Issue 
A&A
Volume 631, November 2019



Article Number  A92  
Number of page(s)  12  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201936259  
Published online  28 October 2019 
From the stellar properties of HD 219134 to the internal compositions of its transiting exoplanets
^{1}
INAF – Osservatorio Astronomico di Brera,
Via E. Bianchi 46,
23807
Merate,
Italy
email: roxanne.ligi@inaf.it
^{2}
Institut of Computational Sciences, University of Zurich,
Winterthurerstrasse 190,
8057
Zurich, Switzerland
^{3}
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire,
CS 34229,
06304
Nice Cedex 4, France
^{4}
Institut Universitaire de France,
103 boulevard SaintMichel,
75005
Paris,
France
^{5}
LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Université Paris 06, Université Paris Diderot,
Sorbonne Paris Cité,
92195
Meudon Cedex, France
^{6}
Univ. Rennes, CNRS, IPR (Institut de Physique de Rennes) – UMR 6251,
35000
Rennes, France
^{7}
Univ. Lyon, Univ. Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR 5574,
69230
SaintGenisLaval, France
Received:
5
July
2019
Accepted:
20
September
2019
Context. The harvest of exoplanet discoveries has opened the area of exoplanet characterisation. But this cannot be achieved without a careful analysis of the host star parameters.
Aims. The system of HD 219134 hosts two transiting exoplanets and at least two additional nontransiting exoplanets. We revisit the properties of this system using direct measurements of the stellar parameters to investigate the composition of the two transiting exoplanets.
Methods. We used the VEGA/CHARA interferometer to measure the angular diameter of HD 219134. We also derived the stellar density from the transits light curves, which finally gives a direct estimate of the mass. This allowed us to infer the mass, radius, and density of the two transiting exoplanets of the system. We then used an inference model to obtain the internal parameters of these two transiting exoplanets.
Results. We measure a stellar radius, density, and mass of R_{⋆} = 0.726 ± 0.014 R_{⊙}, ρ_{⋆} = 1.82 ± 0.19 ρ_{⊙}, and M_{⋆} = 0.696 ± 0.078 M_{⊙}, respectively; there is a correlation of 0.46 between R_{⋆} and M_{⋆}. This new mass is lower than that derived from the C2kSMO stellar evolutionary model, which provides a mass range of 0.755−0.810 (±0.040) M_{⊙}. Moreover, we find that planet b and c have smaller radii than previously estimated of 1.500 ± 0.057 and 1.415 ± 0.049 R_{⊕} respectively; this clearly puts these planets out of the gap in the exoplanetary radii distribution and validates their superEarth nature. Planet b is more massive than planet c, but the former is possibly less dense. We investigate whether this could be caused by partial melting of the mantle and find that tidal heating due to nonzero eccentricity of planet b may be powerful enough.
Conclusions. The system of HD 219134 constitutes a very valuable benchmark for both stellar physics and exoplanetary science. The characterisation of the stellar hosts, and in particular the direct determination of the stellar density, radius, and mass, should be more extensively applied to provide accurate exoplanets properties and calibrate stellar models.
Key words: stars: fundamental parameters / stars: individual: HD 219134 / planetary systems / techniques: interferometric / methods: numerical / planets and satellites: fundamental parameters
© ESO 2019
1 Introduction
The huge harvest of exoplanets discovered by the space telescopes Kepler (Borucki et al. 2010) and CoRoT (Baglin 2003) has led to the understanding that exoplanets are the rule rather than the exception. We have now moved to the era of exoplanet characterisation, and the next challenge is to understand how common rocky planets are and if any are suitable for life. The most interesting exoplanets to study are certainly the transiting exoplanets, as the transit light curve allows us to know the planetary radius. An additional radial velocity (RV) followup provides the planetary mass and thus the planetary density. The three ingredients to estimate planetary bulk composition are then gathered. But this is only true if the stellar radius and mass are known. Up to now, most of transiting exoplanet hosts have been very faint, driven by the search for exoplanets rather than their characterisation, often leading to inaccurate and/or imprecise stellar parameters. This makes the characterisation of the whole exoplanetary system difficult and the determination of the exoplanetary internal structure approximate.
Several methods can be employed to obtain the stellar parameters. Concerning the mass, it is often determined indirectly, as only stars inbinary systems can have their mass directly measured if the system inclination is known. However, if an exoplanet is transiting its host star, the density of the star can be directly inferred from the transit light curve (Seager & MallénOrnelas 2003). Then, in the case of bright stars, the radius can be directly determined using interferometry, which is a high angular resolution technique aimed at measuring the angular diameter of stars with a precision up to a few percent (Baines et al. 2010; Boyajian et al. 2012a,b; Huber et al. 2012; Creevey et al. 2012, 2015; Ligi et al. 2012, 2016, e.g.). The mass can thus be directly computed from the transit and interferometric measurements. This method has recently been used by Crida et al. (2018a,b) to derive the mass of the very bright star 55 Cnc with a precision of 6.6% using the interferometric diameter measured by Ligi et al. (2016) and the density from the transit light curve obtained for 55 Cnc e (Bourrier et al. 2018). This yielded the best characterisation of the transiting superEarth 55 Cnc e so far and a new estimate of its internal composition.
HD 219134 (HIP 114622, GJ 892) is also a bright (V = 5.57) K3V star 6.5 parsecs away from us. Motalebi et al. (2015) first detected four exoplanets around the star from RV measurements using the High Accuracy Radial velocity Planet Searcher for the Northern hemisphere (HARPSN) on the Telescopio Nazionale Galileo (TNG). Moreover, Spitzer timeseries photometric observations allowed the detection of the transit of planet b, leading to the estimate of a rocky composition. The same year, Vogt et al. (2015) claimed the detection of six planets around HD 219134 from the analysis of RV obtained with the HIgh Resolution Echelle Spectrometer (HIRES) on Keck I Observatory and the Levy Spectrograph at the Automated Planet Finder Telescope (Lick Observatory). These authors derived similar periods for planets b, c, and d, the other diverging because of the different Keplerian analysis of the RV signal leading to a different number of planets. Later, Gillon et al. (2017) reported additional Spitzer observations of the system that led to the discovery of the transit of the second innermost planet, HD 219134 c. The two innermost planets seem rocky, but more interestingly, planet c shows a higher density while it has a lower mass than planet b. The detailed planetary data and their relative differences place additional constraints on their interiors with implications to their formation and evolution.
In this paper, we report new observations of HD 219134 using the Visible spEctroGraph and polArimeter (VEGA) instrument on the Center for High Angular Resolution Astronomy (CHARA) interferometric array that led to a new accurate determination of angular diameter of this star (Sect. 2). In Sect. 3, we determine the stellar radius and density, and derive the joint probability density function (PDF) of the stellar mass and radius independently of stellar models. We then use the PDF to compute the new parameters of the two transiting exoplanets and we revisit those of the nontransiting exoplanets in Sect. 4.1. Finally, we derive the internal composition of planets b and c in Sect. 4.2 using a planetary interior model, and we discuss the possible cause of the different densities of planets b and c in Sect. 4.3. We conclude in Sect. 5.
2 Interferometric measurement of the angular diameter with VEGA/CHARA
We used the technique of interferometry to measure the angular diameter of HD 219134. These measurements constitute the first step to determine the other fundamental parameters of this star.
2.1 Observations and data reduction
We observed HD 219134 from 2016 to 2018 using the VEGA/CHARA instrument at visible wavelengths (see Table 1) and medium resolution. The spectrointerferometer VEGA (Mourard et al. 2009; Ligi et al. 2013) is based on the CHARA array (ten Brummelaar et al. 2005), which takes advantage of the six 1 m telescopes distributed in a Yshape to insure wide (u,v) coverage. It can be used at medium (5000) or high spectral resolution (30 000) and with baselines ranging from 34 to 331 m in the two telescope (2T), 3T, or 4T modes. The observations were calibrated following the sequence calibrator – science star – calibrator, and were performed using different configurations (Table 1), mainly in the 2T mode at once to optimise the signaltonoise ratio (S/N) of the observations. The calibrator stars were selected into the SearchCal software^{1} (Table 2), and we used the uniform disc diameter in the R band (UDDR) found in the JSDC2 (Bourgés et al. 2014) or SearchCal (Chelli et al. 2016) catalogue otherwise. However, for conservative reasons, we decided to use an uncertainty of 7% or that given in the JSDC1 (Bonneau et al. 2006) if higher. We selected the calibrators with several criteria: in the neighbourhood of the star, discarding variable stars and multiple systems, and with high squared visibilities, allowing an optimal measurement of the instrumental transfer function. Finally, the data were reduced using the vegadrs pipeline (Mourard et al. 2009, 2011) developed at Observatoire de la Côte d’Azur. For each observation, we selected two nonredundant spectral bands of 20 nm wide centred at 685, 705, or 725 nm in most cases to derive the squared visibility (V^{2}), but the reddest band is sometimes of bad quality or features absorption lines and cannot be used. In total, we collected 36 data points, which are shown in Fig. 1.
