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An Empirical Correlation of Tmax–MWD of Dwarf Novae and the Average White Dwarf Mass in Cataclysmic Variables in the Galactic Bulge

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Published 2018 February 5 • © 2018. The American Astronomical Society. All rights reserved.
, , Citation Zhuo-li Yu et al 2018 ApJ 853 182 DOI 10.3847/1538-4357/aaa47d

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xuxj@nju.edu.cn

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

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  1. Received 2017 November 1
  2. Revised 2017 December 18
  3. Accepted 2017 December 27
  4. Published

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Keywords

Galaxy: bulge; novae, cataclysmic variables; X-rays: binaries

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0004-637X/853/2/182

Abstract

The mean white dwarf (WD) mass in the Galactic bulge cataclysmic variables (CVs) was measured by applying the shock temperature-WD mass correlation of magnetic cataclysmic variables (mCVs) to the Galactic bulge X-ray emission (GBXE) spectra. However, the resulting mean WD mass is lower than that of the local CVs. This discrepancy could be explained by the dominating sources in the GBXE, which are non-mCVs instead of mCVs. In this work, we conduct a thorough investigation of the X-ray spectra of local DNe from the Suzaku archives and derive semi-empirical correlations between the shock temperature Tmax, the flux ratio of Fe xxvi–Lyα to Fe xxv–Heα lines, and WD mass for quiescent, nonmagnetic CVs. By applying these correlations to the GBXE, we derive the average WD mass of CVs in the Galactic bulge to be 0.81 ± 0.07M⊙. This value is consistent with previous optical measurements of WD mass in local CVs.

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1. Introduction

A cataclysmic variable (CV) is a binary star in which a white dwarf (WD) accretes matter from a main-sequence or a sub-giant star via Roche-lobe overflow. Subgroups of CVs include magnetic CVs (mCVs) and nonmagnetic ones based on the magnetic field strengths of the WDs (Warner 1995; Frank et al. 2002). About 10%–20% of CVs are mCVs, the others are non-mCVs, more specifically, dwarf novae (DNe; e.g., Liebert et al. 2003; Pretorius et al. 2013). With X-ray luminosity ∼1030–34 erg s−1, CVs are important X-ray emitters in the Galaxy (e.g., Sazonov et al. 2006; Revnivtsev et al. 2009a; Warwick 2014). In an mCV, for example, intermediate polars (IPs), matter from the companion star is heated by a strong shock and emits X-ray before falling onto the WD surface along the WD magnetic field lines. In a nonmagnetic CV, on the other hand, X-ray photons are mainly from a boundary layer near the WD surface, where accreted matter is heated by either a strong shock or a series of weak shocks (e.g., Frank et al. 2002). For both subclasses of CVs, the X-ray spectra include a continuum from multi-temperature optical-thin thermal plasma and prominent emission lines (e.g., the H-like and He-like Fe lines near 7 keV). The spectra could be well fitted by an absorbed cooling flow model (mkcflow in Xspec; Mushotzky & Szymkowiak 1988) with an additional intrinsic absorption in some cases (Mukai et al. 2003).

The WD in a CV could hardly retain its accreted matter (e.g., Prialnik & Kovetz 1995; Yaron et al. 2005; Liu & Li 2016). As a result, the present WDs in CVs could reflect the initial properties of WDs when the CV was formed, thus they could be used to constrain (binary) star evolution theories. Moreover, CVs are closely related to the possible progenitors of type Ia supernovae, as both of them involve accreting WDs. Traditionally, the WD masses in CVs were measured in the optical/UV band (e.g., Friend et al. 1990; Mason et al. 2001; Zorotovic et al. 2011), and the results suggest an average WD mass of 0.83 ± 0.23 M⊙ in local CVs (e.g., Zorotovic et al. 2011). In recent years, the X-ray spectral fitting method demonstrated its power in that it can directly measure the temperature of the shock-heated matter and imply the WD mass by assuming the strong shock condition (e.g., Frank et al. 2002) in mCVs:

Equation (1)

In the above equation, Tmax is the shock temperature, μ is the mean molecular weight, mH is the mass of the H atom, k is the Boltzmann constant, G is the gravitational constant, M and R are the mass and radius of the WD, respectively. For nonmagnetic CVs, Tmax is half of that value because half of the gravitational energy has been dissipated in the accretion disk. With the additional MWD–RWD relation of WDs (Nauenberg 1972),

Equation (2)

one can derive the WD mass through

Equation (3)

The above correlation has been tested in dozens of local mCVs by various works (e.g., Suleimanov et al. 2005; Yuasa et al. 2010). A recent one was done by Yuasa et al. (2010), where the authors found a mean WD mass of 0.88 ± 0.25M⊙ for local mCVs based on Suzaku observations.

