Optical and Near-Infrared Radial Velocity Content of M Dwarfs: Testing Models with Barnard’s .

Etienne´ Artigau1, Lison Malo1,2, Ren´eDoyon1, Pedro Figueira3,4, Xavier Delfosse5, Nicola Astudillo-Defru6,7 Send correspondence to [email protected]

ABSTRACT High precision radial velocity (RV) measurements have been central in the study of exoplanets during the last two decades, from the early discovery of hot , to the recent mass measure- ments of -sized uncovered by transit surveys. While optical radial-velocity is now a mature field, there is currently a strong effort to push the technique into the near-infrared (nIR) domain (chiefly Y , J, H and K band passes) to probe planetary systems around late-type . The combined lower mass and luminosity of M dwarfs leads to an increased reflex RV signal for planets in the habitable zone compared to -like stars. The estimates on the detectabil- ity of planets rely on various instrumental characteristics, but also on a prior knowledge of the stellar spectrum. While the overall properties of M dwarf spectra have been extensively tested against observations, the same is not true for their detailed line profiles, which leads to signifi- cant uncertainties when converting a given signal-to-noise ratio to a corresponding RV precision as attainable on a given spectrograph. By combining archival CRIRES and HARPS data with ESPaDOnS data of Barnard’s star, we show that state-of-the-art models over-predict the Y and J-band RV content by more than a factor of ∼2, while under-predicting the H and K-band content by half. Subject headings: techniques: radial velocities, instrumentation: spectrographs, methods: data analysis, stars: low-mass

1. Introduction and context to find planets around other stars. The first exo- around a Sun-like star, , was found Radial velocity (RV) measurements at an ever through precise monitoring of its parent star ve- increasing precision have been central to our quest locity (Mayor & Queloz 1995) using the ELODIE spectrograph. Over the first decade following the 1 Institut de Recherche sur les Exoplan`etes (IREx), discovery of 51 Pegasi, radial velocity monitoring D´epartement de Physique, Universit´ede Montr´eal, C.P. 1 6128, Succ. Centre-Ville, Montr´eal,QC, H3C 3J7, Canada accounted for 88% of exoplanet discoveries. With 2 the launch of Corot and Kepler, and thanks to a arXiv:1803.07646v1 [astro-ph.IM] 20 Mar 2018 Canada-france-Hawaii Corporation, 65-1238 Mamala- hoa Hwy, Kamuela, HI, USA, 96743 number of ground-based surveys, the bulk of ex- 3European Southern Observatory, Alonso de Cordova oplanet discoveries now come from transit detec- 3107, Vitacura, Santiago, Chile tion. The upcoming launch of the Transiting Exo- 4 Instituto de Astrof´ısicae Ciˆenciasdo Espa¸co,Univer- planet Survey Satellite (TESS; Ricker et al. 2014) sidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal will provide an even larger sample of short-period 5Universit´e Grenoble Alpes, CNRS, IPAG, 38000 transiting planets around relatively bright stars Grenoble, France over most of the sky. Ground-based RV follow- 6Universidad de Concepci´on, Departamento de As- up of TESS discoveries will require a significant tronom´ıa,Casilla 160-C, Concepci´on,Chile 7 Observatoire de Gen`eve, Universit´ede Gen`eve, 51 ch. 1http://exoplanets.org des Maillettes, 1290 Sauverny, Switzerland

1 investment of observing time. RV searches con- study compared to planets orbiting Sun-like stars. tinue to play a crucial role in the field by discov- The smaller radius of M dwarfs (0.1−0.5R ) leads ering the majority of the planets in the close so- to much deeper transit depths for a given plane- lar neighborhood, which are not transiting, and tary radius ; this not only makes the discovery of in identifying long-period planets, to which tran- transiting planets more likely, but also facilitates sit searches are not sensitive due to the decreasing transit spectroscopy. Planets around M dwarfs in likelihood of transits on wide and sparse- the solar neighborhood will be also the first prior- ness of transits in time. RV is the prime tool to ity target to characterize atmosphere in coupling establish planetary system architectures out to a high contrasts imaging and high spectral resolu- few astronomical units. In most cases RV monitor- tion capacity of ELT (Snellen et al. 2015) or even ing is the only tool currently available to confirm 10-m telescopes (Lovis et al. 2016). The smaller transiting planets, measure their masses, and ulti- radius and lower temperature (2500-3900 K; Ra- mately constrain their bulk . Furthermore, jpurohit et al. 2013) leads to orbital separation RV measurements can lead to the discovery of ad- for habitable zone planets in the range of 0.017 ditional non-transiting planets in systems with a to 0.2 AU, which increases the likelihood of tran- transiting companion (Cloutier et al. 2017). sit and leads to short orbital periods (< 100 days) Only a handful of transiting Earth-sized plan- for these type of planets. Importantly, the tighter ets orbiting Sun-like stars are known due to the and lower-mass primary (0.07-0.5 M ) lead rarity of their occurrence (once a year) and shal- to a much larger radial-velocity signal than for low depth. No such planet has been found by RV a planet around a Sun-like stars. An Earth- surveys and none of the transiting ones can be mass planet around a field M5V star (∼0.15R , followed-up with current RV instrumentation due ∼ 0.1M , 3200 K) receiving the same illumination to the inherent high-precision required (e.g., the as the Earth has an orbital separation of 0.04 AU Earth produces an RV reflex motion on the Sun and an of 9 days and induces a with a semi-amplitude of only 10 cm/s). Charac- radial-velocity signal of 1 m/s. Signals at this terization of Earth-like planets is one of the major level can be detected by state-of-the-art velocime- goals of exoplanet science in the near-future, and ters such as HARPS (Pepe et al. 2000b) or Keck M dwarfs represent a short-cut for finding such HIRES (Vogt et al. 1994). The main drawback fac- planets with existing technologies. The interest of ing existing precision radial velocity (PRV) spec- M dwarf in the quest of habitable worlds is many- trograph arises from the faintness of M dwarfs at fold. Firstly, despite having a very incomplete optical wavelengths. The HARPS M dwarf planet census of nearby M-dwarf planets, transit surveys search (Bonfils et al. 2013) yielded ∼m/s RV pre- provide strong constraints on the occurrence rate cision in 15 min for stars brighter than V = 10. of planets around early M dwarfs. Within the Ke- Despite the fact that M dwarfs largely outnum- pler dataset, Dressing & Charbonneau (2015) de- ber Sun-like stars in the solar neighborhood, there rived an occurrence rate for 1−4 R⊕ planets of 2.5 are only 116 M dwarfs this bright, mostly of early per star, including an average of 0.56 Earth-sized (

