The Fe Line Flux Ratio as a diagnostic of the maximum temperature and the white dwarf mass of Cataclysmic Variables

The flux ratio of Fe XXVI--Ly$\alpha$ to Fe XXV--He$\alpha$ lines ($I_{\rm 7.0}/I_{\rm 6.7}$) is a sensitive indicator of the maximum temperature ($T_{\rm max}$), and therefore the mass of white dwarf stars ($M_{\rm WD}$) in cataclysmic variables (CVs). To examine and calibrate the theoretical $I_{\rm 7.0}/I_{\rm 6.7}$--$T_{\rm max}$--$M_{\rm WD}$ relations, reliable measurements of $T_{\rm max}$ and $I_{\rm 7.0}/I_{\rm 6.7}$ are necessary. In this work, we conduct a thorough investigation on 3--50 keV X-ray spectra of 25 solar neighborhood magnetic and non-magnetic CVs based on archival \textit{NuSTAR} and \textit{Suzaku} observations. The measured $T_{\rm max}$ are compared to the $I_{\rm 7.0}/I_{\rm 6.7}$ and $M_{\rm WD}$. The results show the sampled CVs closely follow the theoretical $I_{\rm 7.0}/I_{\rm 6.7}$--$T_{\rm max}$ relation. Moreover, all the $M_{\rm WD}$ estimated from $I_{\rm 7.0}/I_{\rm 6.7}$ are consistent with the dynamically measured ones. We conclude that $I_{\rm 7.0}/I_{\rm 6.7}$ can be used as a good diagnostic for $T_{\rm max}$ and $M_{\rm WD}$ in both magnetic and non-magnetic CVs.


Introduction
Cataclysmic variables (CVs) are binary stars where a white dwarf (WD) accretes matter from a main-sequence/sub-giant companion via Roche-lobe overflow and/or stellar wind. CVs can be divided into magnetic ones (mCVs) and nonmagnetic ones (non-mCVs) based on the magnetic field strengths of WDs (Warner 1995;Frank et al. 2002). About 20% of CVs are mCVs, including intermediate polars (IPs) and polars; the others are non-mCVs, most of which are dwarf novae (DNe; e.g., Pretorius et al. 2013). CVs are important X-ray emitters in the luminosity range of 10 30-34 erg s −1 , and were proposed to dominate the Galactic Ridge X-ray emission (e.g., Sazonov et al. 2006;Xu et al. 2016). In mCVs, more specifically IPs, matter from the companion star is channeled to magnetic poles of the WDs along the magnetic lines. A standing shock is formed near the surface of the WD, and the post-shock accreted matter is heated to tens of keV and emits X-rays. In nonmagnetic CVs, on the other hand, X-rays are supposed to originate mainly from the boundary layer near the WD surface. The X-ray spectra of CVs in quiescent states can be well fitted with an isobaric absorbed cooling flow model (mkcflow in Xspec; Mushotzky & Szymkowiak 1988;Mukai et al. 2003;Suleimanov et al. 2005) with a Gaussian component to account for the fluorescent Fe Kα line, and additional intrinsic absorption in some cases (Mukai et al. 2003). The measured maximum emission temperature (T max ) of IPs are around several tens of keV, and those of non-mCVs are ∼10 keV.
One of the fundamental questions of CVs is to measure their WD masses. The mass distribution of WDs in CVs are important for star formation and evolution theory itself. It is also closely related to other interesting astrophysical objects like progenitors of SNe Ia, and merging binary WDs, which are supposed to be important gravitational wave emitters. Traditionally, the WD mass in a CV is derived dynamically from the radial velocity curves. This method is model-independent, but sometimes suffers from the uncertainties brought by the unknown inclination angles.
