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Andrew A. West

Publications and source records attributed to Andrew A. West.

At least 19 recordsLinked to original sources

Simulating Crowds in Real Time with Agent-Based Modelling and a Particle Filter

Agent-based modelling is a valuable approach for systems whose behaviour is driven by the interactions between distinct entities. They have shown particular promise as a means of modelling crowds of people in streets, public transport terminals, stadiums, etc. However, the methodology faces a fundamental difficulty: there are no established mechanisms for dynamically incorporating real-time data into models. This limits simulations that are inherently dynamic, such as pedestrian movements, to scenario testing of, for example, the potential impacts of new architectural configurations on movements. This paper begins to address this fundamental gap by demonstrating how a particle filter could be used to incorporate real data into an agent-based model of pedestrian movements at run time. The experiments show that it is indeed possible to use a particle filter to perform online (real time) model optimisation. However, as the number of agents increases, the number of individual particles (and hence the computational complexity) required increases exponentially. By laying the groundwork for the real-time simulation of crowd movements, this paper has implications for the management of complex environments (both nationally and internationally) such as transportation hubs, hospitals, shopping centres, etc.

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2MASS J11151597+1937266: A Young, Dusty, Isolated, Planetary-Mass Object with a Potential Wide Stellar Companion

We present 2MASS J11151597+1937266, a recently identified low-surface gravity L dwarf, classified as an L2$γ$ based on Sloan Digital Sky Survey optical spectroscopy. We confirm this spectral type with near-infrared spectroscopy, which provides further evidence that 2MASS J11151597+1937266 is a low-surface gravity L dwarf. This object also shows significant excess mid-infrared flux, indicative of circumstellar material; and its strong H$α$ emission (EW$_{\mathrm{H}α}=560\pm82$ Å) is an indicator of enhanced magnetic activity or weak accretion. Comparison of its spectral energy distribution to model photospheres yields an effective temperature of $1724^{+184}_{-38}$ K. We also provide a revised distance estimate of $37\pm6$ pc using a spectral type-luminosity relationship for low-surface gravity objects. The 3-dimensional galactic velocities and positions of 2MASS J11151597+1937266 do not match any known young association or moving group. Assuming a probable age in the range of 5-45 Myr, the model-dependent estimated mass of this object is between 7-21 $M_\mathrm{Jup}$, making it a potentially isolated planetary-mass object. We also identify a candidate co-moving, young stellar companion, 2MASS J11131089+2110086.

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Activity and Kinematics of White Dwarf-M Dwarf Binaries from the SUPERBLINK Proper Motion Survey

We present an activity and kinematic analysis of high proper motion white dwarf-M dwarf binaries (WD+dMs) found in the SUPERBLINK survey, 178 of which are new identifications. To identify WD+dMs, we developed a UV-optical-IR color criterion and conducted a spectroscopic survey to confirm each candidate binary. For the newly identified systems, we fit the two components using model white dwarf spectra and M dwarf template spectra to determine physical parameters. We use H$α$ chromospheric emission to examine the magnetic activity of the M dwarf in each system, and investigate how its activity is affected by the presence of a white dwarf companion. We find that the fraction of WD+dM binaries with active M dwarfs is significantly higher than their single M dwarf counterparts at early and mid spectral types. We corroborate previous studies that find high activity fractions at both close and intermediate separations. At more distant separations the binary fraction appears to approach the activity fraction for single M dwarfs. Using derived radial velocities and the proper motions, we calculate 3D space velocities for the WD+dMs in SUPERBLINK. For the entire SUPERBLINK WD+dMs, we find a large vertical velocity dispersion, indicating a dynamically hotter population compared to high proper motion samples of single M dwarfs. We compare the kinematics for systems with active M dwarfs and those with inactive M dwarfs, and find signatures of asymmetric drift in the inactive sample, indicating that they are drawn from an older population.

