APOSTLE: Eleven Transit Observations of TrES3b
Abstract
The Apache Point Survey of Transit Lightcurves of Exoplanets (APOSTLE) observed eleven transits of TrES3b over two years in order to constrain system parameters and look for transit timing and depth variations. We describe an updated analysis protocol for APOSTLE data, including the reduction pipeline, transit model and Markov Chain Monte Carlo analyzer. Our estimates of the system parameters for TrES3b are consistent with previous estimates to within the 2 confidence level. We improved the errors (by 10–30%) on system parameters like the orbital inclination (), impact parameter () and stellar density () compared to previous measurements. The neargrazing nature of the system, and incomplete sampling of some transits, limited our ability to place reliable uncertainties on individual transit depths and hence we do not report strong evidence for variability. Our analysis of the transit timing data show no evidence for transit timing variations and our timing measurements are able to rule out SuperEarth and Gas Giant companions in low order mean motion resonance with TrES3b.
1 Introduction
When an extrasolar planet eclipses its host star, the event is referred to as a transit or primary eclipse. During a transit, observers can detect dips in starlight caused by the planet obscuring a portion of the stellar disk when it passes in front of the star. The transit method applies to those systems where the orbital inclination of a planet is close to (i.e. edgeon) with respect to the observer’s skyplane. The first transit observations were made by Charbonneau et al. (2000). As of August 2012, more than 200 planets (exoplanet.eu) have been detected using the transit method. The search for new exoplanetary systems via transits has advanced to spacebased missions with the launch of the European Space Agency’s (ESA) CoRoT satellite (Fridlund et al., 2006) and NASA’s Kepler mission (Borucki et al., 2010). The objective of transit detection and followup is mainly to catalog and improve measurements of system parameters. Studying the characteristics of extrasolar planetary systems is crucial for developing theories of planet formation that can adequately explain the origin and evolution of all planetary systems (including our own).
The target discussed in this paper, TrES3b is a HotJupiter with one of the shortest orbital periods known (P=1.3 days O’Donovan et al., 2007) among the exoplanets. The planet orbits a Gtype star (T = 5720K), and has a mass and radius of 1.92 and 1.29 , respectively (O’Donovan et al., 2007). Due to its large impact parameter, TrES3b’s transit has more of a vshape than the typical ushape. However, the transit is not grazing and the ability of various observers to consistently measure the transit parameters seem to indicate that the planet’s disk completely enters the stellar disk during the transit (O’Donovan et al., 2007; Sozzetti et al., 2009; Gibson et al., 2009). A blended eclipsing binary, which could account for the vshape, has been ruled out from the radial velocity analysis (O’Donovan et al., 2007). However, the vshape of the transit makes measurements of transit properties challenging at shorter wavelengths, where stellar limbdarkening further degrades the trapezoidal ushape of the transit. The level of insolation received on the dayside of TrES3b due to its proximity to its host star should place it in the class of warm HotJupiters with a temperature inversion in its atmosphere (Hubeny et al., 2003; Burrows et al., 2007; Fortney et al., 2008). However, secondary eclipse observations from the ground and space have shown that the evidence for an inversion layer is not strong (de Mooij & Snellen, 2009; Fressin et al., 2010). TrES3b’s lack of an inversion layer is consistent with the idea that UV radiation from chromospherically active stars (like TrES3b’s parent star) systematically destroys absorbers in the upper planetary atmosphere, thereby preventing the formation of an inversion layer (Knutson et al., 2010).
The ultra closein orbit of TrES3b also makes it a great target for followup transit monitoring. It has been suggested that planets with very short periods are likely to be falling into their host stars (Jackson et al., 2009). Some have looked for transit timing variations indicative of such orbital decay (for e.g. OGLETR56b Adams et al., 2011). TrES3b is similar to OGLETR56b in many regards, including the orbital period ( days), size () and impact parameter (). Though there has been no strong evidence for TTVs for OGLETR56b, it is one object that warrants long term study (Adams et al., 2011). Transit monitoring of TrES3b by other teams has so far confirmed its linear orbital ephemeris (Sozzetti et al., 2009; Gibson et al., 2009). A search for additional transiting companions using the EPOXI mission has yielded no detections, since the likelihood of detecting a planetary sibling in a coplanar orbit with TrES3b is lower for an outer planet given the inclination of the system (Ballard et al., 2011); inner planets are not expected since TrES3b is already quite close to its parent star.
In this paper we report observations of eleven transits of TrES3b, taken as part of the Apache Point Observatory Survey of Transit Lightcurves of Exoplanets (APOSTLE). The APOSTLE program is a followup transit monitoring program designed to obtain highprecision relative photometry on known transiting systems in order to refine measurements of system parameters and transit times (see e.g. GJ 1214b Kundurthy et al., 2011). In 2 we outline our observations, and in 3 we describe the data reduction. The two sections which follow, 4 and 5, outline the transit model and the Markov Chain Monte Carlo (MCMC) analyzer respectively. In 6 we present our estimates of the system parameters for TrES3b and in the subsections 6.1 and 6.2 we present results from our study of transit depth variations (TDVs) and transit timing variations (TTVs). Finally, in 7 we summarize our findings.
2 Observations
TrES3 was observed by APOSTLE over a time span of 2 years between the summer of 2009 and the fall of 2011. All observations of TrES3 were carried out using Agile a highspeed frametransfer photometer (Mukadam et al., 2011), on the ARC^{1}^{1}1Astrophysical Research Consortium 3.5m telescope at Apache Point, New Mexico. The Agile CCD has no dead time, as the charge is transferred to an adjoining array for readout. All observations for TrES3 were made using Agile’s mediumgain, slow readout (100kHZ) mode, with the charge readout at 45sec intervals. The filter used for the observations was the –band similar to the SDSS^{2}^{2}2Sloan Digital SkySurvey filter (with central wavelength, 626nm, Fukugita et al., 1996). This observing filter is bluer than typical filters used for transit observations since Agile is a blue sensitive CCD that is affected by a strong fringe pattern at longer wavelengths (Mukadam et al., 2011). The summary of the 11 –band observations is given in Table 1. Observations were made by adjusting the focus on the secondary mirror to smear the stellar Point Spread Functions (PSFs) across multiple pixels, which minimizes the systematics caused by pixeltopixel wandering of the PSF. The long readout (exposure time) also allowed for a greater count rate that maximized the signaltonoise per image. The count rate was kept below Agile’s nonlinearity limit of 52k ADU and well below its saturation level of 61k ADU by small adjustments to the telescope’s secondary focus.