In the past two decades, X-ray spectroscopy has provided an alternative method to measure the WD masses in CVs. The basic idea is that T max of a quiescent CV can be measured by fitting the X-ray continuum, and is supposed to be closely related to its WD gravitational potential, and therefore the WD mass. Assuming that the accreted matter falls from infinity (which is usually a good approximation), T max can be estimated for mCVs (where μ is the mean molecular weight, m H is the mass of the H atom, k is the Boltzmann constant, G is the gravitational constant, and M and R are the mass and radius of the WD, respectively; see, e.g., Frank et al. 2002), and a = m T 3 16 m k GM R max H , where α=0.65±0.07 for non-mCVs (Yu et al. 2018). In previous works, the T max of several dozens of CVs have been measured via X-ray continuum fitting, and the derived M WD were, in general, consistent with the dynamically determined values (e.g., Suleimanov et al. 2005Suleimanov et al. , 2019Shaw et al. 2018).
However, reliable measurements of T max based on continuum fitting demand high S/N spectra above 10 keV, which is beyond the ability of most present-day X-ray observatories (e.g., Chandra and XMM-Newton). Furthermore, the T max measured this way sometimes depends on the modeling of the intrinsic absorption (e.g., pcfabs or pwab models; Ezuka & Ishida 1999;Mukai 2017), or the treatment of the reflected X-ray photons by the WD surface or the disk (e.g., Shaw et al. 2018). These issues have restricted the application of M WD -T max relation to limited bright CVs.
The flux ratio of Fe XXVI-Lyα (centered at ∼7.0 keV) to Fe XXV-Heα (centered at ∼6.7 keV) emission lines (I 7.0 /I 6.7 ) can be taken as a sensitive diagnostic for T max (Ezuka & Ishida 1999;Xu et al. 2016;Yu et al. 2018). The basic idea is that a higher T max ionizes more Fe atoms to hydrogen-like ions, and thus leads to a higher I 7.0 /I 6.7 (e.g., Ezuka & Ishida 1999). Comparing to the continuum fitting method, the line flux ratio method has two advantages. First, most current instruments have good response in the Fe line energy so that the uncertainties of measured I 7.0 /I 6.7 are usually small. Second, I 7.0 /I 6.7 has less dependence on the continuum shape, and thus could avoid the uncertainties brought by the X-ray continuum. Early works based on this method included Ezuka & Ishida (1999), who investigated a dozen mCVs using ASCA observations. Recently, Xu et al. (2016) and Yu et al. (2018) measured T max and I 7.0 /I 6.7 for a sample of Suzaku observed CVs, and derive the T max -I 7.0 /I 6.7 -M WD relations for solar neighborhood non-mCVs. However, there is still large scattering in their T max -I 7.0 /I 6.7 -M WD relations. For example, SS Cyg had a too high T max for its I 7.0 /I 6.7 . This scattering could be due to the possible systematics associated with the highly uncertain background of the Hard X-ray Detector (HXD) on board Suzaku, as pointed out by Shaw et al. (2018). Further investigation demands higher quality X-ray spectra in the 10-50 keV energy range in order to put tighter constraints on T max .
With the large effective area and the ability to focus hard X-rays up to ∼79 keV (Harrison et al. 2013), NuSTAR is the most suitable instrument for this purpose. As shown in previous works, NuSTAR could provide high S/N spectra above 10 keV for CVs in the solar vicinity, which were used to derive T max values (e.g., Shaw et al. 2018;Suleimanov et al. 2019). Combining NuSTAR and Suzaku observations, we could reliably measure both T max and the I 7.0 /I 6.7 , and test the relations between them.
In this work, we use the NuSTAR and Suzaku observations on CVs in the solar vicinity to make updated I 7.0 /I 6.7 -T max -M WD relations for both IPs and non-mCVs. We describe our data and method in Section 2. We present the results and examine the relations in Section 3, we provide a brief discussion in Section 4 and summarize in Section 5. Throughout this work, we quote errors at a 90% confidence level, unless otherwise stated.

