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Megan Ansdell

Publications and source records attributed to Megan Ansdell.

At least 55 records · Page 3Linked to original sources

New insights into the nature of transition disks from a complete disk survey of the Lupus star forming region

Transition disks with large dust cavities around young stars are promising targets for studying planet formation. Previous studies have revealed the presence of gas cavities inside the dust cavities hinting at recently formed, giant planets. However, many of these studies are biased towards the brightest disks in the nearby star forming regions, and it is not possible to derive reliable statistics that can be compared with exoplanet populations. We present the analysis of 11 transition disks with large cavities (>20 AU radius) from a complete disk survey of the Lupus star forming region, using ALMA Band 7 observations at 0.3" (22-30 AU radius) resolution of the 345 GHz continuum, 13CO and C18O 3-2 observations and the Spectral Energy Distribution of each source. Gas and dust surface density profiles are derived using the physical-chemical modeling code DALI. This is the first study of transition disks of large cavities within a complete disk survey within a star forming region. The dust cavity sizes range from 20-90 AU radius and in three cases, a gas cavity is resolved as well. The deep drops in gas density and large dust cavity sizes are consistent with clearing by giant planets. The fraction of transition disks with large cavities in Lupus is ~11%, which is inconsistent with exoplanet population studies of giant planets at wide orbits. Furthermore, we present a hypothesis of an evolutionary path for large massive disks evolving into transition disks with large cavities.

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Seeing double with K2: Testing re-inflation with two remarkably similar planets around red giant branch stars

Despite more than 20 years since the discovery of the first gas giant planet with an anomalously large radius, the mechanism for planet inflation remains unknown. Here, we report the discovery of EPIC228754001.01, an inflated gas giant planet found with the NASA K2 Mission, and a revised mass for another inflated planet, K2-97b. These planets reside on ~9 day orbits around host stars which recently evolved into red giants. We constrain the irradiation history of these planets using models constrained by asteroseismology and Keck/HIRES spectroscopy and radial velocity measurements. We measure planet radii of 1.31 +\- 0.11 Rjup and and 1.30 +\- 0.07 Rjup, respectively. These radii are typical for planets receiving the current irradiation, but not the former, zero age main sequence irradiation of these planets. This suggests that the current sizes of these planets are directly correlated to their current irradiation. Our precise constraints of the masses and radii of the stars and planets in these systems allow us to constrain the planetary heating efficiency of both systems as 0.03% +0.03%/-0.02%. These results are consistent with a planet re-inflation scenario, but suggest the efficiency of planet re-inflation may be lower than previously theorized. Finally, we discuss the agreement within 10% of stellar masses and radii, and planet masses, radii, and orbital periods of both systems and speculate that this may be due to selection bias in searching for planets around evolved stars.

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Identification of Young Stellar Variables with KELT for K2 I: Campaign 13 Taurus Dippers and Rotators

One of the most well-studied young stellar associations, Taurus-Auriga, will be observed by the extended Kepler mission, K2, in the spring of 2017. K2 Campaign 13 (C13) will be a unique opportunity to study many stars in this young association at high photometric precision and cadence. Using observations from the Kilodegree Extremely Little Telescope (KELT) survey, we identify "dippers", stochastic variables, and periodic variables among K2 C13 target stars. This release of the KELT data (lightcurve data in e-tables) provides the community with long-time baseline observations to assist in the understanding of the more exotic variables in the association. Transient-like phenomena on timescales of months to years are known characteristics in the light curves of young stellar objects, making contextual pre- and post-K2 observations critical to understanding their underlying processes. We are providing a comprehensive set of the KELT light curves for known Taurus-Auriga stars in K2 C13. The combined data sets from K2 and KELT should permit a broad array of investigations related to star formation, stellar variability, and protoplanetary environments.