T#  UTD  Obs. Cond.  Filter  Exp.  Phot. Ap.  RMS (ppm)  %Rej.  Flux Norm.  Error Scaling 

(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 
1  20090514  Clear  r’  45  30  777  2%  1.9179  0.5618 
2  20090613  Poor Weather  r’  45  22  610  13%  1.9083  0.5135 
3  20100322  Clear  r’  45  20  962  1.7788  0.7436  
4  20100516  Clear  r’  45  21  877  1%  1.8954  0.7769 
5  20100602  Poor Weather  r’  45  30  703  1.8670  0.2287  
6  20101012  Clear  r’  45  24  896  1.8832  0.7804  
7  20110324  Clear  r’  45  19  644  1%  1.9809  0.5644 
8  20110427  Poor Weather  r’  45  14  1022  1%  1.8932  0.9787 
9  20110514  Poor Weather  r’  45  15  2960  1.8887  2.8965  
10  20110621  Clear  r’  45  26  1221  1.9227  0.9683  
11  20110824  Clear  r’  45  19  1240  7%  1.8016  0.7597 
(1) Transit Number, (2) Universal Time Date, (3) Observing Conditions, (4) Observing Filter, (5) Exposure Time (seconds) 
(6) Optimal Aperture Radius (pixels), (7) Scatter in the residuals 
(8) % frames rejected due to saturation or other effects, (9) Flux normalization between the target and comparison star 
(10) The factor by which the photometric errors were scaled 
The parent star of TrES3b, GSC 0308900929 (TrES3) is a G type star with a Johnson R magnitude of 12.2. The comparison star used for the relative photometry was USNOB1.0 12750332540 situated 90 away, with Johnson R of 12.9 (Monet et al., 2003), which was the next brightest star in Agile’s FOV. Several studies that use detectors with larger FOVs than Agile typically use many (N 1) comparison stars. For relative aperture photometry, lightcurve precision is limited by the signaltonoise achieved from aperture extraction on the faintest star in the set. The small FOV of Agile meant that most other comparison stars in the field were too faint to provide the adequate signaltonoise on the final lightcurve. PSF photometry is a solution to the problem of using faint comparison stars since the stellar and background components can be constrained accurately. However due to the difficulty in modeling the complex defocused PSF of APOSTLE observations we did not opt for this route. We used a circular aperture for photometry (see details in 3) and hence the target and brightest companion produced the best results. Our uncalibrated differentrial photometry shows that TrES3 was the brighter of the two by a factor of 1.8 in the –band. We also note variability in the uncalibrated flux between TrES3 and its reference star, with the maximum difference being 20% between the highest (#7 UTD 20110324) and lowest (#3 UTD 20100322) values. The observations were made over a variety of observing conditions (Column ‘Obs. Conditions’ in Table 1). The listed nights include both complete and partial transits (where data were lost due to poor weather, or issues with the instrument). Some of these partial transits include UTD 20090514 (), 20090613 () and 20100602 (). Small portions of the ineclipse data were lost for the transits on UTD 20100322 () and UTD 20110427 (), and the night of UTD 20110514 () had exceptionally poor observing conditions (seeing ). Even though these data will affect the fit, we include them in the analysis since they can be used to determine transit times and other system parameters.
3 APOSTLE Pipeline
The APOSTLE project used a customized data reduction pipeline, written in the Interactive Data Language (IDL) to process data from Agile. The pipeline performs standard image processing steps like dark subtraction and flat fielding, but also implements nonlinearity corrections unique to Agile. The pipeline also creates an uncertainty map of the processed images by propagating pixeltopixel errors through each step of the reduction. In addition to the photon counting errors and readnoise from the raw images, the pipeline propagates the variance on the master dark and master flat during the reduction. Errors were also propagated for those pixels where the counts exceeded the nonlinearity threshold (52k counts) using the uncertainties in an empirically derived nonlinearity correction function. Frames where pixels inside a photometric aperture exceeded Agile’s saturation limit of 61k were rejected. Images at the other extreme, where the stars were obscured by clouds, and resulted in low signal to noise measurements were also rejected (i.e. where photometric errors were ppm). The fraction of rejected frames per night is listed in Column (9) ‘Rej.’ in Table 1.
