Ground-based optical transmission spectroscopy of the small, rocky exoplanet GJ 1132b
Hannah Diamond-Lowe, Zachory Berta-Thompson, David Charbonneau, Eliza M.-R. Kempton
DDraft version May 21, 2018
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GROUND-BASED OPTICAL TRANSMISSION SPECTROSCOPYOF THE SMALL, ROCKY EXOPLANET GJ 1132B
Hannah Diamond-Lowe, Zachory Berta-Thompson, David Charbonneau, and Eliza M.-R. Kempton
3, 4 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA Department of Astrophysical and Planetary Sciences, University of Colorado, 2000 Colorado Ave., Boulder, CO 80305, USA Department of Physics, Grinnell College, 1116 8th Avenue, Grinnell, IA 50112, USA Department of Astronomy, University of Maryland, College Park, MD 20742, USA
ABSTRACTTerrestrial Solar System planets either have high mean molecular weight atmospheres, as with Venus, Mars, andEarth, or no atmosphere at all, as with Mercury. We do not have sufficient observational information to know if this istypical of terrestrial planets or a phenomenon unique to the Solar System. The bulk of atmospheric exoplanet studieshave focused on hot Jupiters and Neptunes, but recent discoveries of small, rocky exoplanets transiting small, nearbystars provide targets that are amenable to atmospheric study. GJ 1132b has a radius of 1.2 R ⊕ and a mass of 1.6 M ⊕ ,and orbits an M-dwarf 12 parsecs away from the Solar System. We present results from five transits of GJ 1132b takenwith the Magellan Clay Telescope and the LDSS3C multi-object spectrograph. We jointly fit our five data sets whendetermining the best-fit transit parameters both for the white light curve and wavelength-binned light curves. We binthe light curves into 20 nm wavelength bands to construct the transmission spectrum. Our results disfavor a clear, 10 × solar metallicity atmosphere at 3.7 σ confidence and a 10% H O, 90% H atmosphere at 3.5 σ confidence. Our data areconsistent with a featureless spectrum, implying that GJ 1132b has a high mean molecular weight atmosphere or noatmosphere at all, though we do not account for the possible presence of aerosols. This result is in agreement with theo-retical work which suggests that a planet of GJ 1132b’s mass and insolation should not be able to retain a H envelope. Keywords: planets and satellites: atmospheres – planets and satellites: terrestrial planets – planetsand satellites: individual: GJ 1132b
Corresponding author: Hannah [email protected] a r X i v : . [ a s t r o - ph . E P ] M a y INTRODUCTIONFour years of transit data from the
Kepler missionshowed us that terrestrial planets are common aroundlow mass stars (Dressing & Charbonneau 2013, 2015;Gaidos et al. 2016). The
Kepler data set also led totheories suggesting that some small planets retain hy-drogen and helium envelopes from formation, compris-ing a small fraction of their total masses (Wolfgang &Lopez 2015). These H/He envelopes are subsequentlysculpted by incident extreme ultra-violet (EUV) and X-ray radiation from the host stars which, in the absenceof a strong planetary magnetic field, drives atmosphericescape (Ehrenreich et al. 2015).M dwarfs have extended pre-main sequence phases(Baraffe et al. 2015) and remain chromospherically ac-tive on long timescales (Newton et al. 2017), so it ispossible that terrestrial planets orbiting M dwarfs havebeen stripped of any primordial atmospheres early on(Lopez & Fortney 2013; Luger & Barnes 2015). Forinstance, the terrestrial planets TRAPPIST-1b and corbiting an ultracool dwarf do not exhibit transmissionspectra consistent with a cloud-free low mean molecu-lar weight atmosphere at the level of ≥ σ confidence(de Wit et al. 2016). TRAPPIST-1d, e, and f also donot exhibit evidence for such atmospheres at the level of ≥ σ confidence (de Wit et al. 2018). We might expecta similar result for other small planets in close-orbitsaround cool stars.In this work we use ground-based observations to in-vestigate the idea that terrestrial exoplanets orbiting Mdwarfs do not possess low mean molecular weight at-mospheres. We focus on the terrestrial exoplanet GJ1132b (1.2 R ⊕ , 1.6 M ⊕ ) orbiting a M4.5V dwarf whichis 12 parsecs away from the Solar System. The radiusand mass of GJ 1132b are consistent with an iron andsilicate composition similar to that of Earth and Venus(Berta-Thompson et al. 2015).The surface gravity and estimated atmospheric tem-perature of GJ 1132b mean that a solar composition,hydrogen-dominated atmosphere might be detectablewith ground-based instrumentation. Though we arelooking for the signature of a low mean-molecular weightatmosphere, hydrogen itself is not a strong absorber,making it a difficult to detect via transmission spec-troscopy. Instead, we assume the atmosphere to be well-mixed and search for tracer molecules like water (H O)or methane (CH ), which have large absorption crosssections in the visible to near-infrared wavelengths.Understanding the nature of terrestrial exoplanet at-mospheres will bolster efforts to constrain planet for-mation and atmospheric evolution, and ultimately in-form our search for biosignatures on other worlds. We do not expect life as we know it to exist on the highlyirradiated surface of GJ 1132b, but understanding theatmospheres of hot, rocky planets will contextualize aneventual search for life on cooler, habitable zone exo-planets.Though our current sample size of terrestrial exoplan-ets is small, it is important to understand them in thecontext of the well-studied Solar System inner plan-ets. Whether a terrestrial exoplanet resembles Earthor Venus or Mercury has vast implications for its for-mation history and life-hosting capabilities. Still moreintriguing is the chance to uncover terrestrial planetswith compositions and characteristics unseen in the So-lar System (e.g., Morley et al. 2017).In Section 2 we describe our observations of GJ 1132bin transit. In Section 3 we describe our customized datareduction pipeline and in Section 4 we describe our cus-tomized data analysis pipeline. We present the resultsof this work in Section 5. We discuss the implications ofground-based investigations of terrestrial planet atmo-spheres in Section 6 and conclude with Section 7. OBSERVATIONSA joint program between Harvard and MIT (PIsDiamond-Lowe and Berta-Thompson, respectively) toobserve transits of GJ 1132b received eight nights onthe Magellan II (Clay) Telescope with the LDSS3C multi-object spectrograph at Las Campanas Observa-tory (Stevenson et al. 2016). Of the eight observingopportunities we observed five transits of GJ 1132b andlost the remaining three nights to clouds and high winds.The details of our observing program are presented inTable 1.GJ 1132 (V = 13.49, K = 8.322) is an M4.5V star(Berta-Thompson et al. 2015). In the 4 (cid:48) field of viewof LDSS3C there are no stars of comparable magnitudeor spectral type, so we opted to simultaneously observenine comparison stars which we later used to removetelluric effects from the GJ 1132 spectrum. Of thesecomparison stars one was brighter than GJ 1132 but itsaturated our detector and we were not able to use it inour analysis.Our LDSS3C masks include slits for GJ 1132 and thenine comparison stars. At the time of our observationsthere was a background star 7.3 arcseconds away formGJ 1132; because GJ 1132 is a high proper motion starthis separation will change over time and future ob-servers of GJ 1132 should account for this. We orientedour mask such that the background star did not contam- Table 1.
