Zhuo-Li Yu,Xiao-Jie Xu,and Xiang-Dong Li

School of Astronomy and Space Science and Key Laboratory of Modern Astronomy and Astrophysics,Nanjing University,Nanjing,210093,China;xuxj@nju.edu.cn

Abstract The distribution of the mass of white dwarfs(WDs)is one of the fundamental questions in the field of cataclysmic variables (CVs).In this work,we make a systematical investigation on the WD masses in two subclass of CVs:intermediate polars (IPs) and non-magnetic CVs in the solar vicinity based on the flux ratios of Fe XXVI–Lyα to Fe XXV–Heα emission lines (I7.0/I6.7) from archival XMM-Newton and Suzaku observations.We first verify the(semi-empirical) relations between I7.0/I6.7,the maximum emission temperature (Tmax) and the WD mass (MWD)with the mkcflow model based on the apec description and the latest AtomDB.We then introduce a new spectral model to measure MWD directly based on the above relations.A comparison shows that the derived MWD is consistent with dynamically measured ones.Finally,we obtain the average WD masses of 58 CVs (including 36 IPs and 22 non-magnetic CVs),which is the largest X-ray selected sample.The average WD masses are〈MWD,IP〉=0.81±0.21 M⊙a(bǔ)nd 〈M WD,DN〉 = 0.81 ±0.21M⊙for IPs and non-magnetic CVs,respectively.Theseresults are consistent with previous works.

Key words: (stars:) binaries (including multiple):close–(stars:) novae–cataclysmic variables–X-rays:binaries

1.Introduction

A cataclysmic variable(CV)is a semi-detached binary where a white dwarf (WD) accretes gas from its main sequence or sub-giant companion star via Roche-lobe overflow.Subclasses of CVs include magnetic (mCVs,including polars and intermediate polars) and non-magnetic (non-mCVs,mostly dwarf novae)ones on terms of the magnetic field of WDs.In an intermediate polar (IP),the magnetic field is strong enough to truncate the accretion disk at a certain radius,and channel the accreted gas onto the magnetic poles of the WD along the magnetic lines.A standing shock is then formed above the surface of the WD.The post-shock gas is ionized and emits X-ray photons.For dwarf novae (DNe),the X-ray emission is supposed to be mainly from a boundary layer between the accretion disk and the surface of WD.The observed X-ray luminosities of IPs and DNe in quiescence are between 1030–34erg s-1.Their X-ray spectra can both be well described with an isobaric cooling flow model (mkcflow,e.g.,see Mushotzky&Szymkowiak1988),with a Gaussian component to describe the fluorescent Fe I–Kα line,and sometimes an additional partial absorption component which is suggested to originate from the un-shocked gas above the accretion column or in the accretion curtain (for a recent review of X-ray emission of CVs,see Mukai2017).

The mass distribution of WDs in CVs is important not only for the theory of binary star evolution,but also for other interesting astrophysical objects.For example,massive WDs are closely related to the progenitors of Type Ia supernovae,which are supposed to be WDs reaching or near the Chandrasekhar mass limit.Based on the Ritter &Kolb(2003)'s CV catalog,Zorotovic et al.(2011) obtained a meanMWD=0.83±0.23M⊙for solar vicinity CVs,which is～0.2–0.3M⊙higher than the mean WD masses in single WDs (e.g.,Kepler et al.2007) and pre-CVs (Zorotovic et al.2011).Moreover,the mean WD masses in CVs in the Galactic bulge and Galactic center were determined as～0.8M⊙(Yu et al.2018)and～0.9M⊙(Hailey et al.2016),which are again～0.2–0.3M⊙higher than single WDs and pre-CVs.The physical scenario responsible for the differences has not been fully understood (e.g.,Knigge2006;Knigge et al.2011).

The WD mass distribution in CVs is worth revisiting.First,the CV sample in Zorotovic et al.(2011) was directly taken from Ritter &Kolb(2003)'s catalog,in which the WD masses were measured with various methods by various authors,so the reliability of the measured masses might be a problem.Second,Zorotovic et al.(2011) had to manually select a sample of 32“fiducial” CVs (those with high quality measurements) from a whole sample of 104 sources,thus the sample may include some bias.With X-ray spectroscopy,it is now possible to make a systematic survey on WD masses in CVs in the solar vicinity.The results would surely provide helpful clues to improve our understanding on this topic.