On the other hand, the population properties of WDs in distant CVs remains unclear because neither optical nor X-ray observations could provide a CV sample with a distance of over several hundred parsecs. The Galactic diffuse X-ray background, on the other hand, provides a natural sample for such a study. Discovered more than 30 years ago, the Galactic ridge/bulge X-ray emission (GR/BXE) spans across the Galactic ridge and bulge in the energy range of 2–10 keV (here we follow the definition of Nobukawa et al. 2016, who define the GBXE as regions of $| {l}_{* }| \lt 0\buildrel{\circ}\over{.} 6$, $1^\circ \lt | {b}_{* }| \lt 3^\circ $). The 1Ms Chandra observations on the "Limiting Window" in the Galactic bulge suggested that the GBXE is dominated by discrete sources (Revnivtsev et al. 2009b). The majority of which were proposed to be IPs, with a minor contribution from active binaries (ABs, e.g., Revnivtsev et al. 2009b; Hong et al. 2012). Thus, the composite X-ray spectra of the GBXE contains information from all the CVs (and ABs) in the direction in which they are free from selection effects and form an unbiased CV sample. With the assumption of mCV dominating the GBXE, Yuasa et al. (2012) fitted the E > 15 keV spectra of GBXE with an IP model and showed that the average WD mass in CVs is ${0.66}_{-0.07}^{+0.09}\,{M}_{\odot }$, which is close to 0.5–0.60 M⊙ from INTEGRAL observations (Krivonos et al. 2007; Turler et al. 2010).

However, these results are only marginally consistent, if not inconsistent at all, with optical measurements for local CVs (0.83 ± 0.23 M⊙, Zorotovic et al. 2011) or those of the Galactic center (∼0.9 M⊙, Hailey et al. 2016). In the Galactic center, the CVs in the crowded stellar environment may have experienced dynamical interactions, which preferentially bring massive WDs into binaries, and thus increased the mean WD mass in CVs. But the 0.2 M⊙ discrepancy between local and the GBXE CVs is still confusing. The Tmax–MWD correlation has been tested in dozens of local mCVs and there is no reason why this correlation could not be applied to mCVs in the Galactic bulge. Recently, Xu et al. (2016) proposed that the GBXE in the Fe emission line range (6–8 keV) should be dominated by non-mCVs instead of mCVs. Similar conclusions were also drawn by Nobukawa et al. (2016) and Hailey et al. (2016) by comparing X-ray continuum of different sources and of the Galactic center. Thus, we should apply the Tmax–MWD correlation of non-mCVs to GBXE spectra to make a reliable measurement of average WD mass. Moreover, the accreted matter may exhibit a series of weak shocks in the boundary layer(BL) of non-mCVs, and the Tmax–MWD correlation may deviate significantly from the theoretical formula (Frank et al. 2002). Moreover, for a typical DNe with Tmax ≲ 80 keV, a higher shock temperature usually means that there will be more hydrogen-like Fe ions comparing to helium-like ones. Therefore, the flux ratio of the hydrogen-like Fe line to the helium-like Fe line (I7.0/I6.7), is also sensitive to the shock temperature. For example, Xu et al. (2016) concluded that I7.0/I6.7 increases with increasing Tmax for Tmax < 80 keV by assuming a cooling flow emission model. In this work, we choose to first test the validity of the strong shock assumption based on the X-ray spectroscopy of local non-mCVs and build semi-empirical Tmax–MWD–I7.0/I6.7 correlations. Next, we apply these correlations to the GBXE and constrain the average MWD in the Galactic bulge. We use the Suzaku archived sample of non-mCVs in this work because Suzaku can provide both a relatively self-consistent sample of non-mCVs and well studied data of the GBXE (e.g., Yuasa et al. 2012; Nobukawa et al. 2016). Furthermore, the spectra generally cover 0.3–40 keV, which is suitable for reliable Tmax measurements.

