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Baptiste Lavie

Publications and source records attributed to Baptiste Lavie.

17 recordsLinked to original sources

Uncovering temperate worlds: an updated look at the K2-3 system with ESPRESSO

The K2-3 system is one of the most unique planetary systems as it features three transiting sub-Neptunes in decreasing order of radii, spanning the radius valley. In addition to that, the two outer planets are temperate in nature making this system a perfect laboratory for investigating the nature and formation of small potentially habitable worlds. We present new radial velocity data obtained during the ESPRESSO-GTO programme to add to the archival HARPS, HARPS-N and HIRES data for a total time span of 8 years. This combined dataset significantly improves the masses of the K2-3 planets. We analyse the spectroscopic data with the line-by-line algorithm. We coupled this timeseries with a multi-dimensional Gaussian process framework to disentangle the 40.5-days stellar rotation from the Keplerian signals. We tested models using two activity indicators to ensure the retrieved semi-amplitudes of the planets are not biased by the choice of activity indicator. The multi-dimensional Gaussian process framework, with either activity indicator, yields the following masses: Mb=6.61+/-0.47Me; Mc=2.23+/-0.68Me; Md=2.12+/-1.27Me. Compared to previous works, our detection significance of K2-3b has increased by a factor of two, the mass of K2-3c has slightly decreased and we now have a marginal detection for K2-3d. The positions of the planets in the mass-radius diagram suggest either a H/He-rich or volatile-rich composition for both K2-3b and c, while the nature of K2-3d is ambiguous. A forward atmospheric model suggests that a single visit with JWST can distinguish between a H/He-rich atmosphere from a water-world scenario. This dataset exemplifies the capability of ESPRESSO to detect sub-metre-per-second radial velocity signals for the detection of temperate exoplanets.

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A 43 day transiting Neptune and two 25 day Saturns from TESS, NGTS and ASTEP

Beyond orbital periods of 10 days, there is a dearth of known transiting gas giants. On longer orbits, planets are less affected by their host star, and become ideal probes of planet formation, migration and evolution. We report the discovery of a long period Neptune and two Saturns, each initially identified as single transits in the TESS photometry, and solved through additional transits from ground-based follow-up photometric observations by NGTS and ASTEP. High-resolution radial velocity mass measurements using CORALIE and HARPS confirm their planetary nature. From joint modelling of the photometric and spectroscopic data, we determine an orbital period of $43.12655_{-0.00017}^{+0.00012}~$days, radius of $3.65\pm0.22~\mathrm{R_{\oplus}}$, and mass of $19.1_{-4.5}^{+4.9}~\mathrm{M_{\oplus}}$ for NGTS-34b, making it one of the longest period well-characterized transiting Neptunes. Orbiting a late F-type star, bright in the K-band (Kmag$~\simeq7.9$), it is amenable for cool atmosphere studies using JWST or Ariel. TOI-4940b is a small Saturn on a $25.867811_{-0.000056}^{+0.000058}~$day orbit with a radius of $6.61\pm0.37~\mathrm{R_{\oplus}}$ and an upper mass limit $<89~\mathrm{M_{\oplus}}$. NGTS-35b(=TOI-6669b) is a larger Saturn on a $25.241192\pm0.000022~$day, moderately eccentric orbit ($e = 0.192_{-0.033}^{+0.037}$), with a radius of $10.90\pm0.65~\mathrm{R_{\oplus}}$ and a mass of $152_{-19}^{+22}~\mathrm{M_{\oplus}}$. With an assumed albedo $A=0.3$, each of these planets has an equilibrium temperature below 700K, with NGTS-35b especially cold at $450~$K. These three giants add to the small but growing population of long period planets that can further our understanding of planet formation mechanisms.

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Simulating RISTRETTO: Proxima b detectability in reflected light

