We explore the effects of the deflagration to detonation transition (DDT) density on the production of Ni-56 in thermonuclear supernova explosions (type Ia supernovae). Within the DDT paradigm, the transition density sets the amount of expansion during the deflagration phase of the explosion and therefore the amount of nuclear statistical equilibrium (NSE) material produced. We employ a theoretical framework for a well-controlled statistical study of two-dimensional simulations of thermonuclear supernovae with randomized initial conditions that can, with a particular choice of transition density, produce a similar average and range of Ni-56 masses to those inferred from observations. Within this framework, we utilize a more realistic "simmered" white dwarf progenitor model with a flame model and energetics scheme to calculate the amount of Ni-56 and NSE material synthesized for a suite of simulated explosions in which the transition density is varied in the range 1-3x10^7 g/cc. We find a quadratic dependence of the NSE yield on the log of the transition density, which is determined by the competition between plume rise and stellar expansion. By considering the effect of metallicity on the transition density, we find the NSE yield decreases by 0.055 +/- 0.004 solar masses for a 1 solar metallicity increase evaluated about solar metallicity. For the same change in metallicity, this result translates to a 0.067 +/- 0.004 solar mass decrease in the Ni-56 yield, slightly stronger than that due to the variation in electron fraction from the initial composition. Observations testing the dependence of the yield on metallicity remain somewhat ambiguous, but the dependence we find is comparable to that inferred from some studies.
We study a new mechanism for deflagration to detonation transition in thermonuclear supernovae (SNe Ia), based on the formation of shocks by amplification of sound waves in the steep density gradients of white dwarfs envelopes. Given a large enough jump in density a small pressure and velocity perturbation, produced by the turbulent deflagration, turns into a shock down of the gradient, where it will dissipate and heat up the media. With the right frequency and amplitude the heating can be enough to initiate a detonation, which can propagate backward and up the density gradient. We studied pla…
The delayed detonation model describes the observational properties of the majority of type Ia supernovae very well. Using numerical data from a three-dimensional deflagration model for type Ia supernovae, the intermittency of the turbulent velocity field and its implications on the probability of a deflagration-to-detonation (DDT) transition are investigated. From structure functions of the turbulent velocity fluctuations, we determine intermittency parameters based on the log-normal and the log-Poisson models. On the other hand, the analysis of the turbulent velocity fluctuations in the vici…
The nucleosynthetic characteristics of various explosion mechanisms of Type Ia supernovae (SNe Ia) is explored based on three two-dimensional explosion simulations representing extreme cases: a pure turbulent deflagration, a delayed detonation following an approximately spherical ignition of the initial deflagration, and a delayed detonation arising from a highly asymmetric deflagration ignition. Apart from this initial condition, the deflagration stage is treated in a parameter-free approach. The detonation is initiated when the turbulent burning enters the distributed burning regime. This oc…
First-Principles Turbulence-Driven Deflagration-to-Detonation Transition Mechanism for Near-Chandrasekhar Mass White Dwarf Progenitors
University of Massachusetts Dartmouth（麻省大学达特茅斯分校）；University of Connecticut, School of Mechanical, Aerospace, and Manufacturing Engineering（康涅狄格大学机械、航空航天与制造工程学院）；Naval Research Laboratory, Laboratories for Computational Physics & Fluid Dynamics（海军研究实验室计算物理与流体动力学实验室）；University of Maryland, Physical Sciences Complex, Stadium Drive（马里兰大学）；Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik（海德堡大学天文学中心理论天体物理学研究所）
A promising model for normal Type Ia supernova (SN Ia) explosions are delayed detonations of Chandrasekhar-mass white dwarfs, in which the burning starts out as a subsonic deflagration and turns at a later phase of the explosion into a supersonic detonation. The mechanism of the underlying deflagration-to-detonation transition (DDT) is unknown in detail, but necessary conditions have been determined recently. The region of detonation initiation cannot be spatially resolved in multi-dimensional full-star simulations of the explosion. We develop a subgrid-scale (SGS) model for DDTs in thermonucl…
We describe the detonation mechanism comprising the "Pulsationally Assisted" Gravitationally Confined Detonation (GCD) model of Type Ia supernovae SNe Ia. This model is analogous to the previous GCD model reported in Jordan et al.(2008); however, the chosen initial conditions produce a substantively different detonation mechanism, resulting from a larger energy release during the deflagration phase. The resulting final kinetic energy and nickel-56 yields conform better to observational values than is the case for the "classical" GCD models. In the present class of models, the ignition of a def…
