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<jats:title>Abstract</jats:title> <jats:p>Although blue horizontal branch (BHB) stars are commonly used to trace halo substructure, the stars bluer than (<jats:italic>g</jats:italic> − <jats:italic>r</jats:italic>) &lt; −0.3 are ignored owing to the difficulty in determining their absolute magnitudes. The blue extension of the horizontal branch (HBX) includes BHB tail stars and extreme horizontal branch (EHB) stars. We present a method for identifying HBX stars in the field, using spectra and photometry from the Sloan Digital Sky Survey Data Release 14 (SDSS DR14). We derive an estimate for the absolute magnitudes of BHB tail and EHB stars as a function of color and use this relationship to calculate distances. We identify an overdensity of HBX stars that appears to trace the northern end of the Hercules-Aquila Cloud. We identify three stars that are likely part of a tidal stream, but this is not enough stars to explain the observed overdensity. Combining SDSS data with Gaia DR2 proper motions allows us to show that the majority of the HBX stars in the overdensity are on high-eccentricity orbits similar to those in the Virgo Radial Merger/Gaia–Enceladus/Gaia Sausage structure and that the overdensity of high-eccentricity orbits extends all the way to the Virgo Overdensity. We use stellar kinematics to separate the HBX stars into disk stars and halo stars. The halo stars are primarily on highly eccentric (radial) orbits. The fraction of HBX stars that are EHBs is highest in the disk population and lowest in the low-eccentricity halo stars.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>R Coronae Borealis (RCB) stars are rare hydrogen-deficient carbon-rich variable supergiants thought to be the result of dynamically unstable white dwarf mergers. We attempt to model RCB stars through all the relevant timescales by simulating a merger event in <jats:italic>Octo-tiger</jats:italic>, a 3D adaptive mesh refinement (AMR) hydrodynamics code, and mapping the post-merger object into MESA, a 1D stellar evolution code. We then post-process the nucleosynthesis on a much larger nuclear reaction network to study the enhancement of <jats:italic>s</jats:italic>-process elements. We present models that match observations or previous studies in most surface abundances, isotopic ratios, early evolution, and lifetimes. We also observe similar mixing behavior to previous modeling attempts that result in the partial He-burning products visible on the surface in observations. However, we do note that our subsolar models lack any enhancement in <jats:italic>s</jats:italic>-process elements, which we attribute to a lack of hydrogen in the envelope. We also find that the <jats:sup>16</jats:sup>O/<jats:sup>18</jats:sup>O isotopic ratio is very sensitive to initial hydrogen abundance and increases outside of the acceptable range with a hydrogen mass fraction greater than 10<jats:sup>−4</jats:sup>.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>We present a systematic X-ray and multiwavelength study of a sample of 47 active galactic nuclei (AGNs) with reverberation mapping measurements. This sample includes 21 super-Eddington accreting AGNs and 26 sub-Eddington accreting AGNs. Using high-state observations with simultaneous X-ray and UV/optical measurements, we investigate whether super-Eddington accreting AGNs exhibit different accretion disk–corona connections compared to sub-Eddington accreting AGNs. We find tight correlations between the X-ray-to-UV/optical spectral slope parameter (<jats:italic>α</jats:italic> <jats:sub>OX</jats:sub>) and the monochromatic luminosity at 2500 Å (<jats:italic>L</jats:italic> <jats:sub>2500Å</jats:sub>) for both the super- and sub-Eddington subsamples. The best-fit <jats:italic>α</jats:italic> <jats:sub>OX</jats:sub>–<jats:italic>L</jats:italic> <jats:sub>2500Å</jats:sub> relations are consistent overall, indicating that super-Eddington accreting AGNs are not particularly X-ray weak in general compared to sub-Eddington accreting AGNs. We find dependences of <jats:italic>α</jats:italic> <jats:sub>OX</jats:sub> on both the Eddington ratio (<jats:italic>L</jats:italic> <jats:sub>Bol</jats:sub>/<jats:italic>L</jats:italic> <jats:sub>Edd</jats:sub>) and black hole mass (<jats:italic>M</jats:italic> <jats:sub>BH</jats:sub>) parameters for our full sample. A multivariate linear regression analysis yields <jats:inline-formula> <jats:tex-math> <?CDATA ${\alpha }_{\mathrm{OX}}=-0.13\mathrm{log}({L}_{\mathrm{Bol}}/{L}_{\mathrm{Edd}})-0.10\mathrm{log}{M}_{\mathrm{BH}}-0.69$?