EDP Sciences
Free Access
Issue
A&A
Volume 608, December 2017
Article Number A129
Number of page(s) 62
Section Catalogs and data
DOI https://doi.org/10.1051/0004-6361/201730993
Published online 14 December 2017

© ESO, 2017

1. Introduction

Hot Jupiters are a rare type of exoplanet existing around 0.5 to 1% of solar-type stars (Mayor & Queloz 1995; Howard et al. 2012; Santerne et al. 2016). Their sizes range from ~ 0.7 RJup to around 2 RJup (Sato et al. 2005; Anderson et al. 2010), a range in size which also corresponds to late M dwarfs with masses lower than ~ 0.2 M (Chabrier & Baraffe 1997; Baraffe et al. 1998, 2015; Dotter et al. 2008; Demory et al. 2009; Díaz et al. 2014; Hatzes & Rauer 2015; Chen & Kipping 2017). To further the comparison, gas giants, brown dwarfs, and very low-mass stars also have similar temperatures in addition to similar sizes. Hot Jupiters’ dayside temperatures range from ~ 800 up to ~ 4600 K (Triaud et al. 2015; Gaudi et al. 2017). This means that all those objects share a similar parameter space in colour–magnitude diagrams (Triaud 2014), with several planets re-emitting more flux than some stars. This easily explains how photometric surveys designed to detect transiting gas giants also net within their data a large number of low-mass stellar secondary companions, which we report here. All of our systems were identified with the CORALIE spectrograph, while distinguishing which of many candidates provided by the Wide Angle Search for Planets (WASP1; Pollacco et al. 2006) were indeed planets. Our paper is part of an ongoing investigation into eclipsing binaries with low mass (EBLM) following three previous instalments which focused on four specific targets (Triaud et al. 2013; Gómez Maqueo Chew et al. 2014; von Boetticher et al. 2017).

The main objective of the EBLM project is to empirically measure the mass/radius relationship at the bottom of the main sequence, and compare it to theoretical expectations (Chabrier & Baraffe 1997; Baraffe et al. 1998, 2015; Dotter et al. 2008). This can be done by making careful measurements of the ratio of sizes of the two components during eclipse, and of their mass function thanks to radial velocities. Assuming parameters for the primaries, as is often done for exoplanetary studies, we can derive accurate physical parameters for the secondaries. Information about the primaries will soon be refined thanks to the Gaia parallaxes (de Bruijne 2012), as well as possibly thanks to asteroseismologic measurements collected by the forthcoming Plato (Rauer et al. 2014).

The photometric identification of an eclipsing low-mass star vs. a transiting gas-giant is similar. We will use our sample of low-mass eclipsing secondaries to serve as comparison sample to the gas-giants discovered by WASP. We also aim to revisit the relative frequencies of low-mass stars to planets, to brown dwarfs in order to confirm the presence, extent and dryness of the brown dwarf desert (Marcy & Butler 2000; Grether & Lineweaver 2006; Sahlmann et al. 2011; Ma & Ge 2014). We will compare the stellar and planetary eccentricity/period and eccentricity/mass distributions in order to study tides (Zahn & Bouchet 1989; Mathieu et al. 1990; Mazeh et al. 1997; Terquem et al. 1998; Meibom & Mathieu 2005; Milliman et al. 2014), investigate whether the statistics on the presence of additional perturbing bodies are similar (Tokovinin et al. 2006; Knutson et al. 2014; Neveu-VanMalle et al. 2016), and find out if the spin–orbit misalignments frequently observed in hot Jupiter systems are also observed in binary stars thanks to the Rossiter–McLaughlin effect (Hale 1994; Triaud et al. 2010; Albrecht et al. 2014; Esposito et al. 2014; Lendl et al. 2014; Winn & Fabrycky 2015).

Another important goal is to warn fellow planet hunters operating within the celestial Southern Hemisphere HAT-South (Bakos et al. 2013), KELT (Pepper et al. 2012), ASTEP (Crouzet et al. 2010), NGTS (Wheatley et al. 2013), K2 (Howell et al. 2014), TESS (Ricker et al. 2014) where are located many systems most likely to masquerade as hot Jupiters.

We emphasise that whilst all of these binaries were discovered by WASP photometrically in eclipse, these data are not present in this paper, nor are any secondary radii that may be inferred from them. The low precision and presence of some mis-understood systematics mean that presenting the WASP photometric data on its own would be misleading2. However, taking the time to follow-up each of these photometrically is beyond the manpower of our team. While we intend to follow some systems (and have already for a few), we cannot do everything. This release is therefore an opportunity for the community to help characterise the mass/radius relationship of the smallest stars within our Galaxy by collecting high-quality photometric data during primary and secondary eclipses. In particular, the 34 secondaries with masses below 0.2 M will ultimately double the number of known objects in that part of the mass-radius diagram.

Late M dwarfs form the bulk of the stellar population (Chabrier 2003; Henry et al. 2006). There exist a number of dedicated surveys to discover planets around small stars, such as MEarth (Nutzman & Charbonneau 2008) and Apache (Giacobbe et al. 2012), with new surveys such as SPECULOOS coming online shortly (Gillon et al. 2013). As planets are known to orbit these stars, they will likely reveal the most frequent pathway to planet formation. The bottom of the main sequence is also where it is most optimal to discover and to study the atmospheres of Earth-sized worlds (e.g. de Wit et al. 2016; He et al. 2017), with the recently discovered TRAPPIST-1, a particularly suited example (Gillon et al. 2016, 2017; Luger et al. 2017).

In the following section, we will describe our instrument, how we selected the targets and how the observations were finally obtained. In Sect. 3 we present our data reduction, and the treatment leading to the radial velocities and their uncertainties. Section 4 details how we adjust a Keplerian model to the radial velocities, with Sect. 5 focusing on our model selection when several appear compatible with the data. Our results appear in Sect. 6, and our interpretations in Sect. 7. There are extensive appendices containing tables supporting the main text, as well as a graphical representation of the orbits for each of the 118 systems that we announce here.

thumbnail Fig. 1

Top: timespan of observations over binaries of different periods. The flat distribution means that our sensitivity to tertiary objects is roughly independent of the binary period. Middle: number of orbits covered for each binary. There is no strict minimum although 86% of binaries have been observed over a timespan covering at least 100 orbits. Bottom: number of observations given to each binary. There is a requirement of at least 13 observations.

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2. Observational campaign

2.1. The CORALIE instrument

CORALIE (Queloz et al. 2001b), is a thermally stabilised (but not pressure-stabilised), high-resolution, fibre-fed spectrograph, mounted on the 1.2 m Euler Telescope, a facility belonging to the University of Geneva and installed at ESO’s observatory of La Silla, in Chile. The spectrograph was built on ELODIE’s design (Baranne et al. 1996), which was installed on the 1.93 m at OHP and produced the first radial-velocity detection of an exoplanet (Mayor & Queloz 1995). The wavelength solution is obtained by simultaneously illuminating a CCD detector with the star, and with a thorium-argon calibration lamp (Lovis & Pepe 2007). Radial velocities are extracted by cross-correlating the observed spectrum with a numerical mask. The resulting cross-correlation function (CCF) is fitted with a Gaussian profile whose mean corresponds to the radial velocity.