The rest of the paper is organized as follows: In Section 2, we describe our sample and data analysis methods. We present our results in Section 3. In Section 4 we compare them with existing results and discuss the implications. Finally, in Section 5, we provide a short summary.

2. Observations and Data Reduction

The Suzaku X-ray Observatory contains two types of instruments: one is the X-ray Imaging Spectrometers (XIS, Koyama et al. 2007), the other is the Hard X-ray Detector (HXD, Takahashi et al. 2007). The XIS consists of four sensors: one is made of back-illuminated CCD(XIS-1), and the other three are made of front-illuminated CCDs (XIS-0, 2, 3). XIS-2 suffered catastrophic damage on 2006 November 9 and no useful data have been transferred since then.

We cross-correlate the Suzaku online archive1 with Ritter & Kolb's (2003) CV catalog to search for publicly available observations. A total of 25 observations on 18 DNe were found. The basic information regarding observed sources is listed in Table 1. This sample is certainly not a complete one. Nevertheless, it could provide us with information regarding the shock temperatures of DNe. Previous works based on similar observations only included XIS data (e.g., Byckling et al. 2010; Wada et al. 2017), but the best-fitted Tmax is usually above 10 keV; thus, HXD may be needed to put tighter constraints on Tmax. We include both XIS and HXD data in this work.

Table 1.  Sampled DNe Properties

Source Obs ID Exposure Time Distance MWD Reference
    (ks) (pc) (M⊙)  
SS Cyg1 109015010a 43.2 117.1 ± 6.2 1.1 ± 0.2 Ramsay et al. (2017), Friend et al. (1990)
SS Cyg2 400006010 39.5 117.1 ± 6.2 1.1 ± 0.2 Ramsay et al. (2017), Friend et al. (1990)
SS Cyg3 400007010b 56.0 117.1 ± 6.2 1.1 ± 0.2 Ramsay et al. (2017), Friend et al. (1990)
V893 Sco 401041010 18.5 ${135}_{-33}^{+65}$ 0.89 ± 0.15c Ozdonmez et al. (2015), Mason et al. (2001)
VY Aqr 402043010 25.4 ${89}_{-10}^{+13}$ ⋯ Ozdonmez et al. (2015)
SW UMa 402044010 16.9 159 ± 22 0.71 ± 0.22 Gansicke et al. (2005), Ritter & Kolb (2003)
SS Aur 402045010 19.5 ${167}_{-9}^{+10}$ 1.08 ± 0.4 Ozdonmez et al. (2015), Sion et al. (2008)
BZ UMa 402046010 29.7 ${204}_{-37}^{+59}$ 0.65 ± 0.15c Ozdonmez et al. (2015), Jurcevic et al. (1994)
FL Psc 403039010 33.3 160 ± 40 ⋯ Patterson (2011)
KT Per 403041010d 29.2 ${145}_{-21}^{+31}$ ⋯ Ozdonmez et al. (2015)
GK Per 403081010 30.4 ${477}_{-25}^{+28}$ 0.87 ± 0.24 Harrison et al. (2013), Morales-Rueda et al. (2002)
Z Cam1 404022010d 37.7 219.3 ± 19.5 0.99 ± 0.15 Ramsay et al. (2017), Ritter & Kolb (2003)
Z Cam2 407016010b 35.9 219.3 ± 19.5 0.99 ± 0.15 Ramsay et al. (2017), Ritter & Kolb (2003)
VW Hyi1 406009010b 70.1 46 ± 6 0.67 ± 0.22 Ozdonmez et al. (2015), Hamilton et al. (2011)
VW Hyi2 406009020 16.2 46 ± 6 0.67 ± 0.22 Ozdonmez et al. (2015), Hamilton et al. (2011)
VW Hyi3 406009030 20.1 46 ± 6 0.67 ± 0.22 Ozdonmez et al. (2015), Hamilton et al. (2011)
VW Hyi4 406009040e 16.8 46 ± 6 0.67 ± 0.22 Ozdonmez et al. (2015), Hamilton et al. (2011)
U Gem1 407034010 119.1 100 ± 4 1.2 ± 0.05 Ozdonmez et al. (2015), Ritter & Kolb (2003)
U Gem2 407035010b 50.3 100 ± 4 1.2 ± 0.05 Ozdonmez et al. (2015), Ritter & Kolb (2003)
CH UMa 407043010 45.2 356 ± 47 1.95 ± 0.3 Ozdonmez et al. (2015), Friend et al. (1990)
EK Tra 407044010 77.8 180 0.46 ± 0.1 Gansicke et al. (1997), Mennickent & Arenas (1998)
BF Eri 407045010 32.8 596 ± 79 1.28 ± 0.05 Ozdonmez et al. (2015), Neustroev & Zharikov (2008)
BV Cen 407047010 33.4 344.3 ± 64.8 1.24 ± 0.22 Ramsay et al. (2017), Watson et al. (2007)
V1159 Ori 408029010 200.5 299 ⋯ Ak et al. (2008)
FS Aur 408041010 62.2 246 ± 32 ⋯ Ozdonmez et al. (2015)