Traditionally,the WD mass in a CV is measured by the dynamical method (e.g.,eclipse light curves,radial velocity curves,etc),but the results are suffered from uncertainties,e.g.,inclination angles.Since the last decades,X-ray spectroscopy has provided an alternative way to measure WD masses in IPs with theTmax–MWDrelation.In an IP,the WD mass is related toTmaxin strong shock condition (assuming the accreted gas falls from infinity) with the following equation (Frank et al.2002):

where μ,mH,k,G,MWDandRWDare the mean molecular weight,the mass of H atom,the Boltzmann constant,the gravitational constant,the WD mass and the WD radius,respectively.Combining Equation(1)withTmaxmeasured from the hard(up to 30–50 keV)X-ray continua,and theMWD–RWDrelation(e.g.,Nauenberg1972),various authors have measuredMWDin IPs in the solar vicinity (e.g.,Yuasa et al.2010;Shaw et al.2018;Yu et al.2018;Suleimanov et al.2019;Shaw et al.2020).Yuasa et al.(2010) and Bernardini et al.(2012) further derived the meanMWDvalues in IPs to be 0.88±0.25M⊙a(bǔ)nd 0.86±0.07M⊙,respectively.Similarly,Suleimanov et al.(2019)derived an averageMWD=0.79±0.16M⊙for a sample of 35 IPs observed by NuSTAR and Swift/BAT.

For DNe,there is currently no widely accepted theory on the physics of boundary layer.The accreted gas may be heated either by a strong shock or a series of weak shocks in the boundary layer (Frank et al.2002).Thus,there is no welldefined equations like Equation (1).Recently,Yu et al.(2018)obtained a semi-empirical relation betweenTmaxandMWDfor DNe from X-ray observations of solar vicinity DNe:

where α=0.646±0.069 (for comparison,α=1 under the strong shock assumption).

The hard X-ray continuum method requires spectra with good counting statistics up to 30–50 keV to obtain reliable measurements ofTmaxandMWD.However,the small effective area and/or the high background level of current X-ray telescopes usually lead to low quality hard X-ray spectra.Moreover,the complex,un-modeled intrinsic absorption found in some IPs,and the existence of the X-ray reflection(Mukai2017;Shaw et al.2018,2020) may also lead to deviatedTmax,thus biased WD masses.

The flux ratio of Fe XXVI–Lyα to Fe XXV–Heα lines(I7.0/I6.7) has been suggested as a good diagnostic forTmax,and thusMWDin CVs (e.g.,Ezuka &Ishida1999;Xu et al.2019b).The basic idea is that more helium-like iron ions will be ionized to hydrogen-like ones in higher plasma temperatures,resulting to higher iron flux ratios.The advantage of this line ratio method is that present X-ray telescopes like XMMNewton and Chandra (and Suzaku which stopped working in 2015) have better energy resolution and larger effective area near the iron line (6–7 keV) compared to 30–50 keV hard X-ray energy ranges,enabling reliableI7.0/I6.7measurements.Additionally,instruments which are sensitive in this energy range include XMM-Newton and Chandra,which have good angular resolution,so that individual sources in the Globular cluster and toward the Galactic bulge/center direction could be resolved and investigated (e.g.,Zhu et al.2018;Xu et al.2019a).Moreover,the flux ratio of Fe lines is less affected by intrinsic absorption and reflections.Early work by Ezuka &Ishida (1999) measured theI7.0/I6.7values of solar vicinity CVs to deriveMWDin IPs based on ASCA observations.Recently,Xu et al.(2016) and Yu et al.(2018) suggested the Tmax–I7.0I6.7–MWDrelations for IPs and DNe based on Suzaku observations of solar vicinity CVs,and obtained a mean WD mass of 0.81±0.07M⊙for CVs in the Galactic Bulge.Xu et al.(2019b) further suggested thatI7.0/I6.7can be used as a good diagnostic of WD mass in IPs and DNe based on Suzaku and NuSTAR observations,and suggested the existence of massive (～1.0–1.2M⊙) WDs in CVs in the Galactic center region (Xu et al.2019a).

