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. We find that the intrinsic signal is weakly polarised and only mildly viewing-angle dependent, owing to the overall spherical symmetry of the explosion and the depolarising contribution of iron-group elements dominating the ejecta composition. The interaction with the companion star carves out a cavity in the ejecta and produces a detectable, but modest signal that is significant only at relatively blue wavelengths ($\lesssim$ 5000 $\unicode{x212B}$). In particular, increasingly fainter and redder spectra are predicted for observer orientations further from the cavity, while a modest polarisation signal $P\sim0.2$ per cent is found at blue wavelengths for orientations 30$^\circ$ and 45$^\circ$ away from the cavity. We find a reasonable agreement between the interaction model viewed from these orientations and spectropolarimetric data of SN 2005hk and interpret the maximum-light polarisation signal seen at blue wavelengths for this event as a possible signature of the ejecta-companion interaction. We encourage further polarimetric observations of SNe Iax to test whether our results can be extended and generalised to the whole SN Iax class.
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…
Stellar evolution models predict the existence of hybrid white dwarfs (WDs) with a carbon-oxygen core surrounded by an oxygen-neon mantle. Being born with masses ~1.1 Msun, hybrid WDs in a binary system may easily approach the Chandrasekhar mass (MCh) by accretion and give rise to a thermonuclear explosion. Here, we investigate an off-centre deflagration in a near-MCh hybrid WD under the assumption that nuclear burning only occurs in carbon-rich material. Performing hydrodynamics simulations of the explosion and detailed nucleosynthesis post-processing calculations, we find that only 0.014 Msu…
Type Iax supenovae (SNe Iax) are an important sub-luminous class of SNe Ia. However, their progenitors and explosion mechanism have not been understood yet. It has been suggested that SNe Iax may be produced from weak deflagration explosions of Chandrasekhar-mass white dwarfs (WDs) in binary systems with a helium (He) star donor. In such progenitor systems, the stripped He companion material caused by the ejecta-companion interaction is expected to present some He features in their late-time spectra. However, the detection of He lines in late-time spectra of SNe Iax has not yet been successful…
We present results based on observations of SN 2015H which belongs to the small group of objects similar to SN 2002cx, otherwise known as type Iax supernovae. The availability of deep pre-explosion imaging allowed us to place tight constraints on the explosion epoch. Our observational campaign began approximately one day post-explosion, and extended over a period of about 150 days post maximum light, making it one of the best observed objects of this class to date. We find a peak magnitude of M$_r$ = -17.27 $\pm$ 0.07, and a ($Δm_{15})_r$ = 0.69 $\pm$ 0.04. Comparing our observations to synthe…
We present comprehensive spectroscopic and photometric analyses of the intermediate luminosity Type Iax supernova (SN Iax) 2019muj based on multi-band datasets observed through the framework of the OISTER target-of-opportunity program. SN 2019muj exhibits almost identical characteristics with the subluminous SNe Iax 2008ha and 2010ae in terms of the observed spectral features and the light curve evolution at the early phase, except for the peak luminosity. The long-term observations unveil the flattening light curves at the late time as seen in a luminous SN Iax 2014dt. This can be explained b…
The recently observed diversity of Type Ia supernovae (SNe Ia) has motivated us to conduct the theoretical modeling of SNe Ia for a wide parameter range. In particular, the origin of Type Iax supernovae (SNe Iax) has been obscure. Following our earlier work on the parameter dependence of SN Ia models, we focus on SNe Iax in the present study. For a model of SNe Iax, we adopt the currently leading model of pure turbulent deflagration (PTD) of near-Chandrasekhar mass C+O white dwarfs (WDs). We carry out 2-dimensional hydrodynamical simulations of the propagation of deflagration wave, which leave…