2.2 Angular diameter
The squared visibilities that we obtained (Fig. 1, coloured filled circles) are well spread on the V^{2} curve. We note some dispersion around 0.7 × 10^{8}∕rad (corresponding to the E1E2 configuration) but it is taken into account in the computation of the error on the angular diameter. We also adopted a conservative approach by setting a minimum error of 5% on V^{2} to balance the known possible bias with VEGA (Mourard et al. 2012, 2015).
We used the LITpro software (TallonBosc et al. 2008) to fit our visibility points and derive the angular diameter of HD 219134 and its related uncertainty. Taking a model of uniform disc, we obtained θ_{UD} = 0.980 ± 0.020 millisecond of arc (mas; Table 3). However, this simple representation is not realistic and we thus used a linear limbdarkening (LD) model to refine it, as the LD diameter (θ_{LD}) cannot be directly measured. We must indeed use empirical tables of LD coefficients μ_{λ}, which depend on the effective temperature T_{eff}, gravity log (g), and metallicity [Fe/H] at a given wavelength λ. We used Claret & Bloemen (2011) tables as a start in the R and I band since we observed between 685 and 720 nm, and proceeded on interpolations to obtain a reliable LD coefficient at our wavelength, as described in Ligi et al. (2016). The LD coefficients in Claret & Bloemen (2011) tables are given in steps of 250 K for T_{eff}, 0.5 dex for log (g) and less uniform steps for [Fe/H]. We set a starting value of these parameters to perform our interpolation in between the surrounding values. We searched in the literature previous values of log (g) and [Fe/H] through the SIMBAD database^{2} and calculated the median and standard deviation of the values given there (see selected values in Table A.1 and the medians in Table 3). Beforehand, we eliminated aberrant values and values obtained before the year 2000, to insure recent and probably more reliable estimates. Since many values of the metallicity could be derived from a same data set, and because the uncertainty in the various papers can be higher than our standard deviation (0.05 dex), we set the uncertainty on [Fe/H] to 0.1 dex. Concerning the starting T_{eff} value, we used that fitted through the spectral energy distribution (SED; T_{eff, SED} = 4839 K, Sect. 3.2). Since the star is close by (distance, d = 6.533 ± 0.038 pc, Table 3), we set the reddening to A_{v} = 0.0 ± 0.01 mag. This value is consistent with the extinction given by the Stilism (Lallement et al. 2014) 3D map of the galactic interstellar matter (E(B–V) = 0 ± 0.014) but corresponds to a smaller uncertainty on the extinction (0.0034 mag).
For each filter, we first computed the linear interpolation of the LD coefficients corresponding to the surrounding values of [Fe/H], log (g) and T_{eff} of our star. Wethen averaged the two coefficients coming out from each filter to get a final coefficient. Then, we used the LITpro software to fit our data using a linear LD model while fixing in the model our new LD coefficient. This results in θ_{LD} = 1.035 ± 0.021 mas (2% precision). It has to be noted that using different LD laws does not significantly change the final diameter as we are not sensitive to it in the first lobe of visibility. If we set T_{eff} = 4750 K, log (g) = 4.5 dex, and [Fe/H] = 0.1 dex, a quadratic LD law described by Claret & Bloemen (2011) yields θ_{LD} = 1.047 ± 0.022 mas in the R band (using the LD coefficients a_{1,R} = 0.5850 and b_{2,R} = 0.1393 given in the table) and θ_{LD} = 1.033 ± 0.022 mas in the I band (taking a_{1,I} = 0.4490 and b_{2,I} = 0.1828). Similarly, averaging a_{1,R} and a_{1,I} coefficients on the one hand, and b_{2,R} and b_{2,I} on the other hand, leads to θ_{LD} = 1.040 ± 0.022 mas, and thus a value within the error bars of our first estimate. Our determined angular diameter is smaller than that previously measured with the CHARA Classic beam combiner (1.106 ± 0.007 mas; Boyajian et al. 2012b). Although their visibilities seem more precise, we stress that we obtain higher spatial frequency data, which resolves the star better. The angular diameter derived from the SED θ_{SED} is also very consistent with our measurement (1.04 mas, see Sect. 3.2).
Observing log.
Angular diameters of the calibrators used.
Fig. 1 Squared visibilities obtained with VEGA/CHARA for HD 219134. The different colours represent the data points obtained with different baselines. The solid line represents the model of LD diameter. 
3 Stellar parameters
The new angular diameter constitutes the basis of our analysis. It is now possible to determine the other stellar parameters from our interferometric measurements, and to compare these parameters with those derived from stellar evolution models.
3.1 Radius, density, and mass
The stellar radius is generally derived using the distance and angular diameter as follows: θ_{LD} = 2R_{⋆}/d. As for the mass, Crida et al. (2018a,b) showed the importance of using the correlation between the stellar mass and radius to reduce the possible solutions in the massradius plane. We took the same approach to derive R_{⋆} and M_{⋆}. The PDF of R_{⋆}, called ${f}_{{R}_{\star}}$, can be expressed as a function of the PDF of the observables θ_{LD} (angular diameter) and π (parallax), called f_{θ} and f_{π} respectively. This gives $${f}_{{R}_{\star}}(R)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{R}_{0}}{{R}^{2}}{\displaystyle {\int}_{0}^{\infty}t}\text{\hspace{0.17em}}{f}_{\pi}\left(\frac{{R}_{0}\text{\hspace{0.17em}}t}{R}\right){f}_{\theta}(t)\text{\hspace{0.17em}d}t,$$(1)
where R_{0} is a constant (see Crida et al. 2018a, for the proof). Concerning Gaia parallaxes, Stassun & Torres (2018) have reported that an offset of − 82 ± 33 μas is observed, while Lindegren et al. (2018) have provided −30 μas. In any case, these offsets are within the uncertainty of the parallax for HD 219134 and do not impact significantly our results. As advised by Luri et al. (2018), we only used the parallax and its error given in the Gaia DR2 catalogue (Gaia Collaboration 2016, 2018), keeping in mind the possible offsets and that for such bright stars, there might still be unknown offsets that DR3 and DR4 will provide.
We found R_{⋆} = 0.726 ± 0.014 ρ_{⊙}, which is a lower value than that found by Boyajian et al. (2012b, R_{⋆} = 0.778 ± 0.005 R_{⊙}). Our uncertainty on R_{⋆} is clearly dominated by the uncertainty on the angular diameter because we took the parallax from Gaia DR2, which is very precise (0.06%).
The stellar density ρ_{⋆} can be derived from the transit duration, period and depth (Seager & MallénOrnelas 2003). In our system, we have two transiting exoplanets. We computed the stellar density independently for both transits using the data given by Gillon et al. (2017) and found 1.74 ± 0.22 and 2.04 ± 0.37 ρ_{⊙} for planets b and c, respectively. We note that the density coming from the analysis of the light curve for HD 219134 c is less precise than that of HD 219134 b. This comes from the transit light curves themselves, which are more complete and more precise for planet b. Combining both densities, we obtained 1.82 ± 0.19 ρ_{⊙}, which we use in the rest of our analysis. We computed the uncertainty following a classical propagation of errors and found a value close but different, and with a bigger error bar compared to that given in Gillon et al. (2017). The joint likelihood of M_{⋆} and R_{⋆} can be expressed as $${\mathcal{\text{L}}}_{MR\star}(M,R)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{3}{4\pi {R}^{3}}\times {f}_{{R}_{\star}}(R)\times {f}_{{\rho}_{\star}}\left(\frac{3M}{4\pi {R}^{3}}\right),$$(2)
as described in Crida et al. (2018a) and where ${f}_{{\rho}_{\star}}$ is the PDF of the stellar density (Fig. 2). The calculated correlation coefficient between R_{⋆} and M_{⋆} is 0.46. Our computation yields M_{⋆} = 0.696 ± 0.078 M_{⊙}, which is consistent with the value determined directly from log (g) and R_{⋆} but with a better precision. For reference, other authors derived 0.763 ± 0.076 M_{⊙} (Boyajian et al. 2012b) using the relation by Henry & McCarthy (1993), and 0.81 ± 0.03 M_{⊙} (Gillon et al. 2017)using stellar evolution modelling. In this latter case, the uncertainty corresponds to the internal source of error of the model and is thus underestimated.