ObservationsData Set Date Exposure Time Number of Airmass Seeing a No. [UTC] [s] Exposures Start Middle End [arcsec]1 2016-02-28 06:01:14 – 2016-02-28 09:15:13 12 401 1.109 1.321 1.849 0 .
542 2016-03-04 02:28:11 – 2016-03-04 06:29:56 13 481 1.119 1.055 1.190 0 .
903 2016-03-08 23:50:48 – 2016-03-09 05:41:20 13 694 1.523 1.077 1.136 0 . − .
10* 2016-03-21 – 2016-03-22 − −− −− −− −− − − − .
805 2016-04-21 23:30:33 – 2016-04-22 05:34:25 13 725 1.100 1.113 1.780 0 . − .
01* 2016-05-04 – 2016-05-05 − −− −− −− −− − − − * 2016-05-22 – 2016-05-23 − −− −− −− −− − − − * We were not able to take data on these nights due to poor weather conditions. a On nights 1, 2, and 4 the seeing remained relatively stable throughout the night while on nights 3 and 5 theseeing deteriorated over the course of the observations. inate the dispersed spectrum of GJ 1132. We cut ourslits 10 (cid:48)(cid:48) in width to avoid slit losses and 20 (cid:48)(cid:48) in length toprovide sky background with which to perform our sub-traction (Bean et al. 2010). We also cut identical maskswith 1 (cid:48)(cid:48) wide slits which we used to take wavelengthcalibration arcs during the afternoon prior to each ob-servation.We set the detector binning to 2x2 and the readoutspeed to
Fast (the LDSS3C user manual says this willgive a 13 second read out time but we found it to be 16seconds). We set the gain to
Low , which, along with thereadout speed, gives a gain of 0.6 ADU/electron. Withour observation mask we took biases, darks, quartz flatfields, and a mask image with which to align our starsin the slits during observations. With our 1 (cid:48)(cid:48) mask wetook helium, neon, and argon arcs so that we could de-termine a wavelength solution for each dispersed stellarspectrum. Both during calibrations and observations wekept every detector pixel that we used to perform ouranalysis below 53,000 ADU. As stated in the LDSS3Cuser manual and corroborated by the Las CampanasObservatory instrument specialists, the full pixel wellis 65,536 ADU, but past 53,000 ADU the detector stopscounting photoelectrons linearly.We chose to use the VPH-Red grism which providesa wavelength coverage of 640-1040 nm with a centralwavelength of 850 nm and a linear dispersion of 0.1175nm / pixel (Stevenson et al. 2016). The VPH-Red grismhas a higher resolution than the VPH-all grism, as wellas a higher throughput at redder wavelengths. Usingthe VPH-Red grism allowed us to take longer exposureswithout saturating the detector, while also focusing onthose wavelengths where GJ 1132 is brightest.We took 13 second integrations and achieved a duty cycle of 45%. The VPH-Red grism introduces ordercontamination onto the detector, which we mitigatedwith the OG590 order-blocking filter as advised in theLDSS3C user manual. This filter blocks spectral con-tamination from higher spectral orders but produces in-ternal reflections. (Stevenson et al. (2016) noted thiscontamination and decided against using the OG590 fil-ter.) After inspecting the calibration arc frames duringthe day we decided that the OG590 contamination wasless problematic than the higher-order line contamina-tion. We therefore used the OG590 filter during obser-vation and also while taking our calibration images.We note that our first night of observation (data setnumber 1 in Table 1) differed from the rest for tworeasons. Firstly, we neglected to use the OG590 orderblocking filter, which is why we exposed for 12 secondson this night instead of 13. In spite of this, the or-der contamination was not drastic since GJ 1132 emitsfew photons blue-ward of 700 nm. Secondly, we useda slightly different mask. The first amplifier (C1) ofLDSS3C’s CCD has several columns of bad pixels whichover-lapped with one of our comparison stars. We cuta second, identical mask with the slits slightly shiftedin order to avoid the bad pixels. We did not end upusing this comparison star since the bad pixels near itsaturated and leaked light into its dispersed spectrum.For consistency we exclude this comparison star from allfive data sets when performing the analysis.For all five of our data sets we acquired at least onetransit-durations’s worth of out-of-transit baseline bothbefore and after the transit event with which to estimatethe basline flux and correct for correlated noise in thedata. DATA EXTRACTIONWe transform our raw Magellan/LDSS3C images into1D stellar spectra by running them through our customPython pipeline, mosasaurus . With this pipeline, weperform basic CCD processing on pairs of FITS imagesfrom the two amplifiers on LDSS3C. After subtracting1D biases estimated from the amplifiers’ overscan re-gions, we stitch images together into full frames, usingthe amplifiers’ reported gains to convert from ADU toelectrons. We create median-stacked 2D bias and darkexposures that we subtract from all quartz flat and sci-ence exposures, to remove the baseline level of the read-out electronics and the (very small) dark current accu-mulated during all exposures.To identify and mitigate cosmic ray contamination,we compare each image to the 10 closest images in time.For each pixel, we calculate the median absolute de-viation from the median (MAD), and flag any upwardoutliers that exceed 10 × MAD as cosmic ray hits. Wereplace the flux value in the pixel affected by cosmic rayswith its median value from the immediately surround-ing exposures. We keep track of which pixels have beenmodified in this fashion, so they can be masked out oflater analysis stages if so desired.We cut out a 60 × github.com/zkbt/mosasaurus , v0.0 together in the slit. Having multiple stars in a singleslit is problematic as we would have to combine theirspectra in a large extraction window, which leads to apoor estimate of the sky background and the blendingof different wavelengths from different stars in the sameextracted spectrum. We end up with four comparisonstars for our analysis. Though we use the same compar-ison stars for each night of data the extraction windowsmay vary from night to night. This is because the see-ing conditions on a given night influence the PSF of thestars on the detector. We therefore stand to benefit fromusing different extraction windows for each star for eachdata set.We median-pass filter quartz flat exposures, takenthrough the same wide slits as our science data, by di-viding each pixel by the median of the 20 ×
