HAT-P-57b: A SHORT-PERIOD GIANT PLANET TRANSITING A BRIGHT RAPIDLY ROTATING A8V STAR CONFIRMED VIA DOPPLER TOMOGRAPHY^*

All time-series photometric data that we collected for HAT-P-57 are provided in Table 1. We discuss these observations below.

Table 1.  Differential Photometry of HAT-P-57

BJDa	Magb	${\sigma }_{{\rm{Phot}}}$	Mag(orig)c	Filter	Instrument
(2,400,000+)
54998.84656	0.00470	0.00247	⋯	r	HATNet
55025.96482	0.00396	0.00220	⋯	r	HATNet
55072.80545	−0.00518	0.00237	⋯	r	HATNet
55067.87492	−0.00243	0.00211	⋯	r	HATNet
55021.03497	−0.00126	0.00251	⋯	r	HATNet
54984.05576	−0.00489	0.00211	⋯	r	HATNet
55062.94639	0.00637	0.00208	⋯	r	HATNet
55077.73838	−0.00658	0.00252	⋯	r	HATNet
55067.87893	−0.00097	0.00212	⋯	r	HATNet
55025.96892	0.00174	0.00215	⋯	r	HATNet

Notes.

^aBarycentric Julian Date calculated directly from UTC, without correction for leap seconds. ^bThe out-of-transit level has been subtracted. These values have been corrected for trends simultaneously with the transit fit for the follow-up data. For HATNet trends were filtered before fitting for the transit. ^cRaw magnitude values after correction using comparison stars, but without additional trend-filtering. We only report this value for the KeplerCam observations.

Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VOT) versions of the full table are available.

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2.1.1. Photometric Detection

The star HAT-P-57 was observed by the HATNet wide-field photometric instruments (Bakos et al. 2004) between the nights of UT 2009 May 12 and UT 2009 September 14. A total of 622 observations of a $10\buildrel{\circ}\over{.} 6\times 10\buildrel{\circ}\over{.} 6$ field centered at ${\rm{R.A.}}={06}^{{\rm{h}}}{24}^{{\rm{m}}},\;$ ${\rm{decl.}}\;=\;+30^\circ$ were made with the HAT-5 telescope in Arizona, and 3202 observations of this same field were made with the HAT-9 telescope in Hawaii (the count is after filtering 12 outlier measurements). We used a Sloan r filter and an exposure time of 300 s. Following Bakos et al. (2010) and Kovács et al. (2005), we reduced the images to trend-filtered light curves and searched these for periodic transit signals using the Box-fitting Least Squares algorithm (BLS; Kovács et al. 2002).

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Transits were detected in the light curve of HAT-P-57 with a period of $P=2.4652950\pm 0.0000032$ day. Figure 1 shows the phase-folded light curve together with our best-fit model. This same target has also been included in a list of TEP candidates published by the Super-WASP survey (Lister et al. 2007), and with a similar ephemeris, but was shortly thereafter set aside as a probable binary system based on follow-up spectroscopic observations showing that the star has a very rapid rotation (Collier Cameron et al. 2007). The target has also been independently identified as a TEP candidate by the KELT survey (J. Pepper 2015, private communication; the KELT project is discussed in Siverd et al. 2012). We searched the light curve for additional transit signals or other periodic variations by running BLS and the Generalized Lomb–Scargle (Zechmeister & Kürster 2009) algorithms on the residuals from the best-fit transit model. No additional transit signals were detected, however we do find two periodic signals at frequencies of 1.839 day⁻¹ (or its alias at 0.837 day⁻¹) and 2.030 day⁻¹ (or its alias at 1.026 day⁻¹) with false alarm probabilities of ${10}^{-6.1}$ and ${10}^{-2.1},$ respectively, and with peak-to-peak amplitudes of 1.7 mmag and 1.3 mmag, respectively. The second frequency is identified after fitting and subtracting a sinusoid with a frequency of 1.839 day⁻¹. A further whitening cycle, with the 2.030 day⁻¹ signal also removed, reveals no other significant periods in the data (the highest signal in the final periodogram has a peak-to-peak amplitude of 1.2 mmag). In each case the aliases are of comparable significance to the highest peak in the periodogram, and so we cannot determine whether the true primary frequency is 1.839 day⁻¹ or 0.837 day⁻¹, or whether the true secondary frequency is 2.030 day⁻¹ or 1.026 day⁻¹. Figure 2 shows the periodogram. One or both of these signals may be associated with the rotation period of the star, or they could correspond to gravity modes (all frequencies are too low for p-modes), in which case, given its surface temperature, HAT-P-57 would be a γ Dor-type variable (cf., WASP-33 which shows both δ-Scuti and γ-Dor variations). The temperature and surface gravity inferred for HAT-P-57 in Section 3.1 place it within both the classical δ-Scuti and γ-Dor instability strips (e.g., Rodríguez & Breger 2001, and Handler & Shobbrook 2002, respectively), so pulsational variability is expected for HAT-P-57.

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2.1.2. Photometric Follow-up

Photometric follow-up observations of HAT-P-57 were carried out with KeplerCam on the Fred Lawrence Whipple Observatory (FLWO) 1.2 m telescope. We observed ingress events on the nights of 2010 April 3 and 2012 April 24, in i and g-bands respectively, and a full transit on the night of 2010 June 26 in z-band. The images were reduced to light curves following Bakos et al. (2010), including external parameter decorrelation performed simultaneously with the transit fit to remove systematic trends; the systematics-corrected light curves are shown in Figure 3. The rms of the residuals from our best-fit model varies from 1.4 to 3.4 mmag for these data.

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Additional photometric follow-up observations were carried out with the FLWO 1.2 m on the night of 2015 May 12 covering most of a predicted secondary eclipse event. The observations were performed in z-band and were used to constrain blend scenarios (Section 3.4).

2.1.3. Adaptive Optics (AO) Imaging

We obtained high-resolution imaging of HAT-P-57 on the night of UT 2011 June 22 using the Clio2 near-IR imager (Freed et al. 2004) on the MMT 6.5 m telescope on Mt. Hopkins, in AZ. Observations in H-band and ${L}^{\prime }$-band were made using the AO system. Figure 4 shows the resulting images which reveal the presence of a binary pair of stars located $2\buildrel{\prime\prime}\over{.} 667\pm 0.001$ from HAT-P-57. The pair of stars is resolved into a $0\buildrel{\prime\prime}\over{.} 225\pm 0.002$ binary in the ${L}^{\prime }$ image. In H-band the two objects are not cleanly resolved, but the point-spread function (PSF) is clearly elongated. Applying aperture photometry to the ${L}^{\prime }$ observations we find that the two components have ${\rm{\Delta }}{L}^{\prime }$ magnitudes relative to HAT-P-57 of ${\rm{\Delta }}{L}_{B}^{\prime }=2.72\pm 0.09$ mag and ${\rm{\Delta }}{L}_{C}^{\prime }=3.16\pm 0.10$ mag, respectively. To determine the relative H-band magnitudes we perform PSF fitting fixing the relative positions of the binary components to those from the ${L}^{\prime }$ images and using HAT-P-57 to define the PSF. We find ${\rm{\Delta }}{H}_{B}=2.82\pm 0.10$ mag and ${\rm{\Delta }}{H}_{C}=3.83\pm 0.11$ mag.

