Searching for Possible Exoplanet Transits from BRITE Data through a Machine Learning Technique

For this study, all BRITE data sets of Data Release 2 (R2) and Data Release 3 (R3) were retrieved from BRITE Public Data Archive. These data sets were obtained by five BRITE nano-satellites: blue-filter BRITE-Austria (BAb), red-filter BRITE-Heweliusz (BHr), blue- filter BRITE-Lem (BLb), red-filter BRITE-Toronto, and red-filter Uni-BRITE (UBr). The name of the monitored star, the observational duration, and the names of the employed nanosatellite name are clearly mentioned in the header of each data file. For one particular star, there could be more than one data file. In this case, we would check which nano-satellite was used to obtain the measurements in these data files. If the observations were done by different nano-satellites, they were considered to be separate light curves. If the photometric measurements in several files were done through the same nano-satellite, these measurements were combined to become one light curve.

Our goal was to detect possible transit candidates from a large number of light curves, therefore, to be efficient, we planed to use a simple procedure to reduce the scattering of light curves without going through elaborate corrections (Popowicz et al. 2017). Thus, we chose suitable intervals to bin the data to average out the scattering. We decided to calculate the average flux within a 20-minute bin. This choice of 20 minutes can maximize the scattering reduction but still would have several data points for one transit event because a typical transit duration is about 60 to 200 minutes. The above-binned data was normalized with respect to the averages of observational runs. Then, the standard deviation was determined and the $3\sigma$ outlier data was removed. All light curves were obtained in this study through the above procedure before any further analysis. They were referred to as BRITE light curve hereafter. Fig. 1 shows one such BRITE light curve of the star HD37468 as an example. All the BRITE data sets used in this study are listed in Table 1. In total, there are 35 BRITE light curves.

To search for transit candidates from light curves, the CNN model needs to be trained by both non-transit and transit light curves. The BRITE light curves were used as the training samples of non-transit light curves. The transit signals were injected into the BRITE light curves to produce synthetic transit light curves to be used as training samples.

However, there may be real transits in BRITE light curves, and thus they cannot be used as non-transit training samples. As presented in the next subsection, the BRITE light curves were folded with a huge number of assumed trial periods. For almost all cases, the possible real transit signals were destroyed during the folding process. The probability that the real transit is not destroyed in folded light curves is extremely small. A huge number of training samples were employed, this tiny defect did not affect the training results of CNN models. Thus, the folded BRITE light curves were treated as non-transit light curves in training samples.

The transit signals were modeled by the method in Mandel and Agol (2002). The parameters included the transit period ($p$), the ratio of the orbital semi-major axis to the stellar radius ($a_{rmos}$), the ratio of planet radius to the stellar radius ($R_{rmps}$), the orbital inclination ($i$), the linear limb darkening coefficient ($u_{1}$), and the quadratic limb darkening coefficient ($u_{2}$). As described in the previous section with the four CNN models which correspond to different ranges of transit periods, the parameter $p$ was set to be uniformly distributed between one and two days for Model P12, two and three days for Model P23, three and four days for Model P34, four and five days for Model P45. Other parameters are set to be uniformly distributed within the intervals shown in Table 2.

The transit light curves after injecting these transit signals into the BRITE data are presented as Fig. 2. For convenience, the transit period ($p$) was set at 1 day for the results in Fig. 2. Fig. 2(a)-(c) present the outcomes injecting the transit signal into the BRITE light curve of the star HD68553, and Fig. 2(d)-(f) present the outcomes for the star HD202444. From these plots, it is clear that the transit depth is much more sensitive to the parameter $R_{\rm ps}$.

BRITE is not designed for the detection of exoplanets, so the time for data resolution and monitoring duration is very different from usual transit observations. The main characteristics of BRITE data are the high time resolution within intervals of about 15 minutes and various gaps between these intervals. Thus, the folding is the best way to reveal the possible transit signals.

However, folding can enhance the signals when the folding period is the same as the signal period and can destroy signals when the folding period is different from the signal period. Therefore, we had to conduct folding for different trial periods, and a huge amount of folded light curves could be produced. The CNNs of machine learning then play a very important role in judging which ones are transit candidates.