100 pixelssurrounding it. We then divide each spectrum regionin the time-series by this filtered quartz flat to correctfor the intrinsic pixel-to-pixel inconsistencies of the de-tector. We create a 1D stellar spectrum from each flat-fielded stellar region by summing up the flux in eachcolumn in the spatial direction, accounting for partialpixels at the edges of the extraction aperture. We cre-ate a 1D sky-background spectrum from each flat-fieldedstellar region by fitting a two-degree polynomial to eachcolumn in the spatial direction outside the extractionwindow, and then summing over the column. We thensubtract the sky-background spectrum from the stellarspectrum.We tested an optimal extraction routine as outlined byHorne (1986). We find that this method makes at most a10% improvement in signal-to-noise for the faintest com-parision stars, but does not improve signal-to-noise forGJ 1132 or the brighter comparision stars. Because thefainter comparison stars have a proportionately small in-fluence on the resulting light curve, we use the extractionmethod outlined above and not the optimal extractionroutine.We use the He, Ne, and Ar arcs taken during cali-bration to develop a rough wavelength solution for eachstar. The LDSS3C user manual provides a wavelengthsolution for the VPH-Red grism that gives the pixel posi-tion of prominent features in the He, Ne, and Ar spectra.Using a customized graphical user interface we matchup the features in the provided wavelength solution tothose in each arc, corresponding to our stellar spectrumregions, and create a polynomial wavelength solution foreach star. In practice, this works better for some starsthan others, but it generally lines up the spectra witheach other to within 5 nm.We then choose one exposure of one star as a basisagainst which to cross-correlate all of the exposures of
Figure 1.
Intermediary steps in the mosasaurus open sourceextraction pipeline for multi-object spectrographs. This fig-ure corresponds to a single spectrum of GJ 1132.
Top : Spec-tral trace of GJ 1132 in which the curvature is apparent.Orange lines show the bounds of the extraction aperture.Shaded purple regions are data that we discard when doingour analysis. This includes a region directly beneath the GJ1132 trace that is masking out the spectrum of a faint back-ground star.
Bottom : Extracted 1D raw spectrum of GJ 1132prior to wavelength calibration (light blue line). Also shownis the 1D sky background spectrum which is removed fromthe GJ 1132 spectrum (black line) all the stars in a given data set. We use five prominentfeatures in the spectra in order to perform the cross-correlation: the O doublet (760.5 nm), each line of theCa triplet (849.8, 854.2, and 866.2 nm), and the forestof water lines (about 930-980 nm). We note that the Catriplet is not a telluric feature and so may be risky touse when calibrating the spectra. In this case, all of thestars we observe are in the Sun’s local moving group, andany Doppler shifting of the Ca lines are not detectableat the velocity dispersion of the LDSS3C spectrographand the VPH-Red grism (about 165 km/s/pixel). Giventhe small field-of-view of the instrument we are not con-cerned about the different lines-of-sight to each star.This process reveals that there is both a shifting andstretching of the spectra over the course of the observa-tions. For instance, in data set number 1, the differencebetween the positions of the O doublet and the water Figure 2.
Wavelength-calibrated spectra of GJ 1132 and thefour stars we use to remove telluric features from the GJ 1132spectrum. The vertical dotted lines show the boundaries ofthe wavelength we use to make our transmission spectrum.The comparison stars are all fainter than GJ 1132. By sum-ming the comparison stars’ flux we achieve 71% of GJ 1132’sflux when integrating over the full wavelength bandpass (700-10400 nm). This means that our results are limited by thecombined photon noise of the comparison stars. line forest in the GJ 1132 spectrum increases by a pixelfrom the start of the observation relative to the end.We use this information to apply a second wavelengthsolution for each spectrum in each exposure such thatthey are lined-up with one another in wavelength spaceto within 0.35 nm across all stars and the entire night.This is the final step in achieving 1D spectra which wecan use to make our light curves. DATA ANALYSISTo perform this analysis we constructed the code detrendersaurus . Though it is not generalized fordata sets other than LDSS3C multi-object spectroscopy,the code is fairly modular and some routines may beuseful to others performing similar analyses.4.1. Analyzing transits separately
GJ 1132 is brighter than the four comparison stars(Figure 2, Table 2). We therefore create our light curvesby summing up the flux from the comparison stars anddividing the GJ 1132 spectrum by the summed compar-ison spectrum for each point in the light curve. GJ 1132is still brighter than the summed flux of the four com-parison stars so we are limited by the photon noise ofthe summed comparison star flux. github.com/hdiamondlowe/detrendersaurus, v1.0 Table 2.
Stars used in this workStar RA Dec Flux/GJ 1132 FluxGJ 1132 10:14:50.09 -47:09:17.5 1.0Comp A 10:14:57.51 -47:05:39.9 0.35Comp B 10:14:58.22 -47:09:35.1 0.14Comp C 10:15:05.74 -47:07:43.9 0.11Comp D 10:15:16.26 -47:06:44.3 0.11
Note.
The relative flux column indicates thefull wavelength-band integrated flux of each starrelative to that of GJ 1132. The comparison starlabels in this table correspond to those in Figure 2.GJ 1132 is a high proper motion star. Positions aregiven for an epoch of 2016.3.
