Probing Transit Timing Variation and Its Possible Origin with 12 New Transits of TrES-3b

The observations of 12 new transits of TrES-3b were carried out using the 2 m Himalayan Chandra Telescope (HCT) at the Indian Astronomical Observatory (IAO), Hanle, India; the 1.3 m Devasthal Fast Optical Telescope (DFOT) at the Aryabhatta Research Institute of Observational Sciences (ARIES), Nainital, India; and the 1.25 m AZT-11 telescope at the Crimean Astrophysical Observatory (CrAO) in Nauchny, Crimea. All the transit observation were made in R band, in order to minimize the effects of stellar limb-darkening and color dependent atmospheric extinction, as well as to achieve the high-cadence photometric observations (Holman et al. 2006). The log of our observations is given in Table 1, whereas the specification of telescopes and CCD detectors used are listed in Table 2. As can be seen in Table 1, we have observed the transits of TrES-3b in focused (Run 1–2), slightly defocused (Run 3–5 and Run 7–12), and heavily defocused (Run 6) modes of the telescopes depending on mirror diameter, detector size, and weather conditions. To avoid the saturation of CCD images with longer exposure time, we had to defocus the telescope heavily in the case of observing Run 6. However, the telescope defocusing technique allows us to improve the precision of the photometric observation (see Southworth et al. 2009a, 2009b; Hinse et al. 2015; Maciejewski et al. 2015; Püsküllü et al. 2017).

All the science images of the TrES-3 system taken during each transit event were preprocessed using the standard tasks available within IRAF¹² for trimming, bias-subtraction, and flat-fielding. After the preprocessing, aperture photometry was performed on the TrES-3 and its nearby 2–8 comparison stars, whose brightness and color are similar to those of TrES-3, using the "daophot" task within IRAF. The aperture size was allowed to vary in such a manner that it should give minimum scattering in the out-of-transit (OOT) data. This was usually 2–3 times the FWHM of the stellar point-spread function (PSF), whose range for each transit event is given in Table 1. To select the comparison stars, we followed the procedure given in Jiang et al. (2016) and checked the correlations of OOT flux of TrES-3 with those of its nearby stars present in the same field by calculating the Pearson's correlation coefficient, r. The stars with r > 0.90 were chosen as the comparison stars. These strong correlations show the brightness consistency between the TrES-3 and the chosen comparison stars. For each transit event, several transit light curves were obtained by dividing the flux of TrES-3 by the flux of each comparison star. Further, the light curves were also obtained by dividing the sum of fluxes from the different combinations of the comparison stars to the flux of TrES-3 (see Gibson et al. 2009; Jiang et al. 2016; Patra et al. 2017). It is also ensured here that the OOT flux variations should not correlate with the airmass, indicating the similar colors of TrES-3 and objects represented with the different combinations of comparison stars. The light curve of each transit event is finalized by identifying the object with the best combination of comparison stars that produces minimum rms in the OOT data (see Hoyer et al. 2016b; Turner et al. 2017). The number of comparison stars used to obtain each transit light curve and corresponding OOT rms are also listed in Table 1. It is worth mentioning here that the used aperture size and number of comparison stars are different in each transit event, which may be due to varied sky conditions and change in field orientation (see Gibson et al. 2009; Collins et al. 2017). To remove time-varying atmospheric effects, the transit light curves were normalized by fitting a linear function to OOT data, which leads to OOT flux close to unity. The time stamps as Heliocentric Julian Days (HJD) in the transit light curves were converted to Barycentric Julian Days (BJD) with the time standard Barycentric Dynamical Time (TDB), i.e., TDB-based BJD, using an online tool provided by Eastman et al. (2010).¹³ The normalized transit light curves of the TrES-3 system obtained from our observations, along with their best-fit models and residuals, are shown in Figure 1 (see Section 3 for details).

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In addition to our 12 new transit observations, we have also taken 71 transit light curves from the literature. These include eight transit light curves from Sozzetti et al. (2009), nine from Gibson et al. (2009), one from Colón et al. (2010), four from Lee et al. (2011), five from Jiang et al. (2013), 10 from Kundurthy et al. (2013), seven from Turner et al. (2013), 11 from Vaňko et al. (2013), five from Ricci et al. (2017), and 11 from Püsküllü et al. (2017). In total, 83 transit light curves of the TrES-3 system spanning over more than a decade are included in this work.