2 at the telescope (CARMENES-IR, Quirrenbach Smette et al. (2015), Figure 1); due to these chal- et al. (2010)). Other instrument designs favor a lenges and the current lack of effort to develop far-red design (0.7 − 1.0µm), which is competi- mid-IR velocimetry, we will not attempt to esti- tive with nIR instruments through most of the mate the relative importance of this wavelength M dwarf regime except for very late-M dwarfs domain. Until recently, this was the only M dwarf (e.g., Maroon-X, PARAS, CARMENES-Optical; with a published near-infrared coverage at very Seifahrt et al. (2016); Chakraborty et al. (2014)). high (λ/∆λ > 70 000) resolution. The recent pub- The expected sensitivity of a radial-velocity spec- lication of CARMENES optical and near-infrared trograph depends on its ability to detect the in- spectrum (Reiners et al. 2017) of 324 M dwarfs duced RV Doppler effects at a level orders of mag- over the 0.52−1.71µm domain (g through most of nitudes smaller than the instrument’s resolution H band) largely fills this gap. The present dataset or the natural width of stellar features. also includes the K band (1.95-2.40µm), which In terms of instrumental development, there are is covered by a few pRV spectrograph (SPIRou, numerous stability and calibration challenges (see GIANO, CRIRES+). None of the currently pub- the review by Fischer et al. 2016 and references licly available high-resolution M dwarf spectrum therein), but with intrinsically stable and well- has been cleaned from telluric absorption through characterized instruments, this floor in the per- either the near-simultaneous observation of hot formances can be significantly smaller than astro- stars and/or the combination of exposures taken physical RV jitter and limitation due to the finite at varying barycentric velocity. signal-to-noise ratio (SNR) of observations. In the In an attempt to obtain a realistic estimate ideal case of a slightly active, slowly rotating star, of the RV content of M dwarfs and assess the the limitation is determined by the radial veloc- usefulness of existing models, we present a com- ity information content of the stellar spectra and parison between the spectrum of Barnard’s star the SNR with which is achieved, and the purpose observed with HARPS, ESPaDOnS and CRIRES of the present paper is to assess this limitation (Pepe et al. 2000b; Donati et al. 2006; Kaeufl et al. for various instrument setups. Connes (1985) and 2004) and -ACES models over the op- Bouchy et al. (2001) provide a formal framework tical and near-infrared domain (0.4 − 2.35µm). for determining the ultimate radial velocity preci- We present the properties of Barnard’s star in sion that can be reached for a given spectrum and section 2 and the dataset used to determine its SNR. radial-velocity content in section 3. Results and Modeled M dwarf spectra can be used to de- discrepancies between the observed and model RV rive expected sensitivities for existing and upcom- content are presented in section 4 while section 5 ing nIR precision radial velocity (pRV) spectro- discusses the implication of these results for nIR graphs (e.g., Reiners et al. 2010; Rodler et al. 2011; PRV instruments. Figueira et al. 2016 ). As the radial velocity con- tent of a spectrum scales as the resolution to the 2. Barnard’s star properties 3 power 2 (Pepe et al. 2014), modest errors in stel- lar line profiles or incomplete line lists may lead Barnard’s star, at 1.8 pc from the Sun, is the to significant errors in the estimation of the RV second closest M dwarf after , content of the underlying spectrum. For the cur- and it holds the distinction of being the star rent analysis, we used the spectra of Barnard’s with the highest apparent proper motion (Barnard star in the CRIRES-POP spectral library (Lebzel- 1916). As a 7-10 Gyr thick-disk star, it is ex- ter et al. 2012). These observations cover the Y , pected to be a slow rotator; while its rotation period has not been unambiguously established, J, H, Ks, L and M bands; of specific interest here being the spectral domain, shortward of 2.38 µm, HST guider photometry points toward a period that is amenable to m/s-level precision velocime- of ∼ 130 days Benedict et al. (1998). This pe- try. While the 3 − 5µm domain contains strong riod is in very good agreement with the estima- molecular bandpasses that could be of interest for tion of Astudillo-Defru et al. 2017 through a de- 0 velocimetry, it also suffers from strong telluric ab- termination of log RHK = −5.7. With a 0.2R sorption and increased thermal background (See radius, this corresponds to a v sin i smaller than

