A HARD X-RAY POWER-LAW SPECTRAL CUTOFF IN CENTAURUS X-4

NuSTAR, the first focusing hard X-ray observatory in orbit, was launched in 2012 and operates in the 3–79 keV range (Harrison et al. 2013). It consists of two coaligned telescopes, and the two focal planes (FPMA and FPMB) are each covered by a 2 × 2 array of cadmium-zinc-telluride (CZT) pixel detectors. Our NuSTAR observation of Cen X-4 (ObsID 30001004002) began on 2013 January 20, 20:20 UT and had an elapsed duration of 219 ks, with an on-source exposure time of 114 ks. The source was imaged on detector 0 in both FPMA and FPMB. The data were processed and screened using the standard pipeline for on-axis point sources in the NuSTAR Data Analysis System (nustardas) version 1.2.0, along with the NuSTAR calibration database (CALDB) version 20130509. Light curves and spectra from both FPMA and FPMB were extracted from a circular region centered on the source position with a radius of 75 arcsec.

A detailed background model for the source position in each of the two telescopes was derived by using the nuskybgd tool (Wik et al. 2014) to fit blank sky regions covering the entire field of view for each focal plane. For faint point sources like Cen X-4, this is more accurate than the usual method of simply scaling from the background of a nearby blank sky region because it correctly accounts for the highly nonuniform background gradients across the detectors. In both telescopes, the background was brighter than the source above around 20 keV. The background-subtracted count rates in FPMA and FPMB were (4.51 ± 0.09) × 10⁻² count s⁻¹ and (4.08 ± 0.09) × 10⁻² count s⁻¹, respectively. The NuSTAR spectra were rebinned so that all but the highest energy bin had a background-subtracted significance of at least 10σ. The highest energy bin had 5.2σ significance in FPMA (20–79 keV) and 4.3σ significance in FPMB (17–79 keV), demonstrating that the source was detected beyond 20 keV.

The XMM-Newton observatory, launched in 1999, is a focusing X-ray telescope operating in the 0.3–12 keV range (Jansen et al. 2001). Our XMM-Newton observation of Cen X-4 (ObsID 0692790201) began on 2013 January 21, 13:01 UT and had a duration of 35 ks; this was simultaneous with part of our NuSTAR observation. We used the data from all three coaligned imaging X-ray cameras (EPIC-pn, EPIC-MOS1, and EPIC-MOS2; Strüder et al. 2001; Turner et al. 2001). The cameras were operated in full-frame mode with the thin optical-blocking filter in place, resulting in a time resolution of 73 ms for EPIC-pn and 2.6 s for the MOS cameras. The data were reduced using the XMM-Newton Scientific Analysis System (SAS) v13.0.1, along with the latest calibration files available as of 2013 July 13. We reprocessed the data using epproc and emproc to produce new event files and applied standard event filtering. We identified background flare intervals by constructing a light curve of the 10–12 keV EPIC-pn data using events from the entire field of view and searching for intervals where the count rate exceeded 1 count s⁻¹. We found three short flares; these intervals were removed from our Cen X-4 event lists. The net exposure time was 27 ks for EPIC-pn and 30 ks for each of the EPIC-MOS units.

For all three detectors, we initially extracted light curves and spectra from a circular region centered on the source position with a radius of 43.5 arcsec. For the MOS detectors, we measured the background using a square blank-sky region 3 × 3 arcmin in size. For the EPIC-pn detector, an important consideration is that the outer parts of the field of view include photons that are due to Cu fluorescence in the instrument while the inner parts do not. Because Cen X-4 was in the region without the Cu emission line, we chose a rectangular region near Cen X-4 for determining the EPIC-pn background. The background-subtracted count rates in the 0.3–10 keV range were 2.58 ± 0.01 count s⁻¹ in EPIC-pn, 0.640 ± 0.005 count s⁻¹ in EPIC-MOS1, and 0.619 ± 0.005 count s⁻¹ in EPIC-MOS2.

These count rates are higher than those observed in previous XMM-Newton observations (Cackett et al. 2010). In fact, the count rates in both the EPIC-MOS and EPIC-pn detectors are close to the threshold where photon pileup effects may begin to distort the measured spectra, particularly during the flares.¹⁵ Moreover, our preliminary analysis found that the two EPIC-MOS spectra each have significant systematic differences with the EPIC-pn spectrum above 2 keV. As a precaution, we reextracted both the MOS and pn data from an annular region centered on the source position with an outer radius of 43.5 arcsec and an inner radius of 10 arcsec, thus excluding the core of the point-spread function (PSF) where any pileup would occur (at the expense of reduced counting statistics). The MOS and pn spectra from the annular region are mutually consistent. This annulus-only data set still includes sufficient counts to obtain a good measurement of the soft X-ray spectrum.

The light curves for the NuSTAR and XMM-Newton observations are shown in Figure 1 and the top panel of Figure 2, respectively. (The XMM-Newton light curve uses the full data set, not the annulus-only data.) Both light curves vary significantly on timescales of a few minutes. The XMM-Newton light curve clearly exhibits flaring activity; the strongest of these flares, around MJD 56313.8, is also easily visible in the NuSTAR light curve. The fractional excess rms variability¹⁶ F_var was 37% ± 6% in the NuSTAR light curve and 37% ± 2% in the XMM-Newton/EPIC-pn light curve. For comparison, a value of 73.0% ± 1.5% was measured in 60 short Swift observations made over three months (Bernardini et al. 2013).

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A comparison of the 0.3–1 keV and 2–10 keV XMM-Newton/EPIC-pn light curves shows no evidence for significant spectral changes during these flares, although there is weak evidence for a small change during the flare at MJD 56313.63 (Figure 2, bottom panel; Figure 3). A cross-correlation analysis indicates that the flares in these two bands are simultaneous to within ≲30 s. We searched the NuSTAR light curve for evidence of orbital variability by folding at the 15.1 hr binary period (Shahbaz et al. 2014). No orbital modulation was detected.

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Because there is no evidence for significant spectral variability during the flares (Figure 2, bottom panel; Figure 3), we chose to include the flare intervals for our spectral analysis. We performed joint spectral fits of the NuSTAR and XMM-Newton (annulus) data with XSPEC version 12.8 spectral fitting software (Arnaud 1996) using χ²-minimization. To allow for systematic calibration differences between the different detectors, we included a constant multiplicative factor in the model. This constant was set to unity for XMM-Newton/EPIC-pn but allowed to vary for the other detectors. All other spectral model parameters were tied together across the three instruments. Interstellar absorption was modeled using the tbabs model (Wilms et al. 2000) along with photoionization cross sections from Verner et al. (1996).

We tried two different models for the soft thermal component. We first fit a passively cooling NS atmosphere using the nsatmos model (Heinke et al. 2006), which assumes a pure H atmosphere and a negligible surface magnetic field (B ≲ 10⁸ G) and also includes the effects of surface gravity, heat conduction by electrons, and self-irradiation. In applying nsatmos, we fixed the source distance at 1.2 kpc (Chevalier et al. 1989) and assumed that the atmospheric emission was coming from the entire NS surface. We found that it was not possible to constrain the NS mass M and radius R when both were allowed to vary because a wide range of M-R pairs gave acceptable fits. We therefore fixed the NS mass to M = 1.9 M_☉ (Shahbaz et al. 2014). As an alternative model, we also tried fitting to the synthetic spectra of Zampieri et al. (1995, 2001) for thermal emission from unmagnetized NSs with a pure H atmosphere in the presence of very low accretion rates. These spectra have been implemented as the XSPEC additive table model zamp and are parameterized in terms of the observed accretion luminosity scaled to the Eddington rate (L_∞/L_E), with fixed NS mass M = 1.4 M_☉ and true radius R = 12.4 km. In practice, the model shapes from nsatmos and zamp are essentially identical (see, e.g., Soria et al. 2011), but it is useful to demonstrate that models that explicitly include accretion are consistent with the data.

