arXiv subjects

Orsola De Marco

Publications and source records attributed to Orsola De Marco.

At least 19 recordsLinked to original sources

arXiv2026

Three-dimensional simulations of accretion disks in pre-CE systems -- II. Accretion efficiency and angular momentum transport

Rapid mass transfer preceding a common envelope event is a critical yet poorly understood stage of binary stellar evolution, setting the initial conditions for the common envelope inspiral and for the formation of compact binary stars. We present three-dimensional smoothed particle hydrodynamics simulations of accretion disk formation during this phase using the Phantom code. We model the final $21~yr$ of Roche lobe overflow from a $7~M_{\odot}$ red giant onto a $1.4~M_{\odot}$ neutron star companion, with the mass transfer rate prescribed by a 1D MESA model. Over this interval, the mass transfer rate increases from $1.3\times10^{-4}$ to $1.0\times10^{-1}~M_\odot~yr^{-1}$. An accretion disk forms around the neutron star, reading a mass of $5.0\times10^{-3} ~ M_{\odot}$, a radius of $40~R_{\odot}$ and an aspect ratio H/R$\sim 0.1$ by the end of the simulation. A direct comparison with the grid-based simulation of Juarez-Garcia et al. (2025) shows that the two codes produce disk masses that agree to within 6%. The accretion rate onto the neutron star reaches $5.2\times10^{-3}~M_\odot~yr^{-1}$, corresponding to 14% of the mass injection rate and greatly exceeding the Eddington limit. We demonstrate that this accretion rate is consistent with being driven by turbulent angular momentum transport, with an effective viscosity parameter $α_{\rm eff}~=~0.03-0.06$. Ejecta leaving the binary system carries specific angular momentum approximately 90% that of the $L_{2}$ Lagrange point, equivalent to $\sim10$ times the binary's specific orbital angular momentum. This indicates that $L_{2}$ mass loss efficiently reduces the binary orbital separation.

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arXiv2026

Second Generation Planet Formation in Post-AGB Discs - II: Dust Coagulation and Core Accretion

Observations have established that the circumbinary discs formed around post-asymptotic giant branch (post-AGB) binaries have striking similarities to protoplanetary discs around young stars. In this study, we examine the feasibility of the different stages of growth in the core accretion framework of planet formation in post-AGB discs, in order to assess whether planet formation is possible in these systems. We find that dust coagulation up to mm sizes is possible within the lifetimes of post-AGB discs, consistent with observations. We then investigate the subsequent growth of planetesimals through pebble and planetesimal accretion. If the streaming instability operates in these discs, the resulting planetesimals formed in higher-mass post-AGB discs ($M_{\rm disc}\sim0.1M_\odot$) can enter rapid pebble accretion and reach their isolation mass within the estimated disc lifetime. By contrast, in lower-mass post-AGB discs ($M_{\rm disc}\sim0.01M_\odot$), planetesimal accretion is too inefficient to produce substantial further growth. We also show that the larger aspect ratios of post-AGB discs compared to protoplanetary discs around young stars imply high pebble isolation masses and therefore potential formation of rocky planets with masses extending up to Jupiter masses. After assessing gas accretion and concurrent migration in these discs, we conclude that second generation planet formation in post-AGB discs is theoretically possible within their estimated lifetimes ($10^4$--$10^5$yr) provided that local dust-to-gas ratio enhancements and sufficiently high disc masses are achieved.

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arXiv2026

Dust Formation in Common Envelope Binary Interactions -- III. Lightcurves

Luminous red novae are transient events thought to arise from common envelope binary interactions. In this paper, we perform post-processing light-curve calculations for two, 3D hydrodynamic simulations of common envelope events. These simulations model interactions between 1.7 $M_\odot$ and 3.7 $M_\odot$ asymptotic giant branch stars and a 0.6 $M_\odot$ compact companion, including dust nucleation. In both our simulations, which are carried out for 44 years under adiabatic conditions, we observe a bright, hot peak lasting $3-5$ years, primarily due to the expansion of the photosphere before and during inspiral. Additional peaks can be seen appearing at different times and for different viewing angles, due to the asymmetry of the interaction. Dust forms about $1-3$ years after the beginning of the simulated interaction and shortly afterwards we witness a sharp decline in the bolometric luminosity, followed by a partial recovery and a plateau with an effective temperature of $\sim$400 K. The dust photosphere reaches a size of $\sim$250 au by the end of the simulations, but we predict that between 100 and 200 years, the dust will become optically thin at visible wavelengths, revealing an inner, warmer photosphere. The lightcurves obtained have two, well-quantified, but large uncertainties: insufficient surface resolution primarily affecting the first 1-2 years of the simulated lightcurves and the adiabatic assumption that affects primarily the later years. We finally contextualise our simulations within a group of observed luminous red nova transients, drawing particular attention to the outburst of OGLE-2002-BLG-360 and AT 2025abao, which are the closest match to our simulation.