Notes. The state is determined using the American Association of Variable Star Observers(AAVSO) International Database.

aNo HXD spectrum. bOutburst state. c0.15 M⊙ mass error assumed. dTransitional state. eVery low HXD net counts, HXD data is not used.

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The data processing and spectral analysis are carried out using HEAsoft2 (version 6.17, Arnaud 1996). The event files are reduced with the standard pipeline aepipeline with the latest calibration files (XIS:20160607). For each XIS screening image, we extracted source events from a 200'' circle and background events from a 250''–400'' annulus, excluding regions outside CCD or contaminating sources with the xselect tools. The response files and ancillary files were generated by the xisrmfgen and xissimarfgen, respectively. For every source, we used the addascaspec tool to combine the spectra and response files of XIS-0, XIS-2 (if it exists), and XIS-3. We regroup the spectra so that the signal-to-noise ratio of each spectrum exceeds three. For HXD data, background files were downloaded from the Suzaku background FTP server3 and data were processed using the hxdpinxbpi tool. Due to the low net counts of HXD for most sources, the HXD spectra were grouped so that there were two to three bins at least.

3. Results

3.1. X-Ray Spectroscopy of Quiescent DNe

The X-ray spectra of quiescent DNe could be well fitted with the mkcflow model in Xspec (Mushotzky & Szymkowiak 1988; Mukai et al. 2003). We choose combined energy ranges of 0.3–10.0 keV for XIS and 12.0–50.0 keV for HXD (if it exists). We started by fitting the background-subtracted spectra with the model phabs × (mkcflow + Gaussian), or phabs ×pcfabs × (mkcflow + Gaussian) if additional intrinsic absorption is needed (see, e.g., Byckling et al. 2010; Wada et al. 2017), where phabs, pcfabs, mkcflow, and Gaussian components account for the interstellar foreground absorption, the intrinsic absorption, the emission from the shock-heated plasma, and the Fe i–Kα fluorescent line, respectively. Examples of the best-fitting are plotted in Figure 1, and the results of Tmax are listed in Table 2. The last column of Table 2 shows the flux ratio of Fe xxvi–Lyα and Fe xxv–Heα lines from Xu et al. (2016).

Figure 1.

Figure 1. Examples of broadband DNe spectra with the best-fit models (see Table 2 for details). Obs-Ids and source names are as shown. Black data points stand for spectra from the combination of XIS-0, XIS-2 (if it exists), and XIS-3. Red and green data points represent spectra from XIS-1 and HXD, respectively. Lines are best-fit models. Spectra are rebinned for plotting only.

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Table 2.  Best-fit Tmax and Comparison with Previous Works