We used SExtractor (Bertin & Arnouts, 1996) to derive initial centroids of our defocused stars. This software allows the use of a customized PSF kernel, so we used a ‘donut’shaped detection kernel for our defocused data. Coordinates obtained from SExtractor were then used for circular aperture photometry with the PHOT task in IRAF’s NOAO.DIGIPHOT.APPHOT package. We derived flux estimates from a range of circular apertures with radii between 5–50 pixels, at intervals of 1 pixel, by simply summing the counts in these apertures. An outlier–rejected global median on the frame was used as the sky estimate, which is removed to derive the instrumental flux of the stars. To derive photometric errors, we extracted counts from the error frames using the same centroids and apertures used for photometry on the target frames. The lightcurves are generated by dividing the instrumental flux of the target star by the comparison star and then dividing the entire lightcurve by the median outofeclipse flux level. At this stage there may still be systematic trends in the lightcurve which have not been removed by reduction; for example, differential extinction due to airmass variation or photometric variation due to centroids wandering over pixels of varying sensitivities (e.g. due to small imperfections in the flatfielding). Thus, for each image we also extracted a set of nuisance parameters which are used to compute a correction function (i.e. detrending function). For the TrES3 data we found that, (i) the airmass, (ii) the global median sky, (iii) the centroid positions of the target star, and the sum of counts in the photometric aperture for (iv) the master dark and (v) the master flat, showed trends that corresponded to trends seen in the lightcurves. The airmass was derived from the image headers, while the global sky and the centroid positions are derived from the photometry on the science frames. The sum of counts in the master dark and master flat are derived from photometry on the master dark and flat using the centroids and apertures used for photometry on the science frames. The correction function () is modeled as a linear sum of nuisance parameters as described by the following equation:
(1) 
where are the nuisance parameters, are the corresponding coefficients. The index counts over the number of nuisance parameters , and the index denotes the transit number. The detrending coefficients are chosen by minimizing the between the observed data (), a model function () and correction function,
(2) 
here is the index which counts over the total number of data points (), when all transits are stacked. During photometry we use a set of trial model parameters (based on values from the literature) to remove the transit lightcurve from the data. We use a linear leastsquares minimizer to fit for the coefficients of (Eq. 1), which removes any correlated trends that remain in the modelsubtracted lightcurve. The resulting residuals are used for selecting the optimal aperture from which to extract the photometry. The aperture where the scatter in the residuals is minimized was chosen as the optimal aperture. Smaller apertures do not completely sample the flux from the stars, and at larger apertures one loses signaltonoise due to the fact that one accumulates fewer counts from the star, compared to sky counts. The optimal aperture typically fell between 5–50 pixel radii. We present the complete list of optimal apertures in Column (6) in Table 1. The sizes of the typical apertures were always a few times larger than the typical halfwidth at half maximum of the stellar PSFs. The photometric precision, which is simply the scatter in the residuals, is also listed in Table 1 in Column (7).
T#  TT0  Norm. Fl. Ratio  Err. Norm. Fl. Ratio  Model Data 

(1)  (2)  (3)  (4)  (5) 
1  0.0490181  0.999407424  0.000934904  1.000000000 
1  0.0479764  1.000889277  0.000936546  1.000000000 
1  0.0474555  1.001329344  0.000936130  1.000000000 
1  0.0469347  1.002419716  0.000941452  1.000000000 
1  0.0455226  1.000658752  0.000936362  1.000000000 
1  0.0444809  0.998944209  0.000923458  1.000000000 
1  0.0434392  0.998014984  0.000914333  1.000000000 
1  0.0429184  0.996495746  0.000917120  1.000000000 
1  0.0423975  0.999418076  0.000914379  1.000000000 
1  0.0418767  1.000283709  0.000908789  1.000000000 
.  .  .  .  . 
.  .  .  .  . 
*The data are presented in their entirety as an onlineonly table. 
(1) Transit Number, (2) Time Stamps  Mid Transit Times (BJD) 
(3) Normalized Flux Ratio, (4) Error on Normalized Flux Ratio 
(5) Model Data 
4 Multi Transit Quick
We developed a transit model called MultiTransitQuick (MTQ) in PYTHON, which is based on the analytic lightcurve models presented in Mandel & Agol (2002), and the PYTHON implementation of some of its functions (from EXOFAST by Eastman et al., 2012). The set of transit parameters used by MTQ include the transit duration (), the limbcrossing duration () and the times of midtransit (). The parameters and are the same as and from Carter et al. (2008). The shape of the transit is characterized by the transit depth () and the stellar limbdarkening parameters and . The parameter simply represents the maximum depth of the transit trough at conjunction, which depends on the ratio of the area of the disks of the planet to the star () and the ratio of the diskaveraged stellar intensity at conjunction to the unobscured diskaveraged intensity at conjunction for an impact parameter of zero, . The transit depth is:
(3a)  
the term I(b)/I(0) can be replaced by the quadratic limbdarkening profile (Mandel & Agol, 2002) to give  
(3b) 
where and are the quadratic limbdarkening coefficients (same as and in Mandel & Agol, 2002). The limbdarkening parameters and in MTQ are linear combinations of the quadratic limbdarkening coefficients and , with and . These linear combinations were used while fitting the lightcurves, since it is known that directly fitting for limbdarkening coefficients results in strongly anticorrelated error distributions for various transit parameters (Brown et al., 2001) and severely hinders the chance of Bayesian techniques from converging to accurate values (more in 5).
We wrote two versions of MTQ, one designed to fit transits observed with different filters (MultiFilter), and the other designed to look for variations in the transit depth (MultiDepth). If one makes transit observations using filters of different wavelength, the resulting set of transit lightcurves must be analyzed using a model that accounts for the different limbdarkening profiles, and differing transit depths extant in the data. The standard set of parameters used for MultiFilter version of MTQ is = {, , , , , }, where are the transit times and and are the limbdarkening parameters described in the previous paragraph. The subscripts and are used to denote multiple transits () and multiple filters () respectively. For APOSTLE’s TrES3b data we only observed using one filter, and the number of transit was eleven.