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Figure 5 shows the location of the two binary components, together with HAT-P-57, on a $H-{L}^{\prime }$ versus H CMD. We also show 1.0 Gyr isochrones from the PARSEC model (Bressan et al. 2012) with metallicities of [M/H] = $-0.25$ and [M/H] = $0.0,$ and shifted to the distance of HAT-P-57 inferred in Section 3.1. We show the PARSEC model isochrones, rather than the Y² isochrones which are used in Section 3.1 to determine the physical parameters of HAT-P-57, because the PARSEC models provide a better match to the NIR photometry of M dwarf stars. The three stars are consistent with being on the same isochrone, so we conclude that the binary objects are likely to be physically associated with HAT-P-57. Assuming this is the case, we adopt the names HAT-P-57B and HAT-P-57C for the components of the binary objects, and estimate their masses to be $0.61\pm 0.10$ ${M}_{\odot }$ and $0.53\pm 0.08$ ${M}_{\odot }$, respectively. The $0\buildrel{\prime\prime}\over{.} 225\pm 0.002$ angular separation between HAT-P-57B and HAT-P-57C corresponds to a projected physical separation of $68\pm 3$ AU, and approximate orbital period of $500$ years (assuming the projected separation corresponds to the physical semimajor axis of the orbit), while the $2\buildrel{\prime\prime}\over{.} 667\pm 0.001$ separation between the binary objects and HAT-P-57 corresponds to a projected physical separation of $800\pm 30$ AU and approximate orbital period of 14,000 year.

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Note that although we do not spatially resolve the binary object from HAT-P-57 in any of our photometric light curves, the Doppler tomography observations prove that the transiting component is orbiting the bright A star rather than either of the fainter components (Section 3.3). We also note that even without the Doppler tomography observations we would still be able to draw this conclusion as the binary object is too faint (in the optical), and its components are too red, to be responsible for the 1% transits seen consistently with KeplerCam in the g, i and z-bands. Given the $H-{L}^{\prime }$ colors of the binary objects, we expect HAT-P-57B to be 6.2 mag fainter than HAT-P-57 in g-band, 4.4 mag fainter than HAT-P-57 in i-band, and 4.0 mag fainter in z-band. Even if HAT-P-57B were totally eclipsed, the blended eclipse depth of 0.3% in g would be too shallow to produce the observed 1% deep transits. Moreover, transits in i and z would be significantly deeper than in g, which is not what we observe.

Figure 6 shows the approximate 5σ detection limits for any additional companions to HAT-P-57 in the H and ${L}^{\prime }$-bands as a function of angular separation. These are estimated as five times the standard deviation of the pixel values in circular annuli centered on HAT-P-57, relative to the peak pixel value of HAT-P-57.

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We carried out spectroscopic observations of HAT-P-57 between UT 2010 April 5 and UT 2010 July 1 with the Tillinghast Reflector Echelle Spectrograph (TRES; Fűresz 2008) on the 1.5 m Tillinghast Reflector at FLWO. We also obtained spectra of HAT-P-57 on UT 2010 July 1–3 with the Fiber-fed Echelle Spectrograph (FIES; Telting et al. 2014) on the 2.56 m Nordic Optical Telescope (NOT) at the Observatorio del Roque de los Muchachos, on La Palma, Spain. Additional spectra were obtained using HIRES (Vogt et al. 1994) on the Keck-I 10 m telescope between UT 2010 June 27 and UT 2012 March 10. A total of 24 HIRES observations were collected during this time period, including 14 observations made through the I₂ cell (e.g., Marcy & Butler 1992), and 10 observations without the I₂ cell. These latter observations were obtained on the night of UT 2010 June 27, primarily during a planetary transit (Section 3.3 discusses the analysis of these observations in more detail).

The TRES and FIES observations were reduced to initial RVs, bisector spans (BSs) and stellar atmospheric parameters following Buchhave et al. (2010). Higher precision stellar atmospheric parameters were also measured from these observations using the Stellar Parameter Classification program (SPC; Buchhave et al. 2012). These measurements clearly indicate that HAT-P-57 is a rapidly rotating star with $v\mathrm{sin}i\gtrsim 100$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, and a surface temperature hotter than 7000 K. Due to the very broad absorption lines, however, the surface gravity cannot be reliably determined from the spectra with these techniques. Finally we note that the spectra do not appear to be composite, and also show no large RV variations. The five TRES RVs have an rms scatter of 2.2 $\mathrm{km}\;{{\rm{s}}}^{-1}$, while the three FIES RVs have an rms scatter of 0.35 $\mathrm{km}\;{{\rm{s}}}^{-1}$. The difference in precision is largely due to using a single spectral order to measure the TRES RVs, compared to five orders used for FIES (a multi-order analysis of the TRES data would yield more precise measurements, but given the lower S/N of the spectra compared to those from FIES, we expect the scatter would still exceed that of the FIES RVs).

Wavelength calibrated spectra were extracted from the HIRES echelle images using the reduction pipeline of the California Planet Search team. The 14 I₂-in observations were reduced to relative RVs in the barycentric frame of the Solar System following the method of Butler et al. (1996). For this we made use of the highest S/N out-of-transit I₂-free observation as a template. These are shown phase-folded with the orbital ephemeris in Figure 7. We also measured spectral line BSs from the I₂-free blue orders for 22 of the observations following Torres et al. (2007). Wavelength extracted spectra were not available for two observations and were excluded from the BS analysis. The BS values are also shown in Figure 7. Our procedure for measuring the BSs involves cross-correlating the observed spectra against a synthetic template with atmospheric parameters similar to those measured for HAT-P-57. We used these same cross-correlations to measure the barycentric-corrected RVs for the spectra, including the 10 I₂-free observations made on UT 2010 June 27. Due to the slit-fed nature of HIRES, and the lack of a simultaneously obtained wavelength calibration reference, the RV precision from this CCF method is substantially lower than the precision obtained from the standard I₂ Doppler pipeline. More precise CCF-based RVs were also measured from the 10 I₂-free observations using both the blue and green spectral orders. Table 2 gives the relative RV measurements obtained with the I₂ Doppler pipeline, the RV measurements obtained from the CCFs, and the BSs for the HIRES observations.