It is now possible to derive the mass of WDs in CVs,especially those in non-magnetic ones in the solar vicinity based on the Tmax–I7.0I6.7–MWDrelations.Before that,a thorough examination on these relations with the new atomic database must be made,because the results in previous works were based on the cooling flow model(mkcflow in Xspec)with the older mekal (and thus the old atomic database) emission description.Moreover,the Tmax–I7.0I6.7–MWDrelations could be built into the mkcflow model,so that the spectral fitting can outputMWDdirectly,and save the trouble of comparing the fittedTmaxorI7.0/I6.7with theTmax–MWDandI7.0/I6.7–MWDcurves to deriveMWD.

In this work,we utilize the archival XMM-Newton and Suzaku observations of 58 individual CVs,including 36 IPs and 22 non-mCVs in the solar vicinity to investigate their WD masses.We start by examining theI7.0/I6.7–MWDrelations with the cooling flow model with apec emission description based on the AtomDB.We then introduce a new spectral model with built-in Tmax–I7.0I6.7–MWDrelations to directly output WD masses.We further assessMWDfrom both XMM-Newton and Suzaku observations and obtain the meanMWDof the sampled CVs.Additionally,mCVs and non-mCVs follow the similar distribution of WD mass in the standard CV evolutionary model(Zorotovic&Schreiber2020),but the formation of high magnetic field WDs were suggested to be related to the common envelope evolution (e.g.,Briggs et al.2018),which may lead to a different WD masses between mCVs and nonmCVs.In this work,we will explore the mean WD mass of IPs and non-mCVs and check whether they are consistent with each other.

The rest of this paper is organized as follows.In Section2,we introduce the sample selection and data preparation,and measure theirI7.0/I6.7.In Section3,we update theTmax–MWDrelation of DNe and introduce a new spectral model to measure the WD masses,with which theMWDof sampled CVs are derived.We make a brief discussion in Section4and summarize in Section5.All results measured in this work are shown at 90% confidence level.On the other hand,the dynamically measured masses in CVs are directly taken from the references with 68% confidence level.

2.Observations and Data Analysis

XMM-Newton is chosen as the main instrument in this work because it provides the most observations of CVs in the solar vicinity compared to other X-ray instruments.The XMMNewton observatory contains three X-ray instruments and one Optical Monitor to provide simultaneous X-ray and optical/UV observations.The three X-ray instruments are:European Photon Imaging Camera Metal-Oxide-Silicon (EPIC-MOS),European Photon Imaging Camera-PN (EPIC-PN) and Reflection Grating Spectrometer.EPIC-MOS (MOS1,MOS2) and EPIC-PN provide relatively good spectral resolutions(E/dE～50) at 6.5 keV,which are suitable to measure theI7.0/I6.7.

We start by searching the XMM-Newton archive for observations of CVs in Ritter &Kolb (2003)'s catalog (Final edition 7.24) and of some IPs recently discovered (de Martino et al.2020) within 5′ off-axis angles,and obtain 419 observations on 247 CVs.As the next step,CVs whoseI7.0/I6.7could not be well constrained (see the next paragraph for details) are excluded from the sample (e.g.,AB Dra,TY PsA;some other sources without enough net counts are also excluded due to low Fe abundance,e.g.,V2731 Oph),which results in a sample of 113 observations on 83 CVs including 20 DNe,36 IPs,18 Polars and nine nova-likes.Then we remove CVs in non-quiescent states1The states in most observations are determined using the American Association of Variable Star Observers(AAVSO)International Database.If no data was found in AAVSO,the states are inferred from the light curves from multiple observations of the same source.from the sample,and exclude polars and nova-likes (which may have differentI7.0/I6.7–MWDrelations).Furthermore,EX Hya and GK Per have extremely low magnetospheric radii (Suleimanov et al.2016,2019),which cause lower shock temperatures and lead to lower derived WD masses,so they are removed from the sample.Finally,we get a sample of 48 CVs from XMMNewton,as listed in Table1.We also include a sample of 26 quiescent IPs and DNe observed by Suzaku from Xu et al.(2019b) and Yu et al.(2018),where sources with weak FeXXV-Heα and FeXXVI-Lyα lines are removed.The Suzaku CV sample is listed in Table2.