Stellar parameters of HD 219134.
Fig. 2 Joint likelihood of the radius and mass of the star HD 219134. The 9 plain red contour lines separate 10 equalsized intervals between 0 and the maximum of Eq. (2). 
3.2 Bolometric flux, effective temperature, and luminosity
To derive the T_{eff} of the star wecombined the angular diameter with its bolometric flux F_{bol} using $${T}_{\text{eff}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\left(\frac{4\times {F}_{\text{bol}}}{{\sigma}_{\text{SB}}{\theta}_{\text{LD}}^{2}}\right)}^{0.25},$$(3)
where σ_{SB} is the StefanBoltzmann constant. This implies computation of the bolometric flux, which we derived from the stellar photometry as described in the next subsection.
3.2.1 Bolometric flux
We determine the bolometric flux F_{bol} and its uncertainty in the following way. We retrieved photometric data from the literature made available by the VizieR Photometry tool^{3}. These photometry convertedtoflux measurements were fitted to the BaSeL empirical library of spectra (Lejeune et al. 1997), using a nonlinear leastsquared minimisation algorithm (LevenbergMarquardt). The spectra are characterised by T_{eff}, [M/H], and log (g). To convert these spectra to observed spectra they need to be scaled by (R_{⋆} ∕d) and reddened for interstellar extinction A_{v}. Thus, each model spectrum is characterised by these five parameters. In practice most of these parameters are degenerate, so it is necessary to fix a subset of these. For each minimisation performed we fixed [M/H], log (g), and A_{v} i.e. we only fitted T_{eff} and (R∕d ∝ θ), and then we integrated under the resulting scaled and unreddened empirical spectrum to obtain F_{bol}.
To properly estimate the uncertainties in the parameters we repeated this method 1000 times to obtain a distribution of F_{bol}. Each of these minimisations had different fixed values of [M/H], log (g), and A_{v} obtained by drawing random numbers from gaussian distributions characterised by the following: [M/H] = +0.07 ± 0.10, log (g) = 4.57 ± 0.14, and A_{v} = 0.00 ± 0.01 mag, as discussed in Sect. 2.2. The initial values of T_{eff} and (R∕d) were obtained by drawing them from a random uniform distribution with values between 4100 and 5700 K and between 4.0 and 8.0. Using the resulting distribution of F_{bol}, we calculated F_{bol} = 19.86 ± 0.21 erg s cm^{−2} × 10^{8}. In the same way, we also estimated the T_{eff} from the resulting distributions of the bestfitted T_{eff} (T_{eff, SED}) and the angular diameter (R∕d) converted tounits of mas (θ_{SED}), although these latter two are not used any further in this work.
The bestfit model spectrum is shown in Fig. 3 in red, along with photometric data points in black. Overall, the model fits the data well, except for two points that are above the fit, but removing the two outliers did not change the results. These overfluxes could come from a close star or another undetermined source, although we could not verify these hypotheses.
Fig. 3 Photometric data (black squares) and fitted model (solid red line) from the BaSeL library of spectra. 
3.2.2 Effective temperature and luminosity
We derived the effective temperature T_{eff} from F_{bol} and θ_{LD} using Eq. (3) to obtain 4858 ± 50 K. This is in very good agreement with the T_{eff} determined by Gaia (${4787}_{73}^{+92}$K) and with that determined through the SED fitting (4839 ± 25 K), which has alower uncertainty. We finally obtained the luminosity using the distance and F_{bol} as follows: $${L}_{\star}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4\pi {d}^{2}{F}_{\text{bol}}.$$(4)
The errors on these final parameters were estimated following a classical propagation of errors (see Ligi et al. 2016, for details). The Gaia luminosity is L_{⋆} = 0.30 L_{⊙}, which is in good agreement with our value (L_{⋆} = 0.264 ± 0.004 L_{⊙}) considering the documented possible systematic errors. All final stellar parameters are reported in Table 3.
3.3 Comparison with stellar evolution models
HD 219134 is now a wellcharacterised star thanks to our direct measurements of its radius and density, providing in turn its mass. Therefore, it constitutes a good benchmark to be compared to stellar evolution models.
We thus confront our measurements (mass, radius, and density) to the values that can be inferred from stellar evolution modelling. For that purpose, we have used the C2kSMO^{4} stellar model optimisation pipeline (Lebreton & Goupil 2014) to find the mass, age, and initial metallicity of the stellar model that best fits the luminosity, effective temperature, and surface metallicity (hereafter observational constraints) of HD 219134 given in Table 3. The procedure operates via a LevenbergMarquardt minimisation performed on stellar models calculated onthefly with the Cesam2k (Morel & Lebreton 2008) stellar evolution code (C2kSMO is described in detail in Lebreton & Goupil 2014).
For a given set of input parameters and physics of a stellar model (nuclear reaction rates, equation of state, opacities, atmospheric boundary conditions, convection formalism, and related mixinglength parameter for convection, element diffusion and mixing, solar mixture of heavy elements, initial helium content, etc.), we can therefore infer the mass, age, and initial metallicity of the best model for HD 219134 with internal error bars resulting from the uncertainties on the observational constraints. However, among these inputs, many are still very uncertain or even unknown. Accordingly, to get a reasonable estimate of the accuracy of the results, we performed several model optimisations, each of which correspond to a different set of input physics and parameters. We varied the following – most uncertain – inputs:
Solar mixture
We investigated the effects of using either the GN93 (Grevesse & Noels 1993) or the AGSS09 (Asplund et al. 2009) mixture. However, we point out that, although still widely used, the GN93 mixture is no longer valid. The AGSS09 mixture is based on carefully updated atomic data and on a 3D timedependent hydrodynamic model of the solar atmosphere, while the GN93 mixture was inferred through a 1D model of the solar atmosphere. As discussed by, for example, Nordlund et al. (2009), the 3D model reproduces the observations of the solar atmosphere remarkably well, while the 1D model atmosphere does not. The AGSS09 mixture should therefore be preferred.
Convection description
We used either the classical mixinglength theory (usually referred to as MLT; BöhmVitense 1958) or the Canuto, Goldmann, and Mazzitelli formalism (usually referred to as CGM; Canuto et al. 1996).
Externalboundary conditions
We investigated the effects of using either the approximate Eddington’s grey radiative T −τ law (T is the temperature, τ the optical depth) or the more physical T − τ law extracted from Model Atmospheres in Radiative and Convective Scheme (MARCS) model atmospheres (Gustafsson et al. 2008). Although MARCS models are classical 1D model atmospheres in local thermodynamical equilibrium, they do include convection in the MLT formalism and use uptodate atomic and molecular data (see e.g. Gustafsson et al. 2008). Therefore, these models represent an important progress with respect to the grey law and should be preferred.
Initial helium abundance
This quantity is not accessible through the analysis of stellar spectra because helium lines are not formed in the spectra of cool and tepid stars. It is a majorsource of uncertainty in stellar model calculation. In stellar models the initial helium abundance is generally estimated from the ΔY∕ΔZ galactic enrichment law^{5} to overcome this difficulty. Two different ΔY∕ΔZ values are usually used: the value obtained from solar model calibration^{6} (chosen for instance in the new Bag of Stellar Tracks and Isochrones (BaSTI) stellar model grids; see Hidalgo et al. 2018), which is ≈ 1 or the socalled galactic value, ΔY∕ΔZ ≈ 2 (Casagrande et al. 2007) adopted for instance in the Modules for Experiments in Stellar Astrophysics (MESA) grids by Coelho et al. (2015). The former depends on the input physics of the solar model while the latter is very uncertain (see e.g. Gennaro et al. 2010). On the other hand, the initial helium content can be estimated by modelling stars with available asteroseismic observational constraints. This is the case of 66 stars in the Kepler Legacy sample for which we obtained values of ΔY∕ΔZ in the range 1– 3 with a mean of ${(\Delta Y/\Delta Z)}_{\text{seism}}\approx 2.3$ with the C2kSMO pipeline (see e.g. Silva Aguirre et al. 2017). Since no strong justification of what would be the best choice can be given, we investigated the impact of using the two values ΔY∕ΔZ = 1 and 2 because the latter is also close to the mean Kepler Legacy asteroseismic value ${(\Delta Y/\Delta Z)}_{\text{seism}}$, but keeping in mind this remains the main source of uncertainty in our results.
