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Daniel Patnaude

Publications and source records attributed to Daniel Patnaude.

At least 19 recordsLinked to original sources

Lynx2030 Science Analysis Group: Final Report

The Lynx2030 Science Analysis Group (SAG) was convened to reassess the scientific goals and technical drivers of the Lynx mission concept amid a rapidly evolving astrophysics landscape. Building on the original Lynx Concept Study, the SAG examined how recent discoveries, emerging facilities, and advances in instrumentation influence the scientific opportunities for a next-generation flagship X-ray observatory. Through focused working groups, the SAG investigated the scientific impact of enhanced capabilities: (i) improved angular resolution, (ii) broader bandpass coverage, (iii) an enhanced microcalorimeter, (iv) new capabilities and observing modes, and (v) larger fields of view. Across a broad range of topics, from the formation of the first black holes and the evolution of galaxies to the baryon cycle, compact objects, stellar explosions, multi-messenger astrophysics, and the dynamic high-energy Universe, the SAG finds that the scientific motivation for a Lynx-class observatory remains compelling and, in many areas, has significantly strengthened over the past decade, prominently through JWST's discovery of the "Little Red Dots", likely massive accreting black holes in infant galaxies whose nature is fundamentally an X-ray question. This report shows that modest extensions beyond the original Lynx design reference mission can unlock transformative science while preserving the observatory's core architecture. Powerful current and future facilities such as Roman, Rubin, JWST, SKA, ngVLA, LISA, and NewAthena highlight the unique role a high-angular-resolution, high-throughput X-ray observatory would play in the multi-wavelength and multi-messenger ecosystem of the 2030s and beyond. The findings of the Lynx2030 SAG confirm Lynx's central vision: an unprecedented view of the hot and energetic Universe, enabling discoveries that will define high-energy astrophysics in the coming decades.

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Ardua: Unveiling the Baryon Cycle from Stars to the Cosmic Web

The circumgalactic medium (CGM) -- the multiphase gas reservoirs surrounding galaxies -- remains the least understood component of the baryon cycle governing galaxy growth, despite its central role in the Astro2020 Decadal Survey's priorities. Existing constraints come almost exclusively from pencil-beam absorption spectroscopy, leaving the spatial structure, kinematics, and phase interactions of CGM gas fundamentally unmapped. We present Ardua, a mission concept for NASA's ASTRA Initiative that combines wide-field far-ultraviolet spectroscopy with a Line Emission Mapper (LEM)-derived X-ray microcalorimeter instrument to obtain the first comprehensive emission maps spanning the full CGM temperature range, including cool neutral gas, ionized warm-hot phase gas, and the volume-filling hot corona. By observing more than 50 nearby galaxies comprehensively in the UV and X-ray, Ardua will test competing galaxy formation models, resolve multiphase gas flows and feedback-driven outflows, and extend baryon-cycle science to the intergalactic medium and the environments of exoplanet-hosting stars. Beyond its core CGM/IGM program, Ardua's wide-field, high-sensitivity instruments are designed to serve as a flexible community resource, supporting guest-investigator science across astrophysics. No planned or approved mission is designed to deliver this combined UV/X-ray survey capability.

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Old and Bright: The Remarkable Radio Brightening of the Engine-driven SN 2012au Several Years After Explosion Signals the Birth of a PWN