All observations on sampled CVs are then reprocessed with Science Analysis System (SAS,v16.1.0) software with the latest calibration files.Good time intervals are chosen by removing flares at the energy of ＞10 keV,which are decided by critical values which vary for different observations.The typical critical values are 0.35 cts s-1and 0.8 cts s-1for MOS and PN chips,respectively.For most observations,source events are extracted from a 40″ circular region centered at the source,and backgrounds from a circular,source-free region of the same size on the same chip,respectively.Specifically,the source and background region radii are reduced to 20″–30″ if the MOS CCDs were operated in the Small Window mode or there are contaminated sources.If potential pile-up occurs2Procedure of testing pile-up is from https://www.cosmos.esa.int/web/xmm-newton/sas-thread-epatplot.source counts will be extracted from an annulus with typical inner radius of 5″.The spectra are then regrouped to ensure a signal-to-noise ratio of three per bin at least.

We then measure theI7.0/I6.7of individual CVs by jointly fitting the 5–8 keV background-subtracted spectra from MOS1,MOS2 and PN detectors.The fitting is performed with the model phabs(apec+threeGaussian)in Xspec 12.10.1,where the abundance of apec is set to 0.In this model,the apec component represents the X-ray continuum of the CVs,3We use apec instead of mekal here because the latter has been used in Xu et al.(2016).We also tried to use mkcflow or bremsstrahlung for continuum and find the differences of the resulting I7.0/I6.7 are within 5%.and the threeGaussian model was built specifically to measure theI7.0/I6.7(Xu et al.2016).We fix all line width values to 1.0×10-5keV following Xu et al.(2016),since the spectral resolution is not enough to constrain them.

The fitting results are shown in Table3,where theI7.0/I6.7from Suzaku observations by Xu et al.(2016) and Xu et al.(2019b)are listed for comparison.Two examples of the XMMNewton fitting are plotted in Figure1.

From Tables1and2,16 CVs (including 11 IPs and five DNe) have both XMM-Newton and Suzaku observations.By merging the two samples,we have a final sample of 58 individual CVs,including 36 IPs and 22 DNe.

3.WD Mass Derivation

3.1.Updating the Tmax–MWD Relation for DNe

The previous semi-empiricalTmax–MWDrelation (i.e.,Equation (2)) for DNe was obtained based onTmaxmeasurements with Suzaku observations.ThoseTmaxvalues could be biased due to the limited counting statistics above 10 keV caused by the high background level of the Suzaku HXD detector.With the recently available NuSTAR observations,it is now possible to update theTmax,and thus theTmax–MWDrelation for DNe.Additionally,in previous works (e.g.,Yu et al.2018;Xu et al.2019b),Tmaxwere measured with the mkcflow model with the mekal description with the old atomic database(parameter“switch”set to 1),thus the measuredTmaxmay be different from the ones when using the apec description with AtomDB (parameter “switch” set to 2,where the latest AtomDB is incorporated).We then fit the spectra of Suzaku and NuSTAR observed CVs again by switching the emission description to apec,and summarize the measuredTmaxinTable4,whereTmaxfrom previous works are also listed for comparison.We further re f i t the Tmax–MWDrelation in Equation (2) with the newTmaxusing the orthogonal distance regression (ODR) method,where he mean molecular weight μ is f i xed at 0.6 (e.g.,Byckling et al.2010) andRWDis derived from WD’sMWD–RWDrelation (Nauenberg1972).The f i tting yields anα=0.69±0.06(shown in Figure2)with=0.66,which isconsistent with previous one(α=0.646±0.069),and will be used in the rest of the paper.

Table 2 Observation Log and Dynamically Measured WD Masses of CVs Observed by Suzaku

Figure 1.The best-fit XMM-Newton 5–8 keV spectra of AO Psc and FO Aqr.The black,red and green data points represent spectra from MOS-1,MOS-2 and PN,respectively.Spectra are rebinned for plotting only.

Table 1 Observation Log and Dynamically Measured WD Masses of CVs Observed by XMM-Newton

3.2.Updating the I7.0/I6.7–Tmax and the I7.0/I6.7–MWD Relations

Similar toTmax,the previousI7.0/I6.7–TmaxandI7.0/I6.7–MWDrelations were also based onI7.0/I6.7values measured with the mkc f l ow model with the mekal description.We thus verify these relations with mkc f l ow with the apec description as follows.

