Final Stages Of Stellar Evolution Research Articles

Revealing the ionising star of evolved planetary nebulae

Context. Planetary nebulae represent a late stage of the evolution of stars of ordinary mass. The nucleus of a planetary nebula, a stellar remnant, is a faint star for which it is usually difficult to gather spectroscopic data. The spectral types of these stars cover a wide range, mainly separated into two large groups, those rich in hydrogen and those that are not. In this complex scenery it is of great importance to increase the number of stars with defined spectral types and physical parameters. In this way, we can improve our actual models of stellar evolution. Aims. Our main objective is to increase the number of known white dwarfs that are the ionisation sources of planetary nebulae. Our aim is to obtain the spectral type and physical parameters of these stars, which are to be confronted with models of stellar evolution. In this way, we want to increase our knowledge of the final stages of stellar evolution. Methods. We present Gemini spectra of nine nuclei of faint and extended southern planetary nebulae. The spectra were fitted with models of stellar atmospheres, which allowed the derivation of their atmospheric parameters. In addition, stellar evolution models were used to infer the stars’ mass. Results. We discovered six hydrogen-rich white dwarfs (spectral types DA and DAO; effective temperatures in the range Teff = 60 000–80 000 K). One of the DAO white dwarfs is probably a new member of only a few known post-red-giant-branch central stars. In addition, we found two hot hydrogen-deficient PG 1159 stars (Teff = 120 000 and 160 000 K). Taken together, this represents a significant increase in the number of known white dwarf central stars. We also discovered an early O(H) star (Teff = 90 000 K).

Evolution of β -decay half-lives in stellar environments

$\ensuremath{\beta}$-decay properties of nuclei are investigated within the relativistic nuclear energy density functional framework by varying the temperature and density, conditions relevant to the final stages of stellar evolution. Both thermal and nuclear pairing effects are taken into account in the description of nuclear properties and in the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA) to calculate the relevant allowed and first-forbidden transitions in the $\ensuremath{\beta}$ decay. The temperature and density effects are studied on the $\ensuremath{\beta}$-decay half-lives at temperatures $T=0$--$1.5\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ and at densities $\ensuremath{\rho}{Y}_{e}={10}^{7}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ and ${10}^{9}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$. The relevant Gamow-Teller transitions are also investigated for Ti, Fe, Cd, and Sn isotopic chains at finite-temperatures. We find that the $\ensuremath{\beta}$-decay half-lives increase with increasing density $\ensuremath{\rho}{Y}_{e}$, whereas half-lives generally decrease with increasing temperature. It is shown that the temperature effects decrease the half-lives considerably in nuclei with longer half-lives at zero temperature, while only slight changes for nuclei with short half-lives are obtained. We also show the importance of including the de-excitation transitions in the calculation of the $\ensuremath{\beta}$-decay half-lives at finite-temperatures. Comparing the FT-PNQRPA results with the shell-model calculations for $pf$-shell nuclei, a reasonable agreement is obtained for the temperature dependence of $\ensuremath{\beta}$-decay rates. Finally, large-scale calculations of $\ensuremath{\beta}$-decay half-lives are performed at temperatures ${T}_{9}(\text{K})=5$ and ${T}_{9}(\text{K})=10$ and densities $\ensuremath{\rho}{Y}_{e}={10}^{7}$ and ${10}^{9}\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$ for even-even nuclei in the range $8\ensuremath{\le}Z\ensuremath{\le}82$, relevant for astrophysical nucleosynthesis mechanisms.

Asteroseismology of luminous red giants with Kepler – II. Dependence of mass-loss on pulsations and radiation