We detrend our light curve using decorrelation pa-rameters that either have the same values for all thestars (e.g., airmass) or are associated with GJ 1132 (e.g.,width of the spectral trace). The parameters that areunique to each star have similar values for all stars inthe data set but because we detect the most photonsfrom GJ 1132 its decorrelation parameters have highersignal-to-noise ratios. We create white light curves foreach data set and also bin the light curves from eachdata set into narrow wavelength bands for the purposeof atmospheric characterization. We restrict our analy-sis to the wavelength range common to GJ 1132 and thefour comparison stars, which is 700-1040 nm.We determine which linear combination of decorre-lation parameters are necessary to remove the effectsof correlated noise (discussed in greater detail in Sec-tion 4.2.1). In a given data set we choose a single 20 nmwavelength bin without any prominent stellar, telluric,or atmospheric features (we use 830 - 850 nm) and cal-culate the Bayesian Information Criterion (BIC) valuefor every combination of possible decorrelation param-eters. We check that there is no correlation with thesky-background, as this would imply that we are notproperly removing the sky background during extrac-tion. Once this process is done for all five data sets wetake the union of all the best decorrelation parametersand marginalize over them in all wavelength bins in alldata sets. A list of these parameters, what vectors theydepend on, and how they are derived can be found inTable 3.From the results of a Levenberg-Marquardt minimizerwe run a makeshift Bayesian test in order to determinewhether our five nights of data should be analyzed sep-arately or taken together in a joint fit. For each 20 nm bin in each of the five data sets we compare the resulting χ value for a fit in which the transit depth is allowedto vary to one in which the transit depth is fixed to ainverse-variance weighted depth derived from the fivenights. We account for the change in the number offitted parameters between these two scenarios. We findthat the χ values for the case of the fixed transit depthin a given wavelength bin can be higher, lower, or iden-tical to the case where the transit depth parameter isallowed to vary. In other words, fixing the transit depthdoes not provide a uniformly worse fit. We thereforedecide to fit the five nights of data jointly, allowing thetransit parameters to be shared across all nights.4.2. Analyzing transits jointly
Levenberg-Marquardt fits
In analyzing the transits jointly we must account forthe different uncertainties associated with the individualdata sets, as well as clip outlying data points. We usea three-step Levenberg-Marquardt process to settle oninitial guesses for our parameters to use in a dynamicnested sampler, which will be discussed in further de-tail in Section 4.2.2. To run our Levenberg-Marquardtfits we employ the open-source lmfit package (Newvilleet al. 2016).In the first pass at the Levenberg-Marquardt fit webuild a linear model unique to each night of data follow-ing the formula M ( t ) = S ( t ) T ( t ) (1)where S ( t ) is the systematics model and T ( t ) is the tran-sit model. The systematics model S ( t ) can further bebroken down to S ( t ) = 1 + N (cid:88) n =1 a n p n ( t ) (2)where N is the number of decorrelation parametersused in the fit, a n are the coefficients we are fitting for,and p n are the arrays of decorrelation parameters thatdescribe the correlated systematics in the data, whichare all functions of time. For decorrelation parameterswhich are functions of wavelength we sum over wave-length space corresponding to the wavelength bin weare working in. The decorrelation parameters are ei-ther common to all stars (airmass and rotator angle) orare taken from the GJ 1132 spectral extraction (width,stretch, peak).We build the transit model T ( t ) using the open-source batman code (Kreidberg 2015) and feed in the free tran-sit parameters. The transit parameters that can be Table 3.
Decorrelation parameters used to model data systematicsParameter Vector Descriptionairmass t average airmass of the fieldrotator angle t instrument rotator anglewidth t , star median width across wavelengths of the stellar trace in the cross-dispersion directionstretch t , star wavelength solution coefficient associated with spectrum stretching in the dispersion directionpeak t , λ , star brightness of the brightest pixel in the cross-dispersion directionnormalization t unit array Note.
All parameters are functions of time t . They can also vary by wavelength λ and by star. For allparameters that are star-dependent we use the values associated with GJ 1132 as it has the highest signal-to-noiseratio. shared across the five data sets are the planet-to-starradius ratio R p /R ∗ , period P , inclination i , scaled or-bital distance a/R ∗ , and uncorrelated quadratic limbdarkening coefficients 2 u + u and u − u as used byHolman et al. (2006). The residuals that we calculatefrom dividing our light curves by the linear models areweighted by the calculated photon noise of each dataset.At this stage we fix the uncorrelated quadratic limbdarkening coefficients to values derived from the LimbDarkening Tool Kit ( ldtk ), an open-source packagethat takes in stellar parameters and uncertainties andcalculates the limb darkening coefficients in a givenwavelength range based on the PHOENIX library ofstellar models (Husser et al. 2013; Parviainen & Aigrain2015). During the next stage of analysis (Section 4.2.2)we instead allow the uncorrelated quadratic limb dark-ening parameters to vary within a prior.In the second Levenberg-Marquardt fit we calculatethe MAD of the residuals and clip the 29 data points (forthe white light curve) or ≤
27 points (for the wavelength-binned light curves) that deviate by 5 × the MAD. In thethird Levenberg-Marquardt fit we change the weightingfrom the calculated photon noise to the uncertainties wederive from each night’s data as a result of our secondfit. Levenberg-Marquardt fits with lmfit are inexpen-sive and quick but running a dynamic nested samplercan be expensive if the priors are too wide. Since wederive our sampling priors from the covariance matrixoutput by the Levenberg-Marquardt fit we find it expe-dient to constrain the fit parameters as much as possibleat this stage.4.2.2. Dynamic nested sampling