We derived a new ephemeris for orbital period P and midtransit time T₀ of TrES-3b by fitting a linear ephemeris model,

$$\begin{eqnarray}&&{T}_{m}^{c}(E)={T}_{0}+{EP},\end{eqnarray} \tag{ 1 }$$

to the 83 midtransit times T_m as a function of epoch E given in Table 6 using the emcee MCMC sampler implementation (Foreman-Mackey et al. 2013), where ${T}_{m}^{c}$, E, P, and T₀ are the calculated midtransit time, epoch, orbital period, and midtransit time at E = 0, respectively. To estimate the new linear ephemeris for orbital period P and midtransit time T₀ using the MCMC technique, we assumed a Gaussian likelihood and imposed uniform priors on the parameters P and T₀. The uniform prior used for each parameter is listed in Table 7. We used 100 walkers and then ran 300 steps of every walker as an initial burn-in to adjust the step size for each parameter (e.g., Garhart et al. 2018). Considering the final position of the walkers after the 300 steps as the initial position, we further ran 20,000 steps per walker of MCMC to determine the best-fit parameters of the linear ephemeris model and their uncertainties (e.g., Hoyer et al. 2016a, 2016b). We ensured here the efficient calculation of the well-sampled MCMC chain, because the estimated mean acceptance fraction of ∼0.44 was found to be consistent within the ideal range of 0.2–0.5 (see Foreman-Mackey et al. 2013; Cloutier & Triaud 2016; Stefansson et al. 2017). To assess the convergence of the MCMC chain, we estimated the integrated autocorrelation time of the chain averaged across parameters (Goodman & Weare 2010; Foreman-Mackey et al. 2013) and found its value to be ∼19 steps. This suggests that only after ∼19 steps, the drawn samples of model parameters become independent and start converging toward the reasonable parameter space (Hogg & Foreman-Mackey 2018). By dividing the estimated value of integrated autocorrelation time of ∼19 steps to 20,000 steps per walker of MCMC, we determined the effective number of independent samples to be ∼1052 (see Mede & Brandt 2017), which was found to be larger than its minimum threshold value of 50 per walker set in our MCMC analysis as suggested by the emcee group¹⁵ (see Shinn 2019). From our MCMC analysis, the above obtained characteristics of the acceptance fraction, and the effective number of independent samples calculated using the estimated value of integrated autocorrelation time confirm their acceptability and reliability. This also supports the well performance and convergence of the MCMC chain.

Table 7.  The Uniform Priors and Best-fit Model Parameters

Model	Parameter	Uniform Prior	Best-fit Model Parameter
Linear Ephemeris	${\boldsymbol{P}}$ (days)	(0, 2)	${1.306186172}_{-0.000000052}^{+0.000000052}$
	${T}_{0}$ $({\mathrm{BJD}}_{\mathrm{TDB}})$	(2454184, 2454186)	${2454185.911225}_{-0.000062}^{+0.000062}$
Orbital Decay Ephemeris	Pq (days)	(0, 2)	${1.306186401}_{-0.000000180}^{+0.000000181}$
	${T}_{q0}$ $({\mathrm{BJD}}_{\mathrm{TDB}})$	(2,454,184, 2,454,186)	${2454185.911131}_{-0.000094}^{+0.000094}$
	$\delta P$ (days)a	(−1, 1)	$-{1.702171}_{-1.286600}^{+1.283450}$
Apsidal Precession Ephemeris	Ps (days)	(0, 2)	${1.306186199}_{-0.000000160}^{+0.000000253}$
	Tap0 $({\mathrm{BJD}}_{\mathrm{TDB}})$	(2454184, 2454186)	${2454185.910992}_{-0.000691}^{+0.000424}$
	e	(0, 0.003)	${0.00137}_{-0.00092}^{+0.00105}$
	${\omega }_{0}$ (rad)	(0, 2π)	${2.838}_{-1.510}^{+1.306}$
	$\tfrac{d\omega }{{dE}}$ $(\mathrm{rad}\ {\mathrm{epoch}}^{-1})$	(0, 0.001)	${0.000472}_{-0.000314}^{+0.000323}$

Note.