In order for a double-detonation model to be viable for normal type Ia supernovae, the adverse impact of helium-burning ash on early-time observables has to be avoided, which requires that the helium envelope mass should be at most 0.02 solar mass. Most of the previous studies introduced detonation by artificial hot spots, and therefore the robustness of the spontaneous helium detonation remains uncertain. In the present work, we conduct a self-consistent hydrodynamic study on the spontaneous ignition of the helium envelope in the context of the double-degenerate channel, by applying an ideali…
We refine our previously introduced parameterized model for explosive carbon-oxygen fusion during thermonuclear supernovae (SN Ia) by adding corrections to post-processing of recorded Lagrangian fluid element histories to obtain more accurate isotopic yields. Deflagration and detonation products are verified for propagation in a uniform density medium. A new method is introduced for reconstructing the temperature-density history within the artificially thick model deflagration front. We obtain better than 5\% consistency between the electron capture computed by the burning model and yields fro…
The gravitationally confined detonation (GCD) model has been proposed as a possible explosion mechanism for Type Ia supernovae in the single-degenerate evolution channel. Driven by buoyancy, a deflagration flame rises in a narrow cone towards the surface. For the most part, the flow of the expanding ashes remains radial, but upon reaching the outer, low-pressure layers of the white dwarf, an additional lateral component develops. This makes the deflagration ashes converge again at the opposite side, where the compression heats fuel and a detonation may be launched. To test the GCD explosion mo…
It has long been recognized that the neutrinos detected from the next core-collapse supernova in the Galaxy have the potential to reveal important information about the dynamics of the explosion and the nucleosynthesis conditions as well as allowing us to probe the properties of the neutrino itself. The neutrinos emitted from thermonuclear - type Ia - supernovae also possess the same potential, although these supernovae are dimmer neutrino sources. For the first time, we calculate the time, energy, line of sight, and neutrino-flavor-dependent features of the neutrino signal expected from a thr…
Observational evidences point to a common explosion mechanism of Type Ia supernovae based on a delayed detonation of a white dwarf. Although several scenarios have been proposed and explored by means of one, two, and three-dimensional simulations, the key point still is the understanding of the conditions under which a stable detonation can form in a destabilized white dwarf. One of the possibilities that have been invoked is that an inefficient deflagration leads to the pulsation of a Chandrasekhar-mass white dwarf, followed by formation of an accretion shock around a carbon-oxygen rich core.…
Over the past decades, many explosion scenarios for Type Ia supernovae have been proposed and investigated including various combinations of deflagrations and detonations in white dwarfs of different masses up to the Chandrasekhar mass. One of these is the gravitationally confined detonation model. In this case a weak deflagration burns to the surface, wraps around the bound core, and collides at the antipode. A subsequent detonation is then initiated in the collision area. Since the parameter space for this scenario, that is, varying central densities and ignition geometries, has not been stu…
In a companion paper, Seitenzahl et al. (2013) presented a set of three-dimensional delayed detonation models for thermonuclear explosions of near-Chandrasekhar mass white dwarfs (WDs). Here, we present multi-dimensional radiative transfer simulations that provide synthetic light curves and spectra for those models. The model sequence explores both changes in the strength of the deflagration phase (controlled by the ignition configuration) and the WD central density. In agreement with previous studies, we find that the strength of the deflagration significantly affects the explosion and the ob…
Type Ia supernovae are associated with thermonuclear explosions of white dwarf stars. Combustion processes convert material in nuclear reactions and release the energy required to explode the stars. At the same time, they produce the radioactive species that power radiation and give rise to the formation of the observables. Therefore, the physical mechanism of the combustion processes, as reviewed here, is the key to understand these astrophysical events. Theory establishes two distinct modes of propagation for combustion fronts: subsonic deflagrations and supersonic detonations. Both are assu…
