White Dwarf Mass Research Articles

Stability criterion for white dwarfs in Palatini f(R) gravity

Recent observations of several peculiar over- and under-luminous type Ia supernovae infer indirect evidence for the violation of the Chandrasekhar mass-limit by suggesting the existence of super- and sub-Chandrasekhar limiting mass white dwarfs. In an attempt to explain these phenomena in the context of general relativistic extensions, we study these objects in Palatini $f(R)$ gravity. We obtain the super- and sub-Chandrasekhar limiting masses as well as the dynamical instability criteria for white dwarfs in the given gravitational theory. We further demonstrate that the conventional positivity condition $\partial{M}/\partial{\rho_\text{c}}>0$ with $M$ being the WD's mass with central density $\rho_\text{c}$, is also a valid criterion for stability in Palatini gravity.

Revised Stellar Parameters for V471 Tau, A Post-common Envelope Binary in the Hyades

Abstract V471 Tau is a post-common-envelope binary consisting of an eclipsing DA white dwarf and a K-type main-sequence star in the Hyades star cluster. We analyzed publicly available photometry and spectroscopy of V471 Tau to revise the stellar and orbital parameters of the system. We used archival K2 photometry, archival Hubble Space Telescope spectroscopy, and published radial-velocity measurements of the K-type star. Employing Gaussian processes to fit for rotational modulation of the system flux by the main-sequence star, we recovered the transits of the white dwarf in front of the main-sequence star for the first time. The transits are shallower than would be expected from purely geometric occultations owing to gravitational microlensing during transit, which places an additional constraint on the white-dwarf mass. Our revised mass and radius for the main-sequence star is consistent with single-star evolutionary models given the age and metallicity of the Hyades. However, as noted previously in the literature, the white dwarf is too massive and too hot to be the result of single-star evolution given the age of the Hyades, and may be the product of a merger scenario. We independently estimate the conditions of the system at the time of common envelope that would result in the measured orbital parameters today.

Probing the non-thermal emission geometry of AR Sco via optical phase-resolved polarimetry

ABSTRACT AR Sco is a binary system that contains a white and red dwarf. The rotation rate of the white dwarf (WD) has been observed to slow down, analogous to rotation-powered radio pulsars; it has thus been dubbed a ‘white dwarf pulsar’. We previously fit the traditional radio pulsar rotating vector model to the linearly polarized optical data from this source, constraining the system geometry as well as the WD mass. Using a much more extensive data set, we now explore the application of the same model to binary phase-resolved optical polarimetric data, thought to be the result of non-thermal synchrotron radiation, and derive the magnetic inclination angle α and the observer angle ζ at different orbital phases. We obtain an ∼10° variation in α and ∼30° variation in ζ over the orbital period. The variation patterns in these two parameters is robust, regardless of the binning and epoch of data used. We speculate that the observer is detecting radiation from an asymmetric emission region that is a stable structure over several orbital periods. The success of this simple model lastly implies that the pitch angles of the particles are small and the pulsed, non-thermal emission originates relatively close to the WD surface.

Masses of White Dwarf Binary Companions to Type Ia Supernovae Measured from Runaway Velocities

Abstract The recently proposed “dynamically driven double-degenerate double-detonation” (D6) scenario posits that Type Ia supernovae (SNe) may occur during dynamically unstable mass transfer between two white dwarfs (WDs) in a binary. This scenario predicts that the donor WD may then survive the explosion and be released as a hypervelocity runaway, opening up the exciting possibility of identifying remnant stars from D6 SNe and using them to study the physics of detonations that produce Type Ia SNe. Three candidate D6 runaway objects have been identified in Gaia data. The observable runaway velocity of these remnant objects represents their orbital speed at the time of SN detonation. The orbital dynamics and Roche lobe geometry required in the D6 scenario place specific constraints on the radius and mass of the donor WD that becomes the hypervelocity runaway. In this Letter, we calculate the radii required for D6 donor WDs as a function of the runaway velocity. Using mass–radius relations for WDs, we then constrain the masses of the donor stars as well. With measured velocities for each of the three D6 candidate objects based on Gaia EDR3, this work provides a new probe of the masses and mass ratios in WD binary systems that produce SN detonations and hypervelocity runaways.