^aThe uniform prior for δP is in days, while its best-fit value is in 10⁻¹⁰ days.

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To avoid the strongly correlated parameters, the initial 37 steps (i.e., nearly two times the estimated value of integrated autocorrelation time) were also discarded as a final burn-in from the 20,000 steps per walker of MCMC (see Almenara et al. 2016; David et al. 2018). Finally, the remaining samples of model parameters P and T₀ were used for Bayesian parameter extraction. The 50.0 percentile level (median) of the posterior probability distribution for each model parameter is inferred as the best-fit value, while the 16.0 and 84.0 percentile levels (i.e., 68% credible intervals) of the posterior probability distribution are considered as its lower and upper 1σ uncertainties, respectively.

The corner plot depicting the marginalized 1D and 2D posterior probability distributions for the parameters of the linear ephemeris model is shown in Figure 2. In this plot, the 1D histogram along the diagonal panel shows the posterior probability distribution for each parameter obtained by marginalizing over the other parameter, where the three vertical dashed lines represent the median and 68% credible intervals. The off-diagonal panel shows the marginalized 2D projection of posterior probability distribution for the covariation between the pair of parameters, with 1σ, 2σ, and 3σ contours. Moreover, the solid vertical and horizontal blue lines in the off-diagonal panel show the best-fit model parameters. The symmetric and Gaussian-like posterior probability distribution of each model parameter (diagonal panel), as well as its best-fit value lying within the smooth and smaller in size 1σ contour (off-diagonal panel), indicate the robust fitting of the linear ephemeris model to the transit time data with the MCMC technique. This confirms the reliable estimation of the new linear ephemeris for orbital period P and midtransit time T₀ along with their 1σ uncertainties and are given in Table 7. The derived values of the new ephemeris are consistent with those reported previously in the literature. The value of minimum χ² of this model fit is 150.58. Because the degree of freedom is 81, the ${\chi }_{\mathrm{red}}^{2}(81)$ is found to be 1.859. The Bayesian Information Criterion (BIC = χ² + k $\mathrm{log}N$, where k is the number of free parameters and N is the number of data points) corresponding to the best fit is 159.41. Using the new ephemeris, the timing residuals, (O−C), defined as the difference between observed midtransit times, T_m, and the calculated midtransit times, ${T}_{m}^{c}$, were calculated for each epoch E considered in this work and are also given in Table 6. The timing residual as a function of epoch E is shown in Figure 3, and its rms is found to be ∼ 69.86 s. Because the linear ephemeris model (i.e., null-TTV model) provides a poor fit to the transit time data with ${\chi }_{\mathrm{red}}^{2}\gt 1$, there may be a possibility of TTV in the TrES-3 system.

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To search for periodicity in the timing residuals given in Table 6, we computed a generalized Lomb–Scargle periodogram (GLS; Zechmeister & Kürster 2009) in the frequency domain. The periodogram defined by the resulting spectral power as a function of frequency is shown in Figure 4. In this periodogram, we found the highest power peak (power = 0.1383) at the frequency of 0.043867 cycle/period. The False Alarm Probability (FAP) of 26% for the highest power peak was determined empirically by randomly permuting the timing residuals to the observing epochs using a bootstrap resampling method with 10⁵ trials. As shown in Figure 4, this FAP of highest power peak is found to be far below the threshold levels of FAP = 5% and 1%. This indicates that the presence of possible TTV in the TrES-3 system does not show any signature of periodicity. Because there is no evidence of short-term TTV due to lack of periodicity in the timing residuals, it encouraged us to look for the long-term TTV that may be produced by either orbital decay or apsidal precession phenomenon in the TrES-3 system.

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Because Levrard et al. (2009), Matsumura et al. (2010), and Penev et al. (2018) have predicted TrES-3b to be tidally unstable and migrating inward toward its parent star due to tidal orbital decay, we made an attempt to explore this phenomenon in the TrES-3 system using the transit time data spanning over a decade. For this study, we followed Adams et al. (2010), Blecic et al. (2014), and Jiang et al. (2016) and constructed the orbital decay ephemeris model by adding a quadratic term to the linear ephemeris model. As similar to the linear ephemeris model fit (see Section 4.1), we used the emcee MCMC sampler technique to fit the 83 midtransit times T_m as a function of epoch E given in Table 6 to the following orbital decay ephemeris model:

$$\begin{eqnarray}&&{T}_{q}^{c}(E)={T}_{q0}+{P}_{q}E+\delta P\displaystyle \frac{E(E-1)}{2},\end{eqnarray} \tag{ 2 }$$

where E is the epoch, T_q0 is the midtransit time at E = 0, P_q is the orbital period, δP is the change of orbital period in each orbit, and ${T}_{q}^{c}(E)$ is the calculated midtransit time. To estimate the best-fit values for the parameters P_q, T_q0, and δP of the orbital decay ephemeris model, we considered uniform prior and Gaussian likelihood to sample their posterior probability distributions. The uniform prior used for each parameter is listed in Table 7. The number of walkers and the procedure of discarding some steps of every walker as the initial and final burn-in were exactly the same as those adopted in the linear ephemeris model fit (see Section 4.1). To have a large effective number of independent samples, we ran 32,000 steps per walker of MCMC, which was larger than those considered in the linear ephemeris model fit. From this MCMC analysis, the estimated mean acceptance fraction, the integrated autocorrelation time, and the effective number of independent samples were found to be ∼0.35, ∼30, and ∼1066, respectively. The discussed acceptability and reliability of these estimated parameters in Section 4.1 indicate the good performance and convergence of the MCMC chain. After discarding 60 steps (i.e., nearly two times the estimated value of integrated autocorrelation time) as a final burn-in from 32,000 steps per walker of MCMC, the remaining samples of model parameters ${P}_{q}$, T_q0, and δP were used for Bayesian parameter extraction. The estimated best-fit values of these parameters along with their 1σ uncertainties (i.e., the medians and 68% credible intervals of the posterior probability distributions) are given in Table 7. The corner plot depicting the marginalized 1D and 2D posterior probability distributions for the parameters of the orbital decay ephemeris model is shown in Figure 5. Similar to Figure 2, the marginalized 1D posterior probability distributions for the parameters of the orbital decay ephemeris model are found to be symmetric and Gaussian (diagonal panel). In addition to this, the best-fit model parameters are also found to be lying within the smooth and smaller in size 1σ contour (off-diagonal panel). This indicates that the fitting of the orbital decay ephemeris model to the transit time data is reliable. The minimum χ² of this model fit is 148.24, ${\chi }_{\mathrm{red}}^{2}(80)$ is 1.853, and the value of BIC is 161.41. Using the best-fitted orbital decay ephemeris given in Table 7, the ${T}_{q}^{c}(E)$ was calculated for each epoch E. By subtracting the midtransit times calculated using linear ephemeris, ${T}_{m}^{c}(E)$, from the above estimated ${T}_{q}^{c}(E)$, the timing residual ${T}_{q}^{c}(E)$–${T}_{m}^{c}(E)$ of the orbital decay ephemeris model was obtained and plotted as a function of epoch E with a red dashed curve in Figure 3. The rms of the timing residuals is found to be ∼70.03 s. Using the values of parameters P_q and δP given in Table 7, the decay rate (${\dot{P}}_{q}=\tfrac{\delta P}{{P}_{q}}$, Jiang et al. 2016) of the orbital period of TrES-3b is found to be $\sim -4.1\pm 3.1\ \mathrm{ms}\ {\mathrm{yr}}^{-1}$.

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Ragozzine & Wolf (2009) have already suggested TrES-3b to be a potential candidate to examine the apsidal precession as long as its orbit is at least slightly eccentric with e > 0.003. Therefore, we probed the possibility of this phenomenon in the TrES-3 system by adopting the following apsidal precession ephemeris model derived from Equations (7), (9), and (10) of Patra et al. (2017):

$$\begin{eqnarray}&&{T}_{\mathrm{ap}}^{c}(E)={T}_{\mathrm{ap}0}+{P}_{s}E-\displaystyle \frac{{{eP}}_{s}\cos \left({\omega }_{0}+E\tfrac{d\omega }{{dE}}\right)}{\pi \left(1-\tfrac{\tfrac{d\omega }{{dE}}}{2\pi }\right)},\end{eqnarray} \tag{ 3 }$$

