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Paul Robertson

Publications and source records attributed to Paul Robertson.

At least 19 recordsLinked to original sources

Improving the Precision of Line-by-Line Radial Velocities: A Data-Driven Iterative Algorithm for Spectral Line Selection

Independent analysis of individual spectral lines, or line-by-line (LBL) analyses, can improve upon standard cross-correlation function (CCF) methods for measuring radial velocities (RVs) because they preserve critical information about individual line shape changes that can be caused by stellar activity. In this work, we measure LBL RVs of 3,830 spectral lines across 383 days of NEID solar observations. Our LBL approach achieves an RV RMS of $2.012~\mathrm{m\,s^{-1}}$, which is slightly lower than the $2.129~\mathrm{m\,s^{-1}}$ achieved by a CCF approach using a shared line list. Then, we describe and benchmark several methods for selecting line lists based on line properties such as depth and intrinsic RV scatter. We find that these subsets have a lower RV RMS compared to either the full line list or random subsets of equal size. Motivated by these results, we present FLARES (Filtering Lines for Accurate Radial-velocity Exoplanet Search), an iterative line-selection algorithm. FLARES selects candidate spectral lines with extreme values of multiple line metrics and properties such as depth, signal-to-noise ratio, and detector position, and preferentially rejects lines whose removal produces the largest decrease in the weighted RV scatter. FLARES achieves an RV RMS of $1.122~\mathrm{m\,s^{-1}}$ using just 24 lines and performs better than the benchmark methods. We perform Monte Carlo simulations and show FLARES is robust and reproducible. Comparisons to alternative line lists chosen to have properties similar to the best FLARES-selected lines demonstrate that FLARES is successfully identifying line properties that lead to effective line lists for future extreme-precision RV measurements.

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A Revised Mass and Period for the Habitable Zone super-Earth GJ 3378b: A Planet Straddling the Cosmic Shoreline

The nearby ($d = 7.7$ pc) M4V star GJ~3378 is a target of our radial velocity (RV) exoplanet survey of fully convective stars in the Solar neighborhood with the near-infrared spectrograph HPF on the Hobby-Eberly Telescope (HET) at McDonald Observatory. Recently, Moutou et al.~(2024) announced the discovery of an $m\sin i = 5.26^{+0.94}_{-0.97} M_\oplus$ planet, GJ 3378b, with an orbital period of $24.73 \pm 0.06$ days, based on SPIRou RV data. Here, we present our HPF RVs for GJ 3378, as well as additional Doppler spectroscopy from the extreme precision NEID Spectrometer on the WIYN telescope at Kitt Peak National Observatory. We have analyzed the HPF+NEID RVs jointly with the published RVs from the CARMENES and SPIRou spectrometers. We present an orbital model for GJ 3378b that differs significantly from the Moutou et al.~solution. The joint RV model reduces the orbital period to $P = 21.45 \pm 0.01$d and the minimum mass to $m \sin i = 2.3 \pm 0.4 M_\oplus$. The shortened orbital distance remains within the conservative circumstellar liquid-water habitable zone (HZ), while the reduced mass increases the likelihood that the planet has a terrestrial composition. The revised planet properties place it near the ``cosmic shoreline," where planets in the HZs of M dwarfs may lose their atmospheres due to radiative stripping.

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Searching for GEMS: Three warm Saturns and a super-Jupiter orbiting four early M-dwarfs

We report the confirmation and characterization of four transiting giant planets orbiting early-M dwarfs discovered by the Searching for Giant Exoplanets around M-dwarf Stars (GEMS) survey: TOI-7189 b, TOI-7265B b, TOI-7393 b, and TOI-7394B b. Joint modeling of TESS and ground-based photometry with precision radial velocities from the Habitable-zone Planet Finder and NEID spectrographs yields self-consistent orbital and physical parameters for all systems. The planets have short orbital periods ($P = 1.25-4.17$ days), masses spanning from $0.5\,M_{\rm J}$ to $2.1\,M_{\rm J}$, and radii comparable to Jupiter ($0.95\,R_{\rm J} < R_p < 1.02\,R_{\rm J}$). TOI-7189 b ($0.50\,M_{\rm J}$), TOI-7265B b ($0.71\,M_{\rm J}$), and TOI-7393 b ($0.61\,M_{\rm J}$) are Saturn-like in mass and density, whereas TOI-7394B b is a dense super-Jupiter ($2.10\,M_{\rm J}$, $\rho_p \approx 2.4$ g cm$^{-3}$) on a 1.25-day orbit. All hosts are early-M dwarfs with a narrow range of stellar properties, enabling a controlled comparison of giant-planet outcomes around low-mass stars. Three systems orbit super-solar metallicity stars, while TOI-7393 ($\mathrm{[Fe/H]} = -0.35 \pm 0.16$) is the most metal-poor GEMS host identified to date, and exhibits kinematics approaching the thin/thick-disk transition, suggestive of an older stellar population. Together, these systems reveal substantial diversity in the masses and bulk properties of short-period giant planets orbiting early-M dwarfs, demonstrating that markedly different planetary outcomes can arise around stars with otherwise similar fundamental properties.