The characterization of exoplanet atmospheres is one of the key topics in modern astrophysics. To date, transmission spectroscopy has been the primary method used, but upcoming instruments will lay the foundation for advancing reflected-light spectroscopy. The main challenge in this area of research is the high contrast ratio between the planet and its star. RISTRETTO, a high-resolution integral-field spectrograph designed for ESO's VLT, aims to address these limitations through a combination of extreme AO, coronagraphy, and high-resolution spectroscopy. The goal of this paper is to demonstrate the detectability of the temperate rocky planet Proxima b with RISTRETTO, using realistic end-to-end simulations and a specifically developed data analysis methodology. We created high-resolution star and planet spectra, selecting realistic observational epochs and incorporating the predicted performance of the AO and coronagraphic systems. We implemented noise and spectrograph effects through the Pyechelle spectrograph simulator. We then applied a new methodology to isolate the signal of the planet from the stellar one and proceeded to fit several planetary models in order of increasing complexity. We also introduced a method to determine the sky orientation of the stellar spin axis, which constrains the orientation of the planetary orbit for aligned systems. Assuming an Earth-like atmosphere, our results show that RISTRETTO can detect Proxima b in reflected light in about 55 hours of observing time, offering the ability to characterize the planet orbital inclination, true mass, and broadband albedo. In addition, molecular absorption by O$_2$ and H$_2$O can be detected in about 85 hours of observations. These findings highlight the potential of RISTRETTO to significantly advance the field of exoplanetary science by enabling reflected-light spectroscopy of a sample of nearby exoplanets.

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X-SHYNE: X-Shooter spectra of young exoplanet analogs II. Presentation and analysis of the full library

The X-SHYNE library is a homogeneous sample of 43 medium-resolution (R=8000) infrared (0.3-2.5um) spectra of young (<500Myr), low-mass (<20Mjup), and cold (Teff=600-2000K) isolated brown dwarfs and wide-separation companions observed with the VLT/X-Shooter instrument. To characterize our targets, we performed a global comparative analysis. We first applied a semi-empirical approach. By refining their age and bolometric luminosity, we derived key atmospheric and physical properties, such as Teff, mass, surface gravity (g), and radius, using the evolutionary model COND03. These results were then compared with the results from a synthetic analysis based on three self-consistent atmospheric models. To compare our spectra with these grids we used the Bayesian inference code ForMoSA. We found similar Lbol estimates between both approaches, but an underestimated Teff from the cloudy models, likely due to a lack of absorbers that could dominate the J and H bands of early L. We also observed a discrepancy in the log(g) estimates, which are dispersed between 3.5 and 5.5 dex for mid-L objects. We interpreted this as a bias caused by a range of rotational velocities leading to cloud migration toward equatorial latitudes, combined with a variety of viewing angles that result in different observed atmospheric properties (cloud column densities, atmospheric pressures, etc.). Finally, while providing robust estimates of [M/H] and C/O for individual objects remains challenging, the X-SHYNE library globally suggests solar values, which are consistent with a formation via stellar formation mechanisms. This study highlights the strength of homogeneous datasets in performing comparative analyses, reducing the impact of systematics, and ensuring robust conclusions while avoiding over-interpretation.

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HD 35843: A Sun-like star hosting a long period sub-Neptune and inner super-Earth

We report the discovery and confirmation of two planets orbiting the metal-poor Sun-like star, HD 35843 (TOI 4189). HD 35843 c is a temperate sub-Neptune transiting planet with an orbital period of 46.96 days that was first identified by Planet Hunters TESS. We combine data from TESS and follow-up observations to rule out false-positive scenarios and validate the planet. We then use ESPRESSO radial velocities to confirm the planetary nature and characterize the planet's mass and orbit. Further analysis of these RVs reveals the presence of an additional planet, HD 35843 b, with a period of 9.90 days and a minimum mass of $5.84\pm0.84$ $M_{\oplus}$. For HD 35843 c, a joint photometric and spectroscopic analysis yields a radius of $2.54 \pm 0.08 R_{\oplus}$, a mass of $11.32 \pm 1.60 M_{\oplus}$, and an orbital eccentricity of $e = 0.15\pm0.07$. With a bulk density of $3.80 \pm 0.70$ g/cm$^3$, the planet might be rocky with a substantial H$_2$ atmosphere or it might be a ``water world". With an equilibrium temperature of $\sim$480 K, HD 35843 c is among the coolest $\sim 5\%$ of planets discovered by TESS. Combined with the host star's relative brightness (V= 9.4), HD 35843 c is a promising target for atmospheric characterization that will probe this sparse population of temperate sub-Neptunes.