Our joint fit comprises a minimum of 30 free param-eters – the same six decorrelation parameters (Table 3)to fit for each of the five data sets. In addition to this there can be free transit model parameters, like the tran-sit midpoint for each night or the transit depth, which isshared between the five nights. Which transit parame-ters are free depends on whether or not we are perform-ing a white light curve fit or a wavelength-dependinglight curve fit. With so many free parameters traditionalMarkov Chain Monte Carlo ensemble samplers such as emcee (Foreman-Mackey et al. 2013) are slow and inef-ficient at exploring the parameter space (Huijser et al.2015). We instead use the open-source dynamic nestedsampling package dynesty (J. Speagle, private commu-nication) to estimate our posteriors.The dynesty code samples each free parameter from0 to 1 and so requires a prior transform function tomap the outputs from the sampling onto the parameterspace we want to explore. For all but the uncorrelatedquadratic limb darkening coefficients we set uniform pri-ors on the parameters used to model the systematic andtransit portions of our models. When possible we as-sume the same uniform priors for the transit model pa-rameters as used by Dittmann et al. (2017a). Otherwisewe set uniform priors by taking the estimated 1 σ uncer-tainties from the covariance matrix of our Levenberg-Marquardt fit and multiplying by 25 such that the priorbounds for each parameter are 25 σ from the estimatedparameter value. These wide uniform priors allow foran uninformed, broad parameter space for the samplerto explore.Following the work of Berta et al. (2012) we placeGaussian priors on the uncorrelated quadratic limbdarkening coefficients 2 u + u and u − u . To de-termine what these Gaussian priors should be we firstget the quadratic limb darkening coefficients in eachwavelength bin from ldtk (Parviainen & Aigrain 2015). ldtk has an option to run a Markov chain Monte Carlo github.com/joshspeagle/dynesty (MCMC) with the input stellar parameters and uncer-tainties in order to derive the limb darkening coefficients.We use the samples from the MCMC to calculate ar-rays of uncorrelated parameters using the formulation2 u + u and u − u , where u and u are the quadraticcoefficients derived with ldtk . We calculate the medianand standard deviation of these uncorrelated arrays anduse these values to set the Gaussian priors. These Gaus-sian priors leverage our knowledge of stellar astrophysicswithout having to place complete faith in the accuracyof the stellar models.Also following Berta et al. (2012) we achieve a χ valueof unity by including a rescaling parameter s (Equations2 and 3 of that paper). We automatically marginalizeover this during our dynamic nested sampling by modi-fying our log-likelihood function such that s is a multi-plier of the theoretical uncertainty associated with eachdata point, including all terms that depend on s . Eachdata set has its own value of s associated with it. An s value of unity implies that we are reaching the theoret-ical photon noise limit with our fits, while a value lessthan unity implies an over-fitting of the model to thedata. 4.2.3. White light curve
We jointly fit the white light curve of our five data setsand allow the time of mid-transit δt to vary for eachdata set, along with the shared parameters of the radiusratio R p /R ∗ , period P , inclination i , scaled orbital dis-tance a/R ∗ , and uncorrelated quadratic limb-darkeningcoefficients 2 u + u and u − u . In doing so we lever-age the five nights of data, which have the same transitmodel parameter values, except for the mid-transit time.Where appropriate we adopt the same priors as thosequoted by Dittmann et al. (2017a) (Table 4). The photo-metric bandpass of MEarth is not identical to the wave-length coverage of our white light curves, and so we usestellar models to set Gaussian priors on the uncorre-lated quadratic limb darkening coefficients, as describedin Section 4.2.2.For the time of mid-transit we fit for an offset δt fromthe calculated mid-transit time using the ephemeris T given by Dittmann et al. (2017a): δt = t − ( T + nP ) (3)where n is the number of elapsed transits since theephemeris transit, P is the period, and t is the time ofmid-transit for the n th transit. We fit for the offset δt as opposed to the mid-transit time itself in order to keepthe model coefficients within a few orders of magnitudeof each other. This is optimal for Levenberg-Marquardtfitting with lmfit . Table 4.
White light curve transit model parameter priorsParameter Value Priors δt , [days] − . − . , . a δt , [days] − . − . , . a δt , [days] − . − . , . a δt , [days] − . − . , . a δt , [days] − . − . , . a R p /R ∗ . . , . a P [days] 1 . . , . b i .
68 [85 , b a/R ∗ .
54 [12 , b u + u − − − . ± . c u − u − − − − . ± . c s , , , , . , d a Uniform priors that are 25 × the 1 σ uncertaintiestaken from the lmfit covariance matrix, as describedin Section 4.2.2. The δt parameter is the offsetfrom the calculated time of mid-transit (Equation 3). R p /R ∗ is the planet-to-star radius ratio. b Uniform priors taken from Dittmann et al. (2017a). P is the period, i is the inclination, and a/R ∗ is thescaled orbital distance. c Gaussian priors calculated with ldtk outputs, as de-scribed in Section 4.2.2, given as mean ± standarddeviation. 2 u + u and u − u are the uncorre-lated quadratic limb darkening parameters. In theLevenberg-Marquardt fits these parameters are fixedto the ldtk outputs, but when sampling the param-eter space with dynesty we use the Gaussian priors; dynesty does not require starting values as inputs. d Wide uniform priors set by hand. Each data set hasa rescaling parameter s as described in Section 4.2.2 In our full band-integrated white light curve fit from700 - 1040 nm we see significant features in the residuals.After experimenting with decorrelation parameters andwavelength clipping we conclude that the deep water ab-sorption bands redward of 920 nm are leaving imprintson the white light curve, suggesting changes in precip-itable water vapor in Earth’s atmosphere during some ofour observations. The white light curves presented heredo not include these problematic bands and are insteadintegrated from 700 - 920 nm.At this stage we investigate any transit timing varia-tions by comparing our five derived mid-transit times tothose quoted in the discovery paper (Berta-Thompsonet al. 2015) and subsequent work with MEarth and
Spitzer (Dittmann et al. 2017a) (Figure 3). The mid-transit times from the
Spitzer data set reported by
Figure 3.
Observed minus calculated (O-C) times of mid-transit for GJ 1132b by transit number with 1 σ error barsderived from fitting each transit. Values for MEarth (greendata points) and Spitzer (red data points) are taken fromDittmann et al. (2017a, Table 4). The
Spitzer points are cor-rected here to include leap seconds. Values for the data pre-sented in this work from the Magellan/LDSS3C instrumentare shown in blue. All values were converted to BJD
TDB for the purpose of direct comparison. We use the val-ues of P = 1 . T = 2457184 . Dittmann et al. (2017a) are the BJD − OBS values takenfrom the
Spitzer header files. We correct these values toBJD
TBD , which accounts for leap seconds. We use thevalues of P = 1 . T = 2457184 . TDB multiple times following Eastman et al. (2010),making sure to account for the exposure and read-outtimes. As a test we perform a simple data reduction us-ing only polynomials and the batman transit light curvepackage (i.e., without the detrendersaurus pipeline)and were unable to derive transit times in agreementwith those of MEarth and
Spitzer .This discrepancy does not affect our results with re-spect to the atmospheric analysis since we fix the timesof transit to the best fit values when performing ouratmospheric analysis, and the time of mid-transit doesnot affect the transit depth at the time resolution of ourdata.
Table 5.
