arXiv subjects

Gregory N. Mace

Publications and source records attributed to Gregory N. Mace.

At least 37 records · Page 2Linked to original sources

arXiv2021

A solar C/O and sub-solar metallicity in a hot Jupiter atmosphere

Measurements of the atmospheric carbon (C) and oxygen (O) relative to hydrogen (H) in hot Jupiters (relative to their host stars) provide insight into their formation location and subsequent orbital migration. Hot Jupiters that form beyond the major volatile (H2O/CO/CO2) ice lines and subsequently migrate post disk-dissipation are predicted have atmospheric carbon-to-oxygen ratios (C/O) near 1 and subsolar metallicities, whereas planets that migrate through the disk before dissipation are predicted to be heavily polluted by infalling O-rich icy planetesimals, resulting in C/O < 0.5 and super-solar metallicities. Previous observations of hot Jupiters have been able to provide bounded constraints on either H2O or CO, but not both for the same planet, leaving uncertain the true elemental C and O inventory and subsequent C/O and metallicity determinations. Here we report spectroscopic observations of a typical transiting hot Jupiter, WASP-77Ab. From these, we determine the atmospheric gas volume mixing ratio constraints on both H2O and CO (9.5$\times 10^{-5}$ - 1.5$\times 10^{-4}$ and 1.2$\times 10^{-4}$ - 2.6$\times 10^{-4}$, respectively). From these bounded constraints, we are able to derive the atmospheric C/H (0.35$^{+0.17}_{-0.10}$ $\times$ Solar) and O/H (0.32 $^{+0.12}_{-0.08}$ $\times$ Solar) abundances and the corresponding atmospheric carbon-to-oxygen ratio (C/O=0.59$\pm$0.08; the solar value is 0.55). The sub-solar (C+O)/H (0.33$^{+0.13}_{-0.09}$ $\times$ Solar) is suggestive of a metal-depleted atmosphere relative to what is expected for Jovian-like planets while the near solar value of C/O rules out the disk-free migration/C-rich atmosphere scenario.

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arXiv2021

V899 Mon: a peculiar eruptive young star close to the end of its outburst

V899 Mon is an eruptive young star showing characteristics of both FUors and EXors. It reached a peak brightness in 2010, then briefly faded in 2011, followed by a second outburst. We conducted multi-filter optical photometric monitoring, as well as optical and near-infrared spectroscopic observations of V899 Mon. The light curves and color-magnitude diagrams show that V899 Mon has been gradually fading after its second outburst peak in 2018, but smaller accretion bursts are still happening. Our spectroscopic observations taken with Gemini/IGRINS and VLT/MUSE show a number of emission lines, unlike during the outbursting stage. We used the emission line fluxes to estimate the accretion rate and found that it has significantly decreased compared to the outbursting stage. The mass loss rate is also weakening. Our 2D spectro-astrometric analysis of emission lines recovered jet and disk emission of V899 Mon. We found the emission from permitted metallic lines and the CO bandheads can be modeled well with a disk in Keplerian rotation, which also gives a tight constraint for the dynamical stellar mass of 2 ${M_{\odot}}$. After a discussion of the physical changes that led to the changes in the observed properties of V899 Mon, we suggest this object is finishing its second outburst.

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arXiv2021

Obliquity Constraints on the Planetary-mass Companion HD 106906 b

We constrain the angular momentum architecture of HD 106906, a 13 $\pm$ 2 Myr old system in the ScoCen complex composed of a compact central binary, a widely separated planetary-mass tertiary HD 106906 b, and a debris disk nested between the binary and tertiary orbital planes. We measure the orientations of three vectors: the companion spin axis, companion orbit normal, and disk normal. Using near-IR high-resolution spectra from Gemini/IGRINS, we obtain a projected rotational velocity of $v\sin{i_p}$ = 9.5 $\pm$ 0.2 km/s for HD 106906 b. This measurement together with a published photometric rotation period implies the companion is viewed nearly pole-on, with a line-of-sight spin axis inclination of $i_p$ = 14 $\pm$ 4 degrees or 166 $\pm$ 4 degrees. By contrast, the debris disk is known to be viewed nearly edge-on. The likely misalignment of all three vectors suggests HD 106906 b formed by gravitational instability in a turbulent environment, either in a disk or cloud setting.