Mass limits of the extremely fast-spinning white dwarf CTCV J2056–3014

CTCV J2056–3014 is a nearby cataclysmic variable with an orbital period of approximately 1.76 h at a distance of about 853 light-years from the Earth. Its recently reported X-ray properties suggest that J2056–3014 is an unusual accretion-powered intermediate polar that harbors a fast-spinning white dwarf (WD) with a spin period of 29.6 s. The low X-ray luminosity and the relatively modest accretion rate per unit area suggest that the shock is not occurring near the WD surface. It has been argued that, under these conditions, the maximum temperature of the shock cannot be directly used to determine the mass of the WD (which, under the abovementioned assumptions, would be around 0.46 M⊙). Here, we explore the stability of this rapidly rotating WD using a modern equation of state (EoS) that accounts for electron–ion, electron–electron, and ion–ion interactions. For this EoS, we determine the mass density thresholds for the onset of pycnonuclear fusion reactions and study the impact of microscopic stability and rapid rotation on the structure and stability of WDs, considering them with helium, carbon, oxygen, and neon. From this analysis, we obtain a minimum mass for CTCV J2056–3014 of 0.56 M⊙ and a maximum mass of around 1.38 M⊙. If the mass of CTCV J2056–3014 is close to the lower mass limit, its equatorial radius would be on the order of 104 km due to rapid rotation. Such a radius is significantly larger than that of a nonrotating WD of average mass (0.6 M⊙), which is on the order of 7 × 103 km. The effects on the minimum mass of J2056–3014 due to changes in the temperature and composition of the stellar matter were found to be negligibly small.

Metallicity-dependent nucleosynthetic yields of Type Ia supernovae originating from double detonations of sub-MCh white dwarfs

Double detonations in sub-Chandrasekhar mass carbon-oxygen white dwarfs (WD) with helium shells ares potential explosion mechanisms for Type Ia supernovae. The mechanism consists of a shell detonation and subsequent core detonation. The focus of our study is the effect of the progenitor metallicity on the nucleosynthetic yields. For this, we computed and analyzed a set of 11 different models with varying core and shell masses at four different metallicities each. This results in a total of 44 models at metallicities between 0.01 Z⊙ and 3 Z⊙. Our models show a strong impact of the metallicity in the high-density regime. The presence of 22Ne causes a neutron-excess that shifts the production from 56Ni to stable isotopes such as 54Fe and 58Ni in the α-rich freeze-out regime. The isotopes of the metallicity implementation further serve as seed nuclei for additional reactions in the shell detonation. The production of 55Mn increases with metallicity, confirming the results of previous work. A comparison of elemental ratios relative to iron shows a good match to solar values for some models. Super-solar values are reached for Mn at 3 Z⊙ and solar values in some models at Z⊙. This indicates that the required contribution of Type Ia supernovae originating from Chandrasekhar-mass WDs can be lower than estimated in previous work to reach solar values of [Mn/Fe] at [Fe/H] = 0. Our galactic chemical evolution models suggest that Type Ia supernovae from sub-Chandrasekhar mass white dwarfs, along with core-collapse supernovae, could account for more than 80% of the solar Mn abundance. Using metallicity-dependent Type Ia supernova yields helps to reproduce the upward trend of [Mn/Fe] as a function of metallicity for the solar neighborhood. These chemical evolution predictions, however, depend on the massive star yields adopted in the calculations.

All-Order Full-Coulomb Quantum Spectral Line-Shape Calculations

Understanding how atoms interact with hot dense matter is essential for astrophysical and laboratory plasmas. Interactions in high-density plasmas broaden spectral lines, providing a rare window into interactions that govern, for example, radiation transport in stars. However, up to now, spectral line-shape theories employed at least one of three common approximations: second-order Taylor treatment of broadening operator, dipole-only interactions between atom and plasma, and classical treatment of perturbing electrons. In this Letter, we remove all three approximations simultaneously for the first time and test the importance for two applications: neutral hydrogen and highly ionized magnesium and oxygen. We found 15%-50% change in the spectral line widths, which are sufficient to impact applications including white-dwarf mass determination, stellar-opacity research, and laboratory plasma diagnostics.

