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J. R. Barnes

Publications and source records attributed to J. R. Barnes.

At least 19 recordsLinked to original sources

Three new exoplanet systems from the Dispersed Matter Planet Project

We present a radial velocity analysis of three bright, low-activity stars identified by the Dispersed Matter Planet Project (DMPP). We use a Bayesian framework to compare purely Keplerian models with models incorporating stellar activity via a quasi-periodic Gaussian Process (GP). DMPP-7 (HD 118006) is a slightly evolved star that harbours a single 0.72 Saturn-mass giant ($m_\textrm{p}\sin i$ = 69 M$_\oplus$) with an orbital period of P = 4.93 d. A longer 21 d - 22 d period cannot be conclusively confirmed as a stellar rotation signature rather than a purely Keplerian signal. For HD 67200, which exhibits Ca ii H&K variability, a model with only a GP is strongly favoured over a purely dynamical model. The GP model shows moderate evidence for a single Keplerian with P = 2.67 d. For HD 2134, a 21 d - 32 d rotation period signal is associated with tentative FWHM variability. A model with a GP is not conclusively favoured, but all models considered show moderate evidence for an additional single Keplerian with P = 2.78 d. Despite our target selection favouring near edge-on orbital geometries, we find no evidence for transits in TESS photometry. DMPP-7 b lies at the transition between the high-radius population and the Neptunian ridge and savannah regions. Further observations are required to establish whether the coherent short-period HD 67200 and HD 2134 signals are stellar or dynamical in origin. If planetary, the signals correspond to minimum masses of $m_\textrm{p}\sin i$ = 2.07 M$_\oplus$ and $m_\textrm{p}\sin i$ = 2.86 M$_\oplus$

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RedDots: Magnetic field of the nearby active M dwarf GJ 729, and a search for companions

M dwarfs are prime targets for discovering exoplanets, and the nearest M dwarfs to the Sun provide among the best opportunities for follow-up detailed exoplanet characterization. GJ 729, the seventh closest M dwarf to the Sun, presents significant challenges for exoplanet detection due to its high levels of magnetic activity. To address this, we present a detailed analysis of GJ 729's magnetic field and its variability, followed by a search for exoplanets beneath the activity-induced noise in the stellar radial velocity. The geometry of GJ 729's large-scale magnetic field was reconstructed using new and archival spectropolarimetric data for a total of four epochs spanning 10 years. Results indicate a weak large-scale field ranging from 50 to 145 G, and an evolving non-axisymmetric field geometry that varies from poloidal dominated to a near-equal poloidal-toroidal configuration. We modeled activity-induced radial velocity variations using Gaussian Process Regression and activity diagnostics, and searched for planetary companions using ~90 d of high-cadence spectra taken contemporaneously with the high-precision CARMENES and HARPS spectrographs. Activity-only and activity + Keplerian models offered statistically equivalent fits, with a consistently preferred Keplerian period of ~7 d and amplitude of ~1.9 m/s across a range of activity modeling approaches. This could relate to an Earth-mass or Super-Earth planet, or residual stellar activity with power concentrated at a multiple of the rotation half-period. Our findings provide insight into the magnetic behavior of fully convective M dwarfs, and highlight the potential and challenges of detecting Keplerian RV signatures that are only a fraction of activity amplitudes.

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Starspot activity and surface differential rotation on UX Arietis

We present new Doppler images of the K0 subgiant primary component of the RS CVn-type binary UX Arietis (UX Ari), derived from time-series spectra obtained in November--December of 2017 and 2024. Observations demonstrate that some spectral lines of the K0 IV component exhibit rapid changes on timescales of 1-2 hours, which seem not to be resulting from spot activity, meanwhile other spectral lines show no such fast variations. Through an investigation, we find that the Ca I 6439 $\unicode{x212B}$ profile shows variation that follows the rotational modulation of spots. Using this line as a reference, we derive the least-squares deconvolution (LSD) profile from the selected lines of each spectrum so as to generate a more reliable Doppler image, which is consistent with the shape of the corresponding Ca I 6439 $\unicode{x212B}$ line. The Doppler images are separately reconstructed from the Ca I 6439 $\unicode{x212B}$ and the LSD profiles for each dataset, and the surface maps are in good agreement with each other. All of the surface maps show dominant starspot structure at mid-to-high latitudes with appendages extending to the equator, while their locations differ by about 0.5 in the rotational phase between 2017 and 2024. In 2017 November-December, the main starspot group appears to be spatially associated with a large flare event just half a month later. Through the cross-correlation method, we have derived a weak anti-solar differential rotation for the primary component of UX Ari, while its equator belt is well tidally locked.