We present the results from an extensive broad-band (radio to X-rays) observing campaign of the engine-driven Type Ib SN 2012au in the first 13 years of evolution. The early-time (${\delta}t\leq{190}$ d) radio and X-ray evolution is well-described by conventional models of a forward shock interacting with a wind-like circumstellar medium ($\rho_{\rm{CSM}}\propto{r}^{-2}$). However, starting at $\delta{t}\approx{6.7}$ yr, we detect a significant radio re-brightening. This late-time emission is dominated by a luminous component characterized by a broad and rapidly evolving spectral peak and a shallow optically thin spectral slope, $F_{\nu}\propto{\nu}^{-0.31\pm0.02}$. These properties imply a compact emitting region ($R\lesssim{10}^{16}$ cm) expanding at a remarkably slow velocity ($\lesssim{500}$ km/s) into a high-density environment ($\geq{10}^4 \rm{cm}^{-3}$), accompanied by a hard electron power-law index $p\approx{1.6}$. No soft or hard X-ray emission is detected at any epoch, indicating that high-energy radiation is either strongly absorbed or intrinsically absent. In the context of aspherical shock-CSM interaction models, these observations imply extreme properties of the CSM (geometry, density, total mass) that lack clear astrophysical motivation. Instead, we show that the emergence of radiation from a newborn Pulsar Wind Nebula (PWN) naturally explains the radio spectral evolution and high-energy limits, where the emission is governed by the adiabatic expansion of a relic pair plasma. We conclude that SN 2012au represents the most compelling candidate for a young, newborn PWN discovered to date, a scenario that can be directly tested with pending Very Long Baseline Interferometry (VLBI) observations.

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Single vs. Binary Origin: The Diversity of Stripped-Envelope Supernova Remnants

Core-collapse supernova remnants (CCSNRs) are crucial for understanding the final stages of massive star evolution, as they reflect the imprints of their progenitors' pre-explosion activities. However, the evolution of CCSNRs, particularly those originating from progenitors with high mass-loss rates -- known as stripped-envelope SNRs (SESNRs) -- remains poorly understood. This is largely due to the lack of comprehensive numerical models connecting progenitor stars to their remnants, especially in the context of binarity. In this study, we perform self-consistent simulations of CCSNRs from both single and binary progenitors, utilizing mass-loss histories and supernova ejecta profiles directly derived from stellar evolution and explosion calculations. Our models reveal significant differences in the circumstellar medium (CSM) structures between single and binary progenitors, which drive distinct SNR dynamics and spectral characteristics. We find that binary-stripped progenitors tend to produce SNRs with more monotonic CSM profiles, resulting in smoother shock dynamics and less pronounced X-ray luminosity peaks compared to their single-star counterparts. Additionally, we introduce a new characteristic timescale, $t_{\rm CSM}$, defined by the total mass lost by the progenitor. This timescale effectively scales the evolutionary phases of CCSNRs in complex CSM environments, thereby facilitating the comparison of SESNRs. Given that observed elemental abundances in SNRs reflect the nucleosynthesis yields of the progenitor, our results highlight the importance of considering the dynamical state of SNRs when interpreting observed abundances. This work provides a fiducial framework for future observational and theoretical studies of CCSNRs, particularly regarding the impact of binary evolution.

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Probing the Diversity of Type Ia Supernova Remnants in 3-D Hydrodynamic Simulations with X-ray Spectral Synthesis

Type Ia supernovae (SNe), thermonuclear explosions of white dwarfs in binary systems, are widely used as standard candles owing to the empirical width-luminosity relation of their light curves. Recent theoretical and observational studies indicate a diversity of progenitor systems and explosion mechanisms. In the supernova remnant (SNR) phase, the diversity in Fe-K$\alpha$ centroid energies and line luminosities suggests variations in the underlying explosion mechanisms. X-ray spectra of SNRs, which trace shocked ejecta and the surrounding medium, are crucial diagnostics of progenitor systems and explosion physics. Thanks to recent advances in spectroscopy with XRISM, high-resolution X-ray spectroscopy enables 3-D diagnostics, including line-of-sight velocities. In this study, we perform 3-D hydrodynamic simulations of SNRs from six Type Ia explosion models: two each of pure deflagration, delayed detonation, and double detonation. Each model is evolved for 1000 years in a uniform medium, consistently accounting for non-equilibrium ionization. Our efficient numerical scheme enables systematic parameter surveys in full 3-D. From these models, we synthesize X-ray spectra with $\sim$1 eV resolution, exceeding XRISM/Resolve's spectral resolution. This work presents the first calculation of X-ray spectra for Type Ia SNRs derived from 3-D hydrodynamic simulations that follow the evolution self-consistently from the SN phase into the SNR phase. Our results show inter-model diversity in the X-ray spectra. Asymmetric, red- and blueshifted line profiles arise from the 3-D ejecta distributions. These findings demonstrate that 3-D SNR modeling can reproduce the observed diversity of Type Ia SNRs and provide qualitative constraints on progenitor systems and explosion mechanisms.