In 2007, CORALIE received a major upgrade allowing it to be more efficient and appropriate for the detection of gas-giants orbiting star as faint as V~13 (Wilson et al. 2008; Ségransan et al. 2010). CORALIE has a resolution of order 55 000. Since 2007, we have announced in excess of a 100 transiting planets in collaboration with WASP (e.g. Turner et al. 2016), with several dozens remaining in preparation. Further improvements to the instrument were conducted in November 2014 (change from circular to octagonal fibres), and in April 2015 (wavelength solution now done using a Fabry-Pérot). The first of these two operations produced a small offset in the zero-point of the instrument, of order 10 m s-1, which remains irrelevant for the precision we obtained on the binary star sample but which needs to be accounted for when deriving orbits for stars with planets (Triaud et al. 2017).

2.2. Target selection and observing campaign

thumbnail Fig. 2

Top: coordinates of each of the EBLM binaries (blue circles) and a comparative distribution of the coordinates of the WASP planets (purple triangles). Bottom: histogram of the visual magnitude.

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Stars showing periodic photometric dimmings consistent with the transit of a hot Jupiter are identified by WASP using an algorithm named hunter (Collier Cameron et al. 2007). An analysis of the stellar colours, of their reduced proper-motion and of the duration of the events permits an exclusion of most giant primaries, as well as a preliminary estimate of the stellar radius (R). The depth of the event, () leads to an estimate of the transiter’s size. If its radius is consistent with Rp ≤ 2.1 RJup and no ellipsoidal variation is initially detected then the object is kept and becomes a planet candidate.

The spectroscopic validation of candidates with declination δ< + 10° is done using CORALIE (Triaud 2011). We start with two exposures of 1800 s, timed to be near the expected radial velocity maximum and minimum. Any amount of radial velocity variation is investigated (even if anti-phased) until the nature of the variation is understood (wrong period from WASP, long-period binaries, stellar activity, chance alignment with another eclipsing system, EBLM, etc.). These two spectra are taken on every star except if on the first attempt we detect a secondary set of lines, in which case we classify this object as a double-line binary (SB2), which is no longer observed.

If there is a radial-velocity variation of less than 100 km s-1 between the two first epochs, but in excess of order 56 km s-1, we classify the object temporarily as part of the EBLM project. Systems with lower variations are followed-up intensively as planetary or brown dwarf candidates, and systems above the criterion are discarded. An amplitude of 100 km s-1 corresponds approximately to a secondary mass of 0.6 M, for an orbital period of 15 days about a 1 M primary. These requirements therefore contain all the secondaries that we could possibly be interested in.

Figures 1 and 2 graphically represent several characteristics of this current data release. We decided to include all EBLM candidates for which we had at least 13 radial velocity measurement by 2016-03-14, and where the orbital period derived from radial velocities is consistent with that derived from photometry (i.e. all are confirmed to be eclipsing). Thirteen measurements correspond to the bare minimum necessary to adjust up to two Keplerian models through the data, although most of the time this is not needed. Usually, the 11+ measurements that complement the first pair, were obtained at reduced exposures of 600 and 900 s (since the semi-amplitudes are much larger than planets), and as high a precision is not required. Many systems received more visits for a variety of reasons including: detection of a tertiary companion, testing whether our uncertainties on periods and eccentricities are robustly determined, and a limited attempt to detect circumbinary objects. Observations were spread over more than three years for the majority of systems, which are mostly contained between V magnitudes 9 and 13, just like the hot Jupiters we identified. There is a spread in the timespan between roughly one and eight years, because new targets were provided by WASP for spectroscopic follow-up progressively. The amount of time spent on a given target is roughly independent of the binary period, so as to limit any potential biases. We also present the amount of orbits covered (timespan/P), indicating that a large majority of targets have been monitored for over 100 orbits. This is important for identifying stellar trends like activity, as well as radial velocity drifts induced by a tertiary star. The EBLMs are spread almost uniformly across the Southern skies in declination and right ascension, with the exception of the galactic plane (α~6−7 h and α~16−17 h), which has not been observed by WASP due to heavy stellar crowding.

thumbnail Fig. 3

Top: FWHM of the CCF as a function of binary period and a red dashed trend fitted to the data for P< 8 d. Roughly beyond this period the primary stars are rotating slow enough such that the instrumental broadening dominates the rotational broadening, and truncates any potentially continued trend below 7 km s-1. Bottom: precision of the radial velocity measurements for each binary as a function of binary period. The red dashed line shows the median precision for all binaries within four coarse period bins, chosen for the later study of triple systems in Sect. 6.4.

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Across all 118 targets, we get a median precision of 107 m s-1. We calculated these values by taking the median photon noise error on the measurements for each system, and quadratically adding an extra term, σadd whose estimation is explained in Sect. 3. The precision obtained is not uniform with binary period, as shown in the bottom plot of Fig. 3. The precision of our radial velocity measurements tends to be worse for shorter binary periods, because these stars are forced to rotate synchronously with their orbital periods, owing to tidal forces; this leads to broadened spectral lines. We verify this by plotting the full width at half maximum (FWHM) of the CCF, in the top of Fig. 3 where it can be seen to increase with decreasing binary period. The FWHM that we measure has two dominant components, and is defined as (1)where FWHMrot is the broadening of the absorption lines (and consequently of the CCF) as caused by the rotation of the star, while FWHMinst is the instrumental broadening of CORALIE, which depends on its resolution. For CORALIE, FWHMinst ~ 7, which sets the minimum observable FWHM in Fig. 3. The FWHMrot increases as the orbital period decreases for objects where the primary star’s rotation period is tidally synchronised to the orbital period of its secondary. This effect saturates at FWHM ~ 7 below which we cannot reliably measure the primaries’ rotational broadening.

2.3. Determination of the primaries’ effective temperatures and masses

We have used the empirical colour – effective temperature from Boyajian et al. (2013) to estimate the effective temperatures of the primary stars in these binary systems. We extracted photometry for each target from the following catalogues – BT and VT magnitudes from the Tycho-2 catalogue (Høg et al. 2000); B, V, g, r and i magnitudes from data release 9 of the AAVSO Photometric All Sky Survey (APASS9, Henden et al. 2015); J, H and Ks magnitudes from the Two-micron All Sky Survey (2MASS, Skrutskie et al. 2006); i, J and K magnitudes from the Deep Near-infrared Southern Sky Survey (DENIS, DENIS Consortium 2005). Not all stars have data in all these catalogues.

Our model for the observed photometry has the following parameters – , the apparent g-band magnitude for the star corrected for extinction; Teff the effective temperature; E(B−V), the reddening to the system; σext the additional systematic error added in quadrature to each measurement to account for systematic errors. For each trial combination of these parameters we use the empirical colour – effective temperature relations by Boyajian et al. (2013) to predict the apparent magnitudes for the binary in each of the observed bands. We assume that the contribution from the low-mass companion is negligible at all wavelengths. We used the same transformation between the Johnson and 2MASS photometric systems as Boyajian et al. (2013). We used Cousins IC as an approximation to the DENIS Gunn i band and the 2MASS Ks as an approximation to the DENIS K band (see Fig. 4; Bessell 2005). We used interpolation in Table 3 of Bessell (2000) to transform the Johnson B, V magnitudes to Tycho-2 BT and VT magnitudes. We assume that the extinction in the V band is 3.1 × E(BV). Extinction in the SDSS and 2MASS bands is calculated using Ar = 2.770 × E(BV) from Fiorucci & Munari (2003) and extinction coefficients relative to the r band from Davenport et al. (2014).