Type Iax supernovae (SN Iax), also called SN 2002cx-like supernovae, are the largest class of peculiar white dwarf (thermonuclear) supernovae, with over fifty members known. SN Iax have lower ejecta velocity and lower luminosities, and these parameters span a much wider range, than normal type Ia supernovae (SN Ia). SN Iax are spectroscopically similar to some SN Ia near maximum light, but are unique among all supernovae in their late-time spectra, which never become fully nebular. SN Iax overwhelmingly occur in late-type host galaxies, implying a relatively young population. The SN Iax 2012Z …
Abundance Stratification in Type Iax SN 2020rea with TARDIS
Birla Institute of Technology & Science - Pilani(比拉技术学院-皮拉尼);National Institute of Science Education and Research(国立科学教育与研究机构);Indian Institute of Astrophysics(印度天体物理研究所);Aryabhatta Research Institute of Observational Sciences(阿亚玻达观测科学研究所以)
Unveiling SN 2022eyw: A Bright Member of the Type Iax Supernova Subclass
Indian Institute of Astrophysics(印度天体物理研究所);Pondicherry University(本地治里大学);Department of Physics and Astronomy, Michigan State University(密歇根州立大学物理与天文学系);Tsung-Dao Lee Institute, Shanghai Jiao Tong University(上海交通大学李政道研究所)
SN 2022eyw是Iax型超新星中明亮成员,其光谱和光度学特征表明其起源于部分熄火的Chandrasekhar质量白矮星。
Recent observations have shown a remarkable diversity of observational behaviors and explosion mechanisms in thermonuclear supernovae (SNe). An emerging class of peculiar thermonuclear SNe, called Type Iax, show photometric and spectroscopic behaviors distinct from normal Type Ia. Their origin remains highly controversial, but pure turbulent deflagration of white dwarfs (WDs) has been regarded as the leading formation theory. The large population of Type Iax indicates the existence of unidentified Galactic Type Iax supernova remnants (SNRs). We report evidence that SNR Sgr A East in the Galact…
Investigations of some type Iax supernovae have led to the suggestion that their ejecta must be layered to some degree. Such an ejecta structure has been argued as inconsistent with the well-mixed composition predicted by pure deflagrations. Based on explosion models, we create toy models in which the ejecta are artificially stratified and progressively mixed until a uniform composition is obtained. We find that models that are heavily mixed, containing burned and unburned material at all velocities, produce reasonably good agreement with SN 2012Z, for which a layered structure has been sugges…
The nature of the progenitors and explosion mechanism of Type Iax supernovae (SNe Iax) remain a mystery. The single-degenerate (SD) systems that involve the incomplete pure deflagration explosions of near-Chandrasekhar-mass white dwarfs (WDs) have recently been proposed for producing SNe Iax, in which non-degenerate companions are expected to survive from SN explosions. In this work we concentrate on the main-sequence (MS) donor SD progenitor systems. By mapping the computed companion models from three-dimensional hydrodynamical simulations of ejecta-companion interaction into a one-dimensiona…
We present UV through NIR broad-band photometry, and optical and NIR spectroscopy of Type Iax supernova 2012Z. The data set consists of both early and late-time observations, including the first late phase NIR spectrum obtained for a spectroscopically classified SN Iax. Simple model calculations of its bolometric light curve suggest SN 2012Z produced ~0.3 M_sun of (56)Ni, ejected about a Chandrasekhar mass of material, and had an explosion energy of ~10^51 erg, making it one of the brightest and most energetic SN Iax yet observed. The late phase NIR spectrum of SN 2012Z is found to broadly res…
Recent studies have suggested that type Iax supernovae (SNe Iax) are likely to result from a weak deflagration explosion of a Chandrasekhar-mass white dwarf in a binary system with a helium (He)-star companion. Assuming that most SNe Iax are produced from this scenario, in this work we extend our previous work on the three-dimensional hydrodynamical simulation of ejecta-companion interaction by taking the orbital and spin velocities of the progenitor system into account. We then follow the post-impact evolution of a surviving He-star companion by using the one-dimensional stellar evolution cod…
The most commonly accepted progenitor system for Type Iax supernovae (SNe Iax) is the partial deflagration of a near-Chandrasekhar-mass white dwarf (WD) accreting from a non-degenerate helium donor star, leaving a bound remnant following the explosion. In this paper, we investigate whether the WD remant can undergo multiple SNe during the system's lifetime. We use Modules for Experiments in Astrophysics (MESA) to evolve various single-degenerate binaries to determine which could plausibly undergo multiple SNe Iax due to multiple helium accretion phases. We also investigate the possibility for …
