TRANSIT TIMING VARIATION MEASUREMENTS OF WASP-12b AND QATAR-1b: NO EVIDENCE OF ADDITIONAL PLANETS

The search for Earth-like planets has been underway from the ground since well before the Kepler mission (Borucki et al. 2010) discovered thousands of extrasolar planets, including several Earth-sized planets and smaller. One method used by ground-based observers enables the discovery of additional planets in systems with at least one transiting planet through high-precision measurements of transit timing variations (TTVs) (Agol et al. 2005; Holman & Murray 2005). Agol et al. (2005) show that an Earth-mass planet in 2:1 resonance with a 3 day period transiting hot Jupiter would cause timing variations of ∼3 minutes, which would be accumulated over a year. Ground-based half-meter class telescopes can be used to produce transit timing data with better than 1 minute precision for most planets with host stars brighter than $V\sim 13$.

To properly characterize a TTV signal, good phase coverage and a long baseline of measurements are needed. Ideally, nearly all transits during the baseline period should be observed. This is, of course, nearly impossible from the ground due to the Earth's rotation and weather. Veras et al. (2011) suggests that in most cases at least 50 consecutive transit observations are necessary to have a reasonable chance of characterizing the perturbing planet and its orbit.

Even if a TTV curve is well sampled, it is notoriously difficult to uniquely infer the perturbing planet's mass and orbital parameters strictly from the TTV data (Ford & Holman 2007; Nesvorný & Morbidelli 2008; Payne et al. 2010; Boué et al. 2012; Lithwick et al. 2012). Degeneracies exist in the orbital period, eccentricity, and inclination. RV data, transit duration variations, or systems with multiple transiting planets can help to break the degeneracies.

To date, no convincing TTVs have been reported from ground-based measurements. However, several authors have claimed evidence of low-amplitude periodic TTVs, followed by other authors with differing conclusions. For example, the analysis of WASP-3b TTVs by multiple authors has produced differing conclusions. Maciejewski et al. (2010) analyzed 13 WASP-3b transits and reported a sinusoidal TTV with a period of ∼127 days and peak value of ∼2 minutes. Then, Eibe et al. (2012) found a lower significance in the TTV detection, but a possible detection of transit duration variations. Montalto et al. (2012) studied 38 WASP-3b light curves and found no evidence of periodicity in the TTVs. Small amplitude TTV claims are inconsistent across ground-based timing measurement analyses and, to date, no perturber planet has been confirmed.

On the other hand, the almost continuous Kepler light curves have demonstrated a number of clear TTV detections (e.g., Holman et al. 2010; Ballard et al. 2011; Lissauer et al. 2011; Steffen et al. 2013; Yang et al. 2013). The Kepler TTV data are responsible for the discovery of non-transiting planets and provide confirmation of the planetary nature of the transiting planet(s) and estimates of their masses, without the necessity of RV data.

Steffen et al. (2012) searched six quarters of the Kepler data for planetary companions orbiting near hot-Jupiter planet candidates ($1\leqslant P\leqslant 5$ day) by looking for multiple transiting planets and evidence of TTVs. Special emphasis was given to companions between the 2:1 interior and exterior mean-motion resonances. They found no evidence of companion planets nearby 63 hot-Jupiter candidates. However, five out of 31 warm Jupiter systems ($6.3\leqslant P\leqslant 15.8$ day) do show multiple transiting planets and/or evidence of TTVs.

Nevertheless, using K2 data (Howell et al. 2014), Becker et al. (2015) recently discovered two additional planets (WASP-47c, WASP-47d) transiting the ground-based discovered system hosting the hot-Jupiter planet WASP-47b (Hellier et al. 2012). The K2 data show WASP-47b TTVs with a period of ∼52 days and peak variation of ∼45 s.

WASP-12b and Qatar-1b are transiting hot Jupiters for which previous works have suggested the presence of TTVs indicative of additional bodies in these systems—an Earth-mass planet in WASP-12 (Maciejewski et al. 2013; M13) and a brown-dwarf mass object in Qatar-1 (von Essen et al. 2013; E13). In this work, we present high-precision photometric observations of 23 full WASP-12b and 18 full Qatar-1b transits. We combine all transits from each system with spectroscopic and RV data from the literature and perform a global fit for each system. The results provide refined system parameters for WASP-12b and Qatar-1b and high-precision transit center times for each transit observation.

We describe the WASP-12 and Qatar-1 systems and the previous reports of TTV measurements in Section 2. We describe the data that we utilize, including data from the literature and our extensive new transit observations, in Section 3. We discuss the methods used for careful time keeping in Section 3.3, data reduction in Section 3.4, and for global fitting in Section 3.5. We present the results of this study in Section 4, including the global fits, improved system parameter results, and TTV results. We discuss our TTV results in the context of previous reports in Section 5, and summarize our conclusions in Section 6.

WASP-12b is a hot-Jupiter exoplanet discovered by the WASP survey (Pollacco et al. 2006) and announced by Hebb et al. (2009) (H09). It orbits a $V\sim 11.7$ evolved late-F star and has an orbital period of P = 1.09 day, a semimajor axis of only 3.1 stellar radii, and a highly inflated radius ${R}_{P}=1.8\,{R}_{{\rm{J}}}$. Bergfors et al. (2013) reported a faint (${\rm{\Delta }}{i}^{\prime }=4.03$) elongated object $\sim 1^{\prime\prime}$ from WASP-12. Crossfield et al. (2012) confirmed the detection. Using the Keck telescope, Bechter et al. (2014) resolved the neighboring object and confirmed it to be a binary composed of two M3V stars that are orbiting WASP-12 as part of a hierarchical triple star system. The projected separation of the binary from WASP-12 is ∼300 au, corresponds to a period of several thousand years, so any influence on the orbital dynamics of WASP-12b should be negligible.

Qatar-1b is a hot-Jupiter exoplanet and is the first planet discovered by the Qatar Exoplanet Survey (Alsubai et al. 2013). The planet was announced by Alsubai et al. 2011 (A11). The host star is a $V\sim 12.8$ metal-rich K-dwarf star, and the planet has a circular orbital with period P = 1.42 days, a semimajor axis of ∼6 stellar radii, a mass ${M}_{P}=1.09\,{M}_{{\rm{J}}}$ and a radius ${R}_{P}=1.2\,{R}_{{\rm{J}}}$.

The short orbital periods of WASP-12b and Qatar-1b result in relatively frequent opportunities to observe complete transit events from the ground, which has prompted other groups to conduct detailed studies of the systems. M13 acquired 61 partial or complete transit light curves from 2009 to 2012 from 14 telescopes distributed around the world. They classified 19 of the transits as high quality, based on the rms of the model residuals and transit coverage of at least 75%. Out of the 19 high quality transits, 11 have both ingress and egress coverage and baseline data before and after the transit. Data reduction and differential photometry were performed using differing methods, depending on the originating observatory. Detrending was implemented by fitting a second order polynomial and a fixed WASP-12b transit model from Maciejewski et al. (2011), which was based on one transit. M13 finds tentative evidence for a TTV signal, which has a period of 545 ± 22 days and a semi-amplitude of 59 ± 11 s, and they suggest that the possible perturbing body has a mass of $0.1\,{M}_{{\rm{J}}}$ and a 3.6 day eccentric orbit.