More details on the uncertainties of stellar model inputs and their consequences can be found in Lebreton et al. (2014). To avoid such sources of uncertainties, direct measurements of stellar parameters should be preferred when possible.
Depending on the stellar model input physics and parameters, we obtained a large range of possible ages, between ≈ 0.2 and 9.3 Gyr with large error bars. The range of possible masses is between 0.755 and 0.810 M_{⊙}. The internal error bar on the inferred mass for an optimised stellar model based on a given set of inputs physics and parameters due to the uncertainty on the observational constraints (luminosity, effective temperature, and metallicity) is ≈ ± 0.04 M_{⊙}. This error bar appears to be small. Indeed, in the LevenbergMarquardt minimisation the error bars on the free parameters are obtained as the diagonal coefficients of the inverse of the Hessian matrix and have been shown to be smaller than those provided with other minimisation techniques (see e.g. Silva Aguirre et al. 2017). The inferred stellar radii are in the range 0.727–0.728 R_{⊙} with an internal error bar of ±0.017 R_{⊙}, while the mean densities arein the range 1.96–2.09 (± 0.22) ρ_{⊙}. We chose as reference model for the star that based on the most appropriate input physics as explained in the description above (AGSS09 solar mixture and boundary conditions from MARCS model atmospheres), and the galactic value ΔY∕ΔZ = 2 derived by Casagrande et al. (2007), which is also rather close to the Kepler Legacy seismic mean value ${(\Delta Y/\Delta Z)}_{\text{seism}}$. This particular model has M_{⋆} = 0.755 ± 0.040 M_{⊙} and an age of 9.3 Gyr. Although this mass estimate is higher than the mass we derived from interferometry and transit by ~ 8%, the interval of solutions is consistent with our uncertainties. Similarly, our radius and density are consistent with those derived from the model (0.727 ± 0.017 R_{⊙}, 1.96 ± 0.22 ρ_{⊙}, respectively). We point out that pushing the ΔY∕ΔZ value from 2 to 3 would induce a change of mass from 0.755 to 0.719 M_{⊙}, i.e. closer to the interferometric measure, but with a change in age from 9.3 to 13.8 Gyr, i.e. the age of the Universe; in our opinion this indicates that ΔY∕ΔZ values that are too high are not realistic for this star.
We point out that, as is wellknown in particular in the case of lowmass stars, the ages of stars are very poorly estimated when only the HR diagram parameters and metallicity are known because of degeneracies in the stellar models (see e.g. Lebreton et al. 2014; Ligi et al. 2016). Furthermore, other values of the classical stellar parameters of HD 219134 have been reported in the literature. To see how these reported values can modify our results we optimised stellar models on the basis of the Folsom et al. (2018) results on T_{eff} and [Fe/H] and on L_{⋆} inferred from the SIMBAD HIPPARCOS V magnitude. We obtained a similar range of masses 0.76–0.79M_{⊙}, while the models systematically point towards higher ages 10.2–13.8 Gyr, which is mainly due to the smaller T_{eff} (4756 ± 86 K) derived by Folsom et al. (2018). It is also worth pointing out that, as noted by Johnson et al. (2016), the very high ages inferred from stellar models commonly found in the literature for HD 219134 seem to be in conflict with ages from activity which, although not very precise, span the range ≈ 3–9 Gyr^{7}.
4 Planetary parameters and composition of the transiting exoplanets
The precise and accurate stellar parameters that we have determined allow us to infer the parameters of the transiting exoplanets of the system. It is then possible to derive their internal composition using an inference scheme, and to verify if they stand in a dynamical point of view.
4.1 Radius, density, and mass of the two transiting exoplanets
The two planets HD 219134 b and c transit their host star, and we can thus derive their properties. We computed the planetary radius R_{p} and mass M_{p} of each planet starting from the PDF of the stellar mass and radius. As explained by Crida et al. (2018a) concerning 55 Cnc e, for any M_{p} and M_{⋆}, we can derive the associated semiamplitude of the RV signal K following Kepler’s law, and for any pair of R_{p} and R_{⋆}, we can derive the associated transit depth ΔF. We took theΔF, K, and the period P from Gillon et al. (2017) to calculate the PDF of the planetary mass and radius following the formula (see Sect. 3.1 of Crida et al. 2018a, for more details) : $$\begin{array}{lll}{f}_{\text{p}}({M}_{\text{p}},{R}_{\text{p}})\hfill & \propto \hfill & {\displaystyle \iint \mathrm{exp}}\left(\frac{1}{2}{\left(\frac{K({M}_{\text{p}},{M}_{\star})K}{{\sigma}_{K}}\right)}^{2}\right)\hfill \\ \hfill & \hfill & \times \text{\hspace{0.17em}}\mathrm{exp}\left(\frac{1}{2}{\left(\frac{\Delta F({M}_{\text{p}},{M}_{\star})\Delta F}{{\sigma}_{\Delta F}}\right)}^{2}\right)\hfill \\ \hfill & \hfill & \times \text{\hspace{0.17em}}{{\text{L}}}_{MR\star}({M}_{\star},{R}_{\star})\text{\hspace{0.17em}d}{M}_{\star}\text{\hspace{0.17em}d}{R}_{\star}\text{}.\hfill \end{array}$$(5)
From this joint PDF, we compute the densities of both transiting exoplanets taking into account the correlation between R_{p} and M_{p} (Fig. 4).
The new values of the planetary parameters are given in Table 4. The radii of planets b and c are 1.500 ± 0.057 and 1.415 ± 0.049 R_{⊕}, respectively.Because we find that the star is smaller than initially thought, the two planets appear smaller as well; Gillon et al. (2017) give R_{p} = 1.602 ± 0.055 and 1.511 ± 0.047 R_{⊕}, and M_{p} = 4.74 ± 0.19 and 4.36 ± 0.22 M_{⊕}, for planets b and c, respectively.This enforces the idea that the two planets lie in the superEarth part of the distribution of exoplanetary radii set by Fulton et al. (2017).
Even more interestingly, planet c presents a higher density than planet b, whereas it has smaller mass and radius. From the values in Table 4, we get ρ_{b} ∕ρ_{c} = 0.901 ± 0.157 assuming ρ_{b} and ρ_{c} to be independent variables. But ρ_{b} and ρ_{c} are slightly correlated as they both depend on the stellar parameters. Estimating directly the ratio, the stellar parameters simplify out to $$\frac{{\rho}_{b}}{{\rho}_{c}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{M}_{b}/{R}_{b}^{3}}{{M}_{c}/{R}_{c}^{3}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}{\left(\frac{{P}_{b}}{{P}_{c}}\right)}^{1/3}{\left(\frac{\Delta {F}_{c}}{\Delta {F}_{b}}\right)}^{3/2}\left(\frac{{K}_{b}}{{K}_{c}}\right)\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.905\text{\hspace{0.17em}}\pm \text{\hspace{0.17em}}0.131,$$(6)
where P_{b} and P_{c} are the orbital periods of the planets; we used a standard propagation of error. This is a larger difference than between the Earth and Venus (whose density is 0.944 ρ_{⊕}). A better knowledge of the transit depth would help discriminate between the density ratio and unity. We investigated the causes of this potential disparity in the next section.
We also updated the values of the minimum masses of planets f and d, which as expected we find lower than previous estimates, and of their semimajor axes (Table 4) using Gillon et al. (2017) orbital solutions, as these planets are confirmed by several independent detection. Finally, we determined the habitable zone (HZ) of the star to verify if any of the exoplanets of this system lie in this zone. To compute the HZ, we used the method described by Jones et al. (2006), who adopted a conservative approach of this range of distances. We first computed the critical flux which depends on the T_{eff} of the star, and we derived the inner and outer boundaries of the HZ (see Eqs. (1a)–(2b) of Jones et al. 2006, for details). As a result, we find that the HZ spreads from 0.46 to 0.91 au from the star and that no planet in the system is located in this area.