3 0.08 km/s, a negligible contributor compared to the natural line width (thermal, turbulence) or instrumental. Instrumental broadenings are, at best, on the order of one to a few km/s in the op- tical (e.g., PEPSI, ESPRESSO respectively with resolutions of up to 1.2 and 2.5 km/s; Strassmeier et al. 2015; M´egevand et al. 2014) or 3-4 km/s in the near-infrared. It only shows a modest activ- ity level with occasional flaring activity (Paulson et al. 2006) and its M4V spectral type corresponds Table 1: Physical properties of Barnard’s star. to that of the bulk of nearby M dwarfs; further- more, Barnard’s star spectral type is close to the Other names median of that expected for TESS targets (Sulli- GJ 699, HIP 87937, 2MASS J17574849+0441405 van et al. 2015). Its surface is expected Spectral type M4 Ve1 to be slightly above log g = 5.0 from evolutionary Rotation period ∼130 days2 models (see Figure 1); this value is overall consis- vsin i ≤ 80 m/s2 tent with measurements of field M surface Radius (e.g., S´egransanet al. 2003). Metallicity ([Fe/H]) S´egransanet al. 2003 0.196 ± 0.008 R determination in the litterature range from −0.13 Dawson & De Robertis 2004 0.200 ± 0.008 R to −0.52, for the comparison with the model we Temperature adopt [Fe/H]= −0.5. The properties of Barnard’s S´egransanet al. 2003 3163 ± 65 K star are summarized in Table 1. Dawson & De Robertis 2004 3134 ± 102 K As one of our immediate galactic neighbors, this Boyajian et al. 2012 3230 ± 10 K star has been subject to planet searches through Rojas-Ayala et al. 2012 3266 ± 29 K astrometry (Benedict et al. 1998), direct imaging Neves et al. 2014 3338 ± 110 K (Gauza et al. 2015) and radial velocity (Choi et al. Gaidos et al. 2014 3247 ± 61 K 2013; K¨ursteret al. 2003), but to date no planet Mann et al. 2015 3228 ± 60 K is known around this star and HARPS measure- Adopted Teff 3200 K ments exclude the existence of planets with a mass Metallicity [Fe/H] superior to 5 − 6 M⊕ in its habitable zone (Bonfils Rojas-Ayala et al. 2012 −0.39 ± 0.17 et al. 2013). Neves et al. 2013 −0.52 ± 0.08 Neves et al. 2014 −0.51 ± 0.09 3. Datasets and analysis Gaidos et al. 2014 −0.32 ± 0.08 Mann et al. 2015 −0.40 ± 0.08 3.1. The HARPS dataset Passegger et al. 2016 −0.13 ± 0.11 The HARPS high-resolution spectrum used is Adopted metallicity −0.5 the median-combination of 22 individual spec- Surface gravity (log g) trum obtained during a RV planets search (Bon- S´egransanet al. 2003 5.05 ± 0.09 fils et al. 2013). HARPS (Pepe et al. 2004) Adopted log g 5.0 is a fiber fed spectrograph at the ESO/3.6- 1 m telescope (La Silla, Chile). It covers the Kirkpatrick & McCarthy 1994 2 380 - 680 nm wavelength domain with a resolution Benedict et al. 1998 of λ/δλ ∼ 115 000. We used 104 HARPS spectra from the ESO archive2 to build a high SNR (∼850 per element) template of Barnard’s star. The in- dividual spectra are reprocessed with the latest version of the standard HARPS pipeline (Lovis & Pepe 2007) which uses nightly set of calibration

2IDs 072.C-0488, 183.C-0437

4 CCD readout mode, to get a resolving power of R ∼ 68 000 covering the 3700 to 10 500 A˚ spec- 5.2 10 Gyr 1 Gyr tral domain over 40 grating orders. The integra- 100 Myr tion time was 90 s taken at an airmass of 2.2. 5.0 10 Myr The resolution for ESPaDONS is that of the far-

4.8 red, between the domain covered by HARPS and CRIRES; this value is derived from calibration 4.6

log g (cgs) lamps taken close in time to our observation.

0.1 M 0.2 Msol 0.4 M The raw frame was processed by CFHT QSO 4.4 sol sol 0.6 Msol team using UPENA1.0, an in-house software that 4.2 calls the Libre-ESpRIT pipeline Donati et al. 1997. Libre-ESpRIT performs optimal extrac- 2800 3000 3200 3400 3600 3800 4000 Temperature (K) tion of ESPaDOnS unpolarized (Stokes I) spec- trum of the Star and the Sky fibers following the Fig. 1.— Chabrier et al. (2000) evolutionary mod- procedure described in Donati et al. 1997. In the els for M dwarfs. At a temperature of 3200 K and present analysis, we used the processed subtracted an thick-disk age, Barnard’s star is expected to (Star-Sky) spectrum with a normalized contin- have a log g = 5.1 − 5.2 surface gravity, hence jus- uum. The accurate wavelength solution that ac- tifying the choice for log g = 5.0 atmosphere mod- counts for instrument drifts was measured from els. These values are confirmed by observations strong telluric absorption lines. ESPaDOnS typ- (S´egransanet al. 2003). ically shows drifts well within a resolution ele- . ment, with typical values below 300 m/s. At the time of our observations, the measured drift was −104 m/s. exposures to locate the orders, flat-field the spec- tra (Tungsten lamp illumination), and precisely 3.3. CRIRES-POP spectra determine the wavelength-calibration scale (ThAr lamp exposure). To build the template we shifted CRIRES spectra were drawn from the CRIRES- 3 all de-blazed spectra to the rest frame and re- POP spectral library(Lebzelter et al. 2012). In- sampled them to a common reference wavelength. dividual spectra from the library were analyzed We then computed the median flux per spectral separately and we did not attempt to merge expo- element, where the tellurics are discarded from the sure into a single spectrum as we were interested calculation. As the barycentric Earth radial ve- in the shape of line profiles rather than the bulk locity moves from −26.3 km/s to 26.5 km/s in the SED properties. Each spectra was drawn from the dataset, the stellar template is free from telluric library and correlated against a telluric absorption absorption. This technique is used to produce full spectrum. Slight offsets in the wavelength calibra- templates that are used in cross-correlations for tion (typically < 5 km/s) were corrected. We then RV measurements in M dwarfs that display rich extracted the time of observation from the file molecular bands, rather than binary masks that header and determined the barycentric correction are more appropriate for spectra dominated by for Barnard’s star. The CRIRES-POP dataset atomic lines; see Astudillo-Defru et al. 2015 for has recently been used as a test dataset for tel- further details on the construction of this telluric- luric line subtraction by modeling of absorption free template. (Smette et al. 2015); while a similar approach could have been applied here to extract RV infor- 3.2. The ESPaDOnS dataset mation from a larger spectral domain, we opted for the simpler approach of performing our analy- Optical high-resolution spectroscopy of Barnard’s sis on nearly telluric absorption-free (< 3%) parts star was obtained with ESPaDOnS on 2015, July of the optical and nIR. 30 HST at the Canada-France-Hawaii Telescope (CFHT). Observation was performed using the 3 http://www.univie.ac.at/crirespop/ “Star & Sky” mode combined with the “normal”

