First Light of Engineered Diffusers at the Nordic Optical Telescope Reveal Time Variability in the Optical Eclipse Depth of WASP-12b

TL;DR

This study demonstrates the use of engineered diffusers on the Nordic Optical Telescope for high-precision photometry, revealing time variability in the optical eclipse depth of exoplanet WASP-12b and providing new transmission spectrum data.

Contribution

It introduces the application of engineered diffusers for reliable, high-precision ground-based photometry and reports novel findings on the variability of WASP-12b's eclipse depth.

Findings

Photometric precision within sub-millimagnitude level achieved.

New blue-band transit depth measurement for CoRoT-1b disfavoring Rayleigh slope.

Detected significant variability in WASP-12b's eclipse depth, deviating from HST data.

Abstract

We present the characterization of two engineered diffusers mounted on the 2.5 meter Nordic Optical Telescope, located at Roque de Los Muchachos, Spain. To assess the reliability and the efficiency of the diffusers, we carried out several test observations of two photometric standard stars, along with observations of one primary transit observation of TrES-3b in the red (R-band), one of CoRoT-1b in the blue (B-band), and three secondary eclipses of WASP-12b in V-band. The achieved photometric precision is in all cases within the sub-millimagnitude level for exposures between 25 and 180 seconds. Along a detailed analysis of the functionality of the diffusers, we add a new transit depth measurement in the blue (B-band) to the already observed transmission spectrum of CoRoT-1b, disfavouring a Rayleigh slope. We also report variability of the eclipse depth of WASP-12b in the V-band. For the WASP-12b secondary eclipses, we observe a secondary-depth deviation of about 5-sigma, and a difference of 6-sigma and 2.5-sigma when compared to the values reported by other authors in similar wavelength range determined from Hubble Space Telescope data. We further speculate about the potential physical processes or causes responsible for this observed variability

Tables

Table 1. Table 1: From left to right: the ED number, the diameter (DM) in mm, the diffusing angle, the stellar FWHM, for the diffusers placed in FASU A and FASU B, respectively, and the distances from the upper filter surface to the focal plane (DFP).

ED #	DM	Angle	FWHM (FASU A)	FWHM (FASU B)	DFP
	(mm)	∘	(arcsec)	(arcsec)	(mm)
1	90	0.35	4.8	5.3	100.5
2	90	0.50	6.4	7.5	117.0

Table 2. Table 2: Technical specifications of the EDs as given by their manufacturers, RPC Photonics .

Material	Polymer on replicated on glass substrate
Encircled energy	$>$99% at 2$\times$FWHM (best effort)
Max deviation from top-hat irradiance	1% of spec, 3% requirement in irradiance space, including impact of speckle (best effort)
Substrate size	90 mm +0.05/-0.2 mm diameter
Thickness	3 mm $\pm$0.2 mm thickness (polymer layer adds 0.1 mm to substrate thickness)
Anti-reflection coating (unpatterned surface)	Rabs 1%, wavelength 380-1000 nm, AOI 0-5 degrees (best effort)

Table 3. Table 3: Most relevant parameters describing our observations. From left to right : The date corresponding to the beginning of the local night in years, months, and days (yyyy.mm.dd), the name of the target, the filter in which the observations were performed, the ED used for the observations, the number of data points per light curve, N, the average cadence in seconds, CAD (accounting for readout time), the exposure time, ET, in seconds, the total observing time in hours, T tot , and the airmass range, χ m ​ i ​ n , m ​ a ​ x subscript 𝜒 𝑚 𝑖 𝑛 𝑚 𝑎 𝑥 \chi_{min,max} , showing minimum and maximum values.

Date	Name	Filter	ED #	N	CAD	ET	Ttot	${\chi }_{min,max}$
yyyy.mm.dd					(sec)	(sec)	(hours)
2018.08.16	TrES-3b	Bessel $R$	1	167	36	25	1.67	1.00 - 1.08
2018.12.11	CoRoT-1b	Bessel $B$	1	82	180	162	4.10	1.00 - 1.51
2019.01.01	WASP-12b	Bessel $V$	1	184	84	50	4.30	1.00 - 1.31
2019.01.12	WASP-12b	Bessel $V$	2	473	37	25	4.92	1.00 - 1.34
2019.01.24	WASP-12b	Bessel $V$	1	202	87	60	4.91	1.00 - 1.49

Table 4. Table 4: Transit parameters (TPs) for TrES-3b . From top to bottom the orbital period in days, the mid-transit time in Barycentric Julian Dates, the semi-major axis in stellar radius, the inclination in degrees, the planet-to-star radii ratio, and the linear and quadratic limb darkening coefficients. The mid-transit time is given in BJD TDB minus 2454000.

TPs	O’Donovan et al. (2007)	This work
P (days)	1.30619 $\pm$ 0.00001	adopted
T0 (BJDTDB)	185.9101 $\pm$ 0.0003	4347.4216 $\pm$ 0.0002
a/Rs	6.06 $\pm$ 0.10	6.07 $\pm$ 0.06
i (∘)	82.15 $\pm$ 0.21	81.93 $\pm$ 0.13
Rp/Rs	0.1660 $\pm$ 0.0024	0.1614 $\pm$ 0.0036
u1, u2	-	0.3612, 0.2814

Table 5. Table 5: Transit parameters (TPs) for CoRoT-1b, following the same description as Table  4 .