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RedDots: Multiplanet system around M dwarf GJ 887 in the solar neighborhood

GJ 887 is a bright M dwarf in the solar neighborhood with two currently reported nontransiting exoplanets with periods of $9~\mathrm{d}$ and $21~\mathrm{d,}$ along with an additional unconfirmed signal at $50~\mathrm{d}$. We reanalyzed the system with 101 new HARPS and 12 new ESPRESSO radial velocities (RVs) secured with a cadence to confirm or refute the origin of the $50~\mathrm{d}$ signal. To do so, we searched for signals related to stellar activity in photometric data and spectroscopic indicators. We modeled the stellar activity in the RVs with Gaussian processes (GPs). With the Bayesian analysis, we confirmed a four-planet model, including the two previously known planets at periods of $9.2619\pm0.0005~\mathrm{d}$ and $21.784\pm0.004~\mathrm{d,}$ as well as two newly confirmed exoplanets: an Earth-mass planet, with a $4.42490\pm0.00014~\mathrm{d}$ period and a sub-meter-per-second amplitude, and a super-Earth with a $50.77\pm0.05~\mathrm{d}$ period located in the habitable zone (HZ). This super-Earth is the second closest planet in the HZ, after Proxima Cen b. We found an additional signal in a 2:1 resonance with the $4.4~\mathrm{d}$ planet at $2.21661\pm0.00010~\mathrm{d}$ with an amplitude of $0.37\pm0.09~\mathrm{m/s}$, which could be related to an additional planet. However, other explanations of its origin are also plausible. This signal remains a candidate, as further investigation is required to confirm its true nature. If the signal is caused by a planet, its minimum mass would be half that of Earth. We measured the stellar rotation period with the characteristic periodic timescale of the GP. We found a period of $38.7\pm0.5~\mathrm{d}$, which is consistent with the rotation period determined from photometry and other activity indices.

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The CARMENES search for exoplanets around M dwarfs : Understanding the wavelength dependence of radial velocity measurements

Context. Current exoplanet surveys are focused on detecting small exoplanets orbiting in the liquid-water habitable zones of their host stars. Despite the recent significant advancements in instrumental developments, the current limitation in detecting these exoplanets is the intrinsic variability of the host star itself. Aims. Our aim is to use the full CARMENES guaranteed time observations (GTO) data set spanning more than 8 years of observations of over 350 stars to investigate the wavelength dependence of high-precision radial velocities (RV), as stellar activity features should exhibit a wavelength dependence while the RV variation due to an orbiting planet will be wavelength independent. Methods. We use the chromatic index (CRX) to quantify the slope of the measured RVs as a function of logarithmic wavelength. We investigate the dependence of the CRX in the full CARMENES GTO sample on 24 stellar activity indices in the visible and near-infrared channels of the CARMENES spectrograph and each star's stellar parameters. We also present an updated convective turnover time scaling for the calculation of the stellar Rossby number for M dwarfs. Results. Our results show that approximately 17\% of GTO stars show a strong or a moderate correlation between CRX and RV. We can improve the measured RVs by a factor of up to nearly 4 in rms by subtracting the RV predicted by the CRX-RV correlation from the measured RVs. Mid M dwarfs with moderate rotational velocities, moderate CRX-gradients and quasi-stable activity features have the best rms improvement factors. Conclusions. We conclude that the CRX is a powerful diagnostic in mitigation of stellar activity and the search for low mass rocky planets.

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HD 28471: a near-resonant compact multiplanet system with a possible cold giant planet

We present radial velocity measurements of the star HD 28471, observed by HARPS at the ESO 3.6 m telescope over a baseline of $\sim19$ years. We have searched for planetary companions to HD 28471 using kima, a trans-dimensional diffusive nested sampling algorithm where the number of planetary signals is explored as a free parameter. We detect a compact system of three planets, with signals in the preferred solution corresponding to orbits of $P\sim3.16,~6.12,~\textrm{and }11.68$ d. These planets lie firmly in the super-Earth and sub-Neptune mass regime, with (minimum) masses of $3.7, 5.7, \textrm{and }4.9$ M$_{\oplus}$, respectively. A long-period ($\sim1500$ d) signal is also strongly detected. Assessment of activity indicator periodicities and RV correlations suggests that the three short-period signals are genuine planets, but casts doubt upon the nature of the long-period signal. The origin may be a short stellar magnetic cycle, though additional data are required to fully sample the periodicity without intervening offsets. HD 28471 b exhibits a more eccentric orbit than the other planets, which may be due to dynamical interaction, or a result of RV variation from an as-yet-undetected 4th planet interior to this compact system. The detected planets lie close to a resonant configuration, indicating that the system may retain features of its natal configuration, with convergent migration potentially responsible for evolving the planets onto such short-period orbits.

