VERY LOW MASS STELLAR AND SUBSTELLAR COMPANIONS TO SOLAR-LIKE STARS FROM MARVELS. V. A LOW ECCENTRICITY BROWN DWARF FROM THE DRIEST PART OF THE DESERT, MARVELS-6b

MARVELS is a multi-epoch radial velocity survey designed to detect radial velocity companions around FGK type stars in a magnitude range of (7.6 ⩽ V ⩽ 12). It uses a dispersed fixed-delay interferometer (DFDI; Ge et al. 2009) on the SDSS 2.5 m telescope (Gunn et al. 2006). The DFDI method was introduced for use in a multi-object RV survey by Ge (2002). A single object version of a DFDI instrument was successfully used to detect a hot Jupiter around HD 102195 (Ge et al. 2006). The DFDI instrument principle was described by Ge (2002), Ge et al. (2002), Erskine (2003), van Eyken et al. (2010), and Wang et al. (2011). The MARVELS interferometer delay calibrations were described in Wang et al. (2012a, 2012b).

Each MARVELS observation consists of 60 stars spread across 120 spectra (2 spectra for each star). The MARVELS survey started in 2008 October and ran through 2012 July. It was divided up into two observing sets: year 1–2 fields which were observed from 2008 October till 2011 January and year 3–4 fields which were observed from 2011 January till 2012 July.

The year 1–2 data set consisted of 43 unique fields (11 of which are in the Kepler; Borucki et al. 2010). The year 3–4 field set contained 13 fields (with 1 field overlapping the year 1–2 field set.) This leads to a total of 3,300 unique stars with >18 epochs.

The large number of RV targets required the development of a software package to easily display and characterize the radial velocity curves from the MARVELS survey. One of the primary tools used in this process is an IDL-based Keplerian model fitter known as MPRVFIT.²⁴ MPRVFIT starts with the input of RV data in the form of Julian date, RV, and RV error (the Julian date and RVs can be of any type that is appropriate to the analysis). In the case of MARVELS, the data from the two beams have been combined as described in Fleming et al. (2010). The software then applies a modified version of the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982) described in Cumming (2004). False alarm probabilities were assigned to the highest peaks using the formulation of Baluev (2008). Once a number of high probability peaks are identified, the frequency space between the peak and the next nearest frequency point in the periodogram is sub-divided into 10 frequency steps (on both sides of the peak). A Keplerian model is then fit to those frequencies using MPFIT (Markwardt 2009). MPFIT is a Levenberg–Marquardt non-linear least squares fitter implemented in IDL. The χ² statistic is determined for each fit, and the best chi-squared fit of the grid is retained. Finally, the best fit models are plotted with the data points for easy reference. For the MARVELS survey, the two best fits for each target were displayed, and the candidates were chosen based on the folded and unfolded radial velocity curves, as well as the significance of the periodogram. An example of this periodogram for GSC 03546-01452 can be found in Figure 1.

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Differential RV observations of GSC 03546-01452 were acquired during the first two years of the SDSS-III MARVELS survey. A total of 22 observations were obtained over the course of 565 days. As discussed in Section 2.1.1, the MARVELS survey uses the DFDI technique which introduces an interferometer into the light path. As a result, each 50 minute observation includes two fringed spectra, or beams, one from each arm of the interferometer. Each spectrum spans a wavelength range roughly 500–570 nm with a R ∼ 12, 000 resolution. Each beam is processed individually through the MARVELS pipeline following the methods described in Lee et al. (2011).

The formal errors derived from the MARVELS pipeline are known to be underestimates of the true error. This systematic underestimate can be partially corrected for by using the fact that 60 stars (120 beams) are taken in each observation. Following the method outlined in Fleming et al. (2010), it is expected that most stars in an observation plate are radial velocity stable, so the median rms is a reasonable estimate of the systematic errors. A quality factor (QF) is derived for each beam, which is the ratio of the radial velocity rms to the median formal error bar, and the median QF is found for the plate. The formal errors are multiplied by this QF (2.334 for GSC 03546-01452), resulting in the error bars used for the analysis of these observations. During the RV analysis, discussed in Section 3.2, it was determined that these scaled error bars were themselves overestimated, leading to a reduction of a factor of 0.5794. The final differential RV measurements and final scaled error bars (approximately 40% larger than the formal uncertainties) for GSC 03546-01452 are presented in Table 1.