TPs	Bean (2009)	This work
P (days)	1.5089656 $\pm$ 0.0000060	adopted
T0* (BJDTDB)	159.452879 $\pm$ 0.000068	4464.54058 $\pm$ 0.00018
a/Rs	4.751 $\pm$ 0.045	4.751 $\pm$ 0.036
i (∘)	83.88 $\pm$ 0.29	83.96 $\pm$ 0.18
Rp/Rs	0.1433 $\pm$ 0.0010	0.1419 $\pm$ 0.0019
u1, u2	-	0.5700, 0.2165

Table 6. Table 6: Transit parameters (TPs) for WASP-12b, following the same description as Table  4 .

TPs	Maciejewski et al. (2018)
P (days)	1.09142172 $\pm$ 0.0000000015
T0* (BJDTDB)	508.97694 $\pm$ 0.000005
dP/dE (days/epoch2)	(-9.67 $\pm$ 0.73) $\times$10-10
a/Rs	3.026 $\pm$ 0.02
i (∘)	82.87 $\pm$ 0.4
Rp/Rs	0.1175 $\pm$ 0.0003
u1, u2	0, 0

Table 7. Table 7: From left to right: Date of observation in dd.mm.yyyy, the secondary eclipse depths of WASP-12b, δ 𝛿 \delta , in ppt, their corresponding β 𝛽 \beta values, and the photometric precision of the data, σ phot subscript 𝜎 phot \sigma_{\mathrm{phot}} , in ppt.

Date	$\delta$	$\beta$	${\sigma }_{\mathrm{phot}}$
(dd.mm.yyyy)	(ppt)		(ppt)
2019.01.01	1.16 $\pm$ 0.17	1.09	0.52
2019.01.12	-0.03 $\pm$ 0.17	1.60	0.59
2019.01.24	0.28 $\pm$ 0.10	1.10	0.57

Equations

$$\mathrm{SEM}(\mathrm{t})=(\mathrm{TM}(\mathrm{t})-1.)\times \mathrm{sf}+1$$

$${\hat{\sigma }}_N=\sigma N^{-1/2}[M/(M-1){]}^{1/2}.$$

$${\sigma }_N=\sqrt{\frac{1}{M}\sum _{i=1}^M(<{\hat{\mu }}_i>-{\hat{\mu }}_i{)}^2}.$$

$${\mathrm{T}}_{\mathrm{o},\mathrm{SE}}={\mathrm{T}}_{\mathrm{o}}+\mathrm{P}\times \mathrm{E}+\frac{1}{2}\frac{\mathrm{dP}}{\mathrm{dE}}\times {\mathrm{E}}^2.$$

Full text

††offprints: cessen@phys.au.dk

1 Introduction

Despite the invaluable efforts in the search for exoplanets carried out by ground-based observatories such as WASP (Pollacco et al., 2006), HAT-P (Bakos et al., 2004), and KELT (Pepper et al., 2007), the space-based era of transiting exoplanet searches has completely revolutionized the field of exoplanets. Lead by the Kepler Space Telescope and its sequential mission K2 (Borucki et al., 2010; Howell et al., 2014), the Convection, Rotation and planetary Transits mission (CoRoT, Baglin et al., 2009) and currently NASA’s Transiting Exoplanet Survey Satellite (TESS, Ricker et al., 2015), these space-based facilities have shaped our understanding about exoplanets. However, observatories in space are expensive and generally directly rely on ground-based follow-up observations to vet and confirm the planetary nature of exoplanet candidate systems. For example, planets discovered by TESS closest to the ecliptic will have only $\sim$27 days of continuous monitoring. In consequence, ground-based facilities will play a key role, first in their confirmation, and later on, in their detailed characterization. These follow-up observations will most likely include a confirmation and characterization of their transits in different wavelengths (e.g., Tingley, 2004; Deeg et al., 2009), a detailed analysis of their transit timings (see e.g., von Essen et al., 2018; Freudenthal et al., 2018, for ground-based follow-up of Kepler planetary candidates with transit timing variations), and potentially a characterization of their atmospheres (Kempton et al., 2018). All of these ground-based follow-up observations rely on having access to stable and precise photometric data.

Collecting photometry from the ground as precise as space-based data is a challenging task. Ground-based observations are affected by scintillation, atmospheric effects that change throughout the observing nights such as color-dependent absorption of stellar light, cirrus or clouds passing by, telescope tracking errors, and poor flat-fielding, among others. These are first-order effects directly decreasing the quality of the observations, manifesting as a larger scatter and increased correlated noise in the ground-based light curves (see e.g., Smith et al., 2006; Carter & Winn, 2009; von Essen et al., 2016).