To enable the CNN models to detect transits from those light curves folded with slightly different periods, we also carried out folding with periods that were two-minute shorter and two-minute longer than the transit period, i.e. $p\pm 2/1440$ (day). In Fig. 3, we present both transit light curves and non-transit light curves of two stars after folding. The folding periods for the star HD68553 in Fig. 3(a)-(c) were set to be 1438 minutes, 1440 minutes, and 1442 minutes, respectively. Similarly, the folding periods for the star HD202444 in Fig. 3(d)-(f) were set to be 1438 minutes, 1440 minutes, and 1442 minutes, respectively. Other transit parameters include $a_{\rm os}=20.0$, $R_{\rm ps}=0.07$, $i=88$, and period $p=1{\rm(day)}=1440{\rm(min)}$ for all light curves in Fig. 3.

Here we describe how we obtained the CNN models, i.e. Model P12, P23, P34, and P45, which were used to search for possible transits. Taking Model P12 as an example, we first needed to inject transit signals into BRITE light curves. To decide a transit signal to be injected, we chose a period $p$ through a uniformly distributed random number ranging between one and two days. Concurrently, other parameters $a_{\rm os}$, $R_{\rm ps}$, and $i$ were also randomly chosen from uniform distributions within the intervals listed in Table 2.

Each light curve was folded with the same chosen period $p$. When $p$ is between one day and two days, the number of data points of one light curve is between 72 and 144 (because the bin size was set as 20 min). To make the Model P12 always have the same number of input nodes, we used interpolation to generate 144 data points for all light curves and set the number of input nodes as 144 for Model P12. Each light curve was also folded with a period that is two-minute larger and two-minute smaller than $p$. Thus, three folded light curves were generated from one BRITE light curve. These three were injected with the chosen transit signal and generated another three folded light curves with transit. Therefore, a total of six folded light curves were generated from one BRITE light curve to become a standard input with 144 nodes for Model P12.

The typical number of training samples is usually larger than 10000 for the construction of any CNN models (Pearson et al 2018). Thus, we used a random number to decide which BRITE light curves could be used to generate six folded light curves. All our 35 BRITE light curves were candidates to be picked and this is repeated until there are several $N_{s}$ input light curves ($N_{s}$=6000).

Each of these generated input light curves was used as training samples following the process previously described. The whole process was repeated five times to get five Model P12. Other CNN models, such as Model P23, Model P34, and Model P45 were also produced in the same way. In Table 3, the range of periods, input node numbers, and the accuracies are listed. Besides the accuracy, the results of reliability (the percentage of real transit light curves among those light curves which predicted to have transits) and completeness (the percentage of the light curves that are successfully predicted to have transits among those transit light curves) are also presented in Table 3. Because all CNN models were trained five times, five versions of Model P12, Model P23, Model P34, Model P45 could be generated. The values of accuracy, reliability, and completeness represent the corresponding median values, and the errors are their standard deviations of five versions. All five versions of these CNN models were employed to search for possible transits.

The accuracy of a CNN model is proportional to the sample size but it may get saturated when the sample size is big enough. Thus, to determine the best sample size, it is important to produce the learning curve, i.e. the accuracy of a CNN model as a function of sample size. With $N_{s}=6000$, six different total numbers of light curves, $N_{t}$ = 2$N_{s}$, 4$N_{s}$, 8$N_{s}$, 16$N_{s}$, 32$N_{s}$, and 64$N_{s}$, were used to train Model P12 to obtain the learning curve here.

For each sample size $N_{t}$, five cycles of training, validation, and testing were done with different sample sizes. Finally, five accuracies were obtained. The median value is defined as the accuracy of a particular $N_{t}$, and the standard deviation gives the error bar. The accuracy as a function of $N_{t}$, i.e. the learning curve, is presented in Fig. 4. From Fig. 4, we oberve that the accuracy is stable when the sample size is $N_{t}=32N_{s}$, so the total number of employed light curves is $N_{t}=32N_{s}$, where $N_{s}=6000$, for the construction of CNN models.

After five versions of Model P12 were obtained, they were used to search for possible transits with periods between one and two days from our 35 BRITE light curves. To be ready as an input of Model P12, each BRITE light curve was folded with a folding period $fp=1440+j$ minute, where $j=0,2,4,6,8,...,1440$. Only those detected by all five versions of Model P12 were considered as the transit candidates here. Then, Model P23, Model P34, and Model P45 were employed to search for possible transits with larger periods in the same way.