Data and Analysis
We choose NuSTAR and Suzaku as the main instruments in this work. NuSTAR contains two focal plane modules, FPMA and FPMB, and is capable of focusing X-rays up to ∼79 keV (Harrison et al. 2013), which is suitable to measure T max of CVs. The Suzaku X-ray Observatory operated between 2005 and 2015. It had two types of instruments: the X-ray Imaging Spectrometers (XIS; Koyama et al. 2007), and the HXD (Takahashi et al. 2007). The XIS consists of four sensors: one is made of a 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. The XIS detectors had the spectral resolution of ∼20-50 among the Fe line energy range and are suitable for I 7.0 /I 6.7 measurements.
We select a sample of CVs in the solar vicinity based on archival NuSTAR and Suzaku observations. First, we carefully select CVs in quiescent states from the Suzaku samples of Xu et al. (2016) and Yu et al. (2018) to maximize counting statistics in the Fe line range. The selection results in a sample of 25 CVs, 13 of which (including 5 IPs and 8 non-mCVs) have dynamical mass measurements and 12 (including 11 IPs and 1 non-mCVs) without mass measurements. The observation log of the sampled CVs are listed in Table 1. We further cross-correlate this CV sample with the NuSTAR archive, and find observations of 12 IPs and 2 non-mCVs. The observation log of this subsample is also presented in Table 1. Seven of the 14 NuSTAR observations on sampled CVs have been previously analyzed, including EX Hya, FO Aqr, RX J2133 +5107, NY Lup, TV Col, V1223 Sgr, and V709 Cas (Shaw et al. 2018;Suleimanov et al. 2019). The other seven observations are first analyzed in this work, including BG  Gansicke et al. (1997). m Gilliland (1982). n Watson et al. (2007). o Friend et al. (1990).
CMi, XY Ari, AO Psc, IGR J1719-4100, V2400 Oph, BZ UMa, and SS Cyg. We reduce the NuSTAR data using the NuSTAR Data Analysis Software (NSuTARDAS v1.9.3), packaged with HEASOFT v6.25 and the latest CALDB (version 20190314) files. The data reduction is performed using the standard pipeline (nupipeline command in heasoft) and the cleaned event files are produced. We further use the nuproducts command to generate spectra, as well as the rmf and arf files. For each source, a 100″circular region centered on the source is used to extract the source spectra, and a cocentered annulus with inner and outer radii of 130″and 200″to extract the background spectra. We also vary the radii of the source regions to 70″or 130″, and the the background region to circular regions in the same CCD with the sources, and find that the results are not sensitive to these variations. We then conclude that the spectra extraction procedures are robust. We group all spectra using grppha so that the signal-to-noise ratio of each bin exceeds three.
We reduce the Suzaku data with the standard pipeline aepipeline with the latest calibration files (XIS: 20181010, HXD: 20110913 and XRT: 20110630). For each XIS screening image, we use xselect tools to extract the source events from a 200″circular region (120″circular region if the source is too close to the CCD edges) and background events from a 250″ to 400″annulus, excluding regions outside CCD or contaminating sources. The results are not sensitive to the exact selection of the background, because the sources are all quite bright. For HXD data, the background files are downloaded from the Suzaku background FTP server and the spectra are generated with the hxdpinxbpi tool. All XIS and HXD spectra are regrouped so that the signal-to-noise ratio of each bin exceeded three.
Following Yu et al. (2018) and Shaw et al. (2018), the T max of individual CVs with available NuSTAR observations is measured by fitting the 3-50 keV NuSTAR spectra with an absorbed mkcflow model, pha×(mkcflow+Gaussian), or pha×pcfabs×(mkcflow+Gaussian) if additional absorption is needed. The mkcflow model describes the X-ray emission, and the Gaussian represents the fluorescent Fe Kα lines centering around 6.4 keV, respectively. The pha and pcfabs components describe the foreground and intrinsic absorption of the CV, respectively. The values of T max would vary up to ∼5% if the IPM model 1 was adopted, hence we conclude that the mkcflow model is robust. For CVs without NuSTAR observations, their T max are measured by fitting the 3-50 keV Suzaku spectra with the same model as for the NuSTAR spectra.
