Issue 
A&A
Volume 635, March 2020



Article Number  A117  
Number of page(s)  7  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/202037546  
Published online  23 March 2020 
Impact of tides on the transittiming fits to the TRAPPIST1 system
^{1}
Observatoire de Genève, Université de Genève,
51 chemin des Maillettes,
C1290
Sauverny, Switzerland
email: emeline.bolmont@unige.ch
^{2}
Physikalisches Institut, Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
^{3}
HarvardSmithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
^{4}
Astronomy Department, University of Washington, Seattle, WA 98195, USA
^{5}
Laboratoire d’astrophysique de Bordeaux, Université de Bordeaux, CNRS, B18N, allée Geoffroy SaintHilaire, 33615 Pessac, France
Received:
21
January
2020
Accepted:
5
February
2020
Transit timing variations (TTVs) can be a very efficient way of constraining masses and eccentricities of multiplanet systems. Recent measurements of the TTVs of TRAPPIST1 have led to an estimate of the masses of the planets, enabling an estimate of their densities and their water content. A recent TTV analysis using data obtained in the past two years yields a 34 and 13% increase in mass for TRAPPIST1b and c, respectively. In most studies to date, a Newtonian Nbody model is used to fit the masses of the planets, while sometimes general relativity is accounted for. Using the Posidonius Nbody code, in this paper we show that in the case of the TRAPPIST1 system, nonNewtonian effects might also be relevant to correctly model the dynamics of the system and the resulting TTVs. In particular, using standard values of the tidal Love number k_{2} (accounting for the tidal deformation) and the fluid Love number k_{2f} (accounting for the rotational flattening) leads to differences in the TTVs of TRAPPIST1b and c that are similar to the differences caused by general relativity. We also show that relaxing the values of tidal Love number k_{2} and the fluid Love number k_{2f} can lead to TTVs which differ by as much as a few 10 s on a 3−4yr timescale, which is a potentially observable level. The high values of the Love numbers needed to reach observable levels for the TTVs could be achieved for planets with a liquid ocean, which if detected might then be interpreted as a sign that TRAPPIST1b and TRAPPIST1c could have a liquid magma ocean. For TRAPPIST1 and similar systems the models to fit the TTVs should potentially account for general relativity, for the tidal deformation of the planets, for the rotational deformation of the planets, and to a lesser extent for the rotational deformation of the star, which would add up to 7 × 2 + 1 = 15 additional free parameters in the case of TRAPPIST1.
Key words: planets and satellites: dynamical evolution and stability / planets and satellites: individual: TRAPPIST1 / planetstar interactions
© ESO 2020
1 Introduction
The measurement of transit timing variations (TTVs) in the context of multitransiting planet systems can be a very efficient method for deriving the dynamical parameters of a planetary system, such as mass and eccentricity (see Agol & Fabrycky 2018, for a review). The TRAPPIST1 system has been intensely monitored by TRAPPIST, K2, and Spitzer, which led to estimates of the masses of the planets by Grimm et al. (2018). Recently additional Spitzer observations were obtained thanks to the Spitzer proposal no. 14223 (Agol et al. 2019).
In most studies on TTVs the model used is an Nbody model assuming pointmass and Newtonian dynamics, sometimes with a prescription for general relativity (as in Grimm et al. 2018; Jordán & Bakos 2008; Pál & Kocsis 2008). Theoretical studies have considered the possible impact of tides and quadrupole distortion on transit times (MiraldaEscudé 2002; Heyl & Gladman 2007). However, the influence of tides has never been consistently taken into account in a multiplanet context.
Some studies do take into account tidal decay (e.g., Maciejewski et al. 2018), but decay typically occurs on time scales that are much longer than the typical duration of the observations available for TRAPPIST1. However, tidal forces are not only a dissipative effect (which drives migration and spin evolution), there is also a nondissipative effect which depends on the real part of the complex Love number of degree2, k_{2}, which quantifies the shape of the tidal deformation. This deformation can lead to a precession of the orbit, which can lead to TTVs. In addition, for fast rotating planets, rotational flattening can also drive a precession of the orbit, which in turn can lead to TTVs. These effects have been considered in systems with a single hotJupiter planet (see Ragozzine & Wolf 2009, for a comparative study of each effect). The precession of the orbit leads to observable TTVs, which then can inform the internal structure of the planet through the determination of the Love number.