Although previous 0.5–10 keV X-ray observations of Cen X-4 in quiescence were all well fit by an absorbed passive NS atmosphere plus power-law model (see Cackett et al. 2010, and references therein), the tbabs*(nsatmos+powerlaw) model does not provide a satisfactory fit to the high-energy data in our combined NuSTAR+XMM-Newton data covering the 0.3–79 keV range. The residuals indicate the presence of a spectral break or cutoff in the 10–20 keV range (see Figure 4). We found several different models that provided a good fit for the hard component: a cutoff power law (cutoffpl), a broken power law (bknpower), thermal Comptonization (comptt; Titarchuk 1994; Hua & Titarchuk 1995), and thermal bremsstrahlung (bremss). In each case, this component was combined with tbabs and either nsatmos or zamp. In all cases, the N_H value is consistent with the integrated interstellar value along the line of sight (Dickey & Lockman 1990; Kalberla et al. 2005). We found no evidence for fluorescent Fe line emission in the spectrum. The upper limit on the equivalent width of a narrow Gaussian Fe emission line at 6.4 keV is <120 eV (90%-confidence).

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The spectral fits with nsatmos are summarized in Table 1; those with zamp are summarized in Table 2. A typical fit is shown in Figure 5. The corresponding unfolded spectrum is shown in Figure 6. We are able to obtain reasonably good fits with a wide variety of models. We note that the only spectral shape parameter in the zamp model is the accretion luminosity L_∞ observed at infinity; one must compare this with the measured flux and source distance to check for self-consistency. We find that the best-fit values of L_∞ are roughly consistent with the measured soft flux for the assumed distance of 1.2 kpc.

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Table 1. Spectral Fits with a Passive Neutron Star Atmosphere^a

Parameter	Symbol	Units	Hard Spectral Component
			cutoffpl	bknpow	comptt(disk)	comptt(sphere)	bremss
Absorption and nsatmos parameters
Absorption column	NH	1021 cm−2	0.88(4)	0.89(3)	0.92(4)	0.92(4)	0.87(5)
Temperature (unredshifted)	log T	K	6.16(4)	6.18(1)	6.20(27)	6.19(23)	6.20(10)
Mass (fixed)	M	M☉	1.9	1.9	1.9	1.9	1.9
Radius	R	km	9.2(1.0)	8.6(2.6)	8.5(5.7)	8.6(4.9)	8.1(1.8)
Covering fraction (fixed)	KNS	⋅⋅⋅	1	1	1	1	1
Distance (fixed)	D	kpc	1.2	1.2	1.2	1.2	1.2
Hard spectral component parameters
Normalization	K	10−4	2.3(2)b	2.2(2)b	0.7(1)b	0.7(1)b	4.1(1)c
Photon index	Γ	⋅⋅⋅	1.02(10)	1.26(10)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Cutoff or break energy	Ec, Eb	keV	10.4(1.4)	5.8(3)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Photon index 2	Γ2	⋅⋅⋅	⋅⋅⋅	2.02(5)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Electron temperature	kTe	keV	⋅⋅⋅	⋅⋅⋅	6.4(1.1)	6.4(1.1)	18.2(1.0)
Seed temperature	kT0	keV	⋅⋅⋅	⋅⋅⋅	0.74(8)	0.74(7)	⋅⋅⋅
Scattering optical depth	τes	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	4.0(5)	8.8(1.0)	⋅⋅⋅
Instrument multiplicative constants
NuSTAR/FPMA	CFPMA	⋅⋅⋅	1.04(3)	1.03(3)	1.02(3)	1.02(3)	1.09(3)
NuSTAR/FPMB	CFPMB	⋅⋅⋅	1.07(4)	1.05(4)	1.05(4)	1.05(4)	1.12(4)
XMM/EPIC-pn (fixed)	Cpn	⋅⋅⋅	1	1	1	1	1
XMM/EPIC-MOS1	CMOS1	⋅⋅⋅	1.04(1)	1.04(1)	1.04(1)	1.03(1)	1.04(1)
XMM/EPIC-MOS2	CMOS2	⋅⋅⋅	0.97(1)	0.97(1)	0.97(1)	0.97(1)	0.97(1)
Fit statistic	$\chi ^2_\nu$/dof	⋅⋅⋅	1.155/380	1.123/379	1.130/379	1.129/379	1.167/381

Notes. ^a1σ uncertainties in last digits shown in parentheses. ^bFlux density at 1 keV in units of photons cm⁻² s⁻¹ keV⁻¹. ^c$(3.02\times 10^{-15}/4\pi D^2)\int n_e^2 dV$ in units of cm⁻⁵.

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Table 2. Spectral Fits with an Accreting Neutron-star Atmosphere^a

Parameter	Symbol	Units	Hard Spectral Component
			cutoffpl	bknpow	comptt(disk)	comptt(sphere)	bremss
Absorption and zamp parameters
Absorption column	NH	1021 cm−2	0.89(3)	0.87(6)	0.69(5)	0.69(5)	0.93(5)
Luminosity	log L∞/LE	⋅⋅⋅	−5.15(3)	−5.13(4)	−4.98(3)	−4.98(3)	−5.19(3)
Mass (fixed)	M	M☉	1.4	1.4	1.4	1.4	1.4
Radius (fixed)	R	km	12.4	12.4	12.4	12.4	12.4
Normalization	Kzamp	10−3	1.25(2)b	1.25(2)b	1.39(2)b	1.39(2)b	1.23(2)b
Hard spectral component parameters
Normalization	K	10−4	2.5(2)b	2.3(2)b	0.6(1)b	0.6(1)b	4.2(1)c
Photon index	Γ	⋅⋅⋅	1.08(9)	1.29(7)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Cutoff or break energy	Ec, Eb	keV	11.2(1.5)	5.9(3)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Photon index 2	Γ2	⋅⋅⋅	⋅⋅⋅	2.03(5)	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅
Electron temperature	kTe	keV	⋅⋅⋅	⋅⋅⋅	7.3(1.6)	7.6(1.7)	17.9(1.0)
Seed temperature	kT0	keV	⋅⋅⋅	⋅⋅⋅	0.81(4)	0.81(4)	⋅⋅⋅
Scattering optical depth	τes	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	3.6(6)	7.8(1.2)	⋅⋅⋅
Instrument multiplicative constants
NuSTAR/FPMA	CFPMA	⋅⋅⋅	1.04(3)	1.03(3)	1.01(3)	1.01(3)	1.08(3)
NuSTAR/FPMB	CFPMB	⋅⋅⋅	1.07(3)	1.05(4)	1.04(4)	1.04(4)	1.10(4)
XMM/EPIC-pn (fixed)	Cpn	⋅⋅⋅	1	1	1	1	1
XMM/EPIC-MOS1	CMOS1	⋅⋅⋅	1.04(1)	1.04(1)	1.03(1)	1.03(1)	1.03(1)
XMM/EPIC-MOS2	CMOS2	⋅⋅⋅	0.97(1)	0.97(1)	0.97(1)	0.97(1)	0.97(1)
Fit statistic	$\chi ^2_\nu$/dof	⋅⋅⋅	1.156/380	1.129/379	1.109/379	1.109/379	1.162/381

Notes. ^a1σ uncertainties in last digits shown in parentheses. ^bFlux density at 1 keV in units of photons cm⁻² s⁻¹ keV⁻¹. ^c$(3.02\times 10^{-15}/4\pi D^2)\int n_e^2 dV$ in units of cm⁻⁵.