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arXiv2021

The IGRINS YSO Survey I. Stellar parameters of pre-main sequence stars in Taurus-Auriga

We present fundamental parameters for 110 canonical K- & M-type (1.3$-$0.13$M_\odot$) Taurus-Auriga young stellar objects (YSOs). The analysis produces a simultaneous determination of effective temperature ($T_{\rm eff}$), surface gravity ($\log$ g), magnetic field strength (B), and projected rotational velocity ($v \sin i$). Our method employed synthetic spectra and high-resolution (R$\sim$45,000) near-infrared spectra taken with the Immersion GRating INfrared Spectrometer (IGRINS) to fit specific K-band spectral regions most sensitive to those parameters. The use of these high-resolution spectra reduces the influence of distance uncertainties, reddening, and non-photospheric continuum emission on the parameter determinations. The median total (fit + systematic) uncertainties were 170 K, 0.28 dex, 0.60 kG, 2.5 km s$^{-1}$ for $T_{\rm eff}$, $\log$ g, B, and $v \sin i$, respectively. We determined B for 41 Taurus YSOs (upper limits for the remainder) and find systematic offsets (lower $T_{\rm eff}$, higher $\log$ g and $v \sin i$) in parameters when B is measurable but not considered in the fit. The average $\log$ g for the Class II and Class III objects differs by 0.23$\pm$0.05dex, which is consistent with Class III objects being the more evolved members of the star-forming region. However, the dispersion in $\log$ g is greater than the uncertainties, which highlights how the YSO classification correlates with age ($\log$ g), yet there are exceptionally young (lower $\log$ g) Class III YSOs and relatively old (higher $\log$ g) Class II YSOs with unexplained evolutionary histories. The spectra from this work are provided in an online repository along with TW Hydrae Association (TWA) comparison objects and the model grid used in our analysis.

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arXiv2021

An Improved Near-Infrared Spectrum of the Archetype Y Dwarf WISEP J182831.08+265037.8

We present a Hubble Space Telescope/Wide-Field Camera 3 near infrared spectrum of the archetype Y dwarf WISEP 182831.08+265037.8. The spectrum covers the 0.9-1.7 um wavelength range at a resolving power of lambda/Delta lambda ~180 and is a significant improvement over the previously published spectrum because it covers a broader wavelength range and is uncontaminated by light from a background star. The spectrum is unique for a cool brown dwarf in that the flux peaks in the Y, J, and H band are of near equal intensity in units of f_lambda. We fail to detect any absorption bands of NH_3 in the spectrum, in contrast to the predictions of chemical equilibrium models, but tentatively identify CH_4 as the carrier of an unknown absorption feature centered at 1.015 um. Using previously published ground- and spaced-based photometry, and using a Rayleigh Jeans tail to account for flux emerging longward of 4.5 um, we compute a bolometric luminosity of log (L_bol/L_sun)=-6.50+-0.02 which is significantly lower than previously published results. Finally, we compare the spectrum and photometry to two sets of atmospheric models and find that best overall match to the observed properties of WISEP 182831.08+265037.8 is a ~1 Gyr old binary composed of two T_eff~325 K, ~5 M_Jup brown dwarfs with subsolar [C/O] ratios.

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arXiv2021

Evidence of Accretion Burst: The Viscously Heated Inner Disk of the Embedded Protostar IRAS 16316-1540

Outbursts of young stellar objects occur when the mass accretion rate suddenly increases. However, such outbursts are difficult to detect for deeply embedded protostars due to their thick envelope and the rarity of outbursts. The near-IR spectroscopy is a useful tool to identify ongoing outburst candidates by the characteristic absorption features that indicate a disk origin. However, without high-resolution spectroscopy, the spectra of outburst candidates can be confused with the late-type stars since they have similar spectral features. For the protostar IRAS 16316-1540, the near-IR spectrum has line equivalent widths that are consistent with M-dwarf photospheres. However, our high-resolution IGRINS spectra reveal that the absorption lines have boxy and/or double-peaked profiles, as expected from a disk and not the star. The continuum emission source is likely the hot, optically thick disk, heated by viscous accretion. The projected disk rotation velocity of 41$\pm$5 km s$^{-1}$ corresponds to $\sim 0.1$ AU. Based on the result, we suggest IRAS 16316-1540 as an ongoing outburst candidate. Viscous heating of disks is usually interpreted as evidence for ongoing bursts, which may be more common than previously estimated from low-resolution near-IR spectra.