> </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabe37fieqn1.gif" xlink:type="simple" /> </jats:inline-formula>, with a scatter similar to that of the <jats:italic>α</jats:italic> <jats:sub>OX</jats:sub>–<jats:italic>L</jats:italic> <jats:sub>2500Å</jats:sub> relation. The hard (rest-frame &gt;2 keV) X-ray photon index (Γ) is strongly correlated with <jats:italic>L</jats:italic> <jats:sub>Bol</jats:sub>/<jats:italic>L</jats:italic> <jats:sub>Edd</jats:sub> for the full sample and the super-Eddington subsample, but these two parameters are not significantly correlated for the sub-Eddington subsample. A fraction of super-Eddington accreting AGNs show strong X-ray variability, probably due to small-scale gas absorption, and we highlight the importance of employing high-state (intrinsic) X-ray radiation to study the accretion disk–corona connections in AGNs.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Observational evidence suggests that the majority of stars may have been born in stellar clusters or associations. Within these dense environments, dynamical interactions lead to high rates of close stellar encounters. A variety of recent observational and theoretical indications suggest stellar-mass black holes may be present and play an active dynamical role in stellar clusters of all masses. In this study, we explore the tidal disruption of main-sequence stars by stellar-mass black holes in young star clusters. We compute a suite of over 3000 independent <jats:italic>N</jats:italic>-body simulations that cover a range of cluster mass, metallicity, and half-mass radii. We find stellar-mass black hole tidal disruption events (TDEs) occur at an overall rate of up to roughly 200 Gpc<jats:sup>−3</jats:sup> yr<jats:sup>−1</jats:sup> in young stellar clusters in the local universe. These TDEs are expected to have several characteristic features, namely, fast rise times of order a day, peak X-ray luminosities of at least 10<jats:sup>44</jats:sup> erg s<jats:sup>−1</jats:sup>, and bright optical luminosities (roughly 10<jats:sup>41</jats:sup>–10<jats:sup>44</jats:sup> erg s<jats:sup>−1</jats:sup>) associated with reprocessing by a disk wind. In particular, we show these events share many features in common with the emerging class of Fast Blue Optical Transients.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>We present the largest currently existing subarcsecond 3–5 <jats:italic>μ</jats:italic>m atlas of 119 local (<jats:italic>z</jats:italic> &lt; 0.3) active galactic nuclei (AGNs). This atlas includes AGNs of five subtypes: 22 are Seyfert 1; five are intermediate Seyferts; 46 are Seyfert 2; 26 are low-ionization nuclear emission regions; and 20 are composites/starbursts. Each active galactic nucleus was observed with the Very Large Telescope Infrared Spectrometer and Array Camera (ISAAC) in the <jats:italic>L</jats:italic> and/or <jats:italic>M </jats:italic>bands between 2000 and 2013. We detected at 3<jats:italic>σ</jats:italic> confidence 92 sources in the <jats:italic>L</jats:italic> band and 83 sources in the <jats:italic>M</jats:italic> band. We separated the flux into unresolved nuclear flux and resolved the flux through two-Gaussian fitting. We report the nuclear flux, extended flux, apparent size, and position angle of each source, giving 3<jats:italic>σ</jats:italic> upper limits for sources that are undetected. Using Wide-field Infrared Survey Explorer (WISE) <jats:italic>W1</jats:italic>- and <jats:italic>W2</jats:italic>-band photometry we derived relations predicting the nuclear <jats:italic>L</jats:italic> and <jats:italic>M</jats:italic> fluxes for Sy1 and Sy2 AGNs based on their <jats:italic>W1–W2</jats:italic> color and WISE fluxes. Lastly, we compare the measured mid-infrared colors to those predicted by dusty torus models SKIRTOR, CLUMPY, CAT3D, and CAT3D-WIND, finding the best agreement with the latter. We find that models including polar winds best reproduce the 3–5 <jats:italic>μ</jats:italic>m colors, indicating that it is an important component of dusty torus models. We found that several AGNs are bluer than models predict. We discuss several explanations for this and find that it is most plausibly stellar light contamination within the ISAAC <jats:italic>L</jats:italic>-band nuclear fluxes.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>The Frontier Fields project is an observational campaign targeting six galaxy clusters, with the intention of using the magnification provided by gravitational lensing to study galaxies that are extremely faint or distant. We used the Karl G. Jansky Very Large Array (VLA) at 3 and 6 GHz to observe three Frontier Fields: MACS J0416.1−2403 (<jats:italic>z</jats:italic> = 0.396), MACS J0717.5+3745 (<jats:italic>z</jats:italic> = 0.545), and MACS J1149.5+2223 (<jats:italic>z</jats:italic> = 0.543). The images reach noise levels of ∼1 <jats:italic>μ</jats:italic>Jy beam<jats:sup>−1</jats:sup> with subarcsecond resolution (∼2.5 kpc at <jats:italic>z</jats:italic> = 3), providing a high-resolution view of high-<jats:italic>z</jats:italic> star-forming galaxies that is unbiased by dust obscuration. We generate dual-frequency continuum images at two different resolutions per band, per