The observed sub-class of "superluminous" Type Ia supernovae lacks a convincing theoretical explanation. If the emission of such objects were powered exclusively by radioactive decay of 56Ni formed in the explosion, a progenitor mass close to or even above the Chandrasekhar limit for a non-rotating white dwarf star would be required. Masses significantly exceeding this limit can be supported by differential rotation. We, therefore, explore explosions and predict observables for various scenarios resulting from differentially rotating carbon-oxygen white dwarfs close to their respective limit o…
Nowadays the number of models aimed at explaining the Type Ia supernova phenomenon is high and discriminating between them is a must-do. In this work we explore the influence of rotation in the evolution of the nuclear flame which drives the explosion in the so called gravitational confined detonation models. Assuming that the flame starts in a point-like region slightly above the center of the white dwarf (WD) and adding a moderate amount of angular velocity to the star we follow the evolution of the deflagration using a smoothed particle hydrodynamics code. We find that the results are very …
Delayed detonations of Chandrasekhar-mass white dwarfs (WDs) have been very successful in explaining the spectra, light curves, and the width-luminosity relation of spectroscopically normal Type Ia supernovae (SNe Ia). The ignition of the thermonuclear deflagration flame at the end of the convective carbon "simmering" phase in the core of the WD is still not well understood and much about the ignition kernel distribution remains unknown. Furthermore, the central density at the time of ignition depends on the still uncertain screened carbon fusion reaction rates, the accretion history and cooli…
Calculations of synthetic spectropolarimetry are one means to test multi-dimensional explosion models for Type Ia supernovae. In a recent paper, we demonstrated that the violent merger of a 1.1 and 0.9 M$_{\odot}$ white dwarf binary system is too asymmetric to explain the low polarization levels commonly observed in normal Type Ia supernovae. Here, we present polarization simulations for two alternative scenarios: the sub-Chandrasekhar mass double-detonation and the Chandrasekhar mass delayed-detonation model. Specifically, we study a two-dimensional double-detonation model and a three-dimensi…
Filling The Pockets: The Spherical Nature of 3D Deflagration in Thermonuclear Supernovae
Florida State University（佛罗里达州立大学）；KU Leuven（荷语鲁汶大学）；University of Hawai’i at Manoa（夏威夷大学马诺阿分校）；Planetary Science Institute（行星科学研究所）；Hamburger Sternwarte（汉堡天文台）
Observational evidences point to a common explosion mechanism of Type Ia supernovae based on a delayed detonation of a white dwarf. However, all attempts to find a convincing ignition mechanism based on a delayed detonation in a destabilized, expanding, white dwarf have been elusive so far. One of the possibilities that has been invoked is that an inefficient deflagration leads to pulsation of a Chandrasekhar-mass white dwarf, followed by formation of an accretion shock that confines a carbon-oxygen rich core, while transforming the kinetic energy of the collapsing halo into thermal energy of …
2002cx-like supernovae are a sub-class of sub-luminous Type Ia supernovae. Their light curves and spectra are characterized by distinct features that indicate strong mixing of the explosion ejecta. Pure turbulent deflagrations have been shown to produce such mixed ejecta. Here, we present hydrodynamics, nucleosynthesis and radiative transfer calculations for a 3D full-star deflagration of a Chandrasekhar-mass white dwarf. Our model is able to reproduce the characteristic observational features of SN 2005hk (a proto-typical 2002cx-like supernova), not only in the optical, but also in the near-i…
We study the initiation of the detonation in the gravitationally confined detonation (GCD) model of Type Ia supernovae (SNe Ia). Initiation of the detonation occurs spontaneously in a region where the length scale of the temperature gradient extending from a flow (in which carbon burning is already occurring) into unburned fuel is commensurate to the range of critical length scales which have been derived from 1D simulations that resolve the initiation of a detonation. By increasing the maximum resolution in a truncated cone that encompasses this region, beginning somewhat before initiation of…
Due to the increasing number of observations Type Ia supernovae are nowadays regarded as a heterogeneous class of objects consisting of several subclasses. One of the largest of these is the class of Type Iax supernovae (SNe Iax) which have been suggested to originate from pure deflagrations in CO Chandrasekhar-mass white dwarfs (WDs). Although a few deflagration studies have been carried out, the full diversity of the class is not captured yet. We therefore present a parameter study of single-spot ignited deflagrations with varying ignition locations, central densities, metallicities and comp…