To reach precision photometry from the ground to enable robust characterization of exoplanets, three main alternatives to in-focus observations have been developed that all rely on spreading out the light over a number of detector pixels. The basic idea behind the first technique is to observe with a heavily defocused telescope, spreading the stellar flux over many pixels over the detector (e.g., a Charge-Coupled Device, CCD). Spreading the stellar light over many pixels has the advantage that flat-fielding errors can be better averaged down compared to focused observations, and further minimizes the impact of atmospheric seeing. However, the required longer integration times mean that the sky background level is higher than for in-focus observations. In addition, by defocusing the telescope this often creates irregularly illuminated donut-shaped Point Spread Functions (PSFs) across the detector that change both with seeing and are susceptible to optical aberrations. Telescope defocusing for precision photometry was first proposed by Kjeldsen & Frandsen (1992) and later used e.g., by Southworth et al. (2009) —and many others— to increase the photometric precision of transiting exoplanet light curves. The second technique relies on using Orthagonal-Transfer CCDs (OTCCDs) (Tonry et al., 1997; Howell et al., 2003), devices that can directly shuffle the electrons on the CCD pixels during an exposure to mold a desired PSF output (e.g., a square). Although this technique has been used to obtain sub-millimag photometry from the ground (see in particular Johnson et al., 2009) this technique requires OTCCDs which are expensive and not widely available at different observatories. The third technique uses Engineered Diffusers (EDs) as a way to scramble the directions of light that passes through them, spreading the incoming photons over many pixels on the detector but in a more homogeneous way than with the defocusing technique (Stefansson et al., 2017, 2018a, 2018b). Instead of the canonically donut-shaped PSF often seen in defocused images, EDs produce a broad and stabilized PSF with a homogeneous illumination that is insensitive to seeing effects. At first order, EDs have the power to increase the exposure times before saturation is reached, reducing scintillation, and minimizing flat-fielding errors. A full description of their in-detailed functionality can be found in Section 3.1.4 of Stefansson et al. (2017). Although the use of EDs for precision ground-based photometric applications is a new technique, EDs and/or other structured diffusers have been used for a number of other applications in astronomy, including direct imaging (Lafrenière et al., 2007) and as light scramblers to provide stable and homogeneous illumination for fiber-fed spectrographs (e.g., Halverson et al., 2014). Currently EDs are widely used in several telescopes for precision photometric applications (see Table 1 in Stefansson et al., 2018a).

In this work, we present the photometric characterization of two engineered diffusers mounted at the 2.5 meter Nordic Optical Telescope (NOT)111http://www.not.iac.es/instruments/alfosc/diffuser.html, henceforth ED #1 and ED #2. The two EDs have different diffusing angles to be able to work in crowded and sparse stellar fields, and with different stellar magnitude ranges for reasonable exposure times. To characterize them in detail, we have carried out test observations focused on two photometric standard stars and the transits of three Hot Jupiters, namely TrES-3b (O’Donovan et al., 2007), CoRoT-1b (Barge et al., 2008), and WASP-12b (Hebb et al., 2009). Besides the photometric characterization of the EDs, the collected data allowed us to improve the transit parameters of TrES-3b, to constrain the transmission spectrum of CoRoT-1b, and to detect variability of the eclipse depth of WASP-12b.

The paper is structured as follows. Section 2 presents the Engineered Diffusers and the basic optical setup at NOT, along with a description of their overall performance, the presence of ghosts, and the impact of these in the photometry. Section 3 describes the nature of our observations and our data reduction strategy. Section 4 contains an in-detailed description of our data modelling, to both primary transits and secondary eclipses, along with a brief description of our data detrending. Section 5 shows our results. We wrap up this work in Section 6 with some final remarks.

2 The Engineered Diffusers at the Nordic Optical Telescope

2.1 Optical Setup

For our observations we used the Alhambra Faint Object Spectrograph and Camera (ALFOSC) and the Filter And Shutter Unit (FASU), where two extra filter wheels are located in the converging beam in front of ALFOSC. The detector is a 2k E2V CCD which with the ALFOSC plate scale of 0.2138″/pixel gives a FOV of 6.5′. The focal length of the ALFOSC camera is 131 mm. Read-out times of ALFOSC for 200 Kpix/sec are 15 seconds in 1$\times$1 binning, and 6.5 seconds in 2$\times$2 binning. The 90 mm diameter diffusers, ED #1 and ED #2, were mounted in FASU B, the upper of the two FASU wheels, with the diffuser coating facing towards the incident beam. We determined which side has the diffuser coating by looking at the reflected light coming in at an oblique angle using a torch. Standard broadband $UBVRI$ and SDSS filters, mounted in a third filter wheel in the parallel beam inside ALFOSC, or alternatively any filter mounted in FASU wheel A, can be used in combination with a diffuser in FASU B.

2.2 The Diffusing Pattern and its Size on Sky

The diffusers used in this work were manufactured by RPC Photonics222https://www.rpcphotonics.com/ and were funded by the Instrument Center for Danish Astrophysics. Similarly to Stefansson et al. (2017), the diffusers have a top-hat diffusing pattern and two different diffusing angles of 0.35∘ and 0.5∘. To better visualize the effect of the EDs over point sources, Figure 1 shows ALFOSC’s field of view centered on WASP-12 with and without EDs. The diffusing angles differ by a factor of almost two, so the Full-Width at Half-Maximum (FWHM) of the stellar images on the CCD increases about 30%. The diffusers are coated on one side, which gives a better throughput (about 4% better by suppressing Fresnel losses). Table 1 summarizes the main properties of the EDs in connection to the NOT, and Table 2 shows their technical specifications as given by RPC Photonics. Even though the diffusing angles differ by a factor of almost two, the FWHM of the stellar images on the CCD increases about 30%.

2.3 The Stability of the Diffusing Pattern with Airmass and Seeing

A key factor in achieving high precision photometry with engineered diffusers is the stability of the diffusing pattern when the photometric conditions change during observations as time progresses (see e.g., Stefansson et al., 2017, Figure 3). To quantify this stability we calibrated and aligned our ALFOSC diffuser-assisted observations from January 1st 2019 (184 frames, ED #1) and January 12th 2019 (473 frames, ED #2) and computed normalized East-West cuts of several stars in the ALFOSC field of view. Figure 2 shows the resulting cut-plots for the ED #1 (top) and the ED #2 (bottom).