Once GSC 03546-01452 was determined to be a candidate for additional investigation, spectroscopic observations were conducted with the Spettrografo Alta Risoluzione Galileo (SARG; Gratton et al. 2001) on the 3.6 m Telescopio Nazionale Galileo (TNG) located at Roque de Los Muchachos Observatory (ORM). All the spectra were acquired using the same instrumental configuration: a slit with a sky-projected width of 08 achieving a resolving power of R = 57000; a yellow cross-dispersing grism providing the wavelength range 462 < λ < 792 nm.

A total of 15 spectra were taken (11 with and 4 without the iodine cell inserted in the light path). One spectrum with the iodine cell proved to have too low signal-to-noise ratio (S/N), leaving only 10 spectra suitable for differential velocity measurements. The spectra were processed using the standard IRAF²⁵ Echelle reduction packages. The S/N per resolution element at 550 nm ranges from 50 to 130. The spectra without I2 lines served as a reference for measuring the relative radial velocities of the 10 spectra with superimposed I2 absorption lines. The technique adopted to derive RV values is described in Marcy & Butler (1992). For details on how the technique was implemented on SARG spectra, see Fleming et al. (2012). As is the case with the MARVELS RV data, the formal uncertainties for the TNG/SARG data were overestimated, and so the error bars were thus reduced by a factor of 0.2224. The final differential RV and reduced error bars are listed in Table 1.

All 15 spectra taken with the TNG/SARG instrument were used to calculate absolute RVs. The stellar spectra were cross-correlated with a high resolution solar spectrum.²⁶ The cross-correlation was done using the SARG red CCD spectrum with a wavelength coverage of 6200–8000 Å. Due to the non-simultaneity of the ThAr wavelength calibration exposures, it is possible that the slit illumination varied between the science spectra and the calibration spectra, which could result in the RV measurement being affected by systematic errors. To partially mitigate this error source, we calculated the cross-correlation function of the telluric lines (Griffin & Griffin 1973) around 6900 Å with a numerical mask. This exercise results in an RV correction of a few hundred m s⁻¹. This correction plus the barycentric correction were applied to the radial velocities and the resulting values are shown in Table 2.

To validate this method, observations of the RV standard star HD3765 were used. The standard was observed at 8 epochs with a mean result of −63,155 m s⁻¹ ± 128 m s⁻¹. This result is within a 1σ agreement with the mean 34 observations taken from the ELODIE archive of −63,286 ± 49 m s⁻¹. For these observations, the systematics caused by the non-simultaneity of the ThAr wavelength calibration is the dominant error source, so we adopt ±128 m s⁻¹ for the error in these measurements.

We used the Hereford Arizona Observatory (HAO), a private facility in southern Arizona (observatory code G95 in the IAU Minor Planet Center), to measure multi-band, absolute photometry of GSC 03546-01452. HAO employs a 14 inch Meade Schmidt-Cassegrain (model LX200GPS) telescope, fork-mounted on an equatorial wedge located in a dome. The telescope's CCD is an SBIG ST-10XME with a KAF-3200ME detector. A 10-position filter wheel accommodates SDSS and Johnson/Cousins filter sets. GSC 03546-01452 and standard stars were observed with Johnson B and SDSS u', g', r', i' filters (Fukugita et al. 1996). For the Johnson–Kron–Cousins bands, standard stars are taken from the list published by Landolt & Uomoto (2007) and Landolt (2009). For the SDSS bands, standard stars are taken from the list published by Smith et al. (2002). Observations were conducted on three dates: 2010 April 24 and 25 and on 2010 May 7. Between 23 and 75 standard stars (Landolt and SDSS) were used to establish the transformations to the standard photometric systems.

We obtained 44 nights of observations of GSC 03546-01452 with the 16 inch Keeler RCX-400 Meade telescope at the Allegheny Observatory, University of Pittsburgh. The observations span 18 months from 2010 May through 2011 November and were all made through a Cousins R filter onto an SBIG KAF-6303E/LE 2048 × 3072 pixel CCD with a pixel scale of 057 pixel⁻¹ for a total FoV of 195 by 292. Exposure times ranged from 30 to 150 s depending on conditions, with a median exposure time of 75 s.

Standard bias and dark subtraction and flatfield calibrations were preformed on the raw data, and the images were astrometrically calibrated to the Two Micron All Sky Survey (2MASS) Point-Source Catalog (Skrutskie et al. 2006). Typical seeing conditions were between 3''and 4'' with a median seeing of 35. Photometry was accomplished using a 10-pixel (57) circular aperture and subtracting the estimated sky flux based on a 15–20 pixel (86–114) radius sky annulus. Typical 1σ uncertainties were 5 mmag. Relative photometry was calculated relative to the average counts from two reference stars in the image with J2000 coordinates: (1) 19:11:34.733 +48:34:54.77; and (2) 19:11:56.335 +48:22:55.38.