It has been proposed that Type Ia supernovae (SNe Ia) that are normal in their spectra and brightness can be explained by a double detonation that ignites first in a helium shell on the surface of the white dwarf (WD). This proposition is supported by the satisfactory match between simulated explosions of sub-Chandrasekhar-mass WDs with no surface He layer and observations of normal SNe Ia. However, previous calculations of He-ignited double detonations have required either artificial removal of the He shell ashes or extreme enrichment of the surface He layer in order to obtain normal SNe Ia. …
The aim of this work is to identify and explain the necessary conditions required for an energetic explosion of a Chandrasekhar-mass white dwarf. We construct and analyze weakly compressible turbulence models with nuclear burning effects for carbon/oxygen plasma at a density expected for deflagration-to-detonation transition (DDT) to occur. We observe formation of carbon deflagrations and transient carbon detonations at early times. As turbulence becomes increasingly inhomogeneous, sustained carbon detonations are initiated by the Zel'dovich reactivity gradient mechanism. The fuel is suitably …
We present results for a suite of fourteen three-dimensional, high resolution hydrodynamical simulations of delayed-detonation modelsof Type Ia supernova (SN Ia) explosions. This model suite comprises the first set of three-dimensional SN Ia simulations with detailed isotopic yield information. As such, it may serve as a database for Chandrasekhar-mass delayed-detonation model nucleosynthetic yields and for deriving synthetic observables such as spectra and light curves. We employ a physically motivated, stochastic model based on turbulent velocity fluctuations and fuel density to calculate in…
We present theoretical delay times and rates of thermonuclear explosions that are thought to produce Type Ia supernovae, including the double-detonation sub-Chandrasekhar mass model, using the population synthesis binary evolution code StarTrack. If detonations of sub-Chandrasekhar mass carbon-oxygen white dwarfs following a detonation in an accumulated layer of helium on the white dwarf's surface ("double-detonation" models) are able to produce thermonuclear explosions which are characteristically similar to those of SNe Ia, then these sub-Chandrasekhar mass explosions may account for at leas…
In the classical picture, electron-capture supernovae and the accretion-induced collapse of oxygen-neon white dwarfs (ONeWDs) undergo an oxygen deflagration phase before gravitational collapse produces a neutron star (NS). These types of core collapse events are postulated to explain several astronomical phenomena. In this work, the deflagration phase is simulated for the first time using multidimensional hydrodynamics, with the aim of gaining new insight into the explosive deaths of $8-10~M_\odot$ stars and ONeWDs that accrete material from a binary companion star. The main aim is to determin…
The double-detonation explosion scenario of Type Ia supernovae has gained increased support from the SN Ia community as a viable progenitor model, making it a promising candidate alongside the well-known single degenerate and double degenerate scenarios. We present delay times of double-detonation SNe, in which a sub-Chandrasekhar mass carbon-oxygen white dwarf accretes non-dynamically from a helium-rich companion. One of the main uncertainties in quantifying SN rates from double-detonations is the (assumed) retention efficiency of He-rich matter. Therefore, we implement a new prescription for…
We show that converging spherical and cylindrical shock waves may ignite a detonation wave in a combustible medium, provided the radius at which the shocks become strong exceeds a critical radius, R_c. An approximate analytic expression for R_c is derived for an ideal gas equation of state and a simple (power-law-Arrhenius) reaction law, and shown to reproduce the results of numerical solutions. For typical acetylene-air experiments we find R_c~0.1 mm (spherical) and R_c~1 mm (cylindrical). We suggest that the deflagration to detonation transition (DDT) observed in these systems may be due to …
Upon formation, degenerate He core white dwarfs are surrounded by a radiative H-rich layer primarily supported by ideal gas pressure. In this Letter, we examine the effect of this H-rich layer on mass transfer in He+C/O double white dwarf binaries that will eventually merge and possibly yield a Type Ia supernova (SN Ia) in the double detonation scenario. Because its thermal profile and equation of state differ from the underlying He core, the H-rich layer is transferred stably onto the C/O white dwarf prior to the He core's tidal disruption. We find that this material is ejected from the binar…