ABSTRACT Mass-loss by red giants is an important process to understand the final stages of stellar evolution and the chemical enrichment of the interstellar medium. Mass-loss rates are thought to be controlled by pulsation-enhanced dust-driven outflows. Here, we investigate the relationships between mass-loss, pulsations, and radiation, using 3213 luminous Kepler red giants and 13 5000 ASAS–SN semiregulars and Miras. Mass-loss rates are traced by infrared colours using 2MASS and Wide-field Infrared Survey Explorer(WISE) and by observed-to-model WISE fluxes, and are also estimated using dust mass-loss rates from literature assuming a typical gas-to-dust mass ratio of 400. To specify the pulsations, we extract the period and height of the highest peak in the power spectrum of oscillation. Absolute magnitudes are obtained from the 2MASS Ks band and the Gaia DR2 parallaxes. Our results follow. (i) Substantial mass-loss sets in at pulsation periods above ∼60 and ∼100 d, corresponding to Asymptotic-Giant-Branch stars at the base of the period-luminosity sequences C′ and C. (ii) The mass-loss rate starts to rapidly increase in semiregulars for which the luminosity is just above the red-giant-branch tip and gradually plateaus to a level similar to that of Miras. (iii) The mass-loss rates in Miras do not depend on luminosity, consistent with pulsation-enhanced dust-driven winds. (iv) The accumulated mass-loss on the red giant branch consistent with asteroseismic predictions reduces the masses of red-clump stars by 6.3 per cent, less than the typical uncertainty on their asteroseismic masses. Thus mass-loss is currently not a limitation of stellar age estimates for galactic archaeology studies.

The sensitivity of presupernova neutrinos to stellar evolution models

ABSTRACT We examine the sensitivity of neutrino emission to stellar evolution models for a 15 M⊙ progenitor, paying particular attention to a phase prior to the collapse. We demonstrate that the number luminosities in both electron-type neutrinos (νe) and their antipartners ($\bar{\nu }_\mathrm{ e}$) differ by more than an order of magnitude by changing spatial resolutions and nuclear network sizes on stellar evolution models. We also develop a phenomenological model to capture the essential trend of the diversity, in which neutrino luminosities are expressed as a function of central density, temperature, and electron fraction. In the analysis, we show that the neutrino luminosity can be well characterized by these central quantities. This analysis also reveals that the most influential quantity to the time evolution of νe luminosity is matter density, while it is temperature for $\bar{\nu }_\mathrm{ e}$. These qualitative trends will be useful and applicable to constrain the physical states of progenitors at the final stages of stellar evolution from future neutrino observations, although more detailed systematic studies including various mass progenitors are required to assess the applicability.

The spectra of evolved stars at 20–25 GHz: Tracing circumstellar chemistry during the asymptotic giant branch to planetary nebula transition

Abstract We report an unbiased radio line survey towards the circumstellar envelopes of evolved stars at the frequency range from 20 to 25 GHz, aiming to obtain a more complete unbiased picture of the chemical evolution in the final stages of stellar evolution. The observation sample includes the asymptotic giant branch (AGB) star IRC +10216, the proto-planetary nebulae (PPNs) CRL 2688 and CRL 618, and the young planetary nebula (PN) NGC 7027, representing an evolutionary sequence spanning about 10000 years. Rotational transitions from cyanopolyyne chains and inversion lines from ammonia are detected in the AGB star and PPNs, while the PN displays several recombination lines. The different spectral behaviors of these evolved stars clearly reflect the evolution of circumstellar chemistry during the AGB–PPN–PN transitions.

Supernova lightCURVE POPulation Synthesis II: Validation against supernovae with an observed progenitor

Abstract We use the results of a supernova light-curve population synthesis to predict the range of possible supernova light curves arising from a population of single-star progenitors that lead to type IIP supernovae. We calculate multiple models varying the initial mass, explosion energy, nickel mass and nickel mixing and then compare these to type IIP supernovae with detailed light curve data and pre-explosion imaging progenitor constraints. Where a good fit is obtained to observations, we are able to achieve initial progenitor and nickel mass estimates from the supernova lightcurve that are comparable in precision to those obtained from progenitor imaging. For 2 of the 11 IIP supernovae considered our fits are poor, indicating that more progenitor models should be included in our synthesis or that our assumptions, regarding factors such as stellar mass loss rates or the rapid final stages of stellar evolution, may need to be revisited in certain cases. Using the results of our analysis we are able to show that most of the type IIP supernovae have an explosion energy of the order of log(Eexp/ergs) = 50.52 ± 0.10 and that both the amount of nickel in the supernovae and the amount of mixing may have a dependence on initial progenitor mass.

