Constraining the Accretion Geometry of the Intermediate Polar EX Hya Using NuSTAR, Swift, and Chandra Observations

In magnetic cataclysmic variables (CVs) the primary is a highly magnetized ($B\gtrsim {10}^{6}\,{\rm{G}}$) white dwarf (WD) whose field controls the accretion flow close to the WD, leading to shock and accretion columns that radiate chiefly in X-rays. The shock temperature kT_sh is determined by the pre-shock velocity and is of the order of 10–50 keV in magnetic CVs. The post-shock plasma must further decelerate and cool before it can settle onto the WD. Thus, the height of the shock (h_sh) is determined by equating the plasma cooling time with the remaining travel time from the shock front to the WD surface (Aizu 1973). Since the X-ray cooling time is inversely proportional to the density of the post-shock plasma, h_sh is small if the accretion rate per unit area (or specific accretion rate) is high. If h_sh is a small fraction of the white dwarf radius (R_WD), and if the accretion flow can be considered to be freefalling from infinity, then kT_sh is an immediate indicator of the white dwarf mass (M_WD). This works well for most intermediate polars (IPs). However, if the specific accretion rate is low, h_sh may not be negligible. This would reduce the freefall velocity above the shock, and hence kT_sh. Also, if accretion is from a truncated disk with a small inner radius R_in, the pre-shock velocity is set by the freefall condition from R_in, requiring a different correction (Suleimanov et al. 2005; Luna et al. 2015).

The subject of this Letter, EX Hya, is a unique IP that has raised several important, and still unresolved questions. One is the very nature of its accretion flow. A standard, Keplerian, partial disk cannot be present in this system if the WD is in spin equilibrium, because R_in (given the long spin period, 67 minutes, relative to the orbital period, 98 minutes) would be so large as to violate the physical condition for the formation of a disk (King & Lasota 1991). Either the WD is far out of equilibrium, or EX Hya possesses a diamagnetic blob/ring type structure between the magnetosphere and L1 (King & Wynn 1999; Norton et al. 2008).

The other major unresolved question is why the X-ray spectrum of EX Hya is so soft. The combination of partial eclipse and optical spectroscopy has led to an estimate of M_WD = 0.78 ± 0.03 M_⊙ (Echevarria et al. 2016), implying kT_sh ∼ 35 keV in the Aizu picture, while X-ray measurements are consistently below ∼20 keV (see, e.g., Luna et al. 2015). This can be resolved by either having a small R_in or a large h_sh.

In EX Hya, h_sh has been inferred through indirect arguments and some of them have yielded opposing answers. Allan et al. (1998) studied the spin modulation and the partial eclipse in the ASCA data, and argued for a tall (${h}_{\mathrm{sh}}\sim 1$ R_WD) shock as the explanation for the spin modulation, and also a large (R_in > 6.1 R_WD) inner disk radius. A smaller R_in would result in the accretion disk blocking our view of the lower pole. Other arguments supporting that ${{\rm{R}}}_{\mathrm{in}}\,\gt \,$ a few R_WD and that thush_sh is a non-negligible fraction of R_WD are: (i) the equilibrium spin period is expected to be close to the Keplerian period at R_in; for small values of R_in, the spin period would be smaller than the observed 67 m; and (ii) Hellier et al. (1987) analyzed an extensive set of optical spectra and found three components in the line profile: a narrow S-wave component; a double-peaked component; and a broad, spin-modulated component. Measuring the width at spin phase 0.5 allowed them to put an upper limit on the high-velocity extent of the double-peaked, presumably accreting disk, concluding that for a 0.78 M_WD, R_in is about 10 R_WD.

On the other hand, Revnivtsev et al. (2011) and Semena et al. (2014) modeled the break frequency in the power spectrum of stochastic variability and proposed R_in = 2.7 R_WD. Belle et al. (2003), Suleimanov et al. (2016) and Echevarria et al. (2016) also derived small R_in using other methods. This small R_in would reduce kT_sh: this was the solution preferred by Luna et al. (2015), who analyzed high-quality, half a megasecond, Chandra HETG data on EX Hya and found a number of discrepancies with respect to the standard models of X-ray emission. A tall shock would imply that gravity adds heat to the cooling flow region and the magnetic field geometry adds heat by magnetic pressure. Both mechanisms should modify the emission measure distribution of the cooling plasma, increasing both the flux of H- and He-like lines. Such models did not match the Chandra/HETG observations.

