Qatar Exoplanet Survey: Qatar-6b—A Grazing Transiting Hot Jupiter

The survey data were collected with the Qatar Exoplanet Survey (QES) hosted by the New Mexico Skies Observatory⁹ located at Mayhill, NM, USA. A full description of QES can be found in our previous publications, e.g., Alsubai et al. (2013), Alsubai et al. (2017). For completeness, here we give the main survey characteristics. QES uses two overlapping wide field 135 mm (f/2.0) and 200 mm (f/2.0) telephoto lenses, along with four 400 mm (f/2.8) telephoto lenses, mosaicked to image an 11° × 11° field on the sky simultaneously at three different pixel scales—12, 9, and 4 arcsec, respectively, for the three different focal length lenses. All lenses are equipped with FLI ProLine PL6801 cameras, with KAF-1680E 4 k × 4 k detectors. Exposure times are 60 s for each of the four CCDs attached to the 400 mm lenses, 45 s for the CCD equipped with the 200 mm lens, and 30 s, for the CCD equipped with the 135 mm lens. The combination of large aperture lenses and high survey angular resolution allows QES to reach 1% photometric precision up to 13.5–14.0 mag stars.

The data were reduced with the QES pipeline, which performs bias-correction, dark-current subtraction, and flat-fielding in the standard fashion, while photometric measurements are extracted using the image subtraction algorithm by Bramich (2008); a more detailed description of the pipeline is given in Alsubai et al. (2013).

The output light curves are ingested into the QES archive and subjected to a combination of the Trend Filtering Algorithm (TFA; Kovács et al. 2005) and the Doha algorithm (Mislis et al. 2017) to model and remove systematic patterns of correlated noise. Transit-like events are identified using the Box Least-Squares algorithm (BLS) of Kovács et al. (2002), during a candidate's search on the archive light curves following the procedure described in Collier Cameron et al. (2006). We note that the initial candidate selection is an automatic procedure, but the final vetting is done by eye. The BLS algorithm provided a tentative ephemeris, which was used to phase-fold the discovery light curve shown in Figure 1. The discovery light curve of Qatar-6b contains 2324 data points obtained from 2012 March to July.

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The host of Qatar-6b is a V = 11.44 mag (B = 12.34 mag) high proper motion star (TYC 1484-434-1, 2MASS J14485047+2209093, henceforth designated Qatar-6) of spectral type close to K2V. A detailed discussion of stellar parameters based on the analysis of our follow-up spectra and on the available photometry is presented in Section 3.1. Here, we note only that the host star spectral type is initially estimated from a multi-color (V-, J-, H-, and K-bands) fit to the magnitudes, using a standard Random-Forest classification algorithm, trained with ∼2000 standards with spectral types ranging from early A to late M.

Follow-up photometric observations for Qatar-6b were collected with the 1.2 m telescope at the Fred L. Whipple Observatory (FLWO; Mount Hopkins, Arizona) using KeplerCam, a single 4 K × 4 K CCD that covers an area of 23' × 23' on the sky. The target was observed on four occasions—(1) on the night of 2017 January 30, through Sloan i filter; (2) on 2017 May 5 (Sloan i); (3) on 2017 May 19 (Sloan r); and (4) on 2017 June 9 (Sloan z).

Two additional transits were obtained using the Near Infrared Transiting ExoplanetS telescope (NITES; McCormac et al. 2014) located at the Observatorio del Roque de los Muchachos (ORM; La Palma, Canary Islands) on 2017 March 9 and on 2017 March 23. On both nights, a total of 354 30 s images were obtained with a Johnson–Bessel V-band filter. The data were reduced in Python using ccdproc (Craig et al. 2015). A master bias, dark and flat was created using the standard process on each night. A minimum of 21 of each frame was used in each master calibration frame. Non-variable nearby comparison stars were selected by hand and aperture photometry extracted using sep (Bertin & Arnouts 1996; Barbary 2016).

Figure 2 shows the follow-up light curves together with the model fits described in Section 3.5.