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Using the tbabs*(nsatmos+bremss) model, we find an average (absorbed) flux of 3.8 × 10⁻¹² erg cm⁻² s⁻¹ in the soft (0.3–3 keV) band and 3.3 × 10⁻¹² erg cm⁻² s⁻¹ in the hard (3–79 keV) band. The unabsorbed soft luminosity was $L_{\rm soft}=6.6\times 10^{32} D^2_{\rm kpc}$ erg s⁻¹ (0.3–3 keV), the unabsorbed hard luminosity was $L_{\rm hard}=4.0\times 10^{32} D^2_{\rm kpc}$ erg s⁻¹ (3–79 keV), and the total unabsorbed luminosity was $L_x=1.1\times 10^{33} D^2_{\rm kpc}$ erg s⁻¹ (0.3–79 keV), where D_kpc is the source distance in kiloparsecs. For comparison with previous observations, the 0.5–10 keV unabsorbed luminosity was $6.4\times 10^{32} D^2_{\rm kpc}$ erg s⁻¹. This is the highest 0.5–10 keV luminosity ever measured from Cen X-4 in quiescence. The next highest observation was 2.6 times fainter in 2001 (Campana et al. 2004b), and the faintest observation was 11.5 times fainter in 2009 (Cackett et al. 2010). The thermal (NS atmosphere) contribution to the 0.5–10 keV luminosity in our observation is 59%. This is consistent with previous observations, where the thermal fraction has always been 50%–60% (Cackett et al. 2010; Bernardini et al. 2013).

One way to explain a cutoff power-law spectrum is thermal Comptonization, where soft thermal seed photons (like those from the NS atmosphere) are Compton scattered from a corona of hot electrons. Menou & McClintock (2001) have previously argued that Comptonization of the NS atmosphere photons cannot account for the 1–10 keV power law observed in Cen X-4 because the observed luminosity in the power-law component is too high relative to the soft luminosity. We show here that a thermal Comptonization model for the hard spectral cutoff in Cen X-4 is not physically self-consistent, despite providing an acceptable fit to our observed spectrum. For simplicity, we consider the spherical geometry case; the results for a disk geometry are qualitatively similar.

Our comptt fits yielded an electron temperature of kT_e = 6–8 keV and an electron scattering optical depth of τ_es = 8–9 for a spherical geometry (see Tables 1 and 2). This is an unusually low kT_e for a Comptonized plasma, reflecting the observed spectral cutoff. It is likewise an unusually high τ_es; this follows from the fact that kT_e and τ_es are inversely proportional for a fixed power-law index Γ (Titarchuk & Lyubarskij 1995). The optical depth in a corona of radius r_c is related to the electron density n_e by

$$\begin{equation} \tau _{\rm es} = \sigma _T \int _R^{r_c} n_e(r) dr \,, \end{equation} \tag{ 3 }$$

where σ_T is the Thomson cross section. As we see below, it is our high value of τ_es that makes it difficult to find a physically self-consistent Comptonization model.

4.1.1. Comptonization above the NS Atmosphere or the NS Magnetosphere

We begin by considering the possibility of a hot, optically thin layer above the NS atmosphere. Based on our measured τ_es, the electron density in the layer would be $n_e = 1.4\times 10^{22}\,h_3^{-1}$ cm⁻³, where h = 10³ h₃ cm is the thickness of the layer. This relatively dense scattering layer would itself be a source of significant thermal bremsstrahlung emission, with kT_e ≃ 7 keV. Despite the large scattering optical depth τ_es, the effective optical depth of the medium at 2 keV (including both scattering and free–free absorption) is only $\tau _{\rm eff}\simeq \sqrt{\tau _{\rm es}\tau _{\rm ff}}=0.4\, h_3^{-1/2}$ (Rybicki & Lightman 1979), where $\tau _{\rm ff}=0.02\,h_3^{-1}$ is the free–free absorption optical depth at 2 keV. (At higher energies, τ_ff will be even smaller.) We can thus use the emissivity for optically thin thermal bremsstrahlung to compute the expected luminosity from the dense scattering layer, $3.6\times 10^{36}\,h_3^{-1}$ erg s⁻¹ $\approx 10^{-2}\, h_3^{-1}\,L_E$. This is orders of magnitude brighter than what we observe and can be ruled out.

We also consider a corona around the NS magnetosphere. In X-ray quiescence, we can scale the magnetospheric radius r_m to the corotation radius (where the Keplerian and stellar angular velocities are equal), which is given by

$$\begin{equation} r_{\rm co} = \left(\frac{GMP^2}{4\pi ^2}\right)^{1/3} = 39\,P_{\rm 3\,ms}^{2/3}\,M_{1.9}^{1/3}\, {\rm km} \,, \end{equation} \tag{ 4 }$$

where we have written the (unknown) NS spin period as P = 3 P_3 ms ms. We assume a spherically symmetric corona with inner radius r_co and scale size h ∼ r_co. Then, following the calculations in the previous paragraph, we find an electron density of n_e ≃ 4 × 10¹⁸ (h/r_co)⁻¹ cm⁻³, an effective optical depth of $\tau _{\rm eff}\simeq \sqrt{\tau _{\rm es}\tau _{\rm ff}}\simeq 0.008\, (h/r_{\rm co})^{-1/2}$ at 2 keV, and a predicted bremsstrahlung luminosity from the scattering corona of 4 × 10³⁵ (h/r_co)⁻¹ erg s⁻¹ ≈10⁻³ (h/r_co)⁻¹ L_E. This is, again, orders of magnitude brighter than what we observe and can be ruled out.

4.1.2. Comptonization in the Accretion Flow

We next consider Comptonization in a RIAF-like spherical accretion flow with a radially uniform mass inflow rate $\dot{M}$ at infall velocity $v_r = \eta \sqrt{GM/r}$ (where η ⩽ 1). The electron density will vary with distance r from the NS as

$$\begin{equation} n_e(r) = \frac{\dot{M}_{\rm NS}}{4\pi \eta \mu m_p (GMr^3)^{1/2}}\;, \end{equation} \tag{ 5 }$$

where μ is the mean molecular weight and m_p is the proton mass. We then find that the optical depth cannot exceed

$$\begin{equation} \tau _{\rm es,max} \approx 10^{-3} \left(\frac{\eta }{0.1}\right)^{-1} \left(\frac{\dot{M}_{\rm NS}/\dot{M}_E}{4\times 10^{-6}}\right), \end{equation} \tag{ 6 }$$

independent of r_c. This is orders of magnitude smaller than our comptt fit values for τ_es.