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RedDots: Planetary masses in the GJ1061 system from planet-planet interaction

GJ1061 is a very nearby M star hosting three low-mass temperate planets detected from radial velocity variations. The close to 4:2:1 period commensurability of the planets, the available long-term monitoring of the system and new very high-precision radial velocity measurements from ESPRESSO enable the determination of masses from the planet-planet interaction. Using nested sampling, we derived parameter distributions for a co-planar configuration. The three planets (Mb =1.07 +- 0.11M_Earth, Pb =3.2073 +- 0.0003 d, Mc=1.76 +- 0.13M_Earth, Pc=6.6821 +- 0.0008 d, Md =1.55 +- 0.17M_Earth, Pd =13.066 +- 0.002 d) are potentially all rocky with equilibrium temperatures between 360 K and 240 K. This makes the GJ1061 system one of the prime targets for future ground or space based instruments suitable for a direct detection of the planetary atmospheres.

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RV-exoplanet eccentricities: Good, Beta, and Best

We examine the eccentricity distribution(s) of radial velocity detected exoplanets. Previously, the eccentricity distribution was found to be described well by a Beta distribution with shape parameters $a=0.867, b=3.03$. Increasing the sample size by a factor of 2.25, we find that the CDF regression method now prefers a mixture model of Rayleigh + Exponential distributions over the Beta distribution, with an increase in Bayesian evidence of $Δ\ln{Z}\sim 77$ ($12.6\,σ$). Using PDF regression, the eccentricity distribution is best described by a Gamma distribution, with a Rayleigh + Exponential mixture a close second. The mixture model parameters, $α= 0.68\pm0.05, λ=3.32\pm0.25, σ=0.11\pm0.01$, are consistent between methods. We corroborate findings that exoplanet eccentricities are drawn from independent parent distributions when splitting the sample by period, mass, and multiplicity. Systems with a known outer massive companion provide no positive evidence for an eccentricity distribution distinct from those without. We quantitatively show M-dwarf hosted planets share a common eccentricity distribution with those orbiting FGK-type stars. We release our python code, eccentriciPy, which allows bespoke tailoring of the input archive to create more relevant priors for particular problems in RV planet discovery and characterisation. We re-characterised example planets using either traditional Beta, or updated priors, finding differences for recovery of low-amplitude multi-signal systems. We explore the effects of a variety of prior choices. The accurate determination of small but non-zero eccentricity values has wide-ranging implications for modelling the structure and evolution of planets and their atmospheres due to the energy dissipated by tidal flexing.

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Further study of starspot activity and measurement of differential rotation for SZ Piscium

We present a series of 9 Doppler images of the magnetically active K component of the RS CVn-type binary SZ Psc, based on the high-resolution spectroscopic data collected from 2014 to 2018. We apply least-squares deconvolution to all spectra to extract the average profiles with high signal-to-noise ratios (SNRs) for Doppler imaging. The surface maps of the K subgiant show starspots widely distributed along latitude and longitude. A prominent, non-axisymmetric polar spot around phase 0 is revealed by all images with sufficient phase coverage, which may be a stable feature on the K component. The starspots evolve in a time scale of one month. We have determined the surface shear rate of the K component from the starspot maps reconstructed 10 days apart in 2017 Nov--Dec, through the cross-correlation method. The surface differential rotation parameters are $Ω_{eq} = 1.591 \pm 0.002$ rad d$^{-1}$ and $ΔΩ= 0.035 \pm 0.003$ rad d$^{-1}$. The absorption lines contributed from the tertiary component are detected in all LSD profiles of SZ Psc, and we measure the radial velocity of the binary system and the tertiary component to derive an elliptical orbit with a period of $1530 \pm 3$ days and a mass of $0.75 \pm 0.06$ M$\odot$ for the tertiary component.