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The role of the secondary white dwarf in a double-degenerate double-detonation explosion, in the supernova remnant phase

Type Ia supernovae (SNe) are believed to be thermonuclear explosions of white dwarf (WD) stars, but their progenitor systems and explosion mechanisms are still unclear. Here we focus on double degenerate systems, where two WDs are interacting, and on the double detonation mechanism, where a detonation of a helium shell triggers a detonation of the carbon-oxygen core of the primary WD. We take the results from three-dimensional SN simulations of Pakmor et al 2022 (arXiv:2203.14990) and carry them into the supernova remnant (SNR) phase, until 1500 yr after the explosion. We reveal signatures of the SN imprinted in the SNR morphology. We confirm the impact of a companion on the SNR: its presence induces a conical shadow in the ejecta, that is long lived. Its intersection with the shocked shell is visible in projection as a ring, an ellipse, or a bar, depending on the orientation. New, we test the case of a nested explosion model, in which the explosion of the primary induces the secondary to also explode. As the explosion of the secondary WD is weaker only the primary outer ejecta interact with the ambient medium and form the main SNR shell. The secondary inner ejecta collide with the reverse shock, which enhances the density and thus the X-ray emissivity. The composition at the points of impact is peculiar, since what is revealed are the outer layers from the inner ejecta. This effect can be probed with spatially-resolved X-ray spectroscopy of young SNRs.

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The first radio view of a type Ibn supernova in SN 2023fyq: Understanding the mass-loss history in the last decade before the explosion

Supernovae that interact with hydrogen-poor, helium-rich circumstellar material (CSM), known as Type Ibn supernovae (SNe Ibn), present a unique opportunity to probe mass-loss processes in massive stars. In this work, we report the first radio detection of a SN Ibn, SN 2023fyq, and characterize the mass-loss history of its stellar progenitor using the radio and X-ray observations obtained over 18 months post-explosion. We find that the radio emission from 58--185 days is best modeled by synchrotron radiation attenuated by free-free absorption from a CSM of density $\sim$ $10^{-18}$ g/$\rm{cm^{3}}$ ($\sim 10^{6} \mathrm{\rho_{ISM}}$) at a radius of $10^{16}$ cm, corresponding to a mass-loss rate of $\sim$ $4 \times 10^{-3} \ \mathrm{M_{\odot} \ yr^{-1}}$ (for a wind velocity of 1700 km/s from optical spectroscopy) from 0.7 to 3 years before the explosion. This timescale is consistent with the time frame over which pre-explosion optical outbursts were observed. However, our late-time observations at 525 days post-explosion yield non-detections, and the 3$\sigma$ upper limits (along with an X-ray non-detection) allow us to infer lower-density CSM at $2\times 10^{16}$ cm with $\rm{\dot{M}}$ $< 2.5\times 10^{-3} \ \mathrm{M_{\odot} \ yr^{-1}}$. These results suggest a shell-like CSM from at most $4 \times 10^{15}$ to $2 \times 10^{16}$ cm ($\sim 10^{5} R_{\rm{\odot}}$) with an elevated CSM density (0.004 $\mathrm{M_{\odot} \ yr^{-1}}$) that is roughly consistent with predictions from a merger model for this object. Future radio observations of a larger sample of SNe Ibn will provide key details on the extent and density of their helium-rich CSM.