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Table 2.  Radial Velocities and Bisector Span Measurements of HAT-P-57 from Keck-I/HIRES

BJDa	RV ${{\rm{I}}}_{2}$ b	${\sigma }_{\mathrm{RV}\ {{\rm{I}}}_{2}}$ c	RV CCF Bd	${\sigma }_{{\rm{RV}}\;{\rm{CCF}}\;{\rm{B}}}$	RV CCF B + Ge	${\sigma }_{{\rm{RV}}\;{\rm{CCF}}\;{\rm{B\ +\ G}}}$	BS	${\sigma }_{{\rm{BS}}}$	Phasef
(2,455,000+)	(${\rm{m}}\;{{\rm{s}}}^{-1}$)	(${\rm{m}}\;{{\rm{s}}}^{-1}$)	($\mathrm{km}\;{{\rm{s}}}^{-1}$)	($\mathrm{km}\;{{\rm{s}}}^{-1}$)	($\mathrm{km}\;{{\rm{s}}}^{-1}$)	($\mathrm{km}\;{{\rm{s}}}^{-1}$)	(${\rm{m}}\;{{\rm{s}}}^{-1}$)	(${\rm{m}}\;{{\rm{s}}}^{-1}$)
374.81081	⋯	⋯	−8.45	0.97	−7.30	0.74	206	108	0.003
374.82136	⋯	⋯	−9.75	1.02	−8.49	0.81	338	77	0.008
374.83442	⋯	⋯	−9.49	0.99	−8.40	0.77	261	66	0.013
374.84418	⋯	⋯	−10.87	0.86	−9.56	0.74	256	58	0.017
374.85378	⋯	⋯	−10.31	0.91	−9.03	0.75	−3	72	0.021
374.86293	⋯	⋯	−9.41	0.97	−8.42	0.75	10	60	0.024
374.87329	⋯	⋯	−8.92	0.97	−7.90	0.74	−46	43	0.029
374.88075	⋯	⋯	−7.84	0.83	−6.75	0.67	−102	70	0.032
374.89223	⋯	⋯	−8.13	0.95	−6.85	0.79	−210	70	0.036
375.04019	⋯	⋯	−7.45	1.09	−6.74	0.77	−116	53	0.096
375.04462	−492	45	−7.99	1.08	⋯	⋯	28	64	0.098
375.85754	−626	48	−8.68	0.99	⋯	⋯	−114	70	0.428
467.82924	−22	49	−8.66	1.00	⋯	⋯	−39	90	0.734
607.15061	319	52	−9.34	1.00	⋯	⋯	48	36	0.247
608.13252	80	53	−9.03	1.06	⋯	⋯	−109	79	0.646
612.15141	−104	57	−7.94	0.90	⋯	⋯	−201	66	0.276
613.16261	206	54	−7.09	1.01	⋯	⋯	−259	53	0.686
699.91700	−185	51	−7.66	0.93	⋯	⋯	−169	93	0.876
701.11852	368	54	−9.37	0.96	⋯	⋯	350	57	0.364
703.89871	84	48	−9.36	0.86	⋯	⋯	−93	85	0.491
705.89808	130	50	−9.17	0.86	⋯	⋯	49	65	0.302
707.87969	351	58	−6.93	1.04	⋯	⋯	−83	95	0.106
879.73744	−31	64	⋯	⋯	⋯	⋯	⋯	⋯	0.817
997.07656	−171	95	⋯	⋯	⋯	⋯	⋯	⋯	0.413

Notes.

^aBarycentric Julian Date calculated directly from UTC, without correction for leap seconds. ^bRVs computed using the I₂ method. The zero-point of these velocities is arbitrary. An overall offset fitted to these velocities in Section 3 has not been subtracted. Spectra obtained without the I₂-cell in do not have an RV measurement listed in this column. ^cInternal errors excluding the component of astrophysical jitter considered in Section 3. ^dRVs computed using the CCF method, applied only to the I₂-free blue spectral orders. Note that the units here are $\mathrm{km}\;{{\rm{s}}}^{-1}$ rather than ${\rm{m}}\;{{\rm{s}}}^{-1}.$ ^eRVs computed using the CCF method, applied to the blue and green spectral orders. Observations obtained with the I₂-cell in do not have a measurement listed here. ^fOrbital phase, with phase zero corresponding to mid-transit.

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The CCF-based RVs and the BSs measured from the in-transit HIRES observations both show evidence of the Rossiter–McLaughlin effect (Figure 8). Note that the sense of the variation seen in both indicators is consistent. For the BS values plotted, we are using the definition

$$\begin{eqnarray}&&{BS}={{RV}}_{{\rm{CCF,min}}}-{{RV}}_{{\rm{CCF,max}}}\end{eqnarray} \tag{ 1 }$$

where ${{RV}}_{{\rm{CCF,min}}}$ is the bisector velocity at the low CCF value (i.e., in the wings of the absorption line), while ${{RV}}_{{\rm{CCF,max}}}$ is the bisector velocity at the high CCF value (in the core of the line). A positive BS indicates that the core of the line is blueshifted compared to the wings of the line. Thus with this definition, we expect the BS and RV variations to be anti-correlated. Rather than attempting to measure the projected spin–orbit alignment angle λ from these observations we perform a Doppler tomography analysis of the line profile distortions in Section 3.3. In Figure 8 we also show the approximate expected RV variation due to the RM effect calculated using the ARoME package (Boué et al. 2013) for the maximum posterior probability solution determined from the line profile modeling. This model underpredicts the anomalous Doppler shift, and if we attempt to fit the model directly to the observations then the results require a very high value for $v\mathrm{sin}i$ (175 $\mathrm{km}\;{{\rm{s}}}^{-1}$), which is completely inconsistent with the width of the line profiles seen in the HIRES spectra. The model, however, is not applicable to very high rotation rates ($v\mathrm{sin}i\gt 20$ $\mathrm{km}\;{{\rm{s}}}^{-1}$), and so such a discrepancy is not unexpected.

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We measured the stellar atmospheric parameters for HAT-P-57 in two ways. First we analyzed both the HIRES I₂-free observations and the FIES observations with SPC. For the second method we performed a ${\chi }^{2}$ comparison of synthetic templates from the Pollux database (Palacios et al. 2010) to the HIRES observations.

The two HIRES orders covered by the SPC library yield substantially different results for the temperature, metallicity and surface gravity. For one of the orders we find ${T}_{{\rm{eff}}}=6620$ K, $\mathrm{log}g=3.39,$ [M/H] = $-1.08$ and $v\mathrm{sin}i\;=101.7$ $\mathrm{km}\;{{\rm{s}}}^{-1}$. The normalized CCF has a peak height of 0.978 indicating a good match between the observations and the synthetic template. For the other order we find ${T}_{{\rm{eff}}}=8450$ K, $\mathrm{log}g=4.37,$ [M/H] = $0.01$ and $v\mathrm{sin}i\;=102.3$ $\mathrm{km}\;{{\rm{s}}}^{-1}$. The CCF in this case has a peak height of 0.984, again indicating a good match. When the surface gravity is fixed to 4.20, based on the Y² stellar evolution models, and using the transit-based stellar density and the effective temperature and metallicities estimated from the spectra (Section 3.2; note that we find that the surface gravity from this analysis is quite well constrained despite the very large initial uncertainty in the temperature and metallicity), we find temperatures of ${T}_{{\rm{eff}}}=7250$ K and ${T}_{{\rm{eff}}}=8280$ K, and metallicities of [M/H] = $-0.70$ and = $-0.03$ from the respective orders. Due to the lack of consistency between the two orders, we conclude that the spectral overlap between HIRES and the SPC library is too small, in this case, for a reliable determination of the atmospheric parameters.