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Identifying activity induced RV periodicities and correlations using Central Line Moments

The radial velocity (RV) method of exoplanet detection requires mitigation of nuisance signals arising from stellar activity. Using analytic cool and facular spot models, we explore the use of central line moments (CLMs) for recovering and monitoring rotation induced RV variability. Different spot distribution patterns, photosphere-spot contrast ratios and the presence or absence of the convective blueshift lead to differences in CLM signals between M dwarfs and G dwarfs. Harmonics of the rotation period are often recovered with the highest power in standard periodogram analyses. By contrast, we show the true stellar rotation may be more reliably recovered with string length minimisation. For solar minimum activity levels, recovery of the stellar rotation signal from CLMs is found to require unfeasibly high signal-to-noise observations. The stellar rotation period can be recovered at solar maximum activity levels from CLMs for reasonable cross-correlation function (CCF) signal-to-noise ratios $> 1000 - 5000$. The CLMs can be used to recover and monitor stellar activity through their mutual correlations and correlations with RV and bisector inverse span. The skewness of a CCF, a measure of asymmetry, is described by the third CLM, $M_3$. Our noise-free simulations indicate the linear RV vs $M_3$ correlation is up to 10 per cent higher than the RV vs bisector inverse span correlation. We find a corresponding $\sim 5$ per cent increase in linear correlation for CARMENES observations of the M star, AU Mic. We also assess the effectiveness of the time derivative of the second CLM, $M_2$, for monitoring stellar activity.

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RedDots: Limits on habitable and undetected planets orbiting nearby stars GJ 832, GJ 674, and Ross 128

Aims. Using HARPS spectroscopic data obtained by the RedDots campaign, as well as archival data from HARPS and CARMENES, supplemented with ASH2 and T90 photometry, we aim to search for additional planets around the three M dwarfs GJ 832, GJ 674, and Ross 128. We also aim to determine limits on possible undetected, habitable planets. We investigate (i) the reliability of the recovered orbital eccentricities and (ii) the reliability of Bayesian evidence as a diagnostic for selecting the best model. Methods. We employed Markov-chain Monte Carlo, nested sampling, and Gaussian process (GP) analyses to fit a total of 20 different models. We used the residuals to create grids for injection-recovery simulations to obtain detection limits on potentially undiscovered planets. Results. Our refined orbital elements for GJ 832 b, GJ 674 b, and Ross 128 b confirm (GJ 832, GJ 674) or increase (Ross 128) prior eccentricity determinations. No additional planets were found in any of the systems. The detection limits obtained for all three systems are between 30 and 50 cm/s for orbital periods in the range of 1 to 10 000 days. Using N-body simulations, we find that undiscovered secondary planets are unlikely (Ross 128) or incapable (GJ 674) of having caused the observed eccentricities of the known planets. We find that the eccentricity of GJ 832 b is not significantly different from zero. Conclusions. GJ 832 b, GJ 674 b, and Ross 128 b retain their status as hosting lonely and (for the latter two) eccentric planets. Finally, our results show that Bayesian evidence, when used in conjunction with GP, is not a robust diagnostic for selecting the best model in cases of low-activity stars. In such cases, we advise an inspection of the shapes of the posterior distributions and to ensure that relevant simulations are performed to assess the validity of the perceived best model.

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The variable magnetic field of V889 Her and the challenge of detecting exoplanets around young Suns using Gaussian process regression

Discovering exoplanets orbiting young Suns can provide insight into the formation and early evolution of our own solar system, but the extreme magnetic activity of young stars obfuscates exoplanet detection. Here we monitor the long-term magnetic field and chromospheric activity variability of the young solar analogue V889 Her, model the activity-induced radial velocity variations and evaluate the impacts of extreme magnetism on exoplanet detection thresholds. We map the magnetic field and surface brightness for 14 epochs between 2004 and 2019. Our results show potential 3-4 yr variations of the magnetic field which evolves from weak and simple during chromospheric activity minima to strong and complex during activity maxima but without any polarity reversals. A persistent, temporally-varying polar spot coexists with weaker, short-lived lower-latitude spots. Due to their different decay time-scales, significant differential rotation and the limited temporal coverage of our legacy data, we were unable to reliably model the activity-induced radial velocity using Gaussian Process regression. Doppler Imaging can be a useful method for modelling the magnetic activity jitter of extremely active stars using data with large phase gaps. Given our data and using Doppler Imaging to filter activity jitter, we estimate that we could detect Jupiter-mass planets with orbital periods of $\sim$3 d. A longer baseline of continuous observations is the best observing strategy for the detection of exoplanets orbiting highly active stars.