The only way to obtain higher optical depths is to assume that the mass inflow rate $\dot{M}$ varies with r in such a way that only a small amount of mass actually reaches the NS, with the remainder being expelled in some sort of outflow. For convenience, we parameterize $\dot{M}$ as a power law in r:

$$\begin{equation} \dot{M}(r) = \left(\frac{r}{R}\right)^p \dot{M}_{\rm NS}, \end{equation} \tag{ 7 }$$

where R is the NS radius, M_NS is the mass accretion rate onto the NS, and p > 0. (The p = 0 case corresponds to the radially uniform inflow rate that we just dismissed.) This is the same parameterization used in the "ADIOS" (adiabatic inflow–outflow solution) model of Blandford & Begelman (1999). The electron density now varies as

$$\begin{equation} n_e(r) = \frac{2}{\mu \eta \sigma _T R(1+X)}\sqrt{\frac{c^2 R}{GM}} \left(\frac{\dot{M}_{\rm NS}}{\dot{M}_E}\right) \left(\frac{r}{R}\right)^{p-3/2} \end{equation} \tag{ 8 }$$

and the optical depth is

$$\begin{eqnarray} \tau _{\rm es} & = & \frac{2}{\mu \eta (1+X)}\sqrt{\frac{c^2 R}{GM}} \left(\frac{\dot{M}_{\rm NS}}{\dot{M}_E}\right) \nonumber \\ && \times \int _1^{r_c/R} \left(\frac{r}{R}\right)^{p-3/2} d\left(\frac{r}{R}\right) \,. \end{eqnarray} \tag{ 9 }$$

For ionized gas with cosmic abundances, we have μ = 0.6 and X = 0.7. We require that r_c < a, where the binary separation is $a=2.8\times 10^{11}\,M_{1.9}^{1/3}$ cm (Frank et al. 2002; Shahbaz et al. 2014). This is equivalent to requiring that r_c/R ≲ 10⁵. Then taking $\dot{M}_{\rm NS}/\dot{M}_E = 4\times 10^{-6}$ and τ_es = 10, we find 1.4 < p < 1.6 for 0.1 < η < 1. Even larger p values are required for smaller coronas. Such high p values yield high electron densities far from the NS, so the scattering cloud would again itself become a source of significant thermal bremsstrahlung emission. As an example, we consider the p = 1.5 (uniform density) case, for which n_e = 4 × 10¹⁴ cm⁻³. The effective optical depth is τ_eff ∼ 10⁻⁴, so we expect optically thin bremsstrahlung emission. Taking r_c/R = 10⁵ and kT_e = 5 keV, we predict a bremsstrahlung emission measure of $\int n_e^2 dV = 8\times 10^{62}$ cm⁻³, which corresponds to a luminosity of 8.5 × 10³⁹ erg s⁻¹ ≈40 L_E. This is nearly seven orders of magnitude brighter than the observed quiescent emission! We conclude that a thermal Comptonization model is not physically self-consistent for the hard spectral component in Cen X-4 in quiescence and can be ruled out.

Another way of producing a cutoff power-law spectrum is through synchrotron emission. Radio pulsars dissipate their rotational energy via a relativistic wind comprising charged particles and Poynting flux (see Arons 2002; Gaensler & Slane 2006, and references therein). Quiescent NS/LMXB transients can turn on as radio pulsars under some circumstances (Stella et al. 1994); this has recently been observed in at least three cases (see Section 5.3). If Cen X-4 turned on as a radio pulsar during X-ray quiescence, then synchrotron shock emission from the radio pulsar wind interacting with intrabinary material or the interstellar medium (ISM) could produce a power-law X-ray spectrum (Campana et al. 1998a). The synchrotron power-law spectrum will have a high-energy exponential cutoff corresponding to the maximum energy of the electron population accelerated in the shock. This is generally thought to occur at energies of ≳100 keV in pulsar wind shocks. Indeed, PSR J1023+0023, the only quiescent NS/LMXB in which synchrotron shock emission is definitely thought to have been observed, has an unbroken power-law spectrum out to at least 80 keV (Tendulkar et al. 2014). However, as we show below, it is possible to have lower cutoff energies in pulsar wind shocks.

4.2.1. Formalism

We begin by reviewing the formalism for the synchrotron shock scenario developed by Arons & Tavani (1993, hereinafter AT93) for the case of the eclipsing "black widow" millisecond pulsar PSR B1957+20. The pulsar wind is powered by spin-down of the pulsar with an energy loss rate set by magnetic dipole radiation:

$$\begin{equation} \dot{E} = \frac{4\pi ^2 I \dot{P}}{P^3}, \end{equation} \tag{ 10 }$$

where P and $\dot{P}$ are the pulsar spin period and its derivative,¹⁷ and we assume I = 10⁴⁵ g cm² for the NS moment of inertia. Because we do not know P and $\dot{P}$ for Cen X-4, we will scale the surface dipole magnetic field as B = 10⁸ B₈ G and the spin period as P = 3 P_3 ms ms, where we assume

$$\begin{eqnarray} B & = & \left(\frac{3c^3 I P\dot{P}}{8\pi ^2 R^6}\right)^{1/2}\qquad\quad \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} & = & 1.8\times 10^8\, P_{\rm 3\,ms}^{1/2}\, \left(\frac{\dot{P}}{10^{-20}}\right)^{1/2} {\rm G}\,. \end{eqnarray} \tag{ 12 }$$

Thus, for a millisecond pulsar, we have $\dot{P} = 0.3\times 10^{-20}\,B_8^2\,P_{\rm 3\,ms}^{-1}$ and $\dot{E} = 4.8\times 10^{33}\,B_8^2\,P_{\rm 3\,ms}^{-3}$ erg s⁻¹.

Upstream from the shock, the Lorentz factor in the wind is

$$\begin{equation} \gamma _{\rm up} = 9\times 10^4\, \left(\frac{\eta _V}{0.3}\right) \left(\frac{Z}{A}\right) B_8\, P_{\rm 3\,ms}^{-3/2}, \end{equation} \tag{ 13 }$$

where we assume that the ions in the wind have charge Z and mass Am_p and are accelerated to a fraction η_V ∼ 0.3 of the open-field line voltage of the NS. Possible ion values range from protons (Z = A = 1) to partially ionized iron (Z ∼ 3, A = 56). The relative energetic contributions of Poynting flux and ions in the wind upstream of the shock are described by a magnetization parameter (Kennel & Coroniti 1984a):

$$\begin{equation} \sigma = \frac{B_{\rm up}^2}{4\pi \rho _{\rm up}\gamma _{\rm up} c^2}, \end{equation} \tag{ 14 }$$

where B_up and ρ_up are the magnetic field strength and density of the upstream wind. For a particle-dominated wind (like the one in the Crab Nebula), σ ∼ 10⁻³, and for a magnetically dominated wind, σ ≫ 1. The upstream magnetic field strength can then be written as (Kennel & Coroniti 1984b)

$$\begin{equation} B_{\rm up}(r_s) = \left(\frac{\sigma }{1+\sigma }\right)^{1/2} \left(\frac{\dot{E}}{r_s^2\, c\, f_p}\right)^{1/2}\protect, \protect \end{equation} \tag{ 15 }$$

where f_p = ΔΩ_p/4π is the fractional solid angle into which the wind is emitted. From the shock jump conditions, the downstream field strength is (Kennel & Coroniti 1984a)

$$\begin{equation} B_{\rm down} \approx \left\lbrace \begin{array}{@{}ll@{}} 3\,B_{\rm up} & {\rm for\, \sigma\, \ll\, 1}\\ B_{\rm up} & {\rm for\, \sigma\, \gg\, 1}. \end{array} \right. \end{equation} \tag{ 16 }$$

The shock will give rise to a power-law electron population with energy distribution N(γ)∝γ^−s for γ > γ_up, with s ∼ 2. This will, in turn, produce a synchrotron radiation spectrum with photon number index Γ ∼ 1.5 for photon energies above

$$\begin{equation} E_{\rm min} \simeq 0.3\,\gamma _{\rm up}^2 \left(\frac{\hbar e B_{\rm down}}{m_e\, c}\right), \end{equation} \tag{ 17 }$$

where e and m_e are the charge and mass of the electron. If radiative losses are negligible, these power laws will extend up to a cutoff at Lorentz factor

$$\begin{equation} \gamma _m =\left(\frac{A}{Z}\right)\left(\frac{m_p}{m_e}\right)\gamma _{\rm up} = 2\times 10^8\,\left(\frac{\eta _V}{0.3}\right)\, B_8\, P_{\rm 3\,ms}^{-3/2}, \end{equation} \tag{ 18 }$$

and at photon energy

$$\begin{equation} E_c = \gamma _m^2 \left(\frac{\hbar e B_{\rm down}}{m_e\, c}\right). \end{equation} \tag{ 19 }$$

