90f. It is worth noting that although the maximum variations of |$e$| are large, most of them have |$\Delta {\overline{e}}$| smaller than Kepler-90g (see Fig. 2b). As shown in Figs 2(a) and (b), |$\Delta {\overline{e}}$| of Kepler-90h has the smallest variations.

It is important to highlight that Fig. 2(c), which shows the high-eccentricity interval (⁠|$10^{-2} \!-\! 10^{-1}$|⁠), was built with only the surviving systems, which represent 6 per cent of the total. All planets experienced variations in |$e$|⁠, except, as expected, Kepler-90h. These variations show that the minimum and maximum values are constant, close to the average variations observed in all three intervals (Fig. 2). Kepler-90d, Kepler-90e, and Kepler-90f experience a small difference between the maximum and minimum eccentricity values compared to the small and medium intervals of |$e$|⁠.

The minimum and maximum values of |$\log D$| (⁠|$\log D_{\rm min}$| and |$\log D_{\rm max}$|⁠, respectively) in each nominal system are shown in Figs 3(a) and (b), where the letters identify the values of |$\log D_{\rm min}$| and |$\log D_{\rm max}$| for each planet in each nominal system. The average value of |$\log D$| is represented by black circles.

Regarding the behaviour of the average |$\log D$|⁠, Figs 3(a) and (b) showed that the Kepler system analogues can be stable. In Fig. 3(a), it is observed that all the systems remained in the range of |$\log D$| from −11 to −8.8, being identified as stable systems. Nevertheless, in Fig. 3(b), the range of |$\log D$| varies from −10.5 to −7.2, but if we compared the |$\log D$| between the systems, we noted that the system numbered 79 has a |$\log D$| larger.

To verify how stable the nominal systems are, we numerically integrated for a longer time, |$10^5 \, T_{\rm h}$|⁠. The results indicated that all systems with low values of |$e$| (low-eccentricity interval) are stable, i.e. 100 per cent of the systems survived. In the medium-eccentricity interval, 99 per cent of the systems remained stable. As pointed out in Fig. 3(b), the system 79 is unstable, Kepler-90e collides with Kepler-90d in |$5 \times 10^{4} \, {\rm yr}$|⁠. In the high-eccentricity interval, no planets survived up to |$10^5 \, T_{\rm h}$|⁠.

The percentage of the largest (in purple) and the smallest (in green) values of |$\log D$| for each planet can be seen in Fig. 4 for |$10^4 \, T_{\rm h}$| (left side) and |$10^5 \, T_{\rm h}$| (right side). In the low-eccentricity interval (Fig. 4a), in 47 per cent of the analysed sets, Kepler-90e reached the highest values of |$\log D_{\rm max}$|⁠, while in 42 per cent Kepler-90f has the highest values of |$\log D_{\rm max}$|⁠, followed by Kepler-90d (with 10 per cent), Kepler-90i (with 1 per cent), and Kepler-90b (with 1 per cent) (Fig. 4a). This result is expected since these planets (Kepler-90d, Kepler-90e, and Kepler-90f) are close to the outer giant planets (Kepler-90g and Kepler-90h). We also verified that in 75 per cent of the systems, Kepler-90c presents the minimum values of |$\log D_{\rm mim}$|⁠, followed by Kepler-90h (with 13 per cent) and Kepler-90i (with 10 per cent) (Fig. 4a).

Similar results were found in the medium-eccentricity interval (Fig. 4b). In 58 per cent of the systems, Kepler-90f shows the highest value of |$\log D_{\rm max}$|⁠, while in 35 per cent of the systems, Kepler-90e has the highest value of |$\log D_{\rm max}$|⁠, followed by Kepler-90d (with 6 per cent) and Kepler-90i (with 1 per cent). On the other hand, Kepler-90c, Kepler-90h, and Kepler-90i present the lowest values of |$\log D_{\rm mim}$| in 83, 13, and 4 per cent of the systems, respectively. The results for a longer integration period (⁠|$10^5 \, T_{\rm h}$|⁠) are similar to results found for |$10^4 \, T_{\rm h}$|⁠. Most of the results are similar in both intervals of |$e$| and both periods of the numerical integration. For example, Kepler-90c has the smallest values of |$\log D_{\rm min}$|⁠, while Kepler-90f has the highest values of |$\log D_{\rm max}$|⁠, except in Fig. 4(a), although the difference between Kepler-90f and Kepler-90e is small.