When we analyzed the three FIES observations with SPC we find ${T}_{{\rm{eff}}}=6830\pm 120$ K, $\mathrm{log}g=2.95\pm 0.15,$ [M/H] = $-0.82\pm 0.04,$ and $v\mathrm{sin}i=103.7\pm 0.9$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, with a cross-correlation peak-height of 0.914. The uncertainties are the standard deviation of the measurements from the three separate observations. When the surface gravity is fixed to $\mathrm{log}g=4.20,$ we find ${T}_{{\rm{eff}}}=7440\pm 80$ K, [M/H] = $-0.39\pm 0.05,\;$ $v\mathrm{sin}i=102.8\pm 1.1$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, and a cross-correlation peak-height of 0.910. The listed uncertainties reflect the precision, but not the accuracy, of the measurements. In particular they do not account for the degeneracies between the parameters, which are more significant at such high rotation velocities than they are for slower rotating stars where the true SPC errors have been well calibrated.

For an A star like HAT-P-57 the wavelength range of 5050–5360 Å used by SPC does not contain very many good absorption lines for determining the atmospheric parameters, resulting in significant degeneracies between the parameters. We therefore carried out a separate analysis of the Keck/HIRES observations of HAT-P-57, focusing in this case on 18 blue orders covering the wavelength range 3840–4793 Å. This range of the spectrum contains several Hydrogen Balmer lines, whose broad profiles constrain the temperature, as well as many ionized metal lines which are useful for determining both the metallicity and the temperature.

We first normalized the continua of the observed spectra by fitting polynomials in wavelength and order to the numerous continuum regions available for this rapidly rotating star. We then used the Pollux database (Palacios et al. 2010) to obtain a grid of synthetic high resolution spectra generated using the MARCS atmosphere models (Gustafsson et al. 2008). The grid spans the temperature range 6500–8000 K in 250 K steps, metallicities from [M/H] = $-1.0$ to 1.0 in 0.25 dex steps, and surface gravities of $\mathrm{log}g=4.0$ and $\mathrm{log}g=4.5.$ We applied a rotational broadening kernel with $v\mathrm{sin}i=102.8$ $\mathrm{km}\;{{\rm{s}}}^{-1}$, based on the SPC analysis of the FIES data, and assuming a linear limb darkening law with a coefficient of 0.6, to the templates and also applied the same continuum normalization procedure as performed on the observed spectra. Working order by order, we then cross-correlated each template against the observed spectrum to determine the redshift, applied the redshift to the template, interpolated the redshifted template to the wavelength grid of the observations, and measured the ${\chi }^{2}$ difference between the template and observations ignoring the edges of the order where the errors are high and the blaze-function and wavelength solution have systematic errors (the wavelength range to use was determined manually for each order). For each $\mathrm{log}g$ value we fit a polynomial to the ${\chi }^{2}$-[M/H]-${T}_{{\rm{eff}}}$ surface to determine the ${T}_{{\rm{eff}}}$ and [M/H] values which minimize the total ${\chi }^{2}$ for a spectrum. For $\mathrm{log}g=4.0$ we find ${T}_{{\rm{eff}}}=7476\pm 11$ K and [M/H] = $-0.2589\pm 0.0037,$ while for $\mathrm{log}g=4.5$ we find ${T}_{{\rm{eff}}}=7541\pm 10$ K and [M/H] = $-0.2337\pm 0.0039.$ The errors here are the standard deviation of the results, demonstrating that this procedure yields very consistent parameters from observation to observation. The real errors, however, are dominated by systematic errors in the models and the relatively low temperature and metallicity resolution of the grid. For simplicity we adopt the grid resolution for our estimated errors, with ${T}_{{\rm{eff}}}$ = 7500 ± 250 K, and [M/H] = $-0.25\pm 0.25.$ These results are consistent with the parameters estimated from the FIES spectra with SPC, and are the parameters that we adopt for the remainder of this paper.

We note that in appearance the spectra are consistent with a late A or early F classification. Based on the temperature–spectral type scale from Pecaut & Mamajek (2013), a temperature of ${T}_{{\rm{eff}}}=7500$ K corresponds to a spectral type of A8.

The adopted values for ${T}_{{\rm{eff}}}$ and [M/H], together with the transit-based mean stellar density (Section 3.2), were then combined with the Yonsei-Yale (Y²) stellar evolution models (Yi et al. 2001) to determine the mass, radius, luminosity and age of HAT-P-57. Figure 14 compares the measured values of ${T}_{{\rm{eff}}}$ and ${\rho }_{\star }$ to the isochrones. Table 3 lists the observed and derived stellar parameters. We find that HAT-P-57 has a mass of $1.47\pm 0.12$ ${M}_{\odot }$, a radius of $1.500\pm 0.050\;$ ${R}_{\odot }$, and is at a reddening- and blend-corrected distance of $303\pm 13$ pc.

Table 3.  Stellar Parameters for HAT-P-57

Parameter	Value	Source
Identifying Information
R.A. (h:m:s)	${18}^{{\rm{h}}}{18}^{{\rm{m}}}58\buildrel{\rm{s}}\over{.} 32$	2MASS
Decl. (d:m:s)	$+10^\circ 35^{\prime} 50\buildrel{\prime\prime}\over{.} 3$	2MASS
GSC ID	GSC 1014–00973	GSC
2MASS ID	2MASS 18185842 + 1035502	2MASS
SWASP ID	1SWASP J181858.42 + 103550.1	SWASP
Spectroscopic properties
${T}_{\mathrm{eff}\star }$ (K)	$7500\pm 250$	HIRES+Polluxa
${\rm{[Fe/H]}}$	−0.25 ± 0.25	HIRES+Pollux
$v\mathrm{sin}i$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	$102.1\pm 1.3$	HIRESb
${\gamma }_{{\rm{RV}}}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	−5.99 ± 0.35	FIES
Photometric properties
B (mag)	$10.916\pm 0.021$	APASS (via URAT1)
V (mag)	$10.465\pm 0.029$	APASS
I (mag)	$10.007\pm 0.063$	TASS Mark IV
g (mag)	$10.741\pm 0.074$	APASS
r (mag)	$10.371\pm 0.053$	APASS
i (mag)	$10.282\pm 0.034$	APASS
J (mag)	$9.670\pm 0.027$	2MASS
H (mag)	$9.497\pm 0.029$	2MASS
Ks (mag)	$9.433\pm 0.024$	2MASS
Derived properties
${M}_{\star }$ (${M}_{\odot }$)	$1.47\pm 0.12$	Isochrones+${\rho }_{\star }$+HIRES+Polluxc
${R}_{\star }$ (${R}_{\odot }$)	$1.500\pm 0.050$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
${\rho }_{\star }$ (cgs)	${0.615}_{-0.036}^{+0.022}$	Light Curves
$\mathrm{log}{g}_{\star }$ (cgs)	$4.251\pm 0.018$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
${L}_{\star }$ (${L}_{\odot }$)	$6.4\pm 1.1$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
MV (mag)	$2.70\pm 0.19$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
MK (mag,ESO)	$2.13\pm 0.10$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
Age (Gyr)	${1.00}_{-0.51}^{+0.67}$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
AV (mag)d	$0.38\pm 0.12$	Isochrones+${\rho }_{\star }$+HIRES+Pollux
Distance (pc)e	$303\pm 13$	Isochrones+${\rho }_{\star }$+HIRES+Pollux