Ensemble Properties of the White Dwarf Population of the Old, Solar Metallicity Open Star Cluster Messier 67*

Abstract White dwarfs (WDs) are excellent forensic tools for studying end-of-life issues surrounding low- and intermediate-mass stars, and the old, solar metallicity open star cluster Messier 67 is a proven laboratory for the study of stellar evolution for solar-type stars. In this paper, we present a detailed spectroscopic study of brighter (M g ≤ 12.4) WDs in Messier 67, and in combination with previously published proper motion membership determinations, we identify a clean, representative sample of cluster WDs, including 13 members with hydrogen-dominated atmospheres, at least one of which is a candidate double degenerate, and 5 members with helium-dominated atmospheres. Using this sample we test multiple predictions surrounding the final stages of stellar evolution in solar-type stars. In particular, the stochasticity of the integrated mass lost by ∼1.5 solar mass stars is less than 7% of the WD remnant mass. We identify WDs likely resulting from binary evolution, including at least one blue straggler remnant and two helium-core WDs. We observe no evidence of a significant population of helium-core WDs formed by enhanced mass loss on the red giant branch of the cluster. The distribution of WD atmospheric compositions is fully consistent with that in the field, limiting proposed mechanisms for the suppression of helium atmosphere WD formation in star clusters. In short, the WD population of Messier 67 is fully consistent with basic predictions of single- and multiple-star stellar evolution theories for solar metallicity stars.

Photometric Variability of Luminous Blue Variable Stars on Different Time-Scales

AbstractWe have compiled historical observations, spanning ∼100 years, for a dozen of the best-studied LBVs in the Local Group. We described how we prepared structure functions for their light-curves and calculated two parameters (the structure function slope and the characteristic time-scale) to describe the behaviour of the LBVs. The sensitivity of those parameters to the variability behaviour of the stars was tested with a number of photometric data sets. The slope of the structure function may anticorrelate with the time-scale. These types of variable stellar objects are crucial to studies of stellar variability and the final stages of stellar evolution.

Dark gamma-ray bursts

Many theories of dark matter (DM) predict that DM particles can be captured by stars via scattering on ordinary matter. They subsequently condense into a DM core close to the center of the star and eventually annihilate. In this work, we trace DM capture and annihilation rates throughout the life of a massive star and show that this evolution culminates in an intense annihilation burst coincident with the death of the star in a core collapse supernova. The reason is that, along with the stellar interior, also its DM core heats up and contracts, so that the DM density increases rapidly during the final stages of stellar evolution. We argue that, counterintuitively, the annihilation burst is more intense if DM annihilation is a $p$-wave process than for $s$-wave annihilation because in the former case, more DM particles survive until the supernova. If among the DM annihilation products are particles like dark photons that can escape the exploding star and decay to Standard Model particles later, the annihilation burst results in a flash of gamma rays accompanying the supernova. For a galactic supernova, this "dark gamma ray burst" may be observable in CTA.

PROBING FINAL STAGES OF STELLAR EVOLUTION WITH X-RAY OBSERVATIONS OF SN 2013ej

ABSTRACT Massive stars shape their surroundings with mass loss from winds during their lifetimes. Fast ejecta from supernovae (SNe), from these massive stars, shock this circumstellar medium. Emission generated by this interaction provides a window into the final stages of stellar evolution, by probing the history of mass loss from the progenitor. Here we use Chandra and Swift X-ray observations of the type II-P/L SN 2013ej to probe the history of mass loss from its progenitor. We model the observed X-rays as emission from both heated circumstellar matter and SN ejecta. The circumstellar density profile probed by the SN shock reveals a history of steady mass loss during the final 400 years. The inferred mass loss rate of 3 × 10 − 6 M ⊙ yr − 1 ?> points back to a 14 M ⊙ ?> progenitor. Soon after the explosion we find significant absorption of reverse shock emission by a cooling shell. The column depth of this shell observed in absorption provides an independent and consistent measurement of the circumstellar density seen in emission. We also determine the efficiency of cosmic ray acceleration from X-rays produced by Inverse Compton scattering of optical photons by relativistic electrons. Only about 1% of the thermal energy is used to accelerate electrons. Our X-ray observations and modeling provide stringent tests for models of massive stellar evolution and micro-physics of shocks.