Covino et al. (2013) (C13) obtained HARPS-N in-transit high-precision radial velocities to measure the Rossiter–McLaughlin (RM) effect in the Qatar-1b system, and out-of-transit measurements to redetermine the spectroscopic orbit. They found that the orbit is consistent with circular and has a well-aligned spin–orbit angle of $\lambda =-8\buildrel{\circ}\over{.} 4\pm 7\buildrel{\circ}\over{.} 1$. E13 presented a detailed TTV analysis of 26 Qatar-1b transits covering a baseline of 18 months and found evidence for a 190 or 380 day TTV signal with a semi-amplitude of ∼1 minute that could be reproduced by either a weak perturber in resonance with Qatar-1b, or by a massive body in the brown-dwarf regime. Mislis et al. 2015 (DM15) analyzed 12 complete Qatar-1b transits and provided updated system parameters. After reviewing all TTV data available, DM15 suggests further and more precise data to determine if TTVs exist in the system. Croll et al. (2015) presented near-infrared secondary eclipse observations of Qatar-1b thermal emission showing a mid-eclipse time consistent with a circular orbit. Maciejewski et al. 2015 (GM15) analyzed 18 Qatar-1b transits to redetermine the system parameters and found no evidence of periodic TTVs $\gtrsim 1\,\mathrm{minute}$.

TTV analyses in the literature often suffer from a lack of control over the data collection and calibration processes. Since transit observations are needed on many epochs for the detection and characterization of a TTV signal, researchers typically depend on multiple observatories to collect the time-series observations. With different instrumentation and operators at each observatory, it is difficult to obtain homogeneously collected data. Lack of control over guiding precision, other sources of systematics, time synchronization, exposure durations, defocus, calibration procedures, selection of comparison star ensembles, photometric aperture radii, and detrending methods may produce veiled inaccuracies in the derived system parameters. For these reasons, for this study we have opted to utilize our homogeneously obtained observations with one consistent set of instruments and procedures, as discussed in the following section.

3.1. Data from the Literature

3.2. New Observations

We observed 22 complete and one nearly complete high-precision transit of WASP-12b over the time span of 2009 November to 2015 February and 18 complete high-precision transits of Qatar-1b over the time span of 2011 June to 2014 September. All observations were collected using the Moore Observatory Ritchie–Chretien (MORC) 0.6 m telescope. The MORC telescope was manufactured by RC Optical Systems and is operated by the University of Louisville. The instrument features a very robust fork mounting with an absolute Renishaw precision encoder on the polar axis that provides highly accurate free-running tracking of the sky. Unguided exposures of up to ∼5 minutes result in a tracking error of less than $1^{\prime\prime} \mbox{--}3^{\prime\prime}$, depending on the altitude of the object.

An Apogee U16M CCD camera was used to collect the time-series observations. The camera has a Kodak KAF-16803 CCD detector with a 4096 × 4096 array of $9\times 9\,\mu {\rm{m}}$ microlensed pixels, which oversample the point-spread function and facilitate high-precision photometry over a wide dynamic range. The pixel scale is $0\buildrel{\prime\prime}\over{.} 39$ per pixel, which provides a field of view of $26\buildrel{\,\prime}\over{.} 2\times 26\buildrel{\,\prime}\over{.} 2$. The wide field of view helps to improve the differential photometry by offering a wider selection of comparison stars. The oversampled point-spread function improves the photometry by sampling the light from a point source with more pixels (similar to telescope defocusing), improving the dynamic range of possible brightness measurements within an image, and reducing noise resulting from inter-pixel variations combined with imperfect guiding.

The date, exposure time, number of exposures, filter, photometric precision, and the error scaling factor (as determined by multi-EXOFAST; see Section 3.5) of each time-series of observations are listed in Tables 1 and 2 for WASP-12b and Qatar-1b, respectively. All observations were guided from the science images and were defocused to improve photometric precision. We carefully synchronized our timing source (see Section 3.3) and converted to ${\mathrm{BJD}}_{\mathrm{TDB}}$ as discussed in Eastman et al. (2010). All exposure times were 100 s, resulting in a ∼2 minute cadence. All WASP-12b observations were in the Sloan $r^{\prime}$ filter, except for the $g^{\prime}$ observation on UT 2010 January 14, the Clear with Blue Block (CBB; a highpass filter with cutoff at ∼500 nm) observation on UT 2011 December 08, and the V observation on UT 2013 November 11. The $g^{\prime}$ observation was the result of an error in the filter wheel controller. The CBB observation was an attempt to improve the photometric precision, but it caused some of the brighter comparison stars to saturate or nearly saturate, so we reverted to $r^{\prime}$ as the nominal filter for the remaining WASP-12b observations. The V filter was used, because the $r^{\prime}$ filter was not installed in the filter wheel that night due to observations for another program. The first seven Qatar-1b transits were observed with no filter (i.e., open). After a CBB filter was acquired in late 2011, it was used to observe the remaining 11 Qatar-1b transits to reduce the airmass trend in the photometric data. Multi-EXOFAST handles the observations in different filters seamlessly by extracting different limb-darkening coefficients for each filter band (see Section 3.5). All transit observations include coverage of pre-ingress baseline, ingress, flat bottom, egress, and post-egress baseline, except for the WASP-12b light curve on UT 2015 January 01, which has no pre-ingress baseline or time of first contact coverage. A few of the transits have short gaps due to passing clouds or equipment problems.

Table 1.  WASP-12 Photometric Observations

Telescope	UT Date	Filter	#Data	ExpTb	rmsc	Scaled
MORCa	2009 Nov 05	$r^{\prime}$	160	100	1.3	1.32
MORC	2009 Nov 28	$r^{\prime}$	134	100	1.0	1.33
MORC	2010 Jan 13	$r^{\prime}$	123	100	1.1	1.24
MORC	2010 Jan 14	$g^{\prime}$	151	100	1.2	1.46
MORC	2010 Nov 09	$r^{\prime}$	129	100	1.2	1.70
MORC	2010 Nov 10	$r^{\prime}$	119	100	0.9	1.29
MORC	2011 Feb 11	$r^{\prime}$	138	100	0.9	1.22
MORC	2011 Dec 08	CBB	148	100	0.9	1.30
MORC	2012 Feb 27	$r^{\prime}$	167	100	1.0	1.31
MORC	2012 Feb 28	$r^{\prime}$	135	100	1.2	1.43
MORC	2012 Mar 10	$r^{\prime}$	148	100	1.2	1.40
MORC	2012 Nov 18	$r^{\prime}$	135	100	1.0	1.26
MORC	2012 Dec 12	$r^{\prime}$	139	100	1.0	1.09
MORC	2012 Dec 23	$r^{\prime}$	180	100	1.1	1.27
MORC	2013 Jan 05	$r^{\prime}$	160	100	1.1	1.20
MORC	2013 Jan 27	$r^{\prime}$	194	100	1.3	1.07
MORC	2013 Nov 11	V	93	100	1.3	1.15
MORC	2013 Dec 28	$r^{\prime}$	167	100	1.1	1.40
MORC	2014 Jan 20	$r^{\prime}$	181	100	1.1	1.25
MORC	2014 Dec 21	$r^{\prime}$	201	100	1.4	1.34
MORC	2015 Jan 01	$r^{\prime}$	150	100	1.9	1.20
MORC	2015 Feb 06	$r^{\prime}$	217	100	1.3	1.10
MORC	2015 Feb 07	$r^{\prime}$	138	100	1.1	1.20

Notes.