3 ANALYSIS OF MEAN MOTION RESONANCES

From the numerical simulation results, we searched for MMRs between the pairs of planets of each Kepler-90 analogue in low and medium-eccentricity intervals. The initial conditions for each planet were the same as those described in Section 2. 100 Kepler-90 system analogues were analysed for each interval, which gives 200 Kepler-90 systems in total. The numerical simulations were carried out for |$1.2 \times 10^4 \, {\rm yr}$|⁠.

Several first-, second-, and third-order MMRs between the pairs of planets were searched, and also the Laplace resonance. We found that only the pairs of planets formed by Kepler-90b and Kepler-90c and Kepler-90g and Kepler-90h are in the 5:4 MMR and the 3:2 MMR or quasi-resonance, respectively. From now on we will analyse, in detail, the time variation of the critical angles associated with such MMRs. A range of different regimes will be described.

The time variation of the resonant angles for the two pairs of planets, planets Kepler-90b and Kepler-90c and planets Kepler-90g and Kepler-90h, is given by

where |$\lambda$| and |$\lambda ^{\prime }$| are the mean anomalies of the inner and outer planets, respectively, |$p$| is an integer, and |$\varpi$| and |$\varpi ^{\prime }$| are the longitude of the pericentre of the inner and outer planets, respectively.

If an MMR exists for the planetary pairs Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h, the corresponding resonant angle must librate according to equation (3) or (4). On the contrary, we will say that the system will be upon the effect of the resonance, a state that can be classified as quasi-resonance (see discussion in Callegari & Rodríguez 2023). In these cases, we define intermittent behaviour when the critical angle changes from circulation to oscillation or vice versa. The intermittence can also be classified as short (⁠|$s$|⁠), long (⁠|$l$|⁠), and short & long (⁠|$s \, \& \, l$|⁠), depending on the libration period of the resonant angle. It is important to point out that the classification in short and long periods does not have a range of particular periods. It is a qualitative and comparative analysis. The initial conditions of the planets are shown in Table 2.

We also highlight that the behaviour of the critical angle can also appear as a slow circulation or a large amplitude. Examples of this behaviour will be given below.

3.1 Low-eccentricity interval

Fig. 5 shows the time variation of the critical angles |$\phi ^{(1)}_{5{:}4}$| and |$\phi ^{(2)}_{5{:}4}$| involving the planets Kepler-90b–Kepler-90c, and |$\phi ^{(1)}_{3{:}2}$| and |$\phi ^{(2)}_{3{:}2}$| involving the planets Kepler-90g–Kepler-90h. The initial conditions of the planets (which corresponds to Kepler-90 system analogue number 88) are presented in Table 2 for the low-eccentricity interval. In Figs 5(a) and (b), the critical angle |$\phi ^{(1)}_{5{:}4}$| is librating around |$180^{\circ }$|⁠, while in Figs 5(c) and (d), |$\phi ^{(2)}_{5{:}4}$| is librating around 0. The plots in the left and right columns show two time-scales to better visualize the behaviour of the critical angles, 180 and 12 000 yr, respectively. Figs 5(e)–(h) show that planets Kepler-90g and Kepler-90h are in quasi-resonance of the 3:2 MMR, which we classified as ‘large amplitude’, since there are few points located between 0° and |$360^{\circ }$| (out of the libration zone). Most of the time |$\phi ^{(1)}_{3{:}2}$| is librating around 0°, while |$\phi ^{(2)}_{3{:}2}$| is librating around |$180^{\circ }$|⁠.