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Identification of Young Stellar Variables with KELT for K2 II: The Upper Scorpius Association

High-precision photometry from space-based missions such as K2 and TESS enables detailed studies of young star variability. However, because space-based observing campaigns are often short (e.g., 80 days for K2), complementary long-baseline photometric surveys are critical for obtaining a complete understanding of young star variability, which can change on timescales of minutes to years. We therefore present and analyze light curves of members of the Upper Scorpius association made over 5.5 years by the ground-based Kilodegree Extremely Little Telescope (KELT), which complement the high-precision observations of this region taken by K2 during its Campaigns~2 and 15. We show that KELT data accurately identify the periodic signals found with high-precision K2 photometry, demonstrating the power of ground-based surveys in deriving stellar rotation periods of young stars. We also use KELT data to identify sources exhibiting variability that is likely related to circumstellar material and/or stellar activity cycles; these signatures are often unseen in the short-term K2 data, illustrating the importance of long-term monitoring surveys for studying the full range of young star variability. We provide the KELT light curves as electronic tables in an ongoing effort to establish legacy time-series datasets for young stellar clusters.

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ALMA Survey of Lupus Protoplanetary Disks I: Dust and Gas Masses

We present the first high-resolution sub-mm survey of both dust and gas for a large population of protoplanetary disks. Characterizing fundamental properties of protoplanetary disks on a statistical level is critical to understanding how disks evolve into the diverse exoplanet population. We use ALMA to survey 89 protoplanetary disks around stars with $M_{\ast}>0.1~M_{\odot}$ in the young (1--3~Myr), nearby (150--200~pc) Lupus complex. Our observations cover the 890~$μ$m continuum and the $^{13}$CO and C$^{18}$O 3--2 lines. We use the sub-mm continuum to constrain $M_{\rm dust}$ to a few Martian masses (0.2--0.4~$M_{\oplus}$) and the CO isotopologue lines to constrain $M_{\rm gas}$ to roughly a Jupiter mass (assuming ISM-like $\rm {[CO]/[H_2]}$ abundance). Of 89 sources, we detect 62 in continuum, 36 in $^{13}$CO, and 11 in C$^{18}$O at $>3σ$ significance. Stacking individually undetected sources limits their average dust mass to $\lesssim6$ Lunar masses (0.03~$M_{\oplus}$), indicating rapid evolution once disk clearing begins. We find a positive correlation between $M_{\rm dust}$ and $M_{\ast}$, and present the first evidence for a positive correlation between $M_{\rm gas}$ and $M_{\ast}$, which may explain the dependence of giant planet frequency on host star mass. The mean dust mass in Lupus is 3$\times$ higher than in Upper Sco, while the dust mass distributions in Lupus and Taurus are statistically indistinguishable. Most detected disks have $M_{\rm gas}\lesssim1~M_{\rm Jup}$ and gas-to-dust ratios $<100$, assuming ISM-like $\rm {[CO]/[H_2]}$ abundance; unless CO is very depleted, the inferred gas depletion indicates that planet formation is well underway by a few Myr and may explain the unexpected prevalence of super-Earths in the exoplanet population.

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An ALMA Survey of Protoplanetary Disks in the $σ$ Orionis Cluster

The $σ$ Orionis cluster is important for studying protoplanetary disk evolution, as its intermediate age ($\sim$3-5 Myr) is comparable to the median disk lifetime. We use ALMA to conduct a high-sensitivity survey of dust and gas in 92 protoplanetary disks around $σ$ Orionis members with $M_{\ast}\gtrsim0.1 M_{\odot}$. Our observations cover the 1.33 mm continuum and several CO $J=2-1$ lines: out of 92 sources, we detect 37 in the mm continuum and six in $^{12}$CO, three in $^{13}$CO, and none in C$^{18}$O. Using the continuum emission to estimate dust mass, we find only 11 disks with $M_{\rm dust}\gtrsim10 M_{\oplus}$, indicating that after only a few Myr of evolution most disks lack sufficient dust to form giant planet cores. Stacking the individually undetected continuum sources limits their average dust mass to 5$\times$ lower than that of the faintest detected disk, supporting theoretical models that indicate rapid dissipation once disk clearing begins. Comparing the protoplanetary disk population in $σ$ Orionis to those of other star-forming regions supports the steady decline in average dust mass and the steepening of the $M_{\rm dust}$-$M_{\ast}$ relation with age; studying these evolutionary trends can inform the relative importance of different disk processes during key eras of planet formation. External photoevaporation from the central O9 star is influencing disk evolution throughout the region: dust masses clearly decline with decreasing separation from the photoionizing source, and the handful of CO detections exist at projected separations $>1.5$ pc. Collectively, our findings indicate that giant planet formation is inherently rare and/or well underway by a few Myr of age.

