# The Optical Phase Curves of CoRoT-1 b

Andrew Li  No Academic Affiliation111The author is currently a high school
student.
contact email -<EMAIL_ADDRESS>
###### Abstract

Of the three space telescopes launched so far to survey transiting extrasolar
planets, CoRoT is unique in that it was the only one with spectral resolution,
allowing for an extraordinary opportunity to study the reflective properties
of exoplanets at different wavelengths. In this work, I present a systematic
lightcurve analysis of the white-light and chromatic CoRoT lightcurves of
CoRoT-1 in order to search for the secondary eclipse and orbital phase
variation of the transiting extrasolar planet CoRoT-1 b, as well at search for
any chromatic difference in the aforementioned effects. I manage to detect a
significant secondary eclipse in the white lightcurve, and detect the eclipse
marginally in all three of the color channels. However I am only able to
significantly detect the planetary phase variation in the red channel
lightcurve. The retrieved secondary eclipse depth is higher in the blue and
green channels compared to the white and red, suggesting that CoRoT-1 b has a
higher geometric albedo at shorter wavelengths. I also attempt to detect the
secondary eclipse using TESS, but show that the available volume and precision
of the data is not high enough to allow detection of the secondary eclipse.

††facilities: Exoplanet Archive, CoRoT††software: astropy (Astropy
Collaboration et al., 2013, 2018), wotan (Hippke et al. (2019)), ACROM
(Dorado-Daza, 2018)

##  1 Introduction

Since the commencement of observations in February 2006, the Convection,
Rotation, and planetary Transits satellite (CoRoT) continuously monitored
several fields of the night sky for six years in order to search for
transiting extrasolar planets, becoming the first space telescope of its kind
launched for such a purpose. One of the advantages of such continuous
monitoring of transiting exoplanet systems is that it allows for high
precision photometry using relatively small optical components, with the high
optical precision being achieved by the large volume and time baseline of the
data. As a result, the study of the optical phase curves of transiting hot
Jupiter-type exoplanetary systems, which require a high photometric precision,
has benefited enormously from the advent of CoRoT, and its successor survey
telescopes, Kepler and TESS. Specifically, observations of the planetary
secondary eclipse and phase curve are only possible with the photometric
precision afforded by space telescopes.

Phase curves of exoplanets usually are studied at infrared wavelengths, both
because the phase curve has a higher amplitude at infrared wavelengths, and
the fact that observing the infrared phase curves of exoplanets allows for the
modelling of the planet’s atmospheric structure and composition. However, at
optical wavelengths, the phase curve instead provides information about the
reflective properties of the exoplanet, including the geometric albedo of the
planet and the homogeneity or existence of clouds in the planet atmosphere
(Shporer & Hu (2015)).

However, one measurement at one wavelength is usually not enough to model or
constrain the reflective and thermal properties of an exoplanet. Because of
this, numerous telescopes from both space and the ground have been used to
study the secondary eclipses and phase curves of hot Jupiters in the infrared,
allowing for the detailed modelling of exoplanet thermal emission spectra.

The same cannot be said for visible-light measurements, where detailed multi-
wavelength secondary eclipse observations of hot Jupiters in the optical are
few and far between, especially at wavelengths shorter than 500 nm. At the
moment, detailed secondary eclipse observations of exoplanets shortward of 500
nm have only been conducted for nine222The author is aware of four additional
exoplanets with unpublished secondary eclipse observations shortward of 500
nm, including a previous analysis of CoRoT-1 b. exoplanets, with only one of
these attempts resulting in a successful detection.