To quantify the stability of the diffusing pattern, for each pixel across the East-West cut we computed the difference between the maximum and minimum values considering the full observing run, and we divided this difference by its average value. Within the center of the profile, for the ED #1 the maximum differences are well contained below 5% and above 3%, while for the far wings, where photometric apertures are usually too large, these values go up to 8%. For the ED #2, the center of the profile has a variability between 3% and 6%, while the wings change up to 10%. As comparison, we carried out the same exercise to data taken by the NOT with the telescope that was defocused to achieve a point-spread function with a FWHM between 2 and 3″. The observed variability is well above 20%.

To further show the stability of the EDs when seeing and airmass change along a given observing night, we compared the extent of their variability to the change in the FWHM of a given set of science frames. To exemplify this, we have chosen the night where seeing changed more drastically during the whole night, but specially during the time range where our observations took place. Figure 3 shows the mentioned changes with respect to their nightly average values for the night of January 1st, 2019. To produce the figure, seeing measurements were taken from the Galileo National Telescope (TNG), airmass was taken from the header of the images, and the diffused FWHM was computed by averaging the corresponding FWHMs of several stars within ALFOSC field of view, as time progressed. The maximum amplitude of variability for the diffused FWHM corresponds to 3.2% (blue points in Figure 3), which corresponds to a maximum change in profile of 0.8 unbinned pixels. On the contrary, the local seeing (black points in Figure 3) changed more than 100% along the whole night and 75% during our observations. A small Pearson’s correlation coefficient between the diffused FWHM and the differential photometry of -0.2 reflects this stability.

2.4 Efficiency of the Diffusers and Ghosts

To determine the throughput of the EDs, on December 2nd, 2018, and on January $20^{th}$, 2019, we observed two standard star fields found in the Landolt (1992) catalogue, namely RU 152 to characterize the ED #1 in the $U,B,V,R,I$ filters, and SA98 670 for the characterization of the ED #2 in the $U,B,V,R,I$ and SDSS $g\&z$ filters. First, we carried out a set of $U,B,V,R,I$ images to monitor the zero point of ALFOSC, followed by the following sequence of observations for each filter studied: a) filter only, b) filter + Diffuser, c) filter only, and d) filter + Diffuser, i.e., resulting in 4 images per filter. This observing strategy allows us to isolate the throughput of the EDs, effectively correcting for instrument throughput and CCD quantum efficiency variations, as the measurements were done a few seconds apart. For this exercise, we typically used the same integration time as for the zero point monitoring.

The bright isolated stars were selected from a zero point frame and these coordinates were used for the whole sequence. The number of stars analyzed per filter ranged between 14 and 21 for the ED #1, and 24 and 37 for the ED #2. In all cases fluxes (in analog to digital units, ADUs) were measured using IRAF’s apphot task, with an aperture of 30 pixels (approximately 5.7 arcseconds). In most cases, the images with and without diffuser have a similar median count, suggesting that the sky was clear and the night photometric. Figure 4 shows the median throughput for the ED #1 on the top panel, and for ED #2 on the bottom panel in the $U,B,V,R,I$ and the $U,B,g,V,R,I,z$ sets of filters, respectively. The throughput ratios were computed averaging the flux counts from each set of images for the respective filters.

From the figures the ED #1 has a throughput of about 90% in the B, peaking at 92% in the R. The U shows a drop placing the throughput around 80%. The ED #2 has a similar trend but with slightly lower values, peaking at 90% in the red. The drop in the U band is caused due to the inefficiency of the coating on those wavelengths (see Table 2).

Furthermore, from our test images taken with the ED #2, we noted some structure around the brightest stars, specially when the count peaks were above $\sim$60% the total dynamic range of ALFOSC’s CCD. We call these features ghosts. Figure 5 shows the ghosts near bright stars that were captured using different filters. The two sided ghost appear to be stronger in bluer wavelengths than in redder ones. When the star is at the centre of the field of view, the left hand side ghost is stronger than the right. However, this is reversed when the star is close to the edge of the field of view. To produce the figure, the data were overscan subtracted, but not flat field corrected. To estimate the relative intensity between the stellar flux and the ghosts, we measured their respective count levels using IRAF task imexamine. While stellar intensities are given by the counts at the peak of the PSFs, the intensity of the ghosts are estimated averaging a 5$\times$5 pixel square near the maximum of the structure, determined from visual inspection. The ratio between the stellar intensity peak and the averaged ghost counts is approximately 196 for the $U$, 296 for the $B$, 550 for the $V$, 1290 for the $R$, 3500 for the $I$, and 290 for the $g$, revealing the increase of its contribution towards bluer wavelengths. The z seems to show no ghosts. The ratios are in all cases so large, that we do not expect –and do not observe– that these features will have any impact in the quality of the photometry.

3 Observations and Data Reduction

3.1 The Collected Data

In Table 3 we summarize the main characteristics of all our science observations.

On the night of August, 16th, 2018, we observed one primary transit of TrES-3b with ED #1. The host star has a magnitude of R = 11.98 (O’Donovan et al., 2007), making it a relatively bright source for the collecting area of the NOT. The transit is fully covered, with about 15 minutes of out-of-transit data before and after the transit. The night was stable and clear. The data were taken in 1x1 binning using a Bessel $R$ filter, centered at 650 $\pm$ 130 nm. The overall, averaged cadence, accounting for readout time, was 36 seconds, while the exposure time was set to 25 seconds. The night was photometric and with good seeing, slowly and steadily increasing between 0.5 and 1.5 arcseconds along the night.