^aMORC = U. of Louisville Moore Obs. RCOS telescope. ^bExposure time in seconds. ^crms in units of 10⁻³. ^dError scaling factor as determined by multi-EXOFAST.

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Table 2.  Qatar-1 Photometric Observations

Telescope	UT Date	Filter	# Data	ExpTb	rmsc	Scaled
MORCa	2011 Jun 30	Open	109	100	1.2	1.49
MORC	2011 Jul 10	Open	120	100	1.3	1.47
MORC	2011 Aug 16	Open	101	100	1.4	1.49
MORC	2011 Aug 23	Open	116	100	1.1	1.30
MORC	2011 Sep 22	Open	86	100	1.1	1.42
MORC	2011 Oct 09	Open	83	100	1.1	1.22
MORC	2011 Dec 02	Open	92	100	1.3	1.25
MORC	2012 Jun 19	CBB	115	100	1.3	1.38
MORC	2012 Jun 29	CBB	163	100	1.3	1.31
MORC	2012 Aug 02	CBB	138	100	1.6	1.37
MORC	2012 Aug 12	CBB	110	100	1.5	1.53
MORC	2012 Aug 22	CBB	145	100	1.4	1.40
MORC	2012 Oct 25	CBB	97	100	1.4	1.43
MORC	2013 Jul 16	CBB	168	100	1.2	1.18
MORC	2014 Jul 16	CBB	175	100	1.5	1.34
MORC	2014 Jul 23	CBB	164	100	1.6	1.32
MORC	2014 Aug 19	CBB	139	100	1.7	1.25
MORC	2014 Sep 25	CBB	162	100	1.6	1.44

Notes.

^aMORC = U. of Louisville Moore Obs. RCOS telescope. ^bExposure time in seconds. ^crms in units of 10⁻³. ^dError scaling factor as determined by multi-EXOFAST.

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3.3. Time Stamps

3.4. Data Reduction

We used AstroImageJ (AIJ; Collins et al. 2017) to calibrate all images and to extract differential aperture photometry and detrending parameters from the calibrated images. AIJ has been well tested against several other commercial and open source scientific image extraction tools as part of the Kilo-degree Extremely Little Telescope (KELT; Pepper et al. 2003, 2007) transit survey program, and has been found to produce photometric results that are consistent with and often more precise than results from other tools. Calibration was performed using the Data Processor module of AIJ and included bias subtraction, CCD linearity correction, dark subtraction, and flat-field division. Differential photometry was performed using the Multi-Aperture (MA) module of AIJ. Supersets of 16 and 15 comparison stars were selected for WASP-12b and Qatar-1b, respectively, ensuring that each had brightness similar to the target star and had no nearby stars that might blend into the photometric aperture. The AIJ Multi-Plot and Light Curve Fitting modules were used to search the superset of comparison stars for a subset that minimized the best fit transit model residuals for each time-series. The final comparison ensembles were different for each time-series due to differing filters and sky transparencies, but each WASP-12b ensemble had 10–14 comparison stars, and each Qatar-1b ensemble had 6–10 comparison stars. A range of fixed and variable aperture sizes were investigated to find the optimal configuration that minimized transit model residuals for each data set. The result was a variable aperture radius with a scaling factor in the range of $0.9\mbox{--}1.0$ times the average FWHM of the stellar apertures in each WASP-12b image and 1.1 times the average FWHM in the Qatar-1b images.

The AIJ MA module calculates photometric uncertainties, including contributions from Poisson noise in the source and sky background, CCD read-out noise, dark current, and digitization noise, and propagates the uncertainties through the differential photometry calculations. AIJ does not include a noise contribution from atmospheric scintillation. However, multi-EXOFAST scales the photometric noise to ensure the best fit individual transit model has ${\chi }_{\mathrm{red}}^{\,2}$ = 1.0 (see Section 3.5).

The 23 individual WASP-12b detrended and fitted light curves are shown as black symbols in Figure 1. The 18 Qatar-1b individual detrended and fitted light curves are shown as black symbols in Figure 3. The red lines show the global fit models from Sections 4.1.1 and 4.1.2. The black symbols in Figures 2 and 4 show the results after phasing (using the global fit ${T}_{0}$ and P from Tables 3 and 5), combining, and binning all of the light curves for each planetary system in 5 minute intervals. The light curve models are also combined and binned at 5 minute intervals and shown as red lines. The binned light curves are not used for analysis, but rather to show the best combined behavior of each set of transits. The model residuals shown in the bottom panels have an rms of 183 and 255 ppm for WASP-12b and Qatar-1b, respectively. The WASP-12b rms value was calculated over the entire data set, while the Qatar-1b rms value was calculated over the somewhat arbitrarily determined region in phase space from −1.5–2.5 hr, where most light curves contain data to constrain the global model. The rms result from this limited region is a better representation of the precision of the in-transit and nearby baseline data that heavily influence the transit derived global parameters. Each light curve is shifted on the y-axis for clarity.

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Table 3.  WASP-12 System Parameter Median Values and 68% Confidence Intervals