First,we obtain theI7.0/I6.7–Tmaxrelations derived from the mkc f l ow model with apec description following Xu et al.(2019a),and compare them with the observed values in Figure3.The data points are from Table4.Obviously,the observedI7.0/I6.7andTmaxstill follow the updatedI7.0/I6.7–Tmaxrelation.

Table 3 I7.0/I6.7 of XMM-Newton Observed CVs

Table 4 Maximum Emission Temperature (Tmax) and I7.0/I6.7 Measured with NuSTAR and Suzaku data,and Dynamically Determined mass (MWD) of CVs

Second,we examine theI7.0/I6.7–MWDrelations.We plot theI7.0/I6.7and the dynamically measuredMWDof sampled CVs in Figure4.We also plot theI7.0/I6.7–MWDrelations derived by combining Equation (1) or Equation (2) and the mkc f l ow with mekal and apec descriptions in Figure4for comparison.From the f i gure,theI7.0/I6.7–MWDcurves of both the mekal and apec description can well describe the sampled DNe.On the other hand,the IPs are more consistent with the apec description.We suspect that it is because the AtomDB used by apec description works better for the highI7.0/I6.7case,where most IPs are located.It is also worth noticing that aI7.0/I6.7may refer to differentMWD(andTmax) values for different metallicityZ.

3.3.The New Spectral Model to Measure WD Masses

We introduce a new model to replace the threeGaussian model,so that the f i tting of the 5–8 keV spectra could measureTmax(from theI7.0/I6.7–Tmaxrelations shown in Figure3) and outputMWD(from the Tmax–MWDrelations shown in Equations (1) and (2)) directly.The new model are divided to two sub-models:ipmass_line and dnmass_line,according to the different Tmax–MWDrelations(Equations(1)and(2))for IPs and DNe,respectively.In Equations (1) and (2),the mean molecular weight μ is f i xed at 0.6 (e.g.,Byckling et al.2010)andRWDis derived from WD’sMWD–RWDrelation(Nauenberg1972).Thus the model only contains two free parameters:the WD mass (MWD) and the abundance (Z).The latter has to beconstrained first because theI7.0/I6.7–Tmaxrelations are abundance dependent,as shown in Figure3.

Figure 2.The semi-empiricalTm ax –MWD relation(Equation(2))for DNe.The solid curve shows the updatedTm ax –MWD relation with α=0.69 and the dashed curve shows the case with the strong shock assumption (α=1).Points surrounded by green represent Zorotovic et al.(2011)'s fiducial sub-sample of “robust dynamical mass measurements”.

To use this model,we first constrain the uncertainty ranges ofZby fitting the 5–8 keV spectra with the cooling flow model:phabs(mkcflow+Gauss) (the Gaussian components describe the Fe I–Kα lines) with the apec description.Then we only allowZof the new models to vary within the uncertainty ranges derived from previous step (Typical abundance～0.1 and～0.3Z⊙for luminous and dim sources,respectively),and refit the 5–8 keV spectra with phabs(apec+ipmass_line+Gauss) or phabs(apec+dnmass_line+Gauss) for IPs and DNe,respectively.During the fitting the abundance of apec is fixed to 0,just like the case for the threeGaussian model.The outputMWDvalues are summarized in Tables5and6,where the upper limit of all derived WD masses is assumed to 1.44M⊙.

3.4.WD Mass Distribution in CVs

From Table6,theMWDderived from XMM-Newton and Suzaku data for same CVs are consistent with each other.We then build a sample of 58 individual CVs withMWDmeasurements,including 22 DNe and 36 IPs from Tables5and6.We adopt Suzaku measuredMWDif there are measurements from both XMM-Newton and Suzaku,because the latter usually provides higher quality spectra and therefore lower uncertainties.