On the night of December, 11${th}$, 2018, we observed one primary transit of CoRoT-1b using the ED #1. The transit coverage was complete, with 45 minutes and 1 hour of out-of-transit data before and after the transit, respectively. The magnitude of the host star is V = 13.6, and its spectral type is G0V (Guenther et al., 2012). We observed the host star and several comparison stars during primary transit in 1x1 binning using a Bessel $B$ filter (440 $\pm$ 100 nm). To the best of our knowledge, these data correspond to the bluest transit observations of CoRoT-1b carried out to date. Due to the magnitude of the host star, the overall cadence was 180 seconds. The seeing varied in the night between 0.5 and 2″, but most of the time well below 1″. The night was overall photometric.

Finally, on the nights of January 1st, 12th, and 24th, 2019, we observed three secondary eclipses of WASP-12b, the first and last one using the ED #1 and the one in between using the ED #2. In all cases the secondary eclipses are fully covered, with approximately 20 minutes to 1 hour of out-of-eclipse data before and after the events. Following the given dates, the overall cadences were between 43, 37 and 87 seconds. All eclipses were observed with the Bessel $V$ filter, with the main goal to characterize the geometric albedo of the exoplanet (see Section 5.3 for further details).

3.2 Data Reduction

The data reduction is carried out by means of our Differential Photometry Pipelines for Optimum Lightcurves, DIP2OL. The procedure is fully described in von Essen et al. (2018). In brief, the photometric reduction pipeline is based in IRAF’s command language, and carries out normal calibration sequences such as bias and dark subtraction and flatfielding, using the task ccdproc. The science frames are then scrutinized for cosmic rays using IRAF’s task cosmicrays. Afterwards, the calibrated images are aligned using xregister, and counts within several combinations of different apertures and sky rings are computed. The apertures are distributed between 0.5 and 5 times the average stellar PSF FWHM of the night computed directly from the science frames, while the sky rings between 1 and 3. As the PSFs are diffused, it is worth to mention that the seeing we compute is absolutely not representative of the local seeing of the site. For the data detrending (see Section 4) DIP2OL outputs the airmass, the seeing, the (x,y) pixel coordinates of the centroid positions, the background counts determined within the sky rings, and the the integrated master flatfield and master dark counts for each aperture, when available. The last three quantities are computed per measured star. Our data reduction finishes converting the time axis given in Julian dates to Barycentric Julian dates. To do so, we make use of the web tool111http://astroutils.astronomy.ohio-state.edu/time/utc2bjd.html presented in Eastman et al. (2010). For this, we use as input values the geographic coordinates of the NOT, along with its altitude above sea level, and the celestial coordinates of the stars.

4 Data Modelling

4.1 Error Determination for the Fitting Parameters

To determine reliable errors for the parameters fitted in this work, we sample from the posterior-probability distribution using Markov-chain Monte Carlo (MCMC). Our MCMC routines are python-based and make use of PyAstronomy333http://www.hs.uni-hamburg.de/DE/Ins/Per/Czesla/

PyA/PyA/index.html (based on PyMC, Patil et al., 2010; Jones et al., 2001). To compute our best-fit parameters we iterated 100 000 times per transit and eclipse, and discarded a conservative first 20% of the samples as burn-in. To investigate the convergence of the chains we first visually inspected them, and then we divided them in three equal parts, computing the best-fit parameters and their errors in each sub-chain, and verifying consistency among results. In this work, error bars are given at 1-$\sigma$ level (68.3% credibility intervals).

4.2 Primary Transit

As the primary transit model we used the one given by Mandel & Agol (2002). From the transit light curve we can infer the orbital period, P, the mid-transit time, $\mathrm{T_{0}}$, the planet-to-star radii ratio, $\mathrm{R_{p}/R_{s}}$, the distance between planet and star centers in units of stellar radii, $\mathrm{a/R_{s}}$, and the orbital inclination, i, in degrees. We modelled the stellar limb-darkening with a quadratic law, with corresponding limb-darkening coefficients, $\mathrm{u_{1}}$ and $\mathrm{u_{2}}$. The latter were taken from Claret & Bloemen (2011) for the used filters, and closely matching the effective temperature, the metallicity and the surface gravity of each one of our observed stars. For the fitting parameters we always considered uniform priors using as starting values the ones obtained from the literature. The width of the priors was set to $\pm$30-$\sigma^{\prime}$, being $\sigma^{\prime}$ the errors reported by the respective authors.

4.3 Secondary Eclipse

As secondary eclipse model we used a scaled version of Mandel & Agol (2002) transit model. In this case, both linear and quadratic limb-darkening coefficients were set to zero. If the transit model is represented by TM(t), then the secondary eclipse model, SEM(t) has the following expression:

[TABLE]

In the equation, sf denotes the scaling factor. This approach conserves perfectly the transit duration and shape. The eclipse depth is then computed as $(\mathrm{R_{P}/R_{s}})^{2}\times\mathrm{sf}$, and its error computed from error propagation.