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Searching for GEMS: Discovery of the Nearby Post-Common-Envelope Binary System TIC-460388167

Short-period white dwarf+main-sequence binaries are Post-Common-Envelope Binaries (PCEB) that have survived a common envelope phase. Such systems, if detached and eclipsing, enable precise measurements of the constituent stars, providing a unique opportunity to probe the effects of the common envelope phase on the system. We report the discovery of one such nearby (57 pc) system, TIC-460388167, using a combination of multi-band photometric light curves and spectroscopic radial velocities. In addition to eclipses, the system exhibits a continuously variable light curve that we model as a combination of ellipsoidal variations and star spots. We determine a period $P$=0.63596258$\pm$0.00000012 d and inclination $i$=89.0$\pm$0.4 deg. The best-fitting model specifies a white dwarf with T$_1$=7607$\pm$127 K and radius R$_1$=0.0131$\pm$0.0003 $R_\odot$, which is eclipsed by a T$_2$=3151 $\pm$ 59 K, R$_2$=0.327$\pm$0.006 $R_\odot$ M dwarf. The white dwarf mass is 0.61$\pm$0.04 M$_\odot$. We present the first velocity resolved profile for a PCEB secondary and show that the rotation of the M-dwarf is synchronous with the orbital period, as expected. We compare the constituent stars to other PCEB systems and find TIC-460388167A is one of the coolest known white dwarfs in such systems. TIC-460388167 is among the longest period eclipsing PCEB systems known.

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A Century of Radial Velocity and Astrometric Monitoring of 70 Oph AB: New PFS Data and Constraints on Planetary Companions

At a distance of 5.1 pc, the 70 Oph AB binary star system is one of the most favorable targets for future direct imaging and astrometry missions surveying mature, terrestrial planets. We present new radial velocities (RVs) obtained with the Planet Finder Spectrograph (PFS) on the 6.5\,m Magellan II Clay Telescope in Chile. We collected 499 measurements of 70 Oph A and 334 measurements of 70 Oph B during 2023--2025. Combining these data with decades of archival RVs and astrometry, we derive an updated orbital solution for the binary and dynamical masses of $0.88 \pm 0.004\,M_\odot$ and $0.73 \pm 0.003\,M_\odot$ for the primary and secondary components, respectively. We find that the long-term RV variability of both components is consistent with stellar activity modulated by rotation periods, and we detect no coherent planetary signals in either component. We place upper limits on any planets orbiting in the plane of the binary. The 27 yr RV baseline for 70 Oph A excludes Jupiter-mass planets interior to 5 au and reaches a sensitivity of $0.3\,M_{\rm Jup}$ at 1 au or $0.5\,M_{\rm Jup}$ at 2 au. For 70 Oph B, with PFS data we rule out planets more massive than $0.25$--$0.3\,M_{\rm Jup}$ inside 0.5 au. We show that stable S-type orbits around 70 Oph A extend to $\sim2.5$ au, covering the habitable zone. Thus, Saturn-mass planets or smaller on stable orbits in the habitable zone of 70 Oph A are allowed. Overall, our results provide important guidance for future planet searches around this stellar system.