We used emcee (Foreman-Mackey et al. 2013) to sample the posterior probability distribution for our model parameters. We used the reddening maps by Schlafly & Finkbeiner (2011) to estimate the total line-of-sight extinction to each target, E(BV)map. This value is used to impose the following (unnormalised) prior on Δ = E(BV)−E(BV)map: The constant 0.034 is taken from Maxted et al. (2014) and is based on a comparison of E(BV)map to E(BV) from Strömgren photometry for 150 A-type stars.

Finally, primary stellar masses were then estimated by interpolation with in Table B.1 of Gray (2008). The masses we obtain can be found in table C.1 of this paper, as well as values for Teff, and E(BV). The error for the primary mass is calculated by (2)where σTeff is the error in Teff and dmA/ dTeff is calculated from the empirical mass-effective temperature relation (Torres et al. 2010) by generating 1000 values of Teff from a normal distribution with a standard deviation of σTeff. The factor of 0.06 accounts for the scatter measured around the mass-effective temperature relation (Torres et al. 2010).

The determination of our primaries’ masses is currently coarse and a finer spectroscopic analysis will be done, to update the values that we provide here. We provide detailed results on the model parameters so that the masses for the secondaries can be easily updated when this newer, more accurate information on the primaries is finally released.

Finally, the primary stellar radii were also determined based on Gray (2008). The only purpose of these radii in this paper is to calculate an inclination-based uncertainty in the secondary mass measurement, as will be explained in Sect. 4.

3. Treatment of radial velocity data

3.1. Data reduction software

The spectroscopic data were reduced using the CORALIE Data Reduction Software (DRS). The radial velocity information was obtained by removing the instrumental blaze function and cross-correlating each spectrum with a numerical mask corresponding to the spectral type of the primary. The position of all orders were calibrated at the beginning of the night using a tungsten lamp. Masks came in two flavours: G2 and K5. This correlation was compared with the Th-Ar spectrum used as a wavelength-calibration reference (see Baranne et al. 1996; and Pepe et al. 2002, for further information). As the instrument was not pressurised, the wavelength solution changed with variations in atmospheric pressure (approximately equivalent to 100 m s-1 mbar-1). The simultaneous calibration Th-Ar (now Fabry-Pérot), on each science frame, accurately corrects instrumental changes. As a precaution, additional calibrations of the wavelength solution were obtained during the night when a drift in excess of 50 m s-1 is detected.

The CORALIE DRS was built similarly to the DRS for the HARPS, HARPS-North and SOPHIE instruments, and has been shown to achieve remarkable stability, precision and accuracy (e.g. Mayor et al. 2009; Molaro et al. 2013; López-Morales et al. 2014; Motalebi et al. 2015) thanks in part to a revision of the reference lines for thorium and argon by Lovis & Pepe (2007) as well as a better understanding of instrumental systematics (e.g. Dumusque et al. 2015). With a resolving power R = 55 000, we obtained a CCF binned in 0.5 km s-1 increments. The range over which we computed the CCF was adapted to be three times the size of the FWHM of the CCF on each of the spectra. This ensures that wings of the function are accurately determined by the Gaussian model applied to the CCF.

3.2. Calculating error bars

Uncertainties on individual data points were estimated by the DRS from photon noise alone. CORALIE was stable to ~6 m s-1 for many years (Marmier 2014), but a recent change from a circular optical fibre to a octagonal at the end of 2014 improved stability to ~3 m s-1 (Triaud et al. 2017). Given that the majority of the measurements for the EBLM project were taken before the change of fibre, we systematically added 6 m s-1 of noise, quadratically, to the 1σ photon noise uncertainties. In the vast majority of cases the photon noise dominates, so the effect of this correction is minimal.

The majority of spectroscopic observations using CORALIE are for a volume-limited Doppler survey to detect planets around bright, Hipparcos-selected, low v sin i stars (Mayor et al. 2011; Marmier et al. 2013), and the confirmation of the WASP transiting planet candidates (Triaud et al. 2011; Lendl et al. 2014; Neveu-VanMalle et al. 2014). For these two programmes the obtained spectra have high signal-to-noise ratios (S/N > 15), whereas for the EBLM project, since we mostly deal with shortened exposure times, we frequently obtained lower S/N spectra (S/N ~ 3−7). Furthermore many of our primaries spin rapidly. This decreases the S/N we obtain on the peak of the CCF, and affects our radial-velocity precision. Our automated error bar estimation was therefore not very well adapted to this new regime of observations for CORALIE, which led to an under-estimation of measurement uncertainties. We corrected the DRS’s uncertainties using an indicator called the span of the bisector slope.

The span of the bisector slope (or the bisector thereafter) measures the asymmetry of the CCF, which reflects the asymmetry of all absorption lines (Queloz et al. 2001a). It has an uncertainty twice the value of the uncertainty achieved on the radial-velocity (Queloz et al. 2001a; Figueira et al. 2013). The bisector is traditionally used to test whether any detected low-amplitude radial-velocity variation is caused by a translation of the CCF (as expected for a Doppler reflex motion), instead of caused by a change in the shape of the CCF. This can be produced by stellar activity (leading to an anti-correlation; Queloz et al. 2001a), or by the Doppler reflex motion of a blended, secondary set of lines (creating a correlation; Santos et al. 2002). Whilst this is important to discover exoplanets whose signal can be similar in amplitude to a line shape variation, in our case, the EBLM project, the orbital motion is large (> than the FWHM of the CCF) in addition to not being subject to any detection problem. In our case, we can use the dispersion of the bisector to calculate the true uncertainty on our radial-velocities and correct any under-estimation produced by the DRS. We therefore computed an additional noise term, σadd, as (3)where δbis is the rms of the bisector measurements, and σγ is the photon noise error. Once σadd is estimated, we check the procedure by finding the mean of the bisector measurements and measuring that the dispersion is compatible with a , where the error terms on the bisector have been updated by quadratically adding (4)similarly the new errors on the radial velocity measurements become (5)In cases where there is no need for any additional noise term, and hence σadd is set to 0.

3.3. Outlier removal

Several steps were taken to remove outliers. First, all observations with a bisector position more than three interquartile ranges below the first quartile or above the third quartile were automatically removed. Observations such as these with significantly different bisector positions are often indicative of the wrong star accidentally being observed or an anomalously low S/N, generally owing to poor observing conditions. Any bisector variation within the remaining observations was accounted for with the added σbis noise term described in the previous section. After this automated removal procedure a visual inspection was done of all data series. In particular, there was a check for the consistency of the FWHM, as occasionally the wrong star being observed may still result in a coincidentally similar bisector, but different FWHM. Additionally, some targets received Rossiter–McLaughlin observations during eclipses to measure the projected spin-orbit alignment. These results are to be presented in a future paper and are removed from the data analysis in this paper as the Rossiter–McLaughlin anomaly would likely bias the radial-velocity fit if not modelled accurately. For other observations, care was taken to take them out of eclipse.

4. Orbit fitting

Orbits are fitted using the Yorbit software developed at the University of Geneva. It uses a genetic algorithm to scan a broad parameter space and avoiding falling into local minima. This is coupled with a Markov chain Monte Carlo to calculate the final orbital solution. Keplerian orbits are fitted independently with no N body interactions between them, although this is something to be developed in the future. This software has been used in particular in many CORALIE and HARPS radial velocity surveys in the past (e.g. Mayor et al. 2011) and is discussed in more detail in Ségransan et al. (2010) and Bouchy et al. (2016).