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The Gold Standard: Accurate Stellar and Planetary Parameters for Eight Kepler M dwarf Systems Enabled by Parallaxes

We report parallaxes and proper motions from the Hawaii Infrared Parallax Program for eight nearby M dwarf stars with transiting exoplanets discovered by Kepler. We combine our directly measured distances with mass-luminosity and radius-luminosity relationships to significantly improve constraints on the host stars' properties. Our astrometry enables the identification of wide stellar companions to the planet hosts. Within our limited sample, all the multi-transiting planet hosts (three of three) appear to be single stars, while nearly all (four of five) of the systems with a single detected planet have wide stellar companions. By applying strict priors on average stellar density from our updated radius and mass in our transit fitting analysis, we measure the eccentricity probability distributions for each transiting planet. Planets in single-star systems tend to have smaller eccentricities than those in binaries, although this difference is not significant in our small sample. In the case of Kepler-42bcd, where the eccentricities are known to be $\simeq$0, we demonstrate that such systems can serve as powerful tests of M dwarf evolutionary models by working in $L_*-ρ_*$ space. The transit-fit density for Kepler-42bcd is inconsistent with model predictions at 2.1$σ$ (22%), but matches more empirical estimates at 0.2$σ$ (2%), consistent with earlier results showing model radii of M dwarfs are under-inflated. Gaia will provide high precision parallaxes for the entire Kepler M dwarf sample, and TESS will identify more planets transiting nearby, late-type stars, enabling significant improvements in our understanding of the eccentricity distribution of small planets and the parameters of late-type dwarfs.

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Dippers and Dusty Disks Edges: A Unified Model

We revisit the nature of large dips in flux from extinction by dusty circumstellar material that is observed by Kepler for many young stars in the Upper Sco and $ρ$ Oph star formation regions. These young, low-mass "dipper" stars are known to have low accretion rates and primarily host moderately evolved dusty circumstellar disks. Young low mass stars often exhibit rotating star spots that cause quasi-periodic photometric variations. We found no evidence for periods associated with the dips that are different from the star spot rotation period in spectrograms constructed from the light curves. The material causing the dips in most of these light curves must be approximately corotating with the star. We find that disk temperatures computed at the disk corotation radius are cool enough that dust should not sublime. Crude estimates for stellar magnetic field strengths and accretion rates are consistent with magnetospheric truncation near the corotation radius. Magnetospheric truncation models can explain why the dips are associated with material near corotation and how dusty material is lifted out of the midplane to obscure the star which would account for the large fraction of young low mass stars that are dippers. We propose that variations in disk orientation angle, stellar magnetic field dipole tilt axis, and disk accretion rate are underlying parameters accounting for differences in the dipper light curves.

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The Mysterious Dimmings of the T Tauri Star V1334 Tau

We present the discovery of two extended $\sim$0.12 mag dimming events of the weak-lined T-Tauri star V1334. The start of the first event was missed but came to an end in late 2003, and the second began in February 2009, and continues as of November 2016. Since the egress of the current event has not yet been observed, it suggests a period of $>$13 years if this event is periodic. Spectroscopic observations suggest the presence of a small inner disk, although the spectral energy distribution shows no infrared excess. We explore the possibility that the dimming events are caused by an orbiting body (e.g. a disk warp or dust trap), enhanced disk winds, hydrodynamical fluctuations of the inner disk, or a significant increase in the magnetic field flux at the surface of the star. We also find a $\sim$0.32 day periodic photometric signal that persists throughout the 2009 dimming which appears to not be due to ellipsoidal variations from a close stellar companion. High precision photometric observations of V1334 Tau during K2 campaign 13, combined with simultaneous photometric and spectroscopic observations from the ground, will provide crucial information about the photometric variability and its origin.