The second version of MTQ is designed to fit for the depths of each individual transit lightcurve separately. Variations in transit depth can arise for several reasons. Commonly invoked sources of such variations are starspots (cool photospheric regions) and faculae (hot photospheric regions) (Pont et al., 2007; Lanza et al., 2009; Knutson et al., 2011, and many more). The appearance and disappearance of spots on the stellar surface due to rotation or stellar activity cycles would result in variations in the transit depth. Transits occurring across the spotless stellar surface will be shallower than when active regions exist on the visible face of the star (for cool spots). Conversely, if these active regions are hot spots, the transit depths would change in the opposite manner. Other sources for transit depth variations include planetary oblateness, spin precession (Carter & Winn, 2010), planetary rings (Barnes & Fortney, 2004), and satellites (Sartoretti & Schneider, 1999; Tusnski & Valio, 2011). The wide variety of proposed sources of transit depth variation means there may be several degeneracies to resolve if a transit depth variation is indeed detected. Nonetheless, understanding such phenomena may only be possible by establishing statistically significant measurements showing variable transit depths. The multifilter capability of MTQ can be modified to fit for the depth of each transit as a unique parameter. In this case the set of parameters used is = {, , , , , }, where is now fit for each transit instead of each filter. However, the filter corresponding to each depth is still tracked as the model needs to convolve the limbdarkening profile to correctly reproduce the full lightcurve profile. For TrES3b observations only a single filter was used. In addition, one transit is chosen as the “reference” transit, using which MTQ internally computes several transit parameters such as , etc. We picked transit # 7 since it had high photometric precision, and was wellsampled.
In the sections that follow we describe results from our attempt to fit for transit parameters using both the MultiFilter and MultiDepth models. The following section outlines the Markov Chain Monte Carlo analyzer used in conjunction with MTQ.
5 Transit MCMC
Markov Chain Monte Carlo (MCMC) analyzers are now the standard for modeling data on exoplanets (Ford, 2005; Holman et al., 2006; Collier Cameron et al., 2007; Kundurthy et al., 2011; Gazak et al., 2011). We developed an MCMC routine called Transit MCMC (TMCMC), which is designed to work in conjunction with MTQ. However, it can also be used to fit other models and data easily. The core TMCMC routine uses the MetropolisHastings (MH) algorithm, and is based on its implementation for astronomical data, as described by Tegmark et al. (2004) and Ford (2005). The Markov chain is computed by making jumps in parameter space and selecting those jumps which tend toward regions of parameter space at lower . Jumps from the step in the chain are made with the following equation:
(4) 
where and are the vectors of model parameters and their associated stepsizes respectively. The term , referred to as the proposal distribution, is a random number drawn from a normal distribution with a mean of 0 and a variance of . It is customary to use a Gaussian proposal distribution but not mandatory. We modify the MH algorithm slightly by allowing adaptive jumpsize adjustments that optimize the sampling rate of the Markov Chain integrator. The desired acceptance rates are achieved by adjusting the stepsize controller () every 100 accepted steps according to = , where are the number of steps attempted for the last 100 accepted steps and is a scaling factor which is 225 or 434 for singleparameter or multiparameter chains respectively (as noted in Collier Cameron et al., 2007). Given sufficient time, a converged chain will have traversed parameter space such that the ensemble of the points accurately represent the uncertainty distributions of the parameters in set .
5.1 Markov Chains for Mtq
For APOSTLE data sets, we explored system parameters using three different kinds of chains. Two of these were based on the MultiFilter parameter set described in 4; First, with Fixed LimbDarkening Coefficients, and second, with Open LimbDarkening Coefficients. The term limbdarkening coefficient will henceforth be abbreviated as LDC. For the Fixed LDC chains (FLDC), the coefficients were simply fixed to values tabulated for the appropriate observing filter (Claret & Bloemen, 2011). For the Open LDC chains (OLDC), the limbdarkening parameters and are allowed to float. It has been shown that measured limbdarkening coefficients from high precision studies are in good agreement with tabulated values (Brown et al., 2001; Tingley et al., 2006). Another study using spectrophotometry from HST STIS showed that any variations in the estimate of transit parameters due to inaccurate limbdarkening are lower than the 1 uncertainty, with greater disagreement seen at shorter wavelengths (Knutson et al., 2007). Groundbased observations typically cannot achieve the level of photometric precision of spacebased studies, hence constraining the limbdarkening from transit observations is difficult. Subtle inaccuracies in the fit limbdarkening profile can lead to incompatible estimates of system parameters, especially at short wavelengths (Kundurthy et al., 2011). In effect, the purpose of the Fixed LDC and Open LDC chains are simply to compare the differences in the fit limbdarkening coefficients to the tabulated values in the literature. The third type of Markov chain was run on the MultiDepth parameter set described in 4. APOSTLE lightcurves were gathered over a long timebaseline, and statistically significant depth variations seen in the data may help shed light on the various phenomena responsible for depth variations (see 4).
Bounds  Notes 

Nonzero transit duration  
Nonzero limbcrossing duration  
Nonzero transit depth  
Impact parameters less than 1 (primary condition for transit)  
Ensures real values for orbital inclination  
Reasonable limbdarkening coefficients  
Reasonable limbdarkening coefficients 
*applied when parameters were fit (OLDC chains) 
We applied bounds to several transit parameters and combinations of transit parameters in MTQ (as shown in Table 3). Most of these bounds were simply to check if the parameters had values that were physically realistic. For most parameters it was quite rare that the MCMC chains got close to the bounding limits, since the chains spend most of their time near low regions; bestfit parameter values for the TrES3 system are far from any physical bounds. When fitting for the limbdarkening however, the degeneracies proved very significant, and the Markov chains often strayed to unrealistic values. For example and took on values which would suggest limbbrightening rather than limbdarkening, either as a result of noisy data or due to gaps in the ingress or egress portions of the lightcurve. Limbbrightening is considered unphysical for broadband observations. Thus, after every jump (Eq. 4), the vector of proposal parameters is run through a series of subroutines in MTQ, to check if any of the conditions in Table 3 are violated. If any one of these conditions are violated, the vector is discarded and a new vector is generated. TMCMC carries out this check until agreeable values emerge, and then continues with the rest of the algorithm.