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Stellar and orbital characterization of three low mass M dwarf binary stars with dynamical spectroscopy from the Habitable Zone Planet Finder

Theoretical models of low-mass stars continue to be discrepant with observations when used to examine the mass-radius relationship and other physical parameters of individual stars. High-resolution spectroscopy that leads to dynamical measurements of binary stars can directly improve these models. We have been using the Habitable-zone Planet Finder spectrograph to monitor binary stars with M dwarf components. Here, we measure the orbital and stellar parameters for three such systems: LSPM J0515+5911, NLTT 43564, and NLTT 45468. Each system has dozens of spectra obtained over a baseline of several years. None of the systems appear to be eclipsing, so our ability to turn them into true benchmark binaries with purely dynamical measurements is limited. We use literature photometry to estimate each system's spectral energy distribution and utilize models in combination with detection limits of our spectroscopic measurements to probe characteristics of the companions. LSPM J0515+5911 is a double-lined spectroscopic binary with period of $126.948 \pm 0.029$ days and derived minimum masses, $M_1\sin^3i =0.058 \pm 0.002$ $M_\odot$ and $M_2\sin^3i = 0.046 \pm 0.001$ $M_\odot$ for the primary and secondary components, respectively. We solved NLTT 43564 with period of $1877 \pm 24$ days and NLTT 45468 with period of $9.686 \pm 0.001$ days as single lined systems, and modeled the primary masses to be $M_1 = 0.32\pm{0.02}$ $M_\odot$ and $M_1 = 0.35^{+0.02}_{-0.07}$ $M_\odot$, respectively.

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Hipparcos, Gaia, and RVs reveal that the radio emitting F star HD 220242 has an M dwarf companion, a likely source of the radio emission

The detection of circularly polarized, low frequency radio emission offers the tantalizing possibility of the observation of interactions between stars and their possible substellar companions, as well as direct emission from exoplanets. Additional follow up of systems with radio emission is key to understanding the true origin of the emission, since multiple astrophysical mechanisms can plausibly lead to such signals. While nineteen M dwarfs were detected by LOFAR in circular polarization as part of the V-LoTSS survey, HD~220242 is the only F star to have a circularly polarized low frequency radio detection in the same survey. We conducted radial velocity follow up with the Habitable-zone Planet Finder and combined these observations with additional archival RVs and \textit{Hipparcos}-\textit{Gaia} proper motion accelerations to determine that HD~220242 has a stellar companion with P=16.79$\pm$0.04\,yrs and a mass of $0.619\pm0.014$\,M$_\odot$. We use Spectral Energy Distribution fitting and lack of any UV excess to rule out a co-evolved white dwarf companion and confirm that the companion is an M dwarf star. Given that F stars lack the coronal properties to produce such coherent emission, and the companion mass and lack of UV excess are consistent with an M dwarf, the radio emission is most plausibly associated with the companion.

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Are We There Yet? Challenges in Quantifying the Frequency of Earth Analogs in the Habitable Zone

Searching for life elsewhere in the universe is one of the most highly prioritized pursuits in astronomy today. However, the ability to observe evidence of Earth-like life through biosignatures is limited by the number of planets in the solar neighborhood with conditions similar to Earth. The occurrence rate of Earth-like planets in the habitable zones of Sun-like stars, $\eta_{\oplus}$, is therefore crucial for addressing the apparent lack of consensus on its value in the literature. Here we present a review of the current understanding of $\eta_{\oplus}$. We first provide definitions for parameters that contribute to $\eta_{\oplus}$. Then, we discuss the previous and current estimated parameter values and the context of the limitations on the analyses that produced these estimates. We compile an extensive list of the factors that go into any calculation of $\eta_{\oplus}$, and how detection techniques and surveys differ in their sensitivity and ability to accurately constrain $\eta_{\oplus}$. Understanding and refining the value of $\eta_{\oplus}$ is crucial for upcoming missions and telescopes, such as the planned Habitable Worlds Observatory and the Large Interferometer for Exoplanets, which aim to search for biosignatures on exoplanets in the solar neighborhood.