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arXiv2026

Weakened inspirals -- I: High mass ratio common envelope interactions in RGB stars

The common envelope (CE) interaction between an expanding giant star and a compact companion typically leads to a rapid orbital decay, ending in either a merger or the formation of a close binary. However, the existence of post-red giant and post-asymptotic giant branch binaries with separations of 100 to 800 Rsun challenges this standard picture, as these systems appear to have experienced strong interactions without undergoing a classic CE inspiral. In this work, we investigate the effect of high mass ratio, q = M2/M1, on the CE inspiral using three-dimensional hydrodynamical simulations performed with the smoothed particle hydrodynamics code PHANTOM. The primary is a 0.88 Msun, 90 Rsun red giant branch star, while the companion masses span q = 0.68 to 1.5. Higher mass ratios lead to wider post-CE separations, with a maximum of approximately 40 Rsun. The pre-CE mass transfer phase is longer for larger companion masses, and for q greater than or equal to 1 the inspiral becomes significantly more stable, broadly consistent with analytical expectations. This phase is not fully converged with respect to numerical resolution, and higher resolution simulations are expected to further increase its duration and stability. Although higher q systems show enhanced mass loss through the L2 and L3 Lagrange points, we find that circumbinary discs are more likely to form from fallback of bound envelope material. Fallback times are short, of order a few hundred years, and fallback radii lie well outside the binary, between 0.5 and 5 au, where discs are expected to spread efficiently through viscous torques. While high mass ratio systems produce wider post-interaction separations, these remain smaller than those observed. In contrast, fallback-formed discs have properties consistent with observed circumbinary discs.

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arXiv2026

Planetary Nebulae

Planetary nebulae are formed by the matter ejected by low-to-intermediate mass stars (~0.8-8 times the mass of the Sun) towards the end of their lives. As hydrogen and then helium fuel sources run out, stars expand. During these giant phases stars also lose sizable amounts of mass. During the second giant phase, after the exhaustion of core helium, the mass loss is so great that stars lose a large fraction of their mass (50 - 90%), leaving behind a small, hot core, known as a white dwarf, surrounded by a nebula. Planetary nebulae are the result of many processes that shape and alter their ionization structure and chemical composition. The resulting nebula, illuminated by the ultraviolet-rich spectrum of the remnant very hot stellar core, is a spectacle of beauty and science. In this chapter, we show that these objects are invaluable laboratories for astrophysics, astrochemistry, and astromineralogy studies, with impact in many areas of Astronomy.

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arXiv2025

Second Generation Planet Formation in Post-AGB Discs: Testing the Role of Gravitational Instability

Post-asymptotic giant branch (post-AGB) binary stars are evolved systems that host circumbinary discs formed through mass loss during late stage binary interactions. Their structural, morphological, kinematic, and chemical similarities to planet-forming discs suggest that these systems may act as sites of "second generation" planet formation. In this study, we assess whether the disc instability mechanism -- a proposed pathway for rapid, giant planet formation in some protoplanetary discs - can operate in post-AGB discs; motivated by their short lifetimes. Using the Toomre criterion under well motivated assumptions for disc structure and size, mass, and thermal properties, we assess the conditions for gravitational instability. We first benchmark our analytical framework using well studied protoplanetary disc systems (including HL Tauri, Elias 2-27, GQ Lupi) before applying the same analysis to observed post-AGB discs. We find that post-AGB discs are generally gravitationally stable at present, due primarily to their low masses. Using viscous disk theory, we find that the discs were stable against collapse even in the past, when their masses were potentially higher. In contrast, several protoplanetary discs analysed in the same way show that they likely experienced gravitationally unstable phases early on. We also find that higher viscosity parameters are better aligned with expected post-AGB disc lifetimes. Finally, we revisit the planet formation scenario proposed for the post-common envelope system NN Ser, first carried out by Schleicher and Dreizler and we show that gravitational instability could be feasible under specific, high disc mass assumptions. Overall, our results provide the first systematic theoretical assessment of gravitational instability in post-AGB discs, demonstrating that this mechanism is unlikely to dominate second generation planet formation in these systems.

