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At least 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Giant Outer Transiting Exoplanet Mass (GOT ‘EM) Survey. I. Confirmation of an Eccentric, Cool Jupiter with an Interior Earth-sized Planet Orbiting Kepler-1514

Despite the severe bias of the transit method of exoplanet discovery toward short orbital periods, a modest sample of transiting exoplanets with orbital periods greater than 100 days is known. Long-term radial velocity (RV) surveys are pivotal to confirming these signals and generating a set of planetary masses and densities for planets receiving moderate to low irradiation from their host stars. Here we conduct RV observations of Kepler-1514 from the Keck I telescope using the High Resolution Echelle Spectrometer. From these data, we measure the mass of the statistically validated giant (1.108 ± 0.023 R {sub J}) exoplanet Kepler-1514 b with a 218-day orbital period as 5.28 ± 0.22 M {sub J}. The bulk density of this cool (∼390 K) giant planet is 4.82{sub −0.25}{sup +0.26}g cm{sup −3}, consistent with a core supported by electron degeneracy pressure. We also infer an orbital eccentricity of 0.401{sub −0.014}{sup +0.013} from the RV and transit observations, which is consistent with planet–planet scattering and disk cavity migration models. The Kepler-1514 system contains an Earth-size, Kepler Object of Interest on a 10.5-day orbit that we statistically validate against false-positive scenarios, including those involving a neighboring star. The combination of the brightness (V = 11.8) of the host star and the long period, low irradiation, and high density of Kepler-1514 b places this system among a rare group of known exoplanetary systems and as one that is amenable to continued study.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2021

A Stellar Activity F-statistic for Exoplanet Surveys (SAFE)

In the search for planets orbiting distant stars, the presence of stellar activity in the atmospheres of observed stars can obscure the radial velocity signal used to detect such planets. Furthermore, this stellar activity contamination is set by the star itself and cannot simply be avoided with better instrumentation. Various stellar activity indicators have been developed that may correlate with this contamination. We introduce a new stellar activity indicator called the Stellar Activity F-statistic for Exoplanet surveys (SAFE) that has higher statistical power (i.e., probability of detecting a true stellar activity signal) than many traditional stellar activity indicators in a simulation study of an active region on a Sun-like star with a moderate-to-high signal-to-noise ratio. Also through simulation, the SAFE is demonstrated to be associated with the projected area on the visible side of the star covered by active regions. We also demonstrate that the SAFE detects statistically significant stellar activity in most of the spectra for HD 22049, a star known to have high stellar variability. Additionally, the SAFE is calculated for recent observations of the three low-variability stars HD 34411, HD 10700, and HD 3651, the latter of which is known to have a planetary companion. As expected, the SAFE for these three only occasionally detects activity. Furthermore, initial exploration appears to indicate that the SAFE may be useful for disentangling stellar activity signals from planet-induced Doppler shifts.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

TESS Hunt for Young and Maturing Exoplanets (THYME). II. A 17 Myr Old Transiting Hot Jupiter in the Sco-Cen Association

We present the discovery of a transiting hot Jupiter orbiting HIP 67522 (T {sub eff} ∼ 5650 K; M {sub *} ∼ 1.2M {sub ⊙}) in the 10–20 Myr old Sco-Cen OB association. We identified the transits in the TESS data using our custom notch filter planet search pipeline and characterize the system with additional photometry from Spitzer; spectroscopy from SOAR/Goodman, SALT/HRS, LCOGT/NRES, and SMARTS/CHIRON; and speckle imaging from SOAR/HRCam. We model the photometry as a periodic Gaussian process with transits to account for stellar variability and find an orbital period of 6.9596{sub −0.000015}{sup +0.000016} days and radius of 10.02{sub −0.53}{sup +0.54} R {sub ⊕}. We also identify a single transit of an additional candidate planet with radius 8.01{sub −0.71}{sup +0.75} R {sub ⊕} that has an orbital period of ≳23 days. The validated planet HIP 67522b is currently the youngest transiting hot Jupiter discovered and is an ideal candidate for transmission spectroscopy and radial velocity follow-up studies, while also demonstrating that some young giant planets either form in situ at small orbital radii or else migrate promptly from formation sites farther out in the disk.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2011