5 3.4. G¨ottigen spectral library by Phoenix 3.5. Barnard and field M4 photometry For our analysis, we used PHOENIX-ACES In order to derive an RV precision and com- models from the G¨ottigen spectral library4 (Husser pare the relative performance reached with various et al. 2013); these are among the most up-to- bandpasses, one needs to scale the flux with pho- date models available and are expected to better tometric measurement. We used the Mann et al. represent the nIR spectral features. More specif- (2015) grizJHK values for Barnard’s Star, but ically, we used the dataset labeled PHOENIX- no Y -band measurement is available in the liter- ACES-AGSS-COND-2011-HiRes. The model grid ature. We therefore used the mean Y − H color is available with a 100 K temperature step and for M3.5-M5.5 dwarfs in Hillenbrand et al. (2002) 0.5 dex log g and metallicity steps. For the pur- (Y − H = 1.07 ± 0.07, or Y = 5.87 ± 0.07). This pose of comparison with the model, we used a allows the scaling of Ne in equation 1. temperature of 3200 K, a sub-solar metallicity For all comparisons with z = 0 metallicity mod- (−0.5 dex) and log g = 5.0. Comparison with els, we use the mean colors for M4V stars in Mann solar metallicity models (0.0 dex) and low-gravity et al. 2015, excluding Barnard’s star; see numer- models (log g = 4.5) were also performed in order ical values in Table 2. These colors are used to to assess the impact of varying these parameters scale models and estimate the signal to noise ra- on the RV content. The model wavelength grid is tio of a given bandpass relative to J band. As finer than the instrumental resolution, which is a expected for a low-metallicity object (e.g., Bonfils necessary condition to properly re-sample models et al. 2005), Barnard star has slightly bluer opti- on the wavelength grid of the observations, with cal to near-infrared colors than field stars; g − J a sampling ranging from 0.3 to 0.6 km/s. This is and r − J colors being ∼ 0.3 mag bluer. All col- the same dataset as used by Figueira et al. (2016), ors with z, J, H and Ks bands are within 0.1 mag which leads to a better consistency between the of the field M4V. This is overall consistent with two analysis. the results from Bonfils et al. 2005, equation 1, In order to account for the finite resolution of where a 0.5 dex metalicity change corresponds to instruments, before comparison with observations, a 0.27 mag change in V −K. While the strength of models were convolved with the 1-D profile corre- molecular bands has a significant impact on the ra- sponding to that of a circular fiber. The adopted dial velocity content (see Section 4.3), the impact profile corresponds to the profile obtained by col- of color change is relatively modest, a difference lapsing a 2-D circle image onto one axis. For a of 0.3 mag corresponding to a ∼ 15 % difference fiber-fed spectrograph this corresponds to the pro- in SNR in the regime where observations are lim- file of a monochromatic line in the approximation ited by the counting statistics from the source’s where the optical design image quality is signifi- photons. cantly smaller than the diameter of the fiber. One can show that, arithmetically, this profile corre- 3.6. Telluric absorption spectrum sponds to a sin function between 0 and π. This Most of the near-infrared domain suffers from profile is representative of most fiber-fed spectro- absorption by the Earth’s atmosphere. Telluric graphs (e.g., NIRPS, SPIRou, HARPS). As we are absorption superimposes a set of sharp telluric interested in differences between modeled and ob- lines on the stellar spectrum. As the line-of-sight served line profiles, we verified that our results velocity of Barnard’s star changes through the were robust against a change in the assume instru- year by ±32 km/s, this component induces a time- mental line-spread-function. In addition to the varying signal that interferes with precise radial- collapsed-circle profile, We also performed all of velocity measurements. Telluric absorption repre- the analysis presented here with a gaussian profile sents a significant challenge to nIR pRV measure- having the same FWHM as the collapsed-circle ments and is discussed at length elsewhere (e.g., one. All of the conclusions drawn here remain Artigau et al. 2014b; Bean et al. 2010; Seifahrt valid with a gaussian profile. et al. 2010). Here, the main problem with tel- luric absorption in our dataset is that its numerous 4http://phoenix.astro.physik.uni-goettingen.de lines add a significant contribution to the RV con-