METAMORPHOSIS OF SN 2014C: DELAYED INTERACTION BETWEEN A HYDROGEN POOR CORE-COLLAPSE SUPERNOVA AND A NEARBY CIRCUMSTELLAR SHELL

We present optical observations of supernova SN 2014C, which underwent an unprecedented slow metamorphosis from H-poor type Ib to H-rich type IIn over the course of one year. The observed spectroscopic evolution is consistent with the supernova having exploded in a cavity before encountering a massive shell of the progenitor star's stripped hydrogen envelope. Possible origins for the circumstellar shell include a brief Wolf-Rayet fast wind phase that overtook a slower red supergiant wind, eruptive ejection, or confinement of circumstellar material by external influences of neighboring stars. An extended high velocity Halpha absorption feature seen in near-maximum light spectra implies that the progenitor star was not completely stripped of hydrogen at the time of core collapse. Archival pre-explosion Subaru Telescope Suprime-Cam and Hubble Space Telescope Wide Field Planetary Camera 2 images of the region obtained in 2009 show a coincident source that is most likely a compact massive star cluster in NGC 7331 that hosted the progenitor system. By comparing the emission properties of the source with stellar population models that incorporate interacting binary stars we estimate the age of the host cluster to be 30 - 300 Myr, and favor ages closer to 30 Myr in light of relatively strong Halpha emission. SN 2014C is the best-observed member of a class of core-collapse supernovae that fill the gap between events that interact strongly with dense, nearby environments immediately after explosion and those that never show signs of interaction. Better understanding of the frequency and nature of this intermediate population can contribute valuable information about the poorly understood final stages of stellar evolution.

Effects of initial condition spectral content on shock-driven turbulent mixing.

The mixing of materials due to the Richtmyer-Meshkov instability and the ensuing turbulent behavior is of intense interest in a variety of physical systems including inertial confinement fusion, combustion, and the final stages of stellar evolution. Extensive numerical and laboratory studies of shock-driven mixing have demonstrated the rich behavior associated with the onset of turbulence due to the shocks. Here we report on progress in understanding shock-driven mixing at interfaces between fluids of differing densities through three-dimensional (3D) numerical simulations using the rage code in the implicit large eddy simulation context. We consider a shock-tube configuration with a band of high density gas (SF(6)) embedded in low density gas (air). Shocks with a Mach number of 1.26 are passed through SF(6) bands, resulting in transition to turbulence driven by the Richtmyer-Meshkov instability. The system is followed as a rarefaction wave and a reflected secondary shock from the back wall pass through the SF(6) band. We apply a variety of initial perturbations to the interfaces between the two fluids in which the physical standard deviation, wave number range, and the spectral slope of the perturbations are held constant, but the number of modes initially present is varied. By thus decreasing the density of initial spectral modes of the interface, we find that we can achieve as much as 25% less total mixing at late times. This has potential direct implications for the treatment of initial conditions applied to material interfaces in both 3D and reduced dimensionality simulation models.

PRECURSORS PRIOR TO TYPE IIn SUPERNOVA EXPLOSIONS ARE COMMON: PRECURSOR RATES, PROPERTIES, AND CORRELATIONS

There is a growing number of supernovae (SNe), mainly of Type IIn, which present an outburst prior to their presumably final explosion. These precursors may affect the SN display, and are likely related to some poorly charted phenomena in the final stages of stellar evolution. Here we present a sample of 16 SNe IIn for which we have Palomar Transient Factory (PTF) observations obtained prior to the SN explosion. By coadding these images taken prior to the explosion in time bins, we search for precursor events. We find five Type IIn SNe that likely have at least one possible precursor event, three of which are reported here for the first time. For each SN we calculate the control time. Based on this analysis we find that precursor events among SNe IIn are common: at the one-sided 99% confidence level, more than 50% of SNe IIn have at least one pre-explosion outburst that is brighter than absolute magnitude -14, taking place up to 1/3 yr prior to the SN explosion. The average rate of such precursor events during the year prior to the SN explosion is likely larger than one per year, and fainter precursors are possibly even more common. We also find possible correlations between the integrated luminosity of the precursor, and the SN total radiated energy, peak luminosity, and rise time. These correlations are expected if the precursors are mass-ejection events, and the early-time light curve of these SNe is powered by interaction of the SN shock and ejecta with optically thick circumstellar material.