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An empirical template library of stellar spectra for a wide range of spectral classes, luminosity classes, and metallicities using SDSS BOSS spectra

We present a library of empirical stellar spectra created using spectra from the Sloan Digital Sky Survey's Baryon Oscillation Spectroscopic Survey (BOSS). The templates cover spectral types O5 through L3, are binned by metallicity from -2.0 dex through +1.0 dex and are separated into main sequence (dwarf) stars and giant stars. With recently developed M dwarf metallicity indicators, we are able to extend the metallicity bins down through the spectral subtype M8, making this the first empirical library with this degree of temperature \emph{and} metallicity coverage. The wavelength coverage for the templates is from 3650 Angstroms through 10200 Angstroms at a resolution better than R~2000. Using the templates, we identify trends in color space with metallicity and surface gravity, which will be useful for analyzing large data sets from upcoming missions like LSST. Along with the templates, we are releasing a code for automatically (and/or visually) identifying the spectral type and metallicity of a star.

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The Late-Type Extension to MoVeRS (LaTE-MoVeRS): Proper motion verified low-mass stars and brown dwarfs from SDSS, 2MASS, and WISE

We present the Late-Type Extension to the Motion Verified Red Stars (LaTE-MoVeRS) catalog, containing 46,463 photometric late-type ($>$ M5) dwarfs within the Sloan Digital Sky Survey (SDSS) footprint. Proper motions were computed for objects combining astrometry from the SDSS Data Release 12 (DR12), the Two-Micron All-Sky Survey (2MASS) Point Source Catalog (PSC), and the \emph{Wide-field Infrared Survey Explorer} (\emph{WISE}) AllWISE datasets. LaTE-MoVeRS objects were required to have significant proper motion ($μ_\mathrm{tot} \geqslant 2σ_{μ_\mathrm{tot}}$). Using the LaTE-MoVeRS sample and \emph{Gaia} Data Release 1, we estimate \emph{Gaia} will be $\sim$64\% complete for very-low-mass objects ($>$ M5) in comparison to the combined SDSS+2MASS+\emph{WISE} dataset ($i < 21.3$). We computed photometric distances and estimated stellar effective temperatures for the LaTE-MoVeRS catalog. The majority of the dwarfs in the sample have distances $< 150$ pc and $T_\mathrm{eff} < 3000$ K. Thirteen objects were identified within LaTE-MoVeRS with estimated photometric distances within 25 pc that have not been previously identified as nearby objects. We also identified one new object with a large amount of excess mid-infrared flux that has not been previously identified (2MASS J11151597$+$1937266). This object appears to be an L2$γ$ at $\sim$50 pc showing spectroscopic signs of a flaring event (e.g., strong Hydrogen Balmer emission lines). This object does not exhibit kinematics similar to any known kinematic association. The LaTE-MoVeRS catalog is available through SDSS CasJobs and VizieR.

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Linking Stellar Coronal Activity and Rotation at 500 Myr: A Deep Chandra Observation of M37

Empirical calibrations of the stellar age-rotation-activity relation (ARAR) rely on observations of the co-eval populations of stars in open clusters. We used the Chandra X-ray Observatory to study M37, a 500-Myr-old open cluster that has been extensively surveyed for rotation periods ($P_{\rm rot}$). M37 was observed almost continuously for five days, for a total of 440.5 ksec, to measure stellar X-ray luminosities ($L_{\mathrm{X}}$), a proxy for coronal activity, across a wide range of masses. The cluster's membership catalog was revisited to calculate updated membership probabilities from photometric data and each star's distance to the cluster center. The result is a comprehensive sample of 1699 M37 members: 426 with $P_{\rm rot}$, 278 with X-ray detections, and 76 with both. We calculate Rossby numbers, $R_o = P_{\rm rot}/τ$, where $τ$ is the convective turnover time, and ratios of the X-ray-to-bolometric luminosity, $L_{\rm X}/L_{\rm bol}$, to minimize mass dependencies in our characterization of the rotation-coronal activity relation at 500 Myr. We find that fast rotators, for which $R_o<0.09\pm0.01$, show saturated levels of activity, with log($L_{\rm X}/L_{\rm bol}$)$=-3.06\pm0.04$. For $R_o\geq0.09\pm0.01$, activity is unsaturated and follows a power law of the form $R_o^β$, where $β$=$-2.03_{-0.14}^{+0.17}$. This is the largest sample available for analyzing the dependence of coronal emission on rotation for a single-aged population, covering stellar masses in the range 0.4$-$1.3 $M_{\odot}$, $P_{\rm rot}$ in the range 0.4$-$12.8 d, and $L_{\rm X}$ in the range 10$^{28.4-30.5}$ erg s$^{-1}$. Our results make M37 a new benchmark open cluster for calibrating the ARAR at ages of $\approx$500 Myr.