Supernovae of Type Iax (SNe Iax) are an accepted faint subclass of hydrogen-free supernovae. Their origin, the nature of the progenitor systems, however, is an open question. Recent studies suggest that the weak deflagration explosion of a near-Chandrasekhar-mass white dwarf in a binary system with a helium star donor could be the origin of SNe Iax. In this scenario, the helium star donor is expected to survive the explosion. We use the one-dimensional stellar evolution codes \textsc{MESA} and \textsc{Kepler} to follow the post-impact evolution of the surviving helium companion stars. The stel…
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 extremely low-luminosity Type Iax SNe 2022ywf and 2023zgx
University of Szeged(塞格德大学);HUN-REN-SZTE Stellar Astrophysics Research Group(匈牙利研究网络-塞格德大学恒星天体物理学研究组);European Southern Observatory(欧洲南方天文台);Rutgers, the State University of New Jersey(新泽西州立罗格斯大学);Purdue University(普渡大学);Steward Observatory, University of Arizona(亚利桑那大学斯图尔德天文台);National Central University(中央大学);Las Cumbres Observatory(拉斯坎伯斯天文台);University of California, Santa Barbara(加州大学圣塔芭芭拉分校);Astronomical Observatory, University of Warsaw(华沙大学天文台)
We present a detailed spectral analysis of Type Iax SN 2011ay. Our spectra cover epochs between -3 and +19 days with respect to the maximum light in B-band. This time range allows us to employ a so-called abundance tomography technique. The synthetic spectral fitting was made with the 1D Monte Carlo radiative transfer code TARDIS. In this paper, we describe our method to fit multiple epochs with a self-consistent, stratified atmospheric model. We compare our results to previously published SYN++ models and the predictions of different explosion scenarios. Using a fixed density profile (exponen…
Recent studies have argued that the progenitor system of type Iax supernovae must consist of a carbon-oxygen white dwarf accreting from a helium star companion. Based on existing explosion models invoking the pure deflagration of carbon-oxygen white dwarfs, we investigate the likelihood of producing spectral features due to helium in type Iax supernovae. From this scenario, we select those explosion models producing ejecta and $^{56}$Ni masses that are broadly consistent with those estimated for type Iax supernovae (0.014 - 0.478~$M_{\odot}$ and $\sim0.003$ - 0.183~$M_{\odot}$, respectively). …
Recently, a new class of supernovae Ia was discovered: the supernovae Iax; the increasing sample of these objects share common features as lower maximum-light velocities and typically lower peak magnitudes. In our scenario, the progenitors of the SNe Iax are very massive white dwarfs, possibly hybrid C+O+Ne white dwarfs; due to the accretion from a binary companion, they reach the Chandrasekhar mass and undergo a central carbon deflagration, but the deflagration is quenched when it reaches the outer O +Ne layer. This class of SNe Ia are expected to be rarer than standard SNe Ia and do not affe…
We present optical and near-infrared observations of the nearby Type Iax supernova (SN) 2014dt from 14 to 410 days after the maximum light. The velocities of the iron absorption lines in the early phase indicated that SN 2014dt showed slower expansion than the well-observed Type Iax SNe 2002cx, 2005hk and 2012Z. In the late phase, the evolution of the light curve and that of the spectra were considerably slower. The spectral energy distribution kept roughly the same shape after ~100 days, and the bolometric light curve flattened during the same period. These observations suggest the existence …
Discovery of a runaway star likely ejected by a Type Iax Supernova
Universität Potsdam(波茨坦大学);University of Warwick(华威大学);Friedrich-Alexander University Erlangen-Nuremberg(埃尔朗根-纽伦堡大学);Christian-Albrechts-Universität zu Kiel(基尔大学);Max Planck Institut für Astrophysik(马克斯·普朗克天体物理学研究所);University of California, Berkeley(加州大学伯克利分校)
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…
Observational signatures of thermonuclear electron-capture supernovae -- Ne II line strengthening and color evolution as traces of the explosion mechanism
Heidelberger Institut für Theoretische Studien(海德堡理论研究所);Queen’s University Belfast(贝尔法斯特女王大学);Los Alamos National Laboratory(洛斯阿拉莫斯国家实验室);Zentrum für Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik(海德堡大学天体物理中心,理论天体物理研究所);Max-Planck-Institut für Astrophysik(马克斯·普朗克天体物理研究所);Sapienza Università di Roma, Dipartimento di Fisica(罗马智慧大学,物理系);Trinity College Dublin, The University of Dublin(都柏林三一学院,都柏林大学);GSI Helmholtzzentrum für Schwerionenforschung(GSI亥姆霍兹重离子研究中心)