All the identified regimes were summarized in Table 3. In 2 per cent of the systems, the resonant angle |$\phi ^{(1)}_{5{:}4}$| librates around 180° with an amplitude equal to |$70^\circ$|⁠. In 42 per cent of the systems, |$\phi ^{(1)}_{5{:}4}$| has an intermittent behaviour as described as short (⁠|$s$|⁠), long (⁠|$l$|⁠), and |$s$| and |$l$|⁠. In 30 and 26 per cent of the systems, |$\phi ^{(1)}_{5{:}4}$| presents a slow circulation and large amplitude, respectively. Examples of |$s$|⁠, |$l$|⁠, |$s \ {\rm and} \ l$| regimes and slow circulation can be seen in Fig. 6.

The critical angle |$\phi ^{(2)}_{5{:}4}$| for the pair Kepler-90b–Kepler-90c is in quasi-resonance in 93 per cent of the systems, while in 7 per cent, the resonant angle |$\phi ^{(2)}_{5{:}4}$| librates around zero (Figs 5c and d). In 45, 2, and 24 per cent of the systems, the critical angle can be classified as |$s$|⁠, |$l$|⁠, and |$s$| and |$l$|⁠, respectively. We identify that in 9 and 13 per cent of systems, the angle |$\phi ^{(2)}_{5{:}4}$| showed slow-circulation and large-amplitude regimes, respectively.

Regarding the critical angle |$\phi ^{(1)}_{3{:}2}$|⁠, we identify that in 10 per cent of the systems, this angle librates around 180°, and in 90 per cent of the cases, the angles are in quasi-resonance, as shown in Figs 5(e) and (f). In 9 and 11 per cent of the systems, the critical angle presents an intermittent behaviour for short and long time-scales, respectively. |$\phi ^{(1)}_{3{:}2}$| presents large-amplitude and slow-circulation regimes in 62 and 8 per cent of the systems, respectively. The same analysis was carried out for |$\phi ^{(2)}_{3{:}2}$|⁠. Table 3 presents all the different regimes identified.

An important result derived from this set of numerical simulations is that we do not identify the four resonant angles (⁠|$\phi ^{(1,2)}_{5{:}4}$| and |$\phi ^{(1,2)}_{3{:}2}$|⁠) librating in the same Kepler-90 system, in each system only one pair of planets is in MMR. For example, the system illustrated in Fig. 5, |$\phi ^{(1,2)}_{5{:}4}$| is librating while |$\phi ^{(1,2)}_{3{:}2}$| is in quasi-resonance.

3.2 Medium-eccentricity interval

The same analysis was performed for the 100 Kepler-90 analogues in the medium-eccentricity interval. Fig. 6 shows the time variation of the critical angles |$\phi ^{(1)}_{5:4}$|⁠, |$\phi ^{(2)}_{5:4}$|⁠, |$\phi ^{(1)}_{3:2}$|⁠, and |$\phi ^{(2)}_{3:2}$| for the Kepler-90 system analogue number 5.

Fig. 6(a) shows |$\phi ^{(1)}_{5:4}$| in the slow circulation regime during 180 yr, but in Fig. 6(b), during 12 000 yr, the angle showed an intermittent behaviour (we classified as ‘long’). It is important to remark that some critical angles show slow circulation for the whole period of integration, the percentage of systems in this regime are shown in Tables 3 and 4. In Fig. 6(c), the critical angle |$\phi ^{(2)}_{5:4}$| also shows an intermittent behaviour (we classified as ‘short’) librating for about 40 yr, while in Fig. 6(d), after a longer period, this same angle shows a libration period of about 2000 yr. Therefore, we classified it as ‘short and long’. Figs 6(e) and (f) show the critical angles |$\phi ^{(1)}_{3:2}$| and |$\phi ^{(2)}_{3:2}$| in the ‘large-amplitude’ regime.