6 depth between models and observations, and not Table 2: Optical and near-infrared colors of M4.0- the impact of residual telluric absorption on high- M4.9 dwarfs in the Mann et al. 2015 sample and precision velocimetry. Barnard Star. The Y − J color is from Hillen- brand et al. 2002, see section 3.5. As expected for a 3.7. Useful RV domain in the presence of low-metallicity object, Barnard’s star has slightly telluric absorption bluer optical-to-nIR colors compared to field ob- jects of similar spectral type. Masking telluric lines from the stellar spectrum color Field Barnard leads to the rejection of part of the wavelength g − J 5.42 5.13 domain that may otherwise be used for radial ve- r − J 3.91 3.62 locity measurement, provided that efficient sub- i − J 2.34 2.21 traction of the telluric absorption contribution can z − J 1.48 1.44 be performed. Various techniques have been pro- Y − J 0.50 0.50 posed to do so: most using atmosphere models to J − H 0.56 0.49 fit telluric absorption (e.g., Gullikson et al. 2014; H − Ks 0.84 0.76 Smette et al. 2015), observing reference standard stars of B or A spectral type at roughly the same airmass as the observations (Vacca et al. 2003) or empirical modeling without prior knowledge of tel- tent of the spectrum of Barnard’s star. We used luric absorption (Artigau et al. 2014b). a model spectrum from the TAPAS5 (Bertaux Predicting an RV precision as derived from a et al. 2014) for the conditions prevailing at Paranal model spectrum using a given observational setup (airmass of 1, not convolved by an instrumen- in the presence of telluric absorption implies that tal line width, observation date set as January we assume that telluric absorption will be sub- 1st). We included all molecular opacities proposed tracted up to a certain level. A very conservative by the TAPAS interface (Rayleigh, H O, O ,O , 2 3 2 approach would reject all of the domain that is CO , CH and N ), with an ARLETTY atmo- 2 4 2 affected by telluric absorption at any given time spheric model corresponding to typical conditions through the year. Such drastic wavelength do- occuring in Paranal. The sampling of the telluric main rejection is definitely necessary when RV is absorption spectrum ranges from 0.2 to 1 km/s, computed in correlating the stellar spectrum to a which is higher than the resolving power of any of reference which is not exactly similar (for exam- our datasets and allows for an accurate interpola- ple a cross-correlation of the stellar spectra with tion onto the observed wavelength grid. We opted a numerical weighted mask). However, when the to compare only the radial velocity content of both template is similar to the spectra of the star (e.g., the observed and model spectra in domains where median spectrum), only wavelength domain under the atmospheric transmission is 97% or greater. In the telluric lines at the date of the measurement order to assess the impact of having weak telluric should be rejected. This is well demonstrated in absorption lines contaminating our stellar spec- the optical by Artigau et al. 2014b in using RV trum, we computed the model RV content with computation presented in Astudillo-Defru et al. and without multiplying by the TAPAS telluric 2015 for an early M and a K dwarf. transmission model. The impact of weak (< 3% absorption) telluric lines affects the RV content of Whether this holds in the near-infrared remains the stellar model at the 1% level and is deemed to be confirmed. As shown in Artigau et al. negligible in the current analysis. (2014b) for r-band HARPS observations of an M dwarf, domain with up to 10% telluric absorp- The exact amount of RV content that can be tion can be used for m/s RV measurements with a recovered in the presence of telluric absorption proper library of hot star observations. We there- and its impact on the ultimate RV precision is fore use this threshold for our RV precision predic- a non-trivial problem (e.g., section 3.7). Here tions in Section 5, but the aforementioned caveats we are interested in comparing line profile and apply. To illustrate that our conclusion are only mildly dependent on the exact threshold used for 5http://ether.ipsl.jussieu.fr/tapas/ telluric absorption masking, we also computed the

7 RV precision for a much more conservative telluric content between models or observations absorption rejection threshold of < 2%. In order and models. When comparing the RV precision to compare the same wavelength domains, models that one can reach assuming a SNR within a given were offset in radial velocity to match to match bandpass, one needs to properly scale the flux (i.e., that of Barnard’s star before masking telluric ab- the Ne term in equation 1) with actual photomet- sorption and computing the radial velocity content ric measurement from the target. As Q is a sum density (Q; see section 3.8). over a given wavelength domain and we are are considering in the spectral distribution of RV con- 3.8. Numerical formalism tent, we will express Q integrated over short wave- length domains. The notation Q therefore in- In the analysis, we follow the prescription of ∆λ/λ dicates a sum of Q for a running ∆λ/λ domain. Bouchy et al. (2001). This work evaluates the ultimate precision to which a velocity shift can 4. Results be determined in a well-sampled spectrum at high signal-to-noise ratio. The RV precision is related 4.1. Barnard’s star RV density content to the quality factor Q through the relation : The RV content of Barnard’s star spectrum c was measured from HARPS, ESPaDOnS and σRV = √ , (1) Q N e CRIRES. The Q value is only computed for telluric-free regions as described in Section 3.6. where c is the velocity of light and N the num- e The summation as expressed in Equation 3 is ber of electrons collected per resolution element, performed over ∆λ/λ = 0.2% domains through assuming that observations are photon-noise lim- the optical and nIR domain; derived empirical ited (i.e., the effective readout noise per resolution and modeled values are showed in Figure 2. The element is much smaller than the photon noise, √ Q values are globally consistent between models given by N ). As we are interested in compar- e and observations in the optical (riz bandpasses). ing the RV precision predicted by models with that Near-infrared Q values are much more discrepant, of observational data, only the Q value is relevant with Y and J-band values being over-estimated here as N is assumed to be the same. We there- e by models and H and K values under-estimated. fore set : From these values, one can determine a median 1 correction to be applied to models to predict σRV ∝ . (2) the RV precision reachable in the photon-limited Q regime for all optical to near-infrared bandpasses. Following Bouchy et al. 2001 notation, Q is The correction corresponds to the flux-weighted mean ratio of Qobserved/Qmodel. A correction fac- pΣW (i) tor of 0.5 would correspond to an equivalent in- Q = (3) p crease of a factor of 2 in RV error for a given SNR. ΣA0(i) The precision of the measurement worsens signif- with icantly, and this corresponds to a a four-fold loss in observing efficiency (i.e., assuming that signal- λ2(i)(∂A (i)/∂λ(i))2 W (i) = 0 ; (4) to-noise increases as the square root of integration A0(i) time). Correction values larger than one corre- spond to an improvement in precision; the RV A (i) and λ(i) respectively denote the flux at a 0 precision expressed in m/s decreases. Table 3 and given (i) resolution element and the wavelength of Figure 3 provides the corresponding relative Q fac- that resolution element. Q is independent of flux, tors Q /Q correction for grizY JHK and represents the density of RV content; conver- observed model bandpasses as well as the values derived for differ- sion into an actual RV precision therefore only de- ent stellar models. pends of the total flux (Ne). Assuming that the underlying SED is similar within the bandpass of interest, one can therefore directly compare the ratio of Q values to assess the differences in RV

8 105 g r i z Y J H K % 2 . 104 0 Q

HARPS, Barnard HARPS, model CRIRES, Barnard CRIRES, model ESPADONS, Barnard ESPADONS, model 103 6 × 10 1 100 2 × 100 Wavelength ( m)

g r i z Y J 2.00 ) l e

d 1.75 o m (

1.50 % 2 . 0 1.25 Q / ) d

e 1.00 v r e

s 0.75 b o (

% 0.50 2 . 0

Q HARPS 0.25 CRIRES ESPADONS, Barnard H K 0.00 6 × 10 1 100 2 × 100 Wavelength ( m)

Fig. 2.— (Top) Measured RV content of Barnard’s star over the optical and near-infrared domain. Overall measured (blue) and model (red) RV density are well matched blueward of ∼ 1µm. The agreement is poorer in the near-infrared domain with an over-prediction of RV content in Y and J bands and an under-prediction in H and K. (Bottom) Ratio of observed to model Q0.2% values. Areas unusable for RV measurements because of strong telluric absorption are filled in light blue.