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DMPP-4: Candidate sub-Neptune mass planets orbiting a naked-eye star

We present radial velocity measurements of the very bright ($V\sim5.7$) nearby F star, DMPP-4 (HD 184960). The anomalously low Ca II H&K emission suggests mass loss from planets orbiting a low activity host star. Periodic radial velocity variability with $\sim 10$ ms$^{-1}$ amplitude is found to persist over a $>4$ year timescale. Although the non-simultaneous photometric variability in four TESS sectors supports the view of an inactive star, we identify periodic photometric signals and also find spectroscopic evidence for stellar activity. We used a posterior sampling algorithm that includes the number of Keplerian signals, $N_\textrm{p}$, as a free parameter to test and compare (1) purely Keplerian models (2) a Keplerian model with linear activity correlation and (3) Keplerian models with Gaussian processes. A preferred model, with one Keplerian and quasi-periodic Gaussian process indicates a planet with a period of $P_\textrm{b} = 3.4982^{+0.0015}_{-0.0027}$ d and corresponding minimum mass of $m_\textrm{b}\,\textrm{sin}\,i = 12.2^{+1.8}_{-1.9}$ M$_\oplus$. Without further high time resolution observations over a longer timescale, we cannot definitively rule out the purely Keplerian model with 2 candidates planets with $P_\textrm{b} = 2.4570^{+0.0026}_{-0.0462}$ d, minimum mass $m_\textrm{b}\,\textrm{sin}\,i = 8.0^{+1.1}_{-1.5}$ M$_\oplus$ and $P_\textrm{c} = 5.4196^{+0.6766}_{-0.0030}$ d and corresponding minimum mass of $m_\textrm{b}\,\textrm{sin}\,i = 12.2^{+1.4}_{-1.6}$ M$_\oplus$. The candidate planets lie in the region below the lower-envelope of the Neptune Desert. Continued mass loss may originate from the highly irradiated planets or from an as yet undetected body in the system.

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A Machine Learning approach for correcting radial velocities using physical observables

Precision radial velocity (RV) measurements continue to be a key tool to detect and characterise extrasolar planets. While instrumental precision keeps improving, stellar activity remains a barrier to obtain reliable measurements below 1-2 m/s accuracy. Using simulations and real data, we investigate the capabilities of a Deep Neural Network approach to produce activity free Doppler measurements of stars. As case studies we use observations of two known stars (Eps Eridani and AUMicroscopii), both with clear signals of activity induced RV variability. Synthetic data using the starsim code are generated for the observables (inputs) and the resulting RV signal (labels), and used to train a Deep Neural Network algorithm. We identify an architecture consisting of convolutional and fully connected layers that is adequate to the task. The indices investigated are mean line-profile parameters (width, bisector, contrast) and multi-band photometry. We demonstrate that the RV-independent approach can drastically reduce spurious Doppler variability from known physical effects such as spots, rotation and convective blueshift. We identify the combinations of activity indices with most predictive power. When applied to real observations, we observe a good match of the correction with the observed variability, but we also find that the noise reduction is not as good as in the simulations, probably due to the lack of detail in the simulated physics. We demonstrate that a model-driven machine learning approach is sufficient to clean Doppler signals from activity induced variability for well known physical effects. There are dozens of known activity related observables whose inversion power remains unexplored indicating that the use of additional indicators, more complete models, and more observations with optimised sampling strategies can lead to significant improvements in our detrending capabilities.

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Detailed stellar activity analysis and modelling of GJ 832: Reassessment of the putative habitable zone planet GJ 832c

