# A Ground-based Optical Transmission Spectrum of WASP-6b

^{1}, University of Exeter

^{2}, California Institute of Technology

^{3}, University of Colorado Boulder

^{4}, Sloan Fellows

^{5}, Princeton University

^{6}, University of California, Santa Cruz

^{7}, Harvard University

^{8}, Keele University

^{9}, Swiss National Science Foundation

^{10}, Massachusetts Institute of Technology

^{11}

## Summary (6 min read)

### 1. INTRODUCTION

- Due to their fortuitous geometry, transiting exoplanets allow the determination of physical properties that are inaccessible or hard to reach for non-transiting systems.
- One of the most exciting possibilities enabled by the transiting geometry is to measure atmospheric properties of exoplanets without the need to resolve them from their parent star through the technique of transmission spectroscopy.
- 13 Fellow of the Swiss National Science Foundation.
- A recent study of all the transmission spectra available for HD 189733, spanning the range from 0.32 to 24 μm, points to a spectrum dominated by Rayleigh scattering over the visible and near-infrared range, with the only detected feature being a narrow resonance line of Na (Pont et al. 2013).
- All of these studies have used highresolution spectroscopy or narrowband photometry to specifically target resonant lines of alkali elements.

### 2. OBSERVATIONS

- The transmission spectrum of WASP-6b was obtained performing multi-object differential spectrophotometry with the Inamori-Magellan Areal Camera and Spectrograph (IMACS; Dressler et al. 2011) mounted on the 6.5 m Baade telescope at Las Campanas Observatory.
- The seven comparison stars the authors used are listed in Table 1.
- Observing such masks is necessary in order to produce well-defined lines that are then used to define the wavelength solution.
- The extracted spectra of WASP-6 and the seven comparison stars the authors used are shown for a typical exposure in Figure 1.
- Besides the variation due to varying airmass (and the transit for WASP-6), there were periods with strongly varying levels of transparency concentrated in the period of time 0–2 hr after mid-transit.

### 3.1. Background and Sky Subtraction

- After subtracting the median value of the overscan region to every image, an initial trace of each spectrum was obtained by calculating the centroid of each row, which are perpendicular to the dispersion direction.
- Each row was then divided into three regions: a central region, which contains the bulk of the light of the star; a middle, on-slit region, which is dominated by sky continuum and line emission; and an outer, out-of-slit region, which contains a smooth background outside the slit arising from, e.g., scattered light.
- The middle and outer regions have components on each side of the spectrum.
- The outermost region was used to determine a smooth background that varies slowly along the dispersion direction.
- The estimated ci (one per row) was then subtracted from the central and middle regions in order to obtain a spectrum where only the stellar contribution remains.

### 3.2. Fine Tracing and Spectrum Extraction

- The background- and sky-subtracted spectrum was traced by an algorithm that cross-correlates each slice perpendicular to the wavelength direction with a Gaussian in order to find the spectral trace.
- The centers of the trace were then fitted robustly with a fourth-order polynomial.
- With the trace in hand, the spectrum was extracted by using a simple extraction procedure, i.e., summing the flux on each row ±15 pixels from the trace position at that row.
- The authors also tried optimal extraction (Marsh 1989), but it led to additional systematic effects when analyzing the light curves,15 and in any case optimal extraction is not expected to give significant gains over simple extraction at the high signal-to-noise levels they are working with here.
- The authors therefore used the non-flat-fielded spectra in the present work.

### 3.3. Wavelength Calibration

- The extracted spectra were calibrated using NeHeAr lamps taken at the start of the night.
- The wavelength solution was obtained by the following iterative procedure: pixel centers of lines with known wavelengths were obtained by fitting Gaussians to them, and then all the pixel centers, along with the known wavelengths of the lines, were fitted by a sixth-order Chebyshev polynomial.
- The authors checked the absolute deviation of each line from the fit and removed the most deviant one from their sample, repeating the fit without it.
- This process was iterated, removing one line at a time, until an rms of less than 2000 m s−1 was obtained.
- The procedure explained in the preceding paragraph served to wavelength-calibrate the first spectrum of the night closest in time to the NeHeAr lamps.

