arXiv subjects

M. Zechmeister

Publications and source records attributed to M. Zechmeister.

At least 91 records · Page 5Linked to original sources

arXiv2020

The CARMENES search for exoplanets around M dwarfs. Variability of the He I line at 10830 Å

The He I infrared (IR) triplet at 10830 Åis known as an activity indicator in solar-type stars and has become a primary diagnostic in exoplanetary transmission spectroscopy. He I lines are a tracer of the stellar extreme-ultraviolet irradiation from the transition region and corona. We study the variability of the He I IR triplet lines in a spectral time series of 319 M~dwarf stars that was obtained with the CARMENES high-resolution optical and near-infrared spectrograph at Calar Alto. We detect He I IR line variability in 18% of our sample stars, all of which show H$α$ in emission. Therefore, we find detectable He I IR variability in 78% of the sub-sample of stars with H$α$ emission. Detectable variability is strongly concentrated in the latest spectral sub-types, where the He I IR lines during quiescence are typically weak. The fraction of stars with detectable He I IR variation remains lower than 10% for stars earlier than M3.0 V, while it exceeds 30% for the later spectral sub-types. Flares are accompanied by particularly pronounced line variations, including strongly broadened lines with red and blue asymmetries. However, we also find evidence for enhanced He I IR absorption, which is potentially associated with increased high-energy irradiation levels at flare onset. Generally, He I IR and H$α$ line variations tend to be correlated, with H$α$ being the most sensitive indicator in terms of pseudo-equivalent width variation. This makes the He I IR triplet a favourable target for planetary transmission spectroscopy.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. Measuring precise radial velocities in the near infrared: the example of the super-Earth CD Cet b

The high-resolution, dual channel, visible and near-infrared spectrograph CARMENES offers exciting opportunities for stellar and exoplanetary research on M dwarfs. In this work we address the challenge of reaching the highest radial velocity precision possible with a complex, actively cooled, cryogenic instrument, such as the near-infrared channel. We describe the performance of the instrument and the work flow used to derive precise Doppler measurements from the spectra. The capability of both CARMENES channels to detect small exoplanets is demonstrated with the example of the nearby M5.0 V star CD Cet (GJ 1057), around which we announce a super-Earth ($4.0\pm0.4\,M_\oplus$) companion on a 2.29 d orbit.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. The He I infrared triplet lines in PHOENIX models of M2-3 V stars

The He I infrared (IR) line at a vacuum wavelength of 10833 A is a diagnostic for the investigation of atmospheres of stars and planets orbiting them. For the first time, we study the behavior of the He I IR line in a set of chromospheric models for M-dwarf stars, whose much denser chromospheres may favor collisions for the level population over photoionization and recombination, which are believed to be dominant in solar-type stars. For this purpose, we use published PHOENIX models for stars of spectral types M2 V and M3 V and also compute new series of models with different levels of activity following an ansatz developed for the case of the Sun. We perform a detailed analysis of the behavior of the He I IR line within these models. We evaluate the line in relation to other chromospheric lines and also the influence of the extreme ultraviolet (EUV) radiation field. The analysis of the He I IR line strengths as a function of the respective EUV radiation field strengths suggests that the mechanism of photoionization and recombination is necessary to form the line for inactive models, while collisions start to play a role in our most active models. Moreover, the published model set, which is optimized in the ranges of the Na I D2, H$α$, and the bluest Ca II IR triplet line, gives an adequate prediction of the He I IR line for most stars of the stellar sample. Because especially the most inactive stars with weak He I IR lines are fit worst by our models, it seems that our assumption of a 100% filling factor of a single inactive component no longer holds for these stars.