Source Id Tmax(keV)a ${\chi }_{\nu }^{2}(\mathrm{DOF})$ Model Tmax(keV)b Tmax(keV)c I7.0/I6.7 d MWD(M⊙)e
SS Cyg 400006010 ${42.1}_{-1.0}^{+1.0}$ 1.09(4513) 1 ${52.5}_{-0.7}^{+1.1}$ ${41.99}_{-0.76}^{+1.20}$ ⋯ ⋯
V893 Sco 401041010 ${15.7}_{-1.0}^{+1.1}$ 0.91(2767) 2 ${19.2}_{-0.6}^{+0.6}$ ${19.32}_{-1.40}^{+1.29}$ ${0.37}_{-0.06}^{+0.06}$ ${0.84}_{-0.11}^{+0.13}$
VY Aqr 402043010 ${18.9}_{-2.9}^{+3.8}$ 0.93(502) 1 ${18.4}_{-1.2}^{+2.3}$ ${16.47}_{-2.22}^{+2.68}$ ${0}_{-0}^{+0.01}$ ⋯
SW UMa 402044010 ${7.8}_{-0.5}^{+0.6}$ 0.94(658) 1 ${7.5}_{-0.3}^{+0.4}$ ${8.33}_{-0.99}^{+0.62}$ ⋯ ⋯
SS Aur 402045010 ${26.3}_{-2.9}^{+2.9}$ 0.91(919) 1 ${26.5}_{-1.2}^{+2.1}$ ${23.47}_{-3.02}^{+4.01}$ ${0.56}_{-0.18}^{+0.23}$ ${1.03}_{-0.23}^{+0.23}$
BZ UMa 402046010 ${13.6}_{-0.9}^{+0.9}$ 1.00(1186) 1 ${13.7}_{-0.4}^{+0.6}$ ${13.71}_{-0.81}^{+1.38}$ ${0.40}_{-0.15}^{+0.17}$ ${0.87}_{-0.21}^{+0.23}$
FL Psc 403039010 ${17.2}_{-4.4}^{+5.6}$ 0.99(144) 1 ${15.0}_{-2.5}^{+1.7}$ ${14.43}_{-2.69}^{+4.36}$ ${0.10}_{-0.10}^{+0.57}$ < 1.20
KT Per 403041010 ${13.8}_{-1.5}^{+1.3}$ 0.92(1146) 1 ${14.5}_{-0.5}^{+0.4}$ ⋯ ${0.10}_{-0.10}^{+0.18}$ <0.80
GK Per 403081010 ${62.0}_{-14.1}^{+14.5}$ 0.94(1681) 2 ⋯ ⋯ ${0.94}_{-0.45}^{+0.76}$ >0.90
Z Cam 404022010 ${25.7}_{-1.0}^{+1.4}$ 1.10(3864) 2 ${27.6}_{-0.5}^{+0.3}$ ${25.76}_{-2.39}^{+5.16}$ ${0.58}_{-0.05}^{+0.05}$ ${1.05}_{-0.11}^{+0.10}$
VW Hyi1 406009020f ${8.7}_{-0.3}^{+0.2}$ 1.06(1518) 1 ${9.3}_{-0.1}^{+0.1}$ ⋯ ${0.04}_{-0.05}^{+0.05}$ <0.55
VW Hyi2 406009030 ${9.7}_{-0.6}^{+0.4}$ 1.01(1524) 1 ${10.0}_{-0.1}^{+0.1}$ ⋯ ${0.21}_{-0.06}^{+0.08}$ ${0.66}_{-0.12}^{+0.14}$
VW Hyi3 406009040 ${9.2}_{-0.47}^{+0.55}$ 0.98(1441) 1 ${9.8}_{-0.1}^{+0.1}$ ⋯ ${0.19}_{-0.08}^{+0.08}$ ${0.63}_{-0.13}^{+0.15}$
U Gem 407034010 ${26.9}_{-0.7}^{+0.6}$ 1.07(3387) 1 ${26.2}_{-0.6}^{+1.0}$ ${25.82}_{-1.43}^{+1.98}$ ${0.68}_{-0.08}^{+0.08}$ ${1.13}_{-0.14}^{+0.12}$
CH UMa 407043010 ${14.3}_{-0.9}^{+0.7}$ 0.99(1697) 1 ${15.0}_{-2.5}^{+1.7}$ ⋯ ⋯ ⋯
EK Tra 407044010 ${10.4}_{-0.4}^{+0.5}$ 1.07(2425) 1 ${12.4}_{-0.2}^{+0.1}$ ⋯ ${0.16}_{-0.08}^{+0.08}$ ${0.60}_{-0.14}^{+0.15}$
BF Eri 407045010 ${7.1}_{-0.5}^{+0.5}$ 0.99(697) 1 ${10.2}_{-0.4}^{+0.9}$ ⋯ ${0}_{-0}^{+0.31}$ <0.83
BV Cen 407047010 ${25.1}_{-2.4}^{+2.1}$ 1.04(2866) 2 ${27.5}_{-0.7}^{+0.7}$ ⋯ ${0.53}_{-0.08}^{+0.09}$ ${1.00}_{-0.14}^{+0.15}$
V1159 Ori 408029010 ${9.29}_{-0.40}^{+0.74}$ 1.01(1855) 1 ⋯ ⋯ ${0.37}_{-0.06}^{+0.06}$ ${0.84}_{-0.11}^{+0.13}$
FS Aur 408041010 ${21.1}_{-1.6}^{+2.0}$ 0.96(1703) 1 ⋯ ⋯ ⋯ ⋯