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arXiv2025

Central-star extinctions towards planetary nebulae

Planetary nebulae trace the hottest and most luminous phase of evolution of solar-type stars. We use these hot, bright stars to investigate extinctions towards a complete sample of 262 confirmed PNe with large angular diameters, which have the most reliable photometry and hottest central stars. For 162 of these PNe, we identify central stars, produce spectral energy distributions from survey data using PySSED, then fit reddened model spectra to the observed photometry to obtain extinctions accurate down to $E(B-V)$ of $\pm 0.02$ mag. The fitting is performed by Nelder-Mead $χ^2$ minimisation, with uncertainties evaluated through MCMC. The catalogue of stellar temperatures is updated for our sample for the calculation of luminosities. The extinctions agree well with interstellar extinction. We find evidence of circumnebular extinction for one PN, and evaluate its effect on the planetary nebulae luminosity function. Four new close binaries are identified from the spectral energy distributions. The binary fraction in the full sample is between 23% and 36%. We use our compiled data to evaluate the quality of the central star identifications in the literature. Three objects in our sample have previously been classified as post-RGB systems but we find that their parameters may also be consistent with post-AGB evolution.

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arXiv2025

Three-dimensional simulations of accretion disks in pre-CE systems

Before a binary system enters into a common envelope (CE) phase, accretion from the primary star onto the companion star through Roche Lobe overflow (RLOF) will lead to the formation of an accretion disk, which may generate jets. Accretion before and during the CE may alter the outcome of the interaction. Previous studies have considered different aspects of this physical mechanism. Here we study the properties of an accretion disk formed via 3D hydrodynamic simulations of the RLOF mass transfer between a 7 M$_\odot$, red supergiant star and a 1.4 M$_\odot$, neutron star companion. We simulate only the volume around the companion for improved resolution. We use a 1D implicit MESA simulation of the evolution of the system during 30,000 years between the on-set of the RLOF and the CE to guide the binary parameters and the mass-transfer rate, while we simulate only 21 years of the last part of the RLOF in 3D using an ideal gas isothermal equation of state. We expect that a pre-CE disk under these parameters will have a mass of $\sim 5\times 10^{-3}$ M$_\odot$ and a radius of $\sim$40 R$_\odot$ with a scale height of $\sim$5 R$_\odot$. The temperature profile of the disk is shallower than that predicted by the formalism of Shakura and Sunyaev, but more reasonable cooling physics would need to be included. We stress test these results with respect to a number of physical and numerical parameters, as well as simulation choices, and we expect them to be reasonable within a factor of a few for the mass and 15% for the radius. We also contextualize our results within those presented in the literature, in particular with respect to the dimensionality of simulations and the adiabatic index. We discuss the measured accretion rate in the context of the Shakura and Sunyaev formalism and debate the viscous mechanisms at play, finishing with a list of prospects for future work.

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arXiv2025

Modelling the impact of circumbinary disk accretion on post-AGB binary evolution and surface chemistry