Fig. 4 Joint likelihood of the planetary mass and radius for planet b (green longdashed line) and planet c (yellow solid line). The 9 contour lines separate 10 equalsized intervals between 0 and the maximum of f_{p} (M_{p}, R_{p}). The dashed lines show the isodensities corresponding to the mean densities of planets b and c. 
4.2 Internal compositions
The new mass and radius estimates allowed us to investigate the planetary interiors. Interestingly, there is a 10% density difference between the two planets (see Eq. (6)), which are otherwise very similar in mass. Although the uncertainty in the density ratio allows for no interior difference between the planets, it is worth investigating what could cause this possible difference, which is larger than that between the Earth and Venus with a probability of about 70% (or stated differently, 30% chance for a difference of less than 5%). In fact, Eq. (6) suggests that there is a 50% chance that planet b is more than 10% less dense than planet c.
The lower density of planet b can be associated with secondary atmospheres or a rock composition that is enriched in very refractory elements (Dorn et al. 2018; Dorn & Heng 2018). Recently, Bower et al. (2019) demonstrated that fully or partially molten mantle material can lower the bulk density of superEarth up to 13%. Therefore, a difference between the planetary densities may also be due to different melt fractions in both planets. In Sect. 4.2, we investigate this additional scenario and discuss its implications.
We start by solving an inference problem, for which we use the data of mass, radius (Sect. 4.1), stellar irradiation, and stellar abundances (Table 5) to infer the possible structures and compositions of both planets. Stellar abundances of rockforming elements (e.g. Fe, Mg, Si) are used as proxies for the rocky interiors to reduce interior degeneracy as proposed by Dorn et al. (2015). The differences between both planet interiors may provide evidence of their different formation or evolution history.
Parameters of the innermost exoplanets of the system HD 219134.
Median stellar abundances of HD 219134 from Hypatia catalog (Hinkel et al. 2014) after the outliers and duplicate studies were removed.
4.2.1 Inference scheme
We used theinference scheme of Dorn et al. (2017), which calculates possible interiors and their confidence ranges. Our assumptions for the interior model are similar to those in Dorn et al. (2017) and are summarised in the following. Since these two planets are smaller than ~1.8 R_{⊕}, which is suggested to be the boundary between superEarths and miniNeptunes (Fulton et al. 2017), we consider that the planets are made of ironrich cores, silicate mantles, and terrestrialtype atmospheres. In addition to following Dorn et al. (2017), we also allowed for some reduction of the mantle density as caused by a high melt fraction.
The interior parameters comprise:

core size r_{core};

size of rocky interior r_{core+mantle};

mantle composition (i.e. Fe/Si_{mantle}, Mg/Si_{mantle});

reduction factor of mantle density f_{mantle};

pressure imposed by gas envelope P_{env};

temperature of gas envelope parametrised by α (see Eq. (10));

mean molecular weight of gas envelope μ.
The prior distributions of the interior parameters used in this study are stated in Table 6.
Our interior model uses a selfconsistent thermodynamic model for solid state interiors from Dorn et al. (2017). For any given set of interior parameters, this model allows us to calculate the respective mass, radius, and bulk abundances and to compare them to the actual observed data. The thermodynamic model comprises the equation of state (EoS) of pure iron by Bouchet et al. (2013) and of the light alloy FeSi by Hakim et al. (2018), assuming 2.5% of FeSi similar to Earth’s core. For the silicatemantle, we used the model by Connolly (2009) to compute equilibrium mineralogy and density profiles given the database of Stixrude & LithgowBertelloni (2011). We allowed for a reduction of mantle densities as caused by the presence of melt. Unfortunately, the knowledge of EoS of melts is limited for pressures that occur in superEarths (e.g. Spaulding et al. 2012; Bolis et al. 2016; Wolf & Bower 2018). Therefore, we decided to use a very simplified approach in that we used a fudge factor f_{mantle} that reduces the mantle density ρ_{mantle} in each grid layer i by ρ_{mantle,i} × (1 − f_{mantle}).
For the gas layer, we used a simplified atmospheric model for a thin, isothermal atmosphere in hydrostatic equilibrium and ideal gas behaviour, which is calculated using the scaleheight model (model II in Dorn et al. 2017). The model parameters that parametrise the gas layer and that we aim to constrain are the pressure at the bottom of the gas layer P_{env}, the mean molecular weight μ, and the mean temperature (parametrised by α, see below). The thickness of the opaque gas layer d_{env} is given by $${d}_{\text{env}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}H\mathrm{ln}\frac{{P}_{\text{env}}}{{P}_{\text{out}}},$$(7)
where the amount of opaque scale heights H is determined by the ratio of P_{env} and P_{out}. The quantity P_{out} is the pressure level at the optical photosphere for a transit geometry that we fix to 20 mbar (Fortney et al. 2007). We allowed a maximum pressure P_{env} equivalent to a Venuslike atmosphere (i.e. 100 bar). The scale height H is expressed by $$H\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{{T}_{\text{env\hspace{0.17em}}}{R}^{*}}{{g}_{\text{env\hspace{0.17em}}}\mu},$$(8)
where g_{env} and T_{env} are gravity at the bottom of the atmosphere and mean atmospheric temperature, respectively. The quantity R^{*} is the universal gas constant (8.3144598 J mol^{−1} K^{−1}) and μ the mean molecular weight. The mass of the atmosphere m_{env} is directly related to the pressure P_{env} as $${m}_{\text{env}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4\pi {P}_{\text{env}}\frac{{({R}_{\text{p}}{d}_{\text{env}})}^{2}}{{g}_{\text{env}}},$$(9)
where R_{p} − d_{env} is the radius at the bottom of the atmosphere.
The atmosphere’s constant temperature is defined as $${T}_{\text{env}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\alpha {T}_{\text{eff}}\sqrt{\frac{{R}_{\star}}{2a}}\text{\hspace{0.17em}},$$(10)
where a is the semimajor axis. The factor α accounts for possible cooling and warming of the atmosphere and can vary between 0.5 and 1, which is equivalent to the observed range of albedos among solar system bodies (0.05 for asteroids up to 0.96 for Eris). The upper limit of 1 is verified against the estimated α_{max} (see Appendix A in Dorn et al. 2017), which takes possible greenhouse warming into account.
Prior ranges for interior parameters.
4.2.2 Inference results
Figure 5 summarises the interior estimates. Both planets have mantle compositions and core sizes that fit bulk density and the stellar abundance constraint. The core fraction of both planets is close to that of Venus and Earth (${({r}_{\text{core}}/{r}_{\text{core+mantle}})}_{\oplus}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}0.53$), which validates their denomination as superEarths. Compared to planet c, the lower density of 10% of planet b is associated with a slightly smaller core (by 10%) and higher f_{mantle} (by 45%), which indicates that a significantly stronger reduction of mantle density is plausible given the data. The estimates of f_{mantle} for planet b and c are ${0.073}_{0.05}^{+0.06}$ and ${0.05}_{0.04}^{+0.06}$, respectively. Factors of f_{mantle} up to 0.25 can be associated with high melt fractions (for Earthsized planets). Similar values can be achieved when the mantle composition is enriched by very refractory elements (i.e. Al, Ca).
It should be noted that differences between the interiors are small, since uncertainties on bulk densities are relatively large. The data allow for no difference in bulk densities. However, a significant (more than 5%) difference exists with 70% probability. In this work, we used an interior model that allows us to quantify any possible difference in the rocky interiors of both planets. We assumed that any volatile layer is limited to a 100 bar atmosphere (similar to Venus) at maximum. Further arguments are necessary to evaluate whether a difference between the rocky interiors, specifically the mantle densities, can exist.
Nonetheless, because Bower et al. (2019) demonstrated that for Earthsized planets a fully molten mantle is 25% less dense than a solidified mantle, this possibility must be considered, and it is interesting to investigate whether planet b could be less dense because partially molten. Heating by irradiation from the host star would not be enough; the blackbody equilibrium temperature for this planet is 1036 K. Nevertheless, in the next subsection, we discuss a possible dynamical origin for the possible difference between HD 219134 b and c.