White light curve derived transit model values,compared to Dittmann et al. (2017a)Parameter Value (this work) Value (D17a) δt , [days] − . +0 . − . − − − δt , [days] − . +0 . − . − − − δt , [days] − . +0 . − . − − − δt , [days] − . +0 . − . − − − δt , [days] − . +0 . − . − − − R p /R ∗ . +0 . − . . +0 . − . P [days] 1 . +0 . − . . +0 . − . i [deg] 88 . +0 . − . . +0 . − . a/R ∗ . +1 . − . . +0 . − . u + u . +0 . − . − − − u − u − . +0 . − . − − − s . +0 . − . − − − s . +0 . − . − − − s . +0 . − . − − − s . +0 . − . − − − s . +0 . − . − − − We compare our derived values of the planet-to-starradius ratio R p /R ∗ , period P , inclination i , and scaledorbital distance a/R ∗ to those reported by Dittmannet al. (2017a) and find that our results are in agreement(Table 5). We present the raw white light curves, jointly-fit white light curve, time-binned white light curve, andtime-binned white light curve residuals in Figure 4.4.2.4. Wavelength-binned light curves
We investigate the atmosphere of GJ 1132b by creat-ing a transmission spectrum. We divide our light curvesinto 20 nm wavelength bins and jointly fit for R p /R ∗ and the uncorrelated quadratic limb darkening coeffi-cients 2 u + u and u − u , along with the systematicparameters for each respective data set. We fix the timesof mid-transit t for each night to the values determinedfrom the white light curve fit. We fix the values of P , i , and a/R ∗ in our binned wavelength fits to those re-ported by Dittmann et al. (2017a) as these are derivedfrom a higher resolution Spitzer time-series.Our joint fit produces single values for the radius ra-tio R p /R ∗ and the uncorrelated quadratic limb darken-ing coefficients 2 u + u and u − u for each wavelengthbin, but each of the five data sets has its own linear fitto the systematics in the light curve. In order to makemore meaningful comparisons between the systematicparameters in a given data set we scale each of them by0 Figure 4.
Panel a : Raw white light curves integrated from700 - 920 nm from each of the five data sets with models over-plotted in grey. The systematic parameters for these modelsare unique to each data set but the transit parameters arefree and shared jointly between the data sets. The derivedvalues for the transit parameters are given in Table 5.
Panelb : Unbinned white light curves from the five data sets withthe systematics component of the models divided out. Theover-plotted black line is the transit model.
Panel c : Whitelight curve binned in time at a 3-minute cadence. The over-plotted black line is the transit model.
Panel d : Residualsafter dividing the systematics models and subtracting thetransit models from the raw white light curves and binningat a 3-minute cadence.
Figure 5.
Derived parameters in each of the 17 20 nm wave-length bins used in the transmission spectrum. Our joint fitproduces a single value for the transit depth and the two un-correlated quadratic limb-darkening parameters, along with1 σ uncertainties, for each wavelength bin (top 3 panels). Wealso independently fit for the coefficients associated with thedecorrelation parameters (Table 3) in each data set in ev-ery wavelength bin (bottom 6 panels, colors correspond tothose in Figure 4). We do not see correlations between thecoefficients and the transit depth as a function of wavelengthbin. RESULTSIn Figure 6 we present our light curves after divid-ing out the systematic models for each data set. Thewavelength boundaries, RMS, transit depth, and me-dian factor of the expected photon noise limit for eachwavelength bin are given in Table 6. According to Fig-ure 2 of Stefansson et al. (2017), our observations ofGJ 1132 are limited by the photon noise so we did notestimate the scintillation noise for the analysis. Includ-ing scintillation noise would not change the resultingtransit depths but it would decrease our values in thefinal column of Table 6.Across the 17 wavelength bands we achieve a mediantransit depth error of 90 ppm. We compare this to 80ppm for two GJ 1132b transits with the
Spitzer µ mchannal and 55 ppm with 25 MEarth transits in itsphotometric band (Dittmann et al. 2017a).We present our transmission spectrum in Figure 7and compare it to two sets of four model transmissionspectra generated by the Exo-Transmit open sourcecode (Kempton et al. 2017). As inputs we use customdouble-grey temperature-pressure profiles and associ-ated equation-of-state files as well as GJ 1132b’s surfacegravity and radius at 1 bar of atmosphere and GJ 1132’sstellar radius (Miller-Ricci et al. 2009; Miller-Ricci &Fortney 2010). The 1 bar planet radius is smaller thanthe transit radius by an amount that depends on theatmospheric composition, temperature, and gravity. Asthese values are uncertain we allow the 1 bar planetradius to float in order to achieve the best transmissionmodel fits to our data. Changing the 1 bar planet ra-dius alters the amplitude of the model features as wellas the overall depth of the model. The
Spitzer datafrom Dittmann et al. (2017a) can resolve the ingressand egress of a transit of GJ 1132b so we adopt thestellar mass and radius quoted in that paper in orderto create the temperature-pressure profiles and modeltransmission spectra.One set of four model transmission spectra assumessolar elemental abundances (dominant in H and He)with metallicities that are 1, 10, 100, and 1000 × so-lar by volume. In these solar composition atmospheresthe dominant sources of opacity that contribute to thetransmission features are CH and H O, with modestcontributions from NH , H S, and K. Higher metallicityatmospheres have higher opacities, which strengthen the
Table 6.
Best-Fit Transit DepthsWavelength RMS Transit Depth × Expected[˚A] (ppm) % Noise a − . ± .
010 1 . − . ± .
010 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 1 . − . ± .
009 2 . − . ± .
008 1 . − . ± .
009 1 . − . ± .
010 1 . − . ± .
010 2 . − . ± .
010 1 . − . ± .
012 1 . − . ± .
016 1 . a Though we are jointly fitting the five data sets wecan estimate the expected photon noise limit and re-sulting RMS for each data set separately. This columnrepresents the median of the five resulting RMS val-ues divided by the expected photon noise for each dataset. These values are similar to the average s valuesthat we fit for for each night and for each wavelengthbin. It should be noted that we do not include a clac-ulation of the scintillation noise, so these values areconservative. model features, but also higher mean molecular weights,which dampen the model features. These competing ef-fects are the reason why they highest amplitude featuresare associated with the 10 × solar metallicity model.The other set of four model transmission spectra as-sume H and H O atmospheres where H O makes up1, 10, 50, and 100% of the atmosphere by volume. Thesolar composition models account for collision-inducedabsorption but the H /H O do not. Given how flat themodel transmission spectra are this should not impactthe results. All models assume a clear atmosphere (i.e.,no aerosols).We also compare the GJ 1132b transmission spectrumto a flat line at the inverse-variance weighted-averagetransit depth and to a 1-degree polynomial fit to thetransit depths. The wavelength bin-averaged values for2
Figure 6.
