Star-planet Tidal Interactions Research Articles

Rocky planet formation in compact disks around M dwarfs

Due to the improvements in radial velocity and transit techniques, we know that rocky or rocky-icy planets, in particular close-in super-Earths in compact configurations, are the most common ones around M dwarfs. On the other hand, thanks to the high angular resolution of ALMA we know that many disks around very low-mass stars (between 0.1 and 0.5 M$_ are rather compact and small (without observable substructures and radius less than 20 au), which favors the idea of an efficient radial drift that could enhance planet formation in the terrestrial zone. Our aim was to investigate the potential formation paths of the observed close-in rocky exoplanet population around M dwarfs, especially close-in super-Earths, assuming that planet formation could take place in compact disks with an efficient dust radial drift. We developed N-body simulations that include a sample of embryos growing by pebble accretion exposed to planet-disk interactions, star-planet tidal interactions, and general relativistic corrections that include the evolution of the luminosity, radius, and rotational period of the star. For a star of 0.1 M$_ we considered different gas disk viscosities and initial embryo distributions. We also explored planet formation by pebble accretion around stars of 0.3 M$_ and 0.5 M$_ Lastly, for each stellar mass, we ran simulations that include a sample of embryos growing by planetesimal accretion instead of pebble accretion. Our main result is that the sample of simulated planets that grow by pebble accretion in a gas disk with low viscosity ($ $) can reproduce the close-in low-mass exoplanet population around M dwarfs in terms of multiplicity, mass, and semi-major axis. Furthermore, we found that a gas disk with high viscosity ($ $), and thus lower pebble accretion rates, cannot reproduce the observed planet masses as no planet more massive than 0.5 M$_ could be formed in our simulations. In addition, we show that planetesimal accretion favors the formation of smaller planets than pebble accretion does. Whether this planet population truly exists remains unknown with the current instrumental sensitivity. Rocky planet formation around M dwarfs can take place in compact and small dust disks driven by an efficient radial drift in a gas disk with low viscosity ($ $). This result points toward a new approach in the direction of the disk conditions needed for rocky planet formation around very low-mass stars.

TOI-1268b: The youngest hot Saturn-mass transiting exoplanet

We report the discovery of TOI-1268b, a transiting Saturn-mass planet from the TESS space mission. With an age of less than 1 Gyr, derived from various age indicators, TOI-1268b is the youngest Saturn-mass planet known to date; it contributes to the small sample of well-characterised young planets. It has an orbital period of P = 8.1577080 ± 0.0000044 days, and transits an early K-dwarf star with a mass of M* = 0.96 ± 0.04 M⊙, a radius of R* = 0.92 ± 0.06 R⊙, an effective temperature of Teff = 5300 ± 100 K, and a metallicity of 0.36 ± 0.06 dex. By combining TESS photometry with high-resolution spectra acquired with the Tull spectrograph at the McDonald Observatory, and the high-resolution spectrographs at the Tautenburg and Ondřejov Observatories, we measured a planetary mass of Mp = 96.4 ± 8.3 M⊕ and a radius of Rp = 9.1 ± 0.6 R⊕. TOI-1268 is an ideal system for studying the role of star-planet tidal interactions for non-inflated Saturn-mass planets. We used system parameters derived in this paper to constrain the planet’s tidal quality factor to the range of 104.5–5.3. When compared with the sample of other non-inflated Saturn-mass planets, TOI-1268b is one of the best candidates for transmission spectroscopy studies.

Transit Timing Variation of XO-3b: Evidence for Tidal Evolution of Hot Jupiter with High Eccentricity

Observed transit timing variation (TTV) potentially reveals the period decay caused by star-planet tidal interaction which can explain the orbital migration of hot Jupiters. We report the TTV of XO-3b, using TESS observed timings and archival timings. We generate a photometric pipeline to produce light curves from raw TESS images and find the difference between our pipeline and TESS PDC is negligible for timing analysis. TESS timing presents a shift of 17.6 minutes (80σ), earlier than the prediction from the previous ephemeris. The best linear fit for all timings available gives a Bayesian Information Criterion (BIC) value of 439. A quadratic function is a better model with a BIC of 56. The period derivative obtained from a quadratic function is −6.2 × 10−9 ± 2.9 × 10−10 per orbit, indicating an orbital decay timescale 1.4 Myr. We find that the orbital period decay can be well explained by tidal interaction. The “modified tidal quality factor” would be 1.8 × 104 ± 8 × 102 if we assume the decay is due to the tide in the planet; whereas would be 1.5 × 105 ± 6 × 103 if tidal dissipation is predominantly in the star. The precession model is another possible origin to explain the observed TTVs. We note that the follow-up observations of occultation timing and radial velocity monitoring are needed for fully discriminating the different models.