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Zodiacal Exoplanets in Time (ZEIT) IV: seven transiting planets in the Praesepe cluster

Open clusters and young stellar associations are attractive sites to search for planets and to test theories of planet formation, migration, and evolution. We present our search for, and characterization of, transiting planets in the ~800 Myr old Praesepe (Beehive, M44) Cluster from K2 light curves. We identify seven planet candidates, six of which we statistically validate to be real planets, the last of which requires more data. For each host star we obtain high-resolution NIR spectra to measure its projected rotational broadening and radial velocity, the latter of which we use to confirm cluster membership. We combine low-resolution spectra with the known cluster distance and metallicity to provide precise temperatures, masses, radii, and luminosities for the host stars. Combining our measurements of rotational broadening, rotation periods, and our derived stellar radii, we show that all planetary orbits are consistent with alignment to their host star's rotation. We fit the K2 light curves, including priors on stellar density to put constraints on the planetary eccentricities, all of which are consistent with zero. The difference between the number of planets found in Praesepe and Hyades (8 planets) and a similar dataset for Pleiades (0 planets, ~125 Myr) suggests a trend with age, but may be due to incompleteness of current search pipelines for younger, faster-rotating stars. We see increasing evidence that some planets continue to lose atmosphere past 800 Myr, as now two planets at this age have radii significantly larger than their older counterparts from Kepler.

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Lupus disks with faint CO isotopologues: low gas/dust or large carbon depletion?

An era has started in which gas and dust can be observed independently in protoplanetary disks, thanks to the recent surveys with ALMA. The first near-complete high-resolution disk survey in both dust and gas in a single star-forming region has been carried out in Lupus, finding surprisingly low gas/dust ratios. The goal of this work is to fully exploit CO isotopologues observations in Lupus, comparing them with physical-chemical model results, in order to obtain gas masses for a large number of disks. We have employed physical-chemical models to analyze continuum and CO isotopologues observations of Lupus disks, including isotope-selective processes and freeze-out. Employing also the ALMA 13CO-only detections, disk gas masses have been calculated for a total of 34 sources, expanding the sample of 10 disks studied by Ansdell et al. (2016), where also C18O was detected. We confirm that overall gas-masses are very low, often smaller than 1 $M_{\rm J}$, if volatile carbon is not depleted. Accordingly, global gas/dust ratios predominantly between 1 and 10. Low CO-based gas masses and gas/dust ratios may indicate rapid loss of gas, or alternatively chemical evolution, e.g. via sequestering of carbon from CO to more complex molecules, or carbon locked up in larger bodies. Current ALMA observations cannot distinguish between these two hypotheses. We have simulated both scenarios, but chemical model results do not allow us to rule out one of the two. Assuming that all Lupus disks have evolved mainly due to viscous processes over the past few Myr, the observed correlation between the current mass accretion rate and dust mass found by Manara et al. (2016) implies a constant gas-to-dust ratio, which is close to 100 based on the observed $M_{\rm disk}/\dot{M}_{\rm acc}$ ratio. This in turn points to a scenario in which carbon depletion is responsible for the low CO isotopologue line luminosities.

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Zodiacal Exoplanets in Time (ZEIT) III: A short-period planet orbiting a pre-main-sequence star in the Upper Scorpius OB Association

We confirm and characterize a close-in ($P_{\rm{orb}}$ = 5.425 days), super-Neptune sized ($5.04^{+0.34}_{-0.37}$ Earth radii) planet transiting K2-33 (2MASS J16101473-1919095), a late-type (M3) pre-main sequence (11 Myr-old) star in the Upper Scorpius subgroup of the Scorpius-Centaurus OB association. The host star has the kinematics of a member of the Upper Scorpius OB association, and its spectrum contains lithium absorption, an unambiguous sign of youth (<20 Myr) in late-type dwarfs. We combine photometry from K2 and the ground-based MEarth project to refine the planet's properties and constrain the host star's density. We determine \name's bolometric flux and effective temperature from moderate resolution spectra. By utilizing isochrones that include the effects of magnetic fields, we derive a precise radius (6-7%) and mass (16%) for the host star, and a stellar age consistent with the established value for Upper Scorpius. Follow-up high-resolution imaging and Doppler spectroscopy confirm that the transiting object is not a stellar companion or a background eclipsing binary blended with the target. The shape of the transit, the constancy of the transit depth and periodicity over 1.5 years, and the independence with wavelength rules out stellar variability, or a dust cloud or debris disk partially occulting the star as the source of the signal; we conclude it must instead be planetary in origin. The existence of K2-33b suggests close-in planets can form in situ or migrate within $\sim 10$ Myr, e.g., via interactions with a disk, and that long-timescale dynamical migration such as by Lidov-Kozai or planet-planet scattering is not responsible for all short-period planets.

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The Enigmatic and Ephemeral M Dwarf System KOI 6705: Cheshire Cat or Wild Goose?