The folding periods, in minutes of all the transit candidates detected by our CNN models, are listed in Table 4. The first column of Table 4 represents the Data ID of BRITE light curves originally assigned in Table 1. The second column is the transit period detected by Model P12. The third, 4th, and 5th columns are the transit periods detected by Model P23, Model P34, and Model P45, respectively. For example, for light curve BAb-1, three transits were detected. They represent transits with periods of 2160 minutes, 2892 minutes, and 6338 minutes. These three transits are put in different rows as they correspond to different periods. For light curve BAb-4, only one transit was detected. However, many more transits were detected for the light curve BHr-1. In particular, the transit with a period of 1714 minutes was detected repeatedly by three CNN models, Model P12, Model P23, and Model P34. In addition, for the light curve BLb-2, the transit with a period of 1440 min was also detected by more than one CNN models. Therefore, among all detected transit candidates, these two have larger probabilities to be real transits. However, one must be cautioned that 1440 min is exactly one day. Though BRITE is a space instrument, it might still suffer from various issues related to stability. Therefore, the transit for 1440 min could be a false one. The confirmation is out of the scope of this paper and is left for future further studies. The other transit candidates were only detected, possibly due to week signals, by one CNN model and we decided to leave them to be investigated when there are more high-precision data available in the future.

Let us discuss these two systems whose light curves exhibit real detected transits here. The star HD37468, which gives light curve BHr-1, is also named as $\sigma$ Ori. It is actually a system with several stars. Among these stars, $\sigma$ Ori A and $\sigma$ Ori B form a visual binary with an orbital period of about 160 years. In addition, $\sigma$ Ori A is a binary star itself, comprising $\sigma$ Ori Aa and $\sigma$ Ori Ab, and their orbital period is 143 days. The detected 1714 min photometrical variation is definitely not due to the orbital motion of the binary $\sigma$ Ori Aa and $\sigma$ Ori Ab.

These three stars appear very close to each other (within 0.26 arcsec) in the sky, and are not resolved by the BRITE telescope. Considering the stability, there may be a certain level of mass difference between three stars, which must be the most massive one which dominates the observed flux. It is likely that the 1714 min periodical variation could be due to an exoplanet orbiting around the most massive star.

Though the system seems to be complicated, the photometric transit parameters could still be obtained as an estimation, assuming that the most massive star dominates the observed flux and the planet is moving around the most massive one. Here the model in Mandel $\&$ Agol (2002) is employed to fit the observational light curve through a Markov-Chain Monte-Carlo (MCMC) sampling process. Note that the limb-darkening effect is ignored for the occultation part of the light-curve model. The orbital period was fixed at 1714 minutes, the inclination was fixed to $i=90$, the limb-darkening coefficients were fixed to the values mentioned in Table 2. Other five parameters, including the mid-transit time ($t_{\rm tra}$), the mid-occultation time ($t_{\rm occ}$), the ratio of the orbital semi-major axis to the stellar radius ($a_{\rm os}$), the ratio of planetary radius to the stellar radius ($R_{\rm ps}$), and the ratio of planetary flux to stellar flux ($f_{\rm ps}$) were treated as MCMC samples.

The MCMC sampling code MC3, provided by Cubillos et al. (2017), was used to generate seven chains of $10^{5}$ samples. The best-fit model is then obtained with a value of reduced chi-square $\chi^{2}_{\rm red}=2.39$. In Fig. 5, the observational data is shown as points and the best-fit model is presented as the solid line. Fig. 6 presents the two-dimensional and one-dimensional projections of MCMC posterior distributions, where the dashed lines indicate the best-fit model.

The star HD186882, which gives light curve BLb-2, is also named as $\delta$ Cyg. It is a binary system with components $\delta$ Cyg A and $\delta$ Cyg B. Their separation is about 157 AU and they orbit around each other with a period 780 years. In the sky, their angular separation is about 3 arcsec, thus they cannot be resolved by the BRITE telescope. Their combined apparent magnitude was observed to be nearly 2.87. The primary star $\delta$ Cyg A is much brighter than the companion star $\delta$ Cyg B. The detected transit with a period of 1440 min could be due to a close-in exoplanet or a sub-stellar object orbiting around $\delta$ Cyg A.

The photometric transit parameters could be obtained similarly as described above. As there is no occultation part for the light curve here, with a fixed orbital period of 1440 min, fixed inclination, and fixed limb-darkening coefficients as before, only three parameters, $t_{\rm tra}$, $a_{\rm os}$, $R_{\rm ps}$, are needed for the transit model. Using the same MCMC sampling process, the best-fit model was then obtained with a value of reduced chi-square $\chi^{2}_{\rm red}=4.10$. The best-fit model and the observational light curve are shown in Fig. 7. The MCMC posterior distributions of these three transit parameters are presented in Fig. 8.