We report observations of the Type Iax supernova (SN Iax) 2012Z at optical and near-infrared wavelengths from immediately after the explosion until $\sim$ $260$ days after the maximum luminosity using the Optical and Infrared Synergetic Telescopes for Education and Research (OISTER) Target-of-Opportunity (ToO) program and the Subaru telescope. We found that the near-infrared (NIR) light curve evolutions and color evolutions are similar to those of SNe Iax 2005hk and 2008ha. The NIR absolute magnitudes ($M_{J}\sim-18.1$ mag and $M_{H}\sim-18.3$ mag) and the rate of decline of the light curve ($…
We present multi-wavelength follow-up observations of the ATLAS discovered faint Iax supernova SN 2020kyg that peaked at an absolute magnitude of $M_g \approx -14.9 \pm 0.2$, making it another member of the faint Iax supernova population. The bolometric light curve requires only $\approx 7 \times 10^{-3}$ M$_{\odot}$ of radioactive $^{56}$Ni, with an ejected mass of $M_{\rm ej} \sim 0.4$ M$_{\odot}$ and a low kinetic energy of $E \approx 0.05 \pm 0.02 \times 10^{51}$ erg. We construct a homogeneous volume-limited sample of 902 transients observed by ATLAS within 100 Mpc during a 3.5 year span.…
Photometry and Spectroscopy of SN 2024pxl: A Luminosity Link Among Type Iax Supernovae
Indian Institute of Astrophysics(印度天体物理研究所);Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)(天体物理学交叉探索与研究中心);Department of Physics and Astronomy, Rutgers, the State University of New Jersey(新泽西州立罗格斯大学物理与天文学系);Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Brera(国家天体物理研究所布雷拉天文台);Department of Astronomy, University of California, Berkeley(加州大学伯克利分校天文学系);Gemini Observatory/NSF’s NOIRLab(吉米尼观测站/美国国家科学基金会NOIRLab);Las Cumbres Observatory(拉斯坎布雷斯观测站);Department of Physics, University of California, Santa Barbara(加州大学圣塔芭芭拉分校物理系);Department of Astronomy and Astrophysics, University of California, Santa Cruz(加州大学圣克鲁斯分校天文学与天体物理学系);School of Physics, The University of Melbourne(墨尔本大学物理学院)
SN 2024pxl的光度学和光谱学研究揭示其介于高光度和低光度Iax型超新星之间的特性,建立了该类超新星的光度关联。
We present optical photometric and spectroscopic analysis of a Type Iax supernova (SN) 2020rea situated at the brighter luminosity end of Type Iax supernovae (SNe). The light curve decline rate of SN~2020rea is $Δ$m$_{15}$(g) = 1.31$\pm$0.08 mag which is similar to SNe 2012Z and 2005hk. Modelling the pseudo bolometric light curve with a radiation diffusion model yields a mass of $^{56}$Ni of 0.13$\pm$0.01 M$_{\odot}$ and an ejecta mass of 0.77$^{+0.11}_{-0.21}$ M$_{\odot}$. Spectral features of SN~2020rea during the photospheric phase show good resemblance with SN 2012Z. TARDIS modelling of th…
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)
Tomography analysis of the intermediate-luminosity Type Iax SN 2024pxl
University of Szeged(塞格德大学);HUN-REN-SZTE Stellar Astrophysics Research Group(匈牙利研究网络-塞格德大学恒星天体物理研究组);Indian Institute of Astrophysics(印度天体物理研究所);Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)(天体物理学跨学科探索与研究中心);Rutgers, the State University of New Jersey(新泽西州立罗格斯大学);Istituto Nazionale di Astrofisica, Osservatorio Astronomico di Brera(意大利国家天体物理研究所,布雷拉天文台);Department of Astronomy, University of California, Berkeley(加州大学伯克利分校天文学系);European Southern Observatory(欧洲南方天文台);Gemini Observatory/NSF’s NOIRLab(吉米尼天文台/美国国家科学基金会NOIRLab);Las Cumbres Observatory(拉斯孔布伦斯天文台);Department of Physics, University of California, Santa Barbara(加州大学圣塔芭芭拉分校物理系)
Recent observational evidence has demonstrated that white dwarf (WD) mergers are a highly efficient mechanism for mass accretion onto WDs in the galaxy. In this paper, we show that WD mergers naturally produce highly-magnetized, uniformly-rotating WDs, including a substantial population within a narrow mass range close to the Chandrasekhar mass ($M_{\rm Ch}$). These near-$M_{\rm Ch}$ WD mergers subsequently undergo rapid spin up and compression on a $\sim 10^2$ yr timescale, either leading to central ignition and a normal SN Ia via the DDT mechanism, or alternatively to a failed detonation and…
We present optical UBVRI photometry and low-to-medium resolution spectroscopic observations of type Iax SN 2020sck spanning -5.5 d to +67 d from maximum light in the B-band. From the photometric analysis we find $Δm_{\rm{B}}$(15) = 2.03$\pm$0.05 mag and $M_{\rm{B}}$=-17.81$\pm$0.22 mag. Radiation diffusion model fit to the quasi-bolometric light curve indicates 0.13$\pm$0.02 $M_{\odot}$ of $^{56}$Ni and 0.34 $M_{\rm \odot}$ of ejecta are synthesized in the explosion. Comparing the observed quasi-bolometric light curve with angle-averaged bolometric light curve of three-dimensional pure deflagr…