cluster, and derive catalogs totaling 1966 compact radio sources. Components within the areas of Hubble Space Telescope and Subaru observations are cross-matched, providing host galaxy identifications for 1296 of them. We detect 13 moderately lensed (2.1 &lt; <jats:italic>μ</jats:italic> &lt; 6.5) sources, one of which has a demagnified peak brightness of 0.9 <jats:italic>μ</jats:italic>Jy beam<jats:sup>−1</jats:sup>, making it a candidate for the faintest radio source ever detected. There are 66 radio sources exhibiting complex morphologies, and 58 of these have host galaxy identifications. We reveal that MACS J1149.5+2223 is not a cluster with a double relic, as the western candidate relic is resolved as a double-lobed radio galaxy associated with a foreground elliptical at <jats:italic>z</jats:italic> = 0.24. The VLA Frontier Fields project is a public legacy survey. The image and catalog products from this work are freely available.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>Filamentary structures are recognized as a fundamental component of interstellar molecular clouds in observations made by the Herschel satellite. These filaments, especially massive filaments, often extend in a direction perpendicular to the interstellar magnetic field. Furthermore, the filaments sometimes have an apparently negative temperature gradient—that is, their temperatures decrease toward their centers. In this paper, we study the magnetohydrostatic equilibrium state of negative-indexed polytropic gas with the magnetic field running perpendicular to the axis of the filament. The model is controlled by four parameters: center-to-surface density ratio (<jats:italic>ρ</jats:italic> <jats:sub> <jats:italic>c</jats:italic> </jats:sub>/<jats:italic>ρ</jats:italic> <jats:sub> <jats:italic>s</jats:italic> </jats:sub>), plasma <jats:italic>β</jats:italic> of the surrounding gas, radius of the parent cloud <jats:inline-formula> <jats:tex-math> <?CDATA ${R}_{0}^{{\prime} }$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mo accent="true">′</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabea7aieqn1.gif" xlink:type="simple" /> </jats:inline-formula> normalized by the scale height, and the polytropic index <jats:italic>N</jats:italic>. The steepness of the temperature gradient is represented by <jats:italic>N</jats:italic>. We found that the envelope of the column density profile becomes shallow when the temperature gradient is large. This reconciles the inconsistency between the observed profiles and those expected from the isothermal models. We compared the maximum line mass (mass per unit length), above which there is no equilibrium, with that of the isothermal nonmagnetized filament. We obtained an empirical formula to express the maximum line mass of a magnetized polytropic filament as <jats:inline-formula> <jats:tex-math> <?CDATA ${\lambda }_{{\rm{m}}{\rm{a}}{\rm{x}}}\simeq {\left[{\left({\lambda }_{0,{\rm{m}}{\rm{a}}{\rm{x}}}(N)/{M}_{\odot }{\rm{p}}{{\rm{c}}}^{-1}\right)}^{2}+{\left[5.9{\left(1.0+1.2/N\right)}^{1/2}\left({{\rm{\Phi }}}_{{\rm{c}}{\rm{l}}}/1\mu {\rm{G}}{\rm{p}}{\rm{c}}\right)\right]}^{2}\right]}^{1/2}$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mrow /> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">x</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>≃</mml:mo> <mml:msup> <mml:mrow> <mml:mfenced close="]" open="["> <mml:mrow> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:msub> <mml:mi>λ</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">a</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">x</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> <mml:mspace width="0.25em" /> <mml:mrow /> <mml:mrow> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mn>5.9</mml:mn> <mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mn>1.0</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1.2</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow /> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow /> <mml:mi>μ</mml:mi> <mml:mrow /> <mml:mrow> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> <mml:mspace width="0.25em" /> <mml:mrow> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabea7aieqn2.gif" xlink:type="simple" /> </jats:inline-formula> <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> pc<jats:sup>−1</jats:sup>, where <jats:inline-formula> <jats:tex-math> <?CDATA ${\lambda }_{0,\max }(N)$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>λ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>max</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabea7aieqn3.gif" xlink:type="simple" /> </jats:inline-formula> represents the maximum line mass of the nonmagnetized filament and Φ<jats:sub>cl</jats:sub> indicates one-half of the magnetic flux threading the filament per unit length. Although the negative-indexed polytrope makes the maximum line mass decrease, compared with that of the isothermal model, a magnetic field threading the filament increases the line mass.