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Stellar X-ray variability and planetary evolution in the DS Tucanae system

We present an analysis of four Chandra observations of the 45 Myr old DS Tuc binary system. We observed X-ray variability of both stars on timescales from hours to months, including two strong X-ray flares from star A. The implied flaring rates are in agreement with past observations made with XMM-Newton, though these rates remain imprecise due to the relatively short total observation time. We find a clear, monotonic decline in the quiescent level of the star by a factor 1.8 across eight months, suggesting stellar variability that might be due to an activity cycle. If proven through future observations, DS Tuc A would be the youngest star for which a coronal activity cycle has been confirmed. The variation in our flux measurements across the four visits is also consistent with the scatter in empirical stellar X-ray relationships with Rossby number. In simulations of the possible evolution of the currently super-Neptune-sized planet DS Tuc Ab, we find a range of scenarios for the planet once it reaches a typical field age of 5 Gyr, from Neptune-size down to a completely stripped super-Earth. Improved constraints on the planet's mass in the future would significantly narrow these possibilities. We advocate for further Chandra observations to better constrain the variability of this important system.

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HD 21520 b: a warm sub-Neptune transiting a bright G dwarf

We report the discovery and validation of HD 21520 b, a transiting planet found with TESS and orbiting a bright G dwarf (V=9.2, $T_{eff} = 5871 \pm 62$ K, $R_{\star} = 1.04\pm 0.02\, R_{\odot}$). HD 21520 b was originally alerted as a system (TOI-4320) consisting of two planet candidates with periods of 703.6 and 46.4 days. However, our analysis supports instead a single-planet system with an orbital period of $25.1292\pm0.0001$ days and radius of $2.70 \pm 0.09\, R_{\oplus}$. Three full transits in sectors 4, 30 and 31 match this period and have transit depths and durations in agreement with each other, as does a partial transit in sector 3. We also observe transits using CHEOPS and LCOGT. SOAR and Gemini high-resolution imaging do not indicate the presence of any nearby companions, and MINERVA-Australis and CORALIE radial velocities rule out an on-target spectroscopic binary. Additionally, we use ESPRESSO radial velocities to obtain a tentative mass measurement of $7.9^{+3.2}_{-3.0}\, M_{\oplus}$, with a 3-$σ$ upper limit of 17.7 $M_{\oplus}$. Due to the bright nature of its host and likely significant gas envelope of the planet, HD 21520 b is a promising candidate for further mass measurements and for atmospheric characterization.

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Discovery of two warm mini-Neptunes with contrasting densities orbiting the young K3V star TOI-815

We present the discovery and characterization of two warm mini-Neptunes transiting the K3V star TOI-815 in a K-M binary system. Analysis of the spectra and rotation period reveal it to be a young star with an age of $200^{+400}_{-200}$Myr. TOI-815b has a 11.2-day period and a radius of 2.94$\pm$0.05$\it{R_{\rm\mathrm{\oplus}}}$ with transits observed by TESS, CHEOPS, ASTEP, and LCOGT. The outer planet, TOI-815c, has a radius of 2.62$\pm$0.10$\it{R_{\rm\mathrm{\oplus}}}$, based on observations of three non-consecutive transits with TESS, while targeted CHEOPS photometry and radial velocity follow-up with ESPRESSO were required to confirm the 35-day period. ESPRESSO confirmed the planetary nature of both planets and measured masses of 7.6$\pm$1.5 $\it{M_{\rm \mathrm{\oplus}}}$ ($ρ_\mathrm{P}$=1.64$^{+0.33}_{-0.31}$gcm$^{-3}$) and 23.5$\pm$2.4$\it{M_{\rm\mathrm{\oplus}}}$ ($ρ_\mathrm{P}$=7.2$^{+1.1}_{-1.0}$gcm$^{-3}$) respectively. Thus, the planets have very different masses, unlike the usual similarity of masses in compact multi-planet systems. Moreover, our statistical analysis of mini-Neptunes orbiting FGK stars suggests that weakly irradiated planets tend to have higher bulk densities compared to those suffering strong irradiation. This could be ascribed to their cooler atmospheres, which are more compressed and denser. Internal structure modeling of TOI-815b suggests it likely has a H-He atmosphere constituting a few percent of the total planet mass, or higher if the planet is assumed to have no water. In contrast, the measured mass and radius of TOI-815c can be explained without invoking any atmosphere, challenging planetary formation theories. Finally, we infer from our measurements that the star is viewed close to pole-on, which implies a spin-orbit misalignment at the 3$σ$ level.