We confirm a 0.995 d periodic planetary transit-like signal, KOI 6705.01, in the Kepler lightcurve of the star KIC 6423922. Optical and infrared spectra show that this star is a mid M-type dwarf with an effective temperature $= 3327 \pm 60$K, metallicity [Fe/H] $= -0.08 \pm 0.10$, radius $= 0.31 \pm 0.03 R_{\odot}$, and mass $= 0.28 \pm 0.05 M_{\odot}$. The star is $\approx 70$ pc away and its space motion, rotation period, and lack of H$α$ emission indicate it is an older member of the "thin disk" population. On the other hand, the star exhibits excess infrared emission suggesting a dust disk more typical of a very young star. If the KOI 6705.01 signal is produced by a planet, the transit depth of 60 ppm means its radius is only $0.26^{+0.034}_{-0.029} R_{\oplus}$, or about the size of the Moon. However, the duration ($\gtrsim 3$~hr) and time variation of KOI 6705.01 are anomalous: the signal was undetected in the first two years of the mission and increased through the latter two years. These characteristics require implausible orbits and material properties for any planet and rule out such an explanation, although a dust cloud is possible. We excluded several false positive scenarios including background stars, scattered light from stars that are nearby on the sky, and electronic cross-talk between detector readout channels. We find the most likely explanation to be that KOI 6705.01 is a false positive created by charge transfer inefficiency in a detector column on which KIC 6423922 and a 1.99 d eclipsing binary both happened to fall.

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A SCUBA-2 850-$μ$m survey of circumstellar disks in the $λ$ Orionis cluster

We present results from an 850-$μ$m survey of the $\sim$ 5 Myr old $λ$ Orionis star-forming region. We used the SCUBA-2 camera on the James Clerk Maxwell Telescope to survey a $\sim$0.5-diameter circular region containing 36 (out of 59) cluster members with infrared excesses indicative of circumstellar disks. We detected only one object at $>3σ$ significance, the Herbig Ae star HD 245185, with a flux density of $\sim74$ mJy beam$^{-1}$ corresponding to a dust mass of $\sim150$ M$_{\oplus}$. Stacking the individually undetected sources did not produce a significant mean signal but gives an upper limit on the average dust mass for $λ$ Orionis disks of $\sim3$ M$_{\oplus}$. Our follow-up observations of HD 245185 with the Submillimeter Array found weak CO 2--1 line emission with an integrated flux of $\sim170$ mJy km s$^{-1}$ but no $^{13}$CO or C$^{18}$O isotopologue emission at 30 mJy km s$^{-1}$ sensitivity, suggesting a gas mass of $\lesssim1$ M$_{\rm Jup}$. The implied gas-to-dust ratio is thus $\gtrsim50$ times lower than the canonical interstellar medium value, setting HD 245185 apart from other Herbig Ae disks of similar age, which have been found to be gas rich; as HD 245185 also shows signs of accretion, we may be catching it in the final phases of disk clearing. Our study of the $λ$ Orionis cluster places quantitative constraints on planet formation timescales, indicating that at $\sim5$ Myr the average disk no longer has sufficient dust and gas to form giant planets and perhaps even super Earths; the bulk material has been mostly dispersed or is locked in pebbles/planetesimals larger than a few mm in size.

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The Near-Ultraviolet Luminosity Function of Young, Early M-Type Dwarf Stars

Planets orbiting within the close-in habitable zones of M dwarf stars will be exposed to elevated high-energy radiation driven by strong magneto-hydrodynamic dynamos during stellar youth. Near-ultraviolet (NUV) irradiation can erode and alter the chemistry of planetary atmospheres, and a quantitative description of the evolution of NUV emission from M dwarfs is needed when modeling these effects. We investigated the NUV luminosity evolution of early M-type dwarfs by cross-correlating the Lepine & Gaidos (2011) catalog of bright M dwarfs with the GALEX catalog of NUV (1771-2831A) sources. Of the 4805 sources with GALEX counterparts, 797 have NUV emission significantly (> 2.5 sigma) in excess of an empirical basal level. We inspected these candidate active stars using visible-wavelength spectra, high-resolution adaptive optics imaging, time-series photometry, and literature searches to identify cases where the elevated NUV emission is due to unresolved background sources or stellar companions; we estimated the overall occurrence of these "false positives" as ~16%. We constructed a NUV luminosity function that accounted for false positives, detection biases of the source catalogs, and GALEX upper limits. We found the NUV luminosity function to be inconsistent with predictions from a constant star-formation rate and simplified age-activity relation defined by a two-parameter power law.