Table 4 quantifies all the regimes found in 100 Kepler-90 systems. In 16 and 13 per cent the angles |$\phi _{5:4}^{(1)}$| and |$\phi _{5:4}^{(2)}$| circulate, respectively. We verified that the percentages of the systems where the planets are in quasi-resonance are in 72 and 62 per cent the angles |$\phi _{5:4}^{(1)}$| and |$\phi _{5:4}^{(2)}$| show slow circulation, respectively, while |$\phi _{5:4}^{(1)}$| presents an intermittent behaviour for short and long time-scales in 11 and 1 per cent of the systems, respectively. In 23 and 2 per cent of the systems, |$\phi _{5:4}^{(2)}$| presents an intermittent behaviour in short and short & long time-scales, respectively. We did not identify any critical angle in a large-amplitude regime. It is important to point out that for the medium-eccentricity interval, in none of the Kepler-90 systems, the pair Kepler-90b–Kepler-90c is in 5:4 MMR.

For the 3:2 MMR between the pair Kepler-90g–Kepler-90h, in 9 per cent of systems |$\phi _{3:2}^{(1)}$| librates around zero; in 59 per cent of the systems, |$\phi _{3:2}^{(1)}$| shows an intermittent behaviour; in 56 and 3 per cent, the angle shows intermittence in short and long time-scales, respectively. We also identify that in 22 and 10 per cent of the systems, the angles are in slow-circulation and large-amplitude regimes, respectively. According to our results summarized in Table 4, |$\phi _{3:2}^{(2)}$| presents the following regimes: in 6 per cent of the systems, the angle librates around |$180^{\circ }$|⁠, and in 2 per cent, it circulates. |$\phi _{3:2}^{(2)}$| shows an intermittent behaviour in 58 per cent of the systems (57 per cent in short and 1 per cent in long periods). Large amplitude and slow circulation in 5 and 29 per cent of the systems, respectively.

In summary, Tables 3 and 4 give a better picture of the different regimes observed for the critical angles of the pairs Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h in the two eccentricity intervals.

3.3 Orbital evolution of the planetary pairs Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h

According to the results shown in the last subsections, we identify two planetary pairs, Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h, in resonance or in quasi-resonance effects of the 5:4 and 3:2 MMRs, respectively. This behaviour occurs in both intervals, low and medium values of the eccentricity.

In the first interval, Fig. 7 shows the evolution of the semimajor axis (Figs 7a and b) and eccentricity (Figs 7c and d) for the pairs Kepler-90b–Kepler-90c and Kepler-90g–Kepler-90h, respectively. The four eccentricities, with initial values of about |$10^{-4}$|⁠, varied by one and two orders of magnitude. This system is the same system analysed in Fig. 5 (Kepler-90 analogue number 88).

Figs 7(e)–(h) were already presented in Figs 5(b), (d), (f), and (h), showing the evolution of |$\phi ^{(1,2)}_{5:4}$| and |$\phi ^{(1,2)}_{3:2}$|⁠, respectively.

Figs 7(i) and (j) illustrate the time evolution of |$\Delta \varpi _{\rm bc}$| and |$\Delta \varpi _{\rm gh}$|⁠, respectively. |$\Delta \varpi _{\rm bc}$| measures the angular displacement of the longitudes of the pericentre. If the oscillation of |$\Delta \varpi$| is around 0, it means that the orbits are approximately aligned, but in the case of an oscillation around |$180^\circ$|⁠, they can be identified as approximately anti-aligned.

Fig. 7(i) shows |$\Delta \varpi _{\rm bc}$| oscillating around |$180^\circ$| with small variations in the amplitude of the oscillation. It indicates that the orbits of the pair Kepler-90b–Kepler-90c are defined as orbits anti-aligned (Fig. 7i). Also, we observe that in two periods of time, the amplitude is smaller. In the first period, from 2000 to 4000 yr, and in the second period from 10 500 to 11 500 yr, the amplitudes are close to |${\sim} 45^\circ$| and |${\sim} 22^\circ$|⁠, respectively, trending to an exact 5:4 MMR.