### 4. MODELING FRAMEWORK

- The largest departure from this idealized model in their observations will be given by systematic effects arising from atmospheric and instrumental effects, which are assumed to act multiplicatively on their signals.
- 15 Optimal extraction assumes that the profile along the wavelength direction is smooth enough to be approximated by a low-order polynomial.
- In particular, the authors found that fringing in the reddest part of the spectra induces fluctuations in the extracted flux with wavelength due to the inadequacy of the smoothness assumption.

### 4.1.1. Estimation of Systematic Effects via Principal Component Analysis of the Comparison Stars

- These perturbation signals have in common that they arise from the same physical and instrumental sources.
- The authors model this by assuming that a given perturbation signal is in fact a linear combination of a set of signals si(t), which represent the different instrumental and atmospheric effects affecting all of their light curves, i.e., Sk(t) = K∑ i=1 αk,isi(t).
- (2) Note that this model for the perturbation signal so far includes the popular linear and polynomial trends (e.g., si(t) = t i).
- A way of obtaining those signals is by assuming that the si(t) are uncorrelated random variables, in which case these signals are easily estimated by performing a principal component analysis (PCA) of the mean-subtracted light curves of the comparison stars.
- As written in Equation (3), the authors cannot separate si(t) from k(t), and in general the principal components will have contributions from both terms.

### 4.1.2. Selecting the Number of Principal Components

- The authors need therefore to determine an optimal number of principal components to describe the perturbation signal, taking into consideration that there is noise present in the light curves of the comparison stars and, thus, some of the principal components obtained are mostly noise.
- The authors will determine the optimal number of components as the minimum number of components that are able to achieve the best predictive power allowed by the maximum set of N components available.
- K-fold cross-validation is a procedure that estimates prediction error, i.e., how well a model predicts out-of-sample data.
- This procedure is repeated for all folds.
- These l principal components, which are a set of light curves {si}li=1, are their estimates of the systematic effects, and the authors use the out-of-transit part of the light curve of WASP-6 as the validation data by fitting it with the {si}li=1.

### 4.2. Joint Parameter Estimation for Transit and Stochastic Components

- In the past sub-sections the authors set up an estimation process for the signal given in Equation (2) using PCA.
- It remains to specify a model for the stochastic signal that the authors have termed noise, i.e., the (t) term in Equation (1).
- Nonetheless, this is of no consequence as the authors just aim to model the residuals after the time series has been modeled with the {si}li=1.
- While the authors still call this term (t) in what follows, one should bear in mind this subtlety.
- In the present work the authors consider three different models: a white-noise model, where the covariance matrix is assumed to be diagonal; a flicker-noise model; and ARMA(p, q) models, where the structure of the covariance is determined via the parameters p and q (see Section 4.2.2 below for the definition of ARMA(p, q) models).

### 4.2.1. Flicker Noise Model

- Flicker noise is known to arise in many astrophysical time series (Press 1978).
- An efficient set of algorithms for its implementation in MCMC algorithms was proposed recently by Carter & Winn (2009).
- The basic idea of this implementation is to assume that the noise is made up of two components: an uncorrelated Gaussian process of constant variance and a correlated Gaussian process that follows this flicker-noise model.
- These two components are parametrized by σw and σr , characterizing the white and correlated noise components, respectively.
- A wavelet transform of the residuals takes the problem into the wavelet basis where flicker noise is nearly uncorrelated, making the problem analytically and computationally more tractable.

### 4.2.2. ARMA Noise Model

- ARMA models have been in use in the statistical literature for a long time with a very broad range of different applications (Brockwell & Davis 1991).
- Scargle 1981; Koen & Lombard 1993), these noise models have not been used so far for transit light curves to the best of their knowledge.
- In order to fit ARMA models to the residuals via an MCMC algorithm, the authors need the likelihood function of the model given that the residuals follow an ARMA(p, q) model.