Wide-Field Infrared Survey Telescope (WFIRST) Interim Report

The New Worlds, New Horizons (NWNH) in Astronomy and Astrophysics 2010 Decadal Survey prioritized the community consensus for ground-based and space-based observatories. Recognizing that many of the community s key questions could be answered with a wide-field infrared survey telescope in space, and that the decade would be one of budget austerity, WFIRST was top ranked in the large space mission category. In addition to the powerful new science that could be accomplished with a wide-field infrared telescope, the WFIRST mission was determined to be both technologically ready and only a small fraction of the cost of previous flagship missions, such as HST or JWST. In response to the top ranking by the community, NASA formed the WFIRST Science Definition Team (SDT) and Project Office. The SDT was charged with fleshing out the NWNH scientific requirements to a greater level of detail. NWNH evaluated the risk and cost of the JDEM-Omega mission design, as submitted by NASA, and stated that it should serve as the basis for the WFIRST mission. The SDT and Project Office were charged with developing a mission optimized for achieving the science goals laid out by the NWNH re-port. The SDT and Project Office opted to use the JDEM-Omega hardware configuration as an initial start-ing point for the hardware implementation. JDEM-Omega and WFIRST both have an infrared imager with a filter wheel, as well as counter-dispersed moderate resolution spectrometers. The primary advantage of space observations is being above the Earth's atmosphere, which absorbs, scatters, warps and emits light. Observing from above the atmosphere enables WFIRST to obtain precision infrared measurements of the shapes of galaxies for weak lensing, infrared light-curves of supernovae and exoplanet microlensing events with low systematic errors, and infrared measurements of the H hydrogen line to be cleanly detected in the 1<z<2 redshift range important for baryon acoustic oscillation (BAO) dark energy measurements. The Infrared Astronomical Satellite (IRAS), the Cosmic Background Explorer (COBE), Herschel, Spitzer, and Wide-field Infrared Sur-vey Explorer (WISE) are all space missions that have produced stunning new scientific advances by going to space to observe in the infrared. This interim report describes progress as of June 2011 on developing a requirements flowdown and an evaluation of scientific performance. An Interim Design Reference Mission (IDRM) configuration is presented that is based on the specifications of NWNH with some refinements to optimize the design in accordance with the new scientific requirements. Analysis of this WFIRST IDRM concept is in progress to ensure the capability of the observatory is compatible with the science requirements. The SDT and Project will continue to refine the mission concept over the coming year as design, analysis and simulation work are completed, resulting in the SDT s WFIRST Design Reference Mission (DRM) by the end of 2012.

Green, J.↗
DOE OSTI2021

β Cephei Pulsations in the High-mass Eclipsing System CW Cephei

CW Cep is an early B-type eclipsing binary with mass measurement precisions better than 1%. We report the discovery of pulsation signatures in the Transiting Exoplanet Survey Satellite time-series data of the system observed during Sectors 17 and 18. Our binary modeling indicates that the target star is a partially eclipsing detached system with masses of 12.95 M {sub ⊙} and 11.88 M {sub ⊙} and radii of 5.52 R {sub ⊙} and 5.09 R {sub ⊙} in an eccentric orbit of e = 0.0305. The distance to the eclipsing system, 928 ± 36 pc, is much more precise than the Gaia distance of 962 ± 453 pc. Applying multifrequency analyses to the residual light curve in the outside-eclipse part, we detected 13 significant signals in two frequency regions. Six frequencies below 1 day{sup −1} appeared to be mostly orbital harmonic and combination terms, or sidelobes due to insufficient removal of the binary effects. In contrast, seven frequencies clustered around 2.73 day{sup −1} and 5.34 day{sup −1} could be considered β Cep–type pulsations. Our results represent the second discovery of β Cep pulsations present in double-lined eclipsing binaries with precise masses and, hence, CW Cep serves as an important test bed for the asteroseismic modeling of high-mass stars.