Notes.

^aHIRES+Pollux = Based on a ${\chi }^{2}$ comparison between the extracted HIRES spectra and synthetics MARCS model atmosphere spectra (Gustafsson et al. 2008) from the Pollux database (Palacios et al. 2010) as discussed in Section 3.1. ^bBased on modeling the spectral line profiles as discussed in Section 3.3. ^cIsochrones+${\rho }_{\star }$+HIRES+Pollux = Based on the Y² isochrones (Yi et al. 2001), the stellar density used as a luminosity indicator, and the atmospheric parameter results. ^dTotal $V$ band extinction to the star determined by comparing the catalog broad-band photometry listed in the table to the expected magnitudes from the Isochrones+${\rho }_{\star }$+HIRES+Pollux model for the star, and accounting for blending from the known binary located $2\buildrel{\prime\prime}\over{.} 7$ away from HAT-P-57. We use the Cardelli et al. (1989) extinction law. ^eDistance based on a comparison of the measured photometric magnitudes for HAT-P-57, corrected for blending from HAT-P-57A and HAT-P-57B and for reddening, to the predicted magnitudes from the stellar evolution models.

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We modeled the trend-filtered HATNet light curve and the KeplerCam light curves of HAT-P-57, together with the Keck/HIRES I₂ RVs following the methods described by Bakos et al. (2010) and Hartman et al. (2012). The light curves were fit using a Mandel & Agol (2002) transit model with quadratic limb darkening coefficients fixed to the values adopted from Claret (2004). For the KeplerCam light curves we allowed for a quadratic trend in hour angle, and linear trends in three parameters describing the shape of the PSF. For the HATNet light curve we included a dilution factor to account for distortion of the transit signal due to the filtering procedure, and blending from neighboring stars in the low spatial resolution HATNet images. The RVs were included in the fit and modeled using a circular Keplerian orbit. The RV "jitter" term was also varied in the fit following Hartman et al. (2014). Note that there is no evidence that the low cadence I₂ RV observations are correlated in time, justifying the assumption of uncorrelated RV jitter. Although no orbital variation is detected, this procedure allows us to place an upper limit on the RV semiamplitude, and hence on the mass of HAT-P-57b.

To account for blending in the KeplerCam light curves from HAT-P-57B and HAT-P-57C we included contamination factors in each bandpass. We allowed these factors to vary in the fit with Gaussian priors which were estimated based on the measured ${\rm{\Delta }}H$ and ${\rm{\Delta }}{L}^{\prime }$ magnitudes, together with the PARSEC isochrones, and assuming the binary is physically associated with HAT-P-57. The adopted priors are listed in Table 4.

Table 4.  MCMC State Variables and Priors

Parameter	Prior
Light curve+RV curve analysis
${T}_{c,0}$ (days)a	uniform
${T}_{c,888}$ (days)a	uniform
$\zeta /{R}_{\star }$	uniform
${R}_{p}$/${R}_{\star }$	uniform
b2	uniform with $0\leqslant {b}^{2}\leqslant 1$
K ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	uniform with $K\geqslant 0$
${\gamma }_{{\rm{rel}}}$ HIRES	uniform
RV jitter HIRES	uniform with ${\sigma }_{{\rm{jitter}}}\geqslant 0$
${f}_{{\rm{blend,HN,}}\;{\rm{r}}}$ b	uniform with $0\leqslant {f}_{{\rm{blend,HN}}}\leqslant 1$
${f}_{{\rm{blend,g}}}$ b	N(0.9980, 0.0026)c with $0\leqslant {f}_{{\rm{blend,g}}}\leqslant 1$
${f}_{{\rm{blend,i}}}$ b	N(0.9836, 0.0088)c with $0\leqslant {f}_{{\rm{blend,i}}}\leqslant 1$
${f}_{{\rm{blend,z}}}$ b	N(0.973, 0.011)c with $0\leqslant {f}_{{\rm{blend,z}}}\leqslant 1$
m0d	uniform, linearly optimized
cEPDe	uniform, linearly optimized
Line profile analysis
Tc (days)	N(2455113.48127, 0.00048)c
P (days)	N(2.4652950, 0.0000032)c
$a/{R}_{\star }$	N(5.825, 0.090)c
${R}_{p}$/${R}_{\star }$	N(0.0968, 0.0015)c
b2	N(0.051, 0.023)c with $0\leqslant {b}^{2}\leqslant 1$
λ (days)	uniform $-180\leqslant \lambda \lt 180$
$v\mathrm{sin}i$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	uniform
${\rm{\Delta }}{v}_{0}$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)f	uniform
${u}_{1}^{\prime }$	uniform subject to $0\leqslant {u}_{1}\leqslant 1$
${u}_{2}^{\prime }$	uniform subject to $0\leqslant {u}_{2}\leqslant 1$
ρg	$\propto {e}^{-\rho /200}$ with $0\leqslant \rho \leqslant 200$
Ag	$\propto {e}^{-A/0.0000002}$
αg	$\propto 1/\alpha$ with $\alpha \gt 0$
${a}_{{\rm{LP}}}$ h	uniform
${c}_{{\rm{LP}}}$ i	uniform

Notes.