The I 7.0 /I 6.7 values of individual CVs are adopted from Xu et al. (2016) except XY Ari. The I 7.0 /I 6.7 of XY Ari is remeasured to be 0.94±0.2, which is consistent with the recent XMM-Newton observations (Zengin Camurdan et al. 2018), and is higher than the value (0.62 ± 0.10) obtained by Xu et al. (2016).
The theoretical I 7.0 /I 6.7 -T max -M WD relations are derived separately for IPs and non-mCVs, following Xu et al. (2016) and Yu et al. (2018). Briefly, we generate a series of simulated spectra, by using the mkcflow model and assigning different T max (hence M WD ) values. The simulated spectra are then fitted, the corresponding I 7.0 /I 6.7 measured, in the exact same way as for the real spectra analyzed above. Table 2 summarizes the fitting results of individual CVs, also listed are the WD masses measured dynamically (if available) and those derived from I 7.0 /I 6.7 and T max . In general, the model fitting is acceptable, judged by the c n 2 values. We present in Figure 1 the 3-50 keV NuSTAR spectra, together with the bestfitted models, for two CVs (BG CMi and SS Cyg) as an example. Figures 2 and 3 show I 7.0 /I 6.7 versus T max , and I 7.0 /I 6.7 versus dynamical M WD of the sampled sources, respectively. EX Hya and BV Cen are not included in Figure 3 due to multiple dynamical M WD values. The I 7.0 /I 6.7 -T max and I 7.0 /I 6.7 -M WD relations predicted by the mkcflow model are plotted as solid and dashed curves in both figures, which are to be contrasted with sampled CVs. We present the predicted relations for 0.1 and 1 solar abundances in both figures to cover different populations of CVs following Yu et al. (2018), e.g., those in the solar neighborhood/Galactic bulge and near the Galactic center, respectively. It can be seen that the individual CVs generally follow the predicted I 7.0 /I 6.7 -T max and I 7.0 /I 6.7 -M WD relations in wide ranges (0.2-1.0 for I 7.0 /I 6.7 and 10-60 keV for T max , respectively), especially those of a subsolar metallicity (Z = 0.1). This might reflect the relatively low metallicity of the sample CVs (with a mean Z ∼ 0.3, see Nobukawa et al. 2016). We then conclude that I 7.0 /I 6.7 is a good indicator of T max .

Comparison to Previous Studies and Limitations
Various studies on solar neighborhood CVs have been carried out previously using different instruments.  Table 2, the T max in this work are, in general, consistent with previous measurements (Suleimanov et al. 2005(Suleimanov et al. , 2019Shaw et al. 2018;Yu et al. 2018 Byckling et al. 2010). We speculate that the differences result from the uncertain background of Suzaku HXD, which was used in these previous works, as suggested by Shaw et al. (2018). Actually, unlike the old values, the new T max of SS Cyg closely follow the I 7.0 /I 6.7 -T max relation (see Figure 2, also see Figure 2 of Yu et al. 2018). This consistency further shows the advantage of I 7.0 /I 6.7 to derive T max comparing to the continuum fitting method.
The limitations in this work are addressed as follows. First, the reflection component and the magnetospheric radius of WDs were not considered when fitting the continuum in this work, which may add uncertainties to measured T max values, as discussed by Suleimanov et al. (2019) and Shaw et al. (2018). The modeling of the intrinsic absorption of IPs may also affect the measured T max (e.g., Mukai et al. 2003;Mukai 2017). All these factors may add complications to measured T max . Further Notes.Sources in bold font were observed by both NuSTAR and Suzaku, others were observed by Suzaku only. T max from previous works are also listed for comparison. N H and N H, pc represent foreground and partial covering absorption column density, respectively. C.F. is the covering fraction. a T max from this work. b T max from previous works, including b1: T max derived from the M WD values from Suleimanov et al. (2019), assuming the accreted matter falls from infinity; b2: T max from Shaw et al. (2018); b3: T max from Yu et al. (2018). c M WD derived from I 7.0 /I 6.7 of this work. d M WD derived from T max of this work. e Dynamically measured M WD . f The magnetospheric radius is taken into consideration when deriving M WD of EX Hya from its I 7.0 /I 6.7 and T max values (Suleimanov et al. 2019).