An observational determination of h_sh is essential for deciding which picture is correct. Because of the accretion geometry, we expect the X-rays emitted in the post-shock region to be reflected back into our line of sight by the WD surface, producing a detectable Compton hump at energies above 10 keV (e.g., Hayashi et al. 2018). The detection of the reflection feature in three IPs has been possible only recently with NuSTAR (Mukai et al. 2015). Constraining the presence of a reflection component in the hard X-ray spectrum of EX Hya would allow us to determine h_sh. A tall shock implies little-to-no reflection with a weak Fe Kα 6.4 keV fluorescence line and small or non-existent spin modulation above a few keV, since photoelectric absorption on the order of 10²² cm⁻² cannot affect the light curves at these energies (the measured ${N}_{H}$ is even lower) and the shock region would not be hidden by the WD body. On the other hand, a negligible shock height would imply a strong reflection amplitude and a strong Fe Kα line, while the spin modulation would be almost entirely due to absorption and thus the pulsed fraction of the light curves should be a strong function of the energy. Somewhere in the middle, a shock height of a non-negligible fraction of the WD radius (∼0.1−0.5 R_WD) would imply a moderate reflection amplitude (less than 1), potentially detectable with NuSTAR. The modulation of the low-energy X-rays (E ∼ 5–10 keV) would be mostly due to occultation of the accretion column by the body of the WD, while the expected modulation at higher energies would be entirely due to occultation, so the pulsed fraction at high energies should not be a function of energy. To perform this test, we have observed EX Hya with NuSTAR.

There is an additional reason why it is important to perform a NuSTAR observation of EX Hya. While it was proposed to be the counterpart of a Uhuru source from the early days of X-ray astronomy (Warner 1972), and it is indeed the brightest source in the traditional (0.5–10 keV) X-ray band among all CVs, it is not the only bright X-ray source in this region of the sky. The interacting cluster of galaxies, Abell 3528, consists of two X-ray bright subclusters (Gastaldello et al. 2003), located ∼29 arcmin from EX Hya. Therefore, the possibility of contamination must be kept in mind for interpreting any non-imaging X-ray observations of EX Hya. This means, in part, that there have been no reliable observations of EX Hya above 10 keV until now.

In this Letter, we present new contemporaneous NuSTAR and Swift observations of EX Hya, and new analysis of archival Chandra observation. We present conclusive arguments for an absence of reflection in the X-ray spectrum, implying that the height of the shock must be an appreciable fraction of the size of the WD. In Section 2 we detail the reduction of NuSTAR, Swift, and Chandra data, while Section 3 presents the results from the spectral and timing analysis. Finally, Section 4 presents a discussion about the implications of the non-detection of reflection for the structure of the accretion column.

We observed EX Hya with NuSTAR on 2016-06-05 for 24.8 ks. The data were reduced using the NuSTAR Data Analysis Software as part of HEASOFT 6.21 and filtered using standard filters given that the observation was not affected by abnormal solar activity. Using the tool nuproducts we extracted source spectra, baricenter-corrected source event files, and light curves from a circular region centered on the SIMBAD coordinates, $\alpha ={12}^{{\rm{h}}}$ 52^m 185, $\delta =-29^\circ 16^{\prime} 16^{\prime\prime}$ in the FPMA chip and α = 12^h 52^m 238, δ = −29° 14' 551 in the FPMB chip, with a 30'' radius. For the background, we choose an annular region with inner and outer radii of 110'' and 220'', respectively, and centered on the respective source coordinates.

A Swift observation was obtained almost simultaneously with NuSTAR, with 1.8 ks exposure time. We extracted source X-ray spectra from a circular region with a radius of 20 pixels centered on the SIMBAD coordinates. We extracted background events from an annular region with inner and outer radii of 25 and 40 pixels, respectively. We built the ancillary matrix (ARF) using the tool xrtmkarf and used the swxpc0to12s6_20130101v014.rmf response matrix provided by the Swift calibration team.