However, if radiative losses are significant, then the electron population will extend only to γ_s < γ_m. To find γ_s, we compare the acceleration time (AT93)

$$\begin{equation} t_{\rm acc} = \left(\frac{A}{Z}\right)\frac{\gamma _{\rm up}\, m_p\, c}{e B_{\rm down}} \protect \protect \end{equation} \tag{ 20 }$$

and the synchrotron loss time

$$\begin{equation} t_s(\gamma) = \frac{6\pi m_e\, c}{\sigma _T\, B_{\rm down}^2\,\gamma } \protect \protect \end{equation} \tag{ 21 }$$

and solve t_acc = t_s(γ_s). If γ_s < γ_m, then radiative losses are important, and the photon power law will only extend up to

$$\begin{equation} E_c = \gamma _s^2 \left(\frac{\hbar e B_{\rm down}}{m_e\, c}\right). \end{equation} \tag{ 22 }$$

Given the above formalism, we now consider two possible sites for the pulsar wind shock location.

4.2.2. Synchrotron Shock in the ISM

If the shock occurs where the pulsar wind is confined by ram pressure in the ISM, then the shock radius will be (Kulkarni & Hester 1988, AT93)

$$\begin{eqnarray} r_s & = & \left(\frac{\dot{E}}{4\pi f_p\, c\, m_p\, n\, v_p^2}\right)^{1/2}\qquad \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} & = & 4\times 10^{15}\,f_p^{-1/2}\, n_1^{-1}\, v_{200}^{-2}\, B_8\, P_{\rm 3\,ms}^{-3/2} {\rm cm}, \end{eqnarray} \tag{ 24 }$$

where n = n₁ cm⁻³ is the ISM particle density and v_p = 200 v₂₀₀ km s⁻¹ is the pulsar space velocity (Shahbaz et al. 2014). At this large distance from the NS, we assume that σ ≪ 1 (Arons 2002). Then, the upstream magnetic field strength is

$$\begin{equation} B_{\rm up}(r_s) = 3\times 10^{-6}\ \left(\frac{\sigma }{10^{-3}}\right)^{1/2} n_1\, v_{200}^2\, {\rm G}, \end{equation} \tag{ 25 }$$

and the magnetic field downstream of the shock is B_down ≈ 3B_up. Radiative losses are negligible in such a weak field, so the expected synchrotron power-law spectrum will extend from

$$\begin{eqnarray} E_{\rm min} &=& 3\times 10^{-7}\, \left(\frac{\sigma }{10^{-3}}\right)^{1/2} \left(\frac{\eta _V}{0.3}\right)^2 \left(\frac{Z}{A}\right)^2 \nonumber \\ && \times \, n_1\, v_{200}^2\, B_8^2\, P_{\rm 3\,ms}^{-3}\, {\rm keV} \end{eqnarray} \tag{ 26 }$$

to a cutoff at

$$\begin{eqnarray} E_c &= & 3.3\, \left(\frac{\sigma }{10^{-3}}\right)^{1/2} \left(\frac{\eta _V}{0.3}\right)^2\, n_1\, v_{200}^2\nonumber \\ && \times \, B_8^2\, P_{\rm 3\,ms}^{-3} \,{\rm keV}. \end{eqnarray} \tag{ 27 }$$

The ISM shock model produces a cutoff energy consistent with our observed spectrum for a reasonable range of pulsar parameters, as shown in shaded region (a) of Figure 7. However, the predicted synchrotron emission corresponding to that region is orders of magnitude weaker than our observed power-law luminosity (∼10³² erg s⁻¹). Scaling to P and B values that lie in the central strip of region (a) in Figure 7, the expected 0.3–20 keV synchrotron luminosity from the ISM shock is (AT93, Equation (16))

$$\begin{eqnarray} L_s & \simeq & 2\times 10^{27}\ \left(\frac{\sigma }{10^{-3}}\right) \left(\frac{\epsilon _a}{0.2}\right) \left(\frac{\eta _V}{0.3}\right) \left(\frac{n_1}{f_p^3}\right)^{1/2}\, v_{200} \nonumber \\ && \times \ \left(\frac{f_{\rm band}}{0.8}\right)\, B_8^4\, \left(\frac{P}{{\rm 2 \,ms}}\right)^{-6}\, {\rm erg\,s}^{-1}, \end{eqnarray} \tag{ 28 }$$

where _a is the conversion efficiency of pulsar wind luminosity into particle acceleration in the shock, and f_band is the fraction of the bolometric synchrotron luminosity that lies in the 0.3–20 keV band:

$$\begin{equation} f_{\rm band} \simeq \left[\frac{\min ({\rm 20\, keV},E_c)}{E_c}\right]^{1/2} - \left[\frac{{\rm 0.3\, keV}}{E_c}\right]^{1/2}. \end{equation} \tag{ 29 }$$

Note that the contours of constant E_c and constant L_s have the same slope in Figure 7, so the numerical prefactor in Equation (28) characterizes the entire length of the shaded region. We conclude that synchrotron emission from a shock in the ISM cannot explain the observed hard X-ray spectrum in Cen X-4.

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4.2.3. Intrabinary Shock

Another possibility is that the pulsar wind collides with material lost from the binary companion or the accretion flow at a shock radius r_s ∼ a, where the magnetic field is considerably stronger than in the ISM case. In PSR J1023+0038, the intrabinary shock was modeled as occurring near the inner Lagrangian (L₁) point (Bogdanov et al. 2011); in Cen X-4, the L₁ point lies at r ≈ 0.67a. It is unclear whether the pulsar wind would be particle dominated or not in an intrabinary shock relatively close to the pulsar. Arons (2002) points out that theory predicts σ ≫ 1 at the pulsar light cylinder, but that observations indicate σ ≪ 1 in observed pulsar wind nebulae produced by ISM shocks, and that it is not clear how or where the transition from one regime to the other occurs. We note that Bogdanov et al. (2011) inferred σ ≫ 1 for the intrabinary shock in PSR J1023+0038. We therefore consider both the σ ≪ 1 and σ ≫ 1 cases here.¹⁸

Scaling to the binary separation in Equation (15), the upstream magnetic field is now

$$\begin{eqnarray} B_{\rm up}(r_s) &=& \left(\frac{\sigma }{1+\sigma }\right)^{1/2} \left(\frac{\dot{E}}{a^2\,c\,f_p}\right)^{1/2} \left(\frac{r_s}{a}\right)^{-1}\qquad \end{eqnarray} \tag{ 30 }$$

$$\begin{eqnarray} &=& 0.04\ \left(\frac{\sigma }{10^{-3}}\right)^{1/2} f_p^{-1/2}\, \left(\frac{r_s}{a}\right)^{-1}\, \left(\frac{a}{a_0}\right)^{-1}\, \nonumber \\ && \times \ B_8\, P_{\rm 3\,ms}^{-3/2} {\rm G},\quad {\rm for} {\sigma \ll 1} \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} & = & 1.4\ f_p^{-1/2}\, \left(\frac{r_s}{a}\right)^{-1}\, \left(\frac{a}{a_0}\right)^{-1}\, \nonumber \\ && \times \ B_8\, P_{\rm 3\,ms}^{-3/2}{\rm G},\quad {\rm for} {\sigma \gg 1}, \end{eqnarray} \tag{ 32 }$$