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arXiv2020

A He I upper atmosphere around the warm Neptune GJ 3470b

High resolution transit spectroscopy has proven to be a reliable technique for the characterization of the chemical composition of exoplanet atmospheres. Taking advantage of the broad spectral coverage of the CARMENES spectrograph, we initiated a survey aimed at characterizing a broad range of planetary systems. Here, we report our observations of three transits of \tplanet with CARMENES in search of \het\ absorption. On one of the nights, the He~{\sc i} region was heavily contaminated by OH$^-$ telluric emission and, thus, it was not useful for our purposes. The remaining two nights had a very different signal-to-noise ratio (S/N) due to weather. They both indicate the presence of \het\ absorption in the transmission spectrum of \tplanet, although a statistically valid detection can only be claimed for the night with higher S/N. For that night, we retrieved a 1.5$\pm$0.3\% absorption depth, translating into a $R_p(λ)/R_p = 1.15\pm 0.14$ at this wavelength. Spectro-photometric light curves for this same night also indicate the presence of extra absorption during the planetary transit with a consistent absorption depth. The \het\ absorption is modeled in detail using a radiative transfer code, and the results of our modeling efforts are compared to the observations. We find that the mass-loss rate, \mlr, is confined to a range of 3\,$\times\,10^{10}$\,\gs\ for $T$ = 6000\,K to 10\,$\times\,10^{10}$\,\gs\ for $T$ = 9000\,K. We discuss the physical mechanisms and implications of the He~{\sc i} detection in \tplanet and put it in context as compared to similar detections and non-detections in other Neptune-size planets. We also present improved stellar and planetary parameter determinations based on our visible and near-infrared observations.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. A super-Earth planet orbiting HD 79211 (GJ 338 B)

We report on radial velocity time series for two M0.0V stars, GJ338B and GJ338A, using the CARMENES spectrograph, complemented by ground-telescope photometry from Las Cumbres and Sierra Nevada observatories. We aim to explore the presence of small planets in tight orbits using the spectroscopic radial velocity technique. We obtained 159 and 70 radial velocity measurements of GJ338B and A, respectively, with the CARMENES visible channel. We also compiled additional relative radial velocity measurements from the literature and a collection of astrometric data that cover 200 a of observations to solve for the binary orbit. We found dynamical masses of 0.64$\pm$0.07M$_\odot$ for GJ338B and 0.69$\pm$0.07M$_\odot$ for GJ338A. The CARMENES radial velocity periodograms show significant peaks at 16.61$\pm$0.04 d (GJ338B) and 16.3$^{+3.5}_{-1.3}$ d (GJ338A), which have counterparts at the same frequencies in CARMENES activity indicators and photometric light curves. We attribute these to stellar rotation. GJ338B shows two additional, significant signals at 8.27$\pm$0.01 and 24.45$\pm$0.02 d, with no obvious counterparts in the stellar activity indices. The former is likely the first harmonic of the star's rotation, while we ascribe the latter to the existence of a super-Earth planet with a minimum mass of 10.27$^{+1.47}_{-1.38}$$M_{\oplus}$ orbiting GJ338B. GJ338B b lies inside the inner boundary of the habitable zone around its parent star. It is one of the least massive planets ever found around any member of stellar binaries. The masses, spectral types, brightnesses, and even the rotational periods are very similar for both stars, which are likely coeval and formed from the same molecular cloud, yet they differ in the architecture of their planetary systems.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. Radial velocities and activity indicators from cross-correlation functions with weighted binary masks

For years, the standard procedure to measure radial velocities (RVs) of spectral observations consisted in cross-correlating the spectra with a binary mask, that is, a simple stellar template that contains information on the position and strength of stellar absorption lines. The cross-correlation function (CCF) profiles also provide several indicators of stellar activity. We present a methodology to first build weighted binary masks and, second, to compute the CCF of spectral observations with these masks from which we derive radial velocities and activity indicators. These methods are implemented in a python code that is publicly available. To build the masks, we selected a large number of sharp absorption lines based on the profile of the minima present in high signal-to-noise ratio (S/N) spectrum templates built from observations of reference stars. We computed the CCFs of observed spectra and derived RVs and the following three standard activity indicators: full-width-at-half-maximum as well as contrast and bisector inverse slope.We applied our methodology to CARMENES high-resolution spectra and obtain RV and activity indicator time series of more than 300 M dwarf stars observed for the main CARMENES survey. Compared with the standard CARMENES template matching pipeline, in general we obtain more precise RVs in the cases where the template used in the standard pipeline did not have enough S/N. We also show the behaviour of the three activity indicators for the active star YZ CMi and estimate the absolute RV of the M dwarfs analysed using the CCF RVs.