Table 1

Models ranked by ascending complexity.

Calculating the secondary mass

A single Keplerian orbit is characterised by six parameters. There is more than one way to paramaterise this problem. The ones provided by Yorbit are: period, P, semi-amplitude, K, eccentricity, e, time of periapsis passage, T0, mass function, f(m), and argument of periapsis, ω. For each of these parameters the Yorbit calculates 1σ error bars using 5000 Monte Carlo simulations. These parameters are determined independently of the mass of the primary star, mA.

Since these are single-line binaries it is not possible to directly measure the primary and secondary masses. The only mass quantity which we directly measure is the mass function. We must instead use a primary mass inferred from the models described in Sect. 2.3. The secondary mass is then calculated from the mass function by solving the following equation numerically: (6)When evaluating Eq. (6)we take I = 90°, since our binaries are eclipsing. To calculate the error of mB we use the fact that our binaries all have small mass ratios, allowing us to simplify Eq. (6)to , for which the error calculation becomes (7)The error in sinI stems from us not precisely characterising the eclipse impact parameter and hence inclination using the WASP photometry. Based on possible eclipse geometries, the inclination uncertainty is δsinI = RA/a. This is a less than 20% contribution to the relative uncertainty in mB.

The semi-major axis is calculated using Kepler’s third law, (8)and the error is calculated as (9)The precision in the semi-major axis is always significantly worse than that of the period, due to the uncertainty in the stellar masses.

5. Model selection

We now describe the models which we have fitted to each star and how we choose the most appropriate one.

5.1. Ten different models applied to the data

For each system we try to fit various models to the spectroscopic data and calculate the goodness of fit. The models tested are:

  • 1.

    k1: a single Keplerian orbit;

  • 2.

    k1d1: a single Keplerian plus a linear drift;

  • 3.

    k1d2: a single Keplerian plus a quadratic drift;

  • 4.

    k1d3: a single Keplerian plus a cubic drift;

  • 5.

    k2: two Keplerians.

The drift terms are indicative of an outer third body that is causing the radial velocities to deviate from a single Keplerian over time. Whether we require a linear, quadratic or cubic fit is function of the amplitude of the radial-velocity signal induced by the third body and also the temporal fraction of its orbit covered. When the tertiary orbit is well-covered (≳ 30%), a second Keplerian is generally a better fit. For each of the cases we further tested the goodness of fit with both the binary eccentricity found by Yorbit, and with a forced circular orbit. This means that in total, we adjusted and can test ten different models on our data.

While forcing a circular model, two parameters are dropped: eccentricity, e, and argument of periapsis, ω. We denote eccentric and circular models using the parentheses (ecc) and (circ), respectively. The ten models are split into “base” models – k1, k1d1 and k1d2 – and “complex” models – k1d3 and k2, where sometimes the number of measurements approaches the number of degrees of freedom. This distinction is used in the model selection procedure. Note that for a two-Keplerian fit we only ever force the inner binary orbit to be circular, not the tertiary body. In Table 1 we rank all ten models in ascending order of complexity (number of parameters). Ticks are used to indicate the orbital parameters used in each fit, including linear (lin), quadratic (quad) and cubic drift coefficients.

5.2. Using the BIC to select between models

thumbnail Fig. 4

Residuals (O–C) of the radial velocity fit to J0543-57 of five different models with increasing complexity and improved goodness of fit from top to bottom. A k2 (circ) model was ultimately chosen according to our procedure.

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We use the Bayesian information criterion (BIC; Schwarz 1978) to select the model that provides the optimal balance between goodness of fit and complexity. We assume that the errors in our radial velocity measurements are independent and identically distributed following a normal distribution, so that the BIC can be calculated using (10)where χ2 is the weighted sum of the square of the residuals, k is the number of model parameters and nobs is the number of observations. The BIC is constructed to naturally penalise models that are unnecessarily complex and not justified by the data (otherwise known as Ockham’s razor). Whenever choosing between one model and the next most complex (in terms of the number of parameters) we demand that the BIC increases by at least 6 in order to justify the added complexity. This is deemed “strong” evidence in the literature (Kass & Raftery 1995).

Our model selection procedure follows a forward method, where we start with the simplest model and move up in complexity. The steps are as followed:

  • 1.

    Calculate the BIC for the simplest model: a circular singleKeplerian.

  • 2.

    Calculate the BIC for subsequent base models with increasingnumbers of parameters, as denoted by the order in Table 1.

  • 3.

    Whenever we want to jump from one model to the one with the next highest number of parameters, we demand that the BIC improves (i.e. decreases) by at least 6.

  • 4.

    Note that k1 (ecc) and k1d2 (circ) both have six parameters. When choosing between those two models simply the smallest BIC is chosen.

  • 5.

    In some situations the next most complex model may only marginally improve the BIC (i.e. not by 6) but the more complex model after that may be a significant improvement. In these exceptional circumstances one may “jump” to the well-fitting model two ranks of complexity above by improving the BIC by a factor of 2 × 6 = 12, or in general n × 6 where n is the number of ranks of complexity you want to move up.

  • 6.

    For the base model chosen according to the BIC we calculate the reduced χ2 statistic, , where (nobsk) is the number of degrees of freedom. For a good fit to the data we expect .

  • 7.

    If this value of then we consider the simple model to be a sufficient fit to the data and do not test any others. This conservative approach helps avoid over-fitting.

  • 8.

    Alternatively, if we then test more complex models: k1d3 and k2 (both eccentric and circular). These models are then treated with the same model selection procedure as before. In some cases complex models are tested but a base model is ultimately still chosen.

  • 9.

    An exception to the above procedure comes in the case of heightened stellar activity. This activity can cause variation in the radial velocity measurements that may be confused for another physical body in the system. This occurs in two cases: J0021-16 and J2025-45. These binaries have for the base models of 3.49 and 7.09, respectively, but we do not test more complex models and instead manually assign an appropriate, simpler model. These individual cases are discussed further in Sect. 6.5.

Figure 4 shows our procedure in action on the residuals obtained after removing the most likely parameters for a set model. In this particular case adding a second Keplerian visibly improves the goodness of fit, which also happens in the BIC values. In Table A.1 we show the data pertaining to the model selection. The BIC of the selected model is highlighted in bold font. For most systems the simple models tested yielded a and hence no models of further complexity were needed. In Table A.1 we count the number of binaries fitted to each of the ten models. In the appendices, Table B.1 contains the orbital parameters for all of the binaries, taken from the chosen model according to the BIC. For the four binaries where a k2 model was selected we provide the orbital parameters of the tertiary body in Table F.1.

5.3. Providing upper limits on undetected nested parameters

In reality, no orbit is exactly circular, meaning that the true physical model ought to be eccentric even though statistically that extra degree of complexity in the model is not formally detected. To remedy the issue we provide estimates for upper limits to the eccentricity and the coefficient of linear drift, along with the selected model values, in Table B.1. The values we provide were estimated using the model of higher complexity on that particular parameter. We provide values at 67% confidence by taking the fitted value and adding the 1σ uncertainty. We do the same for the upper limit of the linear drift coefficient for binaries where a single Keplerian fit was chosen.