Parameter	Units	Value
Stellar Parameters:
M*	Mass ($\,{M}_{\odot }$)	${1.434}_{-0.090}^{+0.11}$
R*	Radius ($\,{R}_{\odot }$)	${1.657}_{-0.044}^{+0.046}$
L*	Luminosity ($\,{L}_{\odot }$)	${4.05}_{-0.53}^{+0.54}$
${\rho }_{* }$	Density (cgs)	${0.446}_{-0.014}^{+0.015}$
$\mathrm{log}{g}_{* }$	Surface gravity (cgs)	${4.157}_{-0.012}^{+0.013}$
${T}_{\mathrm{eff}}$	Effective temperature (K)	${6360}_{-140}^{+130}$
$[\mathrm{Fe}/{\rm{H}}]$	Metallicity	${0.33}_{-0.17}^{+0.14}$
λ	Spin–orbit alignment (degrees)	$-{90}_{-26}^{+23}$
Planetary Parameters:
a	Semimajor axis (au)	${0.02340}_{-0.00050}^{+0.00056}$
MP	Mass ($\,{M}_{{\rm{J}}}$)	${1.470}_{-0.069}^{+0.076}$
RP	Radius ($\,{R}_{{\rm{J}}}$)	${1.900}_{-0.055}^{+0.057}$
${\rho }_{P}$	Density (cgs)	${0.266}_{-0.014}^{+0.015}$
$\mathrm{log}{g}_{P}$	Surface gravity	3.004 ± 0.015
Teq	Equilibrium temperature (K)	${2580}_{-62}^{+58}$
Θ	Safronov number	${0.02520}_{-0.00084}^{+0.00087}$
$\langle F\rangle$	Incident flux (109 erg s−1 cm−2)	10.06 ± 0.94
Primary Transit Parameters:
${T}_{0}$	Linear ephemeris from transits (${\mathrm{BJD}}_{\mathrm{TDB}}$)	$2456176.668258\pm 7.7650773\times {10}^{-5}$
P	Linear eph. period from transits (days)	$1.0914203\pm 1.4432653\times {10}^{-7}$
${R}_{P}/{R}_{* }$	Radius of the planet in stellar radii	${0.11785}_{-0.00054}^{+0.00053}$
$a/{R}_{* }$	Semimajor axis in stellar radii	${3.039}_{-0.033}^{+0.034}$
i	Inclination (degrees)	${83.37}_{-0.64}^{+0.72}$
b	Impact parameter	${0.351}_{-0.034}^{+0.030}$
δ	Transit depth	0.01389 ± 0.00013
TFWHM	FWHM duration (days)	0.10958 ± 0.00023
τ	Ingress/egress duration (days)	0.01526 ± 0.00044
T14	Total duration (days)	${0.12483}_{-0.00043}^{+0.00044}$
PT	A priori non-grazing transit probability	${0.2903}_{-0.0031}^{+0.0030}$
${P}_{T,G}$	A priori transit probability	${0.3679}_{-0.0042}^{+0.0041}$
Secondary Eclipse Parameter:
TS	Time of eclipse (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2456176.12256 ± 0.00020
RV Parameters:
K	RV semi-amplitude (m s−1)	226.4 ± 4.1
e	RV eccentricity	0.0 (FIXED)
${K}_{\mathrm{RM}}$	RM amplitude (m s−1)	${30.99}_{-0.29}^{+0.28}$
${M}_{P}\sin i$	Minimum mass ($\,{M}_{{\rm{J}}}$)	${1.460}_{-0.068}^{+0.075}$
${M}_{P}/{M}_{* }$	Mass ratio	0.000977 ± 0.000028
${\gamma }_{\mathrm{SOPHIE}09}$	m s−1	72.4 ± 3.6
${\gamma }_{\mathrm{SOPHIE}10}$	m s−1	19083.0 ± 3.1
Limb-darkening Coefficients:
${u}_{1\mathrm{Kepler}(\mathrm{for}\mathrm{Open})}$	Linear Limb-darkening	${0.322}_{-0.017}^{+0.018}$
${u}_{2\mathrm{Kepler}(\mathrm{for}\mathrm{Open})}$	Quadratic Limb-darkening	${0.3124}_{-0.0098}^{+0.0091}$
${u}_{1\mathrm{Sloang}}$	Linear Limb-darkening	${0.481}_{-0.023}^{+0.024}$
${u}_{2\mathrm{Sloang}}$	Quadratic Limb-darkening	${0.271}_{-0.015}^{+0.014}$
${u}_{1\mathrm{Sloanr}}$	Linear Limb-darkening	${0.312}_{-0.017}^{+0.018}$
${u}_{2\mathrm{Sloanr}}$	Quadratic Limb-darkening	${0.3264}_{-0.0097}^{+0.0088}$
u1V	Linear Limb-darkening	${0.382}_{-0.018}^{+0.019}$
u2V	Quadratic Limb-darkening	${0.3080}_{-0.0100}^{+0.0096}$

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3.5. Global Fitting with Multi-EXOFAST

4.1. Global Fits

4.1.1. WASP-12b

The WASP-12b global fit included our 23 individual light curves and the SOPHIE RV data from H09 and Husnoo et al. (2011) and spectroscopic priors of ${T}_{\mathrm{eff}}=6300\pm 200$ K and $[\mathrm{Fe}/{\rm{H}}]=0.3\pm 0.15$ from H09. A spectroscopic prior was not imposed on $\mathrm{log}{g}_{\star }$, since the value derived exclusively from the light curves should be more accurate than the spectroscopic value (e.g., Mortier et al. 2013, 2014). Multi-EXOFAST scaled the H09 RV errors by 1.45 and the Husnoo et al. (2011) RV errors by 2.59. The photometric scaling factors are listed in Table 1 and range from 1.07–1.70. The orbital eccentricity was fixed to zero, since secondary eclipse observations by Campo et al. (2011) and Croll et al. (2011) and RV observations and analysis by Husnoo et al. (2011) place tight constraints on the circularity of the planet's orbit. The parameter $\,v\sin {i}_{\star }$ was constrained to be within 1 m s⁻¹ of 2200 m s⁻¹ (the nominal value from spectroscopy in H09), since the RM data do not provide a good constraint on the value. Without a tight $\,v\sin {i}_{\star }$ prior, the model fits would not converge.

The $\sim 1^{\prime\prime}$ stellar binary discovered by Bergfors et al. (2013) is blended with WASP-12 in all of the MORC photometry. The binary is four magnitudes fainter than WASP-12 in the $i^{\prime}$ band, so the ratio of the blended flux to the WASP-12 flux is 1.026 in the $i^{\prime}$ band, and would be slightly less in the $r^{\prime}$ band used for most of the MORC observations. Using the blended flux will, therefore, underestimate the true transit depth and ${R}_{P}/{R}_{* }$ by ∼2.6% and ∼1.2%, respectively. Based on the nominal blended transit depth of ∼14 mmag from the literature, the non-blended depth would increase by ∼0.35 mmag. Any other parameters dependent on the transit depth will also be affected. Since typical M_* and R_* uncertainties from stellar models are ∼10% (e.g., Basu et al. 2012) and those uncertainties are not accounted for in this or other typical works, we do not attempt to account for the flux from the blended binary in the global system fit.

The phased RVs are shown in Figure 5 and a close-up of the in-transit RM data is shown in Figure 6. The SOPHIE RVs from H09 are shown as black symbols, and the SOPHIE RVs from Husnoo et al. (2011) are shown as green squares. The RM and RV models from the global fit are shown as red lines.

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4.1.2. Qatar-1b

The Qatar-1b global fit included our 23 individual light curves, TRES RV data from A11, and HARPS-N RV and RM data, and spectroscopic priors of ${T}_{\mathrm{eff}}=4910\pm 100$ K, $[\mathrm{Fe}/{\rm{H}}]=0.2\pm 0.1$, and $\,v\sin {i}_{\star }=1700\pm 300$ m s⁻¹ from C13. A spectroscopic prior was not imposed on $\mathrm{log}{g}_{\star }$, as discussed in Section 4.1.1. Multi-EXOFAST scaled the C13 RV errors by 1.93 and the A11 RV errors by 1.59. The photometric scaling factors are listed in Table 2 and range from 1.18–1.53. The orbital eccentricity was fixed to zero, since secondary eclipse and RV observations are consistent with a circular orbit as discussed in Section 1. The phased RVs are shown in Figure 7 and a close-up of the in-transit RM data is shown in Figure 8. The HARPS-N RVs and RMs from C13 are shown as circular black symbols and square green symbols, respectively. The TRES RVs from A11 are shown as blue triangles. The RM and RV models from the global fit are shown as red lines.

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4.2. Improved System Parameters

4.2.1. WASP-12b

In addition to the original WASP-12b system parameter analysis by H09, Chan et al. 2011, 2012 (C12), Southworth et al. (2012) (S12), and M13 have performed follow-up analyses of the WASP-12b system. The follow-up analyses use various tools to fit the transit data and determine parameter errors, and they use differing methods to constrain M_* and R_*, but none self-consistently perform a global system analysis based on the RV data, spectroscopy, light curves, and stellar models. All of the analyses except M13 included two to four light curves. M13 simultaneously fits 19 light curves to model the system parameters, but no RV or spectroscopic parameters were included in the fit.