The SuperWASP survey (Pollacco et al. 2006) is a wide-angle transiting planet survey that monitors the brightness of millions of stars. The survey uses a visual broad band filter that covers the wavelength range from 400–700 nm. For details on reduction techniques and survey design please consult Pollacco et al. (2006). The SuperWASP photometry for GSC 03546-01452 consists of 8823 points spanning just over 4 yr from 2004 May to 2008 August.

We also acquired adaptive optics (AO) observations of GSC 03546-01452 to assess its multiplicity at wide separations. Images were obtained on UT 2011 August 31 using NIRC2 (PI: Keith Matthews) at the 10 m Keck II telescope (Wizinowich et al. 2000). GSC 03546-01452 (V = 11.7) is sufficiently bright to serve as its own natural guide star. Our observations consist of 9 dithered images (10 coadds per frame, 0.5 s per coadd) taken with the K' filter (λ_c = 2.12 μm). We used NIRC2's narrow camera setting, which has a plate scale of 10 mas pixel⁻¹, to provide fine spatial sampling of the instrument point-spread function (PSF). Raw frames were processed by cleaning hot pixels, flat-fielding, subtracting background noise from the sky and instrument optics, and aligning and coadding the results. No off-axis sources were noticed in individual frames or the final processed image. Figure 2 shows the contrast levels generated by the observations. Our diffraction-limited images rule out the presence of companions down to ΔK' = 5.7, 7.1, 7.5 mag at angular separations of 025, 05, and 10, respectively.

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Lucky imaging (LI) involves acquiring large numbers of observations with a rapid cadence. A subset of these images are then shifted and stacked in order to produce nearly-diffraction-limited images. GSC 03546-01452 was observed over 3 nights separated by 2 yr using FastCam (Oscoz et al. 2008) on the 1.5 m TCS telescope at Observatorio del Teide in Spain. The LI frames were acquired on 2010 October 9, 2011 August 25 and 2012 September 11 in the I band and spanning ∼21 × 21 arcsec² on sky.

The 2010 October 11 observing run had 100,000 frames with 50 ms per frame, the 2011 August 25 run had 60,000 frames with 60 ms per frame, and the 2012 September 11 run had 75,000 frames at 60 ms per frame. The data were processed using a custom IDL software pipeline. After identifying corrupted frames due to cosmic rays, electronic glitches, etc., the remaining frames are bias corrected and flat fielded.

Lucky image selection is applied using a variety of selection thresholds based on the brightest pixel (BP) method. This method involves selecting frames for the final combined LI image based on the BP in that frame. The selected BP must be below a specified brightness threshold to avoid selecting cosmic rays or other non-speckle features. As a further check, the BP must be consistent with the expected energy distribution from a diffraction speckle under the assumption of a diffraction-limited PSF. The BP's of each frame are then sorted from brightest to faintest, and the best X% are then shifted and added to generate a final combined image. Several values of X, the LI threshold, are tried until the best combined image is generated. The combined images using different LI thresholds are shown for each of the three observing runs in Figure 3. The 2012 September 11 run has a better quality image than the other two runs because of an instrument upgraded to a higher sensitivity CCD. The best LI threshold was determined to be 50%, which is what we used for the rest of the analysis. This results in a total exposure time for each of the final combined images from the three runs in order of date 2500, 1800, and 2250 s, respectively.

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In Figure 4, we show the 3σ detectability curves computed out to 8'' from GSC 03546-01452. We follow the same procedure as in Femenía et al. (2011) to compute these curves: at a given angular distance ρ from GSC 03546-01452, we identify all possible sets of small boxes of a size larger but comparable to the FWHM of the PSF (i.e., 5 × 5 pixel boxes). Only regions of the image showing structures easily recognizable as spikes due to diffraction of the telescope spider and/or artifacts on the read-out of the detector are ignored. For each of the valid boxes on the arc at angular distance ρ the standard deviation of the image pixels within the 5-pixel boxes is computed. The value assigned to the 3σ detectability curve at ρ is three times the mean value from the standard deviations of all the eligible boxes at ρ. This procedure, using each of the LI % thresholding values provides a detectability curve, while the envelope of all the family of curves for a given night yields the best possible detectability curve to be extracted from the whole data set.