SOLES I: The Spin–Orbit Alignment of K2-140 b

Abstract Obliquity measurements for stars hosting relatively long-period giant planets with weak star-planet tidal interactions may play a key role in distinguishing between formation theories for shorter-period hot Jupiters. Few such obliquity measurements have been made to date due to the relatively small sample of known wide-orbiting, transiting Jovian-mass planets and the challenging nature of these targets, which tend to have long transit durations and orbit faint stars. We report a measurement of the Rossiter–McLaughlin effect across the transit of K2-140 b, a Jupiter-mass planet with period P = 6.57 days orbiting a V = 12.6 star. We find that K2-140 is an aligned system with projected spin–orbit angle λ = 0.5° ± 9.7°, suggesting a dynamically cool formation history. This observation builds toward a population of tidally detached giant planet spin–orbit angles that will enable a direct comparison with the distribution of close-orbiting hot-Jupiter orbital configurations, elucidating the prevalent formation mechanisms of each group.

TATOO: Tidal-chronology standalone tool to estimate the age of massive close-in planetary systems

Context. The presence of a massive close-in planet with an orbital period of a few days or less around a low-mass star can possibly result in a strong variation in the properties of the central star. Indeed, star-planet tidal interactions generate exchanges of angular momentum that can result in tidal spin-up. This effect could then lead to gyrochronological ages biased towards younger ages. Aims. This article provides the community with TATOO, a standalone tool based on tidal-chronology, with which to estimate the age of a massive close-in planetary system using only its observed properties: mass of the planet and the star, stellar rotation, and planetary orbital periods. Methods. I used a star-planet tidal evolution numerical code to create a large multi-parametric grid of the evolution of synthetic star-planet systems. Furthermore, using the tidal-chronology technique, I employed a 3D interpolation method to provide a fairly precise age estimate of any given planetary system composed of one massive close-in planet. Results. About half of the planetary systems investigated in this work are subject to tidal spin-up bias. I pointed out that this bias linearly scales with the ratio between rotation and orbital period, making this quantity a useful proxy to rapidly investigate whether tidal-chronology needs to be used. Moreover, while being model dependent, TATOO can also be used even if no rotational departure is present. In that case, it gives results in agreement with the classical gyrochronological analysis. Conclusions. TATOO is a useful tool specifically designed for massive close-in planetary systems that can also be used as a classical gyrochronological tool. For now it is the only publicly available software to estimate the age of massive close-in planetary systems subject to tidal spin-up. In that sense, tidal-chronology can be seen as a first order correction of the impact of tidal interaction on gyrochronology.

NGTS-10b: the shortest period hot Jupiter yet discovered

ABSTRACT We report the discovery of a new ultrashort period (USP) transiting hot Jupiter from the Next Generation Transit Survey (NGTS). NGTS-10b has a mass and radius of $2.162\, ^{+0.092}_{-0.107}$ MJ and $1.205\, ^{+0.117}_{-0.083}$ RJ and orbits its host star with a period of 0.7668944 ± 0.0000003 d, making it the shortest period hot Jupiter yet discovered. The host is a 10.4 ± 2.5 Gyr old K5V star (Teff = 4400 ± 100 K) of Solar metallicity ([Fe/H] = −0.02 ± 0.12 dex) showing moderate signs of stellar activity. NGTS-10b joins a short list of USP Jupiters that are prime candidates for the study of star–planet tidal interactions. NGTS-10b orbits its host at just 1.46 ± 0.18 Roche radii, and we calculate a median remaining inspiral time of 38 Myr and a potentially measurable orbital period decay of 7 s over the coming decade, assuming a stellar tidal quality factor $Q^{\prime }_{\rm s}$ =2 × 107.

Does magnetic field impact tidal dynamics inside the convective zone of low-mass stars along their evolution?