However, these effects are usually never taken into account when investigating the TTVs of multiplanetary systems. We show that in the context of TRAPPIST1 (Gillon et al. 2017, 2016), the inclusion of tidal forces may lead to an observable TTV signal. In contrast with TRAPPIST1 d–h, planets b and c are in proximity to a higherorder resonance (increasing the frequency of the TTV pattern modulation; see Agol et al. 2005) and exhibit small TTV amplitudes (2–5 min); both effects inflate the uncertainties on the masses and eccentricities, as shown in Grimm et al. (2018). Interestingly, a recent TTV analysis using data obtained in the past two years yields a 34% increase in mass for TRAPPIST1b and a 13% increase in mass for TRAPPIST1c (in prep.) compared to Grimm et al. (2018). These mass increases of the two inner planets drew our attention to physical processes that could impact the planet physical and orbital parameters on secular timescales. As the parameters for the other planets have remained relatively insensitive to the addition of new data, two hypotheses remain. A first possibility is that the changing masses are due to an incomplete sampling of the TTV pattern that should resolve as new data are included. A second possibility is that dynamical models are missing physical processes that impact the closein planets more strongly, such as tides and rotational flattening.
We show in this paper that the precession caused by general relativity, by tidal deformation, and by rotational flattening could lead to significantly different TTVs for the two inner planets of TRAPPIST1.
2 Simulation setup
We use POSIDONIUS^{1} v2019.07.30 (BlancoCuaresma & Bolmont 2017; BlancoCuaresma & Bolmont, in prep.), an Nbody code which allows users to take into account additional forces and torques: tidal forces and torques, rotational flattening forces and torques, and general relativity (Bolmont et al. 2015). As in Bolmont et al. (2015), tides are computed between a planet and the star independently of the other planets, and the planet–planet tides are not taken into account (which is justified, see Hay & Matsuyama 2019). In Posidonius, we use the integrator IAS15 (Rein & Spiegel 2015) to compute the evolution of the system for 1500 days, which is approximately the time range available from all the observations collected from the system, and we fix the maximum time step allowed at 0.01 day = 14 min. We tested the convergence of our code with time steps of 0.005 day and 0.001 day, and find that the transit timings are stable to a precision of better than 10^{−6} s.
2.1 Tidal model
Posidonius accounts for equilibrium tides following the prescription of Bolmont et al. (2015), which is an implementation of the constant timelag model (Mignard 1979; Hut 1981; Eggleton et al. 1998). The equilibrium tide is the result of the hydrostatic adjustment of a body; instead, the dynamical tide is the tidal response corresponding to the propagation of waves, for example inertial waves in the convective region of stars (see Zahn 1975) or gravitoinertial waves in a planetary liquid layer (see AuclairDesrotour et al. 2019).
We review here the expressions for the tidal force and torques. Let us consider a star, defined by its mass M_{⋆}, its radius R_{⋆}, its degree2 potential Love number k_{2,⋆}, its (constant) time lag Δτ_{⋆}, and its spin vector Ω_{⋆}. Let us consider one planet, j, orbiting the star at a distance r_{j}. The planet is defined by its mass , its radius , its degree2 potential Love number , its (constant) time lag , and its spin vector .