5.1.1 Executing and Analyzing Chains
By varying the entire vector of model parameters, and applying a single stepsize modifier (Eq. 4), we run the risk of using mismatched stepsizes and hence undersampling the posterior distributions of some parameters. As mentioned before, wellconstructed chains will properly sample posterior distributions given the correct acceptance rate (Gelman et al., 2003). The key to properly constructing a chain is to choose the relative starting stepsizes, for the parameter ensemble, such that they all roam high and low probability regions of parameter space at roughly the same rate. Determining a reasonable set of starting stepsizes is done by running short exploratory Markov chains (40,000 steps) for each model parameter (holding all others fixed). If these exploratory chains have not stabilized to the optimal acceptance rate of (as noted by Gelman et al., 2003, for single parameter chains) at the end of 40,000 steps, the chain is run until this rate is achieved. The jump sizes near the end of stabilized exploratory chains are then used as the starting steps for the multiparameter chains. These multiparameter chains will also be referred to as ‘long chains’ from now on.
For each transiting system, we ran long chains of steps from two different starting locations for each model scenario: Fixed LDC, Open LDC and MultiDepth/Fixed LDC. After completion we (1) cropped the initial stages of these chains to remove the burnin phase, where the chain is far from the bestfit region, and (2) we exclude the stage where the chain is far from the optimal acceptance rate of 23 5, as noted for multiparameter chains (Gelman et al., 2003). We run three types of postprocessing on the chains after cropping: (a) We compute the ranked and unranked correlations in the chains of every fit parameter with respect to the others. These statistics provide an estimate of the level of degeneracy between parameters in a given model. The next postprocessing steps are two commonly used diagnostics to check for chain convergence, namely (b) computing the autocorrelation lengths and (c) the GelmanRubin R̂static values (Gelman & Rubin, 1992).
In a Markov chain, since a given step is only dependent on the preceding step, sections in the chain may show trends (i.e. are correlated). The correlation length signifies the interval at which sampled points in a Markov chain will be uncorrelated. The effective length is the total number of points in the chain divided by the correlation length. Short correlation lengths (i.e. large effective lengths) indicate that the MCMC ensemble represents a statistically significant and hence more precise sampling of the posterior distribution (Tegmark et al., 2004). Effective lengths steps are commonly considered to be satisfactory. The R̂statistic is computed using multiple chains of the same parameter set which have different initial conditions. An R̂statistic close to 1 (to within 10) indicates that all chains have converged, cover approximately the same region of parameter space, and that the relevant parameter space has been sufficiently explored. Results from a chain are deemed useful if the autocorrelation and GelmanRubin conditions have been met.
5.2 Comparing with the Transit Analysis Package
It has been noted by Carter & Winn (2009) that transit lightcurves lacking any significant “defects” or artificial trends can have correlated noise buried within the overall scatter, which is invisible to visual examination. They test a wavelet based rednoise model on simulated transit lightcurves with and without artificial rednoise and find that models which do not fit for rednoise are subject to inaccuracies in transit parameters on the order of 23 and tend to have underestimated errors by up to 30. For transit timing studies, poor estimates such as these are cause for concern, since smaller errors and large deviations from the expected time can easily lead to false claims of TTVs. The detrending routine within TMCMC removes longterm trends related to instrumental or other nuisance parameters. We visually examined the residuals of APOSTLE lightcurves (those observed during good conditions), and noted that the data may still have lowlevel correlated noise, even after detrending. The Transit Analysis Package (TAP Gazak et al., 2011) implements the rednoise model of Carter & Winn (2009). We ran separate fits of transit parameters using TAP on the detrended lightcurves from our TMCMC fit.
The typical TAP parameter set is: = {, , , , , }, where and are the whitenoise and rednoise levels for transits, respectively. The TAP package does not fit for the period using the transit times, and often yields poor estimates of the period, so we fixed the period to that from TMCMC fits. The limbdarkening was fixed to values from the literature. The orbital eccentricity and argument of periastron were kept fixed at 0 for TrES3b.
One must also note a pitfall of the Carter & Winn (2009) rednoise model. Their fitting function expects the whitenoise and rednoise components of a given lightcurve to be stationary, i.e. there are no temporal variations in the Gaussian whitenoise level and the rednoise’s powerspectrum amplitude with time. They note that more elaborate noise models may be required to account for the fact that real data do not conform to these requirements. Typically the scatter is increased (due to lower signaltonoise) at the portion of the lightcurve where the sky brightness is higher (like data taken close to twilight). Variable observing conditions may alter not just the whitenoise properties of a lightcurve but also result in rednoise that is more complex than what can be described by Carter & Winn (2009)’s wavelet model. In addition, at the 1000 ppm level, stellar variability is not understood; it is reasonable to assume that the target and comparison may each contribute to the correlated noise in the lightcurve. In spite of these caveats, the TAP rednoise analysis serves as a good secondary check to results derived using TMCMC (which does not account for rednoise).
6 System Parameters
As described in 5.1.1 we ran three chains on lightcurves of TrES3b: using the parameter set with (1) Fixed LDC, (2) Open LDC, and (3) using the parameter set with Fixed LDC. The parameter sets are described in 4. All TrES3b chains use , and include the transit times for the 11 transits. Since all TrES3b observations were taken in the –band, the Fixed LDC and MultiDepth Fixed LDC chains fit for 2 LDCs for the –band. The MultiDepth chain was set up similarly, but with 11 free parameters for the transit depths. Post processing statistics and other data for these chains are listed in Table 4. The column ‘’, ‘Chain Length’, ‘Corr. Length’ and ‘Eff. Length’ list the number of free parameters, the length of the cropped chain, the correlation and effective lengths, respectively. All chains were run for approximately 2 million steps, but about 100,000 of the initial steps were removed to account for “burnin” and selection rate stabilization. The Open LDC and MultiDepth chains both have quite low effective lengths indicating poor Markov chain statistics. Only the Fixed LDC, model satisfies the condition of a well sampled posterior distribution (effective length is ). The final two columns list the goodness of fit (i.e. lowest in the MCMC ensemble) and Degreesoffreedom (DOF) from the respective chain. Parameters from all chains had GelmanRubin R̂statistics close to 1 indicating that the parameter space was covered evenly (though the OLDC and MultiDepth FLDC chains were not sampled finely enough, based on the autocorrelation data).