The system parameter values and uncertainties from a self-consistent global fit to two RV data sets, 23 MORC light curves, spectroscopic ${T}_{\mathrm{eff}}$ and $[\mathrm{Fe}/{\rm{H}}]$, the Demarque et al. (2004) stellar models, and the Claret & Bloemen (2011) stellar limb-darkening models are shown in Table 3, and a comparison to values from the literature is shown in Table 4. Since M13 used a modeling tool that fits only light curves, many physical parameter values were not reported and are, thus, not listed in column M13 of the table. Our transit derived parameter values agree with the M13 values to well within $1\sigma$, except for the highly precise orbital periods that differ by $3\sigma$. This difference is discussed further in Section 4.3.1.

Table 4.  Comparison of WASP-12 System Parameters to Literature Values

Parameter	This Work	M13	S12	C12	H09
Stellar Parameters:
M*	1.434 ± 0.11	⋯	1.38 ± 0.18	1.36 ± 0.14	1.35 ± 0.14
R*	1.657 ± 0.046	⋯	1.619 ± 0.076	1.595 ± 0.071	1.57 ± 0.07
$\mathrm{log}{g}_{\star }$	4.157 ± 0.013	⋯	4.159 ± 0.023	4.164 ± 0.029	4.17 ± 0.03
${\rho }_{* }$	0.446 ± 0.015	0.444 ± 0.01	0.458 ± 0.023	0.475 ± 0.038	0.493 ± 0.04
Planetary Parameters:
MP	1.470 ± 0.076	⋯	1.43 ± 0.13	1.403 ± 0.099	1.41 ± 0.10
RP	1.900 ± 0.057	1.86 ± 0.09	1.825 ± 0.091	1.732 ± 0.092	1.79 ± 0.09
$\mathrm{log}{g}_{{\rm{P}}}$	3.004 ± 0.015	⋯	3.027 ± 0.023	3.069 ± 0.031	2.99 ± 0.03
${\rho }_{P}$	0.266 ± 0.015	⋯	0.292 ± 0.025	0.340 ± 0.039	0.318 ± 0.04
Teq	2580 ± 62	⋯	2523 ± 45	⋯	2516 ± 36
Orbital Parameters:
P	$1.0914203(1)$	$1.0914209(2)$	⋯	$1.0914222(11)$	$1.0914230(30)$
a	0.02340 ± 0.00056	⋯	0.02309 ± 0.00096	0.02293 ± 0.00078	0.0229 ± 0.0008
Transit Parameters:
${R}_{P}/{R}_{* }$	0.11785 ± 0.00054	0.1173 ± 0.0005	0.1158 ± 0.005	0.1119 ± 0.002	0.1175 ± 0.0008
$a/{R}_{* }$	3.039 ± 0.034	3.033 ± 0.022	3.067 ± 0.05	3.105 ± 0.082	⋯
i	83.37 ± 0.72	82.96 ± 0.50	83.3 ± 1.1	86.2 ± 3.0	83.1 ± 1.4

Note. Parameter variable names and units are the same as in Table 3, and a value enclosed in parentheses is the uncertainty in the same number of last digits, M13—Maciejewski et al. (2013), S12—Southworth et al. (2012), C12—Chan et al. (2011, 2012), H09—Hebb et al. (2009).

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The orbital semimajor axis, a, from this work agrees with the other studies within $\sim 1\sigma$. All of the transit derived parameters agree within $1\sigma$ as well, except the C12 ${R}_{P}/{R}_{* }$ value is $2-3\sigma$ lower than the values listed from the other studies, which causes significant differences in some of the derived physical parameters. The C12 inclination value is higher by $1\sigma$ compared to the other studies. It is possible that one or both of the transits included in their analysis had a systematic that was not compensated for, or there could have been an astrophysical anomaly, so the C12 results are excluded from the remainder of the comparison discussion.

The transit derived values ${R}_{P}/{R}_{* }$, $a/{R}_{* }$, and i agree with the other three studies within $1\sigma$. The stellar values M_*, R_*, $\mathrm{log}{g}_{\star }$, and ${\rho }_{* }$ all agree within $1\sigma$, although M_* and R_* from this work are at the upper end of the $1\sigma$ range of the other studies. The planetary parameter values M_P, R_P, and T_EQ from this work are also higher by $\sim 1\sigma$, and $\mathrm{log}{g}_{{\rm{P}}}$ and ${\rho }_{P}$ are smaller by $\sim 1\sigma$.

The presence of a correlated non-periodic component in the SOPHIE RVs is especially obvious during the in-transit sequence (Figure 6), which was also noticed and discussed in Husnoo et al. (2011). SOPHIE is a spectrograph that is optimized for precise radial velocity measurements in the "High Resolution" mode. However, WASP-12b is on the faint end of the range of the 1.93 m telescope and was measured in the "High Efficiency" mode, which is less optimized for RV measurements. Husnoo et al. (2011) discuss that the RV anomaly could be due to instrumental noise, stellar variability, or a planetary companion that is unaccounted for in the system. Given their experience with SOPHIE in "High Efficiency" mode, they conclude that instrumental noise is the likely cause. The impact on this study is that the value of spin–orbit angle, $\lambda =-{90}_{-26}^{+23}$, derived from the global fit, may not be reliable.

4.2.2. Qatar-1b

In addition to the original Qatar-1b system analysis by A11, multi-light curve analyses have been performed by C13, E13, DM15, and GM15. The follow-up analyses use various tools to fit the transit data and determine parameter errors, and they use differing methods to constrain M_* and R_*, but none self-consistently perform a global system analysis based on the RV data, spectroscopy, light curves, and stellar models.

The system parameter values and uncertainties from a self-consistent global fit to two RV data sets, 18 MORC light curves, spectroscopic ${T}_{\mathrm{eff}}$, $[\mathrm{Fe}/{\rm{H}}]$, and $\,v\sin {i}_{\star }$, the Demarque et al. (2004) stellar models, and the Claret & Bloemen (2011) stellar limb-darkening models are shown in Table 5, and a comparison to values from the literature is shown in Table 6. Since E13 only reported values determined exclusively from the transit model fit, physical parameter values are not listed in column E13 of the table.