47 OTHER INSTRUMENTATION↗
NASA NTRS2019

LUVOIR: Engineering Design & Technology Overview

The Large Ultraviolet / Optical / Infrared Surveyor (LUVOIR) is one of four large mission concepts being study by NASA in preparation for the 2020 Decadal Survey in Astronomy and Astrophysics. Over the past three and a half years, the LUVOIR Science and Technology Definition Team (STDT) and Study Office at NASA's Goddard Space Flight Center have developed a broad, compelling science case for LUVOIR, and detailed engineering point designs to achieve it. In this paper, we provide an overview of the two LUVOIR concepts: LUVOIR-A, a 15-m segmented, obscured aperture observatory, and LUVOIR-B, an 8-m segmented, unobscured aperture observatory. Both versions of LUVOIR cover a broad spectral range between 100 nm and 2.5 µm with a suite of imagers and spectrographs, including the High Definition Imager (HDI), the LUVOIR Ultraviolet Multi-object Spectrograph (LUMOS), and the Extreme Coronagraph for Living Planetary Systems (ECLIPS). LUVOIR-A will carry an additional fourth instrument, Pollux, a high-resolution UV spectropolarimeter being studied by the Centre National d'Etudes Spatiales (CNES). Both versions of LUVOIR are also designed to be serviceable and upgradeable. We will also provide a summary of the LUVOIR technology development program, which identifies critical enabling technologies, enhancing technologies, and a development plan to mature those technologies to TRL 6.

technology↗
DOE OSTI2020

Effects of radiation pressure on the evaporative wind of HD 209458b

ABSTRACT The role of radiation pressure in shaping exoplanet photoevaporation remains a topic of contention. Radiation pressure from the exoplanet’s host star has been proposed as a mechanism to drive the escaping atmosphere into a ‘cometary’ tail and explain the high velocities observed in systems where mass-loss is occurring. In this paper, we present results from high-resolution 3D hydrodynamic simulations of a planet similar to HD 209458b. We self-consistently launch a wind flowing outwards from the planet by calculating the ionization and heating resulting from incident high-energy radiation, and account for radiation pressure. We first present a simplified calculation, setting a limit on the Lyman-α flux required to drive the photoevaporated planetary material to larger radii and line-of-sight velocities. We then present the results of our simulations, which confirm the limits determined by our analytic calculation. We thus demonstrate that, within the limits of our hydrodynamic simulation and for the Lyman-α fluxes expected for HD 209458, radiation pressure is unlikely to significantly affect photoevaporative winds or to explain the high velocities at which wind material is observed, though further possibilities remain to be investigated.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

TESS Reveals HD 118203 b to be a Transiting Planet

The exoplanet HD 118203 b, orbiting a bright (V = 8.05) host star, was discovered using the radial velocity method by da Silva et al., but was not previously known to transit. Transiting Exoplanet Survey Satellite (TESS) photometry has revealed that this planet transits its host star. Nine planetary transits were observed by TESS, allowing us to measure the radius of the planet to be 1.136{sub −0.028}{sup +0.029}R{sub J}, and to calculate the planet mass to be 2.166{sub −0.079}{sup +0.074}M{sub J}. The host star is slightly evolved with an effective temperature of T{sub eff}=5683{sub −85}{sup +84} K and a surface gravity of log g=3.889{sub 0.018}{sup 0.017}. With an orbital period of 6.134985{sub −0.000030}{sup +0.000029} days and an eccentricity of 0.314 ± 0.017, the planet occupies a transitional regime between circularized hot Jupiters and more dynamically active planets at longer orbital periods. The host star is among the 10 brightest known to have transiting giant planets, providing opportunities for both planetary atmospheric and asteroseismic studies.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

TIC 278956474: Two Close Binaries in One Young Quadruple System Identified by TESS