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arXiv2020

Is there NaI in the atmosphere of HD 209458b? Effect of the centre-to-limb variation and Rossiter-McLaughlin effect in transmission spectroscopy studies

HD 209458b was the first transiting planet discovered, and the first for which an atmosphere, in particular NaI, was detected. With time, it has become one of the most frequently studied planets, with a large diversity of atmospheric studies using low- and high-resolution spectroscopy. Here, we present transit spectroscopy observations of HD 209458b using the HARPS-N and CARMENES spectrographs. We fit the Rossiter-McLaughlin effect by combining radial velocity data from both instruments (nine transits in total), measuring a projected spin-orbit angle of $-1.6\pm0.3~{\rm deg}$. We also present the analysis of high-resolution transmission spectroscopy around the NaI region at $590~{\rm nm}$, using a total of five transit observations. In contrast to previous studies where atmospheric NaI absorption is detected, we find that for all of the nights, whether individually or combined, the transmission spectra can be explained by the combination of the centre-to-limb variation and the Rossiter-McLaughlin effect. This is also observed in the time-evolution maps and transmission light curves, but at lower signal-to-noise ratio. Other strong lines such as H$α$, CaII IRT, the MgI triplet region, and KI D1 are analysed, and are also consistent with the modelled effects, without considering any contribution from the exoplanet atmosphere. Thus, the transmission spectrum reveals no detectableNaI absorption in HD 209458b. We discuss how previous pioneering studies of this benchmark object may have overlooked these effects. While for some star-planet systems these effects are small, for other planetary atmospheres the results reported in the literature may require revision.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. Characterization of the nearby ultra-compact multiplanetary system YZ Ceti

The nearby ultra-compact multiplanetary system YZ Ceti consists of at least three planets. The orbital period of each planet is the subject of discussion in the literature due to strong aliasing in the radial velocity data. The stellar activity of this M dwarf also hampers significantly the derivation of the planetary parameters. With an additional 229 radial velocity measurements obtained since the discovery publication, we reanalyze the YZ Ceti system and resolve the alias issues. We use model comparison in the framework of Bayesian statistics and periodogram simulations based on a method by Dawson and Fabrycky to resolve the aliases. We discuss additional signals in the RV data, and derive the planetary parameters by simultaneously modeling the stellar activity with a Gaussian process regression model. To constrain the planetary parameters further we apply a stability analysis on our ensemble of Keplerian fits. We resolve the aliases: the three planets orbit the star with periods of $2.02$ d, $3.06$ d, and $4.66$ d. We also investigate an effect of the stellar rotational signal on the derivation of the planetary parameters, in particular the eccentricity of the innermost planet. Using photometry we determine the stellar rotational period to be close to $68$ d. From the absence of a transit event with TESS, we derive an upper limit of the inclination of $i_\mathrm{max} = 87.43$ deg. YZ Ceti is a prime example of a system where strong aliasing hindered the determination of the orbital periods of exoplanets. Additionally, stellar activity influences the derivation of planetary parameters and modeling them correctly is important for the reliable estimation of the orbital parameters in this specific compact system. Stability considerations then allow additional constraints to be placed on the planetary parameters.

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arXiv2020

The CARMENES search for exoplanets around M dwarfs. Dynamical characterization of the multiple planet system GJ 1148 and prospects of habitable exomoons around GJ 1148 b

Context. GJ 1148 is an M-dwarf star hosting a planetary system composed of two Saturn-mass planets in eccentric orbits with periods of 41.38 and 532.02 days. Aims. We reanalyze the orbital configuration and dynamics of the GJ 1148 multi-planetary system based on new precise radial velocity (RV) measurements taken with CARMENES. Methods. We combined new and archival precise Doppler measurements from CARMENES with those available from HIRES for GJ 1148 and modeled these data with a self-consistent dynamical model. We studied the orbital dynamics of the system using the secular theory and direct N-body integrations. The prospects of potentially habitable moons around GJ 1148 b were examined. Results. The refined dynamical analyses show that the GJ 1148 system is long-term stable in a large phase-space of orbital parameters with an orbital configuration suggesting apsidal alignment, but not in any particular high-order mean-motion resonant commensurability. GJ 1148 b orbits inside the optimistic habitable zone (HZ). We find only a narrow stability region around the planet where exomoons can exist. However, in this stable region exomoons exhibit quick orbital decay due to tidal interaction with the planet. Conclusions. The GJ 1148 planetary system is a very rare M-dwarf planetary system consisting of a pair of gas giants, the inner of which resides in the HZ. We conclude that habitable exomoons around GJ 1148 b are very unlikely to exist.