Chain  Model Vector  N  Chain Length  Corr. Length  Eff Length  DOF  

FLDC  14  1,900,001  318  5,974  2503.67  2575  
OLDC  16  1,900,001  4,996  380  2519.14  2573  
MDFLDC  24  1,900,001  2,866  662  2692.94  2565 
As noted previously in Kundurthy et al. (2011) the –band limbdarkening is difficult to characterize using transit models. This result is seen again given the fact that the OLDC chain has the worst autocorrelation statistics. In addition the fact that TrES3b has a neargrazing transit makes constraining the transit shape more challenging. The MultiDepth models also did not converge, which could be due to the fact that several of the transits are not completely sampled during the eclipse event (see Figure 1).
Parameter  FLDC  OLDC  Unit 
MTQ Parameters  
0.02100.0004  0.0212  days  
0.03830.0002  0.03850.0010  days  
0.02510.0002  0.02550.0003    
(0.6767)  0.73710.1279    
(0.3008)  0.17820.4955    
Derived Parameters  
0.16520.0009  0.16490.0015    
b  0.8360.003  0.8370.008   
5.970.03  5.91    
81.950.06  81.86  
28.710.14  28.42  days  
2.360.03  2.29  g/cc  
P (1.3062 days +)  114121  110922  millisec 
Transit Depths  Value  Units  Value  Units  
0.02490.0007    0.16760.0025    
0.02610.0009    0.17110.0030    
0.02760.0007    0.17540.0024    
0.02560.0005    0.16960.0021    
0.02370.0010    0.16380.0034    
0.02610.0006    0.17100.0024    
0.02590.0005    0.17050.0021    
0.02690.0006    0.17340.0024    
0.02740.0012    0.17500.0038    
0.02540.0005    0.16900.0022    
0.02770.0007    0.17580.0025    
Other MTQ Parameters  
Parameter  Value  Units  Parameter  Value  Units 
0.02320.0007  days  0.17580.0025  days  
(0.6767)    0.17580.0025    
Derived Parameters  
b  0.8580.005    5.960.03    
81.720.09  28.660.15  days  
2.350.04  g/cc       
The resulting bestfit parameter estimates are listed in Table 5 for the MultiFilter models, and in Table 6 for the MultiDepth models. These tables also list the derived system parameters. The transformation between the MTQ parameters to the derived system parameters are described in Carter et al. (2008) and Kundurthy et al. (2011). Contour plots showing the joint probability distributions (JPDs) for the fit and derived parameters are shown in Figures 2 and 3 respectively.
It is interesting to note that there is a clear degeneracy in the MTQ parameters as seen by the correlations in the posterior distributions of three parameters (, and ) shown in Figure 2. From the impact parameter ( = 0.84) we note, as previous studies have, that TrES3b has a neargrazing transit. The correlation in the JPDs seen in Figure 2 can be understood from Equation 3b and the following equation (Carter et al., 2008):
(5) 
Rearranging these two expressions gives an expression where . Carter et al. (2008) have shown that when (i.e. the transit is close to grazing), the fraction rises and the covariances between D, and rapidly deviate from zero. This degeneracy may be the primary reason why there is disagreement between the estimates of the system parameters for TrES3b reported by various groups (see Table 7). It may also explain the low statistical significance of the Open LDC and MultiDepth/Fixed LDC chains, since Markov Chains take longer to converge when there are correlations between model parameters.
System parameters agree with previously published values in the literature, as seen by the overlap of the uncertainties in the JPD plot (Figure 3). The errors from TMCMC on the orbital inclination (), impact parameter () and stellar density () are smaller than the previous best measurements of Sozzetti et al. (2009) by factors of 2.5, 3.5 and 2 respectively. However, since the TMCMC+MTQ analysis does not include rednoise analysis the errors presented in Table 6 and Figure 3 are underestimates (Carter & Winn, 2009). More conservative constraints were placed on a subset of these system parameters using the TAP package. Comparisons of some parameters and their uncertainties are presented in 7.
Parameter  TMCMC  TAP  S09  G09  C11  Units 

5.970.03  5.890.05  5.930.06    6.010.84    
81.950.06  81.590.14  81.850.16  81.730.13  81.990.30  
b  0.8360.003  0.8610.007  0.8400.010  0.8520.013     
2.360.03  2.260.06  2.300.07      g/cc 
TMCMC & TAP values are from independent analysis of APOSTLE lightcurves 
S09  (Sozzetti et al., 2009), G09  (Gibson et al., 2009), C11  (Christiansen et al., 2011) 
It is clear that using TAP on the APOSTLE dataset and accounting for rednoise provides more conservative estimates of the system parameters. TAP errors are 10–30% smaller than those reported by Sozzetti et al. (2009). Sozzetti et al. (2009) and Gibson et al. (2009) account for correlated noise by scaling the photometric errors and generally have comparable uncertainties to the APOSTLE analysis with TAP. One must note that improved system parameters derived from transit measurements, like , can be used to place constraints on the age of the system (Sozzetti et al., 2009; Southworth, 2010), which in turn can be used to understand the evolutionary history of the exoplanetary atmosphere. As noted by Carter & Winn (2009), we confirm that correlated noise in lightcurves is an obstacle for obtaining more precise estimates of system parameters.
6.1 Transit Depth Analysis
The autocorrelation data indicate that the MultiDepth/Fixed LDC did not converge and hence errors for the parameters from this chain are unreliable. However, we can still study the results from the MultiDepth fit assuming that the bestfit values are accurate. Since the GelmanRubin test indicate that the parameter space was fully traversed, the bestfit points (median values) from the chain are likely to be close to values that may have resulted from a converged chain.