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Hot Exoplanet Atmospheres Resolved with Transit Spectroscopy (HEARTS) VII. Detection of sodium on the long-transiting inflated sub-Saturn KELT-11 b

KELT-11b is an inflated sub-Saturn with a hot atmosphere and that orbits a bright evolved subgiant star, making it a prime choice for atmospheric characterization, but that transits its host star for more than seven hours. We observed this system in series of three consecutive nights with the HARPS spectrograph and report on the analysis of the transmission spectrum obtained from this dataset. Our results highlight the potential for independent observations of a long-transiting planet over consecutive nights. Our study reveals a sodium excess absorption of $0.28 \pm 0.05 %$ and $0.50 \pm 0.06 %$ in the Na D1 and D2 lines, respectively. This corresponds to 1.44 and 1.69 times the white-light planet radius in the line cores. Wind pattern modeling tends to prefer day-to-night side winds with no vertical winds, which is surprising considering the planet bloatedness. The modeling of the Rossiter-Mclaughlin effect yields a significantly misaligned orbit, with a projected spin-orbit angle of $λ = -77.86^{+2.36}_{-2.26}{}^\circ$. The characteristics of KELT-11 b, notably its extreme scale height and long transit, make it an ideal and unique target for next-generation telescopes. Our results as well as recent findings from HST, TESS, and CHEOPS observations could make KELT-11 b a benchmark exoplanet in atmospheric characterization.

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A medium-resolution spectrum of the exoplanet HIP 65426 b

Medium-resolution integral-field spectrographs (IFS) coupled with adaptive-optics such as Keck/OSIRIS, VLT/MUSE, or SINFONI are appearing as a new avenue for enhancing the detection and characterization capabilities of young, gas giant exoplanets at large heliocentric distances (>5 au). We analyzed K-band VLT/SINFONI medium-resolution (R_lambda~5577) observations of the young giant exoplanet HIP 65426 b. Our dedicated IFS data analysis toolkit (TExTRIS) optimized the cube building, star registration, and allowed for the extraction of the planet spectrum. A Bayesian inference with the nested sampling algorithm coupled with the self-consistent forward atmospheric models BT-SETTL15 and Exo-REM using the ForMoSA tool yields Teff=1560 +/- 100K, log(g)<4.40dex, [M/H]=0.05 +/- 0.22dex, and an upper limit on the C/O ratio (<0.55). The object is also re-detected with the so-called "molecular mapping" technique. The technique yields consistent atmospheric parameters, but the loss of the planet pseudo-continuum in the process degrades or modifies the constraints on these parameters. The solar to sub-solar C/O ratio suggests an enrichment by solids at formation if the planet was formed beyond the water snowline (>20 au) by core-accretion. However, a formation by gravitational instability can not be ruled out. The metallicity is compatible with the bulk enrichment of massive Jovian planets from the Bern planet population models. Finally, we measure a radial velocity of 26 +/- 15km/s compatible with our revised measurement on the star. This is the fourth imaged exoplanet for which a radial velocity can be evaluated, illustrating the potential of such observations for assessing the coevolution of imaged systems belonging to star forming regions, such as HIP 65426.

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Nightside condensation of iron in an ultra-hot giant exoplanet

Ultra-hot giant exoplanets receive thousands of times Earth's insolation. Their high-temperature atmospheres (>2,000 K) are ideal laboratories for studying extreme planetary climates and chemistry. Daysides are predicted to be cloud-free, dominated by atomic species and substantially hotter than nightsides. Atoms are expected to recombine into molecules over the nightside, resulting in different day-night chemistry. While metallic elements and a large temperature contrast have been observed, no chemical gradient has been measured across the surface of such an exoplanet. Different atmospheric chemistry between the day-to-night ("evening") and night-to-day ("morning") terminators could, however, be revealed as an asymmetric absorption signature during transit. Here, we report the detection of an asymmetric atmospheric signature in the ultra-hot exoplanet WASP-76b. We spectrally and temporally resolve this signature thanks to the combination of high-dispersion spectroscopy with a large photon-collecting area. The absorption signal, attributed to neutral iron, is blueshifted by -11+/-0.7 km s-1 on the trailing limb, which can be explained by a combination of planetary rotation and wind blowing from the hot dayside. In contrast, no signal arises from the nightside close to the morning terminator, showing that atomic iron is not absorbing starlight there. Iron must thus condense during its journey across the nightside.