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arXiv2019

The CARMENES search for exoplanets around M dwarfs. The He I triplet at 10830 $\mathrmÅ\,$ across the M dwarf sequence

The He I infrared (IR) triplet at 10830 AA is an important activity indicator for the Sun and in solar-type stars, however, it has rarely been studied in relation to M dwarfs to date. In this study, we use the time-averaged spectra of 319 single stars with spectral types ranging from M0.0 V to M9.0 V obtained with the CARMENES high resolution optical and near-infrared spectrograph at Calar Alto to study the properties of the He I IR triplet lines. In quiescence, we find the triplet in absorption with a decrease of the measured pseudo equivalent width (pEW) towards later sub-types. For stars later than M5.0 V, the He I triplet becomes undetectable in our study. This dependence on effective temperature may be related to a change in chromospheric conditions along the M dwarf sequence. When an emission in the triplet is observed, we attribute it to flaring. The absence of emission during quiescence is consistent with line formation by photo-ionisation and recombination, while flare emission may be caused by collisions within dense material. The He I triplet tends to increase in depth according to increasing activity levels, ultimately becoming filled in; however, we do not find a correlation between the pEW(He IR) and X-ray properties. This behaviour may be attributed to the absence of very inactive stars ($L_{\rm X}/L_{\rm bol}$ < -5.5) in our sample or to the complex behaviour with regard to increasing depth and filling in.

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arXiv2019

Ionized calcium in the atmospheres of two ultra-hot exoplanets WASP-33b and KELT-9b

Ultra-hot Jupiters are emerging as a new class of exoplanets. Studying their chemical compositions and temperature structures will improve the understanding of their mass loss rate as well as their formation and evolution. We present the detection of ionized calcium in the two hottest giant exoplanets - KELT-9b and WASP-33b. By utilizing transit datasets from CARMENES and HARPS-N observations, we achieved high confidence level detections of Ca II using the cross-correlation method. We further obtain the transmission spectra around the individual lines of the Ca II H&K doublet and the near-infrared triplet, and measure their line profiles. The Ca II H&K lines have an average line depth of 2.02 $\pm$ 0.17 % (effective radius of 1.56 Rp) for WASP-33b and an average line depth of 0.78 $\pm$ 0.04 % (effective radius of 1.47 Rp) for KELT-9b, which indicates that the absorptions are from very high upper atmosphere layers close to the planetary Roche lobes. The observed Ca II lines are significantly deeper than the predicted values from the hydrostatic models. Such a discrepancy is probably a result of hydrodynamic outflow that transports a significant amount of Ca II into the upper atmosphere. The prominent Ca II detection with the lack of significant Ca I detection implies that calcium is mostly ionized in the upper atmospheres of the two planets.

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arXiv2019

A giant exoplanet orbiting a very low-mass star challenges planet formation models

Statistical analyses from exoplanet surveys around low-mass stars indicate that super-Earth and Neptune-mass planets are more frequent than gas giants around such stars, in agreement with core accretion theory of planet formation. Using precise radial velocities derived from visual and near-infrared spectra, we report the discovery of a giant planet with a minimum mass of 0.46 Jupiter masses in an eccentric 204-day orbit around the very low-mass star GJ 3512. Dynamical models show that the high eccentricity of the orbit is most likely explained from planet-planet interactions. The reported planetary system challenges current formation theories and puts stringent constraints on the accretion and migration rates of planet formation and evolution models, indicating that disc instability may be more efficient in forming planets than previously thought.