Constraining the evolution of cataclysmic variables via the masses and accretion rates of their underlying white dwarfs

ABSTRACT We report on the masses (MWD), effective temperatures ($\rm{T_\mathrm{eff}}$), and secular mean accretion rates ($\langle \dot{M} \rangle$) of 43 cataclysmic variable (CV) white dwarfs, 42 of which were obtained from the combined analysis of their Hubble Space Telescope ultraviolet data with the parallaxes provided by the Early Third Data Release of the Gaia space mission, and one from the white dwarf gravitational redshift. Our results double the number of CV white dwarfs with an accurate mass measurement, bringing the total census to 89 systems. From the study of the mass distribution, we derive $\langle M_\mathrm{WD} \rangle = 0.81^{+0.16}_{-0.20}\, \mathrm{M_\odot }$, in perfect agreement with previous results, and find no evidence of any evolution of the mass with orbital period. Moreover, we identify five systems with MWD < 0.5 M⊙, which are most likely representative of helium-core white dwarfs, showing that these CVs are present in the overall population. We reveal the presence of an anticorrelation between the average accretion rates and the white dwarf masses for the systems below the $2\!-\!3\,$ h period gap. Since $\langle \dot{M} \rangle$ reflects the rate of system angular momentum loss, this correlation suggests the presence of an additional mechanism of angular momentum loss that is more efficient at low white dwarf masses. This is the fundamental concept of the recently proposed empirical prescription of consequential angular momentum loss (eCAML) and our results provide observational support for it, although we also highlight how its current recipe needs to be refined to better reproduce the observed scatter in $\rm{T_\mathrm{eff}}$ and $\langle \dot{M} \rangle$, and the presence of helium-core white dwarfs.

White Dwarfs in a Uniform Sphere Approximation, with General Relativity Effects Taken into Account

The limiting mass of cold white dwarfs was first calculated by E. Stoner in an approximate model of a uniform star and was soon reduced by ~20% in papers by S. Chandrasekhar and L. D. Landau based on an exact solution of the equations for the stellar equilibrium. Here we examine uniform models of white dwarfs taking general relativity effects and the influence of finite temperature into account. Solutions are obtained in the form of finite analytic formulas and, for masses differing by no more than ~20% from the exact solutions, found by numerical integration of the differential equations for the stellar equilibrium.

The Structure and Stability of Massive Hot White Dwarfs

Abstract We investigate the structure and stability against radial oscillations, pycnonuclear reactions, and inverse β-decay of hot white dwarfs. We consider the fluid matter to be made up of nucleons and electrons confined in a Wigner–Seitz cell surrounded by free photons. It is considered that the temperature depends on the mass density considering the presence of an isothermal core. We find that the temperature produces remarkable effects on the equilibrium and radial stability of white dwarfs. The stable equilibrium configuration results are compared with those for white dwarfs estimated from the Extreme Ultraviolet Explorer survey and the Sloan Digital Sky Survey. We derive masses, radii, and central temperatures for the most massive white dwarfs according to the surface gravity and effective temperature reported by the surveys. We note that these massive stars are in the mass region where general relativity effects are important. These stars are near the threshold of instabilities due to radial oscillations, pycnonuclear reactions, and inverse β-decay. Regarding the radial stability of these stars as a function of the temperature, we find that it decreases with the increment of central temperature. We also find that the maximum-mass point and the zero eigenfrequencies of the fundamental mode are determined at the same central energy density. Regarding low-temperature stars, pycnonuclear reactions occur in similar central energy densities, and the central energy density threshold for inverse β-decay is not modified. For T c ≥ 1.0 × 108 [K], the onset of radial instability is attained before pycnonuclear reaction and inverse β-decay.

Multidimensional Radiative Transfer Calculations of Double Detonations of Sub-Chandrasekhar-mass White Dwarfs

Abstract 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 (WD), 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 WDs 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 multidimensional radiative transfer calculations of thin-shell double-detonation models, exploring a range of WD and helium-shell masses. We find broad agreement with the observed light curves and spectra of nonpeculiar Type Ia supernovae, from subluminous to overluminous subtypes, providing evidence that double detonations of sub-Chandrasekhar-mass WDs produce the bulk of observed Type Ia supernovae. Some discrepancies in spectral velocities and colors persist, but these may be brought into agreement by future calculations that include more accurate initial conditions and radiation transport physics.