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Although there are no companions detected within 7'', we do find a possible companion slightly outside this region at a separation of 77. This companion is significantly dimmer than GSC 03546-01452 with a ΔI ≈ 7.9 mag. This object was only detected in the 2010 October 11 and 2012 September 11 campaigns. The visual companion was not detected in the 2012 August 25 campaign due to the orientation of the camera on the sky. The possible companion can be seen in Figure 3. A discussion of the likelihood that this object is actually a bound companion to GSC 03546-01452 can be found in Section 3.5.

The ARCES moderate resolution spectrum (Section 2.3) was analyzed by two independent analysis pipelines. We refer to these pipeline results as the "IAC" (Instituto de Astrofísica de Canarias) and "BPG" (Brazilian Participation Group) results. Briefly, both methods are based on the principals of Fe i and Fe ii excitation and ionization equilibria. Both techniques employ the 2002 version of the MOOG code (Sneden 1973), but use different line lists, model atmosphere grids, equivalent width measurements, and convergence criteria. For a complete description of the analysis process, please refer to Wisniewski et al. (2012).

Our IAC analysis used 204 Fe i lines and 20 Fe ii lines, and resulted in the following parameters: T_eff = 5502 ± 100, log g = 4.21 ± 0.58, and [Fe/H] =+0.31 ± 0.16. The BPG analysis used 61 Fe i lines and 6 Fe ii lines resulting in these parameters: T_eff = 5652 ± 75, log g = 4.46 ± 0.16, and [Fe/H] =+0.44 ± 0.10. Since the values for both methods are consistent to within 1σ, they were combined using a weighted average as described in Wisniewski et al. (2012). This calculation resulted in the final spectroscopic parameter values of T_eff = 5598 ± 63, log g = 4.44 ± 0.17, and [Fe/H] =+0.40 ± 0.09. These values are recorded in Table 3, and are used for the remaining analysis.

Table 3. Host Star Properties GSC 03546-01452

Parameter	Value	1σ Uncertainty	Source
GSC1.1 Name	3546-01452	⋅⋅⋅	Lasker et al. (2008)
GSC2.3 Name	N2EA000110	⋅⋅⋅	Lasker et al. (2008)
KIC Name	11022130	⋅⋅⋅	Brown et al. (2011)
2MASS Name	J19113252+4830436	⋅⋅⋅	Cutri et al. (2003)
α (J2000) (deg)	287.885735	026	Lasker et al. (2008)
δ (J2000) (deg)	48.512117	025	Lasker et al. (2008)
μα (mas yr−1)	−28.6	2.3	Zacharias et al. (2013)
μδ (mas yr−1)	−8.7	1.7	Zacharias et al. (2013)
Teff (K)	5598	63	This work
log g (cgs)	4.44	0.17	This work
[Fe/H] (dex)	+0.40	0.09	This work
vmic (km s−1)	0.38	0.17	This work
vrotsin i (km s−1)	<9.0	⋅⋅⋅	This work
vsystemic (km s−1)	−13.799	0.034	This work
AV (mag)	0.1	0.1	This work
Distance (pc)	219	21	This work
M* (M☉)	1.11	0.11	This work
R* (R☉)	1.06	0.23	This work
galNUV	18.320	0.100	Morrissey et al. (2007)a
B	12.593	0.020	HAO (This work)a
V	11.799	0.010	Derived from HAO
RC	11.335	0.012	Derived from HAO
IC	10.967	0.012	Derived from HAO
SDSS u'	13.483	0.020	HAO (This work) a
SDSS g'	12.156	0.012	HAO (This work)a
SDSS r'	11.560	0.010	HAO (This work)a
SDSS i'	11.400	0.010	HAO (This work)a
2MASS J	10.46	0.030	Cutri et al. (2003)a
2MASS H	10.07	0.030	Cutri et al. (2003)a
2MASS KS	10.010	0.03	Cutri et al. (2003)a
WISE1	12.650	0.023	Wright et al. (2010)a
WISE2	13.344	0.020	Wright et al. (2010)a
WISE3	15.181	0.032	Wright et al. (2010)a

Note. ^aThese passbands were used to generate SED in Figure 5.

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From these values of T_eff, log g, and [Fe/H] and their errors, and using the Torres et al. (2010) relations, including intrinsic scatter and the uncertainties in the polynomial coefficients and covariances, we find that the mean and rms of the mass and radius of the host are M_* = 1.11 ± 0.11 M_☉ and R_* = 1.06 ± 0.23 R_☉ The median and 68% confidence intervals are $M_*= 1.11 _{-0.10}^{+0.11}\, M_\odot$ and $R_*=1.03_{-0.19}^{+0.26}\, R_\odot$.