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arXiv2019

The CARMENES search for exoplanets around M dwarfs. Two temperate Earth-mass planet candidates around Teegarden's Star

Context. Teegarden's Star is the brightest and one of the nearest ultra-cool dwarfs in the solar neighbourhood. For its late spectral type (M7.0V), the star shows relatively little activity and is a prime target for near-infrared radial velocity surveys such as CARMENES. Aims. As part of the CARMENES search for exoplanets around M dwarfs, we obtained more than 200 radial-velocity measurements of Teegarden's Star and analysed them for planetary signals. Methods. We find periodic variability in the radial velocities of Teegarden's Star. We also studied photometric measurements to rule out stellar brightness variations mimicking planetary signals. Results. We find evidence for two planet candidates, each with $1.1M_\oplus$ minimum mass, orbiting at periods of 4.91 and 11.4 d, respectively. No evidence for planetary transits could be found in archival and follow-up photometry. Small photometric variability is suggestive of slow rotation and old age. Conclusions. The two planets are among the lowest-mass planets discovered so far, and they are the first Earth-mass planets around an ultra-cool dwarf for which the masses have been determined using radial velocities.

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arXiv2019

Water vapor detection in the transmission spectra of HD 209458 b with the CARMENES NIR channel

Aims: We aim at detecting H$_2$O in the atmosphere of the hot Jupiter HD 209458 b and perform a multi-band study in the near infrared with CARMENES. Methods: The H$_2$O absorption lines from the planet's atmosphere are Doppler-shifted due to the large change in its radial velocity during transit. This shift is of the order of tens of km s$^{-1}$, whilst the Earth's telluric and the stellar lines can be considered quasi-static. We took advantage of this to remove the telluric and stellar lines using SYSREM, a principal component analysis algorithm. The residual spectra contain the signal from thousands of planetary molecular lines well below the noise level. We retrieve this information by cross-correlating the spectra with models of the atmospheric absorption. Results: We find evidence of H$_2$O in HD 209458 b with a signal-to-noise ratio (S/N) of 6.4. The signal is blueshifted by --5.2 $^{+2.6}_{-1.3}$ km s$^{-1}$, which, despite the error bars, is a firm indication of day-to-night winds at the terminator of this hot Jupiter. Additionally, we performed a multi-band study for the detection of H$_2$O individually from the three NIR bands covered by CARMENES. We detect H$_2$O from its 1.0 $μ$m band with a S/N of 5.8, and also find hints from the 1.15 $μ$m band, with a low S/N of 2.8. No clear planetary signal is found from the 1.4 $μ$m band. Conclusions: Our significant signal from the 1.0 $μ$m band in HD 209458 b represents the first detection of H$_2$O from this band, the bluest one to date. The unfavorable observational conditions might be the reason for the inconclusive detection from the stronger 1.15 and 1.4 $μ$m bands. H$_2$O is detected from the 1.0 $μ$m band in HD 209458 b, but hardly in HD 189733 b, which supports a stronger aerosol extinction in the latter.

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arXiv2019

Red Dots: A temperate 1.5 Earth-mass planet in a compact multi-terrestrial planet system around GJ1061

Small low-mass stars are favourable targets for the detection of rocky habitable planets. In particular, planetary systems in the solar neighbourhood are interesting and suitable for precise characterisation. The Red Dots campaigns seek to discover rocky planets orbiting nearby low-mass stars. The 2018 campaign targeted GJ 1061, which is the 20$^{\rm th}$ nearest star to the Sun. For three consecutive months we obtained nightly, high-precision radial velocity measurements with the HARPS spectrograph. We analysed these data together with archival HARPS data. We report the detection of three planet candidates with periods of $3.204\pm 0.001$, $6.689\pm 0.005$ and $13.03\pm 0.03$ days, which is close to 1:2:4 period commensurability. After several considerations related to the properties of the noise and sampling, we conclude that a 4$^{\rm th}$ signal is most likely explained by stellar rotation, although it may be due to a planet. The proposed three-planet system (and the potential four-planet solution) is long-term dynamically stable. Planet-planet gravitational interactions are below our current detection threshold. The minimum masses of the three planets range from $1.4\pm 0.2$ to $1.8\pm 0.3$ M$_\oplus$. Planet d, with $m \sin i = 1.68\pm0.25$ M$_\oplus$, receives a similar amount of energy as Earth receives from the Sun. Consequently it lies within the liquid-water habitable zone of the star and has a similar equilibrium temperature to Earth. GJ 1061 has very similar properties to Proxima Centauri but activity indices point to lower levels of stellar activity.