4.3 Possible origin of a partial mantle melt for HD 219134 b
Large melt fractions may be sustained on planet b by tidal heating. In the case of synchronous rotation with spinorbit alignment, which is likely for closein planets such as HD 219134 b, tidal dissipation acts only on planets on eccentric orbits around the star. The power is given by (see e.g. Lainey et al. 2009) $$\dot{E}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{21}{2}\frac{{k}_{2}}{Q}\frac{{(\omega {R}_{\text{p}})}^{5}}{G}{e}^{2},$$(11)
where k_{2} is the Love number and Q the quality factor of the planet of radius R_{p} and spin or orbital frequency ω. The key parameter $\frac{{k}_{2}}{Q}$ depends on the internal properties of the body^{8}. The dissipated energy Ė heats the planet and damps the eccentricity of the orbit, ultimately leading to its circularisation and a reduction of the semimajor axis. To maintain tidal heating, the orbital eccentricity must be excited by the interaction with other secondary objects, as is the case for Jupiter’s moon Io for instance. In order to investigate if tidal heating on planet b is sufficient enough, we ran numerical simulations of the planetary system using the Nbody code SyMBA (Duncan et al. 1998).
To build our initial conditions, we took the e, ϖ, orbital periods, K, and midtransit time from Gillon et al. (2017). They measure a nonzero eccentricity for planets c, f, and d, but not for planet b, whose eccentricity is fixed to zero to fit the other orbital parameters. They do not provide data for the outermost two planets g and h, but the long orbital periods of these planets make them unlikely to affect the inner four planets, and their orbital parameters suffer larger uncertainty so we neglect them in our simulations. We find that the eccentricity of planet b is excited by the other planets. In absence of dissipation, the system is stable for at least 1 Gyr, and e_{b} oscillates freely between 0 and 0.13 with a period of a few thousand years^{9}.
Introducing dissipation in planets b and c, the eccentricities are damped and e_{b} settles to a regime where it oscillates between 0.01 and 0.06. The energy loss is balanced by an inward drift of the planets, mainly planet b. We note that the final value of the eccentricity is independent of the assumed value for k_{2} ∕Q, only the timescale of the evolution and inward drift are proportional to k_{2} ∕Q. Because of this dissipation, the period ratio P_{c}∕P_{b} increases with time, and it is possible that this ratio (which is now 2.19) was smaller than 2, so that the 2:1 mean motion resonance was crossed recently. To check the effect of this phenomenon, we start planets c and especially b slightly out of their present position, inside the 2:1 mean motion resonance. Crossing the resonance at 14.6 Myrs kicks the eccentricities of planets b and c, but this is quickly damped and the eccentricity of planet b ends up oscillating between 0.005 and 0.037 with a period of ~3000 yr when it reaches its present semimajor axis at 73 Myrs, as shown in Fig. 6. Meanwhile, e_{c} converges to 0.025 (while Gillon et al. 2017, find 0.062 ± 0.039). We checked that again, k_{2}∕Q has little influence on the final behaviour of the eccentricities, although the speed at which the resonance is crossed matters.[23.5pc]
Using Eq. (11), 0.005 < e_{b} < 0.037 gives a total power for the tidal heating oscillating in $5.6308\times {10}^{16}\times \left(\frac{{k}_{2}/Q}{0.025}\right)$ W for planet b and around $2.1\times {10}^{16}\times \left(\frac{{k}_{2}/Q}{0.025}\right)$ W for planet c. For reference, tidal heating in Io is of the order of 10^{14} W (Lainey et al. 2009)so that, assuming k_{2}∕Q = 0.025 like for Earth, planet b receives at least 2 and up to 100 times more tidal heating per mass unit than Io (and almost 300 with e_{b} = 0.06). In contrast, because in Eq. (11) the term ${(\omega {R}_{p})}^{5}$ is 70 times smaller for planet c than for planet b, all other parameters being equal, it should be heated much less. We find that it gets a bit less tidal heating than Io per mass unit, so it isunlikely to melt even partially. In the end, the idea of a partial (if not total) melt of the mantle of HD 219134 b to explain its possibly lower density than planet c is strongly supported by dynamics. A refinement of the parameters of the system and a complete stability analysis would help but are beyond the scope of this paper.
Fig. 5 Onedimensional marginalised posteriors of interior parameters: thickness of atmosphere (d_{env}), size of rocky interior (r_{core+mantle}), core size (r_{core}), fudge factor f_{mantle}, and mantle composition (Fe/Si_{mantle} and Mg/Si_{mantle}). The prior distribution is shown in dashed lines (except for d_{env}, for which no explicit prior is defined), while the posterior distribution is shown in solid lines for planets b (red) and c (blue). 
5 Summary and conclusions
We present a new analysis of the exoplanetary system HD 219134. We observed the star with the VEGA/CHARA interferometer and measured an angular diameter of 1.035 ± 0.021 mas and a radius of 0.726 ± 0.014 R_{⊙}. This radius is not significantly affected by the Gaia offset, but new values from the DR3 or DR4 will allow us to refine R_{⋆}. We used the transit parameters from Gillon et al. (2017) to measure the stellar density (1.82 ± 0.19 ρ_{⊙}) directly, and we directly derived from these two measurements the stellar mass (0.696 ± 0.078M_{⊙}) and the correlation between M_{⋆} and R_{⋆} (0.46). We compare our parameters with those obtained with C2kSMO and find that the range of masses is compatible with the directly measured mass, although the best model gives a mass 8% higher than the directly measured mass. This corresponds to an age of 9.3 Gyr, but a large range of ages is possible (0.2–9.3 Gyr). Similarly, previous indirect determinations of M_{⋆} show higher values than our measurement (see e.g. Boyajian et al. 2012b; Gillon et al. 2017), but it has to be noted that they are based on a larger R_{⋆}.
The system includes two transiting exoplanets, HD 219134 b and c, for which we reassess the parameters. Using our new R_{⋆} and M_{⋆}, we computed the PDF of the planetary masses and radii, which we find lower than previous estimates (since previous stellar parameters were higher), and the correlations between M_{p} and R_{p}. These new values clearly validate the superEarth nature of the two planets by putting them out of the gap in the exoplanetary radii distribution noticed by Fulton et al. (2017). We could thus derive the densities of the planets, which appear to differ by 10%, although these values are possibly identical within the error bars (70% chance that the difference is more than 5%). More interestingly, planet b has a lower density than planet c despite its higher mass. Using Dorn et al. (2017) inference scheme, we show that this difference in density can be attributed to a slightly smaller core and/or a significantly lower mantle density. The latter might be due to a molten fraction. Tidal heating might be the cause of such a melting, as we investigated using the SyMBA Nbody code. Excited by the other planets, the eccentricity of planets b and c reaches ~ 0.02 with tidal dissipation. This could lead to considerable heating for planet b (100 times more than on Io per mass unit, possibly leading to partial melting of the mantle), while planet c is too far from the star for tidal heating to be more intense than on Io. Hence, despite their possible density difference, planets b and c may have the same composition, as expected in all standard planet formation models.
The system of HD 219134 constitutes a benchmark case for both stellar and planetary sciences. Our direct estimation of the stellar radius and mass directly impacts the planetary parameters. Although within the error bars of the mass coming from C2kSMO, our new mass changes the planetary mass and the possibilities of interior structures compared to the possible solutions using the stellar models. Improving the precision of the transit light curves of the two planets would allow us to reduce the uncertainty on the stellar density, hence on the stellar mass. It would reduce the uncertainty on the planetary parameters even more, potentially answering the question of the density ratio of the two transiting superEarths. More generally, measuring the stellar radius and density as we have done in this work is the most direct method to infer stellar (hence planetary) parameters and should be more extensively used; this approach will certainly be possible within the Transiting Exoplanet Survey Satellite (TESS) and PLAnetary Transits and Oscillations of stars (PLATO) missions era.
Fig. 6 Evolution of the inner two planets of the HD 219134 system under dissipation with k_{2} ∕Q = 0.025 for both planets, and in presence of planets f and d of Gillon et al. (2017). Top panel: semimajor axis of planet b (red curve), location of the 1:2 mean motion resonance with planet c (green curve, that is a_{c}∕2^{2∕3}), and present semimajor axis of planet b (blue horizontal line). Bottom panel: eccentricities of planets b (red) and c (green). 