4.4 Data Detrending

Our detrending strategy is fully described in von Essen et al. (2018). In brief, for the detrending model we consider a linear combination of seeing, airmass, centroid positions of the stars involved in the differential photometry, integrated counts over flats and darks on the selected aperture, when available, and integrated sky counts for the selected sky ring. To choose the combination of parameters required to properly detrend the data without over-fitting them, we take into consideration the joint minimization of four statistical indicators: the reduced-$\chi^{2}$ statistic, the Bayesian Information Criterion, the standard deviation of the residual light curves enlarged by the number of fitting parameters, and the Cash statistic. During our fitting procedure, at each iteration the detrending coefficients are computed from simple inversion techniques, considering the randomly chosen primary transit or secondary eclipse parameters as fixed. Then, the combined model of detrending times primary transit/secondary eclipse models undergo the $\chi^{2}$ minimization associated to our MCMC strategy. In consequence, the uncertainties of the detrending coefficients propagate into the uncertainties of the primary transit and secondary eclipse fitting coefficients.

In addition to the detrending strategy produced in von Essen et al. (2018), here we add a further checkup to verify that the detrending does not have any impact of significance in the derived fitting parameters. In the case of the secondary eclipses, where the depth is intrinsically low and thus data might suffer most from over-fitting, once we determined our best-fit secondary eclipse depth we verified its proper convergence by means of a $\chi^{2}$ map. For the map, we created a grid of secondary eclipse depths within a reasonable range. We assumed that each one of these values were the ones reproducing our data best, and we computed the coefficients of the linear combination of the detrending model. With these assigned, we computed $\chi^{2}$ and plotted its distribution as a function of the eclipse depth. If the detrending process was properly produced and the MCMC chains converged to their absolute minima, the $\chi^{2}$ map should have a unique minimum around the best-fit value. The resulting $\chi^{2}$ maps are given in Section 5.

4.5 Noise Treatment

To test the extent to which our data are affected by correlated noise (Pont et al., 2006), we followed the approach described in Carter & Winn (2009) and von Essen et al. (2018). After carrying out the MCMC fit, we computed the residual light curves by dividing the photometric data by the best-fit model, and from the residuals we computed the $\beta$ factor. Briefly, the residuals are divided into M bins of N averaged data points. This average accounts for changes in exposure time that might be needed to compensate for changes in airmass or transparency during the observing runs. If the data show no correlated noise, then the noise contained in the residual light curves should follow the expectation of independent random numbers:

[TABLE]

where $\sigma$ is the standard deviation of the unbinned residual light curve, while $\sigma_{N}$ corresponds to the standard deviation of the data binned with N averaged data points per bin:

[TABLE]

Here, $\hat{\mu_{i}}$ corresponds to the mean value of the residuals per bin, i, and $<\hat{\mu_{i}}>$ is the mean value of the means. If some amount of correlated noise is present in the data, then $\sigma_{N}$ and $\hat{\sigma}_{N}$ should differ by a factor, called $\beta_{N}$. To compute the $\beta$ value, we have averaged all the $\beta_{N}$’s obtained from bins that are as large as 0.8, 0.9, 1, 1.1 and 1.2 times the duration of ingress. When $\beta$ was found to be larger than 1, we have enlarged the individual photometric error bars by this factor, and carried out the MCMC fitting process again.

5 Results

5.1 TrES-3b

To determine our best-fit parameters from the data collected during the primary transit of TrES-3b, as starting values for the transit parameters we used the ones found in O’Donovan et al. (2007), their Table 3. Since we have only one transit light curve, the orbital period was considered as fixed using the value reported in the literature. We then fitted the mid-transit time, the distance between planet and star centers in units of stellar radii, the inclination, and the planet-to-star radii ratio. Our derived results, along with 1-$\sigma$ error bars, are listed in Table 4. Our values are fully consistent with the ones reported by O’Donovan et al. (2007), with smaller uncertainties. Figure 6 shows the raw data in grey squares, the combined model in blue continuous lines, the detrended data in black circles, and the best-fit transit model in red continuous line. The standard deviation of the residual light curve for an exposure time of 25 seconds is 0.5 parts-per-thousand (ppt), and the corresponding $\beta$ value is very close to 1. It is worth to mention that this target was also observed by Stefansson et al. (2017) using their own ED with a similar diffusing angle (0.34 versus 0.35 degrees). For their test observations they used the 3.5 meter Astrophysical Research Consortium Telescope, located at Apache Point Observatory, United States of America. Even though their collecting area is significantly larger than the one of the NOT, their achieved precision is 0.75 ppt at a 32.5 second cadence. To our knowledge, the only other ground-based transit light curve of TrES-3b more precise than the one reported here can be found in Parviainen et al. (2016). The authors observed TrES-3b with the 10 meter Gran Telescopio Canarias and the OSIRIS low-resolution spectrograph, achieving in their white light curve a 0.5 ppt scatter for 12 seconds exposure.

5.2 CoRoT-1b

For CoRoT-1b, as initial values for our MCMC fit we used the parameters found in Bean (2009), their Table 1 and 3, which in turn were derived from a global fit of 36 transit light curves taken with the CoRoT space based telescope, 20 of them taken in long cadence (512 second exposures corresponding to a 0.681 ppt photometric precision) and the remaining ones taken in CoRoT’s ”short runs” (32 seconds with a photometric precision of 2.724 ppt, see Baglin et al., 2006, for the definition of long cadence and short run).