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>To investigate the growth history of galaxies, we measure the rest-frame radio, ultraviolet (UV), and optical sizes of 98 radio-selected, star-forming galaxies (SFGs) distributed over 0.3 ≲ <jats:italic>z</jats:italic> ≲ 3 with a median stellar mass of <jats:inline-formula> <jats:tex-math> <?CDATA $\mathrm{log}({M}_{\star }/{M}_{\odot })\approx 10.4$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⋆</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≈</mml:mo> <mml:mn>10.4</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabe876ieqn1.gif" xlink:type="simple" /> </jats:inline-formula>. We compare the size of galaxy stellar disks, traced by rest-frame optical emission, relative to the overall extent of star formation activity that is traced by radio continuum emission. Galaxies in our sample are identified in three <jats:italic>Hubble </jats:italic>Frontier Fields: MACS J0416.1−2403, MACS J0717.5+3745, and MACS J1149.5+2223. Radio continuum sizes are derived from 3 and 6 GHz radio images (≲0.″6 resolution, ≈0.9 <jats:italic>μ</jats:italic>Jy beam<jats:sup>−1</jats:sup> noise level) from the Karl G. Jansky Very Large Array. Rest-frame UV and optical sizes are derived using observations from the <jats:italic>Hubble Space Telescope</jats:italic> and the Advanced Camera for Surveys and Wide Field Camera 3 instruments. We find no clear dependence between the 3 GHz radio size and stellar mass of SFGs, which contrasts with the positive correlation between the UV/optical size and stellar mass of galaxies. Focusing on SFGs with <jats:inline-formula> <jats:tex-math> <?CDATA $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10$?> </jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⋆</mml:mo> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>⊙</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>10</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="apjabe876ieqn2.gif" xlink:type="simple" /> </jats:inline-formula>, we find that the radio/UV/optical emission tends to be more compact in galaxies with high star formation rates (≳100 <jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub> yr<jats:sup>−1</jats:sup>), suggesting that a central, compact starburst (and/or an active galactic nucleus) resides in the most luminous galaxies of our sample. We also find that the physical radio/UV/optical size of radio-selected SFGs with log(<jats:italic>M</jats:italic> <jats:sub>⋆</jats:sub>/<jats:italic>M</jats:italic> <jats:sub>⊙</jats:sub>) &gt; 10 increases by a factor of 1.5–2 from <jats:italic>z</jats:italic> ≈ 3 to <jats:italic>z</jats:italic> ≈ 0.3, yet the radio emission remains two to three times more compact than that from the UV/optical. These findings indicate that these massive, radio-selected SFGs at 0.3 ≲ <jats:italic>z</jats:italic> ≲ 3 tend to harbor centrally enhanced star formation activity relative to their outer disks.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>By combining the Large Sky Area Multi-object Fiber Spectroscopic Telescope and Gaia data, we investigate the vertical phase mixing across the Galactic disk. Our results confirm the existence of phase space snail shells (or phase spirals) from 6–12 kpc. We find that grouping stars by the guiding radius (<jats:italic>R</jats:italic> <jats:sub> <jats:italic>g</jats:italic> </jats:sub>), instead of the present radius (<jats:italic>R</jats:italic>), further enhances the snail shell signal in the following aspects: (1) clarity of the snail shell shape is increased; (2) more wraps of the snail shell can be seen; (3) the phase spaces are less affected by the lack of stars closer to the disk midplane due to extinction; and (4) the phase space snail shell is amplified in greater radial ranges. Compared to the <jats:italic>R</jats:italic>-based snail shell, the quantitatively measured shapes are similar, except that the <jats:italic>R</jats:italic> <jats:sub> <jats:italic>g</jats:italic> </jats:sub>-based snail shells show more wraps with a better contrast. These lines of evidence lead to the conclusion that the guiding radius (angular momentum) is a fundamental parameter tracing the phase space snail shell across the Galactic disk. Results of our test particle simulations with impulse approximation verify that particles grouped according to <jats:italic>R</jats:italic> <jats:sub> <jats:italic>g</jats:italic> </jats:sub> reveal well-defined and sharper snail shell features. By comparing the radial profiles of the snail shell pitch angle between the observation and simulation, the external perturbation can be constrained to ∼500−700 Myr ago. For future vertical phase mixing study, it is recommended to use the guiding radius with additional constraints on orbital hotness (ellipticity) to improve the clarity of the phase snail.</jats:p>