We have identified a quadruple system with two close eclipsing binaries in Transiting Exoplanet Survey Satellite (TESS) data. The object is unresolved in Gaia and appears as a single source at parallax 1.08 ± 0.01 mas. Both binaries have observable primary and secondary eclipses and were monitored throughout TESS Cycle 1 (sectors 1–13), falling within the TESS Continuous Viewing Zone. In one eclipsing binary (P = 5.488 days), the smaller star is completely occluded by the larger star during the secondary eclipse; in the other (P = 5.674 days) both eclipses are grazing. Using these data, spectroscopy, speckle photometry, spectral energy distribution analysis, and evolutionary stellar tracks, we have constrained the masses and radii of the four stars in the two eclipsing binaries. The Li i equivalent width indicates an age of 10–50 Myr and, with an outer period of 858{sub −5}{sup +7} days, our analysis indicates this is one of the most compact young 2 + 2 quadruple systems known.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

On the Chemical Abundance of HR 8799 and the Planet c

Comparing chemical abundances of a planet and the host star reveals the origin and formation pathway of the planet. Stellar abundance is measured with high-resolution spectroscopy. Planet abundance, on the other hand, is usually inferred from low-resolution data. For directly imaged exoplanets, the data are available from a slew of high-contrast imaging/spectroscopy instruments. Here, we study the chemical abundance of HR 8799 and its planet c. We measure stellar abundance using LBT/PEPSI (R = 120,000) and archival HARPS data: stellar [C/H], [O/H], and C/O are 0.11 ± 0.12, 0.12 ± 0.14, and 0.54{sub −0.09}{sup +0.12}, all consistent with solar values. We conduct atmospheric retrieval using newly obtained Subaru/CHARIS data together with archival Gemini/GPI and Keck/OSIRIS data. We model the planet spectrum with petitRADTRANS and conduct retrieval using PyMultiNest. Retrieved planetary abundance can vary by ∼0.5 dex, from sub-stellar to stellar C and O abundances. The variation depends on whether strong priors are chosen to ensure a reasonable planet mass. Moreover, comparison with previous works also reveals inconsistency in abundance measurements. We discuss potential issues that can cause the inconsistency, e.g., systematics in individual data sets and different assumptions in the physics and chemistry in retrieval. We conclude that no robust retrieval can be obtained unless the issues are fully resolved.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Flare Statistics for Young Stars from a Convolutional Neural Network Analysis of TESS Data

All-sky photometric time-series missions have allowed for the monitoring of thousands of young (t age < 800 Myr) stars in order to understand the evolution of stellar activity. In this work, we developed a convolutional neural network (CNN), stella, specifically trained to find flares in Transiting Exoplanet Survey Satellite (TESS) short-cadence data. We applied the network to 3200 young stars in order to evaluate flare rates as a function of age and spectral type. The CNN takes a few seconds to identify flares on a single light curve. We also measured rotation periods for 1500 of our targets and find that flares of all amplitudes are present across all spot phases, suggesting high spot coverage across the entire surface. Additionally, flare rates and amplitudes decrease for stars t age > 50 Myr across all temperatures T eff ≥ 4000 K, while stars from 2300 ≤ T eff < 4000 K show no evolution across 800 Myr. Stars of T eff ≤ 4000 K also show higher flare rates and amplitudes across all ages. We investigate the effects of high flare rates on photoevaporative atmospheric mass loss for young planets. In the presence of flares, planets lose 4%–7% more atmosphere over the first 1 Gyr. stella is an open-source Python toolkit hosted on GitHub and PyPI.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

High-contrast Thermal Infrared Spectroscopy with ALES: The 3–4 μm Spectrum of κ Andromedae b