EX Hya was observed with Chandra using the ACIS-S/HETG combination for 496 ks and the spectral analysis has been already described in Luna et al. (2010; see also Luna et al. 2015). For the present study, the events arrival times were barycentrically corrected using the axbary script and filtered to extract event arrival times from the source in the energy regions of interest (strongest emission lines), selecting only HEG and MEG ± 1 orders.

3.1. Spectral Model

3.2. The Uncontaminated Hard X-Ray Flux

3.3. Timing Analysis: Power Spectrum and Pulsed Fraction

In order to study the dependence of the WD spin-pulsed fraction with energy from the photon arrival times we calculated the ${Z}_{1}^{2}$ (Rayleigh) statistic (Buccheri et al. 1983) as a function of frequency in the range 0.00022 Hz $\lt \,f\,\lt 0.0003$ Hz (the WD spin period is 67.02696576 minutes or 0.00024865614 Hz; Mauche et al. 2009) with a step ${\rm{\Delta }}f=1.0/(T\times q$), where T is the exposure time and q is the oversampling factor, which we took to be equal to 1000. As the Swift data do not cover a single spin period, we did not include them in this analysis. The value of ${Z}_{1}^{2}$ needed to detect a pulsation with a probability $P=2.699\times {10}^{-3}$ (3σ detection) is ${Z}_{1}^{2}\gt 2\mathrm{ln}(\tfrac{T{\rm{\Delta }}f}{P})$. If the peak in the power spectrum is due to nearly sinusoidal modulations, the pulsed fraction is $p={p}_{\mathrm{obs}}({N}_{S}+{N}_{B}){N}_{S}^{-1}$, where N_S and N_B are the number of source and background counts and p_obs is the observed pulsed fraction uncorrected by the background. However, as EX Hya is a very bright X-ray source, the background contribution is negligible and the pulsed fraction can be expressed as $p\simeq \sqrt{2{Z}_{1}^{2}}{N}_{S}^{-1/2}\pm 2{N}_{S}^{-1/2}$.

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In Figure 2 we plot the pulsed fraction of those spin periods detected with >3σ significance in the energies of the strongest emission lines as observed in the Chandra/HETG spectra (and whose fluxes were measured in Luna et al. 2015) and the broad energy bands of 3–6, 6–9, and 9–12 keV in the NuSTAR data. Modulation at the spin period is detected up to energies of less than ∼12 keV in the NuSTAR data. We found that there is a dependence of the pulsed fraction with energy. For energies of less than about 1 keV, the pulsed fraction seems to be constant with energy. On the other hand, for energies greater than 1 keV, the pulsed fraction decays with energy. The low absorption cannot be responsible for the modulation at these energies The origin of this effect remains a mystery. The non-detection of pulsation at energies greater than ∼12 keV indicates that the height of the shock is at least greater than 1R_WD but not big enough to get the lower pole occulted by the inner region of the accretion disk.

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We can qualitatively understand the energy dependence of spin modulation amplitude as a consequence of the height dependence of both the physical condition and the visibility. In the post-shock region, the temperature is at its highest near the shock and declines toward the white dwarf surface. The density, on the other hand, is at its lowest near the shock and increases near the surface. Continuum photons of energy E originate from regions where ${kT}\gtrsim E;$ line photons originate from a limited range of temperatures (e.g., Ne X lines require temperatures of the order of 0.54 keV; neither regions that are too cool or too hot contribute significantly). Thus, photon energy plotted along the x-axis of Figure 2 is a proxy for the origin of these photons within the post-shock region. If the shock is tall, of the order of ${h}_{\mathrm{sh}}\sim 1$ R_WD, then the highest temperature continuum will escape self-occultation almost completely. The lowest energy lines are emitted only near (but still above) the white dwarf surface. When we view the poles at a right angle, both poles are visible; half a spin cycle later, most of the lower pole is behind the body of the white dwarf, with a small residual that depends on the geometrical extent of the accretion footpoint. In Figure 3, we show the result of a proof-of-concept simulation, in which light curves for uniform emission regions with a limited range of h_sh have been simulated. Following the two-stage process explained in Mukai (1999), the arc-shaped accretion footpoints were calculated assuming a rigid magnetic dipole with a magnetic colatitude of 5°, accreting uniformly from a transition region at the inner disk edge at 9–10 R_WD (each pole accreting from an 180° azimuth). A quantitative model including the location and shape of the threading region, and the resulting shock structure, is beyond the scope of this paper.