Post-asymptotic giant branch (post-AGB) binaries are surrounded by dusty circumbinary disks, and exhibit unexpected orbital properties resulting from poorly understood binary interaction processes. Re-accreted gas from the circumbinary disk alters the photospheric chemistry of the post-AGB star, producing a characteristic underabundance of refractory elements that correlates with condensation temperature $\unicode{x2013}$a phenomenon known as chemical depletion. This work investigates how re-accretion from a disk drives chemical depletion, and the impact accreted matter has on post-AGB evolution. We used the MESA code to evolve 0.55 and 0.60 M$_{\odot}$ post-AGB stars with the accretion of refractory element-depleted gas from a circumbinary disk. Our study adopts observationally-constrained initial accretion rates and disk masses to reproduce the chemical depletion patterns of six well-studied post-AGB binary stars: EP Lyr, HP Lyr, IRAS 17038-4815, IRAS 09144-4933, HD 131356, and SX Cen. We find high accretion rates ($>\,$10$^{-7}$ M$_{\odot}$yr$^{-1}$) and large disk masses ($\geq\,$10$^{-2}$ M$_{\odot}$) necessary to reproduce observed depletion, particularly in higher-mass, hotter post-AGB stars (T$_{\textrm{eff}}\geq$ 6000 K). A slower evolution (lower core mass) is required to reproduce cooler (T$_{\textrm{eff}}\leq$ 5000 K) depleted post-AGB stars. Rapid accretion significantly impacts post-AGB evolution, stalling stars at cooler effective temperatures and extending post-AGB lifetimes by factors of around 3 to 10. Despite this, extended post-AGB timescales remain within or below the planetary nebula (PN) visibility timescale, suggesting accretion cannot account for the observed lack of ionised PNe in post-AGB binaries. Our findings constrain accretion-flow parameters and advance our understanding of disk-binary interactions in post-AGB systems.

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arXiv2025

The Structure of the Molecular Envelope of the Ring Nebula (NGC 6720)

We present the first interferometric imaging of molecular line emission from the Ring Nebula, NGC~6720, in the form of Submillimeter Array (SMA) observations of CO $J=2\rightarrow 1$ emission. The SMA $^{12}$CO(2--1) mapping data, with $\sim$3$''$ spatial resolution and 2 km s$^{-1}$ velocity resolution, provide an unprecedentedly detailed, 3D view of the Ring's clumpy molecular envelope. The morphology of the velocity-integrated SMA $^{12}$CO(2--1) image closely resembles those of near-IR H$_2$ and PAH emission in JWST/NIRCam imaging of NGC~6720, with the molecular gas forming a geometrically thin layer surrounding the ionized gas imaged by HST and JWST. A simple, geometrical model of the $^{12}$CO(2--1) data shows that the intrinsic structure of NGC~6720's molecular envelope closely resembles a truncated, triaxial ellipsoid that is viewed close to pole-on, and that the dynamical age of the molecular envelope is $\sim$6000 yr. The SMA $^{12}$CO(2--1) data furthermore reveal that filamentary features seen projected in the Ring's interior in JWST imaging are in fact fast-moving polar knots or bullets with radial velocities of $\pm$45--50 km s$^{-1}$ relative to systemic, and that the hot progenitor star remnant is positioned at the precise geometric center of the clumpy, ellipsoidal molecular shell. We assert that the Ring's molecular envelope was formed via a relatively sudden, AGB-terminating mass ejection event $\sim$6000 yr ago, and that this ellipsoidal envelope was then punctured by fast, collimated polar outflows resulting from interactions between the progenitor and one or more companion stars. Such an evolutionary scenario may describe most molecule-rich, ``Ring-like'' planetary nebulae.

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arXiv2024

Hydrodynamic simulations of WD-WD mergers and the origin of RCB stars

We study the properties of double white dwarf (DWD) mergers by performing hydrodynamic simulations using the new and improved adaptive mesh refinement code Octo-Tiger. We follow the orbital evolution of DWD systems of mass ratio q=0.7 for tens of orbits until and after the merger to investigate them as a possible origin for R Coronae Borealis (RCB) type stars. We reproduce previous results, finding that during the merger, the Helium WD donor star is tidally disrupted within 20-80 minutes since the beginning of the simulation onto the accretor Carbon-Oxygen WD, creating a high temperature shell around the accretor. We investigate the possible Helium burning in this shell and the merged object's general structure. Specifically, we are interested in the amount of Oxygen-16 dredged-up to the hot shell and the amount of Oxygen-18 produced. This is critical as the discovery of very low Oxygen-16 to Oxygen-18 ratios in RCB stars pointed out the merger scenario as a favorable explanation for their origin. A small amount of hydrogen in the donor may help keep the Oxygen-16 to Oxygen-18 ratios within observational bounds, even if moderate dredge-up from the accretor occurs. In addition, we perform a resolution study to reconcile the difference found in the amount of Oxygen-16 dredge-up between smoothed-particle hydrodynamics and grid-based simulations.