where a₀ = 2.8 × 10¹¹ cm. The downstream field B_down is given by Equation (16). From Equation (17), the lower energy bound for the synchrotron power law is

$$\begin{eqnarray} E_{\rm min} &=& 0.003\ \left(\frac{\sigma }{10^{-3}}\right)^{1/2} f_p^{-1/2}\, \left(\frac{\eta _V}{0.3}\right)^2 \left(\frac{r_s}{a}\right)^{-1} \left(\frac{a}{a_0}\right)^{-1} \nonumber \\ && \times \ \left(\frac{Z}{A}\right)^2 \, B_8^3\, P_{\rm 3\,ms}^{-9/2} \,{\rm keV,\quad\, for \,\sigma \ll 1} \end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray} &=& 0.04\ f_p^{-1/2}\, \left(\frac{\eta _V}{0.3}\right)^2 \left(\frac{r_s}{a}\right)^{-1} \left(\frac{a}{a_0}\right)^{-1} \nonumber \\ && \times \ \left(\frac{Z}{A}\right)^2 B_8^3\, P_{\rm 3\,ms}^{-9/2} {\rm keV,\,\quad\, for\, \sigma\, \gg\, 1}. \end{eqnarray} \tag{ 34 }$$

The upper bound depends on whether or not radiative losses are important, which is set by the ratio

$$\begin{eqnarray} \frac{\gamma _s}{\gamma _m} & = & 36\ \left(\frac{\sigma }{10^{-3}}\right)^{-1/2} f_p^{1/2}\, \left(\frac{\eta _V}{0.3}\right)^{-2} \left(\frac{r_s}{a}\right) \, \left(\frac{a}{a_0}\right) \nonumber \\ && \times \ B_8^{-3}\, P_{\rm 3\,ms}^{9/2},\quad {\rm for\, \sigma\, \ll\, 1} \end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray} &=& 3\ f_p^{1/2}\, \left(\frac{\eta _V}{0.3}\right)^{-2} \left(\frac{r_s}{a}\right) \, \left(\frac{a}{a_0}\right) \nonumber \\ && \times \ B_8^{-3}\, P_{\rm 3\,ms}^{9/2},\quad {\rm for\, \sigma\, \gg\, 1}. \end{eqnarray} \tag{ 36 }$$

From Equations (19) and (22), the cutoff energy for the σ ≪ 1 case is then

$$\begin{eqnarray} E_c &=& 4.5\times 10^4\ \left(\frac{\sigma }{10^{-3}}\right)^{1/2} f_p^{-1/2} \left(\frac{\eta _V}{0.3}\right)^2 \left(\frac{r_s}{a}\right)^{-1} \nonumber \\ && \times \ \left(\frac{a}{a_0}\right)^{-1} B_8^3\, P_{\rm 3\,ms}^{-9/2} {\rm keV,\,\quad\, for\, \gamma _s\,>\,\gamma _m} \end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray} &=& 5.6\times 10^7\ \left(\frac{\sigma }{10^{-3}}\right)^{-1/2} f_p^{1/2} \left(\frac{\eta _V}{0.3}\right)^{-2} \left(\frac{r_s}{a}\right)\nonumber \\ && \times \ \left(\frac{a}{a_0}\right)\, B_8^{-3}\, P_{\rm 3\,ms}^{9/2} {\rm keV,\,\quad\, for\, \gamma _s\,<\,\gamma _m}, \end{eqnarray} \tag{ 38 }$$

and for the σ ≫ 1 case it is

$$\begin{eqnarray} E_c &=& 5.2\times 10^5\ f_p^{-1/2} \left(\frac{\eta _V}{0.3}\right)^2 \left(\frac{r_s}{a}\right)^{-1} \, \left(\frac{a}{a_0}\right)^{-1}\nonumber \\ && \times \ B_8^3\, P_{\rm 3\,ms}^{-9/2} {\rm keV,\,\quad\, for\, \gamma _s\,>\,\gamma _m} \end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray} &=& 4.7\times 10^6\ f_p^{1/2} \left(\frac{\eta _V}{0.3}\right)^{-2} \left(\frac{r_s}{a}\right) \left(\frac{a}{a_0}\right) \nonumber \\ && \times \ B_8^{-3}\, P_{\rm 3\,ms}^{9/2} {\rm keV,\,\quad\, for\, \gamma _s\,<\,\gamma _m}. \end{eqnarray} \tag{ 40 }$$

The intrabinary shock model with γ_s > γ_m is able to produce a cutoff energy consistent with our observed spectrum for a reasonable range of pulsar parameters, as shown in shaded regions (b) and (c) of Figure 7. However, again, the predicted synchrotron luminosity corresponding to those regions is orders of magnitude smaller than our observed power-law luminosity. The expected 0.3–20 keV synchrotron luminosity from an intrabinary shock is (AT93, Equation (24))

$$\begin{equation} L_s \simeq \epsilon _a\, \dot{E}\, f_{\rm band}\, f_{\rm geom}\protect, \protect \end{equation} \tag{ 41 }$$

where f_geom is the fraction of the pulsar wind intercepted by intrabinary material. By analogy to PSR J1023+0038, we expect f_geom to be in the range 0.01–0.1 (Bogdanov et al. 2011). Scaling to P and B values inside the shaded regions, we thus have

$$\begin{eqnarray} L_s & \simeq & 2\times 10^{29}\, \left(\frac{f_{\rm band}}{0.8}\right)\, \left(\frac{\epsilon _a}{0.2}\right) \left(\frac{f_{\rm geom}}{0.05}\right) \nonumber \\ && \times \ \left(\frac{B}{{3\times 10^8 {\rm G}}}\right)^2 \left(\frac{P}{{\rm 40\, ms}}\right)^{-3} {\rm erg\,s^{-1}} \protect \protect \end{eqnarray} \tag{ 42 }$$

for the γ_s > γ_m, σ ≪ 1 case [region (b)], and

$$\begin{eqnarray} L_s & \simeq & 3\times 10^{28}\, \left(\frac{f_{\rm band}}{0.8}\right)\, \left(\frac{\epsilon _a}{0.2}\right) \left(\frac{f_{\rm geom}}{0.05}\right) \nonumber \\ && \quad \times \ \left(\frac{B}{{3\times 10^8 {\rm G}}}\right)^2 \left(\frac{P}{{\rm 70\, ms}}\right)^{-3} {\rm erg\,s^{-1}}, \end{eqnarray} \tag{ 43 }$$

for the γ_s > γ_m, σ ≫ 1 case [region (c)]. The cases where radiative losses dominate (γ_s < γ_m) only give low enough cutoff energies for an unphysical set of pulsar parameters (P ≲ 1 ms with B ≳ 10⁹ G, beyond the upper left corner of Figure 7), and they predict a synchrotron luminosity at least two orders of magnitude larger than what we observe. Taken together, we conclude that synchrotron emission from an intrabinary shock cannot explain the cutoff power-law spectrum in Cen X-4.

A third way of modeling our cutoff spectrum is through thermal bremsstrahlung emission from an optically thin cloud of hot electrons. Our bremsstrahlung fits (see Tables 1 and 2) yielded an electron temperature kT_e =18 keV and an emission measure

$$\begin{equation} \int n_e^2\, dV = 1.6\times 10^{55}\, D_{\rm kpc}^2 {\rm \ cm}^{-3}\,. \end{equation} \tag{ 44 }$$

We examine two different possibilities for the emission site.