In Table 5 we list our best-fit transit parameters along with 1-$\sigma$ errors, compared to the ones reported by the authors. All values are consistent within uncertainties to the ones reported by Bean (2009). Figure 7 shows the detrended transit photometry in black circles, along with our best-fit transit model in red continuous line. The standard deviation of the residual light curve, for an exposure time of 162 seconds, is 0.65 ppt. The corresponding $\beta$ value is close to 1. Considering that our exposure time is three times smaller than CoRoT’s long cadence, our derived photometric precision is significantly superior than CoRoT’s.

As previously mentioned, our data were taken with the ED #1 in simultaneous to the Bessel $B$ filter. In consequence, our derived planet-to-star radii ratio is, to date, the bluest point characterized in CoRoT-1b’s transmission spectrum. Figure 8 shows its complete transmission spectrum. A non-chromatic planet-star radius ratio results in a $\chi^{2}$ of 15.79 with 11 degrees of freedom (equivalently, a reduced $\chi^{2}$ of 1.43) and a probability $P$ of 0.1. Thus, the current set of observations do not show significant variations in Rp/Rs. CoRot-1b has an atmospheric pressure scale height of $\sim$ 700 km. Models of hot Jupiter transmission spectra predict variations of $\mathrm{R_{P}/R_{S}}$ in low spectral resolution within approximately $\pm$ 2 scale heights (Fortney et al., 2010; Schlawin et al., 2014). The mean uncertainty of the existent measurement for $\mathrm{R_{P}/R_{S}}$, and also the uncertainty of the new measurement of this work, roughly correspond to this value of two scale heights. In consequence, the data are not precise enough for a detailed transmission model fitting. Nonetheless, the new $B$ band data point is lower than the mean by $\sim$ 2.5 scale heights. The optical transmission spectra of many other Hot Jupiters show an increase of the $\mathrm{R_{P}/R_{S}}$ values towards short wavelengths by about a few scale heights due to scattering (Sing et al., 2016; Mallonn & Wakeford, 2017). Thus, the observations reported here disfavor a typical scattering signature in the transmission spectrum of CoRoT-1b. If confirmed by follow-up observations, it might be indicative for a grey absorption by clouds composed of rather large condensate particles (Wakeford & Sing, 2015).

5.3 WASP-12b

The three eclipses of WASP-12b were fitted independently from each other, using as fitting strategy the one described in previous sections. For the scaling factor of the primary transit we allowed both positive (physical) and negative (nonphysical) values. To fit the scaling factor we considered uniform priors corresponding to a reasonable secondary eclipse depth of $\pm$3 ppt. For the transit parameters we adopted the values recently reported by Maciejewski et al. (2018), particularly for an orbital period that allows for the observed orbital decay. The mid-eclipse times for our observed secondary eclipses, $\mathrm{T_{o,SE}}$, were computed from a quadratic ephemeris:

[TABLE]

E corresponds to the epoch of the observation with respect to the mid-transit time of reference, $\mathrm{T_{o}}$. The values of $\mathrm{T_{o}}$, P and $\mathrm{\frac{dP}{dE}}$ can be found in Table 6, along with other relevant transit parameters fully adopted in this work. In Table 7 we report the best-fit secondary eclipse depths for the three dates, along with other information that reflects the quality of our data and fit. The left panels of Figure 9 shows our resulting secondary eclipse light curves for the mentioned nights, and the right panels show our $\chi^{2}$ maps for each one of them, compared to the MCMC best-fit values and errors at a 2-$\sigma$ level. In all cases, the detrending parameters that were always favoured were the x,y centroid positions of the stars and the airmass. The first set of parameters does not come as a surprise, as the telescope was passing very close to the zenith, and its there where the stability of the tracking is not optimal. Because of this, a $\sim$20 minute gap is clearly seen during the first and second nights. This data gap is mostly produced by our 5-$\sigma$ clipping algorithm that takes place during the construction of the raw light curves, as these data points naturally have large photometric uncertainties. Originally, some sparse data points were left within the gap, but we noticed that including them in the fitting procedure made the number of detrending parameters reach their maximum to account for this noise. In consequence, to minimize the number of detrending components we discarded the few points left inside the gap, and carried out our usual fitting procedure with the minimum possible detrending components. Finally, extinction of light crossing our atmosphere is an effect that is inherent from the diffusers, and thus is found in our detrending components as well.

WASP-12 is a triple star system with a separation of only 1″between the primary component and its low-mass companions (Bechter et al., 2014). In the $V$ band, the flux contribution of both companions amounts to about 0.9% (Crossfield et al., 2012) and is fully included in our photometric apertures. The measured eclipse depth of Table 7 are diluted by this amount and need to be corrected by 0.9% upwards. However, we report the uncorrected, diluted values because the correction is one order of magnitude smaller than the given uncertainties.

The depth of the first eclipse measurement deviates by about 5 $\sigma$ from the depth of the second and third event. The latter are in agreement at the 2 $\sigma$ level. Observations of the secondary eclipse of the same target in an overlapping wavelength regime have been published by Bell et al. (2017). The $V$ band corresponds to their two reddest wavelength bins, which resulted in a depth of 0.06 $\pm$ 0.1 ppt when averaged. This value is in good agreement to the result of our second event, however, our first and third event results deviate by 6 $\sigma$ and 2.5 $\sigma$, respectively.