### 4.2.3. Stochastic Model Selection

- Given the three proposed noise models for the stochastic signal (t), it remains to define which of the three affords a better description of the data, taking into account the trade-off between the complexity of the proposed model and its goodness of fit.
- There are several criteria for model selection; a comprehensive comparison between different criteria has been done recently by Vehtari & Ojanen (2012).
- The main conclusion is that, despite the fact that many model selection criteria have good asymptotic behavior under the constraints that are explicit when deriving them, there is no “perfect model selection” criterion, and there is a need to compare the different methods in the finite-sample case.
- Following this philosophy, the authors compare in this work the results of the AIC (“An Information Criterion”; Akaike 1974), the BIC (“Bayesian Information Criterion”; Schwarz 1978), the DIC (“Deviance Information Criterion”; Spiegelhalter et al. 2002), and the DICA, a modified version of DIC with a proposal for bias correction (Ando 2012).

### 5. LIGHT-CURVE ANALYSIS

- From the initial 10 comparison stars, only seven were used to correct for systematic effects.
- One star was eliminated on the grounds of having significantly less flux than the rest, and the other two due to not having the whole spectral range of interest recorded in the CCD.
- Given the seven comparison stars, the authors applied PCA to the mean-subtracted time series in order to obtain an estimate of the perturbation signals.
- The authors describe now the construction and analysis of the white-light transit light curve and the light curves for 20 wavelength bins.
- 17 ARMA(p, q) models have been considered recently in the modeling of radial velocity data (Tuomi et al. 2013).

### 5.1. White-light Transit Light Curve

- In order to obtain the white-light transit light curve of WASP-6, the authors summed the signal over the wavelength range 4718–8879 Å for the target and the comparison stars.
- The authors adopted priors for the white-light transit analysis are detailed in Table 2.
- The results of the MCMC fits assuming a white Gaussian noise model, an ARMA(2, 2) noise model, and a 1/f noise model for the residuals are shown in Figure 6, and a summary of the values of the information criteria for each of their MCMC fits is shown in Table 3.
- As opposed to deterministic components, the stochastic components cannot just be predicted given the times ti of the observations, as the authors only know the distribution of expected values once they know the parameters ({θ1, θ2, φ1, φ2, σw} for ARMA(2, 2), {σr, σw} for flicker noise, and σw for white Gaussian noise).
- The authors select the model parameters fitted using the 1/f noise model, which are quoted in Table 4, as the best estimates from now on.

### 5.2. WASP-6b Transmission Spectrum

- Priors were the same as the white-light analysis for parameters for μ1, σr , and σw, and the MCMC chains were set up similarly except that a thinning value of 103 was used.
- Boundaries that lie in the pseudo-continuum of the WASP-6 spectrum, as boundaries in steep parts of the spectrum such as spectral lines would in principle maximize redistribution of flux between adjacent bins under the changing seeing conditions that set the spectral resolution in their setup.
- For a given spectral bin, the number of principal components was selected separately because different systematics may be dependent on wavelength, and therefore the number of principal components needed may change.
- Figure 7 shows the baseline-subtracted data along with the best-fit transit model at different wavelengths, and Table 5 tabulates the transit parameters from the MCMC analysis for each wavelength bin.

### 5.3. Limits on the Contribution of Unocculted Stellar Spots

- As pointed out in several works (e.g., Pont et al. 2008; Sing et al. 2011b), stellar spots—both occulted and unocculted during transit—can affect the transmission spectrum.
- The decrease of flux during transit with respect to the out-of-transit flux F0 is given by (ΔF/F0) = k2 (neglecting any emission from the planet).
- The authors used the method described in Maxted et al. (2011) to look for periodic variations due to spots in the light curves of WASP6 from the WASP archive (Pollacco et al. 2006).
- The light curves typically contain ∼4500 observations with a baseline of 200 nights.

### 6. THE TRANSMISSION SPECTRUM: ANALYSIS

- The main feature of the transmission spectrum shown in Figure 8 is a general sloping trend with Rp/R∗ becoming smaller for longer wavelengths.
- The clear atmosphere models fail to give a better match to the spectrum due to the lack in the latter of evidence for the broad features expected around Na and K.
- The AIC for the scattering model assuming Gaussian noise given by the known error bars gives −115.3, while the values for the T = 1000 and T = 1500 clear atmosphere models are −97.2 and −90.9, respectively, providing a very significant preference for the scattering model.
- In order to assess the potential impact of correlations in the wavelength direction, the authors compute the partial autocorrelation function (PACF) for the residuals in the wavelength dimension.