Let us define and as the dissipative part and the nondissipative part, respectively, of the force exerted on planet j due to the planetary tide as (Bolmont et al. 2015)
where is the unit vector r_{j}∕r_{j} and is a vector collinear with the orbital angular momentum of planet j, the norm of which is equal to the time derivative of the true anomaly. Let us define F_{diss,⋆} and F_{nodiss,⋆} as the dissipative part and the nondissipative part, respectively, of the force exerted on planet j due to the stellar tide as
The total force as a result of the tides acting on a planet j is therefore given by the sum of these contributions (Bolmont et al. 2015) (5)
2.2 Rotational flattening model
To account for rotational flattening, we also follow here the prescription of Bolmont et al. (2015), which assumes that the deformation due to the rotational flattening results in a triaxial ellipsoid symmetric with respect to the rotation axis (Murray & Dermott 1999). This deformation is quantified by a parameter, J_{2}, which depends on the radius, mass, and spin of the body and on the potential Love number of degree2 for a perfectly fluid body (which we call here the fluid Love number; Correia & Rodríguez 2013). We define this parameter for planet j and the star as
Let us define the force exerted on planet j due to the rotational flattening of planet j and F_{rot,⋆} the force exerted on planet j due to the rotational flattening of the star as (Murray & Dermott 1999; Correia et al. 2011)
where C_{⋆} and are defined as follows:
The resulting force on planet j due to the rotational deformation of both the star and planet j is the sum of the two contributions: (12)
2.3 General relativity
We use three different prescriptions for general relativity: Kidder (1995), which is the one used in MercuryT (Bolmont et al. 2015; Anderson et al. 1975; and Newhall et al. 1983).
The prescription of Kidder (1995) was designed for two bodies, and Posidonius takes into account the postNewtonian, the spinorbit, and the post^{2}Newtonian contributions to the total acceleration (Eqs. (2.2b)–(2.2d), respectively), as well as the spin precession equations for both bodies (Eqs. (2.4a) and (2.4b)).
The prescription of Anderson et al. (1975) accounts for the postNewtonian acceleration of two bodies. We refer the reader to Eq. (12) in Anderson et al. (1975) where the expression of this acceleration is given.
The prescription of Newhall et al. (1983) is more complete insofar as it accounts for the postNewtonian effect between all bodies. We refer to Eq. (1) in Newhall et al. (1983), which gives the pointmass acceleration. Posidonius accounts for this acceleration, except for the last term which accounts for the perturbation of five solar system asteroids.
3 Transit timing variations
We performed simulations of the TRAPPIST1 system switching on and off these various effects: the effect of the planetary tide (by varying the Love number k_{2,p} and the time lag τ_{p}), the effect of the stellar tide (by varying the Love number k_{2,⋆}), the effect of the rotational flattening of the planets (by varying the fluid Love number k_{2f,p}), the effect of the rotational flattening of the star (by varying the fluid Love number k_{2f,⋆}), and the effect of general relativity. We list in Table 1 the reference values of the parameters varied here, and we refer the reader to Appendix A for the parameters that remain constant in our simulations, including the initial orbital elements for the planets. We tested the three different prescriptions of the general relativity introduced in Sect. 2.3. They gave very similar results so that in the following we compare the other effects with respect to the simulations performed using the prescription of Kidder (1995).
The planetary reference values were taken to be representative of the Earth; in particular, the quantity k_{2,p} Δτ_{p} is equal to 213 s (Neron de Surgy & Laskar 1997). The stellar reference values were chosen to be representative of fully convective M dwarfs (Bolmont et al. 2015).
For all the simulations we performed, we calculated the transit timing variations (Agol & Fabrycky 2018) as follows: (i) for each transit we find the time of the transit midtime by performing an interpolation to find the precise time a given planet crosses a reference direction. This corresponds to the “Observed transit time” O; (ii) we evaluate the “Calculated transit times” C by performing a linear fit^{2} of the transit times calculated in step i) over the total number of transits; (iii) we calculate the difference O–C to obtain the TTVs as a function of the epoch (or transit number).
To quantify the impact of each additional effect on the simulated TTVs, we compute the difference between the TTVs calculated taking into account an additional effect and the TTVs obtained for a Newtonian Nbody integration.
Reference values for the parameters we vary in this study.
4 Influence of each effect on the TTVs
We performed a set of six simulations of the TRAPPIST1 system to test the impact of the additional effects listed in Sect. 2, and we compare each with a Newtonian Nbody simulation. One after another, we explored the effect of each parameter using the reference values in Table 1and general relativity. We tested the influence of the dissipative part of the planetary tide by assuming for all planets, with all the other parameters set to zero. We repeated the operation for the nondissipative part of the planetary tide (through k_{2,p}, equal for all planets), the rotational flattening of the planets (through k_{2f,p}, equal for all planets), the nondissipative part of the stellar tide (through k_{2,⋆}), the rotational flattening of the star (through k_{2f,⋆}), and for general relativity.