We present the discovery and optical follow-up of the faintest supernova-like transient known. The event (SN 2019gsc) was discovered in a star-forming host at 53\,Mpc by ATLAS. A detailed multi-colour light curve was gathered with Pan-STARRS1 and follow-up spectroscopy was obtained with the NOT and Gemini-North. The spectra near maximum light show narrow features at low velocities of 3000 to 4000 km s$^{-1}$, similar to the extremely low luminosity SNe 2010ae and 2008ha, and the light curve displays a similar fast decline (\dmr $0.91 \pm 0.10$ mag). SNe 2010ae and 2008ha have been classified a…
SN 2025qe: A case study of diversity in low-luminosity thermonuclear explosions
Indian Institute of Astrophysics(印度天体物理研究所);Pondicherry University(本地治里大学);National Central University(中央大学);Sardar Vallabhbhai National Institute of Technology(萨达尔·瓦拉巴伊国立理工学院);Indian Institute of Technology Bombay(孟买印度理工学院)
We present the discovery of ZTF 21aaoryiz/SN 2021fcg -- an extremely low-luminosity Type Iax supernova. SN 2021fcg was discovered by the Zwicky Transient Facility in the star-forming galaxy IC0512 at a distance of $\approx$ 27 Mpc. It reached a peak absolute magnitude of $M_{r} =$ $-12.66\pm0.20$ mag, making it the least luminous thermonuclear supernova discovered to date. The E(B-V) contribution from the underlying host galaxy is unconstrained. However, even if it were as large as 0.5 mag, the peak absolute magnitude would be $M_{r} = -13.78\pm0.20$ mag -- still consistent with being the lowe…
We present two-dimensional hydrodynamics simulations of near-Chandrasekhar mass white dwarf (WD) models for Type Ia supernovae (SNe Ia) using the turbulent deflagration model with deflagration-detonation transition (DDT). We perform a parameter survey for 41 models to study the effects of the initial central density (i.e., WD mass), metallicity, flame shape, DDT criteria, and turbulent flame formula for a much wider parameter space than earlier studies. The final isotopic abundances of $^{11}$C to $^{91}$Tc in these simulations are obtained by post-process nucleosynthesis calculations. The sur…
Recent extensive observations of Type Ia Supernovae (SNe Ia) have revealed the existence of a diversity of SNe Ia, including SNe Iax. We introduce two possible channels in the single degenerate scenario: 1) double detonations in sub-Chandrasekhar (Ch) mass CO white dwarfs (WDs), where a thin He envelope is developed with relatively low accretion rates after He novae even at low metallicities, and 2) carbon deflagrations in Ch-mass possibly hybrid C+O+Ne WDs, where WD winds occur at [Fe/H] ~ -2.5 at high accretion rates. These subclasses of SNe Ia are rarer than `normal' SNe Ia and do not affec…
Helium accretion induced explosions in CO white dwarfs (WDs) are considered promising candidates for a number of observed types of stellar transients, including supernovae (SNe) of Type Ia and Type Iax. However, a clear favorite outcome has not yet emerged. We explore the conditions of helium ignition in the white dwarf and the final fates of helium star-WD binaries as function of their initial orbital periods and component masses. We compute 274 model binary systems with the Binary Evolution Code (BEC), where both components are fully resolved. Stellar and orbital evolution is computed simult…
In our quest to identify the progenitors of Type Ia supernovae (SNe Ia), we first update the nucleosynthesis yields both for near-Chandrasekhar (Ch) and sub-Ch mass white dwarfs (WDs), for a wide range of metallicity, with our two-dimensional hydrodynamical code and the latest nuclear reaction rates. We then include the yields in our galactic chemical evolution code to predict the evolution of elemental abundances in the solar neighborhood and dwarf spheroidal (dSph) galaxies: Fornax, Sculptor, Sextans, and Carina. In the observations of the solar neighborhood stars, Mn shows an opposite trend…
I review new studies of type Ia supernovae (SNe Ia) from 2019, and use these to improve the comparison between the five binary SN Ia scenarios. New low polarisation measurements solidify the claim that most SN Ia explosions are globally spherically symmetric (clumps are possible). Explosions by dynamical processes, like explosions that take place during a merger process of two white dwarfs (WDs) in the double degenerate (DD) scenario, or during an accretion process in the double detonation (DDet) scenario and in the single degenerate (SD) scenario, lead to non-spherical explosions, in contradi…