^aThe times of transit center for event number 0 (the first transit covered by our light curves) and event number 888 (the last transit covered by our light curves). ^bScaling factors for each filter applied to the fractional stellar flux blocked by the planet to account for dilution from HAT-P-57B and HAT-P-57C in all of the light curves, and to account for over-filtering in the HATNet data. ^cHere $N(\mu ,\sigma )$ corresponds to a normal distribution with mean μ and standard deviation σ. For the blend factors these are determined based on the measured magnitudes of HAT-P-57B and HAT-P-57C together with the PARSEC isochrones. For the line profile parameters these are determined from the posterior distributions for each parameter from the light curve and RV curve analysis. ^dOut-of-transit magnitude. One such parameter is used for each light curve in the analysis. For computational efficiency these parameters are optimized via linear least squares at each step in the MCMC. ^eEPD coefficients used to remove quadratic variations in time, or variations that are correlated with changes in the shape of the PSF. Five such parameters are used for each of the KeplerCam light curves. For computational efficiency these parameters are optimized via linear least squares at each step in the MCMC. ^fCenter velocity of the line profile. One such parameter is used for each profile analyzed. ^gNoise model parameters discussed in Section 3.3. ^hParameter scaling the depth of the line profile. One such parameter is used for each profile analyzed. ⁱThe continuum level of the line profile. One such parameter is used for each profile analyzed.

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We used a Differential Evolution Markov Chain Monte Carlo (DEMCMC) simulation (ter Braak 2006) to explore the likelihood function and produce posterior distributions for all varied parameters. The parameters that we varied, together with their adopted priors, are listed in Table 4. The resulting Markov Chains were combined with the chains of stellar parameters produced in Section 3.1 to determine the radius, semimajor axis, and other physical and orbital parameters for HAT-P-57b. In particular we find that HAT-P-57b has a radius of ${R}_{P}=1.413\pm 0.054$ ${R}_{{\rm{J}}}$, and we place a 95% confidence upper limit on its mass of ${M}_{P}\lt 1.85$ ${M}_{{\rm{J}}}$. Table 5 lists these and other parameters for HAT-P-57b.

Table 5.  Parameters for the Transiting Planet HAT-P-57 ${\rm{b}}$

Parameter	Valuea	Parameter	Valuea
Light curve parameters
P (days)	$2.4652950\pm 0.0000032$	Tc (${\rm{BJD}}$)b	$2455113.48127\pm 0.00048$
T14 (days)b	$0.14578\pm 0.00080$	${T}_{12}={T}_{34}$ (days)b	$0.01325\pm 0.00054$
$a/{R}_{\star }$	${5.825}_{-0.116}^{+0.069}$	$\zeta /{R}_{\star }$ c	$15.099\pm 0.071$
${R}_{p}$/${R}_{\star }$	$0.0968\pm 0.0015$	b2	${0.031}_{-0.023}^{+0.040}$
$b\equiv a\mathrm{cos}i/{R}_{\star }$	${0.177}_{-0.084}^{+0.090}$	i (deg)	$88.26\pm 0.85$
Line profile parameters
λ(deg)d	$-16\buildrel{\circ}\over{.} 7\lt \lambda \lt 3\buildrel{\circ}\over{.} 3$ or $27\buildrel{\circ}\over{.} 6\lt \lambda \lt 57\buildrel{\circ}\over{.} 4$	$v\mathrm{sin}i$ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)	$102.1\pm 1.3$
ρ ($\mathrm{km}\;{{\rm{s}}}^{-1}$)e	$26\pm 17$	${u}_{1}^{\prime }$ f	$0.27\pm 0.20$
${u}_{2}^{\prime }$ f	$0.798\pm 0.049$	u1	$0.84\pm 0.15$
u2	$0.21\pm 0.17$
Limb-darkening coefficientsg
${c}_{1},g$ (linear term)	$0.3401$	${c}_{2},g$ (quadratic term)	$0.3758$
${c}_{1},i$	$0.1526$	${c}_{2},i$	$0.3515$
${c}_{1},r$	$0.2150$	${c}_{2},r$	$0.3764$
${c}_{1},z$	$0.0946$	${c}_{2},z$	$0.3459$
RV parameters
K(${\rm{m}}\;{{\rm{s}}}^{-1}$)h	$\lt 215.2$	ei	0
RV jitter (${\rm{m}}\;{{\rm{s}}}^{-1}$)j	$312\pm 70$
Planetary parameters
${M}_{p}$ (${M}_{{\rm{J}}}$)h	$\lt 1.85$	${R}_{p}$ (${R}_{{\rm{J}}}$)	$1.413\pm 0.054$
a (AU)	$0.0406\pm 0.0011$	${T}_{{\rm{eq}}}$ (K)k	$2200\pm 76$
$\langle F\rangle$ (${10}^{9}$ $\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}$)l	$5.29\pm 0.74$

Notes.

^aThe adopted parameters assume a circular orbit. Based on the Bayesian evidence ratio we find that this model is strongly preferred over a model in which the eccentricity is allowed to vary in the fit. For each parameter we give the median value and 68.3% (1σ) confidence intervals from the posterior distribution. ^bReported times are in Barycentric Julian Date calculated directly from UTC, without correction for leap seconds. ${T}_{c}$: reference epoch of mid transit that minimizes the correlation with the orbital period. ${T}_{14}$: total transit duration, time between first to last contact; ${T}_{12}={T}_{34}$: ingress/egress time, time between first and second, or third and fourth contact. ^cReciprocal of the half duration of the transit used as a jump parameter in our MCMC analysis in place of $a/{R}_{\star }.$ It is related to $a/{R}_{\star }$ by the expression $\zeta /{R}_{\star }=a/{R}_{\star }(2\pi (1+e\mathrm{sin}\omega ))/(P\sqrt{1-{b}^{2}}\sqrt{1-{e}^{2}})$ (Bakos et al. 2010). ^dThe marginalized posterior probability distribution for λ is multimodal. We list the ranges of λ above the 95% confidence level. The two ranges have relative probabilities of 26% and 74%, respectively. ^eThe exponential correlation length scale for describing systematic variations in the line profile (Equation (2)). ^fUncorrelated quadratic limb darkening coefficients from modeling the spectral line profiles. These are related to the usual quadratic limb coefficients u₁ and u₂ via ${u}_{1}=0.576236{u}_{1}^{\prime }+0.81732928{u}_{2}^{\prime }$ and ${u}_{2}=-0.81732928{u}_{1}^{\prime }+0.576236{u}_{2}^{\prime }.$ ^gValues for a quadratic law, adopted from the tabulations by Claret (2004) according to the spectroscopic (SPC) parameters listed in Table 3. ^h95% confidence upper limit. ⁱWe assume a circular orbit for the analysis. ^jError term, either astrophysical or instrumental in origin, added in quadrature to the formal RV errors. This term is varied in the fit assuming a prior inversely proportional to the jitter. ^kPlanet equilibrium temperature averaged over the orbit, calculated assuming a Bond albedo of zero, and that flux is reradiated from the full planet surface. ^lIncoming flux per unit surface area, averaged over the orbit.