The distribution of WD masses in sampled IPs and DNe are plotted in Figure5.From the figure,WD masses in both IPs and DNe are distributed in a wide range,from～0.4–0.5M⊙to～1.2M⊙a(bǔ)nd peaking at～0.8M⊙.There might be hints of a second peak at～1.1–1.2M⊙for DNe,but the statistics is too low to draw any firm conclusion.The mean WD masses in IPs and DNe can be calculated to be 〈MWD,IP〉=0.81±0.21M⊙a(bǔ)nd 〈MWD,DN〉 = 0.81 ±0.21M⊙,respectively.We also examine the distribution of WD masses in IPs and DNe with a Kolmogorov–Smirnov test,and the two-sidedp-value is～0.92.In other words,we do not find systematical differences between the distribution of WD masses in IPs and DNe.

Figure 3.I7.0/I6.7 versus Tmax for CVs.The solid and dashed curves are relations predicted by the mkcflow model with apec description and with Z=1Z⊙a(bǔ)nd Z=0.1Z⊙,respectively.Black and red points represent IPs and DNe,respectively.

4.Discussion

4.1.Comparison with Previous Works

First,we check the reliability of our measuredI7.0/I6.7.At first,iron absorption edge in CVs may influence the measurement ofI7.0/I6.7.We add an iron absorption edge component,where the absorption depth is set to 0.1 at 7.11 keV(Nobukawa et al.2016,mean absorption depth is 0.02±0.01 and 0.08±0.04 for IPs and non-mCVs,respectively),and perform spectral fitting again.The results show that the newI7.0/I6.7are consistent with previous measurements.4No absorption edge component were included in previous measurements,which is equivalent to zero absorption depth here.We further compare theI7.0/I6.7values measured in this work with those in previous works like Ezuka &Ishida (1999) and Rana et al.(2006) using ASCA and Chandra HETG observations,and they are again consistent with each other.Moreover,ourI7.0/I6.7from XMM-Newton observations are consistent with those based on Suzaku observations (Xu et al.2016,2019b),as shown in Table3and Figure6.Considering the fact that these measurements are made with different instruments (XMM-Newton,Suzaku,Chandra–HETG and ASCA) in a time range spanning～20 yr,the consistency indicates the robustness ofI7.0/I6.7,which is one of the essential quality to be used as a diagnostic forMWD.

Figure 4.I7.0/I6.7 versus dynamically determined MWD for sampled CVs.The blue(red)solid curves are the predicted relations for IPs(DNe)from Equations(1)and(2) by the mkcflow model.

Second,we check the possible bias brought by the uncertainties in previous WD mass measurements.Following Zorotovic et al.(2011),we considered WD mass measurements on VW Hyi,U Gem,HT Cas,OY Car,Z Cha as“robust”ones,and manually add 20%systematic errors for other sources.We then perform best-fit for Equation (2),and the result shows α=0.67±0.06,which is consistent with the previous value(α=0.69±0.06),and there is no significant influence onI7.0/I6.7–Tmax–MWDrelations.

Third,we compare our derivedMWDwith those from Suleimanov et al.(2019),who derivedMWDof IPs with the continuum fitting method based on NuSTAR and Swift observations.5We utilize NuSTAR to derive WD masses of IPs with cooling flow model and post-shock region (PSR) model developed by Suleimanov et al.(2016),and the typical WD mass difference is within 5%.There are 25 CVs included in both their and our samples.The results of 21 CVs are consistent with ours except for the other four CVs:MWD=0.67±0.08M⊙for MU Cam,MWD=0.72±0.02M⊙for V1223 Sgr,MWD=1.05±0.04M⊙for NY Lup andMWD=0.72±0.06M⊙for CXOU J171935.8-410053.Our results arein MU Cam,V1223 Sgr,NY Lup and CXOU J171935.8-410053,respectively.The differences may be explained as follows:First,we assume the accreted material falls from infinity,while in reality they could fall from a certain distance,e.g.,the inner radius of the truncated accretion disk,leading to underestimation ofMWD.Second,the Compton hump (～10–30keV) caused by the reflection (Mukai et al.2015) could soften the hard X-ray continuum,which would lead to a lower temperature than ours,thus a lower WD mass.Third,a different local mass accretion rate,fixed at 1 g s-1cm-2in Suleimanov et al.(2019),could affect the hard X-ray continuum (Suleimanov et al.2016),leading to a deviation ofMWD.Similar reasons could also be responsible for the difference between our results and those of Yuasa et al.(2010).Additionally,we notice our derived WD masses have larger errors than those in Suleimanov et al.(2019).The uncertainties of the derived WD masses are mainly related to the uncertainties of the measured flux ratios,which is determined by the counting statistics of the spectra in the Fe line energy range.On the other hand,as discussed in Section1,the method in this work make use of the large archival observations on CV by XMM-Newton and Suzaku which allows a larger sample size compared to previous works using hard X-ray (up to 30–50 keV) observations.