where E is the epoch, ${T}_{\mathrm{ap}}^{c}(E)$ is the calculated midtransit time, P s is the sidereal period, e is the eccentricity of orbit, ω is the argument of periastron, ω₀ is the argument of periastron at epoch zero (E = 0), and $\tfrac{d\omega }{{dE}}$ is the precession rate of periastron. By following the previous two model fits (see Sections 4.1 and 4.3), we used the emcee MCMC sampler technique and fitted the above Equation (3), representing the apsidal precession ephemeris model, to the 83 midtransit times given in Table 6 as a function of epoch E. In this MCMC analysis, the uniform prior and Gaussian likelihood were assumed to sample the posterior probability distributions for the parameters P_s T_ap0, e, ${\omega }_{0}$, and $\tfrac{d\omega }{{dE}}$ of the apsidal precession ephemeris model. The uniform prior used for each of these model parameters is listed in Table 7. We followed the linear ephemeris model fit and used the same number of walkers, as well as adopted exactly the same procedure of discarding some steps of every walker as the initial burn-in. However, we ran 2 × 10⁵ steps per walker of MCMC in order to have a sufficient effective number of independent samples of model parameters. During the initial test runs of model fits, we noticed that when the uniform prior for eccentricity e was assumed to be either (0, 0.05) or (0, 0.01), the 1σ uncertainty in midtransit time T_ap0 was found to be increased by 1 or 2 orders more as compared with those obtained in the previous two model fits with the ${\chi }_{\mathrm{red}}^{2}\gt 6$. After several test runs of model fits, the slightly improved fit was achieved when the uniform prior for e was assumed to be (0, 0.003). In this model fit, the estimated values of mean acceptance fraction, integrated autocorrelation time, and the effective number of independent samples were found to be ∼0.23, ∼782, and ∼255, respectively. The good sampling and convergence of the MCMC chain are suggested by the acceptability and reliability of these estimated parameters discussed in Section 4.1. However, the rate of convergence appears to be slow because of the larger value of integrated autocorrelation time of ∼782 steps as compared with previous two model fits (see Sections 4.1 and 4.3). This suggests that the samples drawn before ∼782 steps have strongly correlated model parameters. To avoid this, the initial 2346 steps (i.e., nearly three times the estimated value of integrated autocorrelation time) were discarded as a final burn-in from the 2 × 10⁵ steps per walker of MCMC. Finally, Bayesian parameter extraction was performed using the remaining samples of model parameters P_s T_ap0, e, ${\omega }_{0}$, and $\tfrac{d\omega }{{dE}}$. The best-fit values of these parameters along with their 1σ uncertainties (i.e., the medians and the 68% credible intervals of the posterior probability distributions) are listed in Table 7. The corner plot depicting the marginalized 1D and 2D posterior probability distributions for the parameters of the apsidal precession ephemeris model is shown in Figure 6. In contrast to Figures 2 and 5, the marginalized 1D posterior probability distribution for each parameter of the apsidal precession ephemeris model does not appear to be symmetric and Gaussian (diagonal panel). Besides this, the marginalized 2D posterior probability distribution for each pair of model parameters seems to be asymmetric and broader, and for most of the cases, the best-fit values of model parameters lie outside the 1σ contour (off-diagonal panel). In this MCMC analysis, the measured values of the model parameters e, ${\omega }_{0}$, and $\tfrac{d\omega }{{dE}}$ appear to be statistically less significant as the estimated 1σ uncertainties in these parameters are found to be large (see Table 7). All these obtained unusual features might be originated due to the nearly circular orbit of TrES-3b with $e={0.00137}_{-0.00092}^{+0.00105}$ (see Table 7), as well as the strongly correlated model parameters. This also indicates that the fitting of the apsidal precession ephemeris model to the transit time data is not very reliable. The minimum χ² of this model fit is 220.27, ${\chi }_{\mathrm{red}}^{2}(78)$ is 2.824, and the value of BIC is 242.39. Using the best-fitted apsidal precession ephemeris given in Table 7, the midtransit time ${T}_{\mathrm{ap}}^{c}(E)$ was calculated for each epoch E. By subtracting the midtransit times calculated using the linear ephemeris, ${T}_{m}^{c}(E)$, from the above estimated ${T}_{\mathrm{ap}}^{c}(E)$, the timing residual ${T}_{\mathrm{ap}}^{c}(E)$–${T}_{m}^{c}(E)$ of the apsidal precession ephemeris model was obtained and plotted as a function of epoch E with the blue dashed curve in Figure 3. The rms of this timing residual is found to be ∼69.91 s.