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Verification of Cas A neutron star cooling rate using Chandra HRC-S observations

The young neutron star (NS) in the Cassiopeia A (Cas A) supernova remnant is a fascinating test for theories of NS cooling. Chandra observations have indicated that its surface temperature is declining rapidly, about 2% per decade, using 20 years of data, if a uniform carbon atmosphere is assumed for the NS. This rapid decline may be caused by the neutrons in the NS core transitioning from a normal to a superfluid state. However, most of the Cas A NS observations were performed by the Chandra ACIS detectors, which suffer complicated systematic effects. Here, we test the cooling of the Cas A NS with Chandra HRC data over 25 years. The Chandra HRC detector has independent systematics, serving as a cross-check. Assuming a fixed hydrogen column density ($N_{\rm H}$), we infer the cooling rate of the Cas A NS to be 0.57$^{+0.26}_{-0.27}$% per decade. Allowing the $N_{\rm H}$ to vary with time (as estimated using ACIS data), the cooling rate is 1.11$^{+0.25}_{-0.28}$% per decade. These cooling rates are smaller than measured using ACIS data, implying systematic uncertainties have not been eradicated from either or both datasets. However, we have verified the decline in the absorbed flux from the Cas A NS using an independent instrument, at $>3\sigma$ level (4.7%$\pm$1.5% over 10 years). Additionally, the weaker cooling rate of Cas A NS inferred from HRC datasets eliminates the tension with the theoretically predicted cooling, and can be explained by the reduced efficiency of the neutrino emission accompanying the Cooper pair breaking and formation process in neutron triplet-state superfluid.

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A Classification Scheme for X-ray Bright Type Ia Supernova Remnants Based on Their Circumstellar Interaction

The parameter space for mass loss in Type Ia supernova progenitors is large, with different progenitor scenarios favoring different mass loss regimes. Here we focus on the impact that uniform and isotropic outflows have on the circumstellar environment of Type Ia supernova progenitors. We vary mass loss rate, wind velocity, and outflow duration, and evolve supernova remnant (SNR) models in this grid of circumstellar structures in order to compare the bulk properties of these models (ages, radii, and \feka\ centroids and luminosities) to observations. We find that roughly 55\% (7/13) of young X-ray bright Type Ia SNRs in the Milky Way and the Large Magellanic Cloud had progenitors that did not substantially modify their surroundings on $\sim$pc scales. This group includes SN Ia with a range of luminosities, and at least one likely product of a double detonation explosion in a sub-Chandrasekhar white dwarf. The other half of our sample can be divided in two distinct classes. A small subset of SNRs ($\sim$15\%, 2/13) have large radii and low \feka\ centroids and are likely expanding into large cavities that might have been excavated by fast ($\sim$1000 km/s), sustained progenitor outflows. The majority of the SNRs that are expanding into a modified medium ($\sim30\%$, 4/13) show evidence for dense material, likely associated with slow ($\sim$10 km/s) progenitor outflows, possibly a byproduct of accretion processes in near-Chandrasekhar white dwarfs spawned by younger stellar populations.

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Deep Optical Images of the Ejecta Nebula Around the Wolf-Rayet Star WR 8 (HD 62910)

We report the results of deep H$\alpha$ and [O III] images of the bright WN6/WC4 Wolf-Rayet star WR~8 (HD~62910). These data show considerably more surrounding nebulosity than seen in prior imaging. The brighter portions of the nebula span $\simeq6'$ in diameter and exhibit considerable fine-scale structure including numerous emission clumps and bright head-tail like features presumably due to the effects of the WR star's stellar winds. Due to the overlap of a relatively bright band of unrelated foreground diffuse interstellar H$\alpha$ emission, WR~8's nebula is best viewed via its [O III] emission. A faint $9' \times 13'$ diffuse outer nebulosity is detected surrounding the nebula's main ring of emission. Comparison of the nebula's optical structure with that seen in WISE 22 $\mu$m data shows a similarly clumpy structure but in a better defined emission shell of thermal continuum from dust. The infrared shell is coincident with the nebula's southern [O III] emissions but is mainly seen in the fainter outer portions of the northern [O III] emission clumps. It is this greater radial distance of dust emission in the nebula's northern areas that leads to a striking off-center position of the WR star in the IR shell.