6. Results

6.1. Summary

In total we analysed 118 eclipsing binaries. Table A.1 shows the number of stars for which each model was selected using the BIC. The results of the model fits to individual stars are given in a series of tables in the appendices to the paper. First, in Table A.1 we demonstrate our model selection procedure based on the BIC and reduced χ2 statistic. The chosen model is given in this table and the BIC for that model is highlighted in bold. For most systems the simple models tested yielded a and hence no models of further complexity were needed. The flag column has three different flags: “drift” indicating that a linear, quadratic or cubic drift was the best fit to the data, “triple” for the four systems where we fitted two Keplerian orbits to the triple star system and “active” for the two systems showing signs of stellar activity.

The orbits of the best-fitting models and their residuals are shown in Appendix E.

Contained in in Table B.1 are the orbital parameters for all of the binaries. For each parameter we show both the measured value and the uncertainty. The uncertainty is the value inside the brackets and corresponds to the final two digits of the measured value. For example, for J0008+02 P = 4.7222907(63) days, which means P = 4.7222907 ± 0.0000063 days. This table includes the calculated primary and secondary masses. More detailed parameters for the primary stars are shown in Table C.1. The J and V magnitudes come from the NOMAD survey and the R magnitude comes from 2MASS. An exception is that for three targets, J1934-42, J1509-10 and J2353-10, no Vmag was available from NOMAD so it was calculated as a function of the primary mass using models by Baraffe et al. (2015) at an age of 1 Gyr.

For the four targets with characterised tertiary orbits we provide their orbital parameters and plots of the radial velocity fits in Appendix F.

Parameters for the secondary stars are shown in Table C.2. The error in the secondary mass is predominantly due to uncertainties in the primary mass and orbital inclination, and not the radial velocity semi-amplitude. Unlike the primary star, which has measured magnitudes, the secondary magnitudes are all calculated using the Baraffe et al. (2015) models. Values for the V, R and J magnitudes are given at ages of 1 Gyr and 5 Gyr. This is because we do not have accurate estimates for the true ages of the systems, although we note that the magnitude difference is small.

Finally, in Table D.1 are various observational parameters for the binaries. The period P and times T0,pri and T0,sec (for the primary and secondary eclipses) are taken from the radial velocity fit, not the WASP photometry.

thumbnail Fig. 5

Top: mass ratio q = mA/mB for each of the binaries including error bars. The fractional error of the secondary is generally similar to that of the primary, and consequently the absolute error for the secondary is invisibly small on this plot for small mB. Red dashed lines correspond to mass ratios of 1, 1/2, 1/3, 1/5 and 1/10. Middle: histogram of the secondary mass. Bottom: histogram of the mass ratio.

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6.2. Primary and secondary masses and magnitudes

We now demonstrate visually some of the results in our sample. In Fig. 5 we show the primary and secondary masses in our sample. It is seen that 60 of our binaries (50% of the sample) have mass ratio q< 0.2, and, 34 (31%) companions have masses mB< 0.2 M. A consequence of these small mass ratios is that our secondary stars are all between 3.1 and 12.6 mag fainter than the primary stars, and hence we only observe a single-line spectroscopic binary. A histogram of this difference in magnitudes is shown in Fig. C.1.

Figure 7 shows a combined WASP/EBLM mass spectrum for objects creating photometric eclipses compatible with sizes < 2.1 RJup. We usually give a “WASP” identifier for all substellar objects (planets and brown dwarfs). We collected all objects with WASP identifiers that are public and were observed with the CORALIE spectrograph and added all stellar companions in this paper. In overall this means 143 substellar objects and the 118 stellar companions presented in this paper. One of our substellar companions, WASP-30 (Triaud et al. 2013) falls within the brown dwarf range.

We compare our preliminary results to the 50 pc mass spectrum shown in Grether & Lineweaver (2006). To do this, we normalise their histogram to our number of substellar objects with masses superior to 1 MJup. From Fig. 7, we see that our mass spectrum covers a broader range in the planetary masses (although at low masses we are most likely incomplete), and does not cover stars as massive (due to the restriction of our survey). Over the common range between Grether & Lineweaver (2006) and us, we have a resolution that is twice better. The results are broadly consistent, but differ in an interesting way. The brown dwarf desert derived from the WASP and EBLM results appears to stretch deeper into the planetary domain than the result of Grether & Lineweaver (2006). We find that in our results, massive gas-giants (~ 3−13 MJup) appear less abundant.

The reason for the discrepancy with the Grether & Lineweaver (2006) results is that their work probed planets on wider orbits a ≲ 10 AU, whereas EBLM and WASP are typically sensitive to a< 0.2 AU. Whatever process(es) is(are) important in shaping the population of hot Jupiters, it(they) favour(s) smaller mass gas-giants. This was also noted in Udry et al. (2003).

The entire EBLM sample contains over 200 binaries (of which only 118 are presented here); the WASP survey is still on-going. Once those results are all published we will revisit this mass spectrum. In particular, in the future we will be able to compare the relative abundance of close hot Jupiters, brown dwarfs and M dwarfs. On this occasion, we will also produce a more thorough analysis and debiasing of the spectrum.

thumbnail Fig. 6

Eccentricity of the eclipsing binary as a function of semi-major axis (top two plots) and period (bottom two plots). Plots b and d are zoomed versions of the a and c, respectively, showing the tightest binaries (Pbin< 12 d) with eccentricities compatible with zero. In the top plot the error in semi-major axis is shown, but this is excluded in the zoomed version for clarity. In the third figure (period vs. eccentricity, not zoomed) we use dashed lines to denote fits using the Meibom & Mathieu function in Eq. (11). The purple dashed line is a fit to all of the data where Pcut = 8.9 days, whilst the red dashed line is a fit to all binaries with M1< 1.3 M and no sign of a tertiary companion. In this latter case Pcut = 7.0 days.

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6.3. Eccentricities

In Fig. 6 we show the eccentricity of our systems as a function of the semi-major axis (top two plots) and period (bottom two plots). For the unzoomed P vs. e plot we show a fit to the data using the Meibom–Mathieu function (Meibom & Mathieu 2005), which is presented in Sect. 7. For both the semi-major axis and period we show zoomed versions of the plots for eccentricities between 0 and 0.022. There are many binaries for which we can constrain their orbits to being circular within these small bounds of eccentricity. Furthermore, there are some binaries for which we actually measure eccentricities that are small but significantly non-zero. This high precision is by virtue of using the CORALIE instrument with planet-finding precision to observe much larger amplitude binaries. The smallest significantly non-zero eccentricity measured is e = 0.00108 ± 0.00017 for J0353+053. The most precisely measured eccentricity is e = 0.051004 ± 0.000086 for J0042-17. These results are expanded upon in Fig. 8a which is a histogram of the eccentricity precision obtained in the EBLM program. For half of our targets we can constrain eccentricities to a precision of 0.0025. For 29% we obtain a precision better than 0.001 and for 14% a precision better than 0.0005.

The periods of our binaries are measured to an even better precision. In Fig. 8b is a histogram of the precision obtained on the binary period, shown in a scale of seconds. Half of the sample have a period measured to better than 1.4 s. As a percentage error, 50% of our targets have their period measured to better than 0.00031% with the worst being 0.047% for J0629-67. Our highly precise periods are a result of both the high-resolution CORALIE instrument and the fact that all orbits have been observed for a timespan of at least 16 Pbin and 86% are observed over a timespan of more than 100 Pbin.