The impact of metallicity on nova populations

ABSTRACT The metallicity of a star affects its evolution in a variety of ways, changing stellar radii, luminosities, lifetimes, and remnant properties. In this work, we use the population synthesis code binary_c to study how metallicity affects novae in the context of binary stellar evolution. We compute a 16-point grid of metallicities ranging from Z = 10−4 to 0.03, presenting distributions of nova white dwarf masses, accretion rates, delay-times, and initial system properties at the two extremes of our 16-point metallicity grid. We find a clear anticorrelation between metallicity and the number of novae produced, with the number of novae at Z = 0.03 roughly half that at Z = 10−4. The white dwarf mass distribution has a strong systematic variation with metallicity, while the shape of the accretion rate distribution is relatively insensitive. We compute a current nova rate of approximately 33 novae per year for the Milky Way, a result consistent with observational estimates relying on extra-Galactic novae but an under-prediction relative to observational estimates relying on Galactic novae. However, the shape of our predicted Galactic white dwarf mass distribution differs significantly to existing observationally derived distributions, likely due to our underlying physical assumptions. In M31, we compute a current nova rate of approximately 36 novae per year, under-predicting the most recent observational estimate of $65^{+15}_{-16}$. Finally, we conclude that when making predictions about currently observable nova rates in spiral galaxies, or stellar environments where star formation has ceased in the distant past, metallicity can likely be considered of secondary importance compared to uncertainties in binary stellar evolution.

AT 2019qyl in NGC 300: Internal Collisions in the Early Outflow from a Very Fast Nova in a Symbiotic Binary* †

Abstract Nova eruptions, thermonuclear explosions on the surfaces of white dwarfs (WDs), are now recognized to be among the most common shock-powered astrophysical transients. We present the early discovery and rapid ultraviolet (UV), optical, and infrared (IR) temporal development of AT 2019qyl, a recent nova in the nearby Sculptor Group galaxy NGC 300. The light curve shows a rapid rise lasting ≲1 day, reaching a peak absolute magnitude of M V = −9.2 mag and a very fast decline, fading by 2 mag over 3.5 days. A steep dropoff in the light curves after 71 days and the rapid decline timescale suggest a low-mass ejection from a massive WD with M WD ≳ 1.2 M ⊙. We present an unprecedented view of the early spectroscopic evolution of such an event. Three spectra prior to the peak reveal a complex, multicomponent outflow giving rise to internal collisions and shocks in the ejecta of an He/N-class nova. We identify a coincident IR-variable counterpart in the extensive preeruption coverage of the transient location and infer the presence of a symbiotic progenitor system with an O-rich asymptotic-giant-branch donor star, as well as evidence for an earlier UV-bright outburst in 2014. We suggest that AT 2019qyl is analogous to the subset of Galactic recurrent novae with red-giant companions such as RS Oph and other embedded nova systems like V407 Cyg. Our observations provide new evidence that internal shocks between multiple, distinct outflow components likely contribute to the generation of the shock-powered emission from such systems.

Breaking the Degeneracy in Magnetic Cataclysmic Variable X-Ray Spectral Modeling Using X-Ray Light Curves

Abstract We present an analysis of mock X-ray spectra and light curves of magnetic cataclysmic variables using an upgraded version of the 3D cyclops code. This 3D representation of the accretion flow allows us to properly model total and partial occultation of the postshock region by the white dwarf as well as the modulation of the X-ray light curves due to the phase-dependent extinction of the preshock region. We carried out detailed postshock region modeling in a four-dimensional parameter space by varying the white dwarf mass and magnetic field strength as well as the magnetosphere radius and the specific accretion rate. To calculate the postshock region temperature and density profiles, we assumed equipartition between ions and electrons; took into account the white dwarf gravitational potential, the finite size of the magnetosphere, and a dipole-like magnetic field geometry; and considered cooling by both bremsstrahlung and cyclotron radiative processes. By investigating the impact of the parameters on the resulting X-ray continuum spectra, we show that there is an inevitable degeneracy in the four-dimensional parameter space investigated here, which compromises X-ray continuum spectral fitting strategies and can lead to incorrect parameter estimates. However, the inclusion of X-ray light curves in different energy ranges can break this degeneracy, and it therefore remains, in principle, possible to use X-ray data to derive fundamental parameters of magnetic cataclysmic variables, which represents an essential step toward understanding their formation and evolution.