Left : Detrended light curves (colored points with each color representing one of the 5 data sets used in this analysis)with best fit transit model over-plotted (black lines). The text states the wavelength range in angstroms covered by the lightcurve directly underneath it.
Right : Residuals given by the detrended light curves minus the products of the best fit systematicsmodels and transit models. For clarity the y-axis labels in both panels are given only for a single light curve, but all light curvesand residuals are plotted on the same scale. For reference, the colors correspond to those in Figure 4 and the transit depths andRMS values for each wavelength bin are given in Table 6. Figure 7.
Transmission spectrum of GJ 1132b with 1 σ error bars derived from a joint fit of the five data sets analyzed in thiswork (both top and bottom ). Top : We compare the GJ 1132b transmission spectrum to four clear, solar composition models at1, 10, 100, and 1000 × solar metallicity by volume. We label the molecular sources of the most prominent features in the modelspectra. Bottom : We compare the GJ 1132b transmission spectrum to four clear, H and H O models where H O makes up1, 10, 50, and 100% of the atmosphere by volume. All features in these models are due to H O. Both figures also compare theGJ 1132b transmission spectrum to a flat line at the inverse-variance weighted-average transit depth (black dashed line) and a1-degree polynomial fit to the transit depths (black dotted line). In the legends of each figure we provide the mean molecularweights of the atmospheres used to create the model transmission spectra and confidences to which the meausred GJ 1132btransmission spectrum disfavors the model atmospheres. The data disfavor low mean molecular weight atmospheres.
Exo-Transmit models are weighted by the recordedcounts of a GJ 1132 spectrum across the same wave-length range. By using an observed spectrum of GJ1132 we account for the difference in relative brightnessof GJ 1132 as a function of wavelength, as well as thetelluric features imprinted on the spectrum. Becauseour wavelength bins are so narrow this weighting is vir-tually indistinguishable from a simple mean across themodel wavelength bins. We use the wavelength bin-averaged values of the model transmission spectra tocalculate the χ values associated with the model fits tothe measured transit depths.Our results disfavor a clear, 1 × solar metallicity at-mosphere at 3.09 σ (99.80%) and a clear, 10 × solarmetallicity atmosphere at 3.7 σ (99.98%) confidence.We disfavor a 10% H O, 90% H atmosphere at 3.5 σ (99.95%) confidence. Our measured transmission spec-trum is consistent with a flat line and with metallicitiesin excess of ∼ × solar or water abundances greaterthan ∼ Spitzer µ m bandpass(Dittmann et al. 2017a), but not in agreement with thephotometric transit depths from the GROND multi-band imager (Southworth et al. 2017). DISCUSSION6.1.
Ground-based detection of terrestrial exoplanetatmospheres
Our data-reduction process highlights the difficultiesof trying to detect terrestrial exoplanet atmospheresfrom the ground. The signal we are looking for is small(a transit depth of 0.24% and an atmospheric varia-tion of 0.02%) and we are not able to reach the pho-ton noise limit (Table 6). One question is whether moredata could disfavor higher mean molecular weight at-mospheres, or if we needed less data to reach the sameconclusions.To answer this question we select a test-case 20 nmwavelength bin, from 830-850 nm, and jointly fit for R p /R ∗ and the uncorrelated quadratic limb darkeningcoefficients 2 u + u and u − u , as we did in our anal-ysis, using 1, 2, 3, 4, and 5 data sets in each fit. We addthe data sets in order of decreasing singal-to-noise: firstdata set 2, then 5, 1, 4, 3. We record the error on thetransit depth after each data set is added to the analysis.We compare these to the errors in transit depth from thefirst data set, scaled by the inverse of the square-root ofthe number of data sets included. As shown in Figure 9, we require all five transits ofGJ 1132b to rule out low mean molecular weight atmo-spheres at high confidence. Theoretically, eight tran-sits are needed to rule out the highest mean molecularweight atmospheres we tested (1000 × solar metallicityand 100% H O), though this is a minimum estimatesince we do not achieve the photon noise limit and there-fore our error bars do not decrease by the square-root ofthe number of data sets included in the analysis.In the coming era of extremely large ground-basedtelescopes (ELTs) detecting and characterizing terres-trial exoplanet atmospheres may be in reach. For exam-ple, the Giant Magellan Telescope (GMT) will have adiameter of 24.5 m, compared to the 6.5 m diameter ofMagellan Clay. This means that the GMT will receiveabout (24 . / . = 14 . > Theoretical atmosphere of GJ 1132b
It would be surprising if a planet with such a smallradius (1.2 R ⊕ ) and high insolation (19 × Earth inso-lation) possessed a low mean molecular weight atmo-sphere. Based on thermal evolution models and extremeultraviolet mass loss, GJ 1132b falls into a class of plan-ets that would be unable to retain a H/He envelope(Lopez & Fortney 2013). There is statistical evidencefrom the
Kepler data set that close-in planets with smallradii ( < R ⊕ ) are rocky and lacking in low-density en-velopes (Rogers 2015; Fulton et al. 2017).5 Figure 8.