Table 1: Previous Secondary Eclipse Measurements with $\lambda<500nm$

Planet |  $\delta_{occ}$ (ppm) | Reference
---|---|---
CoRoT-2 b | - | Snellen et al. (2010)
HD 189733 b | $126_{-37}^{+36}$ | Evans et al. (2013)
KELT-9 b | $-71\pm 84$ | Hooton et al. (2018)
TrES-3 b | $80\pm 90$ | Mallonn et al. (2022)
WASP-12 b | $59\pm 134$ | Sing et al. (2019)
WASP-19 b | $10\pm 280$ | Burton et al. (2012)
WASP-43 b | $<860(3\sigma)$ | Chen et al. (2014b)
| $-70\pm 110$ | Mallonn et al. (2022)
WASP-46 b | $490\pm 300$ | Chen et al. (2014a)
WASP-103 b | $200\pm 160$ | Mallonn et al. (2022)

Note. — Snellen et al. (2010) did not detect the secondary eclipse of CoRoT-2
b in the blue CoRoT channel, but did not provide an upper limit value.

CoRoT-1 b is an exoplanet in a unique position ripe for chromatic analysis of
its secondary eclipses and planetary phase curves. It possesses an orbital
configuration that makes the expected eclipse depth relatively large, and the
continuous white-light and chromatic monitoring of the host star by CoRoT
means that there is ample data to make a detection of the eclipse and phase
curve. Indeed, previous attempts to detect the secondary eclipse have proven
that the eclipse depth and phase curve is large and easily detectable with the
CoRoT data (Alonso et al. (2009), Snellen et al. (2009)). Taking all of the
factors that make this planet conducive to study of its eclipse and phase
curve, I endeavor to try and detect these effects in the white-light and
chromatic CoRoT lightcurves.

In addition to the CoRoT lightcurves of CoRoT-1, CoRoT-1 and its transiting
planet have also been observed by the Transiting Exoplanet Survey Satellite
(TESS). In an effort to add another data point to the chromatic eclipse depths
of CoRoT-1 b, I also attempt to detect the eclipse in TESS data.

##  2 Methods

I use the public N2 CoRoT data for CoRoT-1 in .tbl format, as provided by the
NASA Exoplanet Archive. The white-light and chromatic lightcurves are then
extracted from the lightcurve file. To allow a uniform analysis of all of the
lightcurve, the section with 32s candence was split off from the previous
section with 512s cadence, and binned into bins of 16 points to match the rest
of the data. The resulting lightcurve consists of 7117 datapoints spread
across 54.72 days and 36.26 planetary orbits.

The first and most obvious systematic effect in the CoRoT lightcurves are the
numerous flux discontinuities, or ”jumps”, which are caused by the impact of
high-energy particles onto the spacecraft’s CCD detectors. I designed a fairly
basic but effective way of removing these discontinuities from the lightcurve
by taking the average flux of the lightcurve 0.5 days before and after the
flux jump and subtracting the differences in the means from the part of the
lightcurve that occurs after the jump. The resulting lightcurves were much
smoother than the originals.

Table 2: Adopted Planetary Parameters

Parameter | Value | Reference
---|---|---
$a/R_{*}$ | 4.751 | von Essen et al. (2019)
$R_{P}/R_{*}$ | 0.1419 | von Essen et al. (2019)
$P$ | 1.508968772 d | Ivshina & Winn (2022)
$T_{*}$ | 6355 K | Stassun et al. (2019)
$T_{P,day}$ | 2279 K | Deming et al. (2023)

Note. — The cited $T_{P,day}$ value is the average of the brightness
temperatures in the 3.6 and 4.5 micron bands.

For the TESS lightcurves, I use the SPOC 120s cadence Sector 6 and 33
lightcurves as provided by the download function of the Python package
lightkurve (Lightkurve Collaboration et al. (2018)). I extracted the PDC
lightcurve from these files, as compared to the SAP lightcurve, the
instrumental effects impacting the SAP lightcurve have been removed in the PDC
lightcurve. The lightcurves from each of the two sectors were then joined and
analyzed together. Because the time gap between the two sectors’ data is
large, I do not expect this action to significantly affect the detrending
process or the end results of the analysis. The resulting lightcurve contains
30298 data points, is near continuous over 47.61 days, and covers 31.55
planetary orbits, with a large gap in time between the Sector 6 and Sector 33
data.