Fourthly,we compare theMWDderived from the new models (ipmass_line or dnmass_line) with the dynamical results in Figure7and in Tables5and6.The comparison shows that the derived masses of all the 20 CVs are consistent with the dynamical values.Moreover,we make a quantitative examination on the goodness of the new model derivedMWD.Following Xu et al.(2019b)'s method,we assume a linear relation in the form ofMWD,derived=A×MWD,dynamical+B(For a good relation,we expectA～1 andB～0.)and perform fitting with the ODR method.The best-fitted results are plotted in Figure7to be compared with observed values.For the 11 Suzaku sampled CVs mentioned in Xu et al.(2019b),the bestfit yieldsA=0.86±0.16,B=0.08±0.15 andr2=0.95,which are consistent with the previous values (A=0.97±0.09 andB=0.06±0.09 in Xu et al.(2019b),whereMWD,derivedare derived with previousI7.0/I6.7–MWDrelations).For the combined Suzaku and XMM-Newton sample of 20 CVs,the best-fit yieldsA=0.90±0.15,B=0.07±0.12 andr2=0.93.Both fittings are consistent withA=1 andB=0.Besides,additional 20% systematic errors on non-robust mass measurements are also examined on above linear relation.The best-fit yieldsA=0.90±0.14 andB=0.07±0.12 that is the nearly same as above.6All errors in linear relation are shown with 90% confidence level.With the calibrated relations,we obtain the mean WD masses for IPs and DNe are 0.82±0.23M⊙a(bǔ)nd 0.82±0.23M⊙,both of which are consistent with uncalibrated ones.

Table 5 Masses of WDs Derived from the ipmass_line or the dnmass_line Models and those Dynamically Measured for Suzaku Observed CVs

Table 6 Masses of WDs Derived from ipmass_line or dnmass_line Model and those Dynamically Measured for XMM-Newton Observed CVs

Finally,we compare theMWDdistribution with those from previous works.The comparison shows that our results are consistent with the those by Suleimanov et al.(2019),where〈MWD〉=0.79±0.16M⊙for IPs,and Zorotovic et al.(2011)where 〈MWD〉=0.83±0.23M⊙for CVs.

4.2.Limitations

The results of this work suffer from several limitations,which are discussed as follows.

First,the typical uncertainties ofI7.0/I6.7in this work(～20%–30%) are higher than those of Suzaku observed CVs(～10%–20%,Xu et al.2019b),which is presumably due to the limited counting statistics of XMM-Newton spectra in the Fe line energy ranges.For example,typical spectra of Suzaku observed CVs have more than 1000 bins between 5–8 keV,while those of XMM-Newton observed ones have about several hundred bins.In fact,XMM-Newton sources with better spectral quality do have better constrainedI7.0/I6.7values,e.g.,the uncertainty is～9% for V426 Oph.Further observations on target CVs would improve the situation.

Second,the sample size is still not large enough,and could be biased toward luminous sources.Among the total 20 CVs(Figure4) to derive theI7.0/I6.7–MWDrelations,there are nine new ones in this work and 11 old ones from previous works(Yu et al.2018;Xu et al.2019b).However,the sample size is certainly still not large enough,which should be dealt with in future works.Moreover,luminous sources would have higher chances to be selected as XMM-Newton targets and cause potential bias.We propose observations on less luminous ones(e.g.,those with 2–10 keV luminosity below 1031erg s-1,which are supposed to have accretion rates below 10-11M⊙yr-1) to test both theI7.0/I6.7–MWDrelations and their dependence on accretion rates in the future.Our sample is also lack of WDs more massive than 1.2M⊙,which should be improved by investigating theI7.0/I6.7of CVs,especially DNe with massive WDs.

Third,the maximum WD mass derived from ipmass_line is～1.16M⊙due to the hard limitations of apec description.MWDof more massive WDs could not be derived with this method.