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Due to the very rapid rotation and brightness of HAT-P-57, the in-transit Keck-I/HIRES observations are amenable to Doppler tomography (e.g., Collier Cameron et al. 2010b). Because of the rapid rotation, there are essentially no unblended lines in the spectrum of HAT-P-57. We therefore make use of the Least Squares Deconvolution method (Donati et al. 1997; see also Collier Cameron et al. 2010b who apply this method to observations of WASP-33) to extract the broadening profile from the blended spectrum. Rather than using a list of weighted delta-functions at known spectral features for the line pattern function, as done by Donati et al. (1997), we use the unbroadened MARCS-atmosphere synthetic template which provided the best match (after broadening) to the Keck-I/HIRES spectra (Section 3.1). The deconvolution is done on an order-by-order basis, and we then take the weighted average of 21 orders spanning 4000–5700 Å, excluding those containing deep Hydrogen Balmer lines or Ca ii lines. The average broadening profiles are shown in Figure 9. The residuals from a quadratic limb-darkening rotational profile are shown in Figure 10. The TEP is clearly seen as the dip in Figure 9, or shadow in Figure 10, moving from ${\rm{\Delta }}V\sim 15$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ in the first observation near transit center to ${\rm{\Delta }}V\sim 77$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ shortly before egress.

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To model the broadening profile measurements we use an analytic expression for the rotational broadening kernel of a spherical star with quadratic limb-darkening, undergoing solid-body rotation and transited by a spherical non-luminous planet (see Appendix for the derivation). This fit was done separately from the modeling of the light curves and RVs discussed in Section 3.2, but to ensure that the constraints on the orbital parameters of the planet and its radius relative to the star are incorporated into the line profile fit, we used the posterior parameter distributions from the light curve and RV curve analysis to determine priors on these same parameters for the line profile fit. Our line profile model also depends on the projected angle between the spin axis of the host and the orbital axis of the planet (λ), the maximum projected rotation velocity of the star ($v\mathrm{sin}i$), the mean velocity of the star (γ, which we take to be a free parameter, and independent of the γ velocity measured with other spectrographs or reductions of the Keck-I/HIRES data), the quadratic limb darkening coefficients (u₁ and u₂) and two factors scaling and offsetting the model to match the measurements. For the limb darkening coefficients we vary the combinations ${u}_{1}^{\prime }$ and ${u}_{2}^{\prime }$ in the fit, with ${u}_{1}=0.576236{u}_{1}^{\prime }+0.81732928{u}_{2}^{\prime }$ and ${u}_{2}=-0.81732928{u}_{1}^{\prime }+0.576236{u}_{2}^{\prime },$ which we find to have uncorrelated posterior probability distributions, rather than varying u₁ and u₂ directly. The set of parameters that we vary in this fit, together with the adopted priors, are listed in Table 4.

As seen in Figure 10 the line profile residuals from the physical model exhibit correlated variations that are not associated with the planet. Based on inspecting the continuum region of the line profiles outside of the range shown in Figure 9 we find that the systematic variations are limited to within the line profile, so we conclude that the variations are most likely due to intrinsic stellar variability, evidence for which is also seen in the HATNet photometry (similar variations have also been seen in the line profile of WASP-33, e.g., Collier Cameron et al. 2010b). In order to account for these systematic variations, which is especially important in determining accurate uncertainties for λ and $v\mathrm{sin}i,$ we use a non-diagonal covariance matrix in evaluating the likelihood function. We parameterize the covariance matrix using an exponential model, which we found by inspection to provide a good match to the autocorrelation of the residuals from the physical model. The covariance between points i and j with velocity differences ${\rm{\Delta }}{v}_{i}$ and ${\rm{\Delta }}{v}_{j}$ is taken to be:

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{ij}}=\alpha {\sigma }_{i}^{2}{\delta }_{{ij}}+A\mathrm{exp}(-| {\rm{\Delta }}{v}_{i}-{\rm{\Delta }}{v}_{j}| /\rho )\end{eqnarray} \tag{ 2 }$$

where ${\sigma }_{i}$ is our estimated uncertainty for point i, ${\delta }_{{ij}}$ is the Kronecker delta function, α is a parameter used to scale the uncertainties, and A and ρ are parameters describing the amplitude and length-scale for the covariance (i.e., we are using a Gaussian-process regression, or GP, to model the systematic variations, see Gibson et al. 2012 for a more detailed discussion of this technique). The likelihood of the data in column vector $y$ given the model in column vector ${{\boldsymbol{y}}}_{{\rm{mod}}}$ parameterized by ${\theta }_{1}$ and with covariance matrix ${\boldsymbol{\Sigma }}$ having parameters ${\theta }_{2}$ (i.e., α, A and ρ) is then

$$\begin{eqnarray}\mathrm{log}{\mathcal{L}}({\boldsymbol{y}}| {\theta }_{1},{\theta }_{2}) & = & -\displaystyle \frac{1}{2}{({\boldsymbol{y}}-{{\boldsymbol{y}}}_{\mathrm{mod},{\theta }_{1}})}^{T}{{{\boldsymbol{\Sigma }}}_{{{\boldsymbol{\theta }}}_{{\bf{2}}}}}^{-1}({\boldsymbol{y}}-{{\boldsymbol{y}}}_{\mathrm{mod},{\theta }_{1}})\\ & & -\displaystyle \frac{1}{2}\mathrm{log}| {{\boldsymbol{\Sigma }}}_{{{\boldsymbol{\theta }}}_{{\bf{2}}}}| +C\end{eqnarray} \tag{ 3 }$$

for some constant C. We assume exponential priors for ρ and A and a Jeffreys prior for α.

We run a DEMCMC analysis to explore this likelihood function for the line profiles, and determine the posterior distributions of the parameters. The maximum a posteriori model is shown in Figure 9. Figure 11 shows the marginalized posterior probability distribution for λ, while Figure 12 shows the correlations between λ and $v\mathrm{sin}i$ and between λ and b.

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When the GP is used to model the systematic variations we find a multi-modal posterior distribution for λ, with the ranges $-10\buildrel{\circ}\over{.} 9\lt \lambda \lt -4\buildrel{\circ}\over{.} 1,\;$ $34\buildrel{\circ}\over{.} 2\lt \lambda \lt 44\buildrel{\circ}\over{.} 8,$ and $47\buildrel{\circ}\over{.} 3\lt \lambda \lt 55\buildrel{\circ}\over{.} 9$ having marginal posterior probability above the 68.3% confidence limit, and the ranges $-16\buildrel{\circ}\over{.} 7\lt \lambda \lt 3\buildrel{\circ}\over{.} 3$ and $27\buildrel{\circ}\over{.} 6\lt \lambda \lt 57\buildrel{\circ}\over{.} 4$ having marginal posterior probability above the 95% confidence limit. The relative probabilities of the two modes permitted at 95% confidence ($-16\buildrel{\circ}\over{.} 7\lt \lambda \lt 3\buildrel{\circ}\over{.} 3,$ and $27\buildrel{\circ}\over{.} 6\lt \lambda \lt 57\buildrel{\circ}\over{.} 4$) are 26% and 74%, respectively.