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arXiv2021

The Transition from Diffuse Molecular Gas to Molecular Cloud Material in Taurus

We study four lines of sight that probe the transition from diffuse molecular gas to molecular cloud material in Taurus. Measurements of atomic and molecular absorption are used to infer the distribution of species and the physical conditions toward stars behind the Taurus Molecular Cloud (TMC). New high-resolution spectra at visible and near infrared wavelengths of interstellar Ca II, Ca I, K I, CH, CH^+, C2, CN, and CO toward HD28975 and HD29647 are combined with data at visible wavelengths and published CO results from ultraviolet measurements for HD27778 and HD30122. Gas densities and temperatures are inferred from C2, CN, and CO excitation and CN chemistry. Our results for HD29647 are noteworthy because the CO column density is 10^{18} cm^{-2} while C2 and CO excitation reveals a temperature of 10 K and density about 1000 cm^{-3}, more like conditions found in dark molecular clouds. Similar results arise from our chemical analysis for CN through reactions involving observations of CH, C2, and NH. Enhanced potassium depletion and a reduced CH/H2 column density ratio also suggest the presence of a dark cloud. The directions toward HD27778 and HD30122 probe molecule-rich diffuse clouds, which can be considered CO-dark gas, while the sight line toward HD28975 represents an intermediate case. Maps of dust temperature help refine the description of the material along the four sight lines and provide an estimate of the distance between HD29647 and a clump in the TMC. An Appendix provides results for the direction toward HD26571; this star also probes diffuse molecular gas.

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arXiv2021

IGRINS RV: A Precision RV Pipeline for IGRINS Using Modified Forward Modeling in the Near-Infrared

Application of the radial velocity (RV) technique in the near infrared is valuable because of the diminished impact of stellar activity at longer wavelengths, making it particularly advantageous for the study of late-type stars but also for solar-type objects. In this paper, we present the IGRINS RV open source python pipeline for computing infrared RV measurements from reduced spectra taken with IGRINS, a R ~ 45,000 spectrograph with simultaneous coverage of the H band (1.49--1.80 $μ$m) and K band (1.96--2.46 $μ$m). Using a modified forward modeling technique, we construct high resolution telluric templates from A0 standard observations on a nightly basis to provide a source of common-path wavelength calibration while mitigating the need to mask or correct for telluric absorption. Telluric standard observations are also used to model the variations in instrumental resolution across the detector, including a yearlong period when the K band was defocused. Without any additional instrument hardware, such as a gas cell or laser frequency comb, we are able to achieve precisions of 26.8 $\rm m\,s^{-1}$ in the K band and 31.1 $\rm m\,s^{-1}$ in the H band for narrow-line hosts. These precisions are empirically determined by a monitoring campaign of two RV standard stars as well as the successful retrieval of planet-induced RV signals for both HD 189733 and $τ$ Boo A; furthermore, our results affirm the presence of the Rossiter-McLaughlin effect for HD 189733. The IGRINS RV pipeline extends another important science capability to IGRINS, with publicly available software designed for widespread use.

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arXiv2020

Chemical Compositions of Red Giant Stars from Habitable Zone Planet Finder Spectroscopy

We have used the Habitable Zone Planet Finder (HPF) to gather high resolution, high signal-to-noise near-infrared spectra of 13 field red horizontal-branch (RHB) stars, one open-cluster giant, and one very metal-poor halo red giant. The HPF spectra cover the 0.81$-$1.28 \micron\ wavelength range of the $zyJ$ bands, filling in the gap between the optical (0.4$-$1.0~\micron) and infrared (1.5$-$2.4~\micron) spectra already available for the program stars. We derive abundances of 17 species from LTE-based computations involving equivalent widths and spectrum syntheses, and estimate abundance corrections for the species that are most affected by departures from LTE in RHB stars. Generally good agreement is found between HPF-based metallicities and abundance ratios and those from the optical and infrared spectral regions. Light element transitions dominate the HPF spectra of these red giants, and HPF data can be used to derive abundances from species with poor or no representation in optical spectra (\eg, \species{C}{i}, \species{P}{i}, \species{S}{i}, \species{K}{i}). Attention is drawn to the HPF abundances in two field solar-metallicity RHB stars of special interest: one with an extreme carbon isotope ratio, and one with a rare very large lithium content. The latter star is unique in our sample by exhibiting very strong \species{He}{i} 10830~Å absorption. The abundances of the open cluster giant concur with those derived from other wavelength regions. Detections of \species{C}{i} and \species{S}{i} in HD~122563 are reported, yielding the lowest metallicity determination of [S/Fe] from more than one multiplet.