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arXiv2019

Planetary system around the nearby M dwarf GJ 357 including a transiting, hot, Earth-sized planet optimal for atmospheric characterization

We report the detection of a transiting Earth-size planet around GJ 357, a nearby M2.5V star, using data from the Transiting Exoplanet Survey Satellite (TESS). GJ 357 b (TOI-562.01) is a transiting, hot, Earth-sized planet (Teq=525+-11 K) with a radius of Rb=1.217+-0.084 Re and an orbital period of Pb=3.93 d. Precise stellar radial velocities from CARMENES and PFS, as well as archival data from HIRES, UVES, and HARPS also display a 3.93-day periodicity, confirming the planetary nature and leading to a planetary mass of Mb=1.84+-0.31 Me. In addition to the radial velocity signal for GJ 357 b, more periodicities are present in the data indicating the presence of two further planets in the system: GJ 357 c, with a minimum mass of Mc=3.40+-0.46 Me in a 9.12 d orbit, and GJ 357 d, with a minimum mass of Md=6.1+-1.0 Me in a 55.7 d orbit inside the habitable zone. The host is relatively inactive and exhibits a photometric rotation period of Prot=78+-2 d. GJ 357 b is to date the second closest transiting planet to the Sun, making it a prime target for further investigations such as transmission spectroscopy. Therefore, GJ 357 b represents one of the best terrestrial planets suitable for atmospheric characterization with the upcoming JWST and ground-based ELTs.

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arXiv2019

The CARMENES search for exoplanets around M dwarfs -- Photospheric parameters of target stars from high-resolution spectroscopy. II. Simultaneous multiwavelength range modeling of activity insensitive lines

We present precise photospheric parameters of 282 M dwarfs determined from fitting the most recent version of PHOENIX models to high-resolution CARMENES spectra in the visible (0.52 - 0.96 $μ$m) and near-infrared wavelength range (0.96 - 1.71 $μ$m). With its aim to search for habitable planets around M dwarfs, several planets of different masses have been detected. The characterization of the target sample is important for the ability to derive and constrain the physical properties of any planetary systems that are detected. As a continuation of previous work in this context, we derived the fundamental stellar parameters effective temperature, surface gravity, and metallicity of the CARMENES M-dwarf targets from PHOENIX model fits using a $χ^2$ method. We calculated updated PHOENIX stellar atmosphere models that include a new equation of state to especially account for spectral features of low-temperature stellar atmospheres as well as new atomic and molecular line lists. We show the importance of selecting magnetically insensitive lines for fitting to avoid effects of stellar activity in the line profiles. For the first time, we directly compare stellar parameters derived from multiwavelength range spectra, simultaneously observed for the same star. In comparison with literature values we show that fundamental parameters derived from visible spectra and visible and near-infrared spectra combined are in better agreement than those derived from the same spectra in the near-infrared alone.

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arXiv2019

The CARMENES search for exoplanets around M dwarfs Detection of a mini-Neptune around LSPM J2116+0234 and refinement of orbital parameters of a super-Earth around GJ 686 (BD+18 3421)

Although M dwarfs are known for high levels of stellar activity, they are ideal targets for the search of low-mass exoplanets with the radial velocity (RV) method. We report the discovery of a planetary-mass companion around LSPM J2116+0234 (M3.0 V) and confirm the existence of a planet orbiting GJ 686 (BD+18 3421; M1.0 V). The discovery of the planet around LSPM J2116+0234 is based on CARMENES RV observations in the visual and near-infrared channels. We confirm the planet orbiting around GJ 686 by analyzing the RV data spanning over two decades of observations from CARMENES VIS, HARPS-N, HARPS, and HIRES. We find planetary signals at 14.44 and 15.53 d in the RV data for LSPM J2116+0234 and GJ 686, respectively. Additionally, the RV, photometric time series, and various spectroscopic indicators show hints of variations of 42 d for LSPM J2116+0234 and 37 d for GJ 686, which we attribute to the stellar rotation periods. The orbital parameters of the planets are modeled with Keplerian fits together with correlated noise from the stellar activity. A mini-Neptune with a minimum mass of 11.8 Me orbits LSPM J2116+0234 producing an RV semi-amplitude of 6.19 m/s, while a super-Earth of mass 6.6 Me orbits GJ 686 and produces an RV semi-amplitude of 3.0 m/s. Both LSPM J2116+0234 and GJ 686 have planetary companions populating the regime of exoplanets with masses lower than 15 Me and orbital periods <20 d.

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