Table 5.  Qatar-1 System Parameter Median Values and 68% Confidence Intervals

Parameter	Units	Value
Stellar Parameters:
M*	Mass (M⊙)	${0.838}_{-0.041}^{+0.043}$
R*	Radius (R⊙)	0.803 ± 0.016
L*	Luminosity (L⊙)	${0.365}_{-0.034}^{+0.040}$
${\rho }_{* }$	Density (cgs)	${2.286}_{-0.070}^{+0.074}$
$\mathrm{log}{g}_{* }$	Surface gravity (cgs)	${4.552}_{-0.011}^{+0.012}$
${T}_{\mathrm{eff}}$	Effective temperature (K)	${5013}_{-88}^{+93}$
$[\mathrm{Fe}/{\rm{H}}]$	Metallicity	${0.171}_{-0.094}^{+0.097}$
$v\sin {I}_{* }$	Rotational velocity (m s−1)	1760 ± 210
λ	Spin–orbit alignment (degrees)	$-{7.5}_{-7.6}^{+7.5}$
Planetary Parameters:
a	Semimajor axis (au)	${0.02332}_{-0.00038}^{+0.00040}$
MP	Mass (MJ)	${1.294}_{-0.049}^{+0.052}$
RP	Radius (RJ)	${1.143}_{-0.025}^{+0.026}$
${\rho }_{P}$	Density (cgs)	${1.076}_{-0.053}^{+0.057}$
$\mathrm{log}{g}_{P}$	Surface gravity	3.390 ± 0.015
Teq	Equilibrium temperature (K)	${1418}_{-27}^{+28}$
Θ	Safronov number	0.0629 ± 0.0019
$\langle F\rangle$	Incident flux (109 erg s−1 cm−2)	${0.918}_{-0.069}^{+0.075}$
Primary Transit Parameters:
${T}_{0}$	Linear ephemeris from transits (${\mathrm{BJD}}_{\mathrm{TDB}}$)	$2456234.103218\pm 6.0708415\times {10}^{-5}$
P	Linear eph. period from transits (days)	$1.4200242\pm 2.1728848\times {10}^{-7}$
${R}_{P}/{R}_{* }$	Radius of the planet in stellar radii	${0.14629}_{-0.00064}^{+0.00063}$
$a/{R}_{* }$	Semimajor axis in stellar radii	${6.247}_{-0.065}^{+0.067}$
i	Inclination (degrees)	${84.08}_{-0.15}^{+0.16}$
b	Impact parameter	${0.645}_{-0.011}^{+0.010}$
δ	Transit depth	0.02140 ± 0.00019
TFWHM	FWHM duration (days)	0.05498 ± 0.00020
τ	Ingress/egress duration (days)	${0.01423}_{-0.00038}^{+0.00039}$
T14	Total duration (days)	0.06921 ± 0.00033
PT	A priori non-grazing transit probability	${0.1367}_{-0.0014}^{+0.0013}$
${P}_{T,G}$	A priori transit probability	0.1835 ± 0.0020
Secondary Eclipse Parameter:
TS	Time of eclipse (${\mathrm{BJD}}_{\mathrm{TDB}}$)	2456233.3931 ± 0.0021
RV Parameters:
K	RV semi-amplitude (m s−1)	261.6 ± 5.5
e	RV eccentricity	0.0 (FIXED)
${K}_{\mathrm{RM}}$	RM amplitude (m s−1)	38.5 ± 4.6
${M}_{P}\sin i$	Minimum mass (MJ)	${1.287}_{-0.048}^{+0.052}$
${M}_{P}/{M}_{* }$	Mass ratio	0.001474 ± 0.000039
${\gamma }_{\mathrm{HARPS}}$	m s−1	$-{50.5}_{-6.1}^{+6.0}$
${\gamma }_{{\mathrm{HARPS}}_{{rm}}}$	m s−1	−59.6 ± 2.0
${\gamma }_{\mathrm{TRES}}$	m s−1	118 ± 16
Limb-darkening Coefficients:
${u}_{1\mathrm{Kepler}(\mathrm{for}\mathrm{Open})}$	Linear Limb-darkening	${0.577}_{-0.022}^{+0.021}$
${u}_{2\mathrm{Kepler}(\mathrm{for}\mathrm{Open})}$	Quadratic Limb-darkening	0.148 ± 0.016
${u}_{1R(\mathrm{for}\mathrm{CBB})}$	Linear Limb-darkening	${0.534}_{-0.021}^{+0.020}$
${u}_{2R(\mathrm{for}\mathrm{CBB})}$	Quadratic Limb-darkening	0.177 ± 0.014

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Table 6.  Comparison of Qatar-1 System Parameters to Literature Values

Parameter	This Work	DM15	GM15	C13	E13	A11
Stellar Parameters:
M*	0.838 ± 0.043	0.818 ± 0.047	0.803 ± 0.072	0.850 ± 0.030	⋯	0.850 ± 0.030
R*	0.803 ± 0.016	0.796 ± 0.016	0.782 ± 0.025	0.800 ± 0.050	⋯	0.823 ± 0.025
$\mathrm{log}{g}_{\star }$	4.552 ± 0.012	4.549 ± 0.011	4.556 ± 0.020	4.550 ± 0.100	⋯	4.536 ± 0.024
${\rho }_{* }$	2.286 ± 0.074	2.282 ± 0.065	2.365 ± 0.079	2.281 ± 0.113	⋯	2.140 ± 0.169
Planetary Parameters:
MP	1.294 ± 0.052	1.293 ± 0.052	1.275 ± 0.079	1.330 ± 0.050	⋯	1.090 ± 0.084
RP	1.143 ± 0.026	1.142 ± 0.026	1.136 ± 0.037	1.180 ± 0.090	⋯	1.164 ± 0.045
$\mathrm{log}{g}_{{\rm{P}}}$	3.390 ± 0.015	3.390 ± 0.012	3.409 ± 0.012	3.372 ± 0.024	⋯	3.265 ± 0.045
${\rho }_{P}$	1.076 ± 0.057	1.075 ± 0.048	1.154 ± 0.133	1.061 ± 0.265	⋯	0.915 ± 0.130
Teq	1418 ± 28	1388 ± 29	⋯	1389 ± 39	⋯	1399 ± 42
Orbital Parameters:
P	$1.4200242(2)$	$1.4200259(28)$	$1.4200241(2)$	$1.4200250(7)$	$1.4200246(4)$	$1.4200330(160)$
a	0.02332 ± 0.00040	0.02313 ± 0.00044	0.02298 ± 0.00069	0.02343 ± 0.00120	⋯	0.02343 ± 0.00026
Transit Parameters:
${R}_{P}/{R}_{* }$	0.1463 ± 0.0006	0.1475 ± 0.0009	0.1459 ± 0.0008	0.1513 ± 0.0008	0.1435 ± 0.0008	0.1455 ± 0.0016
$a/{R}_{* }$	6.25 ± 0.07	6.25 ± 0.08	6.32 ± 0.07	6.25 ± 0.10	6.42 ± 0.10	⋯
i	84.08 ± 0.16	84.03 ± 0.16	84.26 ± 0.17	83.82 ± 0.25	84.52 ± 0.24	83.47 ± 0.40

Note. Parameter variable names and units are the same as in Table 5, and a value enclosed in parentheses is the uncertainty in the same number of last digits, DM15—Mislis et al. (2015), GM15—Maciejewski et al. (2015), C13—Covino et al. (2013), E13—von Essen et al. (2013), A11—Alsubai et al. (2011).

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The stellar parameters, M_*, R_*, $\mathrm{log}{g}_{\star }$, and ${\rho }_{* }$ from this work agree with values from all five literature sources to well within $1\sigma$. C13 revised the RV semi-amplitude upward based on new HARPS-N velocities, which resulted in an increase in planet mass. Since we include the new RVs in this global fit, our planetary parameters are best compared to the literature values starting with C13. They all agree to within $1\sigma$ and are almost identical to the DM15 results, except that our T_eq is higher by about $1\sigma$. The orbital and transit parameters agree within $1\sigma$, except ${R}_{P}/{R}_{* }$ from C13 and E13 differ from all others by more than $3\sigma$ and i from E13 is higher by $2\sigma$.

In summary, the parameters from this work agree well with the literature values, except for the few outliers mentioned in the previous paragraph.