The longest period binary for which we measure a zero eccentricity is J1008-29 with a 10.4 day period and e< 0.0016.

The implications of our measured eccentricities as a function of period and semi-major axis are discussed in Sect. 7.

6.4. Triple star systems

According to Table A.1 there are 21 systems fitted with a model other than a single Keplerian (circular or eccentric). All solutions fitted with a drift or second Keplerian are indicative of a third body, most likely a tertiary star but possibly a circumbinary brown dwarf or massive planet. Our overall tertiary rate is 21/118 = 17.8%. In Fig. 9 we plot the percentage of systems with indications of a close tertiary companion as a function of inner binary period (blue solid line). Bin edges are chosen to match the study of Tokovinin et al. (2006): 3, 6, 9, 12, 40. This plot indicates a roughly flat rate of triples as a function of P. This contrasts with the results of Tokovinin et al. (2006) (orange dash-dotted line) in two ways. First, our results are significantly lower at all binary periods. The cause of this is however simple. Our only indicator of a third star is an additional radial-velocity signal, and this is only sensitive to close triples (estimate period range) and can miss tertiary stars with inclinations near zero. By contrast, Tokovinin et al. (2006) was an imaging survey of known spectroscopic binaries, so whilst it may miss some very tight triple systems it is sensitive to a much larger range.

Second, one of the most important results of Tokovinin et al. (2006) was the sharp dependence of tertiary fraction on binary period. This has been interpreted as evidence for the formation of close binaries via Kozai-Lidov cycles followed by tidal circularisation (Lidov 1962; Kozai 1962; Mazeh & Shaham 1979; Fabrycky & Tremaine 2007) and this is different to the flat distribution seen in our raw triple fraction. If the distributions of tertiary periods and masses are uncorrelated with the inner binary period, then on average the slope of the radial velocity drift would be independent of the inner binary period. However, our detectability of this radial-velocity drift is not uniform with binary period in our sample. As shown in the top of Fig. 1, the observing time spent on a given target is not dependent on the binary period, so this does not introduce a bias. However, a significant bias is the trend of decreasing precision with closer binaries. This is due to tidal locking leading to broadened spectral lines, as discussed in Sect. 2.2 and evidenced in Fig. 3. For Pbin< 3 days there are 20 binaries and the median precision obtained is 477 m s-1. Between 3 and 6 days the median precision improves to 118 m s-1. For 6 to 12 day binaries there is a further improvement to 56 m s-1 and for our 10 binaries with a period longer than 12 days we have an excellent median precision of 24 m s-1. This strong bias hurts our ability to detect tertiary companions to very close binaries. This was not shared with Tokovinin et al. (2006), who used imaging.

thumbnail Fig. 7

In dark blue, the observed mass spectrum from the WASP planet survey and the EBLM binaries released in this paper. The orange histogram depicts the results from Grether & Lineweaver (2006), normalised to the number of WASP and EBLM objects heavier than 1 MJup. The vertical grey lines denote the rough mass limits for deuterium burning (13 MJup) and hydrogen burning (80 MJup). There is an evident deficit of objects between these two limits, which corresponds to the realm of brown dwarfs.

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thumbnail Fig. 8

Top: histogram of the precision of the eccentricity for all of our 118 binaries. For models where an eccentricity is favoured it is the 1σ error bar. For models where a circular solution is favoured it is the upper limit, which is equal to the fitted eccentricity value plus the 1σ uncertainty. The vertical red dashed line is the median precision of 0.0025. Bottom: histogram of the precision of the binary period, in seconds. The red dashed line is the median precision of 1.4 s. We emphasise that the period precision is obtained purely by the radial velocity fit, not from the eclipse timing.

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thumbnail Fig. 9

Percentage of our EBLM binaries with a tertiary companion (blue solid line). Binaries are said to have a tertiary companion if the best fitting radial velocity model is not a single Keplerian. For comparison we show the results of the Tokovinin et al. (2006) imaging survey of close spectroscopic binaries (orange dashed line).

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Of the 21 binaries identified as having a tertiary companion in four cases our observations allow a characterisation of the tertiary orbit: J0543-57, J1146-42, J2011-71 and J2046-40. These four triples have outer periods of 3062, 260, 663 and 5584 days, respectively. All orbital parameters are provided in Table F.1. In this appendix we also provide orbits of both the inner binary and outer tertiary and a top-down view. J1146-42 in particular, with three stars within ~ 1 AU (modulo siniC), is a rare tight triple star system. We note that for the best fitting model which is actually the worst in our sample, which may seem surprising given the precision obtained is so high (median 18 m s-1) owing to its brightness (Vmag =10.3) and long inner period (P = 10.47 days). We suggest that the cause of these large residuals is a Newtonian perturbation between the two orbits causing them to become non-Keplerian, which is not accounted for within Yorbit. This arises because the inner and outer orbits are close: aout/ain = 9.1. Our observing timespan of 3.34 yrs and high precision make us sensitive to these perturbations. It is a future task to analyse the orbital dynamics of these close triple systems and try to exploit them to calculate additional parameters in the system, such as the mutual inclination between the two orbits (e.g. Correia et al. 2010).

It is also likely that the ongoing Gaia astrometric survey will provide additional orbital constraints on these triple systems.

thumbnail Fig. 10

Correlation between the bisector and residuals to an eccentric single Keplerian fit for two of our targets showing signs of stellar activity: J0021-16 and J2025-45. This negative linear trend is an indicator of stellar activity.

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6.5. Active systems

Throughout the history of radial-velocity surveys for extra-solar planets, stellar activity has often contrived to confuse observers by creating spurious radial velocity variations that may be mistaken as planets (e.g. Queloz et al. 2001a). Fortunately in a survey of binaries, the amplitude of the Keplerian signal is tens of km s-1, which is higher by orders of magnitude compared to stellar activity. From this perspective there is therefore no doubt about the existence of our binaries. What stellar activity may do, however, is inhibit our ability to detect smaller amplitude effects such as radial-velocity drifts indicative of a third star, as these may be of a low amplitude of only tens of metres per second.

In exoplanet studies, a classic diagnostic for stellar activity is the span of the bisector slope (Queloz et al. 2001a; Figueira et al. 2013), which we use here as well. An anti-correlation between a motion in radial-velocity and in the slope of the bisector indicates a distortion of the absorption lines, caused by stellar activity.

For two of our binaries we see clear signs of stellar activity: J0021-16 and J2025-45. In Fig. 10a we plot for J0021-16 the residuals from a single Keplerian fit and in Fig. 10b for a single Keplerian plus linear drift. When only a single Keplerian is fitted there is a clear linear negative trend between the residuals and the bisector. However, if we look in more detail we see that the more recent points, denoted in purple, have a systematic shift in bisector to the left in comparison with the older points. This indicates that whilst there is stellar activity present throughout all of the observations, there is an additional source of the residuals. When a linear drift is added, we see in Fig. 10b that the latest points in purple now overlap the other points. We conclude that this system has both stellar activity and a tertiary stellar companion inducing a drift. This illustrates the advantage of our long observing baseline, which was 5.34 yr for this target. The case is different for J2025-45, for which there is no discernible difference in the bisector-residuals correlation between the old and new observations. This is not due to a lack of time spent on the target, as the observations span a total of 5.45 yr. We assign a single eccentric Keplerian fit to this target.

