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Appendix A: Radial velocity correction from the constant star HD 185144

We present in this Appendix the SOPHIE radial velocities of the constant star HD 185144. During 2012 and 2013, this constant star was observed systematically on the same nights as the Kepler targets. It was observed using the same instrumental mode, i.e. the High-Efficiency mode (HE), but in ThoSimult mode (with the calibration ThoAr lamp observed simultaneously). This star was also observed in High-Resolution mode (HR), less systematically, in order to control the performance and stability of the spectrograph to search for low-mass planets (Bouchy et al. 2013). We selected here only the observations that have reached a signal-to-noise per pixel in the spectra of at least 50 at 550 nm. The star HD 185144 was observed 137 times in HE during 2012 and 2013. Figure A.1 shows its radial velocity, bisector, and FWHM variations. Those measurements are listed in Table A.6.

Radial velocities of HD 185144 present a rms of 6.9 m s^-1  in HE during both seasons. During 2012 only, this rms was of 8.0 m s^-1  while in 2013 it was of 5.5 m s^-1. These rms are much smaller than the radial velocity uncertainty of the target KOI-1257 (⟨ σ_RV ⟩ = 26 m s^-1). However, even if the rms is relatively small for the required precision of KOI-1257, the constant star was observed to vary in HE by a maximum of 34 m s^-1  in 2012 during a timescale of 20 days, and by 24 m s^-1  in 2013 with a timescale of four days. This corresponds to about one third and one fourth of the radial velocity amplitude of the transiting planet (K = 94 ± 21 m s^-1). Those variations are not well understood but seem to be correlated with the temperature outside the dome. During the summers of 2012 and 2013 at Observatoire de Haute-Provence there was a fast drop in the temperature due to a storm after a week of relatively hot days. This fast drop of the local temperature at the observatory is observed for both seasons

	Fig. A.1 Radial velocities, bisector, and FWHM variations (from top to bottom) of the constant star HD 185144 observed by SOPHIE in HE during 2012 and 2013. We corrected these measurements by using their median value: about 27.79 km s-1  for the RVs, about 8.8 km s-1  for the FWHM, and 3 m s-1  for the bisector.
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in HD 185144 data by a radial velocity variation at the level of ~20 m s^-1  with a timespan of about one week. This effect is not observed in the HR mode. Before the implementation of octagonal fibres in June 2011 (Perruchot et al. 2011), the same effect was observed with an amplitude about five times larger (see Hébrard et al. 2013a). s

The bisector of HD 185144 does not show variation within ~4 m s^-1  during 2012 and 2013. We concluded that SOPHIE bisectors are stable and we did not correct the observed bisectors of KOI-1257.

The observed FWHM of HD 185144 presents a rms of about 15 m s^-1  for 2012 and 2013. The observed pattern is not well understood, but is correlated with the flux of the ThoAr lamp. The peak-to-valley amplitude is of about 60 m s^-1, which is one order of magnitude less than the variation observed for KOI-1257 (at the level of about 600 m s^-1). Since the observations of KOI-1257 were not performed with the simultaneous ThoAr lamp, we decided not to apply a correction to its FWHMs.

Appendix B: Upper-limit constraint in mass from a radial velocity drift

The quadratic radial velocity drift observed in SOPHIE data of KOI-1257 was analysed in Sect. 3.8 assuming a circular orbit. The MCMC analysis converged toward a higher probability of having a short-period brown dwarf rather than a long-period solar-like star. However, it is well known that from a radial velocity drift, it is possible to constrain only the lower-limit in mass of a companion, but not its upper-limit in mass. To understand the result obtained in Sect. 3.8, we performed the following test. We generated synthetic radial velocity data using the SOPHIE observations (the observing times and the radial velocity uncertainty) assuming a pure white noise. We modelled two circular orbits with a period of 3400 days (which corresponds to the most-likely period of the outer orbit in the KOI-1257 system) and a radial-velocity semi-amplitude of 1 km s^-1. The epoch of periastron of the two orbits were chosen in order that the data display either a linear drift or a quadratic drift, the latter being the same as observed in the case of KOI-1257. The synthetic data for the linear and quadratic drift are displayed in the top panels of Figs. B.1 and B.2, respectively.

Fig. B.1 Top panel: synthetic radial velocity dataset (black points) superimposed with the circular orbit model (red line) used to generate the data. This model shows only a linear drift during the timespan of the observations. Bottom panel: the 99.7% confidence region of the posterior distribution for the orbital period versus the radial velocity semi-amplitude (black line). The modelled orbit is marked with the red circle. The upper limit in the radial velocity amplitude (at 10 km s-1) comes from the prior.

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We analysed both datasets as done in Sect. 3.8 but assuming that the system is only described with a circular orbit. We used exactly the same priors for both analyses, using Jeffreys priors for both the orbital period and the radial-velocity semi-amplitude. The other priors were chosen as large and uniform distributions. We show in the bottom panels of Figs. B.1 and B.2 the 99.7% confidence region from the posterior distribution computed through the same MCMC procedure as in Sect. 3.8. The two posterior distributions present a different shape: while the linear-drift dataset (Fig. B.1) provides only a lower-limit in mass of the companion, the quadratic-drift dataset (Fig. B.2) provides both a lower- and an upper-limit in mass. This can be explained easily by considering that it is more likely to observe a significant curvature in the radial velocity data if the orbital period is relatively short. On the other hand, for orbital periods much longer than the timespan of the observations, it is more likely to observe a nearly linear drift. This effect might also be explained by the fact that the second-order polynomial, needed to describe the radial-velocity data, more accurately constrains the orbit of the companion than a first-order polynomial. Then, the constraints in period translate into constraints in radial-velocity amplitude (or companion mass) thanks to the assumption of purely circular orbit.

	Fig. B.2 Same as Fig. B.1, but for the quadratic drift dataset.
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The constraints on the upper-mass of the companion found in Sect. 3.8 therefore come from the assumption of a perfectly circular orbit and the curvature of the drift observed by SOPHIE.

Appendix C: Addditional tables

© ESO, 2014