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Similar to our campaigns for all QES candidates, follow-up spectroscopic observations were obtained with the Tillinghast Reflector Echelle Spectrograph (TRES) on the 1.5 m Tillinghast Reflector at the FLWO. We used TRES in a configuration with the medium fiber, which yields a resolving power of R ∼ 44,000, corresponding to a velocity resolution element of 6.8 km s⁻¹ FWHM. The spectra were extracted using version 2.55 of the code described in Buchhave et al. (2010). The wavelength calibration for each spectrum was established using exposures of a thorium-argon hollow-cathode lamp illuminating the science fiber, obtained immediately before and after each observation of the star.

For Qatar-6, a total of 34 spectra were obtained between 2016 April 13 and May 31 with exposure times ranging from 450 to 1600 s and an average signal-to-noise ratio per resolution element (S/Ne) of ∼36 at the peak of the continuum in the echelle order centered on the Mg b triplet near 519 nm. Relative radial velocities (RV) were derived by cross-correlating each observed spectrum against the strongest exposure of the same star, order by order for a set of echelle orders selected to have good S/Ne and minimal contamination by telluric lines introduced by the Earth's atmosphere. These RVs are reported in Table 1 (with the time values in Barycentric Julian Date in Barycentric Dynamical time, BJD_TDB) and plotted in Figure 3. The observation that was used for the template spectrum has, by definition, a RV of 0.0 km s⁻¹. The error on the template RV is defined as the smallest error of all the other errors. We also derived values for the line profile bisector spans (BS; lower panel in Figure 3), to check for astrophysical phenomena other than orbital motion that might produce a periodic signal in the RVs with the same period as the photometric ephemeris for the transits. The procedures used to determine RVs and BSs are outlined in Buchhave et al. (2010).

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To obtain the absolute center-of-mass velocity for the system (γ), we have to provide an absolute velocity for the observation that was used for the template when deriving the relative velocities. To derive an absolute velocity for that observation, we correlate the Mg b order against the template from the CfA library of synthetic templates that gives the highest peak correlation value. Then we add the relative γ-velocity from the orbital solution and also correct by −0.61 km s⁻¹, mostly because the CfA library does not include the gravitational redshift. This offset has been determined empirically by many observations of IAU Radial Velocity Standard Stars. We quote an uncertainty in the resulting absolute velocity of ±0.1 km s⁻¹, which is an estimate of the residual systematic errors in the IAU Radial Velocity Standard Star system.

To derive the host star characteristics, we analyzed the TRES spectra using the Stellar Parameter Classification (SPC) tool developed by Buchhave et al. (2012). The SPC cross correlates the observed spectrum with a library of synthetic spectra from Kurucz model atmospheres and finds the stellar parameters from a multi-dimensional surface fit to the peak correlation values. We used the ATLAS9 grid of models with the Opacity Distribution Functions from Castelli & Kurucz (2004). The SPC determined stellar parameters—effective temperature (T_eff), metallicity ([m/H]¹⁰ ), surface gravity (logg), and projected rotational velocity v sin i—are given in Table 2. These values are calculated by averaging the stellar parameters derived for each spectrum individually and weighted by the height of the cross-correlation function. T_eff, logg, and [m/H] are then used as input parameters for the Torres relations (Torres et al. 2010) to derive estimates for the stellar mass and radius, yielding M_⋆ = 0.820 M_⊙ and R_⋆ = 0.714 R_⊙.

Table 2.  Basic Observational Parameters of Qatar-6b Host Star and Photometry Used for the SED Fit