Context. The dissipation of the kinetic energy of wave-like tidal flows within the convective envelope of low-mass stars is one of the key physical mechanisms that shapes the orbital and rotational dynamics of short-period exoplanetary systems. Although low-mass stars are magnetically active objects, the question of how the star’s magnetic field impacts large-scale tidal flows and the excitation, propagation and dissipation of tidal waves still remains open. Aims. Our goal is to investigate the impact of stellar magnetism on the forcing of tidal waves, and their propagation and dissipation in the convective envelope of low-mass stars as they evolve. Methods. We have estimated the amplitude of the magnetic contribution to the forcing and dissipation of tidally induced magneto-inertial waves throughout the structural and rotational evolution of low-mass stars (from M to F-type). For this purpose, we have used detailed grids of rotating stellar models computed with the stellar evolution code STAREVOL. The amplitude of dynamo-generated magnetic fields is estimated via physical scaling laws at the base and the top of the convective envelope. Results. We find that the large-scale magnetic field of the star has little influence on the excitation of tidal waves in the case of nearly-circular orbits and coplanar hot-Jupiter planetary systems, but that it has a major impact on the way waves are dissipated. Our results therefore indicate that a full magneto-hydrodynamical treatment of the propagation and dissipation of tidal waves is needed to properly assess the impact of star-planet tidal interactions throughout the evolutionary history of low-mass stars hosting short-period massive planets.

Star-planet tidal interaction and the limits of gyrochronology

Context. Age estimation techniques such as gyrochronology and magnetochronology cannot be applied to stars that have exchanged angular momentum with their close environments. This is especially true for a massive close-in planetary companion (with a period of a few days or less) that could have been strongly impacted by the rotational evolution of the host star, throughout the stellar evolution, through the star-planet tidal interaction. Aims. In this article, we provide the community with a reliable region in which empirical techniques such as gyrochronology can be used with confidence. Methods. We combined a stellar angular momentum evolution code with a planetary orbital evolution code to study in detail the impact of star-planet tidal interaction on the evolution of the surface rotation rate of the star. Results. We show that the interaction of a close-in massive planet with its host star can strongly modify the surface rotation rate of this latter, in most of the cases associated with a planetary engulfment. A modification of the surface rotation period of more than 90% can survive a few hundred Myr after the event and a modification of 10% can last for a few Gyr. In such cases, a gyrochronology analysis of the star would incorrectly make it appear as rejuvenated, thus preventing us from using this method with confidence. To try overcome this issue, we proposed the proof of concept of a new age determination technique that we call the tidal-chronology method, which is based on the observed pair Prot, ⋆–Porb of a given star-planet system, where Prot, ⋆ is the stellar surface rotational period and Porb the planetary orbital period. Conclusions. The gyrochronology technique can only be applied to isolated stars or star-planet systems outside a specific range of Prot, ⋆–Porb. This region tends to expand for increasing stellar and planetary mass. In that forbidden region, or if any planetary engulfment is suspected, gyrochronology should be used with extreme caution, while tidal-chronology could be considered. This technique does not provide a precise age for the system yet; however, it is already an extension of gyrochronology and could be helpful to determine a more precise range of possible ages for planetary systems composed of a star between 0.3 and 1.2 M⊙ and a planet more massive than 1 Mjup initially located at a few hundredths of au from the host star.

The impact of magnetism on tidal dynamics in the convective envelope of low-mass stars

AbstractFor the shortest period exoplanets, star-planet tidal interactions are likely to have played a major role in the ultimate orbital evolution of the planets and on the spin evolution of the host stars. Although low-mass stars are magnetically active objects, the question of how the star’s magnetic field impacts the excitation, propagation and dissipation of tidal waves remains open. We have derived the magnetic contribution to the tidal interaction and estimated its amplitude throughout the structural and rotational evolution of low-mass stars (from K to F-type). We find that the star’s magnetic field has little influence on the excitation of tidal waves in nearly circular and coplanar Hot-Jupiter systems, but that it has a major impact on the way waves are dissipated.

Asteroseismic Determination of the Stellar Rotation Period of the Kepler Transiting Planetary Systems and its Implications for the Spin–Orbit Architecture

Abstract We measure the rotation periods of 19 stars in the Kepler transiting planetary systems, P rot,astero from asteroseismology and P rot,phot from the photometric variation of their light curves. Two stars exhibit two clear peaks in the Lomb–Scargle periodogram, neither of which agrees with the seismic rotation period. Other four systems do not show any clear peak, whose stellar rotation period is impossible to estimate reliably from the photometric variation; their stellar equators may be significantly inclined with respect to the planetary orbital plane. For the remaining 13 systems, P rot,astero and agree within 30%. Interestingly, 3 out of the 13 systems are in the spin–orbit resonant state in which with P orb,b being the orbital period of the innermost planet of each system. The corresponding chance probability is (0.2–4.7)% based on the photometric rotation period data for 464 Kepler transiting planetary systems. While further analysis of stars with reliable rotation periods is required to examine the statistical significance, the spin–orbit resonance between the star and planets, if confirmed, has important implications for the star–planet tidal interaction, in addition to the origin of the spin–orbit (mis-)alignment of transiting planetary systems.