4.3.1. Emission from above the NS Atmosphere

We again consider the possibility of a hot, optically thin layer above the NS atmosphere with geometric thickness h = 10³ h₃ cm. The observed emission measure requires an electron density of

$$\begin{equation} n_e= 3.6\times 10^{19}\,h_3^{-1/2}\,D_{\rm kpc}\,{\rm cm}^{-3}. \end{equation} \tag{ 45 }$$

This implies a free–free absorption optical depth of only τ_ff ≃ 10⁻⁸ at 2 keV, which is self-consistent for optically thin thermal bremsstrahlung emission. In this scenario, both the soft (thermal) emission and the hard (bremsstrahlung) emission are formed in or above the NS atmosphere. This can thus easily account for the fact that the soft and hard emission are observed to vary together on short timescales, with no detectable time lag.

Deufel et al. (2001) calculated the spectrum of an unmagnetized NS atmosphere accreting at low rates through a RIAF flow. Their work differs from the low-$\dot{M}$ case considered by Zampieri et al. (1995, see also Section 3.2) in that it includes Coulomb heating of the atmosphere by energetic protons in the RIAF flow. Interestingly, they predict a hot, optically thin surface layer above the NS atmosphere with kT_e ≈ 50 keV and with significant bremsstrahlung emission expected for a certain range of $\dot{M}_{\rm NS}$, in addition to the soft thermal emission from the NS surface. Their most detailed calculations were made assuming a proton temperature of 0.5kT_vir, where T_vir is the virial temperature. For this case, they found that the bremsstrahlung emission is comparable in strength to the thermal component when $\dot{M}_{\rm ns}\gtrsim 10^{-2} \dot{M}_E$, but that it becomes negligible when $\dot{M}_{\rm ns}\lesssim 10^{-4} \dot{M}_E$. By comparison, our observation of Cen X-4 measured comparable luminosity in the two components when the accretion rate was much lower, $\dot{M}_{\rm ns}\le 3.7\times 10^{-6} \dot{M}_E$. This is a factor of 10⁴ discrepancy with the Deufel et al. (2001) calculation.

One way of reconciling this is to invoke magnetic channeling of the quiescent accretion flow onto the NS polar caps, thus increasing the local accretion rate per unit area. However, this would require very small polar caps with area A_cap ∼ 10⁻⁴ πR². The measured spectral parameters of the soft thermal component in Cen X-4 are not consistent with such small polar caps (see Tables 1 and 2). On the contrary, they suggest that the thermal emission arises from a significant fraction of the stellar surface. Another alternative is to reexamine the assumptions of the Deufel et al. (2001) calculation because the expected bremsstrahlung luminosity must depend on the details of the Coulomb heating of the atmosphere. The authors found that, for a fixed accretion rate, reducing the proton temperature increases the heating in the upper atmosphere because the protons do not penetrate as deeply. However, they did not investigate this case in detail, so it is not clear whether a physically reasonable choice of proton temperature can produce significant bremsstrahlung emission at accretion rates as low as we observed in Cen X-4.

We conclude that bremsstrahlung emission from a hot layer above the NS atmosphere is consistent with our observed spectral cutoff. However, it is not yet clear how to produce sufficient luminosity to match our data based on existing theoretical models for RIAF accretion onto NSs.

4.3.2. Emission from the RIAF Accretion Flow

As another alternative, we again assume that $\dot{M}$ varies with distance from the NS according to Equation (7) and use Equation (8) for n_e. Then, taking μ = 0.6 and X = 0.7 for ionized gas with cosmic abundances, the expected emission measure for a cloud of radius r_c is

$$\begin{equation} \int n_e^2\,dV = \frac{8\pi R}{p\eta ^2\sigma _T^2} \left(\frac{c^2R}{GM}\right) \left(\frac{\dot{M}_{\rm NS}}{\dot{M}_E}\right)^2 \left(\frac{r_c}{R}\right)^{2p} \,. \end{equation} \tag{ 46 }$$

Equating this to Equation (44), we find that

$$\begin{eqnarray} \frac{r_c}{R} & \approx & 10^{4/p}\, p^{1/2p}\, D_{\rm kpc}^{1/p}\, \left(\frac{\eta }{0.1}\right)^{1/p} \nonumber \\ & & \times \left(\frac{\dot{M}_{\rm NS}/\dot{M}_E}{4\times 10^{-6}}\right)^{-1/p}, \end{eqnarray} \tag{ 47 }$$

which sets the scale for the bremsstrahlung emission region in units of the NS radius R. Requiring r_c < a or equivalently r_c/R ≲ 10⁵, we find that p ≳ 0.8. Alternatively, if we require that the cloud lies inside the RIAF transition radius (r_c < r_t or r_c/R ≲ 10⁴), then we find that p ≳ 1. In either case, only a small fraction (r_c/R)^−p ≲ 10⁻⁴ of the mass transferred reaches the NS in quiescence; the rest accumulates at large r or is expelled. In particular, we note that much of the bremsstrahlung emission is coming from electrons at large r. For p = 1, the electron density n_e ranges from 10¹⁴ cm⁻³ near the NS to 10¹² cm⁻³ near r_c.

This scenario is able to self-consistently account for the observed bremsstrahlung luminosity. However, it is challenging to explain the short (≲30 s) lag time observed between the soft and hard flux variability in the X-ray light curve (see Section 3.1). Presumably, the hard flares would arise from mass fluctuations in the RIAF flow, and the soft flares would arise when those fluctuations reach the NS surface. The shortest possible timescale between the soft and hard flares is then the free-fall timescale from r_c,

$$\begin{equation} t_{\rm ff} = 60\, M_{1.9}^{-1/2}\, \left(\frac{r_c}{10^4\,R}\right)^{3/2} {\rm \ s}\,. \end{equation} \tag{ 48 }$$

This is only marginally consistent with our upper limit on the lag time. We conclude that bremsstrahlung emission from the RIAF flow is consistent with our observed spectral cutoff and luminosity, but placing this emission far from the NS is difficult to reconcile with the fact that the soft and hard emission vary together on short timescales.

We have measured a cutoff in the hard X-ray power-law spectrum of Cen X-4 that can be fit with an exponential cutoff at around 10 keV or a bremsstrahlung spectrum with kT_e = 18 keV. This is the first detection of a power-law cutoff in this source class, and it finally permits a more detailed investigation of the origin of the hard component in quiescent NS/LMXBs. We were able to rule out both thermal Comptonization and synchrotron shock emission as the origin of the spectral cutoff. Instead, the hard X-ray spectrum can be understood as bremsstrahlung emission, arising either from a hot, optically thin corona above the NS atmosphere or from hot electrons in an optically thin RIAF. The NS atmosphere scenario has the advantage that it can easily explain why the soft and hard emissions vary together on short timescales, while the RIAF scenario has the advantage that it can self-consistently account for the observed luminosity.