These spectroscopic values also allow us to transform the detection curves from the left panel of Figure 4 into upper limits on the mass of a possible stellar companion undetected at the 3σ level. We start by taking the spectroscopic T_eff from Table 3, and using the tables from Mamajek (2010), we derive absolute V- and I-band magnitudes. We can apply this technique because GSC 03546-01452 has been identified as a main-sequence star by the log g measurement. The M_I and the 3σ detectability curves allow the construction of the M_I versus angular distance, ρ, from the central star. This curve provides an absolute upper I-band limit to any companion, which would also have to be a main-sequence star. From the M_V versus ρ curves we can use empirical mass–luminosity relationships (Henry et al. 1999; Delfosse et al. 2000; Henry 2004; Xia et al. 2008; Xia & Fu 2010) to derive the upper mass limit. These constraints are plotted for all three observation nights in the right panel of Figure 4.

We corroborated the spectroscopic results with a series of photometric tests, including determining the RPM-J statistic from Collier Cameron et al. (2007). For GSC 03546-01452, the RPM-J value was 2.31 and its (J − H) 2MASS color was 0.387, results that are consistent with GSC 03546-01452 being a dwarf.

We also constructed a spectral energy diagram (SED) of GSC 03546-01452 using the Galaxy Evolution Explorer (GALEX) NUV filter (Morrissey et al. 2007); Johnson B and SDSS u',g',r',i' observations from HAO; 2MASS J, H, and K (Cutri et al. 2003); and WISE1, WISE2, and WISE3 (Wright et al. 2010). A summary of the photometric values are presented in Table 3. We used the NextGen model atmosphere grid (Hauschildt et al. 1999) to construct theoretical SEDs. These models were fixed by the spectroscopic values for T_Eff, log g, and [Fe/H] described in Section 3.1.1; the reddening was constrained by the Schlegel et al. (1998) dust maps in the direction of (l, b) = (79339029, 16853379) to be less than A_V = 0.213. The resulting fit to the photometry is shown in Figure 5. The best fitting model has a reduced χ² of 3.91, a negligible reddening of A_V = 0.1 ± 0.1, and a distance of 219 ± 21 pc. There appears to be a slight UV excess, which may be the result of modest stellar activity.

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Using the spectroscopic host star parameters and the mass and radius derived from the Torres et al. (2010) relations, GSC 03546-01452 can be placed onto an evolutionary track. We use the Yonsei-Yale ("Y2") model tracks from Demarque et al. (2004 and references therein), and select the track corresponding to 1.11 M_☉ with a [Fe/H] =+0.40. Figure 6 shows this track with stellar ages marked as blue dots on the track. The dashed lines show tracks for the same metallicity, but for ±0.11 M_☉, which is the 1σ uncertainty on our mass estimate. The gray area shows the region that this family of tracks occupies. The red point shows GSC 03546-01452 with 1σ error bars. From the error bars in log g and T_eff in Figure 6, we can constrain the age of GSC 03546-01452 to being less than approximately 6 Gyr old.

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We can determine the Galactic population membership of GSC 03546-01452 by using the absolute systemic RV = −13.799 ± 0.128 km s⁻¹ (where the 0.128 km s⁻¹ is a conservative estimate of the systematic error from the comparison with RV standard stars), the proper motions from UCAC4 (Zacharias et al. 2013; μ = −28.6 ± 2.3, −8.7 ± 1.7 mas yr⁻¹), and the distance from Table 3 (219 ± 21 pc). We find (U, V, W) = (24.0 ± 2.5, −10.1 ± 1.3, 25.6 ± 3.1) km s⁻¹. This velocity is consistent with membership in the thin disk according to the criteria of Bensby et al. (2003). The space motion velocities were determined using a modification of the IDL routine GAL_UVW, which is ultimately based on the method of Johnson & Soderblom (1987). We adopt the correction for the Sun's motion with respect to the local standard of rest from Coşkunoǧlu et al. (2011), and choose a right-handed coordinate system such that positive U is toward the Galactic Center.