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Using Close White Dwarf + M Dwarf Stellar Pairs to Constrain the Flare Rates in Close Stellar Binaries

We present a study of the statistical flare rates of M dwarfs (dMs) with close white dwarf (WD) companions (WD+dM; typical separations < 1 au). Our previous analysis demonstrated that dMs with close WD companions are more magnetically active than their field counterparts. One likely implication of having a close binary companion is increased stellar rotation through disk-disruption, tidal effects, and/or angular momentum exchange; increased stellar rotation has long been associated with an increase in stellar activity. Previous studies show a strong correlation between dMs that are magnetically active (showing Hα in emission) and the frequency of stellar flare rates. We examine the difference between the flare rates observed in close WD+dM binary systems and field dMs. Our sample consists of a subset of 181 close WD+dM pairs from Morgan et al. (2012) observed in the Sloan Digital Sky Survey Stripe 82, where we obtain multi-epoch observations in the Sloan ugriz-bands. We find an increase in the overall flaring fraction in the close WD+dM pairs (0.09$\pm$0.03%) compared to the field dMs (0.0108$\pm$0.0007%; Kowalski et al. 2009) and a lower flaring fraction for active WD+dMs (0.05$\pm$0.03%) compared to active dMs (0.28$\pm$0.05%; Kowalski et al. 2009). We discuss how our results constrain both the single and binary dM flare rates. Our results also constrain dM multiplicity, our knowledge of the Galactic transient background, and may be important for the habitability of attending planets around dMs with close companions.

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The rotation and Galactic kinematics of mid M dwarfs in the Solar Neighborhood

Rotation is a directly-observable stellar property, and drives magnetic field generation and activity through a magnetic dynamo. Main sequence stars with masses below approximately 0.35Msun (mid-to-late M dwarfs) are fully-convective, and are expected to have a different type of dynamo mechanism than solar-type stars. Measurements of their rotation rates provide insights into these mechanisms, but few rotation periods are available for these stars at field ages. Using photometry from the MEarth transit survey, we measure rotation periods for 387 nearby, mid-to-late M dwarfs in the Northern hemisphere, finding periods from 0.1 to 140 days. The typical detected rotator has stable, sinusoidal photometric modulations at a semi-amplitude of 0.5 to 1%. We find no period-amplitude relation for stars below 0.25Msun and an anti-correlation between period and amplitude for higher-mass M dwarfs. We highlight the existence of older, slowly-rotating stars without Hα emission that nevertheless have strong photometric variability. The Galactic kinematics of our sample is consistent with the local population of G and K dwarfs, and rotators have metallicities characteristic of the Solar Neighborhood. We use the W space velocities and established age-velocity relations to estimate that stars with P<10 days are on average <2 Gyrs, and that those with P>70 days are about 5 Gyrs. The period distribution is mass dependent: as the mass decreases, the slowest rotators at a given mass have longer periods, and the fastest rotators have shorter periods. We find a lack of stars with intermediate rotation periods. [Abridged]

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Motion Verified Red Stars (MoVeRS): A Catalog of Proper Motion Selected Low-mass Stars from WISE, SDSS, and 2MASS

We present a photometric catalog of 8,735,004 proper motion selected low-mass stars (KML-spectral types) within the Sloan Digital Sky Survey (SDSS) footprint, from the combined SDSS Data Release 10 (DR10), Two-Micron All-Sky Survey (2MASS) Point Source Catalog (PSC), and Wide-field Infrared Survey Explorer (WISE) AllWISE catalog. Stars were selected using $r-i$, $i-z$, $r-z$, $z-J$, and $z-W1$ colors, and SDSS, WISE, and 2MASS astrometry was combined to compute proper motions. The resulting 3,518,150 stars were augmented with proper motions for 5,216,854 earlier type stars from the combined SDSS and United States Naval Observatory B1.0 catalog (USNO-B). We used SDSS+USNO-B proper motions to determine the best criteria for selecting a clean sample of stars. Only stars whose proper motions were greater than their $2$$σ$ uncertainty were included. Our Motion Verified Red Stars (MoVeRS) catalog is available through SDSS CasJobs and VizieR.