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arXiv2024

Novel Constraints on Companions to the Helix Nebula Central Star

The Helix is a visually striking and the nearest planetary nebula, yet any companions responsible for its asymmetric morphology have yet to be identified. In 2020, low-amplitude photometric variations with a periodicity of 2.8 d were reported based on Cycle 1 TESS observations. In this work, with the inclusion of two additional sectors, these periodic light curves are compared with lcurve simulations of irradiated companions in such an orbit. Based on the light curve modelling, there are two representative solutions: i) a Jupiter-sized body with 0.102 Rsol and an arbitrarily small orbital inclination i=1 deg, and ii) a 0.021 Rsol exoplanet with i approx. 25 deg, essentially aligned with the Helix Nebular inclination. Irradiated substellar companion models with equilibrium temperature 4970 K are constructed and compared with existing optical spectra and infrared photometry, where Jupiter-sized bodies can be ruled out, but companions modestly larger than Neptune are still allowed. Additionally, any spatially-unresolved companions are constrained based on the multi-wavelength, photometric spectral energy distribution of the central star. No ultracool dwarf companion earlier than around L5 is permitted within roughly 1200 au, leaving only faint white dwarfs and cold brown dwarfs as possible surviving architects of the nebular asymmetries. While a planetary survivor is a tantalizing possibility, it cannot be ruled out that the light curve modulation is stellar in nature, where any substellar companion requires confirmation and may be possible with JWST observations.

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arXiv2024

High-Speed Outflows and Dusty Disks during the AGB to PN Transition: The PANORAMA survey

As mass-losing asymptotic giant branch (AGB) stars evolve to planetary nebulae (PNe), the mass outflow geometries transform from nearly spherical to extreme aspherical. The physical mechanisms governing this transformation are widely believed to be linked to binarity and the associated production of disks and fast jets during transitional (post-AGB) evolutionary stages. We are carrying out a systematic ALMA survey ($P$re-planet$A$ry $N$ebulae high-angular-res$O$lution su$R$vey with $A$L$MA$ or PANORAMA) of a representative sample of bipolar and multipolar post-AGB objects. We have obtained high angular-resolution (0".1-0".4) observations of the CO(3--2) and/or 6--5 emission in order to probe the spatio-kinematic structure of the collimated outflows and the central disk/torii. The results are remarkable, generally showing the presence of bipolar or multipolar high-velocity outflows, dense toroidal waists, and in one case, a geometrically-thin circular ring around the central bipolar nebula. A high degree of point-symmetry characterizes the morphology of the mass ejecta. In this contribution, we present these and other highlights from our survey. We aim to use 2D/3D radiative transfer modeling in order to derive accurate outflow momenta, masses and mass-loss rates for our sample, and build hydrodynamical models that can explain the observed spatio-kinematic structures. These results will then be used to distinguish between different classes of PN-shaping binary interaction models.

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arXiv2024

Dust formation in common envelope binary interactions -- II: 3D simulations with self-consistent dust formation

We performed numerical simulations of the common envelope (CE) interaction between thermally-pulsing asymptotic giant branch (AGB) stars of 1.7~\Msun and 3.7~\Msun, respectively, and a 0.6~\Msun compact companion. We use tabulated equations of state to take into account recombination energy. For the first time, formation and growth of dust is calculated explicitly, using a carbon dust nucleation network with a C/O abundance ratio of 2.5 (by number). The first dust grains appear within $\sim$1--3~yrs after the onset of the CE, forming an optically thick shell at $\sim$10--20~au, growing in thickness and radius to values of $\sim$400--500~au over $\sim$40~yrs, with temperatures around 400~K. Most dust is formed in unbound material, having little effect on mass ejection or orbital evolution. By the end of the simulations, the total dust yield is $\sim8.4\times10^{-3}$~\Msun and $\sim2.2\times10^{-2}$~\Msun for the CE with a 1.7~\Msun and a 3.7~\Msun AGB star, respectively, corresponding to a nucleation efficiency close to 100\%, if no dust destruction mechanism is considered. Despite comparable dust yields to single AGB stars, \textit{in CE ejections the dust forms a thousand times faster, over tens of years as opposed to tens of thousands of years}. This rapid dust formation may account for the shift in the infrared of the spectral energy distribution of some optical transients known as luminous red novae. Simulated dusty CEs support the idea that extreme carbon stars and "water fountains" may be objects observed after a CE event.