We present the first L-band (2.8–4.1 μm) spectroscopy of κ Andromedae b, a ~20 M Jup companion orbiting at 1" projected separation from its B9-type stellar host. We combine our Large Binocular Telescope (LBT) Arizona Lenslets for Exoplanet Spectroscopy (ALES) integral field spectrograph data with measurements from other instruments to analyze the atmosphere and physical characteristics of κ And b. We report a discrepancy of ~20% (2σ) in the L' flux of κ And b when comparing to previously published values. We add an additional L' constraint using an unpublished imaging data set collected in 2013 using the LBT Interferometer/LMIRCam, the instrument in which the ALES module has been built. The LMIRCam measurement is consistent with the ALES measurement, both suggesting a fainter L-band scaling than previous studies. The data, assuming the flux scaling measured by ALES and LMIRCam imaging, are well fit by an L3-type brown dwarf. Atmospheric model fits to measurements spanning 0.9–4.8 μm reveal some tension with the predictions of evolutionary models, but the proper choice of cloud parameters can provide some relief. In particular, models with clouds extending to very low pressures composed of grains ≤1 μm appear to be necessary. If the brighter L' photometry is accurate, there is a hint that subsolar metallicity may be required.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Hubble WFC3 Spectroscopy of the Habitable-zone Super-Earth LHS 1140 b

Atmospheric characterization of temperate, rocky planets is the holy grail of exoplanet studies. These worlds are at the limits of our capabilities with current instrumentation in transmission spectroscopy and challenge our state-of-the-art statistical techniques. Here we present the transmission spectrum of the temperate super-Earth LHS 1140b using the Hubble Space Telescope (HST). The Wide Field Camera 3 (WFC3) G141 grism data of this habitable-zone (T {sub eq} = 235 K) super-Earth (R = 1.7 R {sub ⊕}) shows tentative evidence of water. However, the signal-to-noise ratio, and thus the significance of the detection, is low and stellar contamination models can cause modulation over the spectral band probed. We attempt to correct for contamination using these models and find that, while many still lead to evidence for water, some could provide reasonable fits to the data without the need for molecular absorption although most of these cause features in the visible ground-based data which are nonphysical. Future observations with the James Webb Space Telescope would be capable of confirming, or refuting, this atmospheric detection.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

TOI-677b: A Warm Jupiter (P = 11.2 days) on an Eccentric Orbit Transiting a Late F-type Star

We report the discovery of TOI-677 b, first identified as a candidate in light curves obtained within Sectors 9 and 10 of the Transiting Exoplanet Survey Satellite (TESS) mission and confirmed with radial velocities. TOI-677 b has a mass of M{sub p} = 1.236{sub −0.067}{sup +0.069} M{sub J}, a radius of R{sub P} = 1.170±0.03 R{sub J}, and orbits its bright host star (V = 9.8 mag) with an orbital period of 11.23660±0.00011 d, on an eccentric orbit with e=0.435±0.024. The host star has a mass of M{sub ⋆} = 1.181±0.058 M{sub ⊙}, a radius of R{sub ⋆} = 1.28{sub −0.03}{sup +0.03} R{sub ⊙}, an age of 2.92{sub −0.73}{sup +0.80} Gyr and solar metallicity, properties consistent with a main-sequence late-F star with T{sub eff}=6295±77 K. We find evidence in the radial velocity measurements of a secondary long-term signal, which could be due to an outer companion. The TOI-677 b system is a well-suited target for Rossiter–Mclaughlin observations that can constrain migration mechanisms of close-in giant planets.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Kepler-90: Giant Transit-timing Variations Reveal a Super-puff

Exoplanet transit-timing variations (TTVs) caused by gravitational forces between planets can be used to determine planetary masses and orbital parameters. Most of the observed TTVs are small and sinusoidal in time, leading to degeneracies between the masses and orbital parameters. Here we report a TTV analysis of Kepler-90g and Kepler-90h, which exhibit large TTVs up to 25 hr. With optimization, we find a unique solution that allows us to constrain all of the orbital parameters. The best-fit masses for Kepler-90g and 90h are 15.0{sub −0.8}{sup +0.9} M {sub ⊕} (Earth mass) and 203{sub −5}{sup +5}M{sub ⊕}, respectively, with Kepler-90g having an unusually low apparent density of 0.15 ± 0.05 g cm{sup −3}. The uniqueness of orbital parameter solution enables a long-term dynamical integration, which reveals that although their periods are close to 2:3 orbital resonance, they are not locked in resonance, and the configuration is stable over billions of years. The dynamical history of the system suggests that planet interactions are able to raise the eccentricities and break the resonant lock after the initial formation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Atmospheric Escape From TOI-700 d: Venus versus Earth Analogs