Notes. Model 1: phabs(mkcflow + Gaussian). Model 2: phabs(pcfabs(mkcflow + Gaussian)) Errors show a 90% confidence level.

aThis work, errors show a 90% confidence level. bResults of Wada et al. (2017), errors are one standard deviation. cResults of Byckling et al. (2010), errors show a 90% confidence level. d ${I}_{7.0}/{I}_{6.7}$ refers to Xu et al. (2016). eMWD derived from I7.0/I6.7 using Equation (4). fAn energy band of >16 keV for HXD is used.

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3.2. The Empirical ${M}_{{wd}}\mbox{--}{T}_{{\max }}-{I}_{{7.0}/{6.7}}$ Correlations for DNe

To build a Mwd–Tmax correlation for DNe, we exclude several sources that do not belong to the DNe class. For example, GK Per was proposed to be an IP with DN-like outbursts (see, e.g., Kim et al. 1992), BF Eri was proposed to be an old nova exhibiting "stunted" outbursts (Neustroev & Zharikov 2008). In addition, only quiescent DNe are included in the following analysis because the Tmax in other states would be different from that in the quiescent state due to the altering of the structure of the accretion disk and BL in high accretion rates (Wada et al. 2017). Furthermore, the Obs-ID 109015010 of SS Cyg did not include HXD data, so we only use the results from Obs-ID 400006010 to represent Tmax of SS Cyg. The data points of excluded sources are still plotted in the figures only for reference.

The masses of WDs in sampled DNe are adopted from various works, as listed in Table 1 (see references therein for details). We assign a typical 0.15 M⊙ uncertainty to WD masses of V893 Sco and BZ UMa because the mass errors were not mentioned in the references (this assumption would not affect our results, see Section 4 for details). We further exclude CH UMa from our sample because its mass is higher than the Chandrasekhar limit. At last, we have 11 available data points from nine sources. In Figure 2, we plot Tmax against the WD mass with a theoretical green curve representing the strong shock condition case. It is obvious that the green curve deviates from the data points. We then simply assume a parameter α, so that Tmax follows

Equation (4)

where μ = 0.6 is assumed. The best-fit results show an α = 0.646 ± 0.069, with ${\chi }_{\nu }^{2}(\mathrm{DOF})=2.02(10)$ and an r2 value of 0.69. The fitted curve with error ranges is given as solid and dashed black curves in Figure 2.

Figure 2.

Figure 2. Tmax–MWD correlation. Black points represent normal quiescent DNe. Red points are sources whose mass errors are assumed to be 0.15M⊙. Sources in the transition state are plotted with blue points. Only black and red data points are used for fitting. The solid green curve shows the theoretical Tmax–MWD in the strong shock assumption. The solid and dashed black curves show the best-fitted Tmax–MWD correlation and its error range.

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As discussed in Xu et al. (2016), I7.0/I6.7, the flux ratio of H-like Fe to He-like Fe lines, is a good indicator of Tmax when Tmax is below ∼50 keV. Thus I7.0/I6.7 is also correlated to the WD mass in CVs. In Figures 3 and 4, we plot the measured I7.0/I6.7 against Tmax and MWD. In addition, we simulate the the theoretical I7.0/I6.7 − Tmax and I7.0/I6.7 − MWD (where MWD is converted to Tmax using Equation (4), uncertainty of α has been considered) correlation curves for solar and 0.1 solar metallicities from the mkcflow model. The resulting curves are also plotted in the figures. The data points, in general, are consistent with the semi-empirical correlation curves, except for three sources with low net counts (BF Eri, FL Psc, and VY Aqr) and two sources in the transition state (KT Per and Z Cam). In Table 2, we also list the MWD derived from the I7.0/I6.7 − MWD correlation. In general, the derived WD masses of sampled DNe are all consistent with optical measurements (see Table 1 for details, BF Eri and GK Per are not included for comparison since they are not DNe). We then conclude that DNe in the quiescent state have I7.0/I6.7 − Tmax and I7.0/I6.7 − MWD correlations, which are consistent with theoretical predictions. Moreover, the differences among different abundances is negligible when Tmax is less than ∼15 keV.