7. A discussion on tidal evolution

One of the scientific advantages of having very precise eccentricities and periods is to allow investigations into tidal interactions. A future work is planned to exploit the results of the EBLM survey in more details and to better our understanding tidal evolution in close binaries. For now, we discuss some of the first order implications that our results may have.

Tidal interactions between two close stars have several effects (Zahn 1975, 1977):

  • Synchronisation of the rotation and orbital periods for circularorbits (pseudo-synchronisation for eccentric orbits;Hut 1981).

  • Alignment of the orbital and spin axes of the stars.

  • Circularisation of the orbit.

In the plot of FWHM and precision as a function of the binary period (Fig. 3) we saw evidence for tidal synchronisation, which manifests itself as spectral lines being more broadened with a reduced orbital period. Measurements of the Rossiter–McLaughlin effect, which probe the projected obliquity of the orbit, will serve to help better understand the strength of tidal realignment, and confront theoretical expectations (e.g. Anderson et al. 2017).

These are to be presented in a future paper. Finally, the plots of binary eccentricity as a function of semi-major axis and period in Fig. 6 allow us to probe the circularisation of the orbit. By eye, it is evident that there is a trend of increased eccentricity with semi-major axis and period, which is expected since tides are mostly effective over very short distances. Binaries are not expected to form on primordially circular orbits. Rather, they circularise over time (e.g. Mazeh & Shaham 1979; Fabrycky & Tremaine 2007; Bate 2012). A consequence of this is that one may define a cut-off period, Pcut, above which orbits are found eccentric. Meibom & Mathieu (2005) provided a means of measuring Pcut as follows: (11)where the constants were calculated to be α = 0.35, β = 0.14 and γ = 1.0. The value of α is defined based on the mean eccentricity of all field binaries of periods greater than 50 days being 0.35, whilst β and γ were shown to optimise the fit. In Fig. 6 we include two versions of our fitted Eq. (11). First, we make a fit to all of the data, and calculate Pcut = 8.9 days. Second, we make a fit to only the binaries where MA< 1.3 M and there is no sign of a tertiary companion. This second adjustment yields a more accurate Pcut = 7.0 days. We base our preference on the following. Heavier stars have a radiative outer envelope, rather than a convective one (Pinsonneault et al. 2001). Tidal dissipation in radiative envelopes is less efficient, which causes tidal circularisation to be slower (similar arguments have been invoked in the exoplanet literature; Albrecht et al. 2012; Dawson 2014). Additionally, outer tertiary companions may induce some eccentricity in the inner binary via secular perturbations.

Our Pcut result comes in conflict with other estimates discussed in the literature and compiled in Meibom & Mathieu (2005) with an update in Milliman et al. (2014). Results obtained in the field (Duquennoy & Mayor 1991), the halo (Latham et al. 2002), in M 67 (Mathieu et al. 1990) and NGC 188 (Mathieu et al. 2004), all with ages >1 Gyr, are found with Pcut> 10 days. The only exception is NGC 6819 (Milliman et al. 2014) whose value Pcut = 6.2 ± 1.1 days is consistent with ours. This is also consistent with results on young open clusters (< 100 Myr), where Pcut is found around 8 days (Melo et al. 2001). The reason for disagreement between our value and the bulk of other results on old populations is not presently known. A contributing factor may be that our survey is, by design, biased towards small mass ratio binaries, and the circularisation timescale is dependent on the mass ratio (Zahn 1977, 1978). Past surveys may also suffer from small number statistics and a poorer precision on eccentricities than what we can produce nowadays. This preliminary result is consistent with there only being marginal tidal evolution during the main sequence. We have another 100+ binary systems under observation at the moment and will update our Pcut and analysis once observations on those are completed.

8. Conclusion

We presented the spectroscopic orbits of 118 stellar systems, all eclipsing single-line binaries. We produced that sample in order to map out the sky position of eclipsing systems mimicking transiting hot Jupiters. This will be of great help in the advent of large-scale exoplanet surveys such as TESS and Plato. In addition, this sample can be used as a comparison to hot Jupiters for a host of topics that are detailed in the introduction.

Our release of these systems opens multiple opportunities for further research, for instance to detect tertiary companions with direct imaging, astrometry, or eclipse timing variations. As high-resolution, near-infrared spectrographs are coming online, it will become feasible to transform our single-line binaries into double-line systems and derive accurate masses and radii (Torres & Ribas 2002; Brogi et al. 2012; Rodler et al. 2012). As part of the efforts of our team, we will double the current sample of eclipsing low-mass binaries (we are currently acquiring data to reach a minimum of 13 radial-velocity measurements), to measure some primary eclipses of secondaries with mass < 0.2 M, and to prepare a publication on the Rossiter–McLaughlin effect of 20 of our binaries.

We derived orbital periods with a precision of the order of one second and compute eccentricities well below 1%. This is thanks to long-term observations using a stable, high-precision spectrograph usually employed in the discovery of exoplanets. We used our results to carryout a preliminary investigation of the strength of tidal forces in stars. To a first order, we find that binaries in the field have a similar eccentricity distribution to pre-main sequence binaries. This is consistent with there being marginal tidal evolution over a main sequence lifetime, although further investigation is needed to make a definitive statement.

Ordering our sample as a function of mass, and adding the results with the same instrument from WASP and CORALIE on exoplanets, we construct a preliminary mass spectrum. Its appears to show a deficit of planets with masses > 3 MJup compared to earlier results considering wider orbital separations.

For 21 of our systems there is a significant indication of an outer tertiary companion, most likely stellar in nature and long-period. For four systems we were actually able to characterise the tertiary orbit, including one system with three stars packed within 1 astronomical unit. We will also intensify our monitoring of a subset of our systems in search of circumbinary gas-giants in a connected program known as BEBOP.

Nota Bene. We used the barycentric Julian dates in our analysis. Our results are based on the equatorial solar and Jovian radii and masses taken from Allen’s Astrophysical Quantities (Cox 2000).


2

For all of the published hot Jupiters in WASP and the few EBLM binaries published in Triaud et al. (2013), Gómez Maqueo Chew et al. (2014), von Boetticher et al. (2017) improved transits/eclipses had been obtained with larger telescopes.

3

For J0540-17 our BIC selection algorithm favoured an eccentric solution over a circular one, however the calculated eccentricity is e = 0.00025 ± 0.00055 is smaller than that for J0353+05 but compatible with 0 within 1σ. This is a one off case.

Acknowledgments

We would like to acknowledge our referee for his/her time reading our paper, and providing helpful corrections. We benefitted from enlightening discussions with Dan Bayliss, Corinne Charbonnel, Florian Gallet, Dave Latham, Rosemary Mardling, Bob Mathieu, John Papaloizou, Andrei Tokovinin, and Yanqin Wu. We also thank Dan Fabrycky for providing comments on the manuscript. The authors would like to acknowledge the help and kind attention of the ESO staff at La Silla and the dedication of the many technicians and observers from the University of Geneva for the upkeep of the telescope and acquiring the data that we present here. We would also like to acknowledge that the Euler Swiss Telescope at La Silla is a project funded by the Swiss National Science Foundation (SNSF). Over the time required to collect and analyse the data, AHMJT received funding from the SNSF, the University of Toronto, the University of Cambridge, and the University of Birmingham. D.V.M. is supported by the SNSF. This publication makes use of data products from two projects, which were obtained through the Simbad and VizieR services hosted at the CDS-Strasbourg: The Two Micron All Sky Survey (2MASS), which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation (Skrutskie et al. 2006); the Naval Observatory Merged Astrometric Dataset (NOMAD), which is project of the US Naval Observatory (Monet et al. 2003); the Tycho2 catalog (Høg et al. 2000); and the AAVSO Photometric All-Sky Survey (APASS), funded by the Robert Martin Ayers Sciences Fund (Henden et al. 2015)

References

Appendix A: Model selection using the Bayesian information criterion

Table A.1

Bayesian information criterion (BIC) with selected model in bold.