The mass function (${\cal M}_b$) is the only property of a single-lined radial velocity companion that we can derive that is independent of the properties of the primary. Using the MCMC chain from the joint RV fit, we can derive ${\cal M}_b$ of the companion,

$$\begin{eqnarray} {\cal M}_b &\equiv & \frac{(M_b \sin i)^3}{(M_*+M_b)^2} = K^3 (1-e^2)^{3/2} \frac{P}{2\pi G} \nonumber\\ &=& (2.136 \pm 0.049) \times 10^{-5}\, M_\odot , \end{eqnarray} \tag{ 1 }$$

where the uncertainty is essentially dominated by the uncertainty in K, such that $\sigma _{{\cal M}_b}/{\cal M}_b \sim 3(\sigma _{K}/{K}) = 3\times 0.8\% \sim 2.4\%$.

In order to determine the mass or mass ratio of the secondary, we must estimate the mass of the primary as well as the inclination of the secondary.

To estimate the mass and radius of the primary, we proceed as follows. For each link in the MCMC chain from the joint RV fit, we draw a value of T_eff, log g, and [Fe/H] for the primary from Gaussian distributions, with means and dispersions given in Table 3. We then use the Torres et al. (2010) relations to estimate the mass M_* and radius R_* of the primary, including the intrinsic scatter in these relations.

The minimum mass (i.e., M_b if sin i = 1) and minimum mass ratio of the secondary are:

$$\begin{eqnarray} M_{b,\rm min}&=& 31.7 \pm 2.0\ M_{{\rm Jup}} = 0.0303 \pm 0.0020\, M_\odot , \nonumber\\ q_{{\rm min}} &=& 0.02733 \pm 0.00092. \end{eqnarray} \tag{ 2 }$$

The uncertainties in these estimates are almost entirely explained by the uncertainties in the mass of the primary: $\sigma _{M_b}/M_b \sim (2/3)(\sigma _{M_*}/M_*) = (2/3)\times 9.7\% \sim 6.5\%$, close to the uncertainty in M_{b, min} above (6.4%), and $\sigma _q/q \sim (1/3)(\sigma _{M_*}/M_*) \sim (1/3)\times 9.7\% \sim 3.2\%$, close to the uncertainty in q (3.4%).

The a posteriori distribution of the true mass of the companions given our measurements depends on our prior distribution for the mass of the companion, which is roughly equivalent to our prior on the mass ratio.

If we assume a prior that is uniform in the logarithm of the true mass of the companion, then the posterior distribution of cos i will be uniform. More generally, for other priors, cos i is not uniformly distributed. We adopt priors of the form

$$\begin{equation} \frac{\it dN}{dq} \propto q^{\alpha } , \end{equation} \tag{ 3 }$$

where q is the mass ratio between the companion and the primary, and α = −1 for the uniform logarithmic prior discussed above. To include this prior, for each link of the MCMC chain we draw a value of cos i from a uniform distribution, but then weight the resulting values of the derived parameters for that link (i.e., the companion mass m) by q^{α + 1}.

For α > 0, the a posteriori distribution does not converge. However, we can rule out equal mass ratio companions by the lack of a second set of spectral lines. We therefore assume that the mass ratio cannot be greater than unity; i.e., we give zero weight to inclinations such that q > 1. In doing so, we implicitly assume that the companion is not a stellar remnant.

Figures 9 shows the resulting cumulative a posteriori distributions of the true mass of the companion for α = −1 (uniform logarithmic prior on q), α = 0 (uniform linear prior), and α = 1. For α = −1 and α = 0, the inferred median masses are ∼37 M_Jup and ∼45 M_Jup, respectively. For these priors, we conclude that the companion has a mass below the hydrogen burning limit and thus is a bona fide brown dwarf at 90% and 72% confidence, respectively. For α = 1, the median mass is 172 M_Jup or ∼0.16 M_☉, firmly within the stellar regime. Indeed, for this prior, there is only a ∼32% probability that the companion is a brown dwarf. However, this conclusion is sensitive to the precise form for our constraint that q ⩽ 1. With a more careful analysis it may be possible to rule out a wider range of companion masses based on the lack of evidence for a second set of spectral lines. Furthermore, there is little evidence that the mass function of companions to solar type stars is rising as steeply as α = 1 for minimum masses around 30 M_Jup (Grether & Lineweaver 2006).

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We conclude that the companion to GSC 03546-01452 is most likely a brown dwarf. However, we cannot definitively exclude that it is, in fact, a low-mass stellar companion seen at low inclination.