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MIRACLES: atmospheric characterization of directly imaged planets and substellar companions at 4-5 $μ$m. I. Photometric analysis of $β$ Pic b, HIP 65426 b, PZ Tel B and HD 206893 B

Directly imaged planets and substellar companions are key targets for the characterization of self-luminous atmospheres. Their photometric appearance at 4-5 $μ$m is sensitive to the chemical composition and cloud content of their atmosphere. We aim at systematically characterizing the atmospheres of directly imaged low-mass companions at 4-5 $μ$m. We want to homogeneously process the data and compile a photometric library at thermal wavelengths of these mostly young, low-gravity objects. In this way, we want to find trends related to their spectral type and surface gravity by comparing with isolated brown dwarfs and predictions from atmospheric models. We have used the high-resolution, high-contrast capabilities of NACO at the VLT to directly image the companions of HIP 65426, PZ Tel, and HD 206893 in the NB4.05 and/or $M'$ filters. For the same targets, and additionally $β$ Pic, we have also analyzed six archival VLT/NACO datasets which were taken with the NB3.74, $L'$, NB4.05, and $M'$ filters. The $L'$-NB4.05 and $L'$-$M'$ colors of the studied sample are all red while the NB4.05-$M'$ color is blue for $β$ Pic b, gray for PZ Tel B, and red for HIP 65426 b and HD 206893 B (although typically with low significance). The absolute NB4.05 and $M'$ fluxes of our sample are all larger than those of field dwarfs with similar spectral types. Finally, the surface gravity of $β$ Pic b has been constrained to $\log{g} = 4.17_{-0.13}^{+0.10}$ dex from its photometry and dynamical mass. A red color at 3-4 $μ$m and a blue color at 4-5 $μ$m might be (partially) caused by H$_2$O and CO absorption, respectively. The red characteristics of $β$ Pic b, HIP 65426 b, and HD 206893 B at 3-5$μ$m, as well as their higher fluxes in NB4.05 and $M'$ compared to field dwarfs, indicate that cloud densities are enhanced close to the photosphere as a result of their low surface gravity.

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Orbital misalignment of the Neptune-mass exoplanet GJ 436b with the spin of its cool star

The angle between the spin of a star and its planets' orbital planes traces the history of the planetary system. Exoplanets orbiting close to cool stars are expected to be on circular, aligned orbits because of strong tidal interactions with the stellar convective envelope. Spin-orbit alignment can be measured when the planet transits its star, but such ground-based spectroscopic measurements are challenging for cool, slowly-rotating stars. Here we report the characterization of a planet three-dimensional trajectory around an M dwarf star, derived by mapping the spectrum of the stellar photosphere along the chord transited by the planet. We find that the eccentric orbit of the Neptune-mass exoplanet GJ 436b is nearly perpendicular to the stellar equator. Both eccentricity and misalignment, surprising around a cool star, can result from dynamical interactions (via Kozai migration) with a yet-undetected outer companion. This inward migration of GJ 436b could have triggered the atmospheric escape that now sustains its giant exosphere. Eccentric, misaligned exoplanets orbiting close to cool stars might thus hint at the presence of unseen perturbers and illustrate the diversity of orbital architectures seen in exoplanetary systems.

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Retrieval Analysis of the Emission Spectrum of WASP-12b: Sensitivity of Outcomes to Prior Assumptions and Implications for Formation History

We analyze the emission spectrum of the hot Jupiter WASP-12b using our HELIOS-R retrieval code and HELIOS-K opacity calculator. When interpreting Hubble and Spitzer data, the retrieval outcomes are found to be prior-dominated. When the prior distributions of the molecular abundances are assumed to be log-uniform, the volume mixing ratio of HCN is found to be implausibly high. A VULCAN chemical kinetics model of WASP-12b suggests that chemical equilibrium is a reasonable assumption even when atmospheric mixing is implausibly rigorous. Guided by (exo)planet formation theory, we set Gaussian priors on the elemental abundances of carbon, oxygen and nitrogen with the Gaussian peaks being centered on the measured C/H, O/H and N/H values of the star. By enforcing chemical equilibrium, we find substellar O/H and stellar to slightly superstellar C/H for the dayside atmosphere of WASP-12b. The superstellar carbon-to-oxygen ratio is just above unity, regardless of whether clouds are included in the retrieval analysis, consistent with Madhusudhan et al. (2011). Furthermore, whether a temperature inversion exists in the atmosphere depends on one's assumption for the Gaussian width of the priors. Our retrieved posterior distributions are consistent with the formation of WASP-12b in a solar-composition protoplanetary disk, beyond the water iceline, via gravitational instability or pebble accretion (without core erosion) and migration inwards to its present orbital location via a disk-free mechanism, and are inconsistent with both in-situ formation and core accretion with disk migration, as predicted by Madhusudhan et al. (2017). We predict that the interpretation of James Webb Space Telescope WASP-12b data will not be prior-dominated.