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Discovery of a Nearby Habitable Zone Super-Earth Candidate Amenable to Direct Imaging

We present the discovery of GJ 251 c, a candidate super-Earth orbiting in the Habitable Zone (HZ) of its M dwarf host star. Using high-precision Habitable-zone Planet Finder (HPF) and NEID RVs, in conjunction with archival RVs from the Keck I High Resolution Echelle Spectrometer (HIRES), the Calar Alto high-Resolution search for M dwarfs with Exoearths with Near-infrared and optical Echelle Spectrograph (CARMENES), and the SPectropolarim\`etre InfraROUge (SPIRou), we improve the measured parameters of the known planet, GJ 251 b ($P_{b}$ = 14.2370 days; $m \sin(i)$ = 3.85$^{+0.35}_{-0.33}$ M$_{\oplus}$), and we significantly constrain the minimum mass of GJ 251 c, placing it in a plausibly terrestrial regime (P$_{c}$ = 53.647 $\pm$ 0.044 days; $ m \sin i_{c}$ = 3.84 $\pm$ 0.75 M$_{\oplus}$). Using activity mitigation techniques that leverage chromatic information content, we perform a color-dependent analysis of the system and a detailed comparison of more than 50 models that describe the nature of the planets and stellar activity in the system. Due to GJ 251's proximity to Earth (5.5 pc), next generation, thirty meter class telescopes will likely be able to image terrestrial planets in GJ 251's HZ. In fact, GJ 251 c is currently the best candidate for terrestrial, HZ planet imaging in the Northern Sky.

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Order-by-order Modeling of Exoplanet Radial Velocity Data

Precise radial velocity (RV) measurements are a crucial tool for exoplanet discovery and characterization. Today, the majority of these measurements are derived from Echelle spectra in the optical wavelength region using cross-correlation techniques. Although for certain stars these approaches can produce RVs with sub-1 m~s$^{-1}$ measurement errors, for many others, we are now in a regime where instrumental precision is fundamentally below the intrinsic RV variations of the star that result from astrophysical processes that can be correlated in both time and wavelength. We explore new methods for measuring exoplanet orbital parameters that take advantage of the fact that RV data sets are fundamentally multi-wavelength. By analyzing NEID extremely precise radial velocity (EPRV) data of three known exoplanet systems, we show that fitting a single Keplerian model to multi-wavelength RVs can produce a factor of 1.5 -- 6.8 better $M_p \sin i$ uncertainties compared to fitting RVs that are derived from a weighted average across wavelength.

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Searching for GEMS: TOI-5916 b & TOI-6158 b are two Saturn-density planets orbiting M2 dwarfs

We confirm the planetary nature of (1) TOI-5916 b and (2) TOI-6158 b, two Exoplanets Transiting M-dwarf Stars (GEMS), both discovered by the Transiting Exoplanet Survey Satellite (TESS). Both systems were confirmed with ground-based photometry (Red Buttes Observatory and Swope, respectively) and radial velocity data from the Habitable-zone Planet Finder. Their radii are $R_{1}=11.8^{+0.52}_{-0.51}\text{ }R_{\oplus}$ and $R_{2}=10.4^{+2.70}_{-1.11}\text{ }R_{\oplus}$ and masses are $M_{1}=219\pm28\text{ }M_{\oplus}$ and $M_{2}=135^{+19}_{-18}\text{ }M_{\oplus}$. Both planets have Saturn-like densities ($\rho_{1} = 0.73^{+0.14}_{-0.13}\,\text{g cm}^{-3}$, $\rho_{2} = 0.66^{+0.41}_{-0.23}\,\text{g cm}^{-3}$), which appears to be a growing trend among GEMS systems and, more generally, warm Jupiters. In confirming both of these exoplanets, we add to the growing evidence for a population of Saturn-density planets among the GEMS systems. We also find evidence for a preliminary trend in which GEMS exhibit systematically closer orbits compared to FGK giants.

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Searching for GEMS: The Occurrence of Giant Planets orbiting M-dwarfs within 100 pc

We present results from a systematic search for transiting short-period Giant Exoplanets around M-dwarf Stars (GEMS; $P < 10$ days, $R_p \gtrsim 8~R_\oplus$) within a distance-limited 100 pc sample of $149,316$ M-dwarfs using TESS-Gaia Light Curve (TGLC) data. We describe the development and application of the \textit{TESS-miner} package and associated vetting procedures used in this analysis. To assess detection completeness, we conducted $\sim$72 million injection-recovery tests across $\sim$26,000 stars with an average of $\sim$3 sectors of data per star, subdivided into early-type (M0--M2.5), mid-type (M2.5--M4), and late-type (M4 or later) M-dwarfs. Our pipeline demonstrates high sensitivity across all subtypes within the injection bounds. We estimate the occurrence rates of short-period GEMS as a function of stellar mass, and combine our measured rates with those derived for FGK stars, fitting an exponential trend with stellar mass, consistent with core-accretion theory predictions. We find GEMS occurrence rates of $0.118\% \pm 0.068\%$ for early-type M-dwarfs, $0.153\% \pm 0.069\%$ for mid-type, and $0.036\% \pm 0.024\%$ for late-type M-dwarfs, with a mean rate of $0.068\%\pm0.024\%$ across the full sample. While our search spanned $1.0~\mathrm{days} < P < 10.0$ days, these rates were calculated using planets orbiting with $1.0~\mathrm{days} < P < 5.0$ days. This work establishes the basis for future occurrence rate studies of transiting GEMS.