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arXiv2024

Dust formation during the interaction of binary stars by common envelope

We performed numerical simulations of the common envelope (CE) interaction between two intermediate-mass asymptotic giant branch (AGB) stars and their low-mass companions. For the first time, formation and growth of dust in the envelope is calculated explicitly. We find that the first dust grains appear as early as $\sim$1-3 yrs after the onset of the CE, and are smaller than grains formed later. As the simulations progress, a high-opacity dusty shell forms, resulting in the CE photosphere being up to an order of magnitude larger than it would be without the inclusion of dust. At the end of the simulations, the total dust yield is $0.0082~M_{\odot}$ ($0.022~M_{\odot}$) for a CE with a $1.7~M_{\odot}$ ($3.7~M_{\odot}$) AGB star. Dust formation does not substantially lead to more mass unbinding or substantially alter the orbital evolution.

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arXiv2024

Dust Formation in Common Envelope Binary Interaction -- I: 3D Simulations Using the Bowen Approximation

We carried out 3D smoothed particle hydrodynamics simulations of the common envelope binary interaction using the approximation of Bowen to calculate the dust opacity in order to investigate the resulting dust-driven accelerations. We have simulated two types of binary star: a 1.7 and a 3.7 $M_{\odot}$ thermally-pulsating, asymptotic giant branch stars with a 0.6 $M_{\odot}$ companion. We carried out simulations using both an ideal gas and a tabulated equations of state, with the latter considering the recombination energy of the envelope. We found that the dust-driven wind leads to a relatively small increase in the unbound gas, with the effect being smaller for the tabulated equation of state simulations and for the more massive primary. Dust acceleration does contribute to envelope expansion with only a slightly elongated morphology, if we believe the results from the tabulated equation of state as more reliable. The Bowen opacities in the outer envelopes of the two models, at late times, are large enough that the photosphere of the post-inspiral object is about ten times larger compared to the same without accounting for the dust opacities. As such, the prediction of the appearance of the transient would change substantially if dust is included.

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arXiv2024

The Molecular Exoskeleton of the Ring-like Planetary Nebula NGC 3132

We present Submillimeter Array (SMA) mapping of $^{12}$CO $J=2\rightarrow 1$, $^{13}$CO $J=2\rightarrow 1$, and CN $N=2\rightarrow 1$ emission from the Ring-like planetary nebula (PN) NGC 3132, one of the subjects of JWST Early Release Observation (ERO) near-infrared imaging. The $\sim$5$''$ resolution SMA data demonstrate that the Southern Ring's main, bright, molecule-rich ring is indeed an expanding ring, as opposed to a limb-brightened shell, in terms of its intrinsic (physical) structure. This suggests that NGC 3132 is a bipolar nebula viewed more or less pole-on (inclination $\sim$15--30$^\circ$). The SMA data furthermore reveal that the nebula harbors a second expanding molecular ring that is aligned almost orthogonally to the main, bright molecular ring. We propose that this two-ring structure is the remnant of an ellipsoidal molecular envelope of ejecta that terminated the progenitor star's asymptotic giant branch evolution and was subsequently disrupted by a series of misaligned fast, collimated outflows or jets resulting from interactions between the progenitor and one or more companions.

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arXiv2023

The messy death of a multiple star system and the resulting planetary nebula as observed by JWST

Planetary nebulae (PNe), the ejected envelopes of red giant stars, provide us with a history of the last, mass-losing phases of 90 percent of stars initially more massive than the Sun. Here, we analyse James Webb Space Telescope (JWST) Early Release Observation (ERO) images of the PN NGC3132. A structured, extended H2 halo surrounding an ionised central bubble is imprinted with spiral structures, likely shaped by a low-mass companion orbiting the central star at 40-60 AU. The images also reveal a mid-IR excess at the central star interpreted as a dusty disk, indicative of an interaction with another, closer companion. Including the previously known, A-type visual companion, the progenitor of the NGC3132 PN must have been at least a stellar quartet. The JWST images allow us to generate a model of the illumination, ionisation and hydrodynamics of the molecular halo, demonstrating the power of JWST to investigate complex stellar outflows. Further, new measurements of the A-type visual companion allow us to derive the value for the mass of the progenitor of a central star to date with excellent precision: 2.86+/-0.06 Mo. These results serve as path finders for future JWST observations of PNe providing unique insight into fundamental astrophysical processes including colliding winds, and binary star interactions, with implications for supernovae and gravitational wave systems.

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