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Cassiopeia A's Reverse Shock and its Effects on the Expanding SN Ejecta

Using optical and near-infrared images of the Cassiopeia A (Cas A) supernova remnant covering the time period 1951 to 2022, together with optical spectra of selected filaments, we present an investigation of Cas A's reverse shock velocity and the effects it has on the remnant's metal-rich ejecta. We find the sequence of optical ejecta brightening and the appearance of new optical ejecta indicating the advancement of the remnant's reverse shock in the remnant's main shell has velocities typically between 1000 and 2000 km/s, which is ~1000 km/s less than recent measurements made in X-rays. We further find the reverse shock appears to move much more slowly and is nearly even stationary in the sky frame along the remnant's western limb. However, we do not find the reverse shock to move inward at velocities as large as ~2000 km/s as has been reported. Optical ejecta in Cas A's main emission shell have proper motions indicating outward tangential motions ~3500 - 6000 km/s, with the smaller values preferentially along the remnant's southern regions which we speculate may be partially the cause of the remnant's faint and more slowly evolving southern sections. Following interaction with the reverse shock, ejecta knots exhibit extended mass ablated trails 0.2" - 0.5" in length leading to extended emission indicating reverse shock induced decelerated velocities as large as 1000 km/s. Such ablated material is most prominently seen in higher ionization line emissions, whereas denser parts of ejecta knots show surprisingly little deceleration.

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Late-Time Optical and X-ray Emission Evolution of the Oxygen-Rich SN 1996cr

When the ejecta of supernovae interact with the progenitor star's circumstellar environment, a strong shock is driven back into the ejecta, causing the material to become bright optically and in X-rays. Most notably, as the shock traverses the H-rich envelope, it begins to interact with metal rich material. Thus, continued monitoring of bright and nearby supernovae provides valuable clues about both the progenitor structure and its pre-supernova evolution. Here we present late-time, multi-epoch optical and Chandra} X-ray spectra of the core-collapse supernova SN 1996cr. Magellan IMACS optical spectra taken in July 2017 and August 2021 show a very different spectrum from that seen in 2006 with broad, double-peaked optical emission lines of oxygen, argon, and sulfur with expansion velocities of $\pm 4500$ km s$^{-1}$. Red-shifted emission components are considerably fainter compared to the blue-shifted components, presumably due to internal extinction from dust in the supernova ejecta. Broad $\pm 2400$ km s$^{-1}$ H$\alpha$ is also seen which we infer is shocked progenitor pre-SN mass-loss, H-rich material. Chandra data indicate a slow but steady decline in overall X-ray luminosity, suggesting that the forward shock has broken through any circumstellar shell or torus which is inferred from prior deep Chandra ACIS-S/HETG observations. The X-ray properties are consistent with what is expected from a shock breaking out into a lower density environment. Though originally identified as a SN IIn, based upon late time optical emission line spectra, we argue that the SN 1996cr progenitor was partially or highly stripped, suggesting a SN IIb/Ib.

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A JWST Survey of the Supernova Remnant Cassiopeia A