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A Catalog of GALEX Ultraviolet Emission from Spectroscopically Confirmed M Dwarfs

We present a catalog of GALEX Near-UV (NUV) and Far-UV (FUV) photometry for the Palomar/MSU and SDSS DR7 spectroscopic M dwarf catalogs. The catalog contains NUV measurements matched to 577 spectroscopically confirmed M dwarfs and FUV measurements matched to 150 spectroscopically confirmed M dwarfs. Using these data, we find that NUV and FUV luminosities strongly correlate with Hα emission, a typical indicator of magnetic activity in M dwarfs. We also examine the fraction of M dwarfs with varying degrees of strong line emission at NUV wavelengths. Our results indicate that the frequency of M dwarf NUV emission peaks at intermediate spectral types, with at least ~30% of young M4-M5 dwarfs having some level of activity. For mid-type M dwarfs, we show that NUV emission decreases with distance from the Galactic plane, a proxy for stellar age. Our complete matched source catalog is available online.

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Runaway M Dwarf Candidates from the Sloan Digital Sky Survey

We present a sample of 20 runaway M dwarf candidates (RdMs) within 1 kpc of the Sun whose Galactocentric velocities exceed 400 km s$^{-1}$. The candidates were selected from the SDSS DR7 M Dwarf Catalog of West et al. (2011). Our RdMs have SDSS+USNO-B proper motions that are consistent with those recorded in the PPMXL, LSPM, and combined WISE+SDSS+2MASS catalogs. Sixteen RdMs are classified as dwarfs, while the remaining four RdMs are subdwarfs. We model the Galactic potential using a bulge-disk-halo profile (Kenyon et al. 2008; Brown et al. 2014). Our fastest RdM, with Galactocentric velocity 658.5 $\pm$ 236.9 km s$^{-1}$, is a possible hypervelocity candidate, as it is unbound in 77% of our simulations. About half of our RdMs have kinematics that are consistent with ejection from the Galactic center. Seven of our RdMs have kinematics consistent with an ejection scenario from M31 or M32 to within 2σ, although our distance-limited survey makes such a realization unlikely. No more than four of our RdMs may have originated from the Leo stream. We propose that to within measurement errors, most of our bound RdMs are likely disk runaways or halo objects, and may have been accelerated through a series of multi-body interactions within the Galactic disk or possibly supernovae explosions.

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An Activity-Rotation Relationship and Kinematic Analysis of Nearby Mid-to-Late-type M Dwarfs

Using spectroscopic observations and photometric light curves of 238 nearby M dwarfs from the MEarth exoplanet transit survey, we examine the relationships between magnetic activity (quantified by H-alpha emission), rotation period, and stellar age. Previous attempts to investigate the relationship between magnetic activity and rotation in these stars were hampered by the limited number of M dwarfs with measured rotation periods (and the fact that vsini measurements probe only rapid rotation). However, the photometric data from MEarth allows us to probe a wide range of rotation periods for hundreds of M dwarf stars (from shorter than than one to longer than 100 days). Over all M spectral types that we probe, we find that the presence of magnetic activity is tied to rotation, including for late-type, fully convective M dwarfs. We also find evidence that the fraction of late-type M dwarfs that are active may be higher at longer rotation periods compared to their early-type counterparts, with several active, late-type, slowly rotating stars present in our sample. Additionally, we find that all M dwarfs with rotation periods shorter than 26 days (early-type; M1-M4) and 86 days (late-type; M5-M8) are magnetically active. This potential mismatch suggests that the physical mechanisms that connect stellar rotation to chromospheric heating may be different in fully convective stars. A kinematic analysis suggests that the magnetically active, rapidly rotating stars are consistent with a kinematically young population, while slow-rotators are less active or inactive and appear to belong to an older, dynamically heated stellar population.