Acknowledgements
We thank the anonymous referee who brought useful comments and significantly helped improve our manuscript. R.L. has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement n. 664931. C.D. acknowledges support from the Swiss National Science Foundation under grant PZ00P2_174028. We thank A. Nakajima for her help with the initial conditions of the Nbody simulations. This work is based upon observations obtained with the Georgia State University Center for High Angular Resolution Astronomy Array at Mount Wilson Observatory. The CHARA Array is supported by the National Science Foundation under Grants No. AST1211929 and AST1411654. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This research has made use of the JeanMarie Mariotti Center SearchCal service^{10} codeveloped by LAGRANGE and IPAG, and of CDS Astronomical Databases SIMBAD and VIZIER^{11}. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.
Appendix A Selected log (g) and metallicity from literature
References
 Allende Prieto, C., Barklem, P. S., Lambert, D. L., & Cunha, K. 2004, A&A, 420, 183 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Asplund, M., Grevesse, N., Sauval, A. J., & Scott, P. 2009, ARA&A, 47, 481 [NASA ADS] [CrossRef] [Google Scholar]
 Baglin, A. 2003, Adv. Space Res., 31, 345 [NASA ADS] [CrossRef] [Google Scholar]
 Baines, E. K., Döllinger, M. P., Cusano, F., et al. 2010, ApJ, 710, 1365 [NASA ADS] [CrossRef] [Google Scholar]
 Boeche, C., & Grebel, E. K. 2016, A&A, 587, A2 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 BöhmVitense, E. 1958, Z. Astrophys., 46, 108 [Google Scholar]
 Bolis, R., Morard, G., Vinci, T., et al. 2016, Geophys. Res. Lett., 43, 9475 [NASA ADS] [CrossRef] [Google Scholar]
 Bonneau, D., Clausse, J.M., Delfosse, X., et al. 2006, A&A, 456, 789 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Boro Saikia, S., Marvin, C. J., Jeffers, S. V., et al. 2018, A&A, 616, A108 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Bouchet, J., Mazevet, S., Morard, G., Guyot, F., & Musella, R. 2013, Phys. Rev. B, 87, 094102 [NASA ADS] [CrossRef] [Google Scholar]
 Bourgés, L., Lafrasse, S., Mella, G., et al. 2014, ASP Conf. Ser., 485, 223 [NASA ADS] [Google Scholar]
 Bourrier, V., Dumusque, X., Dorn, C., et al. 2018, A&A, 619, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Bower, D. J., Kitzmann, D., Wolf, A. S., et al. 2019, A&A, in press http://doi.org/10.1051/00046361/201935710 [Google Scholar]
 Boyajian, T. S., McAlister, H. A., van Belle, G., et al. 2012a, ApJ, 746, 101 [NASA ADS] [CrossRef] [Google Scholar]
 Boyajian, T. S., von Braun, K., van Belle, G., et al. 2012b, ApJ, 757, 112 [NASA ADS] [CrossRef] [Google Scholar]
 Canuto, V. M., Goldman, I., & Mazzitelli, I. 1996, ApJ, 473, 550 [NASA ADS] [CrossRef] [Google Scholar]
 Casagrande, L., Flynn, C., Portinari, L., Girardi, L., & Jimenez, R. 2007, MNRAS, 382, 1516 [NASA ADS] [CrossRef] [Google Scholar]
 Chelli, A., Duvert, G., Bourgès, L., et al. 2016, A&A, 589, A112 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Claret, A., & Bloemen, S. 2011, A&A, 529, A75 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Coelho, H. R., Chaplin, W. J., Basu, S., et al. 2015, MNRAS, 451, 3011 [NASA ADS] [CrossRef] [Google Scholar]
 Connolly, J. 2009, Geochem. Geophys. Geosyst., 10, Q10014 [NASA ADS] [CrossRef] [Google Scholar]
 Creevey, O. L., Thévenin, F., Boyajian, T. S., et al. 2012, A&A, 545, A17 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Creevey, O. L., Thévenin, F., Berio, P., et al. 2015, A&A, 575, A26 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Crida, A., Ligi, R., Dorn, C., & Lebreton, Y. 2018a, ApJ, 860, 122 [NASA ADS] [CrossRef] [Google Scholar]
 Crida, A., Ligi, R., Dorn, C., Borsa, F., & Lebreton, Y. 2018b, Res. Notes Am. Astron. Soc., 2, 172 [NASA ADS] [CrossRef] [Google Scholar]
 da Silva, R., Milone, A. d. C., & RochaPinto, H. J. 2015, A&A, 580, A24 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Dorn, C., & Heng, K. 2018, ApJ, 853, 64 [NASA ADS] [CrossRef] [Google Scholar]
 Dorn, C., Khan, A., Heng, K., et al. 2015, A&A, 577, A83 [EDP Sciences] [Google Scholar]
 Dorn, C., Venturini, J., Khan, A., et al. 2017, A&A, 597, A37 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Dorn, C., Harrison, J. H., Bonsor, A., & Hands, T. O. 2018, MNRAS, 484, 712 [Google Scholar]
 Duncan, M. J., Levison, H. F., & Lee, M. H. 1998, AJ, 116, 2067 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Folsom, C. P., Fossati, L., Wood, B. E., et al. 2018, MNRAS, 481, 5286 [NASA ADS] [CrossRef] [Google Scholar]
 Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659, 1661 [NASA ADS] [CrossRef] [Google Scholar]
 Frasca, A., Covino, E., Spezzi, L., et al. 2009, A&A, 508, 1313 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109 [NASA ADS] [CrossRef] [Google Scholar]
 Gaia Collaboration (Prusti, T., et al.) 2016, A&A, 595, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gaia Collaboration (Brown, A. G. A., et al.) 2018, A&A, 616, A1 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gennaro, M., Prada Moroni, P. G., & Degl’Innocenti, S. 2010, A&A, 518, A13 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Gillon, M., Demory, B.O., Van Grootel, V., et al. 2017, Nat. Astron., 1, 0056 [NASA ADS] [CrossRef] [Google Scholar]
 Gray, R. O., Corbally, C. J., Garrison, R. F., McFadden, M. T., & Robinson, P. E. 2003, AJ, 126, 2048 [NASA ADS] [CrossRef] [Google Scholar]
 Grevesse, N., & Noels, A. 1993, in Origin and Evolution of the Elements, eds. N. Prantzos, E. VangioniFlam, & M. Casse (Singapore: World Scientific), 15 [Google Scholar]
 Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Hakim, K., Rivoldini, A., Van Hoolst, T., et al. 2018, Icarus, 313, 61 [NASA ADS] [CrossRef] [Google Scholar]
 Heiter, U., & Luck, R. E. 2003, AJ, 126, 2015 [NASA ADS] [CrossRef] [Google Scholar]
 Henry, T. J., & McCarthy, Jr., D. W. 1993, AJ, 106, 773 [NASA ADS] [CrossRef] [Google Scholar]
 Hidalgo, S. L., Pietrinferni, A., Cassisi, S., et al. 2018, ApJ, 856, 125 [NASA ADS] [CrossRef] [Google Scholar]
 Hinkel, N. R., Timmes, F., Young, P. A., Pagano, M. D., & Turnbull, M. C. 2014, AJ, 148, 54 [NASA ADS] [CrossRef] [Google Scholar]
 Huber, D., Ireland, M. J., Bedding, T. R., et al. 2012, ApJ, 760, 32 [NASA ADS] [CrossRef] [Google Scholar]
 Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, MNRAS, 470, 1750 [NASA ADS] [CrossRef] [Google Scholar]
 Johnson, M. C., Endl, M., Cochran, W. D., et al. 2016, ApJ, 821, 74 [NASA ADS] [CrossRef] [Google Scholar]
 Jones, B. W., Sleep, P. N., & Underwood, D. R. 2006, ApJ, 649, 1010 [NASA ADS] [CrossRef] [Google Scholar]
 Lainey, V., Arlot, J.E., Karatekin, Ö., & van Hoolst, T. 2009, Nature, 459, 957 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Lallement, R., Vergely, J.L., Valette, B., et al. 2014, A&A, 561, A91 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lebreton, Y., & Goupil, M. J. 2014, A&A, 569, A21 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lebreton, Y., Goupil, M. J., & Montalbán, J. 2014, EAS Pub. Ser., 65, 99 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lee, Y. S., Beers, T. C., Allende Prieto, C., et al. 2011, AJ, 141, 90 [NASA ADS] [CrossRef] [Google Scholar]
 Lejeune, T., Cuisinier, F., & Buser, R. 1997, VizieR Online Data Catalog: J/A+A/412/50229 [Google Scholar]