We present initial results from a JWST survey of the youngest Galactic core-collapse supernova remnant Cassiopeia A (Cas A), made up of NIRCam and MIRI imaging mosaics that map emission from the main shell, interior, and surrounding circumstellar/interstellar material (CSM/ISM). We also present four exploratory positions of MIRI/MRS IFU spectroscopy that sample ejecta, CSM, and associated dust from representative shocked and unshocked regions. Surprising discoveries include: 1) a web-like network of unshocked ejecta filaments resolved to 0.01 pc scales exhibiting an overall morphology consistent with turbulent mixing of cool, low-entropy matter from the progenitor's oxygen layer with hot, high-entropy matter heated by neutrino interactions and radioactivity, 2) a thick sheet of dust-dominated emission from shocked CSM seen in projection toward the remnant's interior pockmarked with small (approximately one arcsecond) round holes formed by knots of high-velocity ejecta that have pierced through the CSM and driven expanding tangential shocks, 3) dozens of light echoes with angular sizes between 0.1 arcsecond to 1 arcminute reflecting previously unseen fine-scale structure in the ISM. NIRCam observations place new upper limits on infrared emission from the neutron star in Cas A's center and tightly constrain scenarios involving a possible fallback disk. These JWST survey data and initial findings help address unresolved questions about massive star explosions that have broad implications for the formation and evolution of stellar populations, the metal and dust enrichment of galaxies, and the origin of compact remnant objects.

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Potential of the Cherenkov Telescope Array for studying the young nearby supernova remnants

Modern imaging atmospheric Cherenkov telescopes have extensively observed young nearby supernova remnants (SNRs), with ages of about 1000 years or less, in the very-high-energy (VHE) gamma-ray band. These efforts resulted in the detection of VHE emission from three young SNRs - Cassiopeia A, Tycho, and SN 1006 - and provided significant evidence for emission from the more distant Kepler's SNR. However, many questions on the production of VHE gamma rays in these remnants remain unanswered. Using detailed physical models for Tycho's SNR based on the CR-hydro-NEI code and physically motivated models for the other young nearby remnants, we simulated observations with the Cherenkov Telescope Array (CTA) of these gamma-ray sources. We highlight properties of these remnants accessible for investigation with future CTA observations and discuss which questions are expected to be answered.

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Do Type Ia Supernovae Explode Inside Planetary Nebulae?

The nature of Type Ia Supernova (SN Ia) explosions remains an open issue, with several contending progenitor scenarios actively being considered. One such scenario involves a SN Ia explosion inside a planetary nebula (PN) in the aftermath of a stellar merger triggered by a common envelope (CE) episode. We examine this scenario using hydrodynamic and non-equilibrium ionization simulations of the interaction between the SN ejecta and the PN cocoon into the supernova remnant (SNR) phase, focusing on the impact of the delay between the CE episode and the SN explosion. We compare the bulk dynamics and X-ray spectra of our simulated SNRs to the observed properties of known Type Ia SNRs in the Milky Way and the Magellanic Clouds. We conclude that models where the SN explosion happens in the immediate aftermath of the CE episode (with a delay $\lesssim$1,000 yr) are hard to reconcile with the observations, because the interaction with the dense PN cocoon results in ionization timescales much higher than those found in any known Type Ia SNR. Models with a longer delay between the CE episode and the SN explosion ($\sim$10,000 yr) are closer to the observations, and may be able to explain the bulk properties of some Type Ia SNRs.

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Line Emission Mapper (LEM): Probing the physics of cosmic ecosystems

The Line Emission Mapper (LEM) is an X-ray Probe for the 2030s that will answer the outstanding questions of the Universe's structure formation. It will also provide transformative new observing capabilities for every area of astrophysics, and to heliophysics and planetary physics as well. LEM's main goal is a comprehensive look at the physics of galaxy formation, including stellar and black-hole feedback and flows of baryonic matter into and out of galaxies. These processes are best studied in X-rays, and emission-line mapping is the pressing need in this area. LEM will use a large microcalorimeter array/IFU, covering a 30x30' field with 10" angular resolution, to map the soft X-ray line emission from objects that constitute galactic ecosystems. These include supernova remnants, star-forming regions, superbubbles, galactic outflows (such as the Fermi/eROSITA bubbles in the Milky Way and their analogs in other galaxies), the Circumgalactic Medium in the Milky Way and other galaxies, and the Intergalactic Medium at the outskirts and beyond the confines of galaxies and clusters. LEM's 1-2 eV spectral resolution in the 0.2-2 keV band will make it possible to disentangle the faintest emission lines in those objects from the bright Milky Way foreground, providing groundbreaking measurements of the physics of these plasmas, from temperatures, densities, chemical composition to gas dynamics. While LEM's main focus is on galaxy formation, it will provide transformative capability for all classes of astrophysical objects, from the Earth's magnetosphere, planets and comets to the interstellar medium and X-ray binaries in nearby galaxies, AGN, and cooling gas in galaxy clusters. In addition to pointed observations, LEM will perform a shallow all-sky survey that will dramatically expand the discovery space.