Potentially, systematic noise might mimic the deep ingress and egress features of our first eclipse event. However, we find no abnormal behaviour in any instrumental parameters at this time. Also the very close match of the ingress/egress timings to the predicted times makes an instrumental or systematic origin of these features less likely. The literature offers additional hints for a time-variability of the eclipse depth of WASP-12b. Observations obtained in the Sloan $z^{\prime}$ band at 0.9 $\mu$m by López-Morales et al. (2010) and Föhring et al. (2013) disagreed at the 2.4 $\sigma$ level. Very recently, Hooton et al. (2019) measured two eclipses of WASP-12b in the Sloan i’ band that differed by $\sim$ 3 $\sigma$. Also for another very hot Jupiter, WASP-19b, discrepant results have been published by Burton et al. (2012) and Lendl et al. (2013) that might potentially be explained by variability in time.

We assume that the origin of the different eclipse depths of this work is a variability in time, and want to briefly discuss potential sources. The depth of a secondary eclipse is the sum of light reflected by the planet towards the observer and thermally emitted light from the planet. We use Equation 4 and 5 of Alonso (2018) for a rough estimation of the potential time variability of these two components. The thermal radiation of planet and host star is approximated by black body radiation. For the involved stellar and planetary parameters, we adopt the values of Collins et al. (2017). If we keep the thermal component fixed to a planetary black body radiation of 2900 K (Stevenson et al., 2014), our secondary eclipse values can be explained by a range of the geometric albedo from zero to $\sim$ 0.75. Reflective clouds formed by large-sized condensates might cause a geometric albedo of up to 0.4 (Sudarsky et al., 2000), and we are not aware of any physical mechanisms causing a higher albedo in the dayside atmosphere of WASP-12b, thus a geometric albedo of 0.75 appears unlikely.

If the variability of the eclipse depth is purely explained by an increase in thermal radiation, a planetary temperature of about 4000 K is needed. However, such an increase in planetary temperature by 1000 K would result in even more pronounced depth variations at NIR wavelengths. The rough agreement of all existent NIR secondary eclipse data of WASP-12b, taken at different epochs, to a black body temperature of about 3000 K (e.g. Cowan et al., 2012; Stevenson et al., 2014), makes a significant time variability of the photospheric temperature unlikely.

Could both components act together to produce the large eclipse depth of our first measurement? If we assume the temporary formation of a reflective cloud deck causing a geometric albedo of 0.4, our simple estimation results in an additional need of an increased black body temperature to $\sim$ 3600 K. However, a spontaneous cloud formation would intuitively be associated with a decrease in temperature instead of an increase, therefore it is unclear under which circumstances the reflective light component and the thermal light component can be enhanced at the same time. Additionally, we note that in the dayside of WASP-12b no cloud formation is expected because the very high temperatures do not allow for particle condensation (Wakeford et al., 2017).

The WASP-12 system is known to contain material eroded and blown off from the planetary atmosphere by the extreme stellar irradiation (Fossati et al., 2013). We can only speculate if potentially the variable eclipse depth is not caused by the planetary dayside atmosphere, but by an in-homogeneous flow of escaping material. This material might form temporary clumps near the planet, which scatter a fraction of the star light towards the observer. In this scenario, the deep secondary eclipse would rather be caused by an occultation of the escaping material than by the occultation of the planet.

6 Conclusions

In this work we present the characterization of two brand new engineered diffusers (EDs) mounted at the 2.5 meter Nordic Optical Telescope. The diffusers have two slightly different diffusing angles, spreading the light of the stars differently over the CCD. They can be placed in two different filter wheels and can work simultaneously to several photometric filters placed on a third wheel. To characterize the stability and reliability of the EDs we observed two photometric standard stars. We found the throughput of the diffusers to be superior to 90% in the visible and for redder wavelengths, dropping down to $\sim$75% in the ultraviolet, as expected from their construction and design. Both the core and the wings of the diffused stellar images are quite stable as observations evolve, without significant changes that correlate with seeing or airmass. Furthermore, we observed three exoplanetary systems, namely TrES-3b, CoRoT-1b and WASP-12b. While the first two were observed during one primary transit each, the latter was followed-up during three secondary eclipse events. Since the EDs can be combined to regular photometric filters, our science frames were taken using the Bessel $B$, $V$, and $R$ filters. The achieved photometric precision of 0.5 ppt for a 25 second exposure in TrES-3b, 0.65 ppt for 162 seconds for CoRoT-1b, and between 0.52 and 0.59 ppt for less than one minute cadence for WASP-12b, clearly shows the power of engineered diffusers to achieve high precision photometric data. In addition to this, we present a new planet-to-star radii ratio for CoRoT-1b, to our knowledge the bluest data point taken to date, and we observe a significant variability in the eclipse depth of WASP-12b, as pointed out by other authors before in other optical wavelengths. Within our collected data we observe a deviation within eclipse depths of about 5 $\sigma$. From previous observations in a similar wavelength range, our first and third eclipse depths deviate by 6 $\sigma$ and 2.5 $\sigma$. The reasons causing this variability might be purely speculative.

Acknowledgements.

CvE acknowledges funding for the Stellar Astrophysics Centre, provided by The Danish National Research Foundation (Grant DNRF106), and support from the European Social Fund via the Lithuanian Science Council grant No. 09.3.3-LMT-K-712-01-0103. This work made use of PyAstronomy444https://github.com/sczesla/PyAstronomy. The data presented here were obtained with ALFOSC, which is provided by the Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the University of Copenhagen and NOTSA. GKS acknowledges support by NASA Headquarters under the NASA Earth and Space Science Fellowship Program - Grant NNX16AO28H. JF and SD acknowledge funding from the German Research Foundation (DFG) through grant DR 281/30-1 and DR 281/32-1.