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The NEID Earth Twin Survey. IV. Confirming an 89 d, $m\sin i=10~\mathrm{M_\oplus}$ Planet Orbiting a Nearby Sun-like Star

We present the confirmation of HD 190360 d, a warm ($P=88.690^{+0.051}_{-0.049}~\mathrm{d}$), low-mass ($m\sin i=10.23^{+0.81}_{-0.80}~\mathrm{M_\oplus}$) planet orbiting the nearby ($d=16.0$ pc), Sun-like (G7) star HD 190360. We detect HD 190360 d at high statistical significance even though its radial velocity (RV) semi-amplitude is only $K=1.48\pm0.11~\mathrm{m~s^{-1}}$. Such low-amplitude signals are often challenging to confirm due to potential confusion with low-amplitude stellar signals. The HD 190360 system previously had two known planets: the $1.7~\mathrm{M_J}$ (true mass) HD 190360 b on a $7.9$ yr orbit and the $21~\mathrm{M_\oplus}$ (minimum mass) HD 190360 c on a $17.1$ d orbit. Here, we present an in-depth analysis of the HD 190360 planetary system that comprises more than 30 years of RV measurements and absolute astrometry from the Hipparcos and Gaia spacecrafts. Our analysis uses more than 1400 RVs, including nearly 100 from NEID. The proper motion anomaly as measured by these two astrometric missions solves for the dynamical mass of HD 190360 b and contributes to our understanding of the overall system architecture, while the long baseline of RVs enables the robust characterization of HD 190360 c and confirms the discovery of HD 190360 d.

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An Earth-Sized Planet in a 5.4h Orbit Around a Nearby K dwarf

We present the discovery and confirmation of the ultra-short period (USP) planet TOI-2431 b orbiting a nearby ($d\sim36$ pc) late K star ($T_{\mathrm{eff}}$ = $4109 \pm 28 \, {\rm K}$) using observations from the Transiting Exoplanet Survey Satellite (TESS), precise radial velocities with the NEID and the Habitable-zone Planet Finder (HPF) spectrographs, as well as ground-based high contrast imaging from NESSI. TOI-2431 b has a period of 5 hours and 22 minutes, making it one of the shortest-period exoplanets known to date. TOI-2431 b has a radius of $1.536 \pm 0.033\, \rm{R_\oplus}$, and a mass of $6.2 \pm 1.2\, \rm{M_\oplus}$, suggesting it has a density compatible with an Earth-like composition and, due to its high irradiation, is likely a 'lava-world' with a $T_{\mathrm{eq}}$ = $2063 \pm 30 \, {\rm K}$. We estimate that the current orbital period is only 30% larger than the Roche-limit orbital period, and that it has an expected orbital decay timescale of only $\sim$31 Myr. Finally, due to the brightness of the host star ($V = 10.9$, $K = 7.6$), TOI-2431 b has a high Emission Spectroscopy Metric of 27, making it one of the best USP systems for atmospheric phase-curve analysis.

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TOI-1259Ab: A Warm Jupiter Orbiting a K-dwarf White-Dwarf Binary is on a Well-aligned Orbit