Component masses of young, wide, non-magnetic white dwarf binaries in the Sloan Digital Sky Survey Data Release 7

We present a spectroscopic component analysis of 18 candidate young, wide, non-magnetic, double-degenerate binaries identified from a search of the Sloan Digital Sky Survey Data Release 7 (DR7). All but two pairings are likely to be physical systems. We show SDSS J084952.47+471247.7 + SDSS J084952.87+471249.4 to be a wide DA+DB binary, only the second identified to date. Combining our measurements for the components of 16 new binaries with results for three similar, previously known systems within the DR7, we have constructed a mass distribution for the largest sample to date (38) of white dwarfs in young, wide, non-magnetic, double-degenerate pairings. This is broadly similar in form to that of the isolated field population with a substantial peak around M~0.6 Msun. We identify an excess of ultra-massive white dwarfs and attribute this to the primordial separation distribution of their progenitor systems peaking at relatively larger values and the greater expansion of their binary orbits during the final stages of stellar evolution. We exploit this mass distribution to probe the origins of unusual types of degenerates, confirming a mild preference for the progenitor systems of high-field-magnetic white dwarfs, at least within these binaries, to be associated with early-type stars. Additionally, we consider the 19 systems in the context of the stellar initial mass-final mass relation. None appear to be strongly discordant with current understanding of this relationship.

An edge-on translucent dust disk around the nearest AGB star, L2Puppis

As the nearest known AGB star (d=64pc) and one of the brightest (mK-2), L2 Pup is a particularly interesting benchmark object to monitor the final stages of stellar evolution. We report new lucky imaging observations of this star with the VLT/NACO adaptive optics system in twelve narrow band filters covering the 1.0-4.0 microns wavelength range. These diffraction limited images reveal an extended circumstellar dust lane in front of the star, that exhibits a high opacity in the J band and becomes translucent in the H and K bands. In the L band, extended thermal emission from the dust is detected. We reproduce these observations using Monte-Carlo radiative transfer modeling of a dust disk with the RADMC-3D code. We also present new interferometric observations with the VLTI/VINCI and MIDI instruments. We measure in the K band an upper limit to the limb-darkened angular diameter of theta_LD = 17.9 +/- 1.6 mas, converting to a maximum linear radius of R = 123 +/- 14 Rsun. Considering the geometry of the extended K band emission in the NACO images, this upper limit is probably close to the actual angular diameter of the star. The position of L2 Pup in the Herzsprung-Russell diagram indicates that this star has a mass around 2 Msun and is probably experiencing an early stage of the asymptotic giant branch. We do not detect any stellar companion of L2 Pup in our adaptive optics and interferometric observations, and we attribute its apparent astrometric wobble in the Hipparcos data to variable lighting effects on its circumstellar material. We however do not exclude the presence of a binary companion, as the large loop structure extending to more than 10 AU to the North-East of the disk in our L band images may be the result of interaction between the stellar wind of L2 Pup and a hidden secondary object. The geometric configuration that we propose, with a large dust disk seen almost edge-on, appears particularly favorable to test and develop our understanding of the formation of bipolar nebulae.

On the possibility of using seismic probes to study the core composition in pulsating white dwarfs

AbstractWhite dwarfs correspond to the final stages of stellar evolution of solar‐type stars. In these objects, production of energy by nuclear burning has ended which means that a white dwarf simply cools down over the course of the next billion years. It is now known that white dwarfs spend some of their cooling history in an instability strip. The pulsating white dwarfs with an hydrogen atmosphere (called DAV or ZZ Ceti stars) show non‐radial oscillation modes with periods in the range 100–1200 s. In this work we try to illustrate how the oscillation p‐mode frequencies of idealized white dwarf models change as the result of a different chemical composition in the core, with the ultimate goal of determining the chemical stratification from seismic observations. The presence of acoustic glitches in the internal structure results in a periodic signal in the frequencies. We find that this signal depends on the chemical stratification/composition of the core in a form that can be analytically modelled (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Puffing up early-type galaxies by baryonic mass loss: numerical experiments