We analyze the statistical properties of the turbulent velocity field in the deflagration model for Type Ia supernovae. In particular, we consider the question of whether turbulence is isotropic and consistent with the Kolmogorov theory at small length scales. Using numerical data from a high-resolution simulation of a thermonuclear supernova explosion, spectra of the turbulence energy and velocity structure functions are computed. We show that the turbulent velocity field is isotropic at small length scales and follows a scaling law that is consistent with the Kolmogorov theory until most of …
Detonations in helium-rich envelopes surrounding white dwarfs have garnered attention as triggers of faint thermonuclear ".Ia" supernovae and double detonation Type Ia supernovae. However, recent studies have found that the minimum size of a hotspot that can lead to a helium detonation is comparable to, or even larger than, the white dwarf's pressure scale height, casting doubt on the successful ignition of helium detonations in these systems. In this paper, we examine the previously neglected effects of C/O pollution and a full nuclear reaction network, and we consider hotspots with spatially…
Growing evidence suggests that Type Iax supernovae might be the result of thermonuclear deflagrations of Chandrasekhar-mass white dwarfs in binary systems. We carry out Monte Carlo radiative transfer simulations and predict spectropolarimetric features originating from the supernova explosion and subsequent ejecta interaction with the companion star. Specifically, we calculate viewing-angle dependent flux and polarisation spectra for a 3D model simulating the deflagration of a Chandrasekhar-mass white dwarf and, for a second model, simulating the ejecta interaction with a main-sequence star. W…
Type Ia supernovae are thought to result from thermonuclear explosions of carbon-oxygen white dwarf stars. This model generally explains the observed properties with certain exceptions, like sub-luminous supernovae. Here we discuss the possibility of sub-Chandrasekhar WDs detonating due to the build up of a layer of helium on the C-O WD by accreting from a helium rich companion star to explain observed deviations such as subluminous type Ia. We also detail some of the energetics involved that will make such scenarios plausible.
The progenitor systems and explosion mechanism of Type Ia supernovae are still unknown. Currently favoured progenitors include double-degenerate systems consisting of two carbon-oxygen white dwarfs with thin helium shells. In the double-detonation scenario, violent accretion leads to a helium detonation on the more massive primary white dwarf that turns into a carbon detonation in its core and explodes it. We investigate the fate of the secondary white dwarf, focusing on changes of the ejecta and observables of the explosion if the secondary explodes as well rather than survives. We simulate a…
Incomplete carbon-oxygen detonation with reactions terminating after burning of 12C in the leading 12C + 12C reaction (C-detonation) may occur in the low density outer layers of white dwarfs exploding as Type Ia supernovae (SNIa). Previous studies of carbon-oxygen detonation structure and stability at low densities were performed under the assumption that the "velocity" of a detonation wave derives from complete burning of carbon and oxygen to iron. In fact, at densities <10^6 g/cm3 the detonation in SNIa may release less than a half of the available nuclear energy. In this paper we study basi…
Supernovae of type Ia are thought to arise from the thermonuclear incineration of a carbon-oxygen white dwarf stellar remnant. However, the detailed explosion scenario and stellar evolutionary origin scenario -- or scenarios -- which lead to observed supernovae are still quite uncertain. One of the principal tests of proposed scenarios is comparison with the explosion products inferred, for example, from the spectrum of the supernovae. Making this comparison requires computation of the combustion dynamics and products through simulation of proposed scenarios. Here we discuss two specific propo…
The explosion of sub-Chandrasekhar mass white dwarfs via the double detonation scenario is a potential explanation for type Ia supernovae. In this scenario, a surface detonation in a helium layer initiates a detonation in the underlying carbon/oxygen core leading to an explosion. For a given core mass, a lower bound has been determined on the mass of the helium shell required for dynamical burning during a helium flash, which is a necessary prerequisite for detonation. For a range of core and corresponding minimum helium shell masses, we investigate whether an assumed surface helium detonation…
We investigate whether pure deflagration models of Chandrasekhar-mass carbon-oxygen white dwarf stars can account for one or more subclass of the observed population of Type Ia supernova (SN Ia) explosions. We compute a set of 3D full-star hydrodynamic explosion models, in which the deflagration strength is parametrized using the multispot ignition approach. For each model, we calculate detailed nucleosynthesis yields in a post-processing step with a 384 nuclide nuclear network. We also compute synthetic observables with our 3D Monte Carlo radiative transfer code for comparison with observatio…