Figure 3.

Figure 3. Tmax–I7.0/I6.7 correlation. The red and green curves show the theoretical correlation from the mkcflow model for solar and 0.1 solar metallicity, respectively. The data points are shown in black, blue, and cyan, using Tmax in this work and I7.0/I6.7 in Xu et al. (2016). Points in black, blue, and cyan are the sources in the quiescent state, the sources in the transition state, and sources with relatively low net photon counts, respectively.

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Figure 4.

Figure 4. I7.0/I6.7–MWD correlation. The curves show the semi-empirical correlation derived from Equation (4) and the Tmax–I7.0/I6.7 theoretical correlation. The red and green curves correspond to 1 and 0.1 solar abundances, respectively. Normal DNe are plotted by black data points. Sources with assumed 0.15 M⊙ WD mass error are plotted in red points. The source in the transition state is plotted as the blue point. Points in cyan show that sources have a relatively low net photon counts.

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4. Discussion

4.1. Comparison with Previous Measurements and Bias Estimation

Both Wada et al. (2017) and Byckling et al. (2010) have analyzed some of our sampled observations in their works. We then compare our best-fitting Tmax with theirs in Table 2. It is obvious that most of our results are consistent with theirs, which gives us confidence to make further investigations. The only exception is SS Cyg. Our Tmax (42.1 ± 1.0 keV) is consistent with that of Byckling et al. (2010) but is about 10 keV lower than that of Wada et al. (2017). We suspect the difference may be due to the additional HXD data included in our analysis, because SS Cyg has a Tmax much higher than 10 keV and should be better constrained by HXD data (see Figure 1).

Byckling et al. (2010) have suggested that the correlation between Tmax and MWD in DNe is consistent with the strong shock assumption, equivalent to α = 1 in Equation (4). However, our best-fitting results suggest α = 0.646 ± 0.069. This discrepancy could be due to the different sampled DNe in their work. For example, BV Cen and U Gem are included in this work but not in theirs. These two sources are essential in our analysis because they provide important data points of relatively massive WDs, which could alter the fitting results significantly. Furthermore, Byckling et al. (2010) only plotted the theoretical curve and did not provide a fitting, which may also affect their arguments.

The potential biases in our fitting mostly come from the uncertainties brought by the optical/UV mass measurements and the limited sample size. As shown in Table 1, the typical WD mass error is on the order of ∼0.1–0.2 M⊙. This error is equivalent to a Tmax uncertainty of ∼8–20 keV, which is much greater than the typical Tmax error of ∼1 keV in this work. On the other hand, only 9 out of 18 X-ray sampled DNe have mass measurements, which greatly limits our sample size and could add more uncertainties to our results. Thus, to test and improve the Tmax–MWD–I7.0/6.7 correlation in the future, we need better constrained WD mass values of more CVs. Furthermore, there are multiple WD mass results for several sources, including SS Cyg and BZ UMa. For example, the WD mass in SS Cyg was measured to be 0.81 ± 0.20 M⊙, 1.09 ± 0.19 M⊙, and 1.19 ± 0.02 M⊙ by various authors (e.g., Friend et al. 1990; Bitner et al. 2007). We tested each of them in the fitting, and found that the Tmax–MWD correlation was altered by ∼10%, with a α value from 0.58 to 0.65, respectively. We also tested the effects of excluding sources with assumed mass errors by removing data points of BZ UMa and V893 Sco, and performed fitting again. The new best-fitted α value is 0.63 ± 0.08, which is within 5% of the original value. These changes in α would alter the resulting mean MWD (see Section 4.2) in a ∼5% level, so we will use α = 0.646 ± 0.069 to make further discussion.