Suppressed Far-UV Stellar Activity and Low Planetary Mass Loss in the WASP-18 System*

Abstract WASP-18 hosts a massive, very close-in Jupiter-like planet. Despite its young age (<1 Gyr), the star presents an anomalously low stellar activity level: the measured activity parameter lies slightly below the basal level; there is no significant time-variability in the value; there is no detection of the star in the X-rays. We present results of far-UV observations of WASP-18 obtained with COS on board of Hubble Space Telescope aimed at explaining this anomaly. From the star’s spectral energy distribution, we infer the extinction ( ≈ 0.01 mag) and then the interstellar medium (ISM) column density for a number of ions, concluding that ISM absorption is not the origin of the anomaly. We measure the flux of the four stellar emission features detected in the COS spectrum (C ii, C iii, C iv, Si iv). Comparing the C ii/C iv flux ratio measured for WASP-18 with that derived from spectra of nearby stars with known age, we see that the far-UV spectrum of WASP-18 resembles that of old (>5 Gyr), inactive stars, in stark contrast with its young age. We conclude that WASP-18 has an intrinsically low activity level, possibly caused by star–planet tidal interaction, as suggested by previous studies. Re-scaling the solar irradiance reference spectrum to match the flux of the Si iv line, yields an XUV integrated flux at the planet orbit of 10.2 erg s−1 cm−2. We employ the rescaled XUV solar fluxes to models of the planetary upper atmosphere, deriving an extremely low thermal mass-loss rate of 10−20 M J Gyr−1. For such high-mass planets, thermal escape is not energy limited, but driven by Jeans escape.

Detection of a westward hotspot offset in the atmosphere of hot gas giant CoRoT-2b

Short-period planets exhibit day-night temperature contrasts of hundreds to thousands of degrees K. They also exhibit eastward hotspot offsets whereby the hottest region on the planet is east of the substellar point; this has been widely interpreted as advection of heat due to eastward winds. We present thermal phase observations of the hot Jupiter CoRoT-2b obtained with the IRAC instrument on the Spitzer Space Telescope. These measurements show the most robust detection to date of a westward hotspot offset of 23 $\pm$ 4 degrees, in contrast with the nine other planets with equivalent measurements. The peculiar infrared flux map of CoRoT-2b may result from westward winds due to non-synchronous rotation magnetic effects, or partial cloud coverage, that obscures the emergent flux from the planet's eastern hemisphere. Non-synchronous rotation and magnetic effects may also explain the planet's anomalously large radius. On the other hand, partial cloud coverage could explain the featureless dayside emission spectrum of the planet. If CoRoT-2b is not tidally locked, then it means that our understanding of star-planet tidal interaction is incomplete. If the westward offset is due to magnetic effects, our result represents an opportunity to study an exoplanet's magnetic field. If it has Eastern clouds, then it means that our understanding of large-scale circulation on tidally locked planets is incomplete.

Spin–orbit alignment for KELT-7b and HAT-P-56b via Doppler tomography with TRES

We present Doppler tomographic analyses for the spectroscopic transits of KELT-7b and HAT-P-56b, two hot-Jupiters orbiting rapidly rotating F-dwarf host stars. These include analyses of archival TRES observations for KELT-7b, and a new TRES transit observation of HAT-P-56b. We report spin-orbit aligned geometries for KELT-7b (2.7 +/- 0.6 deg) and HAT-P-56b (8 +/- 2 deg). The host stars KELT-7 and HAT-P-56 are among some of the most rapidly rotating planet-hosting stars known. We examine the tidal re-alignment model for the evolution of the spin-orbit angle in the context of the spin rates of these stars. We find no evidence that the rotation rates of KELT-7 and HAT-P-56 have been modified by star-planet tidal interactions, suggesting that the spin-orbit angle of systems around these hot stars may represent their primordial configuration. In fact, KELT-7 and HAT-P-56 are two of three systems in super-synchronous, spin-orbit aligned states, where the rotation periods of the host stars are faster than the orbital periods of the planets.