The TESS and CoRoT lightcurves were then detrended using the Python package
wotan (Hippke et al. (2019)). In the case of the CoRoT data, I used a biweight
filter with a window size equal to that of the planetary orbital period. This
is chosen specifically because, in this way, the detrending algorithm will
detrend out all long term trends across longer timescales than the planetary
orbital period while keeping any possible planetary phase variation in the
detrended lightcurve. In the case of the TESS data, I initially also processed
the TESS data in the same manner, but discovered that, because of the
faintness of the host star, correlated noise dominated the resulting
lightcurve and its analysis. As a result, I decided to utilize a smaller
window of about 0.7 days. This was too small to detrend over the planetary
transits properly, so the transits in the TESS lightcurves were removed before
the detrending process was applied. This means that the phase curve will be
detrended out of the lightcurve, and that I will no longer be able to search
for it in the TESS data. However, the secondary eclipse should still be
present, since it is a high frequency feature on a scale much smaller than the
detrending window.

The resulting TESS and CoRoT lightcurves were then $\sigma$-clipped to
$3\sigma$, which had the effect of clipping out not only any possible
outliers, but also the planetary transits in the CoRoT lightcurves as well.
After detrending was performed, it became obvious that several low-level flux
modulations and residual incompletely removed discontinuities remained. I
tested a variety of methods to remove these effects, but, in accordance with
the methods of previous analyses such as Snellen et al. (2009), I found that
the best way to account for these effects was to just remove them from the
lightcurve altogether. To do this, I constructed a ”blacklist” of ranges of
data points, determined by a visual inspection of the data, and then
automatically removed these sections from the sigma-clipped data. In the end,
the red and green channel CoRoT lightcurves, as well as the TESS lightcurve,
did not have a blacklist, meaning that I did not find any portions of those
respective lightcurves that warranted removal.

Table 3: CoRoT-1 b Lightcurve Blacklist Color | Interval | Reason for Removal
---|---|---
WHITE | {2594.0, 2596.2} | flux jump
WHITE | {2598.5, 2599.0} | flux jump
WHITE | {2622.0, 2623.0} | undetrended flux variation
WHITE | {2635.5, 2637.7} | flux jump
WHITE | {2640.0, 2640.8} | flux jump
WHITE | {2642.5, 2644.4} | flux jump
BLUE | {2594.0, 2595.0} | flux jump
BLUE | {2598.5, 2599.0} | undetrended flux variation
BLUE | {2637.0, 2637.7} | flux jump
BLUE | {2642.5, 2644.4} | flux jump

After the previously mentioned processing steps, the TESS and CoRoT white
lightcurves are ready to be analyzed for the presence of planetary phase
variations and secondary eclipses. However, the CoRoT color lightcurves need
additional processing. This is because the color channel lightcurves, unlike
the white lightcurve, are not corrected for systematics caused by pointing
drifts of the telescope. Two main systematic effects are visible in the color
channel lightcurves: a 103-min high-amplitude variation cognate with the
satellite orbital period, and a 24-hour low-amplitude sinusoidal variation. I
remove these variations in largely the same way that is done by Snellen et al.
(2009). A sinusoidal function with a period of 24 hours was fitted to the
data, with the resulting lightcurve being the residuals of the fit. Then, the
data was then phase-folded to the 103-min orbital period of the satellite, and
then boxcar-smoothed with a window of 8 datapoints. The resulting curve was
then used to detrend the entire lightcurve so that no systematic variation
remained. I should note that Snellen et al. (2009) find evidence of a $\sim$10
day variation in the data, which they attribute to spot variability and/or
stellar rotation. I do not attempt to find and remove this variability,
because such a large-period variation is likely to have already been removed
by the detrending algorithm, and the estimated effect of such a low-amplitude
variation on the final values is ostensibly small.