Parameter	Description	Value	Source	References
Names
225-118642 (UCAC3), TYC 1484-434-1
SDSS J144850.45+220909.2, 2MASS J14485047+2209093
GALEX J14485047+2209091, WISE J14485047+2209091
Astrometry
α2000	R.A. (J2000)	14h48m5042	GAIA	(1)
δ2000	Decl. (J2000)	+22o09'0940	GAIA	(1)
μα	Proper motion, R.A., mas yr−1	−51.9 ± 1.1	GAIA	(1)
μδ	Proper motion, Decl., mas yr−1	14.8 ± 1.1	GAIA	(1)
Photometry
NUV	GALEX NUV, mag	18.65 ± 0.03	GALEX	(2)
B	Johnson B, mag	12.389 ± 0.046	APASS	(7)
V	Johnson V, mag	11.438 ± 0.080	APASS	(7)
u	Sloan u, mag	13.892 ± 0.02	SDSS	(3)
g	Sloan g, mag	11.845 ± 0.02	SDSS	(4)
r	Sloan r, mag	11.070 ± 0.03	this work
i	Sloan i, mag	10.91 ± 0.05	this work
z	Sloan z, mag	10.76 ± 0.05	this work
J	2MASS J, mag	9.711 ± 0.028	2MASS	(5)
H	2MASS H, mag	9.307	2MASS	(5)
Ks	2MASS Ks, mag	9.225	2MASS	(5)
W1	WISE1, mag	9.018 ± 0.022	WISE	(6)
W2	WISE2, mag	9.078 ± 0.020	WISE	(6)
W3	WISE3, mag	9.059 ± 0.022	WISE	(6)
W4	WISE4, mag	9.405 ± 0.405	WISE	(6)
Spectroscopic parameters
	Spectral type	K2V	this work
Teff	Effective temperature, K	5052 ± 66	this work
log(g*)	Surface gravity, cgs	4.64 ± 0.01	this work
[m/H]	Metallicity	−0.025 ± 0.093	this work
γabs	Systemic velocity, km s−1	−27.864 ± 0.1	this work
v sin i	Rotational velocity, km s−1	2.9 ± 0.5	this work
Prot	Rotation period, days	12.75 ± 1.75	this work
τ	Age, Gyr	1.0 ± 0.5	this work
AV	Extinction, mag	0.093 ± 0.003	S&F2011	(8)
d	Distance, pc	101 ± 6	this work

References. (1) GAIA DR1 http://gea.esac.esa.int/archive/, (2) GALEX http://galex.stsci.edu/, (3) Pickles & Depagne (2010), (4) SDSS DR13 Albareti et al. (2017), (5) 2MASS http://irsa.ipac.caltech.edu/Missions/2mass.html, (6) WISE http://irsa.ipac.caltech.edu/Missions/wise.html (7) APASS https://www.aavso.org/apass (8) Schlafly & Finkbeiner (2011).

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The host star age is an important parameter for understanding the evolution of exoplanetary systems. We estimate the age of Qatar-6 using both the gyrochronology and isochrone fitting methods. For the gyrochronology age, we followed the formalism of Brown (2014) assuming the stellar rotation axis is perpendicular to the orbital plane. There are four equations in Brown's work describing the rotational period–color–age relation; in our case, Equations (2) (3), and (4) give consistent estimates of the star's age, while Equation (1) (based on the B − V color, see Section 3.2) gives about half the value. The uncertainty of the age estimate is entirely dominated by the errors in determining the stellar rotation (±0.5 km s⁻¹) leading to age estimates, τ_gyr, in the range 0.75–1.5 Gyr (Equations (2), (3), and (4)) and 0.4–0.6 Gyr (Equation (1)).

Additionally, using the model isochrones from the Dartmouth database (Dotter et al. 2008) and input parameters from Table 2, we calculate an independent value for the age of the host star as τ_iso ≈ 1.0 Gyr. Furthermore, the global solution for the system parameters presented in Section 3.5, which uses the YY isochrones (Yi et al. 2001), gives a star age of τ_iso = 1.02 ± 0.62 Gyr. In Table 2, we quote the age of the host star as 1 Gyr, which is where all our estimates converge and put the uncertainty conservatively at 0.5 Gyr.

The host star of Qatar-6b has been observed by a number of imaging surveys covering the entire wavelength range 0.23–22 μm, i.e., from the GALEX NUV to the WISE IR bands. The wide wavelength coverage combined with the parameters of the star derived from our spectroscopy provide an excellent base for a robust distance estimate based on a Spectral Energy Distribution (SED) fit.

However, inspection of the available data revealed that the target star is heavily saturated in many of the photometric bands covering the optical region (∼0.4 μm–1.0 μm), particularly in the automated surveys, such as Pan-STARRS (PS1¹¹ ), where the star is saturated to such a degree in all survey bands (g, r, i, z, y), that no magnitude values are available.

In SDSS (DR13, Albareti et al. 2017), measurements of the stellar magnitudes are provided, but there are clearly problems in some of the bands. For example, SDSS quotes a z-band value 2 mag fainter than the quoted i-band value, which is completely unrealistic for a star with T_eff ∼ 5000 K, as indicated by our spectra. Similarly, the u-band value is quoted to be 3 mag fainter than the g-band, which is again not realistic for the type of star we expect and the low extinction in the direction of the object (Schlafly & Finkbeiner 2011).