The evolution of one member of a stellar binary into a white dwarf has been proposed as a mechanism that triggers the formation of close-in gas giant planets. The star's asymmetric mass loss during the AGB stage gives it a "kick" that can initiate Eccentric Lidov-Kozai oscillations, potentially causing a planet around the secondary star to migrate inwards and perturbing the eccentricity and inclination of its orbit. Here we present a measurement of the stellar obliquity of TOI-1259Ab, a gas giant in a close-in orbit around a K star with a white dwarf companion about 1650 au away. By using the NEID spectrograph to detect the Rossiter-McLaughlin effect during the planetary transit, we find the sky-projected obliquity to be $\lambda = 6^{+21}_{-22}\,^\circ$. When combined with estimates of the stellar rotation period, radius, and projected rotation velocity, we find the true 3D obliquity to be $\psi = 24^{+14}_{-12}\,^\circ$ ($\psi < 48^\circ$ at 95% confidence), revealing that the orbit of TOI-1259Ab is well aligned with the star's equatorial plane. Because the planet's orbit is too wide for tidal realignment to be expected, TOI-1259Ab might have formed quiescently in this well-aligned configuration. Alternatively, as we show with dynamical simulations, Eccentric Lidov-Kozai oscillations triggered by the evolution of the binary companion are expected to lead to a low obliquity with a probability of about $\sim$14%.

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The NEID Earth Twin Survey. III. Survey Performance After Three Years on Sky

The NEID Earth Twin Survey (NETS) has been delivering a rich set of precise radial velocity (RV) measurements for 41 bright, nearby main sequence stars. Here, we describe the status of the survey after three years on sky and we present the full set of RV measurements and accompanying stellar activity indicators. We discuss intermediate survey diagnostics, including calibration of the known RV zero point offset introduced following the Contreras fire in 2022 and the identification of an undiagnosed and previously unknown zero point offset in 2021. An analysis of our data set using RVSearch demonstrates that for these target stars, NEID is independently sensitive to nearly all known planets with periods shorter than the NETS observing baseline. We also highlight a number of newly detected RV signals, which present exciting opportunities for future investigations.

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Hundreds of TESS exoplanets might be larger than we thought

The radius of a planet is a fundamental parameter that probes its composition and habitability. Precise radius measurements are typically derived from the fraction of starlight blocked when a planet transits its host star. The wide-field Transiting Exoplanet Survey Satellite (TESS) has discovered hundreds of new exoplanets, but its low angular resolution means that the light from a star hosting a transiting exoplanet can be blended with the light from background stars. If not fully corrected, this extra light can dilute the transit signal and result in a smaller measured planet radius. In a study of hundreds of TESS planet discoveries using deblended light curves from our validated methodology, we show that systematically incorrect planet radii are common in the literature: studies using various public TESS photometry pipelines have underestimated the planet radius by a weighted median of $6.1\% \pm 0.3\%$, leading to a $\sim20\%$ overestimation of planet density. The widespread presence of these biases in the literature has profoundly shaped-and potentially misrepresented-our understanding of the exoplanet population. Addressing these biases will refine the exoplanet mass-radius relation, reshape our understanding of exoplanet atmospheric and bulk composition, and potentially inform prevailing planet formation theories.

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Time-resolved p-mode oscillations for subgiant HD 142091 with NEID at WIYN

Detections of Earth-analog planets in radial velocity observations are limited by stellar astrophysical variability occurring on a variety of timescales. Current state-of-the-art methods to disentangle potential planet signals from intrinsic stellar signals assume that stellar signals introduce asymmetries to the line profiles that can therefore be separated from the pure translational Doppler shifts of planets. Here, we examine this assumption using a time series of resolved stellar p-mode oscillations in HD 142091 ($\kappa$ CrB), as observed on a single night with the NEID spectrograph at 2-minute cadence and with 25 cm/s precision. As an evolved subgiant star, this target has p-mode oscillations that are larger in amplitude (4-8 m/s) and occur on longer timescales (80 min.) than those of typical Sun-like stars of RV surveys, magnifying their corresponding effects on the stellar spectral profile. We show that for HD 142091, p-mode oscillations manifest primarily as pure Doppler shifts in the average line profile -- measured by the cross-correlation function (CCF) -- with "shape-driven" CCF variations as a higher-order effect. Specifically, we find that the amplitude of the shift varies across the CCF bisector, with 10% larger oscillation amplitudes closer to the core of the CCF, and 25% smaller oscillation amplitudes for bisector velocities derived near the wings; we attribute this trend to larger oscillation velocities higher in the stellar atmosphere. Using a line-by-line analysis, we verify that a similar trend is seen as a function of average line depth, with deeper lines showing larger oscillation amplitudes. Finally, we find no evidence that p-mode oscillations have a chromatic dependence across the NEID bandpass beyond that due to intrinsic line depth differences across the spectrum.

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