Figure 4 shows the transit depth vs. transit epoch for 11 –band observations of TrES3b. The overall variations in the –band depth are 0.12% compared to the 0.02% uncertainty in D from the joint fit to depths (FLDC, Table 5). The median depth value (from depth measurements in Table 6 ) is 0.0261, compared to D 0.0251 from the FLDC chain. We note that several of the lightcurves are not completely sampled. We excluded lightcurve # 3, 5, 8, 9 and 11 and recomputed the median depth and scatter to be 0.0257 and 0.04% respectively, which is more consistent with the results from the Fixed LDC chain. However, since the scatter is still larger than the uncertainty in the depth measured by the SingleDepth Fixed LDC chain we cannot completely rule out variability. Since TMCMC does not have rednoise analysis and the transit depth cannot be computed via TAP’s parameter set we also cannot know the level at which the depth uncertainties in the Fixed LDC chains are underestimates. In addition, there are several facts about TrES3 which could support the notion that the depth variability seen in Figure 4 is due to stellar activity. Firstly, the activity index reported for TrES3 classifies it as an active star (Sozzetti et al., 2009; Knutson et al., 2010). Secondly, observations in the –band are known to be affected by spots, due to the fact that the line falls in this wavelength range, and spottostar contrast ratios are enhanced. However, there have been no indications of spotcrossing events in the transit data (a sure sign of stellar activity); this could be due to a large impact parameter and the greater likelihood of starspots being equatorial (assuming the stellar spinaxis is aligned with the skyplane). Thirdly, the possibility that TrES3 is variable is evidenced by the changing fluxratio with its comparison star (Column ‘Flux Norm.’ in Table 1); though this fact could just as easily be explained by variability in the comparison.
In summary, after accounting for incomplete and poor quality lightcurves, the resulting lowlevel variance in transit depth does not warrant a confident claim for the detection of variability in TrES3. More continuous sampling of transit lightcurves, without interruptions would provide better insight into variability. Moreover, as with other parameter measurements, the affect of rednoise on depth measurements needs to be studied further.
6.2 Transit Timing Analysis
Using Transit Timing Variations (TTVs) to look for additional planets was first proposed by Agol et al. (2005) and Holman & Murray (2005); in a system where only one planet is seen in transit, a deviation from the Keplerian period could indicate the presence of additional undetected planets. TTVs on the order of minutes can be produced if an unseen companion lies close to mean motion resonance with the transit planet (Holman et al., 2010; Lissauer et al., 2011a; Ballard et al., 2011; Nesvorný et al., 2012).
We gathered transit times published by Sozzetti et al. (2009), Gibson et al. (2009) and Christiansen et al. (2011), and pooled them alongside TMCMC and TAP measurements of 11 TrES3 transit times. The timestamps of all APOSTLE data were converted to BJD (TDB) in the customized reduction pipeline (Kundurthy et al., 2011). The APOSTLE pipeline’s time conversions have been verified by comparison to the commonly used time conversion routines made available by Eastman et al. (2010). The transit times from the literature were also converted to BJD (TDB) before comparisons were made. The Observed minus Computed (OC) plot is shown in Figure 5. A linear ephemeris was fit to all the data using the equation,
(6) 
resulting in a best fit ephemeris of,
days  
BJD 
with a goodness of fit 204.1 for 52 Degreesoffreedom. The large reduced chisquared = 3.92 indicates that a linear fit does not precisely fit the transit times, and there are significant timing deviations in the data. The largest timing deviation of sec is from the Christiansen et al. (2011) data set. As previously noted, these deviations could either be due to underestimated timing errors or unaccounted timing systematics between different studies. The standard deviation of the OC values is sec, hence the largest timing deviation is very close to being a 3 outlier. Several features of the OC plot seem to indicate that these timing variation are likely due to inconsistencies between the various methods used to derive transit times. For example, Christiansen et al. (2011)’s times are consistently early when compared to the expected transit times (i.e. OCs are all negative). Removing the Christiansen et al. (2011) transit times from the ephemeris fit results in a millisecond change in the fit period, but improves the goodness of fit slightly to = 3.5, with the scatter being 35 seconds and the largest timing deviation being 78 sec, a 2 offset. This fact establishes that there may be discrepancies between the methods used to derive the transit times, and hence we cannot claim any TTVs for TrES3.
In order to compare the ephemerides derived with and without rednoise analysis, we fit for a linear ephemeris to the APOSTLE transit times from TMCMC and TAP respectively (presented in the bottom half of Table 8). The difference between the periods derived for these subsets and the period derived from all available transit times was milliseconds. The reduced s were 7.99 and 1.33 for the TMCMC and TAP subsets respectively, confirming that TAP gives more conservative errors for the transit times thanks to the rednoise analysis. Given the much more robust fit to a linear ephemeris from TAP’s transit times we can rule out any TTVs in the system larger than sec (the scatter in the OC for transit times derived from TAP).
Epoch  T0 (TMCMC)  T0 (TAP)  

2,400,000+ (BJD)  (BJD)  2,400,000+ (BJD)  (BJD)  
597  54965.7046437  0.0001079  54965.7046300  0.0002000 
620  54995.7465390  0.0001326  54995.7465900  0.0001400 
836  55277.8821699  0.0002456  55277.8822500  0.0003600 
878  55332.7425269  0.0001033  55332.7426300  0.0002700 
891  55349.7228375  0.0003216  55349.7230400  0.0003800 
992  55481.6479691  0.0001589  55481.6479900  0.0001600 
1117  55644.9210892  0.0000699  55644.9211900  0.0001800 
1143  55678.8828740  0.0001637  55678.8825200  0.0003000 
1156  55695.8623990  0.0003861  55695.8623500  0.0006600 
1185  55733.7414753  0.0001397  55733.7417000  0.0003300 
1234  55797.7456860  0.0001699  55797.7456600  0.0002900 

Fit  Period (days)  T0 (BJD)  

TMCMC  1.306186240  0.000000181  2454965.7043612  0.0000770 
TAP  1.306186489  0.000000302  2454965.7043416  0.0001118 
For the case when planets are in meanmotion resonance (MMR), Agol et al. (2005) show that the analytic expression, , can roughly estimate the amplitude of the timing deviation (). The quantities , , and are the mass of the unseen perturber, the mass of the transiting planet, the orbital period of the transiting planet and the order of the resonance respectively. For the TrES3 system, we can rule out possible system configurations given the variations seen in the OC result from APOSTLE’s TAP fit. The maximum possible TTV amplitude that could be hidden within the variations seen in APOSTLE transit times are . Using the orbital period from Table 8 and the mass of TrES3b, = 1.91 (Sozzetti et al., 2009), we compute the maximum mass perturber that could exist in the TrES3 system in the 2:1 MMR to be , i.e. additional planets with Mmay exist near the 2:1 MMR. At higher order resonances, this maximum mass (for a possible perturber) is larger.