Appendix B: Binary orbital parameters

Table B.1

Orbital parameters from the selected models.

Appendix C: Parameters for the primary and secondary stars

Table C.1

Observational and calculated parameters of the primary stars.

Table C.2

Parameters of the secondary stars.

thumbnail Fig. C.1

Histogram of the difference between the secondary and primary visual magnitudes, using the data provided in Table D.1. Primary Vmag values come from the NOMAD survey, except for three binaries where they are calculated using models from Baraffe et al. (1998) at an age of 1 Gyr. All secondary Vmag values are calculated using Baraffe models with a 1 Gyr age.

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Appendix D: Observational parameters

Table D.1

Observational parameters of the systems we observed.

Appendix E: Binary orbits and residuals

thumbnail Fig. E.1

Four panels composed of the following: radial-velocities, and model as a function of orbital phase, a colour bar indicating when measurements were obtained, and a lower panel showing the residuals as a function of time. The system name and some of its parameters are provided on top of each panel.

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Appendix F: Triple star systems

Table F.1

Orbital parameters from the selected models for k2 fits.

thumbnail Fig. F.1

Top left panel: inner binary orbit like in Fig. E.1. Top right: outer, tertiary orbit, in phase. Bottom right plot: radial-velocity data and best fit model as a function of time, and the bottom right panel is a top representation of the triple system, where the orbit is located on the right-hand side.

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All Tables

Table 1

Models ranked by ascending complexity.

Table A.1

Bayesian information criterion (BIC) with selected model in bold.

Table B.1

Orbital parameters from the selected models.

Table C.1

Observational and calculated parameters of the primary stars.

Table C.2

Parameters of the secondary stars.

Table D.1

Observational parameters of the systems we observed.

Table F.1

Orbital parameters from the selected models for k2 fits.

All Figures

thumbnail Fig. 1

Top: timespan of observations over binaries of different periods. The flat distribution means that our sensitivity to tertiary objects is roughly independent of the binary period. Middle: number of orbits covered for each binary. There is no strict minimum although 86% of binaries have been observed over a timespan covering at least 100 orbits. Bottom: number of observations given to each binary. There is a requirement of at least 13 observations.

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In the text
thumbnail Fig. 2

Top: coordinates of each of the EBLM binaries (blue circles) and a comparative distribution of the coordinates of the WASP planets (purple triangles). Bottom: histogram of the visual magnitude.

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In the text
thumbnail Fig. 3

Top: FWHM of the CCF as a function of binary period and a red dashed trend fitted to the data for P< 8 d. Roughly beyond this period the primary stars are rotating slow enough such that the instrumental broadening dominates the rotational broadening, and truncates any potentially continued trend below 7 km s-1. Bottom: precision of the radial velocity measurements for each binary as a function of binary period. The red dashed line shows the median precision for all binaries within four coarse period bins, chosen for the later study of triple systems in Sect. 6.4.

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In the text
thumbnail Fig. 4

Residuals (O–C) of the radial velocity fit to J0543-57 of five different models with increasing complexity and improved goodness of fit from top to bottom. A k2 (circ) model was ultimately chosen according to our procedure.

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In the text
thumbnail Fig. 5

Top: mass ratio q = mA/mB for each of the binaries including error bars. The fractional error of the secondary is generally similar to that of the primary, and consequently the absolute error for the secondary is invisibly small on this plot for small mB. Red dashed lines correspond to mass ratios of 1, 1/2, 1/3, 1/5 and 1/10. Middle: histogram of the secondary mass. Bottom: histogram of the mass ratio.

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In the text
thumbnail Fig. 6

Eccentricity of the eclipsing binary as a function of semi-major axis (top two plots) and period (bottom two plots). Plots b and d are zoomed versions of the a and c, respectively, showing the tightest binaries (Pbin< 12 d) with eccentricities compatible with zero. In the top plot the error in semi-major axis is shown, but this is excluded in the zoomed version for clarity. In the third figure (period vs. eccentricity, not zoomed) we use dashed lines to denote fits using the Meibom & Mathieu function in Eq. (11). The purple dashed line is a fit to all of the data where Pcut = 8.9 days, whilst the red dashed line is a fit to all binaries with M1< 1.3 M and no sign of a tertiary companion. In this latter case Pcut = 7.0 days.

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In the text
thumbnail Fig. 7

In dark blue, the observed mass spectrum from the WASP planet survey and the EBLM binaries released in this paper. The orange histogram depicts the results from Grether & Lineweaver (2006), normalised to the number of WASP and EBLM objects heavier than 1 MJup. The vertical grey lines denote the rough mass limits for deuterium burning (13 MJup) and hydrogen burning (80 MJup). There is an evident deficit of objects between these two limits, which corresponds to the realm of brown dwarfs.

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In the text
thumbnail Fig. 8

Top: histogram of the precision of the eccentricity for all of our 118 binaries. For models where an eccentricity is favoured it is the 1σ error bar. For models where a circular solution is favoured it is the upper limit, which is equal to the fitted eccentricity value plus the 1σ uncertainty. The vertical red dashed line is the median precision of 0.0025. Bottom: histogram of the precision of the binary period, in seconds. The red dashed line is the median precision of 1.4 s. We emphasise that the period precision is obtained purely by the radial velocity fit, not from the eclipse timing.

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In the text
thumbnail Fig. 9

Percentage of our EBLM binaries with a tertiary companion (blue solid line). Binaries are said to have a tertiary companion if the best fitting radial velocity model is not a single Keplerian. For comparison we show the results of the Tokovinin et al. (2006) imaging survey of close spectroscopic binaries (orange dashed line).

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In the text
thumbnail Fig. 10

Correlation between the bisector and residuals to an eccentric single Keplerian fit for two of our targets showing signs of stellar activity: J0021-16 and J2025-45. This negative linear trend is an indicator of stellar activity.

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In the text
thumbnail Fig. C.1

Histogram of the difference between the secondary and primary visual magnitudes, using the data provided in Table D.1. Primary Vmag values come from the NOMAD survey, except for three binaries where they are calculated using models from Baraffe et al. (1998) at an age of 1 Gyr. All secondary Vmag values are calculated using Baraffe models with a 1 Gyr age.

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In the text
thumbnail Fig. E.1

Four panels composed of the following: radial-velocities, and model as a function of orbital phase, a colour bar indicating when measurements were obtained, and a lower panel showing the residuals as a function of time. The system name and some of its parameters are provided on top of each panel.

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In the text
thumbnail Fig. F.1

Top left panel: inner binary orbit like in Fig. E.1. Top right: outer, tertiary orbit, in phase. Bottom right plot: radial-velocity data and best fit model as a function of time, and the bottom right panel is a top representation of the triple system, where the orbit is located on the right-hand side.

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In the text

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