### 7. DISCUSSION AND CONCLUSIONS

- The authors have measured the optical transmission spectrum for WASP-6b in the range ≈480–860 nm via differential spectrophotometry using seven comparison stars with IMACS on Magellan.
- The authors fit their transmission spectrum with three different models: two clear transmission spectra from Fortney et al. (2010) and a spectrum caused by pure scattering.
- The potentially prominent role of condensates or hazes in determining the transmission spectra of exoplanets has been apparent from the very first measurement (Charbonneau et al. 2002), and their transmission spectrum of WASP-6b is in line with what seems to be a building trend for transmission spectra with muted features in the optical.
- N.E. is supported by CONICYT-PCHA/Doctorado Nacional, and M.R. is supported by FONDECYT postdoctoral fellowship 3120097.

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##### Citations

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^{2}, University of California, Santa Cruz

^{3}, Tennessee State University

^{4}, University of Arizona

^{5}, University of Oxford

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^{7}, University of Colorado Boulder

^{8}, European Southern Observatory

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...The atmosphere of WASP-6b has recently been probed in the optical regime from the ground by Jordán et al. (2013)....

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...Magellan observation did suffer from changing weather conditions during the second half of the transit event as detailed in Jordán et al. (2013)....

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...Recently, Jordán et al. (2013) reported WASP-6b transmission spectrum with the Magellan IMACS instrument....

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...Thus, the differences between the systematics models employed in Jordán et al. (2013) could explain more than ∼ 80% of the observed offset between the Magellan and HST transmission spectra....

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##### References

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### "A Ground-based Optical Transmission..." refers background or methods or result in this paper

...We scale the models that have a surface gravity of g = 10 m s−2 to match the measured surface gravity of WASP-6b (g = 8.71 m s−2; Gillon et al. 2009) by scaling the spectral features from the base level by 10/g....

[...]

...b Obtained from the values cited in Gillon et al. (2009). The variance of the prior covers more than 3σ of their values. c Obtained from the arithmetic mean between the values cited in Gillon et al. (2009) and Dragomir et al....

[...]

...We scale the models that have a surface gravity of g = 10 m/s2 to match the measured surface gravity of WASP-6b (g = 8.71 m/s2, Gillon et al. 2009) by scaling the spectral features from the base level by 10/g....

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...We assume T = 5400 K, ΔT = −500 K, and other stellar parameters to be the closest available in the model grid to those presented in Gillon et al. (2009). The resulting expected maximum value for δk/k given the constraints on the rotational modulation afforded by the WASP-6 discovery light curve is presented in Figure 9....

[...]

...The potentially prominent role of condensates or hazes in determining the transmission spectra of exoplanets has been apparent from the very first measurement (Charbonneau et al. 2002), and our transmission spectrum of WASP-6b is in line with what seems to be building trend for transmission spectra with muted features in the optical....

[...]

^{1}, University of California, Berkeley

^{2}, San Jose State University

^{3}, Las Cumbres Observatory Global Telescope Network

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^{8}, Lowell Observatory

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^{10}, California Institute of Technology

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^{12}, Lawrence Hall of Science

^{13}, Goddard Space Flight Center

^{14}, United States Department of the Navy

^{15}, Carnegie Institution for Science

^{16}, University of Washington

^{17}, University of Hawaii at Hilo

^{18}, University of California, Santa Cruz

^{19}, Massachusetts Institute of Technology

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^{21}, San Diego State University

^{22}, Southern Connecticut State University

^{23}, Planetary Science Institute

^{24}, Yale University

^{25}, Marshall Space Flight Center

^{26}, The Catholic University of America

^{27}, University of Idaho

^{28}, Villanova University

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...This constant flux is multiplied by the transit signal, f(t; θ), which we describe parametrically using the formalism of Mandel & Agol (2002)....

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