The Initial–Final Mass Relation for Hydrogen-deficient White Dwarfs*

Abstract The initial–final mass relation represents the total mass lost by a star during the entirety of its evolution from the zero age main sequence to the white-dwarf cooling track. The semiempirical initial–final mass relation (IFMR) is largely based on observations of DA white dwarfs, the most common spectral type of white dwarf and the simplest atmosphere to model. We present a first derivation of the semiempirical IFMR for hydrogen-deficient (non-DA) white dwarfs in open star clusters. We identify a possible discrepancy between the DA and non-DA IFMRs, with non-DA white dwarfs ≈0.07 M ⊙ less massive at a given initial mass. Such a discrepancy is unexpected based on theoretical models of non-DA formation and observations of field white dwarf mass distributions. If real, the discrepancy is likely due to enhanced mass loss during the final thermal pulse and renewed post-AGB evolution of the star. However, we are dubious that the mass discrepancy is physical and instead is due to the small sample size, to systematic issues in model atmospheres of non-DAs, and to the uncertain evolutionary history of Procyon B (spectral type DQZ). A significantly larger sample size is needed to test these assertions. In addition, we also present Monte Carlo models of the correlated errors for DA and non-DA white dwarfs in the initial–final mass plane. We find the uncertainties in initial–final mass determinations for individual white dwarfs can be significantly asymmetric, but the recovered functional form of the IFMR is grossly unaffected by the correlated errors.

Resolving dichotomy in compact objects through continuous gravitational waves observation

ABSTRACT More than two dozen soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs) have been detected so far. These are isolated compact objects. Many of them are either found to be associated with supernova remnants or their surface magnetic fields are directly measured, confirming that they are neutron stars (NSs). However, it has been argued that some SGRs and AXPs are highly magnetized white dwarfs (WDs). Meanwhile, the existence of super-Chandrasekhar WDs has remained to be a puzzle. However, not even a single such massive WD has been observed directly. Moreover, some WD pulsars are detected in electromagnetic surveys and some of their masses are still not confirmed. Here, we calculate the signal-to-noise ratio for all these objects, considering different magnetic field configurations and thereby estimate the required time for their detection by various gravitational wave (GW) detectors. For SGRs and AXPs, we show that, if these are NSs, they can hardly be detected by any of the GW detectors, while if they are WDs, big bang Observer (BBO), DECi-hertz Interferometer Gravitational wave Observatory (DECIGO) and Advanced Laser Interferometer Antenna (ALIA) would be able to detect them within a few days to a year of integration, depending on the magnetic field strength and its configuration. Similarly, if a super-Chandrasekhar WD has a dominant toroidal field, we show that even Laser Interferometer Space Antenna (LISA) and TianQin would be able to detect it within one year of integration. We also discuss how GWs can confirm the masses of the WD pulsars.

Birth of the ELMs: a ZTF survey for evolved cataclysmic variables turning into extremely low-mass white dwarfs

ABSTRACT We present a systematic survey for mass-transferring and recently detached cataclysmic variables (CVs) with evolved secondaries, which are progenitors of extremely low mass white dwarfs (ELM WDs), AM CVn systems, and detached ultracompact binaries. We select targets below the main sequence in the Gaia colour–magnitude diagram with ZTF light curves showing large-amplitude ellipsoidal variability and orbital period Porb < 6 h. This yields 51 candidates brighter than G = 18, of which we have obtained many-epoch spectra for 21. We confirm all 21 to be completely or nearly Roche lobe filling close binaries. Thirteen show evidence of ongoing mass transfer, which has likely just ceased in the other eight. Most of the secondaries are hotter than any previously known CV donors, with temperatures 4700 < Teff/K < 8000. Remarkably, all secondaries with $T_{\rm eff} \gtrsim 7000\, \rm K$ appear to be detached, while all cooler secondaries are still mass-transferring. This transition likely marks the temperature where magnetic braking becomes inefficient due to loss of the donor’s convective envelope. Most of the proto-WD secondaries have masses near $0.15\, \rm M_{\odot }$; their companions have masses near $0.8\, \rm M_{\odot }$. We infer a space density of $\sim 60\, \rm kpc^{-3}$, roughly 80 times lower than that of normal CVs and three times lower than that of ELM WDs. The implied Galactic birth rate, $\mathcal {R}\sim 60\, \rm Myr^{-1}$, is half that of AM CVn binaries. Most systems are well-described by mesa models for CVs in which mass transfer begins only as the donor leaves the main sequence. All are predicted to reach minimum periods 5 ≲ Porb min−1 ≲ 30 within a Hubble time, where they will become AM CVn binaries or merge. This sample triples the known evolved CV population and offers broad opportunities for improving understanding of the compact binary population.