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The Brown Dwarf Kinematics Project (BDKP). IV. Radial Velocities of 85 Late-M and L dwarfs with MagE

Radial velocity measurements are presented for 85 late M- and L-type very low mass stars and brown dwarfs obtained with the Magellan Echellette (MagE) spectrograph. Targets primarily have distances within 20 pc of the Sun, with more distant sources selected for their unusual spectral energy distributions. We achieved precisions of 2--3 km/s, and combined these with astrometric and spectrophotometric data to calculate $UVW$ velocities. Most are members of the thin disk of the Galaxy, and velocity dispersions indicate a mean age of 5.2$\pm$0.2 Gyr for sources within 20 pc. We find significantly different kinematic ages between late-M dwarfs (4.0$\pm$0.2 Gyr) and L dwarfs (6.5$\pm$0.4 Gyr) in our sample that are contrary to predictions from prior simulations. This difference appears to be driven by a dispersed population of unusually blue L dwarfs which may be more prevalent in our local volume-limited sample than in deeper magnitude-limited surveys. The L dwarfs exhibit an asymmetric $U$ velocity distribution with a net inward flow, similar to gradients recently detected in local stellar samples. Simulations incorporating brown dwarf evolution and Galactic orbital dynamics are unable to reproduce the velocity asymmetry, suggesting non-axisymmetric perturbations or two distinct L dwarf populations. We also find the L dwarfs to have a kinematic age-activity correlation similar to more massive stars. We identify several sources with low surface gravities, and two new substellar candidate members of nearby young moving groups: the astrometric binary DENIS J08230313$-$4912012AB, a low-probability member of the $β$ Pictoris Moving Group; and 2MASS J15104786-2818174, a moderate-probability member of the 30-50 Myr Argus Association.

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SLoWPoKES-II: 100,000 Wide Binaries Identified in SDSS without Proper Motions

We present the SLoWPoKES-II catalog of low-mass visual binaries identified from the Sloan Digital Sky Survey by matching photometric distances. The candidate pairs are vetted by comparing the stellar density at their respective Galactic positions to Monte Carlo realizations of a simulated Milky Way. In this way, we are able to identify large numbers of bona fide wide binaries without the need of proper motions. 105,537 visual binaries with angular separations of $\sim$1-20", are identified, each with a probability of chance alignment of $\lesssim$5%. This is the largest catalog of bona fide wide binaries to date, and it contains a diversity of systems---in mass, mass ratios, binary separations, metallicity, and evolutionary states---that should facilitate follow-up studies to characterize the properties of M dwarfs and white dwarfs. There is a subtle but definitive suggestion of multiple populations in the physical separation distribution, supporting earlier findings. We suggest that wide binaries are comprised of multiple populations, most likely representing different formation modes. There are 141 M7 or later wide binary candidates, representing a 7-fold increase in the number currently known. These binaries are too wide to have been formed via the ejection mechanism. Finally, we find that ~6% of spectroscopically confirmed M dwarfs are not included in the SDSS STAR catalog; they are misclassified as extended sources due to the presence of a nearby or partially resolved companion. The SLoWPoKES-II catalog is publicly available to the entire community on the world wide web via the Filtergraph data visualization portal.

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Warm Dust around Cool Stars: Field M Dwarfs with WISE 12 or 22 Micron Excess Emission

Using the SDSS DR7 spectroscopic catalog, we searched the WISE AllWISE catalog to investigate the occurrence of warm dust, as inferred from IR excesses, around field M dwarfs (dMs). We developed SDSS/WISE color selection criteria to identify 175 dMs (from 70,841) that show IR flux greater than typical dM photosphere levels at 12 and/or 22 $μ$m, including seven new stars within the Orion OB1 footprint. We characterize the dust populations inferred from each IR excess, and investigate the possibility that these excesses could arise from ultracool binary companions by modeling combined SEDs. Our observed IR fluxes are greater than levels expected from ultracool companions ($>3σ$). We also estimate that the probability the observed IR excesses are due to chance alignments with extragalactic sources is $<$ 0.1%. Using SDSS spectra we measure surface gravity dependent features (K, Na, and CaH 3), and find $<$ 15% of our sample indicate low surface gravities. Examining tracers of youth (H$α$, UV fluxes, and Li absorption), we find $<$ 3% of our sample appear young, indicating we are observing a population of field stars $\gtrsim$ 1 Gyr, likely harboring circumstellar material. We investigate age-dependent properties probed by this sample, studying the disk fraction as a function of Galactic height. The fraction remains small and constant to $|Z| \sim 700$ pc, and then drops, indicating little to no trend with age. Possible explanations for disks around field dMs include: 1) collisions of planetary bodies, 2) tidal disruption of planetary bodies, or 3) failed planet formation.