The evolution of |$\Delta \varpi _{\rm gh}$| (Fig. 7j) shows that the pair Kepler-90g–Kepler-90h is in quasi-resonance. Most of the time |$\Delta \varpi _{\rm gh}$| is librating around |$180^{\circ }$| with small amplitude, although in some periods, the pair Kepler-90g–Kepler-90h escapes from the resonance.

Figs 7(k) and (m) also help to illustrate that Kepler-90b–Kepler-90c is in the 5:4 MMR, while Figs 7(l) and (n) show that Kepler-90g–Kepler-90h is in quasi-resonance, since some points encircle the origin of the system.

The same analysis was carried out for the second eccentricity interval (Fig. 8). Figs 8(a) and (c) show the temporal evolution of |$a$| and |$e$| for the pair Kepler-90b–Kepler-90c, while Figs 8(b) and (d) show the temporal evolution of |$a$| and |$e$| for the pair Kepler-90g–Kepler-90h, respectively. As seen in the first interval, the four values of |$e$| also presented a small variation in their initial values. Figs 8(e)–(h) were already presented in Figs 6(b), (d), (f), and (h), showing the evolution of |$\phi ^{(1,2)}_{5:4}$| and |$\phi ^{(1,2)}_{3:2}$|⁠, respectively. The important results are presented in Figs 8(i) and (j) that show the oscillation of |$\Delta \varpi _{\rm bc}$| and |$\Delta \varpi _{\rm gh}$|⁠, respectively. In this system, Kepler-90b–Kepler-90c are in quasi-resonance, represented by an oscillation of |$\Delta \varpi _{\rm bc}$| around |$180^{\circ }$| in |${\sim} 4000$| and |${\sim} 11\,000$| yr. The plot of |$e_\mathrm{ c} \cos (\Delta \varpi)$| versus |$e_\mathrm{ c} \sin (\Delta \varpi)$| (Figs 8k–m) encircled the origin of the system.

On the other hand, the pair Kepler-90g–Kepler-90h is under the effect of the 3:2 MMR, clearly observed in Fig. 8(j), as we already classified the behaviour of the critical angle as a ‘large amplitude’. |$\Phi ^{(1,2)}_{3:2}$| oscillates around |$180^{\circ }$| most of the time, also visualized in Figs 8(l) and (n).

4 DISCUSSION AND CONCLUSION

Kepler-90 system presents a set of eight planets similar to our Solar system. It is a hierarchical system, the size of the majority of the planets increases as increases their distance to the star. The two close planets (Kepler-90b and Kepler-90c) have sizes similar to Earth, the other three planets (Kepler-90d, Kepler-90e, and Kepler-90f) are super-Earths, and the two outermost planets (Kepler-90g and Kepler-90h) are giant planets. The exception is Kepler-90i that is not so close to the star but has a size similar to our planet. It is a compact system, the outermost planet is located at 1.01 au.

Several papers analysed the dynamics through numerical simulations and secular theory. In general, their results show a stable system for a small value of the eccentricity of the planets. Although Contreras & Boley (2018) found that about 20 per cent of their systems are unstable despite all planets having initial circular orbits. Cabrera et al. (2014) analysed the ratio between the periods of Kepler-90b and Kepler-90c suggesting that this pair could be in a 5:4 MMR, and Kepler-90d, Kepler-90e, and Kepler-90f are near the Laplacian resonance. Contreras & Boley (2018) analysed the time variation of the resonant angles identifying that Kepler-90b and Kepler-90c are near the 5:4 MMR, while Kepler-90g and Kepler-90h are near the 3:2 MMR. These two angles librate with large amplitude around 0° and |$180^{\circ }$|⁠. However, the resonant angle of the three-body resonance circulates in all of their numerical simulations.