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Refined Rotational Period, Pole Solution & Shape Model for (3200) Phaethon

(3200) Phaethon exhibits both comet- and asteroid-like properties, suggesting it could be a rare transitional object such as a dormant comet or previously volatile-rich asteroid. This justifies detailed study of (3200) Phaethon's physical properties, as a better understanding of asteroid-comet transition objects can provide insight into minor body evolution. We therefore acquired time-series photometry of (3200) Phaethon over 15 nights from 1994 to 2013, primarily using the Tektronix 2048x2048 pixel CCD on the University of Hawaii 2.2-m telescope. We utilized light curve inversion to: (1) refine (3200) Phaethon's rotational period to P=3.6032+/-0.0008 h; (2) estimate a rotational pole orientation of lambda=+85+/-13 degrees and beta=-20+/-10 degrees; and (3) derive a shape model. We also used our extensive light curve dataset to estimate the slope parameter of (3200) Phaethon's phase curve as G~0.06, consistent with C-type asteroids. We discuss how this highly oblique pole orientation with a negative ecliptic latitude supports previous evidence for (3200) Phaethon's origin in the inner main asteroid belt as well as the potential for deeply buried volatiles fueling impulsive yet rare cometary outbursts.

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Prospecting in Ultracool Dwarfs: Measuring the Metallicities of Mid- and Late-M Dwarfs

Metallicity is a fundamental parameter that contributes to the physical characteristics of a star. However, the low temperatures and complex molecules present in M dwarf atmospheres make it difficult to measure their metallicities using techniques that have been commonly used for Sun-like stars. Although there has been significant progress in developing empirical methods to measure M dwarf metallicities over the last few years, these techniques have been developed primarily for early- to mid-M dwarfs. We present a method to measure the metallicity of mid- to late-M dwarfs from moderate resolution (R~2000) K-band (2.2 microns) spectra. We calibrate our formula using 44 wide binaries containing an F, G, K, or early M primary of known metallicity and a mid- to late-M dwarf companion. We show that similar features and techniques used for early M dwarfs are still effective for late-M dwarfs. Our revised calibration is accurate to 0.07 dex for M4.5-M9.5 dwarfs with -0.58<[Fe/H]<+0.56 and shows no systematic trends with spectral type, metallicity, or the method used to determine the primary star metallicity. We show that our method gives consistent metallicities for the components of M+M wide binaries. We verify that our new formula works for unresolved binaries by combining spectra of single stars. Lastly, we show that our calibration gives consistent metallicities with the Mann et al. (2013a) study for overlapping (M4-M5) stars, establishing that the two calibrations can be used in combination to determine metallicities across the entire M dwarf sequence.

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Spectro-Thermometry of M dwarfs and their candidate planets: too hot, too cool, or just right?

We use moderate-resolution spectra of nearby late K and M dwarf stars with parallaxes and interferometrically determined radii to refine their effective temperatures, luminosities, and metallicities. We use these revised values to calibrate spectroscopic techniques to infer the fundamental parameters of more distant late-type dwarf stars. We demonstrate that, after masking out poorly modeled regions, the newest version of the PHOENIX atmosphere models accurately reproduce temperatures derived bolometrically. We apply methods to late-type hosts of transiting planet candidates in the Kepler field, and calculate effective temperature, radius, mass, and luminosity with typical errors of 57 K, 7%, 11%, and 13%, respectively. We find systematic offsets between our values and those from previous analyses of the same stars, which we attribute to differences in atmospheric models utilized for each study. We investigate which of the planets in this sample are likely to orbit in the circumstellar habitable zone. We determine that four candidate planets (KOI 854.01, 1298.02, 1686.01, and 2992.01) are inside of or within 1-sigma of a conservative definition of the habitable zone, but that several planets identified by previous analyses are not (e.g. KOI 1422.02 and KOI 2626.01). Only one of the four habitable-zone planets is Earth sized, suggesting a downward revision in the occurrence of such planets around M dwarfs. These findings highlight the importance of measuring accurate stellar parameters when deriving parameters of their orbiting planets.

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