The SuperWASP photometric dataset for GSC 03546-01452 consists of 8823 points spanning just over 4 yr from HJD' = 3128 to 4690 (where HJD' ≡ HJD-2450000). The full, detrended SuperWASP dataset has a relatively high weighted rms of 1.8% and exhibits evidence for systematics. The distribution of residuals from the weighted mean is asymmetric and non-Gaussian, showing long tails containing a much larger number of >3σ outliers than would be expected for a normally-distributed population. We therefore cleaned the SuperWASP data as follows. We first add a trial systematic fractional error in quadrature to the photon noise uncertainties σ_sys. We then compute the error-weighted mean flux, and determine and reject the largest error-normalized outlier from the mean flux. We then recompute the mean flux and scale the uncertainties by a constant factor r to force χ²/dof = 1. We iterate this procedure until no more >4σ outliers remain. We then repeat the entire procedure to determine the value of σ_sys that results in a distribution of normalized residuals that has the smallest rms from the Gaussian expectation. Although 4σ is a slightly larger deviation than we would expect based on the final number of points, we adopt this conservative threshold in order to avoid removing a potential transit signal. We adopt r = 0.60 and σ_sys = 0.011. The cleaned light curve has 8635 data points with an rms of 1.5% and χ²/dof = 1 (by design).

The Allegheny photometric dataset consists of 3990 points spanning roughly a year and a half from HJD' = 5341 to 5884. The weighted rms of the raw light curve is 0.63%. There is mild evidence for systematics errors in this dataset, and thus we repeat the identical procedure as with the SuperWASP data. We choose a more conservative 6σ cut, given the better precision of the Allegheny observations, to prevent us from removing real transit-like variability. We adopt r = 1.04 and σ_sys = 0.0028, with a final rms of 0.4% from 3983 data points.

While continuous variability will not be removed by our procedure, variability in the form of a small number of highly-discrepant points will be masked. This possibility is particularly relevant for transit signatures. Given the typical uncertainties of ∼0.4% for that data set, we do not expect our cleaning procedure to remove outliers that differ by less than ∼2.4% from the mean, roughly corresponding to the typical depth for a transit of a ∼1.5 R_Jup companion given the primary radius of ∼1.03 R_☉.

Finally, we combine all the relative photometry after normalizing each individual data set by its mean weighted flux. Figure 10 shows the combined data set, which consists of 12,618 data points spanning roughly 7.6 yr from HJD' = 3128 to 5884. The weighted rms is 0.66%. The resulting light curve is constant to within the uncertainties over the entire time span. Because the individual data sets sample disjoint times and are not relative to a common set of reference stars, our procedure of normalizing each data set by its mean flux will remove variations on the longest timescales.

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We ran a Lomb–Scargle periodogram on the full photometric dataset, testing periods between 1 and 10³ days. The resulting periodogram shown in Figure 11 does not exhibit any strong peaks. The maximum fitted amplitude over this range is ∼0.1%. The inset presents the periodogram of the combined data set for periods within ∼10 days of the period of the companion. The individual periodograms for the SuperWASP and Allegheny data are also shown. While the SuperWASP data display a local peak at the period of the companion, this feature is not corroborated by the Allegheny data. Although this difference could in principle arise from real variability that is present in the SuperWASP data, but not in the Allegheny data (i.e., evolving spots), given the higher precision of the Allegheny data, we believe it is more likely that the peak in the SuperWASP periodogram is due to low-level systematics in the form of residual correlations on a range of time scales.

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Figure 12 shows the combined light curve, folded at the median period and time of conjunction of the companion (P = 47.8929 ± 0.0063 and T_C = 5023.377) and binned 0.05 in phase. The weighted rms of the binned light curve is ∼0.041%. Although the variations are larger than expected from a constant light curve based on the uncertainties (χ²/dof ∼ 2.6), we again suspect that these are due to systematic errors in the relative photometry. In particular, the folded, binned Allegheny light curve shows a somewhat lower rms of ∼0.016%, and is more consistent with a constant flux with a χ²/dof = 0.4.

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We conclude that there is no strong evidence for variability of GSC 03546-01452 on any time scale we probe. We can robustly constrain the amplitude of any persistent periodic variability to be less than ≲ 0.1% over timescales of ≲ 3 yr, and we can constrain the amplitude of persistent photometric variability at the period of the companion to ≲ 0.02%.