In this work, we study several Kepler-90 analogues to analyse, in detail, how the values of the eccentricity of each planet change the orbital evolution of the system and, consequently, its stability. This analysis was performed through the frequency analysis technique. The system was formed by the star and eight planets for three different intervals of eccentricity. Our results show that for the first interval (⁠|$0 \lt e \lt 10^{-3}$|⁠) the system is stable for |$10^5$| orbital periods of Kepler-90h (the exterior planet), while 99 per cent of the whole set of Kepler-90 analogues, in the medium values of |$e$| (⁠|$10^{-3} \lt e \lt 10^{-2}$|⁠), are also stable. No set of Kepler-90 system with large values of |$e$| (⁠|$10^{-2} \lt e \lt 10^{-1}$|⁠) survived up to |$10^5 \, T_{\rm h}$|⁠. As expected Kepler-90b and Kepler-90c do not suffer a significant influence of the outermost planets, they are closer to the star. The planets located at intermediate distances (Kepler-90i, Kepler-90d, and Kepler-90f) are perturbed by Kepler-90g and Kepler-90h, leading to an increase in their values of |$e$|⁠. None the less, this increase in |$e$| is not large enough to destabilize the system.

We investigated several first- and second-order MMRs, and a few third-order MMRs between the pairs of planets in all Kepler-90 analogues, by computing the evolution of the critical angle. Two pairs of MMRs were identified and analysed in detail: 5:4 MMR between Kepler-90b and Kepler-90c, and 3:2 MMR between Kepler-90g and Kepler-90h. We did not find the Laplace configuration nor 2:1 MMR between the Kepler-90d and Kepler-90f.

We classified the behaviour of the two critical angles when the pair of Kepler-90 planets is under the effects of the resonance (or in quasi-resonance): intermittent time interval (short, long, and short & long), large amplitude, and slow circulation. The intermittent interval classifies those critical angles that librate and oscillate in different periods, as we call short, long, and short & long. The large amplitude classification indicates that the critical angle is most of the time librating, although, in a few periods, the system escapes from the resonance. When the critical angle librates, the planets are in resonance, otherwise, if the critical angle circulates the planets are not in MMR. All these behaviours were illustrated in the last section.

In the first eccentricity interval, two pairs of planets, Kepler-90b–Kepler-90c, and Kepler-90g–Kepler-90h, were found to be in resonance in about 10 and 35 per cent of the systems, respectively. It is important to point out that the two pairs are not in MMR in the same Kepler-90 system, for both intervals, low and medium values of |$e$|⁠. Only 10 per cent of the pair Kepler-90b–Kepler-90c is in 5:4 MMR, while in the second eccentricity interval, no MMR was found between the pair Kepler-90b–Kepler-90c. In the second eccentricity interval in 15 per cent of Kepler-90 systems, the pair Kepler-90g–Kepler-90h is in 3:2 MMR, while the critical angle of the pair Kepler-90b–Kepler-90c undergoes intermittent behaviour.

To sum up our findings. (i) Kepler-90 planets are stable for values of |$e \lt 10^ {-2}$|⁠. As expected the nearest planets to the star are less disturbed by the giant planets, which are located in the exterior region of the system. Although there is a small variation in the values of |$e$| for those planets located between Kepler-90c and Kepler-90g, this variation is not strong enough to destabilize the system. (ii) The results concerning the stable systems statistically indicate that these resonances directly affect most of them. The influence covers a wide range of possibilities. It might be as a strong resonant mean motion coupling added to a coupling of the longitude of the pericentres, with two resonant angles librating, or just a single resonant argument librating. There are also cases of intermittent behaviour, interleaving libration, and circulation of the resonant arguments. In many cases, it occurs the quasi-resonant influence, when at least one of the critical arguments slowly circulates. However, in any of these cases, some effect occurs due to the near commensurability.

ACKNOWLEDGEMENTS

The authors deeply thank the anonymous referee for constructive review. This study was financed in part by the Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Finance Code 001), Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP, Proc. 2016/24561-0), and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Proc. 316991/2023-6 and Proc. 313043/2020-5).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References