Figure 1 shows the results for planets b to d. The top panels of Fig. 1 show the transit timing variations for the three planets for the seven simulations, and the bottom panels show the difference between the TTVs and the TTVs corresponding to the pure Nbody simulation. The different additional effects have a very limited impact on the shape of the TTVs, but computing the difference with the pure Nbody case reveals the amplitude of each effect.
For TRAPPIST1b (T1b), the dominant effects are the nondissipative part of the planetary tide (green in Fig. 1a) and general relativity (pink), respectively accounting for a difference in TTVs of about −0.63 s and 0.56 s after 1500 days. The effect of the rotational flattening of the planets (red) plays a smaller role, but still accounts formore than half the amplitude due to the nondissipative part of the planetary tide with a difference of −0.33 s. The effect of the dissipative part of the planetary tide (orange) and the nondissipative part of the stellar tide (purple) are completely negligible (accounting for a difference ~1 × 10^{−3} and ~ 1 ×10^{−4} s, respectively), which is in agreement with Ragozzine & Wolf (2009). The effect of the rotational flattening of the star (brown) is much smaller (accounting for a difference of −0.057 s), but might contribute to a lesser extent. Accounting for all effects (gray) leads to an absolute difference of −0.45 s at the end of the 1500day simulation. The effects of general relativity and the nondissipative part of the planetary tide almost cancel each other out, while the amplitude is determined by the effect of the rotational flattening of the planet (red curve) and of the star (brown curve).
For TRAPPIST1c (T1c), the dominant effects are the nondissipative part of the planetary tide (accounting for a difference of 1 s, in green in Fig.1b), followed by the effect of the rotational flattening of the planet (0.53 s, in red), followed by the effect of general relativity (−0.44 s, pink). The effect of the rotational flattening of the star accounts for 0.067 s (purple) and the effect of the dissipative part of the planetary tide remains negligible (0.013 s, orange). As with T1b, to reproduce the difference observed when all effects are taken into account (in gray in Fig.1b), we need to account for the nondissipative part of the planetary tide, the rotational flattening of the planet, general relativity, and the rotational flattening of the star to a lesser extent. We note that the precession of the orbits due to the rotational flattening depends on the square of the spin frequency of the considered body (Ragozzine & Wolf 2009). Here we use a rotation period of 3.3 days for TRAPPIST1 (Luger et al. 2017). It is possible that the rotation is slower (as the period distribution of nearby late M dwarfs shows, Newton et al. 2016), in which case the contribution of the rotational flattening of the star would be even less important.
For TRAPPIST1d (T1d), Fig. 1c shows that the dominant effect is general relativity (−2.03 s, in pink). The effect of the nondissipative part of the planetary tide (accounting for −0.81 s, green) and the effect of the rotational flattening of the planet (accounting for −0.43 s, red) should probably also be taken into account. This is also true for all the external planets: the amplitude due to general relativity is much higher, but at the same time not accounting for at least the nondissipative part of the planetary tide and the rotational flattening of the planet leads to small offsets (see Appendix B, Fig B.1).
Fig. 1 Impact of various additional effects on the TTVs of (a) planet b, (b) planet c, and (c) planet d. Top panel: transit timing variations for a pure Nbody simulation (blue); for a simulation using for all planets (orange), using for all planets (green), using for all planets (red), using (purple), using (brown), and where we only consider the general relativity (pink); and for a simulation where all effects are taken into account. Bottom panel: corresponding TTVs differences with the Newtonian Nbody case. 
5 Potential observable effects
We performed simulations for which we varied the potential Love number k_{2,p} and the fluid Love number k_{2f,p} over a wide range. We first treat these parameters as free parameters with no limitations on their value and then we discuss the validity of this approach in Sect. 6.