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H$α$ Emission From Active Equal-mass, Wide M Dwarf Binaries

We identify a sample of near-equal mass wide binary M dwarf systems from the SLoWPoKES catalog of common proper-motion binaries and obtain follow-up observations of their chromospheric activity as measured by the H$α$ emission line. We present optical spectra for both components of 48 candidate M dwarf binaries, confirming their mid-M spectral types. Of those 48 coeval pairs, we find eight with H$α$ emission from both components, three with weak emission in one component and no emission in the other, and 37 with two inactive components. We find that of the eleven pairs with at least one active component, only three follow the net trend of decreasing activity strength $L_{\rm Hα}/L_{\rm bol}$ with later spectral type. The difference in quiescent activity strength between the A and B components is larger than what would be expected based on the small differences in color (mass). For five binaries with two active components, we present 47 hours of time-resolved spectroscopy, observed on the ARC 3.5-m over twelve different nights. For four of the five pairs, the slightly redder (B) component exhibits a higher level of H$α$ emission during the majority of the observations and the redder objects are the only components to flare. The full range of H$α$ emission observed on these variable mid-M dwarfs is comparable to the scatter in H$α$ emission found in single-epoch surveys of mid-M dwarfs, indicating that variability could be a major factor in the spread of observed activity strengths. We also find that variability is independent of both activity strength and spectral type.

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BOSS Ultracool Dwarfs I: Colors and Magnetic Activity of M and L dwarfs

We present the colors and activity of ultracool (M7-L8) dwarfs from the Tenth Data Release of the Sloan Digital Sky Survey (SDSS). We combine previous samples of SDSS M and L dwarfs with new data obtained from the Baryon Oscillation Sky Survey (BOSS) to produce the BOSS Ultracool Dwarf (BUD) sample of 11820 M7-L8 dwarfs. By combining SDSS data with photometry from the Two Micron All Sky Survey and the Wide-Field Infrared Sky Explorer mission, we present ultracool dwarf colors from $i-z$ to $W2-W3$ as a function of spectral type, and extend the SDSS-2MASS-WISE color locus to include ultracool dwarfs. The $i-z$, $i-J$, and $z-J$ colors provide the best indication of spectral type for M7-L3 dwarfs. We also examine ultracool dwarf chromospheric activity through the presence and strength of H$α$ emission. The fraction of active dwarfs rises through the M spectral sequence until it reaches $\sim$90% at spectral type L0. The fraction of active dwarfs then declines to 50% at spectral type L5; no H$α$ emission is observed in the late-L dwarfs in the BUD sample. The fraction of active L0-L5 dwarfs is much higher than previously observed. The strength of activity declines with spectral type from M7 through L3, after which the data do not show a clear trend. Using one-dimensional chromosphere models, we explore the range of filling factors and chromospheric temperature structures that are consistent with H$α$ observations of M0-L7 dwarfs. M dwarf chromospheres have a similar, smoothly varying range of temperature and surface coverage while L dwarf chromospheres are cooler and have smaller filling factors.

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A New Sample of Cool Subdwarfs from SDSS: Properties and Kinematics

We present a new sample of M subdwarfs compiled from the 7th data release of the Sloan Digital Sky Survey. With 3517 new subdwarfs, this new sample significantly increases the number of spectroscopically confirmed low-mass subdwarfs. This catalog also includes 905 extreme and 534 ultra sudwarfs. We present the entire catalog including observed and derived quantities, and template spectra created from co-added subdwarf spectra. We show color-color and reduced proper motion diagrams of the three metallicity classes, which are shown to separate from the disk dwarf population. The extreme and ultra subdwarfs are seen at larger values of reduced proper motion as expected for more dynamically heated populations. We determine 3D kinematics for all of the stars with proper motions. The color-magnitude diagrams show a clear separation of the three metallicity classes with the ultra and extreme subdwarfs being significantly closer to the main sequence than the ordinary subdwarfs. All subdwarfs lie below (fainter) and to the left (bluer) of the main sequence. Based on the average $(U,V,W)$ velocities and their dispersions, the extreme and ultra subdwarfs likely belong to the Galactic halo, while the ordinary subdwarfs are likely part of the old Galactic (or thick) disk. An extensive activity analysis of subdwarfs is performed using H$α$ emission and 208 active subdwarfs are found. We show that while the activity fraction of subdwarfs rises with spectral class and levels off at the latest spectral classes, consistent with the behavior of M dwarfs, the extreme and ultra subdwarfs are basically flat.

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