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As several authors (e.g., Blecic et al. 2014; Hoyer et al. 2016a, 2016b; Patra et al. 2017) have adopted the BIC statistic for obtaining the most plausible model that can represent the transit time data, we also adopted the same procedure and calculated the values of BIC corresponding to minimum values of χ² obtained from linear, orbital decay, and apsidal precession ephemeris model fits. The obtained values of BIC from the orbital decay and apsidal precession ephemeris model fits (see Sections 4.3 and 4.4) favor the orbital decay of TrES-3b rather than the apsidal precession by $\bigtriangleup \mathrm{BIC}=80.98$, corresponding to the approximate Bayes factor of $\exp (\bigtriangleup \mathrm{BIC}/2)=3.84\times {10}^{17}$ (see Blecic et al. 2014; Maciejewski et al. 2016; Patra et al. 2017). This is also justified by the fact that the very low value of eccentricity $e={0.00137}_{-0.00092}^{+0.00105}$, estimated from the apsidal precession ephemeris model fit (see Table 7), would have a marginal effect in the transit parameters determined from the transit light curves (Maciejewski et al. 2016). Moreover, the unusual features obtained while fitting the apsidal precession ephemeris model to the considered transit time data also favor to rule out the possibility of this phenomena in the TrES-3 system (see Section 4.4). In this regard, it is worth mentioning here that the observations of secondary eclipses would be important to provide tight constraints on the eccentricity and apsidal precession rate of the TrES-3 system. Because the values of ${\chi }_{\mathrm{red}}^{2}$ and BIC are not significantly different between the linear and orbital decay ephemeris model fits (see Sections 4.1 and 4.3), it is difficult to find which one of these two models can represent the transit time data considered in this work. However, we prefer the linear ephemeris model over the orbital decay ephemeris model for the presently available transit time data by considering the slightly smaller value of BIC for the former model fit in comparison to the later one, and the measured orbital decay rate is consistent within ∼1.3σ, indicating its statistically less significant estimation (see Section 4.3). In this context, it is noteworthy here that further follow-up observation of transits of TrES-3b may provide the statistically possible measurement of orbital decay rate and thus would be useful to rule in or out this phenomenon.

Assuming that the measured decreasing period of TrES-3b is real and attributed to orbital decay, the modified stellar tidal quality factor (${Q}_{* }^{{\prime} }$) that indicates the efficiency of tidal dissipation in the host star is estimated using the following equation of Maciejewski et al. (2016):

$$\begin{eqnarray}&&{Q}_{* }^{{\prime} }=9{P}_{q}{\dot{P}}_{q}^{-1}\left(\displaystyle \frac{{M}_{p}}{{M}_{* }}\right){\left(\displaystyle \frac{a}{{R}_{* }}\right)}^{-5}\left({\omega }_{* }-\displaystyle \frac{2\pi }{{P}_{q}}\right),\end{eqnarray} \tag{ 4 }$$

where P_q is the orbital period, ${\dot{P}}_{q}$ is the decay rate of orbital period, $\tfrac{{M}_{p}}{{M}_{* }}$ is the mass ratio of planet to star, $\tfrac{a}{{R}_{* }}$ is the ratio of semimajor axis to stellar radius, and ω_* is the rotational frequency of the host star. The values of P_q and ${\dot{P}}_{q}$ are already calculated in Section 4.3. However, the values of $\tfrac{{M}_{p}}{{M}_{* }}$ = 0.001964 and $\tfrac{a}{{R}_{* }}=5.926$ are taken from Sozzetti et al. (2009), whereas ω_* = 0.2294808 day⁻¹ is calculated from the stellar rotation period given in Matsumura et al. (2010). We substituted these values in the above Equation (4) and found the modified stellar tidal quality factor to be ${Q}_{* }^{{\prime} }\sim 1.11\times {10}^{5}$ for TrES-3, which lies within the typical range of ${10}^{5}\mbox{--}{10}^{7}$ reported for the stars hosting the hot-Jupiters (Essick & Weinberg 2016; Penev et al. 2018), and is also of the same order of magnitude as found by Sun et al. (2018). Because the timescale of orbital evolution of hot-Jupiters depends on the efficiency of tidal dissipation in their host stars, we substituted ${Q}_{* }^{{\prime} }\sim 1.11\times {10}^{5}$ and relevant parameters from Sozzetti et al. (2009) in the following equation of Levrard et al. (2009) to calculate the remaining lifetime of TrES-3b (before it collides with its host star):