The recent discovery of an Earth-sized planet (TOI-700 d) in the habitable zone of an early-type M-dwarf by the Transiting Exoplanet Survey Satellite constitutes an important advance. In this Letter, we assess the feasibility of this planet to retain an atmosphere – one of the chief ingredients for surface habitability – over long timescales by employing state-of-the-art magnetohydrodynamic models to simulate the stellar wind and the associated rates of atmospheric escape. We take two major factors into consideration, namely, the planetary atmospheric composition and magnetic field. In all cases, we determine that the atmospheric ion escape rates are potentially a few orders of magnitude higher than the inner Solar system planets, but TOI-700 d is nevertheless capable of retaining a 1 bar atmosphere over gigayear timescales for certain regions of the parameter space. The simulations show that the unmagnetized TOI-700 d with a 1 bar Earth-like atmosphere could be stripped away rather quickly (< 1 gigayear), while the unmagnetized TOI-700 d with a 1 bar CO 2 -dominated atmosphere could persist for many billions of years; we find that the magnetized Earth-like case falls in between these two scenarios. We also discuss the prospects for detecting radio emission of the planet (thereby constraining its magnetic field) and discerning the presence of an atmosphere.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

A Highly Eccentric Warm Jupiter Orbiting TIC 237913194

The orbital parameters of warm Jupiters serve as a record of their formation history, providing constraints on formation scenarios for giant planets on close and intermediate orbits. Here, we report the discovery of TIC 237913194b, detected in full-frame images from Sectors 1 and 2 of the Transiting Exoplanet Survey Satellite (TESS), ground-based photometry (Chilean–Hungarian Automated Telescope, Las Cumbres Observatory Global Telescope), and Fiber-fed Extended Range Optical Spectrograph radial velocity time series. We constrain its mass to M P = 1.942 -0.091 +0.091 M J and its radius to R P = 1.117 -0.047 +0.054 R J , implying a bulk density similar to Neptune’s. It orbits a G-type star (M &lowast; = 1.026 -0.055 +0.057 M ⊙ , V = 12.1 mag) with a period of 15.17 days on one of the most eccentric orbits of all known warm giants (e ≈ 0.58). This extreme dynamical state points to a past interaction with an additional, undetected massive companion. A tidal evolution analysis showed a large tidal dissipation timescale, suggesting that the planet is not a progenitor for a hot Jupiter caught during its high-eccentricity migration. TIC 237913194b further represents an attractive opportunity to study the energy deposition and redistribution in the atmosphere of a warm Jupiter with high eccentricity.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Atmospheric Transmission Spectroscopy of Hot Jupiter KELT-10b using Synthetic Telluric Correction Software

High-resolution spectroscopic visible data were obtained with the Ultraviolet and Visible Echelle Spectrograph on the Very Large Telescope. Our goal was to analyze the data in an effort to detect the presence of sodium in the atmosphere of hot Jupiter exoplanet KELT-10b, as well as characterize the orbit of the planet via the Rossiter-McLaughlin effect. Eighty spectra were collected during a single transit of KELT-10b. After standard spectroscopic calibration using ESO-Reflex, the synthetic telluric modeling software molecfit was applied to remove terrestrial atmospheric effects, and to refine the wavelength calibration. Sodium is recognized by its characteristic absorption doublet located at 5895.924 and 5889.951 Å, which can be seen in the planet atmosphere transmission spectrum and through excess absorption during the transit. The radial velocity of the host star was analyzed by measuring the average shift of absorption features from spectrum to spectrum. Our results indicate a sodium detection in the planet transmission spectrum with a line contrast of 0.66% and 0.43% ± 0.09% for the sodium DII and DI lines, respectively. Excess absorption measurements agree to within one half combined standard deviation between the planet transmission spectrum (0.143% ± 0.020%, a 7σ detection) and during the time series (0.124% ± 0.034%, a 3.6σ detection) in a band 1.25 Å wide. The wavelength grid corrections provided by molecfit were insufficient to determine radial velocities and measure the Rossiter-McLaughlin effect.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