To rectify this situation, we used our follow-up observations with KeplerCam taken in the Sloan r-, i-, and z-bands to measure the magnitudes ourselves. This was achieved in the following fashion: for each band, we selected images based on two criteria for the stellar profile, (i) FWHM ≤ 18 (≤20 for the r-band) and (ii) ellipticity ≤0.1. These limits were a compromise between the actual data quality and the need to have an adequate number of images on the one hand; and the requirement that the best images are selected on the other. The above cuts typically left us with 20%–30% of the total images in each filter. These images were subsequently aligned and co-added, creating three master frames with equivalent exposure times of 12, 11, and 23 minutes for the r-, i-, and z-band, respectively.

We then measured the magnitude of our target in each of the three master images through differential photometry using 11 stars in its immediate vicinity, which were not saturated in the SDSS survey. Our measurements compare well with the spectrally matched magnitudes of Tyco 2 stars (Pickles & Depagne 2010) in the r- and i-bands, and are within 0.2 mag for the z-band. In the r-band, the SDSS estimate, the spectrally matched value, and our measurement are within 0.1 mag, and the g-band magnitudes given by SDSS and Pickles & Depagne (2010) are indistinguishable. For this reason, for the SED analysis, we used the u and g magnitudes from Pickles & Depagne (2010) and our measurements for the r-, i-, and z-bands.

In the B- and V-bands, independent measurements are available from the APASS survey and can also be derived from the Tycho B_T and V_T magnitudes using the Bessell (2000) relations. The TASS photometric catalog (Droege et al. 2006) gives an additional independent V-band estimate. Most notably, the three V-band values are consistent to within 0.03 mag, but the B-band measurements from APASS and Tycho differ by 0.44 mag. We note here, that the Tycho measurements have substantial errors (${\sigma }_{{B}_{{\rm{T}}}}=0.302$ and ${\sigma }_{{V}_{{\rm{T}}}}=0.124$) leading to a very uncertain B − V color. Because of stated large uncertainties in the Tycho B_T and V_T magnitudes, we adopted the B and V APASS values as representative of the true brightness of the host star in these bands. We note also that the spectrally matched B and V magnitudes from Pickles & Depagne (2010) are very close to the APASS values.

Table 2 presents the basic observational—astrometric, photometric, and spectroscopic—parameters of Qatar-6b host star. We used these broadband measurements combined with the stellar radius R_⋆ derived from the Torres relations to fit an SED using the NextGen library of theoretical models and solve for the extinction and distance. We imposed a prior of maximum extinction A_V = 0.1 mag as suggested by the Galactic dust reddening maps (Schlafly & Finkbeiner 2011). Figure 4 shows that the SED of a T_eff ∼ 5000 K main-sequence star fits well the broadband photometry for a distance d = 101 pc and minimal extinction (A_V = 0.05 mag) with a possible exception of the flux in the GALEX NUV band.

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We note that the distance determined from the SED fit compares well with the estimated photometric distance. Assuming an absolute magnitude for the host star M_V = 6.19 (K2V; Pecaut & Mamajek 2013), extinction A_V = 0.05, and apparent magnitude V = 11.438 ± 0.08 (Table 2) yields photometric distance d_phot = 101–104 pc.

Inspection of our KeplerCam follow-up data readily revealed a neighboring object, approximately 45 due south from the primary star. This object is naturally blended with the primary in the QES survey images because of the coarse pixel size, but it is clearly visible and detected in all surveys with good spatial resolution imaging data (PanSTARRS, SDSS, and 2MASS). An SDSS composite image of our target star and the neighboring object is shown in Figure 5.

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In the SDSS, the object is classified as a galaxy and even a photometric redshift is given in the catalog. However, the object is located in the wings of a heavily saturated PSF and given the problems with estimating correct magnitudes of the primary star in at least some of the survey bands, the quoted values for this neighboring object are also suspect.