Context. Gliese 832 (GJ 832) is an M2V star hosting a massive planet on a decade-long orbit, GJ 832b, discovered by radial velocity (RV). Later, a super Earth or mini-Neptune orbiting within the stellar habitable zone was reported (GJ 832c). The recently determined stellar rotation period (45.7 $\pm$ 9.3 days) is close to the orbital period of putative planet c (35.68 $\pm$ 0.03 days). Aims. We aim to confirm or dismiss the planetary nature of the RV signature attributed to GJ 832c, by adding 119 new RV data points, new photometric data, and an analysis of the spectroscopic stellar activity indicators. Additionally, we update the orbital parameters of the planetary system and search for additional signals. Methods. We performed a frequency content analysis of the RVs to search for periodic and stable signals. Radial velocity time series were modelled with Keplerians and Gaussian process (GP) regressions alongside activity indicators to subsequently compare them within a Bayesian framework. Results. We updated the stellar rotational period of GJ 832 from activity indicators, obtaining $37.5^{+1.4}_{-1.5}$ days, improving the precision by a factor of 6. The new photometric data are in agreement with this value. We detected an RV signal near 18 days (FAP < 4.6%), which is half of the stellar rotation period. Two Keplerians alone fail at modelling GJ 832b and a second planet with a 35-day orbital period. Moreover, the Bayesian evidence from the GP analysis of the RV data with simultaneous activity indices prefers a model without a second Keplerian, therefore negating the existence of planet c.

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The CARMENES search for exoplanets around M dwarfs: Benchmarking the impact of activity in high-precision radial velocity measurements

Current exoplanet surveys using the radial velocity (RV) technique are targeting M dwarfs because any habitable zone terrestrial-mass planets will induce a high RV and orbit on shorter periods than for more massive stars. One of the main caveats is that M dwarfs show a wide range of activity levels from inactive to very active, which can induce an asymmetry in the line profiles and, consequently, a spurious RV measurement. We aim to benchmark the impact of stellar activity on high-precision RV measurements using regular-cadence CARMENES visible and near-infrared observations of the active M3.5 dwarf EV Lac. We used the newly developed technique of low-resolution Doppler imaging to determine the centre-of-light, or spot-induced RV component, for eight observational epochs. We confirm a statistically significant and strong correlation between the independently measured centre-of-light and the chromatic index, which is a measure of the amplitude variation with wavelength of the RVs. We also find circular closed-loop relations of several activity indices with RV for a subset of data that covers only several rotation periods. We also investigate the implications of large phase gaps in the periodograms of activity indicators. Finally, by removing the spot-induced RV component we improve the planet-mass sensitivity by a factor of at least three. We conclude that for active M stars, a regular-cadence observing strategy is the most efficient way to identify and eliminate sources of correlated noise.

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A multiple planet system of super-Earths orbiting the brightest red dwarf star GJ887

The nearest exoplanets to the Sun are our best possibilities for detailed characterization. We report the discovery of a compact multi-planet system of super-Earths orbiting the nearby red dwarf GJ 887, using radial velocity measurements. The planets have orbital periods of 9.3 and 21.8~days. Assuming an Earth-like albedo, the equilibrium temperature of the 21.8 day planet is approx 350 K; which is interior, but close to the inner edge, of the liquid-water habitable zone. We also detect a further unconfirmed signal with a period of 50 days which could correspond to a third super-Earth in a more temperate orbit. GJ 887 is an unusually magnetically quiet red dwarf with a photometric variability below 500 parts-per-million, making its planets amenable to phase-resolved photometric characterization.

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Doppler imaging and differential rotation of $σ^{2}$ Coronae Borealis using SONG

We present new Doppler images of both components of the double-lined binary $σ^{2}$ CrB, based on the high-resolution spectroscopic data collected during 11 nights in 2015 March--April. The observed spectra form two independent data sets with sufficient phase coverage. We apply the least-squares deconvolution to all observed spectra to obtain high signal-to-noise mean profiles, from which we derive the Doppler images of both components of $σ^{2}$ CrB simultaneously. The surfaces of both F9 and G0 components are dominated by pronounced polar spots. The F9 component exhibits a weak spot at latitude 30$^{\circ}$ and its mid-to-low latitudes are relatively featureless. The G0 star shows an extended spot structure at latitude 30$^{\circ}$, and its surface spot coverage is larger than that of the F9 star, which suggests a higher level of magnetic activity. With the cross-correlation method, we derive a solar-like surface differential rotation on the G0 star of $σ^{2}$ CrB for the first time, and the surface shear rate is $ΔΩ= 0.180 \pm 0.004$ rad d$^{-1}$ and $α= ΔΩ/ Ω_{eq} = 0.032 \pm 0.001$. We do not obtain a clear surface shear law for the F9 star due to the lack of mid-to-low latitude features, but detect a systematic longitude shift of high-latitude spots, which indicates a slower rotation with respect to the co-rotating frame.

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