After all detrending was accomplished, the lightcurves were then phase-folded
to the planetary orbital period. I then tested fitting two different models to
the CoRoT lightcurves - one a pure phase curve and eclipse model, and the
other including the eclipse and all three phase curve effects, reflection and
thermal emission, ellipsoidal variation, and Doppler beaming. Note that I do
not expect to detect the ellipsoidal variation and Doppler beaming in the
lightcurves, since the expected amplitude is quite small compared to the
photometric precision of the lightcurves, but instead use the ellipsoidal
variation and doppler beaming amplitudes as ”correlated noise detrending
functions”. The rationale behind this is that, due to the nature of the color
channel lightcurves, it is likely that significant correlated noise remains in
the lightcurves. The ”ellipsoidal variation” and ”Doppler beaming” curves
would therefore be able to eliminate the two sources of noise most likely to
influence the phase curve amplitude: noise at phases 0.25 and 0.75, and noise
causing the first half of the phase curve to be higher than the second half.
After the fitting was complete, a visual inspection of the fit was done, and
based on this visual inspection I adopted the ”reflection-only” model for the
white lightcurve and the red and green color lightcurves. For the blue-channel
lightcurve, the ”3-effect” model provided a better fit, so I adopted this
model for the blue channel lightcurve instead. In the case of the TESS
lightcurve, where I do not expect any contribution from a putative phase curve
signal due to the nature of the lightcurve detrending, I simply adopted an
eclipse-only fit.

The fitting was done using a DLS (damped least-squares) method, and the error
estimation was done using a residual-shuffling bootstrap method similar to the
one used in Alonso et al. (2009). I should note that such a method does not
account for additional uncertainty due to correlated noise, and as a result I
expect that my derived error bars are marginally underestimated. Ideally,
analysis of this data should be done using an MCMC (Markov-chain Monte Carlo)
parameter and uncertainty estimation method, which is more robust to different
sources of uncertainty than the types of bootstrap methods described here, but
unfortunately, due to the limited computational resources of the author, I am
unable to perform an MCMC analysis at this time. Despite the limitations of my
analysis, a visual inspection of the fitted model suggest that the analysis is
robust.

##  3 Results

From my analysis, I clearly detect the secondary eclipse in the white
(4.44$\sigma$) channel data, and marginally detect the eclipse in the three
color channels red (2.72$\sigma$), green (2.99$\sigma$), and blue
(2.44$\sigma$). The eclipse is not detected in the TESS data, and I am only
able to set a $1\sigma$ upper limit of 223 ppm for a possible eclipse depth.
As for the planetary phase variation, I clearly detect the planetary phase
variation only in the red channel (3.01$\sigma$), with only marginal
detections in the white (2.50$\sigma$), green (1.23$\sigma$), and blue
(2.41$\sigma$) channels. As mentioned before, I do not expect to detect the
planetary phase variation in the TESS lightcurve due to the nature of the
detrending algorithm. According to Parviainen et al. (2013), the flux
contamination in the CoRoT lightcurves is negligible, and the TESS PDC
lightcurves come pre-adjusted for contamination, so no further correction was
required.

To verify that the detected eclipse signal is real, I shifted the eclipse
model in units 0.01 in phase between phases 0.25 and 0.75, and fitted each
model with its own corresponding eclipse depth. The resulting array of values
show a spike in the eclipse depth at phase 0.5 and a lack of periodic
variability elsewhere, suggesting that my detection of the eclipse depth is
real and not due to correlated noise or undetrended systematics.

Table 4: CoRoT-1 b Phase Curve Fits Color |  $\delta_{occ}$ (ppm) |  $A_{Refl}$ (ppm)
---|---|---
WHITE | 192 ±43 | 56 ±22
RED | 147 ±54 | 82 ±27
GREEN | 334 ±112 | 58 ±47
BLUE | 323 ±132 | 110 ±46
TESS |  $<223$ ($1\sigma$) |
Figure 1: CoRoT white-light phase curve of CoRoT-1 b.  Figure 2: CoRoT red
channel phase curve of CoRoT-1 b.  Figure 3: CoRoT green channel phase curve
of CoRoT-1 b.  Figure 4: CoRoT blue channel phase curve of CoRoT-1 b.  Figure
5: TESS white-light phase curve of CoRoT-1 b.