To address this problem, we used the master KeplerCam frames from the follow-up observations described above to measure the neighbor/primary flux ratio. We employed two methods: (1) a PSF, determined from several isolated stars in close proximity to the target was aligned with the primary, scaled and subtracted; (2) a postage-stamp image centered on the primary was cut out, flipped along the y-axis, then aligned and subtracted from the original. Both methods allowed us to isolate the neighboring object and measure its flux using aperture photometry. This flux was then compared with the flux from the primary measured in matching aperture radii. To avoid substantial contamination from residuals from the subtraction, we used 3 and 4 pixel aperture radius. We measured flux ratios of F_neighbor/F_primary = 0.0049, 0.0146, 0.0266 in the Sloan r-, i- and z-bands respectively, corresponding to magnitude differences of Δm = 5.80, 4.59, and 3.94 mag. In addition, we note that at the limit of KeplerCam images resolution (18), we do not detect departure from the PSF.

To further investigate the nature of the neighboring object, we used the NIRC2 infrared camera behind the Keck II NGS AO system on 2017 August 3 to obtain differential near-infrared photometry of the companion and to examine its shape. We operated NIRC2 in its 9.95 mas pixel⁻¹ mode (Service et al. 2016), which results in a field of view of ∼10''. We used the J, H and K filters and obtained 2-point dithered images, for sky subtraction, with total exposure times of 40 s in each filter. We calibrated the images with dome flat-fields taken during the day. Figure 6 shows half of the sky subtracted NIRC2 AO frame (dithered position 1).

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The AO images show, without a doubt, the companion object is a star—the first two Airy rings of the diffraction limited PSF are clearly visible in all three filters. The flux ratio was measured separately from the two dithered positions and the results were averaged. We note that flux ratios determined from the two dithered positions are constant to within the measurements uncertainties for all aperture radii between R_ap = 03 − 07. The estimated values in the J-, H-, and K-bands are F_neighbor/F_primary = 0.0382, 0.0438, 0.0512, respectively, corresponding to magnitude differences Δm = 3.54, 3.40, and 3.19 mag.

From the flux ratio measurements described above, we derive the apparent magnitudes of the companion object—17.02, 15.49, 14.69, 13.26, 13.71, 12.41 in the r, i, z, J, H, and K filters, respectively. The estimated uncertainties are 0.05–0.1 mag in the Sloan bands, dominated by the image subtraction residuals, and 0.05–0.08 mag in the NIR filters. The observed broadband magnitudes constrain well the shape of the SED and indicate the companion is a late M star with T_eff ≈ 3200 K. Unfortunately, we do not have a independent handle on the stellar radius that will allow us to determine the distance to the companion. If we assume it is an MS star, the R_⋆ can be estimated using the semi-empirical T_eff–radius relation derived by Mann et al. (2015). The nominal value from that relation is R_⋆ = 0.25 R_⊙, corresponding to a distance to the neighbor D ≈ 90 pc. There is, however, a significant uncertainty (∼15%) associated with the T_eff–radius relation leading to a substantial spread in distances. It is interesting to note that for R_⋆ = 0.28 R_⊙, i.e., only 1σ away from the nominal value, the estimated distance to the companion is the same as to the primary, D ≈ 100 pc (see Figure 4) in which case the projected distance between the two stars would be ∼450 au.

To further test whether the two stars are possibly associated, we place them on a color–magnitude diagram (CMD) using the Gaia G (DR1; Gaia collaboration 2016) and 2MASS J magnitudes. Gaia separates the two stars cleanly and lists their magnitudes as 11.076 and 15.982, respectively. For the J-band, we use the 2MASS magnitude of the primary and our estimate of the companion magnitude based on the Keck AO images. Figure 7 shows the CMD for the two stars and a 1 Gyr theoretical isochrone from the Padova database (Marigo et al. 2017). The CMD clearly shows our estimates of the primary star parameters and the distance are close to the theoretically expected values for a 1 Gyr old star. The companion object, on the other hand, projects about 3σ away from the theoretical isochrone. At face value, the CMD does not support the association scenario; however, given the uncertainties and assumptions about the close-by star, we cannot draw any definite conclusion. Further investigation of possible association is beyond the scopes of this paper and can be carried out much more reliably once the proper motion of both stars is measured by Gaia.

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Eclipsing binaries, both projected and gravitationally bound, are the main source of false-positive detections in transit surveys. Fortunately, this is not the case for Qatar-6b for two reasons. First, being more that 5 mag fainter, the companion object cannot produce the observed transit depth of ∼2% in the r- and V-bands. Second, the observed RVs, which are measured from the part of the spectrum covering the Sloan g- and r-bands, can only come from the primary star, as the companion contributes <1% to the combined light in this wavelength range. Thus, the observed RV curve reflects the orbital motion of the primary star only and it is at the same period as the transits.