As in Sect. 4, we always assume the same value of the Love numbers for all planets. We vary the parameter k_{2,p} from to ; the impact on the TTV differences with the pure Nbody case can be seen in the top panel of Fig. 2a for T1b and in the top panel of Fig. 2b for T1c.
Considering the highest value of the tidal Love number leads to a difference in TTVs after 1500 days of −31.7 s for T1b and 50.7 s for T1c. The amplitude of these effects are then comparable to the precision achievable today on the observed TTVs of the two inner planets. If the tidal Love number could reach such high values, the effect of the nondissipative part of the planetary tide (the tidal deformation of T1b and T1c) would therefore be detectable.
Similarly, we vary the parameter k_{2f,p} from to , and the impact on the TTV differences with the pure Nbody case can be seen in the bottom panel of Fig. 2a for T1b and in the bottom panel of Fig. 2b for T1c. Assuming the highest value of the fluid Love number leads to a difference inTTVs after 1500 days of 17.3 s for T1b and 27.7 s for T1c.
Considering all the important effects given in Sect. 4 and relaxing the range of possible Love numbers might therefore be a way to settle the question of the increasing masses of T1b and T1c and to settle the two hypotheses given in the introduction: is it a sampling problem or are we missing dynamical processes? The answer to this question depends upon the potential degeneracy of these effects with varying the Nbody parameters, and on the duration and precision of the transit timing measurements. We do not explore these effects here.
6 Discussion
We showed that for systems like TRAPPIST1 the effect of the tidal deformation of the planets (through the planetary tidal Love number k_{2,p}), the effect of the rotational deformation of the planets (through the planetary fluid Love number k_{2f,p}), and the effect of the rotational flattening of the star (through the stellar fluid Love number k_{2f,⋆}) can impact the TTVs of the two inner planets at the same order of magnitude as the general relativity, if we assume “standard” values for these parameters. We also showed that the tidal dissipation (responsible for the misalignment which drives long term tidal evolution) does not significantly impact the TTVs of the system over the short observation time that we simulated. By relaxing the assumptions on the planetary tidal and fluid Love numbers, we also showed that a high Love number can lead to differences in TTV on the order of ~ 10 s. This difference is potentially observable with the current precision we have on the transit timings, unless there is significant degeneracy with other Nbody parameters.
However, it is commonly accepted that a tidal Love number cannot exceed 1.5, which corresponds to a homogeneous body. This means that the physical range of our study should encompass maximum values, which are . Limiting ourselves to this value would entail a difference in TTVs for T1b of less than 2.5 s, which is below the precision we can achieve today. On the other hand, it is known that if a planet has a liquid layer (liquid water ocean, or liquid magma ocean), the response of the body becomes more complex; in particular, it becomes highly dependent on the excitation frequency. Specifically, if a frequency excites a resonant mode of the ocean, the tidal response can be much higher than a homogeneous rockyplanet model would predict (see, e.g., AuclairDesrotour et al. 2019). Investigating this aspect consistently will require us to generalize the tidal formalism used here to account for the frequency dependence of the dynamical tide (e.g., using the formalism of Kaula 1961).
That is why we think that we might need to perform a TTV analysis of the TRAPPIST1 system accounting for the various physical processes described here with no particular preconception about the values of the parameters for the planetary Love numbers. If the TTVs are reproduced by having a T1b planet with a high Love number, this could be a sign for a liquid layer on the planet, possibly a magma ocean given the flux it receives and the tidal heat flux it might generate (e.g., Turbet et al. 2018; Makarov et al. 2018).
While a difference of a few ~10 s is potentially observable, it could be interpreted as a system with slightly different planetary masses and periods by a classical TTV retrieval code. Our group is thus currently investigating if these effects could be picked up with such a retrieval code and, if so, under what conditions (e.g., duration of the observations, precision of the timings). We are also working on implementing these effects in the TTV analysis pipeline and plan to revisit the analysis with the additional parameters mentioned earlier (Grimm et al., in prep.).