9 whether the results described here hold for dif- ferent choices of model parameters. The previous g r i z Y J H K results, i.e. that the RV content is over-estimated 2.0 in Y and J and is under-estimated in H and K, remains true if one of the above parameters is 1.5 changed to one of the extremes of the plausible physical range. Table 3 and Figure 3 compile the

1.0 correction factor that needs to be applied on the

Correction factor RV precision at a given SNR for grizY JHK band- Nominal 0.5 Solar passes as derived from our dataset. The nominal logg = 5.5 T=3400 correction applies to a log g = 5.0, −0.5 dex metal- T=3000 licity and Teff = 3200 K, and corresponds to the 0.0 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 Wavelength ( m) nominal model values shown in Figure 3. The val- ues derived when using slightly different models differ, but the overall conclusions remain valid. As Fig. 3.— Correction factors (Q /Q ) observed model shown in Figure 4, models at solar metallicity have for the RV precision. The various models tested a higher Q value in the H and K, leading to a more are described in Section 4.2. modest correction than for sub-solar metallicity at the same Teff . Table 3: Multiplicative correction factors to be ap- When assuming a higher surface gravity (log g = plied on the RV precision derived from stellar mod- 5.5), the correction becomes more important in H els. These values correspond to the square-root of and K, but has little impact for other bandpasses. the flux-weighted mean Q ratio between observa- Assuming an effective temperature that is hotter tion and models for each bandpass. The nominal or cooler by 200 K (i.e., a larger difference than values are for a comparison with the default model suggested by any recent literature value, see Ta- described here, but we also explore the impact of ble 1) has little impact on our results. The exact other physical parameter choices. values for the correction are therefore only slightly Nominal Variants affected by the choice of physical parameters as- [Fe/H] −0.5 0.0 sumed for Barnard’s star and the main conclusions log g 5.0 5.5 remain unchanged. T (K) 3200 3400 3000 eff 4.3. Qualitative assessment of RV content g 0.66 0.63 0.69 0.82 0.51 differences r 0.82 0.76 0.84 1.08 0.60 i 0.94 0.81 0.99 1.26 0.73 The results we detailed in section 4.1 show a z 1.27 1.09 1.20 1.82 0.96 significant difference between predicted and ob- Y 0.29 0.30 0.27 0.30 0.25 served RV content for Barnard’s star in YJHK J 0.38 0.40 0.31 0.54 0.37 bands. The difference should lead to notable dif- H 1.37 0.95 1.82 1.42 1.23 ferences in a direct visual comparison of observed K 1.47 1.06 2.00 1.66 1.27 and model spectrum. Figure 5 represents two re- gions of the J and H bands, chosen due to the abundance of sharp lines. The over-estimation in the J band can be traced to deeper and sharper 4.2. Correction value dependence on model predicted lines than observed. As mentioned ear- choice lier, the RV content is proportional to the power 3 of the full-width at half-maximum (FWHM) of The exact physical parameters of Barnard’s star 2 (metallicity, effective temperature, surface grav- lines, so modest differences in line shape leads to ity) have been measured by several groups and significant differences in the predicted RV preci- modest discrepancies exist in the literature (See sion. In H band, numerous lines are observed Table 1). It is therefore important to assess but not predicted, which is most-likely due to in- complete line lists, as suggested by Figueira et al.

10 dwarfs and one cannot directly measure an effec- tive line shape directly with isolated lines as can be done for earlier-type stars. We therefore de- 6 logg=5.0, z = -0.5 (Barnard) termined the auto-correlation of the spectrum for logg=5.0, z = 0.0 (Field) 5 logg=4.5, z = 0.0 (Young) telluric-free parts of grizJHK bands. The auto- correlation profile of the stellar spectrum is di- 4 rectly linked to the mean line profile, both instru-

% mental and physical. From the auto-correlation 5 3 Q profile we recovered the effective mean line pro- 2 files (see Figure 6) for both observed and model spectra. In J and H, the full-width at half max- 1 imum of the line profile is ∼5 km/s, while models

0 predict significantly narrower lines in J. This is consistent with the results displayed in Figure 5, where numerous lines are deeper and narrower in 1.4 models than they are in the observed spectrum, thus leading to an over-estimation of the RV con- 1.2 tent in J. In the optical domain and K band, the o i

t agreement between the observed and model pro- a r

1.0

% files is remarkable. The only notable difference 5 Q between models and observations are the broader 0.8 line wings in the i and z bands.

0.6 5. Discussion 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 Wavelength ( m) The results presented here allow one to empir- ically correct RV content predictions from mod- els. The extent of the validity of these correction Fig. 4.— Density of RV content for the optical factors, both in effective temperature and surface and near-infrared domain. The top panel shows gravity, remain to be established with an anal- ysis comparable to the one presented here, but the RV density Q5% for 3 models; the nominal Barnard’s star model (red), field mid-M at so- spanning a range of spectral types. If we as- lar metallicity (green) and a low-surface gravity, sume that the Qobserved/Qmodel ratios measured young, M dwarf (blue). All 3 models are nor- for Barnard’s star hold at a solar metallicity, one can predict the RV precision that will be achiev- malized to the Q5% value of the ”Barnard’s star” model in J. The i-band (∼ 0.7µm) contains the able for mid-Ms observed by upcoming nIR RV highest RV density content, which favors instru- spectrographs. ments observing in the far-red (See section 5). We assume that a bandpass contribution to the −2 Solar-metallicity M dwarfs are expected to have RV budget scales as σRV. Two bandpasses that a higher RV content than Barnard’s star in the provide a σRV = 1.4 m/s contribute as much as near-infrared. The bottom panel shows in red the a single band for which a σRV = 1 m/s measure- Q density for the ”Barnard’s star” and in blue the ment is possible in the same amount of time. Fig- ”Young” models normalized to the ”Field” model. ure 7 shows the RV precision per bandpass that is reached for a Teff = 3200 K model in 3 metal- licity and surface gravity scenarios. The relative (2016). It is noteworthy that the RV content is contribution of Y and J to the near-infrared RV better determined in the optical and far-red, a content budget is predicted to be much smaller wavelength domain that has historically received than models suggest. For an instrument covering more attention. YJH at R ∼ 100 000 (e.g., NIRPS), predict that Lines are blended at all wavelengths for M Y , J and H contribute respectively 39%, 42% and