For a uniform distribution in cos i (corresponding to a prior that is uniform in the logarithm of the mass), the secondary transit probability is ∼1.8%. The duration for a central transit is ∼R_*P/(πa) ∼ 6.6 hr, and the depth for a Jupiter-radius body is δ ∼ (R_b/R_*)² = 1.0%(R_b/R_Jup)², where R_b is the radius of the companion, and we have adopted the median value for the primary radius of R_* = 1.03 R_☉. Thus, the expected S/N of a transit in the combined photometric dataset assuming uniform phase coverage is

$$\begin{equation} {{\rm S/N}} \sim N^{1/2} \left(\frac{R_*}{\pi a}\right)^{1/2} \frac{\delta }{\sigma }\sim 13 \left(\frac{R_b}{R_{{\rm Jup}}}\right)^2 , \end{equation} \tag{ 4 }$$

where N = 12618 is the number of data points and σ ∼ 0.66% is the rms of the combined light curve. Thus we would expect to be able to robustly detect or exclude transits from companions with radii as small as ∼0.9 R_Jup at 10σ if the transit phase is well covered by the photometric data. However, given the long period of the companion, uniform phase coverage is not necessarily a good approximation.

Therefore, in order to account for the actual sampling of the photometric data, we perform a quantitative search for transit signals using a modified Monte Carlo method. This method is similar to that described in Fleming et al. (2012). We briefly review the details here. We start with the distributions of the relevant radial velocity fit parameters: the period P, semiamplitude K, eccentricity e, and time of conjunction T_C. These are obtained from the MCMC chain determined by fitting of the joint radial velocity data set as described in 3.2. For each link in this MCMC chain, we also draw values for T_eff, log g, and [Fe/H] for the primary from Gaussian distributions, with means and dispersions given in Table 3. We then use the Torres et al. (2010) relations to estimate the mass M_* and radius R_* of the primary, including the intrinsic scatter in these relations. We then draw a value of cos i from a uniform distribution, assuming a prior on the companion mass M_b that is uniform in log M_b. The values of P, K, e, M_*, R_* and i are then used to determine the secondary mass M_b, semimajor axis a, and impact parameter of the secondary orbit b ≡ acos i/R_*. From the impact parameter, we determine if the companion transits (b ⩽ 1), and if so, we determine the properties of the light curve using the routines of Mandel & Agol (2002) for a given companion radius R_b. We assume quadratic limb darkening, adopting coefficients appropriate for the R band from Claret & Hauschildt (2003), assuming solar metallicity and the central values of T_eff and log g listed in Table 3. Finally, we fit the predicted transit light curve to the combined photometric data, and compute the Δχ² between the constant flux fit and the transit model. We perform this calculation for every step in the Markov Chain, and finally repeat this procedure for a range of companion radii.

We find that a best-fit transit model has Δχ² = −9.0 relative to a constant fit. We do not consider this result to be significant, as we find a larger improvement Δχ² = −28.1 when we consider signals with the same ephemeris and shape as transits, but corresponding to positive deviations (i.e., "anti-transits," see Burke et al. 2006). Although formally statistically significant, both the transit and anti-transit signals are likely caused by residual systematics in the cleaned photometric data set, and thus we conclude there is no evidence for a transit signal in the photometric data.

This procedure can be used to determine the confidence with which we can rule out transits of a companion with a given radius. Specifically, the confidence with which we can exclude a companion with a given R_b is simply the fraction of the steps in the Markov Chain where the companion transits, which produces a transit with a Δχ² relative to the constant fit that is greater than some value of Δχ². We consider three different thresholds of Δχ² = 25, 49, and 100. The resulting cumulative probability distributions are shown in Figure 13. Given the values of Δχ² found for anti-transits, we conservatively consider thresholds of Δχ² ≳ 49 to be robust. For this value, we can exclude companions with R_b ≳ 0.7 R_Jup with 50% confidence (i.e., for 50% of the trials), and companions with R_b ≳ 1.2R_J with 90% confidence. However, there is a long tail toward large companion radii, arising from the conflation of the long period of the companion, uncertainties in the ephemeris, and imperfect phase coverage. Therefore, we are unable to exclude transits at >95% confidence even for R_b ≳ 1.5. For the minimum companion mass of ∼32 M_Jup, Baraffe et al. (2003) predict radii of ∼0.9–1 R_Jup for ages between 0.5 and 5 Gyr. Therefore, we cannot definitively exclude the possibility that the companion transits.

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As discussed above, our 6σ clipping procedure would in principle remove ≳ 2.4% transit signatures from the Allegheny dataset arising from a large R_b ≳ 1.5 R_Jup companion. We therefore repeated the transit search on the Allegheny dataset with 10σ clipping, but again find no significant transits.