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arXiv2020

High-resolution spectroscopic monitoring observations of FU Orionis-type object, V960 Mon

We present the results of high-resolution (R $\ge$ 30,000) optical and near-infrared spectroscopic monitoring observations of a FU Orionis-type object, V960 Mon, which underwent an outburst in 2014 November. We have monitored this object with the Bohyunsan Optical Echelle Spectrograph (BOES) and the Immersion GRating INfrared Spectrograph (IGRINS) since 2014 December. Various features produced by a wind, disk, and outflow/jet were detected. The wind features varied over time and continually weakened after the outburst. We detected double-peaked line profiles in the optical and near-infrared, and the line widths tend to decrease with increasing wavelength, indicative of Keplerian disk rotation. The disk features in the optical and near-infrared spectra fit well with G-type and K-type stellar spectra convolved with a kernel to account for the maximum projected disk rotation velocity of about 40.3$\pm$3.8 km s$^{-1}$ and 36.3$\pm$3.9 km s$^{-1}$, respectively. We also report the detection of [S II] and H$_{2}$ emission lines, which are jet/outflow tracers and rarely found in FUors.

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arXiv2020

IGRINS Slit-Viewing Camera Software

We have developed observation control software for the Immersion GRating INfrared Spectrometer (IGRINS) slit-viewing camera module, which maintains the position of an astronomical target on the spectroscopic slit. It is composed of several packages that monitor and control the system, acquire the images, and compensate for the tracking error by sending tracking feedback information to the telescope control system. For efficient development and maintenance of each software package, we have applied software engineering methods, i.e., a spiral software development with model-based design. It is not trivial to define the shape and center of astronomical object point spread functions (PSFs), which do not have symmetric Gaussian profiles in short exposure (<4 s) guiding images. Efforts to determine the PSF centroid are additionally complicated by the core saturation of bright guide stars. We have applied both a two-dimensional Gaussian fitting algorithm (2DGA) and center balancing algorithm (CBA) to identify an appropriate method for IGRINS in the near-infrared K-band. The CBA derives the expected center position along the slit-width by referencing the spillover flux ratio of the PSF wings on both sides of the slit. In this research, we have compared the accuracy and reliability of the CBA to the 2DGA by using data from IGRINS commissioning observations at McDonald Observatory. We find that the performance of each algorithm depends on the brightness of the targets and the seeing conditions, with the CBA performing better in typical observing scenarios. The algorithms and test results we present can be utilized with future spectroscopic slit observations in various observing conditions and for a variety of spectrograph designs.

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arXiv2019

The Mean Magnetic Field Strength of CI Tau

We present a blind comparison of two methods to measure the mean surface magnetic field strength of the classical T Tauri star CI Tau based on Zeeman broadening of sensitive spectral lines. Our approach takes advantage of the greater Zeeman broadening at near-infrared compared to optical wavelengths. We analyze a high signal-to-noise, high spectral resolution spectrum from 1.5-2.5um observed with IGRINS (Immersion GRating INfrared Spectrometer) on the Discovery Channel Telescope. Both stellar parameterization with MoogStokes (which assumes an uniform magnetic field) and modeling with SYNTHMAG (which includes a distribution of magnetic field strengths) yield consistent measurements for the mean magnetic field strength of CI Tau is B of ~2.2 kG. This value is typical compared with measurements for other young T Tauri stars and provides an important contribution to the existing sample given it is the only known developed planetary system hosted by a young classical T Tauri star. Moreover, we potentially identify an interesting and suggestive trend when plotting the effective temperature and the mean magnetic field strength of T Tauri stars. While a larger sample is needed for confirmation, this trend appears only for a subset of the sample, which may have implications regarding the magnetic field generation.