11 Sample J-band spectrum, RV precision ratio : 2.69 Sample H-band spectrum, RV precision ratio : 0.62

1.1 1.1

1.0 1.0

0.9 0.9

0.8 0.8 Barnard spectrum Barnard spectrum Model Model 0.7 Telluric : rejected domain 0.7 Telluric : rejected domain Telluric : useful domain Telluric : useful domain 0.6 0.6 Normalized flux and Transmission + 0.1 1.195 1.196 1.197 1.198 1.199 Normalized flux and Transmission + 0.1 1.688 1.689 1.690 1.691 1.692 1.693 1.694 Wavelength ( m) Wavelength ( m)

Fig. 5.— Spectrum of Barnard’s star from the CRIRES dataset (blue) and models (green) in representative regions of the J (top) and H (bottom) bands. The offset telluric absorption spectrum is shown, with regions included (teal) and excluded (red with cyan background shading) from the determination of Q. Within the J band, a large set of lines have over-estimated depth compared to estimations of the RV content. Within H, a numerous lines appear to be missing from models, leading to an under-estimation of the RV content. For the J-band sample spectrum domain shown here, the RV precision ratio is 2.69; for that wavelength domain, the RV precision limit reachable at a given SNR will be degraded by that amount.

g-band sample line profile (HARPS) r-band sample line profile (HARPS) i-band sample line profile (ESPaDOnS) z-band sample line profile (ESPaDOnS)

1.0 1.0 1.0 1.0

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

Effective mean line shape 0.0 Effective mean line shape 0.0 Effective mean line shape 0.0 Effective mean line shape 0.0

Observed; FWHM = 3.6 km/s Observed; FWHM = 3.4 km/s Observed; FWHM = 4.9 km/s Observed; FWHM = 5.5 km/s 0.2 Model; FWHM = 3.3 km/s 0.2 Model; FWHM = 3.3 km/s 0.2 Model; FWHM = 4.9 km/s 0.2 Model; FWHM = 5.3 km/s

10 5 0 5 10 10 5 0 5 10 10 5 0 5 10 10 5 0 5 10 v (km/s) v (km/s) v (km/s) v (km/s)

J-band sample line profile H-band sample line profile K-band sample line profile

1.0 1.0 1.0

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

Effective mean line shape 0.0 Effective mean line shape 0.0 Effective mean line shape 0.0 Observed; FWHM = 5.8 km/s Observed; FWHM = 5.8 km/s Observed; FWHM = 4.7 km/s 0.2 Model; FWHM = 3.4 km/s 0.2 Model; FWHM = 5.2 km/s 0.2 Model; FWHM = 4.6 km/s

10 5 0 5 10 10 5 0 5 10 10 5 0 5 10 v (km/s) v (km/s) v (km/s)

Fig. 6.— Effective mean line profile derived from auto-correlation function of a sample region of grizJHK bands. The profiles are remarkably well matched to the observations, except for J band where much broader profile partially explains the lower RV content compared to models.

12 19% of the RV budget. After correction and at and the performances of recently commissioned in- solar metallicity, the relative fraction is 7%, 14% struments is likely to improve in the future. Fur- and 79%. thermore, depending on the wavelength domain For an instrument covering YJHK domain probed, the sensitivity to stellar activity will dif- (e.g., SPIRou, GIANO), the relative contribution fer; infrared spectrographs being advantaged, for of Y and J is even smaller. Models suggest a sim- that matter, relative to optical and far-red PRV ilar contributions from all bandpasses (30%, 31%, instruments (e.g., Barnes et al. (2011)). Our com- 20%, 18%) but the correction described here leads parison therefore only applies to the photon-noise to a much larger relative contribution longward of contribution in the complete RV error budget, at 1.5 µm (3%, 6%, 45%, 47%) for solar metallicity. a common flux level. Overall, H and K move from a 38% to a 94% con- We confirm earlier results (e.g., Reiners et al. tribution to the RV budget. (2010), Seifahrt et al. (2016)) that spectrographs The importance of H and K band relative to covering the far-red (griz bands) outperform an Y and J implies that an RV spectrograph that instrument covering the YJHK domain at the observes within a single photometric bandpass at same spectral resolution. In the far-red, the higher a time such as CRIRES+ will be nearly as ef- RV content density compensates for the lower flux. ficient in H as a similar instrument that would Interestingly, in such a spectrograph, the i band cover the entire YJH domain. The inclusion of is more important than z despite the red i − z K in an instrument such as SPIRou nearly leads color of M dwarfs. Qualitatively, this can be seen to a doubling of the RV content. These results in Figure 2, where i band has a higher Q value cast a doubt on the conclusion by Rodler et al. than z. Reiners et al. (2017) presents an analysis 2011 that concludes that for M9 and L dwarfs, of 324 M dwarf spectrum in order to assess their the most important contribution the the RV con- radial-velocity content and its wavelength depen- tent came respectively from Y and J. Admittedly dency. There are notable differences between the our measurement of the RV content of Barnard’s present analysis of Barnard’s star spectrum and star concerns an object ∼1000 K hotter, but if the that of the representative mid-M shown (e.g., Fig- missing opacities in H and K are also present in ure 7 there-in and in particular the M3.5 Luytens very-late-Ms and Ls, then these results will need star). In our analysis, the relative contribution of to be revisited. Our results also underline the lim- J and H bands differs significantly while in Rein- itations of works such as Reiners et al. (2010) and ers et al. (2017), the two bands lead to comparable Figueira et al. (2016) who, being based on stel- RV accuracies. Similarly to our results, Figueira lar models very similar to the ones presented here, et al. (2016) predicted a precision much worse for were affected by important systematic errors in J than for H; for the M3 model, λ/∆λ = 80000, the RV estimates. v sin i = 1 and optimal telluric subtraction, the Having derived correction values for all pho- RV accuracy predicted varies from 5 m/s in H and tometric bandpasses, we can predict the perfor- 16.5 m/s in J (See Table A.1 there-in). As pointed mance for different spectrograph’s resolution and in Reiners et al. (2017), residual telluric absorp- nIR domain coverage. We explore the various tion may lead to an increase RV content in their scenario corresponding to existing and under de- dataset. Residual telluric absorption is also sug- velopment PRV spectrographs. Table 5 provides gested as an explanation for the mismatch between the RV precision reached for a common set of as- the RV-content based prediction of the RV uncer- sumptions regarding the target star. As for the tainties and the measured values. above calculation, we assume a mean SNR of 100 The lack of M dwarf spectral libraries cover- per ∆λ = 3 km/s at the center of J band. We ing the entire near-infrared at high-resolution until did not attempt to provide an exhaustive com- very recently incited previous authors to use mod- parison of the performances of RV spectrographs, els to predict RV content, which, in itself, adds an effort that would be much beyond the scope some uncertainties in the interpretation of results. of the current paper. PRV spectrographs are in- The recent publication of a sample of spectrum stalled on telescope of differing diameter, their obtained with CARMENES (Reiners et al. 2017) overall throughput and intrinsic stability differ partially fills this gap. A need nonetheless remains