Evolution of binaries containing a hot subdwarf and a white dwarf to double white dwarfs, and double detonation supernovae with hypervelocity runaway stars
Max Planck Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；Center for Astrophysics | Harvard & Smithsonian（哈佛-史密森尼天体物理中心）
(Abridged) The explosion mechanism of electron-capture supernovae (ECSNe) remains equivocal. We attempt to constrain the explosion mechanism (neutron-star-forming implosion or thermonuclear explosion) and the frequency of occurrence of ECSNe using nucleosynthesis simulations of the latter scenario, population synthesis, the solar abundance distribution, pre-solar meteoritic oxide grain isotopic ratio measurements and the white dwarf mass-radius relation. Tracer particles from 3d hydrodynamic simulations were post-processed with a large nuclear reaction network in order to determine the complet…
The double-detonation explosion model has been considered a candidate for explaining astrophysical transients with a wide range of luminosities. In this model, a carbon-oxygen white dwarf star explodes following detonation of a surface layer of helium. One potential signature of this explosion mechanism is the presence of unburned helium in the outer ejecta, left over from the surface helium layer. In this paper we present simple approximations to estimate the optical depths of important He I lines in the ejecta of double-detonation models. We use these approximations to compute synthetic spec…
We present a systematic study of the pure deflagration model of Type Ia supernovae using three-dimensional, high-resolution, full-star hydrodynamical simulations, nucleosynthetic yields calculated using Lagrangian tracer particles, and light curves calculated using radiation transport. We evaluate the simulations by comparing their predicted light curves with many observed SNe Ia using the SALT2 data-driven model and find that the simulations may correspond to under-luminous SNe Iax. We explore the effects of the initial conditions on our results by varying the number of randomly selected igni…
The progenitor system(s) and the explosion mechanism(s) of Type Ia supernovae (SNe Ia) are still under debate. Non-electromagnetic observables, in particular gravitational waves and neutrino emission, of thermonuclear supernovae are a complementary window to light curves and spectra for studying these enigmatic objects. A leading model for SNe Ia is the thermonuclear incineration of a near-Chandrasekhar mass carbon-oxygen white dwarf star in a "delayed-detonation". We calculate a three-dimensional hydrodynamic explosion for the N100 delayed-detonation model extensively discussed in the literat…
Study of the double detonation Type Ia supernova scenario, in which a helium shell detonation triggers a carbon core detonation in a sub-Chandrasekhar-mass white dwarf, has experienced a resurgence in the past decade. New evolutionary scenarios and a better understanding of which nuclear reactions are essential have allowed for successful explosions in white dwarfs with much thinner helium shells than in the original, decades-old incarnation of the double detonation scenario. In this paper, we present the first suite of light curves and spectra from multi-dimensional radiative transfer calcula…
During the early evolution of an AM CVn system, helium is accreted onto the surface of a white dwarf under conditions suitable for unstable thermonuclear ignition. The turbulent motions induced by the convective burning phase in the He envelope become strong enough to influence the propagation of burning fronts and may result in the onset of a detonation. Such an outcome would yield radioactive isotopes and a faint rapidly rising thermonuclear ".Ia" supernova. In this paper, we present hydrodynamic explosion models and observable outcomes of these He shell detonations for a range of initial co…
This article presents the study of the effects of magnetic fields on non-distributed nuclear burning fronts as a possible solution to a fundamental problem for the thermonuclear explosion of a Chandrasekhar mass ($M_{Ch}$) white dwarf (WD), the currently favored scenario for the majority of Type Ia SNe (SNe~Ia). All existing 3D hydrodynamical simulations predict strong global mixing of the burning products due to Rayleigh-Taylor (RT) instabilities, which is in contradiction with observations. As a first step and to study the flame physics we present a set of computational magneto-hydrodynamic …
The delayed-detonation explosion mechanism applied to a Chandrasekhar-mass white dwarf offers a very attractive model to explain the inferred characteristics of Type Ia supernovae (SNe Ia). The resulting ejecta are chemically stratified, have the same mass and roughly the same asymptotic kinetic energy, but exhibit a range in 56Ni mass. We investigate the contemporaneous photometric and spectroscopic properties of a sequence of delayed-detonation models, characterized by 56Ni masses between 0.18 and 0.81 Msun. Starting at 1d after explosion, we perform the full non-LTE, time-dependent radiativ…