##  4 Discussion

###  4.1 Significance of Results

For each of the four CoRoT color channels, I measure the significance of the
differences between the phase curve amplitude and the secondary eclipse depth
(or in other terms, the nightside flux) to be 1.30$\sigma$ (white),
0.23$\sigma$ (red), 1.49$\sigma$ (green), and 0.64$\sigma$ (blue). The
apparently significant detection of the nightside flux in the white and green
channels is probably not indicative of real nightside emission, as such
emission would not be detectable at the wavelengths of the white and the green
channels. Instead, the difference probably indicates that the apparent phase
curve signal has been partially masked by unmodeled systematics and other
effects caused by pointing drifts of the CoRoT spacecraft, and that these
values of the phase curve amplitudes should not be taken at their face value.
Thus, I can only confidently conclude that I have detected the phase variation
in the red channel, and marginally in the blue channel, while in the white
lightcurve I probably have also detected the phase variation of the planet,
but that I cannot determine the true value of the phase amplitude due to
residual systematic effects. In the case of the green channel, any possible
phase variation has been completely wiped out by the noise in the data.

My derived white-light eclipse depth is consistent with the value found by
Alonso et al. (2009), while my red channel eclipse depth is marginally larger
than the value found by Snellen et al. (2009), but is still somewhat
consistent. I attribute these differences to the difference in data
processing.

###  4.2 Solving the CoRoT Color Channels

Calculations of the wavelength ranges of the channels is required to place
these measurements into their proper context. The CoRoT color channels do not
correspond to any conventional photometric system, and are actually different
for each star. This is because the color channels are formed by a small bi-
prism that is placed in front of the Planet-Finder channel, which disperses
the light slightly. A photometric mask is then applied to the dispersed flux,
and based on the relative intensities of the photons and the position of the
star on the CCD, the three color channels are created. While the exact ratio
of red to green to blue flux is nontrivial to calculate, Rabello Soares et al.
(2022) use empirically calculated values to approximate the ”average”
wavelength properties of the color channels, which I use for the purposes of
this analysis. To calculate the wavelength limits of the color channels, I
multiply a calculated Planck curve of the star by the CoRoT CCD response
function from Auvergne et al. (2009), then took the wavelength cutoff with the
bluest 22% of the flux to be the blue channel cutoff, and the wavelength
cutoff with the reddest 63% of the flux to be the red channel cutoff. From
these calculations, I calculate the blue wavelength cutoff to be 502 nm, and
the red wavelength cutoff to be 561 nm, and the corresponding effective
wavelengths of the white, blue, green, and red channels to be 623 nm, 456 nm,
533 nm, and 703 nm. This is in agreement with Snellen et al. (2009), who find
an effective wavelength of the red channel for CoRoT-1 of 710 nm and a cutoff
of 560 nm.

It would also be appropriate to conduct an analysis of the effective
wavelength of the TESS lightcurve, but given that I do not detect the
secondary eclipse in the TESS data, and the fact that most other publications
have not adjusted the TESS bandpass to account for the stellar spectrum, I
decide not to proceed for consistency with other literature analyses of TESS
eclipses. In any case, the potential impact on the geometric albedo upper
limit due to this decision is expected to be small due to the relatively wide
bandpass of the TESS cameras.

Figure 6: CoRoT white-light and chromatic secondary eclipse depths.