13 for a near-infrared spectral cleaned form tel- sity of Hawaii. EA and RD acknowledge financial luric absorption, either through modeling and/or support from the National Science and Engineer- a combination of multiple observations obtained ing Research Council of Canada and the Trot- at sufficiently different barycentric velocities. tier Family Foundation. PF acknowledges sup- port by Funda¸c˜aopara a Ciˆenciae a Tecnologia (FCT) through Investigador FCT contract of ref-

14 erence IF/ 01037/2013, and POPH/FSE (EC) by Model, Barnard Corrected, Barnard FEDER funding through the program “Programa 12 Model, Solar Corrected, Solar Operacional de Factores de Competitividade - 10 COMPETE”. PF further acknowledges support from FCT in the form of an exploratory project of 8 reference IF/ 01037/2013CP1191/CT0001. XD 6 acknowledges the support of the PNP (Pro- gramme national de plan´etologie) and of the

RV accuracy (m/s) 4 Labex OSUG@2020.

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14 Table 4: Radial velocity achievable for a Teff = 3200 K M dwarf assuming a median SNR=100 in J, for a λ/∆λ = 105 resolution element, derived from models with log g = 5.0 and −0.5 dex metallicity (Barnard) and, log g = 5.0 and 0.0 dex metallicity (Solar). The corrected values derived from dataset detailed here are given. Values have been computed for the domain within each bandpass with a telluric absorption < 10% (nominal) and < 2% (conservative). Absorption < 10% Absorption < 2% Barnard Solar Barnard Solar Band Model Corr. Model Corr. Model Corr. Model Corr. (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) g 2.84 4.30 2.71 4.11 2.84 4.30 2.71 4.11 r 2.78 3.41 2.58 3.16 2.79 3.41 2.58 3.16 i 1.73 1.84 1.49 1.59 1.74 1.85 1.51 1.60 z 3.31 2.60 2.84 2.23 3.27 2.57 2.75 2.16 Y 3.49 11.90 3.56 12.15 3.47 11.84 3.55 12.11 J 3.35 8.72 3.52 9.15 3.27 8.51 3.45 8.96 H 4.99 3.64 3.45 2.51 5.14 3.75 3.54 2.58 K 5.34 3.63 3.86 2.63 6.29 4.28 4.51 3.07

Table 5: Same as Table 4, but comparing different instrument resolution and wavelength coverage. The approximate correspondence with existing and planned instrument is also given. These value compare instrumental setups at a common flux level (i.e., SNR=100 in J for an element of λ/∆λ = 105) and only account for difference in the radial velocity content contribution to the error budget. Not taken into account are the intrinsic instrument stability, difference in throughput, sensitivity to stellar activity, etc. Scenarios are designated by the bandpasses they cover and the spectral resolution expressed in thousand (e.g., YJH100) corresponds to a 0.98 − 1.8µm coverage at R=100 000. Absorption < 10% Absorption < 2% Barnard Solar Barnard Solar Scenario Model Corr. Model Corr. Model Corr. Model Corr. (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) (m/s) h100a 4.99 3.64 3.45 2.51 5.14 3.75 3.54 2.58 k100b 5.34 3.63 3.86 2.63 6.29 4.28 4.51 3.07 yjh70c 3.01 4.07 2.69 2.91 3.00 4.16 2.70 2.98 yjh80d 2.60 3.65 2.36 2.64 2.58 3.73 2.37 2.70 yjh100e 2.18 3.23 2.03 2.38 2.16 3.29 2.03 2.43 yjhk50f 3.35 3.48 2.81 2.51 3.44 3.78 2.92 2.71 yjhk70g 2.73 2.97 2.33 2.15 2.79 3.23 2.42 2.32 riz80h 1.62 1.66 1.44 1.46 1.62 1.66 1.44 1.46 gr100i 1.99 2.67 1.87 2.50 1.99 2.67 1.87 2.50 gr150j 1.68 2.25 1.53 2.05 1.68 2.25 1.53 2.05

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17