The transmission spectrum of GJ 1132b from this work (blue points) with 1 σ error bars in the context of other GJ1132b transit data. The dashed line is the inverse-variance weighted average of these transit depths. We plot the photometrictransit depths from the MEarth survey (green point) and the Spitzer µ m bandpass (red point) from Dittmann et al. (2017a),as well as the photometric transit depths in g , r , i , z , J , H , and K bands (purple points) from Southworth et al. (2017). Schaefer et al. (2016) ran models that couple GJ1132b’s atmosphere and interior, allowing for oxygen ex-change between the two. They determine that the mostlikely atmosphere for GJ 1132b is a tenuous one domi-nated by abiotic molecular oxygen (O ).This arises as follows: water (H O) in the GJ 1132batmosphere is photolysed by the intense UV radiationfrom the GJ 1132 host star. The hydrogen escapes tospace, taking some oxygen with it, but the different es-cape rates along with uptake by the interior mean thatsome oxygen can combine to form O and remain inthe planet’s atmosphere (Schaefer et al. 2016). Furthermodeling that includes additional atmospheric gassessuch as N and CO would be of interest.If the atmosphere of GJ 1132b is dominated by O thiswould be difficult to detect with any currently existinginstrumentation. Not only is the mean molecular weightof O relatively high ( µ = 32) but it also has few spectro-scopic features. Fortunately the photolysis of O leadsto the production of ozone (O ). Given the asymme-try of this molecule it produces higher-amplitude spec-troscopic features and is more amenable to detection.An atmosphere around GJ 1132b may be dominatedby other molecules. We see examples in the Solar Sys-tem of small bodies with high mean-molecular weight atmospheres other than Earth’s. Venus, for instance,has a thick atmosphere of CO ( µ = 44) and Titan hastraceable CH ( µ = 16). These molecules have manyprominent spectroscopic features and these atmosphereswould be detectable on GJ 1132b in transmission withinstruments aboard the James Webb Space Telescope (JWST) with 10 transits, according to online predictivetools like
PandExo (Batalha et al. 2017; Morley et al.2017). They may also be detectable in transmission withthe GMT though the predictive tools are not yet avail-able to test this. Other observing strategies, such astaking emission spectra, will also be useful in constrain-ing the atmospheric properties.With its 19 × Earth insolation and small radius it islikely that GJ 1132b has a high mean molecular weightatmosphere or atmosphere at all. The same can be saidfor many of the TRAPPIST-1 planets (Gillon et al. 2017;de Wit et al. 2018). Terrestrial planets farther from theirhost stars may fare better. LHS 1140b recieves 0.46 × Earth insolation and has a high surface gravity; it there-fore may not experience the same rates of atmosphericescape (Dittmann et al. 2017b).6.3.
Searching for more terrestrial exoplanets number of datasets included in fit t r a n s i t d e p t h e rr o r [ pp m ] disfavor low mean molecularweight atmospehres at
1 confidence2 confidence3 confidence disfavor high mean molecularweight atmospehres at
3 confidencecalculated erroractual error
Figure 9.
Transit depth error as a function of the number ofdata sets included in the analysis. The blue line and circlesshows how the transit depth error decreases when performingthe analysis with additional data sets. The black line andsquares shows the transit depth error of our first data setdivided by the square-root of the number of data sets usedin the analysis. We extend the calculated error to investigatewhat would happen if we captured more than five transitsof GJ 1132b. The dashed horizontal lines denote the transitdepth error that would disfavor low mean molecular weightatmospheres (10 × solar metallicity and 10% H O, 90% H )at 1, 2, and 3 σ . We require all five transits to disfavor thelow mean molecular weight atmospheres we tested. We theo-retically require eight transits to rule out higher mean molec-ular weight atmospheres (1000 × solar metallicity and 100%H O), though likely more given that we do not reach thephoton noise limit.
Perhaps the terrestrial planets with the most acces-sible atmospheres have not yet been discovered. TheGJ 1132, LHS 1140, and TRAPPIST-1 systems are allabout 12 parsecs away (Berta-Thompson et al. 2015;Dittmann et al. 2017b; Gillon et al. 2017, respectively).Dressing & Charbonneau (2015) investigated the oc-currence rate of planets around nearby M dwarfs usingthe full
Kepler survey and found a cummulative occu-rance rate of 2.5 ± R ⊕ ) per M dwarf,with periods less than 200 days. So there may be stillundiscovered small exoplanets that would be amenableto atmospheric detection and characterization.M dwarfs, with their small sizes, high occurrencerates, and close-in habitable zones, are now the targetsof several dedicated transit and radial velocity surveysthat aim to identify planets amenable to atmosphericfollow-up. Notable transit surveys include MEarth andTRAPPIST (Irwin et al. 2015; Gillon et al. 2013), withSPECULOOS and TESS waiting to come online shortly(Burdanov et al. 2017; Ricker et al. 2015). Radial veloc-ity surveys focusing on M dwarfs stand to make more detections since they are not as limited by a planet’sinclination. Though many of the planets discovered bythis method will not transit, their atmospheres may beamenable to phase curve (Koll & Abbot 2016; Kreidberg& Loeb 2016) or high-resolution spectroscopic (Snellenet al. 2013) observations. The radial velocity surveys(listed by their acronyms) focused on M dwarfs that areeither currently taking data or in the production phaseinclude CARMENES (Quirrenbach et al. 2010), HZPF(Mahadevan et al. 2010), MAROON-X (Seifahrt et al.2016), NEID (Schwab et al. 2016), NIRPS (Bouchy et al.2017), and SPIRou (Artigau et al. 2014). CONCLUSIONWe investigate whether or not the small, rocky terres-trial exoplanet GJ 1132b possesses a low mean molec-ular weight ( µ ∼
2) atmosphere using ground-basedtelescopes and instrumentation to construct a transmis-sion spectrum. Our analysis disfavors a clear, 10 × solarmetallicity and a clear 10% H O at high confidence. GJ1132b likely possesses a high mean molecular weight ordepleted atmosphere.While we search for new terrestrial exoplanets weshould also continue to learn more about the GJ 1132batmosphere. Obtaining transits with HST/WFC3 willallow us to confirm the results from this work, especiallysince space-based telescopes do not have to contend withtelluric water features. Morley et al. (2017) suggest thatGJ 1132b is the most amenable planet of its kind, cur-rently known, for observation in secondary eclipse withJWST. Small, rocky exoplanets like GJ 1132b challengeour limits of detection and characterization but alsopresent the most exciting opportunities for comparativeplanetology with the Solar System terrestrial exoplan-ets, including Earth.This paper includes data gathered with the 6.5 m Mag-ellan Telescopes located at Las Campanas Observa-tory, Chile. We thank the contributors to the LDSS3Cproject, the telescope operators and staff at Las Cam-panas Observatory, and the writers and contributorsof the open-source software used in this work. Wealso thank Mercedes Lopez-Morales, Robin Wordsworth,Dimitar Sasselov, Laura Kreidberg, Kevin Stevenson,and Josh Speagle for helpful comments and conversa-tions. H.D.-L. recognizes support from the NationalScience Foundation Graduate Research Fellowship Pro-gram (grant number DGE1144152). Z.K.B.-T. acknowl-edges support from the MIT Torres Fellowship for Ex-oplanetary Research. E.M.-R.K. was supported by theNational Science Foundation under CAREER Grant No.71654295 and by the Research Corporation for ScienceAdvancement through their Cottrell Scholar program.This publication was made possible through the supportof a grant from the John Templeton Foundation. Theopinions expressed here are those of the authors and donot necessarily reflect the views of the John TempletonFoundation.
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