$$\begin{eqnarray}&&{T}_{\mathrm{remain}}=\displaystyle \frac{1}{48}\displaystyle \frac{{Q}_{* }^{{\prime} }}{n}\left(\displaystyle \frac{{M}_{* }}{{M}_{p}}\right){\left(\displaystyle \frac{a}{{R}_{* }}\right)}^{5},\end{eqnarray} \tag{ 5 }$$

where $n=\tfrac{2\pi }{{P}_{q}}$ is the frequency of mean orbital motion of the planet and found its value to be T_remain ∼ 4.9 Myr. To estimate the expected shift in the transit arrival time of TrES-3b due to its decaying orbit, we used the following equation of Birkby et al. (2014):

$$\begin{eqnarray}&&{T}_{\mathrm{shift}}=\displaystyle \frac{1}{2}{T}^{2}\left(\displaystyle \frac{{dn}}{{dT}}\right)\left(\displaystyle \frac{{P}_{q}}{2\pi }\right),\end{eqnarray} \tag{ 6 }$$

where $\tfrac{{dn}}{{dT}}$ is the current rate of change of frequency of mean orbital motion of the planet whose expression can be obtained in terms of ${Q}_{* }^{{\prime} }$ by substituting ${\dot{P}}_{q}=\tfrac{{P}_{q}^{2}}{2\pi }\left(\tfrac{{dn}}{{dT}}\right)$ in the above Equation (4). For the calculated value of $\tfrac{{dn}}{{dT}}\sim 6.429\times {10}^{-20}$ ${rad}\ {s}^{-2}$ corresponding to ${Q}_{* }^{{\prime} }\sim 1.11\times {10}^{5}$, the expected shift in the transit arrival time of TrES-3b after 11 yr (T = 11 yr) is found to be T_shift ∼ 69.55 s. This value of T_shift is fully consistent with the rms of the obtained timing residuals shown in Figure 3. If ${Q}_{* }^{{\prime} }\sim 1.11\times {10}^{5}$ measured from our timing analysis is maintained for another five years (i.e., in a total of 16 yr monitoring), one can expect T_shift ∼147 s, which can be confirmed from the further follow-up observations. However, if ${Q}_{* }^{{\prime} }\sim {10}^{6}$, the expected T_shift is only ∼16 s, which appears to be difficult to detect.

Ragozzine & Wolf (2009) showed for the TrES-3 system that the contribution from the planet's tidal deformation to the theoretical apsidal precession rate is larger as compared to that from the star's tidal deformation and general relativity. The rate of this precession is proportional to the second order planetary Love number (k_p), a dimensionless parameter that gives information about the interior density profile of the planets. To calculate k_p for TrES-3b, we assumed that the observed precession rate is real and adopted the following equation of Patra et al. (2017):

$$\begin{eqnarray}&&\displaystyle \frac{d\omega }{{dE}}=15\pi {k}_{p}\left(\displaystyle \frac{{M}_{* }}{{M}_{p}}\right){\left(\displaystyle \frac{{R}_{p}}{a}\right)}^{5}.\end{eqnarray} \tag{ 7 }$$

Using the value of $\tfrac{d\omega }{{dE}}$ calculated from the timing analysis in Section 4.4 and the other relevant parameters from Sozzetti et al. (2009), we found the value of k_p to be 1.15 ± 0.32. Although the estimated value of k_p is larger than that of Jupiter (${{\boldsymbol{k}}}_{{\boldsymbol{p}}}=0.59$: Wahl et al. 2016), its measurement appears to be statistically less significant due to the larger value of uncertainty. The model parameters e, T_ap0, ${\omega }_{0}$, and $\tfrac{d\omega }{{dE}}$ have strongly correlated errors, so this may be the reason behind the larger value of uncertainty in the estimation of k_p (see Bouma et al. 2019). To provide more tight constraints on the estimation of k_p, future secondary eclipse observations would be required.