To better determine the transiting planet ephemeris, we used the data from the follow-up photometric curves. As described in a previous section, we have observed six transits of Qatar-6b between 2017 January 30 and June 9, spanning 37 orbital cycles of the system. First, all observational time stamps were placed on the BJD_TDB system and each light curve was rebinned to a uniform cadence of 2 minutes for the KeplerCam observations and 2.5 minutes for the Meade LX200GPS observations, to reduce the error bars on individual points. When binning, the errors were propagated assuming data points were uncorrelated and we checked that this level of binning did not affect derived parameters. Each individual light curve was then fitted with a transit model using EXOFAST (Eastman et al. 2013). We note that EXOFAST uses a wavelength dependent quadratic limb darkening prescription and uncertainties of the fitting parameters are estimated via Markov Chain Monte Carlo (MCMC) minimization.

The transit central times T_C and their uncertainties were estimated from the model fits and we calculated the best ephemeris by fitting a straight line through all of the points. Figure 8 shows the fit and the residuals from the linear ephemeris, and the measurements of T_C and their uncertainties are summarized in Table 3. The final orbital ephemeris is expressed as

$$\begin{eqnarray}&&{T}_{C}=2457784.03270(49)+3.506195(18)E\end{eqnarray} \tag{ 1 }$$

where E is the number of cycles after the reference epoch, which we take to be the y-intercept of the linear fit, and the numbers in parenthesis denote the uncertainty of the last two digits of the preceding coefficient. Here, the reference epoch and the period are in days on the BJD_TDB scale, and their uncertainties correspond to 42 s, and 2 s, respectively. The quality of the fit as measured by χ² = 1.11 indicates that a linear ephemeris is a good match to the available measurements given the quality of the observations. Thus, an investigation of possible transit timing variations (TTV) would require data of much superior quality.

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To determine the physical parameters of the planetary system, we run a global solution of the available RV and transit photometric data using version 2 of EXOFAST—a full re-write of the original package, designed to simultaneously fit RV and/or transit data for multiple planets, transits, and RV sources (J. Eastman 2017, private communication). Further description of EXOFASTv2 and similarities and differences with the original version can be found in Eastman (2017) and Rodriguez et al. (2017). One major difference, which is important in our case, is that EXOFASTv2 uses the YY isochrones (Yi et al. 2001) to model the star, instead of the Torres relations (Torres et al. 2010, see also Eastman et al. 2016 for a more detailed description).

The global fit of Qatar-6b includes the RV measurements listed in Table 1 and the six follow-up photometric light curves shown in Figure 2. The stellar parameters (T_eff, logg, [m/H]) determined from the spectroscopic analysis (Section 3.1) and the orbital period and reference epoch measured from the analysis of the transit central times (Section 3.4) are fed as initial parameters for the global fit.

The spectroscopically determined stellar parameters and the comparison with the theoretical isochrones suggest that the host star is well positioned on the main sequence. Quadratic limb darkening coefficients (LDCs) were interpolated from the Claret & Bloemen (2011) tables for each filter and used as initial input. The LDCs were left free to vary, but we imposed a Gaussian prior around the interpolated value with an uncertainty of 0.05. An additional Gaussian prior was imposed on the stellar age, with a mean value of 1 Gyr and an uncertainty of 0.5 Gyr, based on our analysis in Section 3.1, while an upper-bound uniform prior of A_V < 0.1 was imposed on the visual extinction, as suggested by the Galactic dust reddening maps (Schlafly & Finkbeiner 2011).

With respect to the eccentricity, using the equations from Leconte et al. (2010) and Jackson et al. (2008), we calculated the tidal circularization timescale for the system. We used the values from Table 4 of M_⋆, R_⋆, M_P, R_P, assuming tidal quality factors of Q_⋆ = 10^6.5 and Q_P = 10^5.5. The rotation period given in Table 4, P_rot = 12.75 days, is calculated using the stellar radius from our solution, the v sin i from the spectra, and assuming the stellar rotation axis and the planet orbit are coplanar. The timescale for circularization is τ_circ = 0.08 Gyr, significantly lower than the estimated age of the host star and, thus, we expect the planet orbit to have circularized. Additionally, our RV data is not of high enough quality to allow investigation of (any potential) small departures from circularity. For these reasons, in fitting the Qatar-6b data set, we only considered a circular orbit and kept the eccentricity fixed to zero.