investigations on these issues are necessary to improve the T max -I 7.0 /I 6.7 -M WD relations. Second, the sample size is still small. As the currently best available CV sample, our sample only includes 25 CVs (only 11 of which have dynamical M WD measurements), which is obviously statistically incomplete, and might be biased to relatively bright sources. Moreover, our sample lacks WDs more massive than 1.2 M e , which could restrict the application of the relations to less massive WDs. The derived T max -I 7.0 /I 6.7 -M WD relations should be checked against less luminous CVs, and CVs with more massive WDs in the future.
Third, the dynamical mass uncertainties are large. EX Hya and BV Cen have multiple, inconsistent dynamical mass measurements, so they have to be excluded from the analysis. For the other 11 CVs presented in Figure 4, the typical error range of optically determined M WD is ∼0.1-0.2 M e (see Table 1), which is already comparable, if not greater than those of M WD derived from I 7.0 /I 6.7 and T max (∼0.05-0.1 M e , see Table 2). As a result, the uncertainties in the I 7.0 /I 6.7 -M WD and I 7.0 /I 6.7 -T max relations are dominated by the dynamically measured M WD values. Furthermore, careful calibrations of the dynamical WD masses in CVs may be necessary, because the presence of the "hot spot" or the noncircular motions in the outer accretion disk could distort the radial velocity curves of the optical emission and absorption lines (Marsh et al. 1987;Hessman et al. 1989). More reliable WD masses measurements are needed to improve the I 7.0 /I 6.7 -M WD relations.
4.2. I 7.0 /I 6.7 as a Diagnostic of T max and M WD of CVs Judged from Tables 1 and 2, I 7.0 /I 6.7 is a good indicator of T max ; however, is it also a good diagnostic for M WD ? To address this issue, we compare the M WD derived from I 7.0 /I 6.7 (assuming 0.1 solar abundance) and T max to the dynamically measured values for both IPs and non-mCVs in Figure 4. It is obvious that all M WD derived from I 7.0 /I 6.7 are consistent with the dynamical measured values. On the other hand, although M WD derived from T max in general show smaller uncertainties, there is one CV, EK TrA, whose derived M WD is not consistent with the dynamical value.
To quantify the goodness of the derived M WD , we assume the following linear relation M WD,derived =A×M WD,dynamical +B and perform fitting for derived M WD . The best-fit yields A=0.97±0.09 and B=0.06±0.09, with c = n 1.6 2 and r 2 =0.94 for I 7.0 /I 6.7 derived M WD , and A=0.62±0.02 and B=0.38±0.03, with c = n 8.0 2 and r 2 =0.91 for T max derived M WD . Judged from the fitting results, I 7.0 /I 6.7 derived M WD are more consistent with the optical ones. This comparison does not necessarily imply that I 7.0 /I 6.7 is intrinsically a better indicator of M WD compared to T max , since the latter may be biased due to data quality and continuum modeling. Nevertheless, based on the current data, this comparison suggests that I 7.0 /I 6.7 is a good diagnostic of the M WD of both IPs and non-mCVs compared to T max .

Summary
We have systematically analyzed NuSTAR and Suzaku observations in a sample of 25 solar neighborhood CVs, including 16 IPs and 9 non-mCVs to investigate their T max -I 7.0 /I 6.7 -M WD relations. Our main results can be summarized as follows: (a) The measured T max are in general consistent with previous results except SS Cyg, which shows a lower temperature (26.9 ± 1.4 keV) compared to previous results (∼40-50 keV). (b) I 7.0 /I 6.7 of both IPs and non-mCVs follow the theoretical I 7.0 /I 6.7 -T max relation, which covers a wide I 7.0 /I 6.7 range of ∼0.1-1.0, and a wide T max range of ∼10-60 keV. (c) The M WD derived from I 7.0 /I 6.7 are more consistent with the dynamically measured values compared to those derived from T max , showing that I 7.0 /I 6.7 is a good diagnostic of M WD in CVs.