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arXiv2019

Compact Disks in a High Resolution ALMA Survey of Dust Structures in the Taurus Molecular Cloud

We present a high-resolution ($\sim0.''12$, $\sim16$ au, mean sensitivity of $50~μ$Jy~beam$^{-1}$ at 225 GHz) snapshot survey of 32 protoplanetary disks around young stars with spectral type earlier than M3 in the Taurus star-forming region using Atacama Large Millimeter Array (ALMA). This sample includes most mid-infrared excess members that were not previously imaged at high spatial resolution, excluding close binaries and highly extincted objects, thereby providing a more representative look at disk properties at 1--2 Myr. Our 1.3 mm continuum maps reveal 12 disks with prominent dust gaps and rings, 2 of which are around primary stars in wide binaries, and 20 disks with no resolved features at the observed resolution (hereafter smooth disks), 8 of which are around the primary star in wide binaries. The smooth disks were classified based on their lack of resolved substructures, but their most prominent property is that they are all compact with small effective emission radii ($R_{\rm eff,95\%} \lesssim 50$ au). In contrast, all disks with $R_{\rm eff,95\%}$ of at least 55 au in our sample show detectable substructures. Nevertheless, their inner emission cores (inside the resolved gaps) have similar peak brightness, power law profiles, and transition radii to the compact smooth disks, so the primary difference between these two categories is the lack of outer substructures in the latter. These compact disks may lose their outer disk through fast radial drift without dust trapping, or they might be born with small sizes. The compact dust disks, as well as the inner disk cores of extended ring disks, that look smooth at the current resolution will likely show small-scale or low-contrast substructures at higher resolution. The correlation between disk size and disk luminosity correlation demonstrates that some of the compact disks are optically thick at millimeter wavelengths.

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arXiv2019

Effective Temperatures of Low-Mass Stars from High-Resolution H-band Spectroscopy

High-resolution, near-infrared spectra will be the primary tool for finding and characterizing Earth-like planets around low-mass stars. Yet, the properties of exoplanets can not be precisely determined without accurate and precise measurements of the host star. Spectra obtained with the Immersion GRating INfrared Spectrometer (IGRINS) simultaneously provide diagnostics for most stellar parameters, but the first step in any analysis is the determination of the effective temperature. Here we report the calibration of high-resolution H-band spectra to accurately determine effective temperature for stars between 4000-3000 K ($\sim$K8--M5) using absorption line depths of Fe I, OH, and Al I. The field star sample used here contains 254 K and M stars with temperatures derived using BT-Settl synthetic spectra. We use 106 stars with precise temperatures in the literature to calibrate our method with typical errors of about 140 K, and systematic uncertainties less than $\sim$120 K. For the broadest applicability, we present T$_{\rm eff}$--line-depth-ratio relationships, which we test on 12 members of the TW Hydrae Association and at spectral resolving powers between $\sim$10,000--120,000. These ratios offer a simple but accurate measure of effective temperature in cool stars that is distance and reddening independent.

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arXiv2018

Preliminary Trigonometric Parallaxes of 184 Late-T and Y Dwarfs and an Analysis of the Field Substellar Mass Function into the "Planetary" Mass Regime

We present preliminary trigonometric parallaxes of 184 late-T and Y dwarfs using observations from Spitzer (143), USNO (18), NTT (14), and UKIRT (9). To complete the 20-pc census of $\ge$T6 dwarfs, we combine these measurements with previously published trigonometric parallaxes for an additional 44 objects and spectrophotometric distance estimates for another 7. For these 235 objects, we estimate temperatures, sift into five 150K-wide $T_{\rm eff}$ bins covering the range 300-1050K, determine the completeness limit for each, and compute space densities. To anchor the high-mass end of the brown dwarf mass spectrum, we compile a list of early- to mid-L dwarfs within 20 pc. We run simulations using various functional forms of the mass function passed through two different sets of evolutionary code to compute predicted distributions in $T_{\rm eff}$. The best fit of these predictions to our L, T, and Y observations is a simple power-law model with $α\approx 0.6$ (where $dN/dM \propto M^{-α}$), meaning that the slope of the field substellar mass function is in rough agreement with that found for brown dwarfs in nearby star forming regions and young clusters. Furthermore, we find that published versions of the log-normal form do not predict the steady rise seen in the space densities from 1050K to 350K. We also find that the low-mass cutoff to formation, if one exists, is lower than $\sim$5 $M_{Jup}$, which corroborates findings in young, nearby moving groups and implies that extremely low-mass objects have been forming over the lifetime of the Milky Way.