Fig. 2 Differences in TTVs with the pure Nbody case for (a) planet b and (b) planet c. Top panel: fixing the planetary fluid Love number and the dissipation to their reference values (see Table 1), the potential Love number k_{2,p} is varied between 1 and 50. Bottom panel: fixing the planetary Love number and the dissipation to their reference values, the potential Love number k_{2f,p} is varied between 1 and 50. 
Acknowledgements
The authors would like to thank the anonymous referee for helping improving the manuscript. This work has been carried out within the framework of the NCCR PlanetS supported by the Swiss National Science Foundation. B.O.D. acknowledges support from the Swiss National Science Foundation (PP00P2_163967). P.A.D. acknowledges financial support from the European Research Council via the Consolidator grant EXOKLEIN (grant number 771620). F.S. acknowledges support from the CNRS/INSU PNP (Programme National de Planétologie). This research has made use of NASA’s Astrophysics Data System.
Appendix A Initial conditions for the TRAPPIST1 simulations
To ensure the reproducibility of our simulations, we give here the exact initial conditions we took for the system. Table A.1 gives the stellar parameters used for the integration of the system. The stellar mass and radius come from Gillon et al. (2017) and the rotation comes from Luger et al. (2017). The value of the radius of gyration squared (Hut 1981) comes from Bolmont et al. (2015) and should be typical of a fully convective dwarf.
Stellar parameters.
Table A.2 gives the masses and radii of the planets as well as the initial orbital elements. We consider that all planets have the same radius of gyration squared (where this quantity is related to the moment of inertia ). We consider that all planets have a zero obliquity (angle between the direction perpendicular to the orbital plane and the rotation axis of the planet) and that they are tidally locked (see discussion in Luger et al. 2017).
Masses, radii, and initial orbital elements used for the dynamical simulations of the TRAPPIST1 system.
To perform the integration of the system, we used POSIDONIUS v2019.07.30^{3}. This versionwas slightly altered to be able to fix a maximum time step size (0.01 day). The initial conditions can be found in Bolmont et al. (2020)^{4}.
Appendix B Transit timing variations for the four outer planets of TRAPPIST1
As in Figs. 1 and B.1 shows the difference in TTVs with the Nbody case for seven different simulations (Nbody and simulations with additional effects) for TRAPPIST1e to TRAPPIST1h. General relativity is the dominant effect, but the nondissipative part of the planetary tidal force (via k_{2,p}) and the rotational flattening of the planets (via k_{2f,p}) are still contributing marginally. Only for TRAPPIST1h is general relativity the only relevant process to account for.
Fig. B.1 As the bottom panels of Fig. 1, but for (a) TRAPPIST1e, (b) TRAPPIST1f, (c) TRAPPIST1g, and (d) TRAPPIST1h. The general relativity is the dominant effect, but the planetary deformation (due to tides or rotation) is not quite completely negligible, except for TRAPPIST1h. 
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All Tables
Masses, radii, and initial orbital elements used for the dynamical simulations of the TRAPPIST1 system.
All Figures
Fig. 1 Impact of various additional effects on the TTVs of (a) planet b, (b) planet c, and (c) planet d. Top panel: transit timing variations for a pure Nbody simulation (blue); for a simulation using for all planets (orange), using for all planets (green), using for all planets (red), using (purple), using (brown), and where we only consider the general relativity (pink); and for a simulation where all effects are taken into account. Bottom panel: corresponding TTVs differences with the Newtonian Nbody case. 

In the text 
Fig. 2 Differences in TTVs with the pure Nbody case for (a) planet b and (b) planet c. Top panel: fixing the planetary fluid Love number and the dissipation to their reference values (see Table 1), the potential Love number k_{2,p} is varied between 1 and 50. Bottom panel: fixing the planetary Love number and the dissipation to their reference values, the potential Love number k_{2f,p} is varied between 1 and 50. 

In the text 
Fig. B.1 As the bottom panels of Fig. 1, but for (a) TRAPPIST1e, (b) TRAPPIST1f, (c) TRAPPIST1g, and (d) TRAPPIST1h. The general relativity is the dominant effect, but the planetary deformation (due to tides or rotation) is not quite completely negligible, except for TRAPPIST1h. 

In the text 
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Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.