4.2. Average Mass of WDs in DNe in the Galactic Bulge

To imply the mean WD mass in the GBXE CVs, we need a reliable measurement of Tmax. However, the traditional method of deriving Tmax by X-ray continuum fitting involves a major complication in the GBXE case. The GBXE continuum is a composition of multiple classes of sources (e.g., mCVs, non-mCVs, and ABs, Revnivtsev et al. 2009b), and it is difficult to quantify the contribution of DNe alone. For example, in the energy range above 10–15 keV, the GBXE would be a mixture of both mCV and non-mCV emissions; in the energy range below 2–4 keV, emission from ABs could be important (e.g., Nobukawa et al. 2016; Xu et al. 2016). Thus neither the hard X-ray band nor the broadband spectroscopy could give a reliable measurement of DNe Tmax. On the other hand, in the energy range containing He- and H-like Fe lines (6–8 keV), the GBXE spectra should be dominated by DNe (Xu et al. 2016). So we can make use of the Fe line flux ratio I7.0/I6.7 as an indicator of DNe Tmax, and imply MWD directly. To do so, we adopt the GBXE Fe line ratio value of I7.0/I6.7 = 0.34 ± 0.02 from Nobukawa et al. (2016). This value, according to the semi-empirical curves in Figures 3 and 4, corresponds to a temperature of Tmax = 12 ± 0.6 keV and a mean WD mass of $\langle {M}_{\mathrm{WD}}\rangle =0.81\pm 0.07\,{M}_{\odot }$, respectively. This result is consistent with local value of 0.83 ± 0.23 M⊙ by Zorotovic et al. (2011).

The above measurement of MWD depends sensitively on the precise I7.0/I6.7 value of DNe in GBXE. Recently, Nobukawa et al. (2016) concluded that, in the energy range of 5–10 keV, the flux percentage of mCVs, non-mCVs, and ABs in GBXE are ∼3%, 67% ± 6%, and 30% ± 3%, respectively. Thus, the contribution of Fe lines from mCVs could be ignored. The contribution of Fe lines from ABs, on the other hand, should be considered. Although a quantitative investigation of Fe xxv–Heα properties of ABs combined with their luminosity function is beyond the scope of this work, we could give a qualitative analysis as follows. In general, ABs have relatively lower Tmax, thus they emit more Fe xxv–Heα photons and less Fe xxvi–Lyα photons compared to DNe, so the existence of ABs will reduce the I7.0/I6.7 value of GBXE. In another words, the real I7.0/I6.7 of DNe in GBXE should be higher than the measured GBXE value (0.34 ± 0.02). As a result, our measured $\langle {M}_{\mathrm{WD}}\rangle =0.81\pm 0.07\,{M}_{\odot }$ should be considered to be the lower limit of the average WD mass of DNe in the GBXE. This result is still consistent with Zorotovic et al. (2011).

Comparing to previous works of GBXE mean MWD, the mean WD mass of 0.81 ± 0.07 M⊙ is higher than Yuasa et al.'s (2012) GBXE result of ∼0.66 M⊙. This discrepancy could be explained as follows: The Yuasa et al. (2012) work assumed an mCV origin of GBXE, and implied the mCV WD mass by >15 keV GBXE X-ray continuum fitting. As discussed above, the GBXE in this energy range should include a contribution from both mCVs and non-mCVs. Because non-mCVs usually have lower Tmax than mCVs, Yuasa et al.'s (2012) fitting would result in a relatively lower Tmax, and therefore an MWD lower than local mCVs (∼0.9 M⊙, Yuasa et al. 2010). Interestingly, directly applying the DNe Tmax–MWD correlation to the above hard X-ray continuum would instead overestimate the average WD mass to ∼1.15 M⊙ (which is much higher than the local value), because such a method would ignore the contribution of mCVs. The continuum fitting method can give robust measurements of MWD of local CVs, but for the GBXE, the contribution of various classes of sources must be quantified (e.g., a reliable luminosity function has to be built) before this method could be applied.

5. Summary

We have analyzed observations of DNe from Suzaku archives and presented an empirical formula of Tmax–MWD of DNe, which could help us estimate the WD mass of DNe in the quiescent state through X-ray observations. We also present the I7.0/I6.7–Tmax and I7.0/I6.7–MWD correlation of DNe, and imply the average WD mass of DNe in the Galactic bulge to be $\langle {M}_{\mathrm{WD}}\rangle =0.81\pm 0.07\,{M}_{\odot }$. This result is consistent with measurements of local WDs in CVs.

The authors thank the anonymous referee for constructive comments that helped improve this paper. This work is supported by National Science Foundation of China through grants NSFC-11303015, the National Key Research and Development Program of China (2016YFA0400803), NSFC-11773015, NSFC-11133001, and NSFC-11333004.

Footnotes

10.3847/1538-4357/aaa47d