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A Multi-Wavelength View on the Rapidly-Evolving Supernova 2018ivc: An Analog of SN IIb 1993J but Powered Primarily by Circumstellar Interaction

SN 2018ivc is an unusual type II supernova (SN II). It is a variant of SNe IIL, which might represent a transitional case between SNe IIP with a massive H-rich envelope, and IIb with only a small amount of the H-rich envelope. However, SN 2018ivc shows an optical light curve evolution more complicated than canonical SNe IIL. In this paper, we present the results of prompt follow-up observations of SN 2018ivc with the Atacama Large Millimeter/submillimeter Array (ALMA). Its synchrotron emission is similar to that of SN IIb 1993J, suggesting that it is intrinsically an SN IIb-like explosion of a He star with a modest (~0.5 - 1 Msun) extended H-rich envelope. Its radio, optical, and X-ray light curves are explained primarily by the interaction between the SN ejecta and the circumstellar material (CSM); we thus suggest that it is a rare example (and the first involving the `canonical' SN IIb ejecta) for which the multi-wavelength emission is powered mainly by the SN-CSM interaction. The inner CSM density, reflecting the progenitor activity in the final decade, is comparable to that of SN IIb 2013cu that showed a flash spectral feature. The outer CSM density, and therefore the mass-loss rate in the final ~200 years, is larger than that of SN 1993J by a factor of ~5. We suggest that SN 2018ivc represents a missing link between SNe IIP and IIb/Ib/Ic in the binary evolution scenario.

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The 3-D X-ray Ejecta Structure of Tycho's Supernova Remnant

We present our velocity measurements of 59 clumpy, metal-rich ejecta knots in the supernova remnant (SNR) of SN 1572 (Tycho). We use our 450 ks Chandra High Energy Transmission Grating Spectrometer observation to measure the Doppler shift of the He-like Si K$α$ line-center wavelength emitted from these knots to find their line-of-sight (radial) velocities ($v_r$). We find $v_r$ up to $\sim$ 5500 km s$^{-1}$, with roughly consistent speeds between blueshifted and redshifted ejecta knots. We also measure the proper motions (PMs) for our sample based on archival Chandra Advanced CCD Imaging Spectrometer data taken from 2003, 2009, and 2015. We estimate PMs up to 0$"$.35 yr$^{-1}$, which corresponds to a transverse velocity of about 5800 km s$^{-1}$ for the distance of 3.5 kpc to Tycho. Our $v_r$ and transverse velocity measurements imply space velocities of $\sim$ 1900 - 6000 km s$^{-1}$ for the ejecta knots in Tycho. We estimate a new expansion center of R.A.(J2000) = 00$^h$25$^m$18$^s$.725 $\pm$ 1$^s$.157 and decl.(J2000) = +64$^{\circ}$08$'$02$"$.5 $\pm$ 11$"$.2 from our PM measurements, consistent to within $\sim$ 13$"$ of the geometric center. The distribution of space velocities throughout the remnant suggests that the southeast quadrant generally expands faster than the rest of the SNR. We find that blueshifted knots are projected more in the northern shell, while redshifted knots are more in the southern shell. The previously estimated reverse shock position is consistent with most of our estimated ejecta distribution, however some ejecta show deviations from the 1-D picture of the reverse shock.

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