Fourth,the dynamical mass measurements used to calibrate theI7.0/I6.7–MWDrelations may not always be robust.For example,Marsh et al.(1987) and Hessman et al.(1989) suggested that a“hot spot”or the non-circular motions in the outer accretion diskmay occur,which could distort the radial velocity curves and lead to biasedMWDmeasurements.Moreover,there are multiple measurements of WD masses in DNe,which are not always consistent with each other,thus could affect the best-fit of α in Equation (2).For example,we foundMWD=0.81±0.19M⊙(Bitner et al.2007)andMWD=1.1±0.2M⊙(Friend et al.1990)for SS Cyg;MWD=0.89M⊙(Mason et al.2001) andMWD=0.5–0.6M⊙(Matsumoto et al.2000) for V893 Sco,andMWD=0.83±0.1M⊙(Gilliland1982) andMWD=1.24±0.22M⊙(Watson et al.2007) for BV Cen.To test the dependence of theTmax–MWDrelation of DNe on these different measurements,we remove these sources from the sample,and refit Equation(2).The new best-fit yields α=0.70±0.08,which is still consistent with those of both the previous work(0.65±0.07,Yu et al.2018) and this work (0.69±0.06).

Finally,we notice that EK TrA (Leftmost data point in Figure2) seems to be more consistent with the α=1 curvethan the α=0.69 curve in Figure2.It may attribute to possible uncertainties of previous mass measurements,or to other physical reasons (e.g.,the change of accretion pattern and the boundary layer for CVs with orbital periods below 2 h).Tmax–MWDrelation in this work.Further mass measurements Therefore,we still keep α as the only parameter for the would be necessary to distinguish whether EK TrA is a true outlier.

Figure 5.Masses of WDs derived from the ipmass_line and the dnmass_line models for sampled IPs (left panel) and DNe (right panel).

Figure 6.I7.0/I6.7 measured with XMM-Newton data in this work versus those measured with Suzaku data by Xu et al.(2016).The black solid diagonal line shows a 1:1 relation of the I7.0/I6.7 values.

Figure 7.MWD derived from the ipmass_line or the dnmass_line models versus dynamically determined ones.The black(red)data points represent IPs(non-mCVs),the squares and dots represent CVs observed with XMM-Newton and Suzaku in this work,respectively.The solid diagonal line shows a 1:1 relation for the MWD values.The blue dashed line shows the best linear fit to all CVs,and the green dashed line for Suzaku observed CVs only.Points surrounded by green represent CVs from Zorotovic et al.(2011)'s fiducial sub-sample of “robust dynamical mass measurements”.

5.Summary

In this work we carry out a systematic investigation on the WD masses of 58 CVs(including 36 IPs and 22 DNe)observed by XMM-Newton and Suzaku in the solar vicinity based on theI7.0/I6.7–Tmax–MWDrelations using the mkcflow model with apec description and AtomDB.Our main results are summarized as follows:

1.TheTmax–MWDrelationfor DNe is examined with the mkcflow model with the apec description.The results yield α=0.69±0.06,which is consistent with previous works.

2.TheI7.0/I6.7,MWDandTmaxof sampled CVs follow the theoretical (semi-empirical for DNe) relations predicted by the mkcflow model with the apec description.

3.We introduce a new spectral model to measureMWDby building theI7.0/I6.7–Tmax–MWDrelations into the mkcflow model with the apec description.With constraints on the metallicityZ,the fitting of the 5–8 keV spectra of CVs with this model can outputMWDdirectly.

4.Based on the above model,we derive the WD masses of IPs and DNe in the largest X-ray selected sample.The mean WD masses are 〈MWD,IP〉=0.81±0.21M⊙a(bǔ)nd〈MWD,DN〉=0.81 ± 0.21 M⊙for IPs and DNe,respectively.We also do not find significant difference between the two WD mass distributions.These values are consistent with the optical and hard X-ray measurements of WD masses in CVs in the solar vicinity.

Acknowledgments

We thank the anonymous referee for the very useful comments and suggestions which helped to improve this paper greatly.This work is supported by the National Natural Science Foundation of China through Grant 11873029 and the National Key Research and Development Program of China(2016YFA0400803 and 2017YFA0402703).This work is based on observations obtained with XMM-Newton,an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.