The mode peaking at $\lambda =-7\buildrel{\circ}\over{.} 7$ requires a relatively high impact parameter ($b\gtrsim 0.3,$ Figure 12), while the higher λ modes require a lower impact parameter ($b\lesssim 0.2$). This degeneracy is due in part to the small number of line profile observations in which the planet shadow is clearly detected, and the lack of observations prior to transit center. Figure 9 shows both the overall maximum posterior probability model, which has ${\lambda }_{1}=37\buildrel{\circ}\over{.} 6$ and the other two local maxima (${\lambda }_{2}=51\buildrel{\circ}\over{.} 5$ and ${\lambda }_{3}=-7\buildrel{\circ}\over{.} 7$), overplotted on the observed line profiles, while Figure 13 shows the projected geometry for these different orbital configurations. The models yield similar tracks for the planet in velocity space over the time-span of the observations. A longer time-base covering the full transit would allow the track of the planet to be determined, and not just its position in velocity space near transit center, helping to distinguish between these modes. Additionally, higher precision photometric follow-up to provide a tighter constraint on b could also help break the degeneracy. Note, however, that as seen in Figure 13, based solely on Doppler tomography and transit observations, orbits with projected alignment λ and the north pole of the orbit pointing toward the observer are degenerate with orbits having projected alignment $180^\circ -\lambda$ and the south pole of the orbit pointing toward the observer (Fabrycky & Winn 2009).

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The distributions for $v\mathrm{sin}i,$ the limb darkening coefficients, and the correlation length scale ρ are all nearly Gaussian. In particular we find $v\mathrm{sin}i=102.1\pm 1.3$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ and $\rho \;=26\pm 17\;$ $\mathrm{km}\;{{\rm{s}}}^{-1}$.

For the limb darkening coefficients we find ${u}_{1}^{\prime }\;=0.27\pm 0.20$ and ${u}_{2}^{\prime }=0.798\pm 0.049,$ which correspond to ${u}_{1}=0.84\pm 0.15$ and ${u}_{2}=0.21\pm 0.17.$ These differ significantly from the expected coefficients for the g-band from Claret (2004) for a star with the adopted atmospheric parameters of HAT-P-57. The expected values are ${u}_{1,g}=0.3401$ and ${u}_{2,g}=0.3758,$ respectively. Practically speaking, the Claret (2004) coefficents predict a flatter profile in the center of the line and a steeper profile near the edge than what is observed. While errors in the model limb darkening coefficients have been noted based on, for example, Kepler transit observations (e.g., Espinoza & Jordán 2015), the inferred u₁ coefficient based on our observations is much larger than other works have suggested. For example, for Kepler-13A, Müller et al. (2013) find ${u}_{1}=0.308\pm 0.007$ and ${u}_{2}=0.222\pm 0.014$ in the Kepler band-pass. While models predict u₁ to be higher in the bluer band-pass used in the Doppler tomography analysis, the expected difference is much less than what we measure. One possible explanation for the discrepancy is that a low order pulsation mode is distorting the overall line profile shape, leading to incorrect limb darkening estimates. To determine how systematic errors in the limb darkening affect our results, we have also carried out a fit with the quadratic limb darkening coefficients fixed to the Claret (2004) g-band values. We find that the posterior distribution for λ is not significantly affected by the treatment of limb darkening. For $v\mathrm{sin}i,$ on the other hand, we find that the value is more tightly constrained when the limb darkening coefficients are fixed, but that it is still consistent with the results when the coefficients are allowed to vary ($v\mathrm{sin}i=101.69\pm 0.37\;$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ when the coefficients are fixed, compared to $v\mathrm{sin}i=102.1\pm 1.3\;$ $\mathrm{km}\;{{\rm{s}}}^{-1}$ when they are allowed to vary).

For comparison, if we do not include the GP in the modeling, and instead assume zero covariance between points in the line spread function, then we find a bimodal, but much more tightly constrained, distribution for λ, with the ranges $-12\buildrel{\circ}\over{.} 41\lt \lambda \lt -5\buildrel{\circ}\over{.} 78$ and $40\buildrel{\circ}\over{.} 89\lt \lambda \lt 57\buildrel{\circ}\over{.} 32$ having marginal posterior probability above the 68.3% confidence limit, and the ranges $-13\buildrel{\circ}\over{.} 41\lt \lambda \lt -4\buildrel{\circ}\over{.} 17$ and $39\buildrel{\circ}\over{.} 80\lt \lambda \lt 58\buildrel{\circ}\over{.} 23$ having marginal posterior probability above the 95% confidence limit. In this case the two modes have relative probabilities of 55% and 45%, respectively. The constraint on $v\mathrm{sin}i$ is also tighter with $v\mathrm{sin}i=101.46\pm 0.19$ $\mathrm{km}\;{{\rm{s}}}^{-1}$.

Other methods for modeling the systematic variations in the line profiles were also considered. These include using a squared exponential kernel, and using a Fourier series, but we found that the former did not sufficiently account for long range correlations, while the latter had the undesirable effect of suppressing both the overall line shape and the planet signature. It is likely that a more physically motivated model for the systematics that accounts for the wavelike oscillations over the surface of the star, and their propagation in time, may provide a better description of the data, and reduce the uncertainty on λ that results from suppressing the planet signature through over-fitting with the GP. For example, the two-dimensional Fourier filtering technique used by Johnson et al. (2015) in their analysis of WASP-33 may work better at removing the stellar oscillations. The analysis in that case was facilitated by the retrograde motion of WASP-33, leading to a clean separation of the planet shadow and stellar oscillations in Fourier space. In the case of HAT-P-57 the planet is prograde with λ close to zero, and thus difficult to separate from the stellar oscillations.

The detection of the Doppler shadow of HAT-P-57b moving across the rotational broadening profile of HAT-P-57 during a transit, and its consistency in amplitude and width with the ${R}_{p}$/${R}_{\star }$ value measured from the transit light curves, provides confirmation that the transiting object is orbiting the bright rapidly rotating A star whose light dominates the spectrum. This, coupled with the upper limit of $K\lt 215.2$ ${\rm{m}}\;{{\rm{s}}}^{-1}$ on the RV variation of this star, allows us to confirm that this is a TEP system, and not a blended stellar eclipsing binary system.

As an additional check on our conclusion that HAT-P-57 is not a blended stellar eclipsing binary system, we also carried out a blend analysis following Hartman et al. (2012). We find that, based on the photometry alone, all blended stellar eclipsing binary models that we tested provide a fit to the data that has a higher ${\chi }^{2}$ than the best-fit star+planet model. All of these blend models can be rejected with greater than $5\sigma$ confidence in favor of the star+planet model. Moreover, based on the MMT/Clio2 imaging, any unaccounted-for blending companion bright enough to influence the derived parameters of the system must be within $\sim 0\buildrel{\prime\prime}\over{.} 25$ of HAT-P-57 (Figure 6).