7 Conclusions
I.Photometric Precision: APOSTLE monitored TrES3b over a period of two years between 2009 and 2011, gathering 11 –band transit lightcurves. APOSTLE achieved photometric precision between 600–1200ppm (excluding nights with poor seeing). The summary of observational results is presented in Table 1.
II.TrES3b System Parameters: From our analysis of 11 lightcurves of TrES3b, we were able to confirm previous estimates of system parameters for the TrES3 system. Our estimates of derived system parameters in Table 5 show improvements ( 10–30%) from previous measurements. Due to the system’s near grazing transit we note that several free parameters from MTQ showed strong correlations in their JointProbabilities (see Figure 2). Such correlations are not suited for rapid MCMC convergence, yet we were able to produce Markov chains that converged and hence are able to derive statistically significant uncertainties for system parameters; see results from the FLDC chain in Table 5. As previous attempts showed (Kundurthy et al., 2011), the Markov chains where limbdarkening coefficients were set as free parameters failed to converge, and so we report that APOSTLE did not achieve the photometric precision required to constrain limbdarkening coefficients for TrES3.
III.Search for Transit Depth Variations: Variations in transit depth over epoch could be evidence for stellar variability. TrES3 is known to be variable (Sozzetti et al., 2009), hence one might have expected strong variations in the transit depth with epoch. Our uncalibrated flux ratios between the target and comparison show significant variability, yet we cannot rule out variability in the comparison star as the cause. Our MultiDepth fits also show variations in the transit depth over transit epoch (see Figure 4 and Table 6), however we refrain from making a confident assertion on the detection of stellar variability since, (a) the parameters from the MultiDepth chain do not have statistically significant errors as the chain failed to converge, and (b) we cannot rule out inaccuracies due to the incomplete sampling of several transits, and (c) the neargrazing nature of this transit makes constraining the depth of the transit trough more challenging. In fact removing several of the incompletely sampled transits lowers the overall scatter in transit depth measurements.
IV.Search for Transit Timing Variations: The transit timing precisions achieved by APOSTLE easily allow for the detection of TTV signals 1min (see TTV uncertainties in Table 8). This is a direct consequence of the ability of the APOSTLE program to gather lightcurves with high photometric precision from the ground. However, we were unable to detect significant timing variations for TrES3b in our data. We find that fitting for transit parameters while accounting for rednoise provided more conservative error estimates on transit times. Transit times derived using TMCMC showed significant deviations from a linear ephemeris fit. The same data analyzed with analyzed with TAP (Carter & Winn, 2009; Gazak et al., 2011) significantly reduced these deviations. The overall scatter in the OC values from our rednoise analysis was on the order of sec, which rules out planetary companions more massive than 0.66 Mnear the 2:1 MMR, and larger companions near higher order resonances.
Transit times published in the literature are derived using different techniques for fitting transit parameters. A proper analysis of transit times would need a simultaneous analysis of transit lightcurves using a transit model that is (1) suited for Bayesian inference (i.e. with a fairly uncorrelated parameter set, Carter et al., 2008) and (2) a transit model that can adequately account for rednoise in the data (like TAP, Gazak et al., 2011). The catalog of transit times accumulated in the literature and used in this study of TrES3b are not derived from such an analysis. MCMC analysis can also be inefficiently slow given large parameter sets and complex internal checks on a given model. Fast MCMC routines like The MCMC Hammer (ForemanMackey et al., 2012) may in the future, allow for rapid analysis using complex models. We provide the entire set of APOSTLE lightcurves of TrES3b with this publication to allow future projects to apply improved methods to study this system.
Extrapolations from Kepler planetary candidate data seem to indicate that small planets may be ubiquitous (Borucki et al., 2011). Interesting trends that have been noted are, (1) HotJupiters tend be alone, i.e. lacking other transiting planet siblings (Latham et al., 2011; Steffen et al., 2012) and (2) members of multiplanet systems with short period planets (Period 10 days) are more likely to be HotNeptunes (Latham et al., 2011; Lissauer et al., 2011b). This trend seems to reveal the existence of different formation pathways among volatilerich planets. Hence the lack of detections TTVs in TrES3b (a HotJupiter) is consistent with Kepler’s findings.
Acknowledgments
Funding for this work came from NASA Origins grant NNX09AB32G and NSF Career grant 0645416. RB acknowledges funding from NASA Astrobiology Institute’s Virtual Planetary Laboratory lead team, supported by NASA under cooperative agreement No. NNH05ZDA001C. Data presented in this work are based on observations obtained with the Apache Point Observatory 3.5meter telescope, which is owned and operated by the Astrophysical Research Consortium. We would like to thank the APO Staff, APO Engineers, Anjum Mukadam and Russel Owen for helping the APOSTLE program with its observations and instrument characterization. We would like thank S. L. Hawley for scheduling our observations on APO. This work acknowledges the use of parts of J. Eastman’s EXOFAST transit code as part of APOSTLE’s transit model MTQ. We would also like to thank an anonymous referee for providing feedback that helped improve our paper.
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