###  4.3 A Wavelength-Dependent Albedo of CoRoT-1 b?

Provided that the dayside temperature of the planet, determined from infrared
observations of the secondary eclipse, is known, one can calculate the
geometric albedo (corrected for the presence of thermally emitted light) of
the exoplanet under the assumption that the star and the planetary dayside
radiate like black bodies using the following equation:

$A_{g}=\delta_{occ}(\frac{a}{R_{P}})^{2}-\frac{B(\lambda,T_{P,day})}{B(\lambda,T_{*})}(\frac{a}{R_{*}})^{2}$
(1)

I take the derived dayside brightness temperatures from Deming et al. (2023)
and average the 3.6 micron and 4.5 micron temperatures to arrive at an
approximate estimate of the planetary dayside temperature to use in
calculating the albedo. Thus, the thermally corrected albedos of CoRoT-1 b can
be calculated.

Table 5: Geometric Albedos Color | $A_{g}$ |  $3\sigma$ upper limit
---|---|---
WHITE | 0.182 ±0.048 | -
RED | 0.097 ±0.061 | $<0.280$
GREEN | 0.363 ±0.125 | $<0.738$
BLUE | 0.359 ±0.148 | $<0.803$
TESS |  $<0.155$ ($1\sigma$) | $<0.655$

The geometric albedo of CoRoT-1 b appears to be elevated in the green and blue
channels, in stark contrast to most other hot Jupiter-type planets, which
typically have very low albedos (Esteves et al. (2015)). This suggests the
presence of reflective clouds on the dayside of CoRoT-1 b. In addition to the
elevated geometric albedo, the albedo in the green and blue channels is higher
than the albedo in the red channel, as well as the overall white channel
albedo (which predictably is in between the albedos of the red channel and the
green and blue channels), and the $1\sigma$ upper limit of the albedo in the
TESS bandpass. This suggests some chromatic dependence in the geometric albedo
of CoRoT-1 b, possibly due to Rayleigh scattering, or the inclusion of
pigmenting and/or scattering compounds in the clouds. This chromatic
difference in the albedo means that, to the human eye, CoRoT-1 b would likely
appear to be bluish in color. However, I should caution that the significance
of the eclipse depths and albedos I derive here mean that a cloudless and
unreflective atmosphere with a flat reflection spectrum is plausible, since
such a model would be consistent with my results to within $3\sigma$. The
conclusions I draw here are only tentative, and should not be taken as a
confirmation of any phenomena in the atmosphere of CoRoT-1 b.

##  5 Conclusion

In this work, I searched and analyzed white-light and chromatic CoRoT data, as
well as TESS data, in an attempt to find the phase variations and secondary
eclipse of CoRoT-1 b. I detected the secondary eclipse in each of the CoRoT
lightcurves to better than $2.4\sigma$, while in the case of the phase
variations I can only confidently conclude that I have detected the phase
variation in the red channel, and marginally in the blue channel, while in the
white and green channels, I probably have also detected the phase variation of
the planet, but that I cannot determine the true value of the phase amplitude
due to residual systematic effects. In the case of the TESS data, I am unable
to detect the secondary eclipse, while my choice of processing method for the
TESS lightcurve leaves me unable to study the presence of any possible phase
variations. I find tentative evidence that the geometric albedo of CoRoT-1 b
appears to be elevated and chromatically dependent, with the albedo being
higher in the green and blue CoRoT data than in the white CoRoT data, red
CoRoT data, and the TESS data, suggesting that the atmosphere of CoRoT-1 b may
contain a reflective cloud deck and/or exhibits Rayleigh scattering. Given the
marginal significance of my fitted values, I encourage future study of the
CoRoT-1 system to confirm or refute these tentative results.

This research has made use of the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
This research has also made use of Lightkurve, a Python package for Kepler and
TESS data analysis (Lightkurve Collaboration et al. (2018)). I would also like
to thank Hippke et. al. for the development of the wotan Python package, and
D.J. Dorado-Daza for the ACROM set of Python scripts for processing CoRoT
data. Many of the custom Python analysis scripts used in this research were
derived from these two packages, and this research would not have been
possible without each of them.

Figure 7: Best fit secondary eclipse model and residuals for the CoRoT white-
light and color channel lightcurves.

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