4.3. TTV Measurements

4.3.1. WASP-12b

The WASP-12b global system model includes a TTV parameter for each of the 23 light curves, allowing the transit center time at each epoch to differ from the linear ephemeris. The resulting TTV values are listed in Table 7 and a linear plot of TTV versus epoch is shown in Figure 9. The epoch 776 TTV with large uncertainty and shown as a square yellow symbol is from the transit on UT 2015 January 01, which does not have coverage of pre-ingress baseline or time of first contact and has been excluded from the analysis. This data point illustrates the importance of including only complete transits with good baseline in high-precision TTV studies. Ignoring epoch 776, the TTVs have a maximum value of 79 s, a standard deviation of 32.88 s, and reduced chi-squared of ${\chi }_{\mathrm{red}}^{\,2}=1.1$, with respect to the linear ephemeris. The mean of the timing uncertainty is 31 s.

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Table 7.  WASP-12b Transit Times

${T}_{C}\,({\mathrm{BJD}}_{\mathrm{TDB}})$	Epoch	TTV (s)	${\sigma }_{\mathrm{TTV}}$ (s)	TTV/${\sigma }_{\mathrm{TTV}}$
2455140.909815	−949	−48.89	36	−1.33
2455163.830613	−928	35.05	28	1.21
2455209.668946	−886	−79.01	40	−1.93
2455210.761506	−885	19.46	35	0.55
2455509.809713	−611	−63.37	32	−1.95
2455510.902181	−610	27.15	27	0.97
2455603.672606	−525	1.09	25	0.04
2455903.813566	−250	33.41	28	1.19
2455984.577971	−176	−26.91	28	−0.93
2455985.669747	−175	3.82	36	0.10
2455996.583783	−165	−10.62	32	−0.32
2456249.794041	67	53.79	34	1.58
2456273.805140	89	41.01	26	1.57
2456284.718565	99	−26.22	26	−1.00
2456297.816050	111	11.90	26	0.45
2456319.644239	131	−6.87	33	−0.21
2456607.779376	395	8.21	61	0.13
2456654.710469	438	9.90	29	0.33
2456677.630387	459	17.80	28	0.62
2457012.696166	766	−4.41	42	−0.10
2457023.609018a	776	−121.14	71	−1.69
2457059.627132	809	−13.69	30	−0.45
2457060.718388	810	−27.89	31	−0.89
Standard Deviation		32.88	⋯	1.00
Mean		⋯	31	⋯

Note.

^aThis measurement is from the transit on UT 2015 January 01 that does not have coverage of the pre-ingress baseline or time of first contact and has been excluded from the analysis.

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4.3.2. Qatar-1b

The Qatar-1b global system model includes a TTV parameter for each of the 18 light curves, allowing the transit center time at each epoch to differ from the linear ephemeris. The resulting TTV values are listed in Table 8 and a linear plot of TTV versus epoch is shown in the top panel of Figure 13. The TTVs have a maximum absolute value of 47 s, rms $=\,23.32\,{\rm{s}}$, and ${\chi }_{\mathrm{red}}^{\,2}=1.08$, with respect to the linear ephemeris. The mean of the timing uncertainty is 22 s.

Table 8.  Qatar-1b Transit Times

${T}_{C}\,({\mathrm{BJD}}_{\mathrm{TDB}})$	Epoch	TTV (s)	${\sigma }_{\mathrm{TTV}}$ (s)	TTV/${\sigma }_{\mathrm{TTV}}$
2455742.774748	−346	−7.41	19	−0.38
2455752.714992	−339	−0.98	21	−0.05
2455789.635401	−313	−20.08	22	−0.88
2455796.735833	−308	6.78	18	0.36
2455826.556175	−287	−7.64	19	−0.39
2455843.596639	−275	7.33	19	0.37
2455897.557680	−237	17.71	21	0.83
2456097.780697	−96	−16.82	20	−0.83
2456107.721518	−89	39.46	18	2.09
2456141.801755	−65	9.69	25	0.38
2456151.741312	−58	−43.24	24	−1.74
2456161.681666	−51	−27.31	23	−1.17
2456225.583358	−6	24.67	23	1.07
2456489.707616	180	3.14	17	0.18
2456854.653255	437	−47.71	27	−1.70
2456861.754168	442	20.71	28	0.74
2456888.734234	461	−13.37	26	−0.51
2456925.655231	487	18.34	22	0.81
Standard Deviation		23.32	⋯	1.00
Mean		⋯	22	⋯

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Figure 10 shows a Lomb–Scargle periodogram of our WASP-12b TTVs. All periodograms and associated false alarm probabilities in this work were calculated using the Systemic Console package, as described in Meschiari et al. (2009), by loading the epoch values as the x-axis data set and the TTV values and uncertainties as the y-axis data set. All peaks are well below the analytical 10% false alarm probably (FAP) indicated by the short-dashed line. Nevertheless, we investigated the highest power peak at 3.6615 epochs (3.996 days), which is marked in Figure 10 with a down-pointing arrow labeled 3.6615. The phased plot and model are shown in Figure 11. The 3.996 day period is close to 4.0 days, which suggests a systematic related to the Earth's rotation, but we cannot explain a specific possible cause. The sinusoidal fit has an improved ${\chi }_{\mathrm{red}}^{\,2}=0.66$ compared to the linear fit. Based on the difference in the two ${\chi }^{2}$ values and a difference of two degrees of freedom between the linear and sinusoidal fits, ${\rm{\Delta }}{\chi }^{2}$ analysis suggests a 0.6% probability of a chance improvement in the sinusoidal fit. However, the ∼4.0 day peak has an analytical FAP of more than 100%. The semi-amplitude of the TTV signal is 34 s, which is nearly the same as the standard deviation of the TTVs and the mean timing uncertainty. Our interpretation is that the TTV signal is the result of a night-to-night systematic or a chance fit to noise. The 500 epoch signal that was detected and investigated by M13 is marked in Figure 10 with the down-pointing arrow labeled 500 and shows very little power in the periodogram calculated from our data. Time domain searches around 500 epochs confirm the results of our periodogram.

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M13 included TTVs from an early follow-up light curve from the H09 discovery paper, and two more early light curves from C12. Figure 12 shows the TTVs analyzed by M13 with timing errors $\lt 40\,{\rm{s}}$ (square red symbols), after rephasing them to the linear ephemeris derived in this work (solid gray line). The TTVs from this work are indicated by circular blue symbols. The transits at epochs −1522 and −1224 arrive significantly early relative to the ephemeris from this work, but not quite so early relative to the M13 linear ephemeris (dashed gray line).

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Combining all of the TTV data, a fit to the linear ephemerides gives ${\chi }_{\mathrm{red}}^{\,2}=1.83$ (ephemeris from this work) and ${\chi }_{\mathrm{red}}^{\,2}=1.71$ (ephemeris from M13). Excluding the early epochs −1522 and −1224 gives ${\chi }_{\mathrm{red}}^{\,2}=1.48$ (this work) and ${\chi }_{\mathrm{red}}^{\,2}=1.69$ (M13). An even longer time baseline of observations may be required to determine which ephemeris best describes the WASP-12b orbital period.

The periodogram of the combined TTVs was also investigated. Again, there were no peaks with a reasonably low FAP, and in particular, little support was found for the 500 epoch signal in either the frequency or time domain.

The top panel of Figure 14 shows the Lomb–Scargle periodogram of our Qatar-1b TTV data. The down-pointing arrows labeled 133 and 267 (epochs) mark the ∼190 and ∼380 day periods investigated by E13. We find no evidence for those or any other convincing periodic signals in the MORC TTV data.