Tidal inertial waves in differentially rotating convective envelopes of low-mass stars

Star-planet tidal interactions may result in the excitation of inertial waves in the convective region of stars. In low-mass stars, their dissipation plays a prominent role in the long-term orbital evolution of short-period planets. Turbulent convection can sustain differential rotation in their envelope, with an equatorial acceleration (as in the Sun) or deceleration, which can modify the waves' propagation properties. We explore in this first paper the general propagation properties of free linear inertial waves in a differentially rotating homogeneous fluid inside a spherical shell. We assume that the angular velocity background flow depends on the latitudinal coordinate only, close to what is expected in the external convective envelope of low-mass stars. We use i) an analytical approach in the inviscid case to get the dispersion relation, from which we compute the characteristic trajectories along which energy propagates. This allows us to study the existence of attractor cycles and infer the different families of inertial modes; ii) high-resolution numerical calculations based on a spectral method for the viscous problem. We find that modes that propagate in the whole shell (D modes) behave the same way as with solid-body rotation. However, another family of inertial modes exists (DT modes), which can propagate only in a restricted part of the convective zone. Our study shows that they are less common than D modes and that the characteristic rays and shear layers often focus towards a wedge - or point-like attractor. More importantly, we find that for non-axisymmetric oscillation modes, shear layers may cross a corotation resonance with a local accumulation of kinetic energy. Their damping rate scales very differently from what we obtain for standard D modes and we show an example where it is independent of viscosity (Ekman number) in the astrophysical regime in which it is small.

DISK-PLANETS INTERACTIONS AND THE DIVERSITY OF PERIOD RATIOS INKEPLER'S MULTI-PLANETARY SYSTEMS

The Kepler mission is dramatically increasing the number of planets known in multi-planetary systems. Many adjacent planets have orbital period ratios near resonant values, with a tendency to be larger than required for exact first-order mean-motion resonances. This intriguing feature has been shown to be a natural outcome of orbital circularization of resonant planetary pairs due to star-planet tidal interactions. However, this feature holds in multi-planetary systems with periods longer than ten days, for which tidal circularization is unlikely to provide efficient divergent evolution of the planets orbits. Gravitational interactions between planets and their parent protoplanetary disk may instead provide efficient divergent evolution. For a planet pair embedded in a disk, we show that interactions between a planet and the wake of its companion can reverse convergent migration, and significantly increase the period ratio from a near-resonant value. Divergent evolution due to wake-planet interactions is particularly efficient when at least one of the planets opens a partial gap around its orbit. This mechanism could help account for the diversity of period ratios in Kepler's multiple systems comprising super-Earth to sub-jovian planets with periods greater than about ten days. Diversity is also expected for planet pairs massive enough to merge their gap. The efficiency of wake-planet interactions is then much reduced, but convergent migration may stall with a variety of period ratios depending on the density structure in the common gap. This is illustrated for the Kepler-46 system, for which we reproduce the period ratio of Kepler-46b and c.

The GAPS programme with HARPS-N at TNG

A long-term multi-purpose observational programme has started with HARPS-N@TNG aimed to characterise the global architectural properties of exoplanetary systems. In this first paper we fully characterise the transiting system Qatar-1. We exploit HARPS-N high-precision radial velocity measurements obtained during a transit to measure the Rossiter-McLaughlin effect in the Qatar-1 system, and out-of-transit measurements to redetermine the spectroscopic orbit. New photometric transit light-curves are analysed and a spectroscopic characterisation of the host star atmospheric parameters is performed based on various methods (line equivalent width ratios, spectral synthesis, spectral energy distribution). We achieved a significant improvement in the accuracy of the orbital parameters and derived the spin-orbit alignment of the system; this information, combined with the spectroscopic determination of the host star properties, allows us to derive the fundamental physical parameters for star and planet (masses and radii). The orbital solution for the Qatar-1 system is consistent with a circular orbit and the system presents a sky-projected obliquity of lambda = -8.4+-7.1 deg. The planet, with a mass of 1.33+-0.05 M_J, is found to be significantly more massive than previously reported. The host star is confirmed to be metal-rich ([Fe/H]= 0.20+-0.10) and slowly rotating (vsinI = 1.7+-0.3 km/s), though moderately active, as indicated by strong chromospheric emission in the Ca II H&K line cores (logR'_HK about -4.60). The system is well aligned and fits well within the general lambda vs Teff trend. We definitely rule out any significant orbital eccentricity. The evolutionary status of the system is inferred based on gyrochronology, and the present orbital configuration and timescale for orbital decay are discussed in terms of star-planet tidal interactions.