 Ligi, R., Mourard, D., Lagrange, A. M., et al. 2012, A&A, 545, A5 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ligi, R., Mourard, D., Nardetto, N., & Clausse, J.M. 2013, J. Astron. Instrum., 02, 1340003 [NASA ADS] [CrossRef] [Google Scholar]
 Ligi, R., Creevey, O., Mourard, D., et al. 2016, A&A, 586, A94 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Lindegren, L., Hernández, J., Bombrun, A., et al. 2018, A&A, 616, A2 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Luck, R. E., & Heiter, U. 2005, AJ, 129, 1063 [NASA ADS] [CrossRef] [Google Scholar]
 Luck, R. E., & Heiter, U. 2006, AJ, 131, 3069 [NASA ADS] [CrossRef] [Google Scholar]
 Luri, X., Brown, A. G. A., Sarro, L. M., et al. 2018, A&A, 616, A9 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Maldonado, J., Eiroa, C., Villaver, E., Montesinos, B., & Mora, A. 2012, A&A, 541, A40 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Maldonado, J., Eiroa, C., Villaver, E., Montesinos, B., & Mora, A. 2015, A&A, 579, A20 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mamajek, E. E., & Hillenbrand, L. A. 2008, ApJ, 687, 1264 [NASA ADS] [CrossRef] [Google Scholar]
 Mishenina, T. V., Soubiran, C., Kovtyukh, V. V., & Korotin, S. A. 2004, A&A, 418, 551 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mishenina, T. V., Soubiran, C., Kovtyukh, V. V., Katsova, M. M., & Livshits, M. A. 2012, A&A, 547, A106 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mishenina, T. V., Pignatari, M., Korotin, S. A., et al. 2013, A&A, 552, A128 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Morel, P., & Lebreton, Y. 2008, Ap&SS, 316, 61 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Motalebi, F., Udry, S., Gillon, M., et al. 2015, A&A, 584, A72 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mourard, D., Clausse, J. M., Marcotto, A., et al. 2009, A&A, 508, 1073 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mourard, D., Bério, P., Perraut, K., et al. 2011, A&A, 531, A110 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mourard, D., Challouf, M., Ligi, R., et al. 2012, SPIE Conf. Ser., 8445, 84450K [Google Scholar]
 Mourard, D., Monnier, J. D., Meilland, A., et al. 2015, A&A, 577, A51 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Nordlund, Å., Stein, R. F., & Asplund, M. 2009, Liv. Rev. Sol. Phys., 6, 2 [Google Scholar]
 Oja, T. 1993, A&AS, 100, 591 [NASA ADS] [Google Scholar]
 Pichierri, G., Morbidelli, A., & Crida, A. 2018, Celest. Mech. Dyn. Astron., 130, 54 [NASA ADS] [CrossRef] [Google Scholar]
 Prugniel, P., Vauglin, I., & Koleva, M. 2011, A&A, 531, A165 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ramírez, I., Allende Prieto, C., & Lambert, D. L. 2007, A&A, 465, 271 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ramírez, I., Fish, J. R., Lambert, D. L., & Allende Prieto, C. 2012, ApJ, 756, 46 [NASA ADS] [CrossRef] [Google Scholar]
 Ramírez, I., Allende Prieto, C., & Lambert, D. L. 2013, ApJ, 764, 78 [NASA ADS] [CrossRef] [Google Scholar]
 Seager, S., & MallénOrnelas, G. 2003, ApJ, 585, 1038 [NASA ADS] [CrossRef] [Google Scholar]
 Silva Aguirre, V., Lund, M. N., Antia, H. M., et al. 2017, ApJ, 835, 173 [NASA ADS] [CrossRef] [Google Scholar]
 Soubiran, C., Bienaymé, O., Mishenina, T. V., & Kovtyukh, V. V. 2008, A&A, 480, 91 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Spaulding, D., McWilliams, R., Jeanloz, R., et al. 2012, Phys. Rev. Lett., 108, 065701 [NASA ADS] [CrossRef] [Google Scholar]
 Stassun, K. G., & Torres, G. 2018, ApJ, 862, 61 [NASA ADS] [CrossRef] [Google Scholar]
 Stixrude, L., & LithgowBertelloni, C. 2011, Geophys. J. Int., 184, 1180 [NASA ADS] [CrossRef] [Google Scholar]
 TallonBosc, I., Tallon, M., Thiébaut, E., et al. 2008, SPIE Conf. Ser., 7013, 44 [Google Scholar]
 ten Brummelaar, T. A., McAlister, H. A., Ridgway, S. T., et al. 2005, ApJ, 628, 453 [NASA ADS] [CrossRef] [Google Scholar]
 Valenti, J. A., & Fischer, D. A. 2005, ApJS, 159, 141 [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
 Vogt, S. S., Burt, J., Meschiari, S., et al. 2015, ApJ, 814, 12 [NASA ADS] [CrossRef] [Google Scholar]
 Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Wolf, A. S., & Bower, D. J. 2018, Phys. Earth Planet. Inter., 278, 59 [NASA ADS] [CrossRef] [Google Scholar]
Available at http://cdsweb.ustrasbg.fr/
In the solar model calibration process, the evolution of a 1 M_{⊙} model is calculated up to the known solar age. Its initial helium content and mixinglength parameter are fixed by the constraint that at solar age, the model has reached the observed values of the solar radius, luminosity, and surface metallicity.
We estimated this age range from the empirical relation relating the CaII H & K emission index ${R}_{\text{HK}}^{\prime}$ and age derived by Mamajek & Hillenbrand (2008), with the value of ${R}_{\text{HK}}^{\prime}$ measured by Boro Saikia et al. (2018).
Using initial circular orbits, i.e. assuming that the planets were fully formed locally in the protoplanetary disc, we observe no increase of the eccentricities of the four planets in 500 Myrs. This is not compatible with the observations of Gillon et al. (2017); this suggests that these four planets may not have acquired their final mass and/or orbits during the protoplanetary disc phase. A phase of giant impacts or the breaking of a resonance chain (Izidoro et al. 2017; Pichierri et al. 2018) could have happened in the early history of the system.
Available at http://www.jmmc.fr/searchcal
Available at http://cdsweb.ustrasbg.fr/
All Tables
Median stellar abundances of HD 219134 from Hypatia catalog (Hinkel et al. 2014) after the outliers and duplicate studies were removed.
All Figures
Fig. 1 Squared visibilities obtained with VEGA/CHARA for HD 219134. The different colours represent the data points obtained with different baselines. The solid line represents the model of LD diameter. 

In the text 
Fig. 2 Joint likelihood of the radius and mass of the star HD 219134. The 9 plain red contour lines separate 10 equalsized intervals between 0 and the maximum of Eq. (2). 

In the text 
Fig. 3 Photometric data (black squares) and fitted model (solid red line) from the BaSeL library of spectra. 

In the text 
Fig. 4 Joint likelihood of the planetary mass and radius for planet b (green longdashed line) and planet c (yellow solid line). The 9 contour lines separate 10 equalsized intervals between 0 and the maximum of f_{p} (M_{p}, R_{p}). The dashed lines show the isodensities corresponding to the mean densities of planets b and c. 

In the text 
Fig. 5 Onedimensional marginalised posteriors of interior parameters: thickness of atmosphere (d_{env}), size of rocky interior (r_{core+mantle}), core size (r_{core}), fudge factor f_{mantle}, and mantle composition (Fe/Si_{mantle} and Mg/Si_{mantle}). The prior distribution is shown in dashed lines (except for d_{env}, for which no explicit prior is defined), while the posterior distribution is shown in solid lines for planets b (red) and c (blue). 

In the text 
Fig. 6 Evolution of the inner two planets of the HD 219134 system under dissipation with k_{2} ∕Q = 0.025 for both planets, and in presence of planets f and d of Gillon et al. (2017). Top panel: semimajor axis of planet b (red curve), location of the 1:2 mean motion resonance with planet c (green curve, that is a_{c}∕2^{2∕3}), and present semimajor axis of planet b (blue horizontal line). Bottom panel: eccentricities of planets b (red) and c (green). 

In the text 
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