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GM15 reanalyzed light curves from E13 and C13 and combined the results with 18 new Qatar-1b transits. We re-phase the GM15 transit center times based on the refined ephemeris from this work (which is nearly identical to the GM15 ephemeris; see Table 6) and plot the resulting TTVs (square red symbols), along with the TTVs from this work (circular blue symbols), in the bottom panel of Figure 13. With respect to the linear ephemeris model, ${\chi }_{\mathrm{red}}^{\,2}=0.82$ for the combined data. It appears that the TTV uncertainties from GM15 are slightly overestimated, and that the linear model is a good representation of the combined data set, unless the uncertainties from this work and GM15 are significantly overestimated. The bottom panel of Figure 14 shows the Qatar-1b combined data Lomb–Scargle periodogram. The significance of the strongest peak is even less than in the periodogram of our data alone. We find no support for periodic TTVs with semi-amplitude greater than ∼25 s in the Qatar-1b system.

Even for systems without astrophysical TTVs, if correlated noise systematics are dominant in the data, the TTV scatter from a linear ephemeris should be significantly higher than would be expected based on the TTV uncertainties extracted by multi-EXOFAST (see Section 3.5). Tables 7 and 8 show that the standard deviations of our TTVs for WASP-12b and Qatar-1b, respectively, are only 6% larger than the mean of the $1\sigma$ TTV uncertainties. Apparently, correlated noise is minimally impacting our extracted TTV uncertainties, so we expect the same, or even less, impact on the other extracted system parameter uncertainties.

We present the results of self-consistent global fits to 23 WASP-12b light curves and 18 Qatar-1b light curves (that have all been homogeneously reduced starting from the individual images), RV and spectroscopic data from the literature, the Demarque et al. (2004) stellar models, and the Claret & Bloemen (2011) stellar limb-darkening models. We reach space-like photometric precision of 183 and 255 ppm for WASP-12b and Qatar-1b, respectively, per 5 minute bin in the combined light curve model residuals.

The WASP-12b derived system parameters in this work are consistent with values from the literature at a level of $1\sigma$, except for the period of the linear ephemeris compared to M13. M_*, R_*, M_P, R_P, and T_EQ from this work are at the upper end of the $1\sigma$ range of the other studies, while $\mathrm{log}{g}_{{\rm{P}}}$ and ${\rho }_{P}$ are at the lower end of the $1\sigma$ range.

The Qatar-1b system parameters derived in this work are consistent with values from the literature at a level of $\sim 1\sigma$, except for the initial planetary mass underestimate by A11. The values fall near the mean of the literature values, except that T_eq from this work is at the upper end of $1\sigma$ relative to the literature values.

The WASP-12b linear ephemeris from this work differs from the M13 ephemeris by $3\sigma$, so a longer baseline of observations may be required to determine the precise ephemeris. Or, if the two earliest TTV measurements shown in Figure 12 are accurate, a long term TTV signal (of about ∼20 years) may be causing the apparent changes in the shifting linear ephemeris. If the two measurements are found to be inaccurate and are removed from the analysis, the linear ephemeris from this work is a good fit to the combined high-precision timing data from M13 and this work. The $\sim 10 \%$ uncertainty in the stellar models is not accounted for by multi-EXOFAST, so no attempt has been made to correct for the $\sim 1 \%$ underestimate of ${R}_{P}/{R}_{* }$ in the global model. Because of this, our global model parameter values are directly comparable to those in the literature. We do not expect the unaccounted for stellar model uncertainties, or the blended binary, to affect our TTV measurements. Nevertheless, we aim to improve our global modeling capabilities to account for stellar model uncertainties in future works.

Unless multi-EXOFAST is overestimating the parameter errors (which is unlikely given the error scaling values listed in Table 1), the WASP-12b transit center times from this work are well modeled by a linear ephemeris with ${T}_{0}=2456176.668258\,\pm$ $7.7650773\times {10}^{-5}$ and $P=1.0914203\,\pm$ $1.4432653\times {10}^{-7}$, which has ${\chi }_{\mathrm{red}}^{\,2}=1.09$. A sinusoidal fit to the data using the period corresponding to the highest power peak in the periodogram (3.996 days), yields an improved fit with ${\chi }_{\mathrm{red}}^{\,2}=0.66$. However, that peak has an analytical FAP of more than 100%. Our interpretation is that the TTV signal is the result of a night-to-night systematic or a chance fit to noise. Based on the reduced chi-squared value for the linear ephemeris model and the lack of signals in the periodogram, we find no convincing evidence for sinusoidal TTVs with a semi-amplitude of more than ∼35 s in the WASP-12b data from this work alone or in combination with data from the literature.

The Qatar-1b transit center times are also well modeled by a linear ephemeris with ${T}_{0}=2456234.103218$ ± $6.0708415\times {10}^{-5}$ and $P=1.4200242$ ± $2.1728848\times {10}^{-7}$, which has ${\chi }_{\mathrm{red}}^{\,2}=1.08$. A Lomb–Scargle periodogram shows no periodic signals in the TTV data that have an analytical FAP less than 100%. Based on the reduced chi-squared value for the linear ephemeris model and the lack of significant signals in the periodogram, We find no convincing evidence for periodic TTVs with a semi-amplitude of more than ∼25 s in the Qatar-1b data. This interpretation is consistent with the conclusion by MG15.

The lack of significant TTVs in these systems is consistent with the Steffen et al. (2012) study that found no evidence of TTVs in the orbits of Kepler hot-Jupiter planets with $1\leqslant P\leqslant 5$ days. On the other hand, the data are sparsely sampled and it may be possible that short period, low level, or non-sinusoidal TTV signals are lurking in the data. In particular, the WASP-12b timing data between epochs 0 and 200 (the 2012–2013 observing season) seem to show a correlated downward trend. However, the data are too sparse to consider fitting non-sinusoidal TTV signals.

Finally, the results of our extensive new transit observations and systematic re-fitting of the global system parameters has resulted in updated parameters and uncertainties for the WASP-12b and Qatar-1b systems. The WASP-12b system parameter values from this work are consistent with values from previous studies, but typically have $\sim 40 \% \mbox{--}50$% smaller uncertainties than those reported by S12 and C12 as shown in Table 4. Most of the Qatar-1b system parameter values and uncertainties from this work are consistent with values recently reported by DM15 and GM15 as shown in Table 6.

K.A.C. acknowledges support from NASA Kentucky Space Grant Consortium Graduate Fellowships. K.A.C. and K.G.S. acknowledge support from NSF PAARE Grant AST-1358862 and the Vanderbilt Initiative in Data-intensive Astrophysics. We thank the anonymous referee for a thoughtful reading of the manuscript and for useful suggestions. This work has made use of NASA's Astrophysics Data System, the Exoplanet Orbit Database at exoplanets.org, the Extrasolar Planet Encyclopedia at exoplanet.eu (Schneider et al. 2011), and the SIMBAD database operated at CDS, Strasbourg, France. The authors would like to thank Antonio Claret for computing the quadratic limb-darkening coefficients for the CBB and Open filter bands, which were critical to properly fitting many of the light curves presented herein. K.A.C. would like to thank Jason Eastman for insightful TTV and correlated noise discussions and for the development and support of multi-EXOFAST.