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HELIOS-Retrieval: An Open-source, Nested Sampling Atmospheric Retrieval Code, Application to the HR 8799 Exoplanets and Inferred Constraints for Planet Formation

We present an open-source retrieval code named HELIOS-Retrieval (hereafter HELIOS-R), designed to obtain chemical abundances and temperature-pressure profiles from inverting the measured spectra of exoplanetary atmospheres. In the current implementation, we use an exact solution of the radiative transfer equation, in the pure absorption limit, in our forward model, which allows us to analytically integrate over all of the outgoing rays (instead of performing Gaussian quadrature). Two chemistry models are considered: unconstrained chemistry (where the mixing ratios are treated as free parameters) and equilibrium chemistry (enforced via analytical formulae, where only the elemental abundances are free parameters). The nested sampling algorithm allows us to formally implement Occam's Razor based on a comparison of the Bayesian evidence between models. We perform a retrieval analysis on the measured spectra of the HR 8799b, c, d and e directly imaged exoplanets. Chemical equilibrium is disfavored by the Bayesian evidence for HR 8799b, c and d. We find supersolar C/O, C/H and O/H values for the outer HR 8799b and c exoplanets, while the inner HR 8799d and e exoplanets have substellar C/O, substellar C/H and superstellar O/H values. If these retrieved properties are representative of the bulk compositions of the exoplanets, then they are inconsistent with formation via gravitational instability (without late-time accretion) and consistent with a core accretion scenario in which late-time accretion of ices occurred differently for the inner and outer exoplanets. For HR 8799e, we find that spectroscopy in the K band is crucial for constraining C/O and C/H. HELIOS-R is publicly available as part of the Exoclimes Simulation Platform (ESP; www.exoclime.org).

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HELIOS: An Open-source, GPU-accelerated Radiative Transfer Code For Self-consistent Exoplanetary Atmospheres

We present the open-source radiative transfer code named HELIOS, which is constructed for studying exoplanetary atmospheres. In its initial version, the model atmospheres of HELIOS are one-dimensional and plane-parallel, and the equation of radiative transfer is solved in the two-stream approximation with non-isotropic scattering. A small set of the main infrared absorbers is employed, computed with the opacity calculator HELIOS-K and combined using a correlated-$k$ approximation. The molecular abundances originate from validated analytical formulae for equilibrium chemistry. We compare HELIOS with the work of Miller-Ricci & Fortney using a model of GJ 1214b, and perform several tests, where we find: model atmospheres with single-temperature layers struggle to converge to radiative equilibrium; $k$-distribution tables constructed with $\gtrsim 0.01$ cm$^{-1}$ resolution in the opacity function ($ \lesssim 10^3$ points per wavenumber bin) may result in errors $\gtrsim 1$-10 % in the synthetic spectra; and a diffusivity factor of 2 approximates well the exact radiative transfer solution in the limit of pure absorption. We construct "null-hypothesis" models (chemical equilibrium, radiative equilibrium and solar element abundances) for 6 hot Jupiters. We find that the dayside emission spectra of HD 189733b and WASP-43b are consistent with the null hypothesis, while it consistently under-predicts the observed fluxes of WASP-8b, WASP-12b, WASP-14b and WASP-33b. We demonstrate that our results are somewhat insensitive to the choice of stellar models (blackbody, Kurucz or PHOENIX) and metallicity, but are strongly affected by higher carbon-to-oxygen ratios. The code is publicly available as part of the Exoclimes Simulation Platform (ESP; exoclime.net).

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A Non-isothermal Theory for Interpreting Sodium Lines in Transmission Spectra of Exoplanets

We present a theory for interpreting the sodium lines detected in transmission spectra of exoplanetary atmospheres. Previous analyses employed the isothermal approximation and dealt only with the transit radius. By recognising the absorption depth and the transit radius as being independent observables, we develop a theory for jointly interpreting both quantities, which allows us to infer the temperatures and number densities associated with the sodium lines. We are able to treat a non-isothermal situation with a constant temperature gradient. Our novel diagnostics take the form of simple-to-use algebraic formulae and require measurements of the transit radii (and their corresponding absorption depths) at line center and in the line wing for both sodium lines. We apply our diagnostics to the HARPS data of HD 189733b, confirm the upper atmospheric heating reported by Huitson et al. (2012), derive a temperature gradient of $0.4376 \pm 0.0154$ K km$^{-1}$ and find densities $\sim 1$ to $10^4$ cm$^{-3}$.

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