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arXiv2018

Probing Late-type T dwarf J-H Color Outliers for Signs of Age

We present the results of a Keck/NIRSPEC follow-up survey of thirteen late-type T dwarfs (T6-T9), twelve of which have unusually red or blue J-H colors. Previous work suggests that J-H color outliers may represent the high-gravity, low-metallicity (old) and low-gravity, solar-metallicity (young) extremes of the late-type T dwarf population. We use medium-resolution Y- and H-band spectroscopy to probe regions of T dwarf atmospheres that are more sensitive to gravity and metallicity variations than the J band. We find that the spectral morphologies of our sample are largely homogeneous, with peak-normalized, Y- and H-band morphologies consistent with spectral standards. However, three objects stand out as potentially old, with overluminous Y-band spectra compared to their respective spectral standards, and a fourth object stands out as potentially young, with an underluminous Y band. Of these four objects, three have been previously identified as potential metallicity/gravity outliers, including the one object in our sample with a normal J-H color. We fit publicly available atmospheric model grids to our spectra and find that the best-fit physical parameters vary depending on the model used. As we continue to probe the characteristics of the late-T population, differences in synthetic spectra of ~10-20% in the blue wing of the Y band and ~45% at 1.65 microns, for the same physical parameters, must be reconciled. Further development and public availability of nonsolar metallicity models is also recommended. Future progress toward deciphering the impacts of gravity, metallicity, and variability in the late-type T dwarf population will also require high signal-to-noise, multiwavelength and multi-epoch photometry and spectroscopy.

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arXiv2018

IGRINS Spectral Library

We present a library of high-resolution (R $\equiv$ $λ$/$Δ$$λ$ $\sim$ 45,000) and high signal-to-noise ratio (S/N $\geq$ 200) near-infrared spectra for stars of a wide range of spectral types and luminosity classes. The spectra were obtained with the Immersion GRating INfrared Spectrograph (IGRINS) covering the full range of the H (1.496-1.780 $μ$m) and K (2.080-2.460 $μ$m) atmospheric windows. The targets were primarily selected for being MK standard stars covering a wide range of effective temperatures and surface gravities with metallicities close to the Solar value. Currently, the library includes flux-calibrated and telluric-absorption-corrected spectra of 84 stars, with prospects for expansion to provide denser coverage of the parametric space. Throughout the H and K atmospheric windows, we identified spectral lines that are sensitive to $T_\mathrm{eff}$ or $\log g$ and defined corresponding spectral indices. We also provide their equivalent widths. For those indices, we derive empirical relations between the measured equivalent widths and the stellar atmospheric parameters. Therefore, the derived empirical equations can be used to calculate $T_\mathrm{eff}$ and $\log g$ of a star without requiring stellar atmospheric models.

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arXiv2018

Chemical Compositions of Evolved Stars From Near-Infrared IGRINS High-Resolution Spectra. I. Abundances in Three Red Horizontal Branch Stars

We have derived elemental abundances of three field red horizontal branch stars using high-resolution (R$\simeq$ 45,000), high signal-to-noise ratio (S/N $\gtrsim$ 200) $H$ and $K$ band spectra obtained with the Immersion Grating Infrared Spectrograph (IGRINS). We have determined the abundances of 21 elements including $α$ (Mg, Si, Ca, S), odd-Z (Na, Al, P, K), Fe-group (Sc, Ti, Cr, Co, Ni), neutron-capture (Ce, Nd, Yb), and CNO group elements. S, P and K are determined for the first time in these stars. $H$ and $K$ band spectra provide a substantial number of S I lines, which potentially can lead to a more robust exploration of the role of sulfur in the cosmochemical evolution of the Galaxy. We have also derived $^{12}$C/$^{13}$C ratios from synthetic spectra of the first overtone (2$-$0) and (3$-$1) $^{12}$CO and (2$-$0) $^{13}$CO lines near 23440 Å and $^{13}$CO (3$-$1) lines at about 23730 Å. Comparison of our results with the ones obtained from the optical region suggests that the IGRINS high-resolution $H$ and $K$ band spectra offer more internally self-consistent atomic lines of the same species for several elements, especially the $α$ elements. This in turn provides more reliable abundances for the elements with analytical difficulties in the optical spectral range.

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