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The non-detection of a reflection component in the hard X-ray spectrum taken with NuSTAR, the small Fe Kα equivalent width of ∼25 eV, and the absence of spin modulation for energies greater than ∼12 keV implies that h_sh must be an appreciable fraction of the WD radius. This can be used to distinguish between the two possible origins (large h_sh and small R_in) of low kT_sh in EX Hya.

In Figure 4, the top panel shows the reflection amplitude as a function of h_sh for a point-like emitting region. The upper limit on the reflection amplitude of $\lesssim 0.15$ (see Table 1) implies ${h}_{\mathrm{sh}}\lesssim 0.9$ × R_WD. Also, the EW of FeKα implies a reflection amplitude of about 0.15 in the model presented by George & Fabian (1991). The bottom panel in Figure 4 shows a set of four curves of kT_sh as a function of h_sh, for four different assumed values of R_in, for a 0.78 M_⊙ white dwarf, with a horizontal line at 19.7 keV, the kT_sh that we measured. Accounting for the error bar in the kT_sh, the reflection amplitude upper limit, and our simplified reflection model, the data presented here are compatible with a large R_in. Moreover, given the lack of strong reflection signature, a solution of R_in = 2 × R_WD, ${h}_{\mathrm{shock}}=0$ is clearly untenable for EX Hya.

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Table 1.  Spectral Fit Results

Data	Model	${\chi }_{\nu }^{2}$/dof	${{kT}}_{\max }$ [keV]	$\mathrm{Fe}/{\mathrm{Fe}}_{\odot }$	Amplitude
NuSTAR	brems × ref	1.27/212	9.9 ± 0.6	1	$\lesssim 0.29$
NuSTAR	brems × ref	1.27/212	9.9 ± 0.6	0.6	$\lesssim 0.29$
NuSTAR	vmcflow × ref	1.19/347	13.4 ± 0.8	0.6	1.24 ± 0.33
NuSTAR	vmcflow	1.33/348	17.7 ± 0.4	0.6	⋯
NuSTAR	vmcflow	1.06/347	18.2 ± 0.5	0.87 ± 0.05	⋯
NuSTAR	vmcflow × ref	1.06/346	${18.3}_{-0.6}^{+0.4}$	0.87 ± 0.04	$\lesssim 0.08$
NuSTAR + Swift + Chandra	vmcflow × ref	1.17/1900	19.7 ± 0.4	0.88 ± 0.02	$\lesssim 0.15$
NuSTAR + Swift + Chandra	vmcflow	1.17/1989	19.7 ± 0.4	0.88 ± 0.02	⋯

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The inferred large R_in is still much smaller than the co-rotation radius for a WD rotating with a 67-minute period. In the diamagnetic blob scenario proposed by King & Wynn (1999) and Norton et al. (2008), which implies a perturbed accretion flow that deviates from purely Keplerian velocity structure, it is possible that the WD is in spin equilibrium. Note that the magnitude of the spin-up in EX Hya does not stand out among all IPs, most of which are presumably in spin equilibrium. Moreover, in this scenario, the very premises of the model of stochastic variability used by Revnivtsev et al. (2011) and Semena et al. (2014), as well as other determinations of R_in based on strictly Keplerian flows, are suspect. Alternatively, EX Hya may possess a purely Keplerian accretion disk with a large R_in: in this case, the WD is far out of equilibrium, and we have no concrete explanation for the disagreement with the break frequency method. One remaining problem that still needs to be explained is the detailed soft X-ray emission line spectroscopy in the Chandra HETG data.

This research has made use of data obtained with the NuSTAR mission, a project led by the California Institute of Technology (Caltech), managed by the Jet Propulsion Laboratory (JPL) and funded by NASA. M. Orio acknowledges support from NASA grant NNX17AB76G. We acknowledge the Swift team for planning this observation. G.J.M.L. is a member of the CIC-CONICET (Argentina) and acknowledges support from grants PIP-Conicet/2011 #D4598, ANPCYT-PICT 0478/14.

Facilities: NuSTAR - The NuSTAR (Nuclear Spectroscopic Telescope Array) mission, Swift, and Chandra - .