The intermediate polar cataclysmic variable GK Persei 120 years after the nova explosion: a first dynamical mass study

ABSTRACT We present a dynamical study of the intermediate polar and dwarf nova cataclysmic variable GK Persei (Nova Persei 1901) based on a multisite optical spectroscopy and R-band photometry campaign. The radial velocity curve of the evolved donor star has a semi-amplitude $K_2=126.4 \pm 0.9 \, \mathrm{km}\, \mathrm{s}^{-1}$ and an orbital period $P=1.996872 \pm 0.000009 \, \mathrm{d}$. We refine the projected rotational velocity of the donor star to $v_\mathrm{rot} \sin i = 52 \pm 2 \, \mathrm{km}\, \mathrm{s}^{-1}$ that, together with K2, provides a donor star to white dwarf mass ratio q = M2/M1 = 0.38 ± 0.03. We also determine the orbital inclination of the system by modelling the phase-folded ellipsoidal light curve and obtain i = 67° ± 5°. The resulting dynamical masses are $M_{1}=1.03^{+0.16}_{-0.11} \, \mathrm{M}_{\odot }$ and $M_2 = 0.39^{+0.07}_{-0.06} \, \mathrm{M}_{\odot }$ at 68 per cent confidence level. The white dwarf dynamical mass is compared with estimates obtained by modelling the decline light curve of the 1901 nova event and X-ray spectroscopy. The best matching mass estimates come from the nova light curve models and an X-ray data analysis that uses the ratio between the Alfvén radius in quiescence and during dwarf nova outburst.

Super-Chandrasekhar limiting mass white dwarfs as emergent phenomena of noncommutative squashed fuzzy spheres

The indirect evidence for at least a dozen massive white dwarfs (WDs) violating the Chandrasekhar mass limit is considered to be one of the wonderful discoveries in astronomy for more than a decade. Researchers have already proposed a diverse amount of models to explain this astounding phenomenon. However, each of these models always carries some drawbacks. On the other hand, noncommutative geometry is one of the best replicas of quantum gravity, which is yet to be proved from observations. Madore introduced the idea of a fuzzy sphere to describe a formalism of noncommutative geometry. This paper shows that the idea of a squashed fuzzy sphere can self-consistently explain the super-Chandrasekhar limiting mass WDs. We further show that the length scale beyond which the noncommutativity is prominent is an emergent phenomenon, and there is no prerequisite for an ad hoc length scale.

Spatial distribution of the X-ray-emitting plasma of SS Cygni in quiescence and outburst

Abstract We present our analysis of the Suzaku data of SS Cygni (SS Cyg) from 2005 both in quiescence and outburst. A fluorescent iron Kα line bears significant information about the geometry of an X-ray-emitting hot plasma and a cold reflector, such as the surfaces of the white dwarf (WD) and the accretion disk (AD). Our reflection simulation has revealed that the X-ray-emitting hot plasma is located either very close to the WD surface in the boundary layer (BL), with an upper limit radial position of <1.004 times the white dwarf radius (RWD), or near the entrance of the BL where the optically thick AD is truncated at a distance of 1.14–1.27 RWD for the assumed WD mass of 1.19 M⊙ in quiescence. In the latter configuration, the plasma torus is located just above the inner edge of the AD. The result suggests that the accreting matter is heated up close to the maximum temperature immediately after the matter enters the BL. The matter probably expands precipitously at the entrance of the BL and leaves the disk plane to reach a height comparable to the radial distance of the plasma torus from the center of the WD. In outburst, on the other hand, our spectral analysis favors the picture that the optically thick disk reaches the WD surface. In addition, the plasma distributes above the disk like coronae, as suggested by a previous study, and the 90% upper limit of the coronae radial position is 1.2 RWD.