OBLIQUITIES OF HOT JUPITER HOST STARS: EVIDENCE FOR TIDAL INTERACTIONS AND PRIMORDIAL MISALIGNMENTS

We provide evidence that the obliquities of stars with close-in giant planets were initially nearly random, and that the low obliquities that are often observed are a consequence of star-planet tidal interactions. The evidence is based on 14 new measurements of the Rossiter-McLaughlin effect (for the systems HAT-P-6, HAT-P-7, HAT-P-16, HAT-P-24, HAT-P-32, HAT-P-34, WASP-12, WASP-16, WASP-18, WASP-19, WASP-26, WASP-31, Gl 436, and Kepler-8), as well as a critical review of previous observations. The low-obliquity (well-aligned) systems are those for which the expected tidal timescale is short, and likewise the high-obliquity (misaligned and retrograde) systems are those for which the expected timescale is long. At face value, this finding indicates that the origin of hot Jupiters involves dynamical interactions like planet-planet interactions or the Kozai effect that tilt their orbits, rather than inspiraling due to interaction with a protoplanetary disk. We discuss the status of this hypothesis and the observations that are needed for a more definitive conclusion.

Effect of the stellar spin history on the tidal evolution of close-in planets

We investigate how the evolution of the stellar spin rate affects, and is affected by, planets in close orbits, via star-planet tidal interactions. To do this, we used a standard equilibrium tidal model to compute the orbital evolution of single planets orbiting both Sun-like stars and 0.1 M\odot M-dwarfs. We tested two stellar spin evolution profiles, one with fast initial rotation (P=1.2 day) and one with slow initial rotation (P=8 day). We tested the effect of varying the stellar and planetary dissipation and the planet's mass and initial orbital radius. Conclusions: Tidal evolution allows to differentiate the early behaviors of extremely close-in planets orbiting either a rapidly rotating star or a slowly rotating star. The early spin-up of the star allows the close-in planets around fast rotators to survive the early evolution. For planets around M-dwarfs, surviving the early evolution means surviving on Gyr timescales whereas for Sun-like stars the spin-down brings about late mergers of Jupiter planets. In light of this study, we can say that differentiating between one spin evolution from another given the present position of planets can be very tricky. Unless we can observe some markers of former evolution it is nearly impossible to distinguish the two very different spin profiles, let alone intermediate spin profiles. Though some conclusions can still be drawn from statistical distributions of planets around fully convective M-dwarfs. However, if the tidal evolution brings about a merger late in its history it can also entail a noticeable acceleration of the star in late ages, so that it is possible to have old stars that spin rapidly. This raises the question of better constraining the age of stars.

Angular momentum exchange during secular migration of two-planet systems

We investigate the secular dynamics of two-planet coplanar systems evolving under mutual gravitational interactions and dissipative forces. We consider two mechanisms responsible for the planetary migration: star-planet (or planet-satellite) tidal interactions and interactions of a planet with a gaseous disc. We show that each migration mechanism is characterized by a specific law of orbital angular momentum exchange. Calculating stationary solutions of the conservative secular problem and taking into account the orbital angular momentum leakage, we trace the evolutionary routes followed by the planet pairs during the migration process. This procedure allows us to recover the dynamical history of two-planet systems and constrain parameters of the involved physical processes.

Tidal decay and orbital circularization in close-in two-planet systems

The motion of two planets around a Sun-like star under the combined effects of mutual interaction and tidal dissipation is investigated. The secular behaviour of the system is analyzed using two different approaches. First, we solve the exact equations of motion through the numerical simulation of the system evolution. In addition to the orbital decay and circularization, we show that the final configuration of the system is affected by the shrink of the inner orbit. Our second approach consist in the analysis of the stationary solutions of mean equations of motion based on a Hamiltonian formalism. We consider the case of a hot super-Earth planet with a more massive outer companion. As a real example, the CoRoT-7 system is analyzed solving the exact and mean equations of motion. The star-planet tidal interaction produces orbital decay and circularization of the orbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity of CoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in the space of eccentricities can be constructed through the computation of stationary solutions of mean equations including dissipation. The application to CoRoT-7 system shows that the stationary curve agrees with the result of numerical simulations of exact equations. A similar investigation performed in a super-Earth-Jupiter two-planet system shows that the doubly circular state is accelerated when there is a significant orbital migration of the inner planet, in comparison with previous results were migration is neglected.