Table 4 summarizes the physical parameters of the planetary system. The best fit for both radial velocity and photometric light curves comes from the EXOFASTv2 global fit. The Safronov number is not used in the current paper and is provided in Table 4 for completeness, as it may be useful for other studies. A plot of the posterior distributions form the MCMC fit for selected stellar and planetary parameters is presented in Figure 9 to demonstrate the quality of the fit. For the LDCs, the MCMC fits converge very close to the interpolated values from Claret & Bloemen (2011) with typical uncertainty of 0.05.

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The shape of our follow-up transit curves suggested the planet is likely on a grazing transit. For each transiting planetary system, there is a minimum inclination, i_gr, below which the transit is grazing, i.e., part of the planet disk's shadow will always be outside the stellar disk even at maximum coverage. This theoretical limit is given by Equation (2)

$$\begin{eqnarray}&&\cos {i}_{{gr}}=\displaystyle \frac{{R}_{\star }-{R}_{{\rm{p}}}}{a}\end{eqnarray} \tag{ 2 }$$

In the case of Qatar-6b, this limit is i_gr = 8616, using the planetary radius and semimajor axis values from Table 4. On the other hand, the inclination we measure through the global model fit is i = 8606. The transit is just grazing (considering the error bars); thus, the value of the planet radius we measure from the transiting light curves should be very close to its true value (Figure 10). Nevertheless, we need to make clear that, for a grazing transit, the value of the planet radius from the model fit should be taken as a lower limit.

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To a first degree, the shape of a planetary transit light curve is determined by the ratio of the planet radius to the stellar radius, the orbit scale, and the transit impact parameter. Stellar limb darkening (LD), although it is a second order effect, also plays an important role, as it modifies the shape of the light curve and affects the transit depth. In addition, the changes to the shape of the light curve and the depth of the transit are both wavelength- and impact parameter-dependent. In the case of Qatar-6b, our data allow us to check if the observed differences in the transit depth conform with the expected effects of the stellar limb darkening.

The effects of the LD have been the subject of a number of studies. Here, we use the formalism of Csizmadia et al. (2013, see their Appendix B), where the transit depth ΔF/F dependence on the impact parameter and wavelength can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}F}{F}={k}^{2}\displaystyle \frac{1-{u}_{1}(1-\mu )-{u}_{2}{(1-\mu )}^{2}}{1-\tfrac{{u}_{1}}{3}-\tfrac{{u}_{2}}{6}}\end{eqnarray} \tag{ 3 }$$

where k = R_p/R_⋆ is the ratio of planet radius to star radius, and u₁ and u₂ are the linear and quadratic limb darkening coefficients. We also use the common notation $\mu =\cos \gamma$, where γ is the angle between the normal vector of the stellar surface point and the direction to the observer. It is easy to show that, at the time of maximum light loss, $\mu =\sqrt{1-{b}^{2}}$, where b is the impact parameter. Keeping in mind that limb darkening coefficients are wavelength dependent, Equation (3) describes both the impact parameter and wavelength dependence of the transit depth.

The case of Qatar-6b is demonstrated in Figure 11. The left panel shows the expected radial profile of the central transit depth ΔF/F as a function of the impact parameter b, as approximated by Equation (3). For a given value of b, the wavelength dependence of ΔF/F is apparent as different values for curves shown for V-band (0.55 μ, green), Sloan r (0.62 μ, yellow), Sloan i (0.77 μ, red), and Sloan z (0.92 μ, black). The panel illustrates that for central transits (e.g., b ≲ 0.5), the transit depth is expected to decrease with wavelength, while for grazing transits (b ≳ 0.8), it should increase with wavelength. The right panel of Figure 11 shows what we observe in Qatar-6b, and it matches what is expected for a grazing transit. For clarity, only the model fits to the filter light curves are shown in matching colors, as in the left panel. For Qatar-6b, we measure central transit depth values of 0.0178, 0.0180, 0.0182, and 0.0192 for V-, Sloan r-, i-, and z-band filters, respectively. These compare reasonably well with the predicted values of 0.0185, 0.0190, 0.0196, and 0.0201 from Equation (3).

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