Observations performed in the last few years indicate that most massive early-type galaxies (ETGs) observed at redshift z>1 exhibit sizes smaller by a factor of a few than local ETGs of analogous stellar mass. We present numerical simulations of the effect of baryonic mass loss on the structure of a spheroidal stellar system, embedded in a dark matter halo. This process, invoked as a possible explanation of the observed size increase of ETGs since z \sim 2, could be caused either by QSO/starburst driven galactic winds, promptly ejecting from Early Type Galaxies (ETGs) the residual gas and halting star formation (galactic winds), or by stellar mass returned to the ISM in the final stages of stellar evolution. Indeed, we find that a conceivable loss of about 50% of the baryonic mass can produce a significant size increase. However, the puffing up due to galactic winds occurs when the stellar populations are much younger than the estimated ages >0.5 Gyr of compact high-z ETGs. Therefore, while it may have had a role in deciding the final structure of ETGs, it cannot explain the evolution observed so far of their size-mass relation; its signature should be searched for in much younger systems. Conversely, the mass loss due to stellar evolution could cause a relatively modest expansion of passively evolving stellar systems later on, contributing to, without dominating, the observed evolution of their mass-size relationship.

The DODO survey - II. A Gemini direct imaging search for substellar and planetary mass companions around nearby equatorial and Northern hemisphere white dwarfs

The aim of the Degenerate Objects around Degenerate Objects (DODO) survey is to search for very low mass brown dwarfs and extrasolar planets in wide orbits around white dwarfs via direct imaging. The direct detection of such companions would allow the spectroscopic investigation of objects with temperatures much lower (< 500 K) than the coolest brown dwarfs currently observed. These ultra-low mass substellar objects would have spectral types > T8.5 and so could belong to the proposed Y dwarf spectral sequence. The detection of a planet around a white dwarf would prove that such objects can survive the final stages of stellar evolution and place constraints on the frequency of planetary systems around their progenitors (with masses between 1.5 - 8 solar masses, i.e., early B to mid F). This paper presents the results of a multi-epoch J band common proper motion survey of 23 nearby equatorial and northern hemisphere white dwarfs. We rule out the presence of any common proper motion companions, with limiting masses determined from the completeness limit of each observation, to 18 white dwarfs. For the remaining five targets, the motion of the white dwarf is not sufficiently separated from the non-moving background objects in each field. These targets require additional observations to conclusively rule out the presence of any common proper motion companions. From our completeness limits, we tentatively suggest that < 5% of white dwarfs have substellar companions with effective temperatures > 500 K between projected physical separations of 60 - 200 AU.

Chemical Yields from Supernovae and Hypernovae

AbstractWe review the final stages of stellar evolution, supernova properties, and chemical yields as a function of the progenitor's mass. (1) 8 - 10M⊙stars are super-AGB stars when the O+Ne+Mg core collapses due to electron capture. These AGB-supernovae may constitute an SN 2008S-like sub-class of Type IIn supernovae. These stars produce little α-elements and Fe-peak elements, but are important sources of Zn and light p-nuclei. (2) 10 - 90M⊙stars undergo Fe-core collapse. Nucleosynthesis in aspherical explosions is important, as it can well reproduce the abundance patterns observed in extremely metal-poor stars. (3) 90 - 140M⊙stars undergo pulsational nuclear instabilities at various nuclear burning stages, including O and Si-burning. (4) Very massive stars withM≳ 140M⊙either become pair-instability SNe, or undergo core-collapse to form intermediate mass black holes if the mass loss is small enough.

Mass loss from pulsating cool stars

AbstractIt has long been clear that most, if not all, of the mass loss experienced by stars from 0.8 to 8 solar masses occurs near the tip of the AGB and/or the RGB. Evolutionary studies have incorporated empirical mass loss laws but theoretical models suggest quite different dependence of mass loss rate on stellar parameters. We are combining evolutionary model calculations with ISUEVO with mass loss modeling using the Bowen code in a systematic study of final stages of stellar evolution. We mapped the RGB (without steady mass loss) to the “Death Zone” as a function of mixing length, mass, and metallicity. We compared these results with observation data from Origlia. We are investigating a possible mass loss mechanism through companions as a complement to mass loss through pulsation. By the end of the project we expect to provide a reliable prescription for AGB mass loss.