The 18 keV electron temperature for the bremss model is much lower than either the ∼50 keV electron temperature predicted for the hot layer above an NS atmosphere (Deufel et al. 2001) or the ≳100 keV electron temperature expected in a RIAF flow around a BH (Mahadevan & Quataert 1997). This may be due to Compton cooling of the bremsstrahlung electrons by the soft X-ray photons from the NS atmosphere, in which case we would expect T_e to depend upon the soft X-ray luminosity L_soft. The absence of a detectable Compton emission component is not problematic. The Compton radiative power density is

$$\begin{equation} P_C = \sigma _T\,n_e\,\left(\frac{4 kT_e}{m_e c^2}\right) \left(\frac{L_{\rm soft}}{4\pi r^2}\right). \end{equation} \tag{ 49 }$$

The resulting inverse Compton luminosity is

$$\begin{eqnarray} L_{C, {\rm atm}} & = & 5\times 10^{29}\,h_3^{1/2}\, \left(\frac{kT}{{\rm 18\, keV}}\right) \nonumber \\ & & \times \left(\frac{L_{\rm soft}/L_E}{2\times 10^{-6}}\right) {\rm erg\, s\,^{-1}} \end{eqnarray} \tag{ 50 }$$

for cooling above the NS atmosphere, or

$$\begin{eqnarray} L_{C, {\rm RIAF}} & = & 3\times 10^{30}\, \left(\frac{\eta }{0.1}\right)^{-1} \left(\frac{kT}{{\rm 18\, keV}}\right) \left(\frac{\dot{M}_{\rm NS}/\dot{M}_E}{4\times 10^{-6}}\right) \nonumber \\ & & \times \left(\frac{L_{\rm soft}/L_E}{2\times 10^{-6}}\right) {\rm erg\, s^{-1}} \end{eqnarray} \tag{ 51 }$$

for cooling in the accretion flow. In either case, the Compton luminosity is no more than a few percent of the overall source luminosity and thus essentially undetectable in our spectrum.

It is interesting to consider whether the flares observed in the X-ray light curve might be expected to affect the electron temperature (and thus the cutoff energy). The Compton cooling timescale is t_C = (3/2)n_ekT_e/P_C. For emission above the NS atmosphere, this yields

$$\begin{equation} t_{C,{\rm atm}} = 1\times 10^{-2}\, M_{1.9}^{-1}\, \left(\frac{L_{\rm soft}/L_E}{2\times 10^{-6}}\right)^{-1} {\rm s} \,, \end{equation} \tag{ 52 }$$

so the cooling is nearly instantaneous. For this scenario, we might expect to measure spectral changes in the NuSTAR band (≳10 keV) during the flares, although we did not have a sufficiently high signal-to-noise ratio to do this with our observation. By contrast, emission in the RIAF gives

$$\begin{equation} t_{C,{\rm RIAF}} = 1\times 10^{6}\, M_{1.9}^{-1}\, \left(\frac{r}{10^4\, R}\right)^2 \left(\frac{L_{\rm soft}/L_E}{2\times 10^{-6}}\right)^{-1} {\rm s} \,, \end{equation} \tag{ 53 }$$

so in this case short-term flaring behavior will not result in significant Compton cooling of the bremsstrahlung electrons in the RIAF flow. This might provide an avenue for discriminating between the two scenarios.

On longer timescales (months to years), the fact that the slope of the hard X-ray power-law spectrum in Cen X-4 was observed to vary from epoch to epoch over the course of two decades (Cackett et al. 2010) makes it unlikely that the 18 keV bremsstrahlung spectrum we have measured is a constant feature of the source in quiescence. In fact, we can demonstrate that the hard X-ray cutoff energy in Cen X-4 is likely variable by noting that the departure of the hard X-ray spectrum from an unbroken power law above ≃6 keV is evident in our 2013 XMM-Newton spectrum alone, even without including the NuSTAR data. The shape of an 18 keV bremsstrahlung spectrum will show noticeable curvature below 10 keV. On the other hand, all previous deep observations of Cen X-4 in the 0.5–10 keV band are consistent with an unbroken hard X-ray power-law spectrum, indicating a higher cutoff energy for those observations. All of these observations occurred at significantly lower luminosity (see Figure 8). We would expect a lower thermal luminosity to result in reduced Compton cooling of the bremsstrahlung electrons and hence a higher electron temperature, consistent with a higher cutoff energy.

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Moreover, we would expect the 2–10 keV power-law slope to vary systematically with the thermal luminosity L_th. For energy bandpasses far below kT_e, a bremsstrahlung spectrum is a Γ = 1 power law (see, e.g., Rybicki & Lightman 1979); as one approaches the cutoff at kT_e from below, the effective Γ over a fixed bandpass increases as the spectrum begins to gradually roll over. Thus, if we assume that kT_e ≳ 50–100 keV in the absence of Compton cooling, then a Compton-cooled bremsstrahlung model predicts that Γ should be close to 1 at low L_th and should increase as L_th rises and kT_e falls. This relationship will eventually break down when kT_e gets sufficiently low because an unbroken power law no longer provides even a rough fit to a sharp spectral cutoff. The archival data roughly support this picture for $L_{\rm th}\gtrsim 10^{32}\,D_{\rm kpc}^2$ erg s⁻¹. In Figure 8, we plot Γ versus L_th for deep quiescent observations of Cen X-4 made since 1994, including our observation.¹⁹ These observations are listed in Table 3; the archival spectra are collected from Cackett et al. (2010) and Cackett et al. (2013). The trend of the observations in Figure 8 with $L_{\rm th}\gtrsim 10^{32}\,D_{\rm kpc}^2$ erg s⁻¹ is roughly consistent with our expectation for a Compton-cooled bremsstrahlung model.

At the lowest thermal luminosities, however, Γ jumps to higher values. The abrupt change is suggestive of a spectral transition to a different emission mechanism. Given the extremely low luminosities, one might consider coronal X-ray emission from the companion star (Bildsten & Rutledge 2000), but the observed spectral shapes for these observations were not consistent with coronal emission (Cackett et al. 2010, 2013). Instead, we suggest that this may indicate a transition to synchrotron shock emission at the lowest luminosities. We discuss this further in Section 5.3.

As in most quiescent transient NS/LMXBs, a basic requirement for Cen X-4 is that most of the accretion flow does not reach the NS because the inferred $\dot{M}_{\rm NS}$ is substantially smaller than the binary mass transfer rate $\dot{M}_T\sim 0.01 \dot{M}_E$ expected for a Roche-lobe–filling main-sequence donor in a 15.1 hr binary (e.g., King et al. 1996). There may be several mechanisms that contribute to this. First of all, the disk instability model for LMXB transients predicts that most of the accretion flow during X-ray quiescence builds up in the outer accretion disk until a thermal instability ensues, causing an outburst (see Lasota 2001). At low $\dot{M}$, the outer disk will transition into a quasi-spherical RIAF flow at r_t. It has previously been noted that RIAF models for quiescent NS transients require that most of the RIAF flow is somehow prevented from reaching the NS (Asai et al. 1998; Menou et al. 1999). One way of achieving this is the ADIOS-like outflow that we discussed in Section 4.3.2.

Another possibility is that most of the flow reaching the NS magnetosphere is centrifugally inhibited by the magnetic "propeller effect" (Illarionov & Sunyaev 1975; Ustyugova et al. 2006). This occurs when the magnetosphere extends beyond the corotation radius (see Equation (4)). We define the magnetospheric radius r_m as the location where the magnetic and material stresses are equal:

$$\begin{eqnarray} r_m & = & \xi \left(\frac{\mu _m^4}{GM\dot{M}^2}\right)^{1/7}\qquad \end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray} & = & 31\ \xi \,B_8^{4/7}\,M_{1.9}^{-1/7}\,R_{10}^{10/7} \nonumber \\ & & \times \left(\frac{\dot{M}/\dot{M}_E}{0.01}\right)^{-2/7} {\rm km}, \end{eqnarray} \tag{ 55 }$$

where μ_m is the magnetic dipole moment of the NS, ξ is an order unity constant that depends upon the details of the accretion flow near the magnetosphere (see, e.g., Psaltis & Chakrabarty 1999), and the usual R^12/7 scaling is modified by the R dependence of $\dot{M}_E$. In the propeller regime, r_m > r_co. For ordinary thin-disk magnetic accretion, the disk extends to the magnetosphere, and the accretion is entirely shut off in this regime. However, for a RIAF flow onto a millisecond pulsar, r_m will generally lie inside the transition radius r_t, so the flow onto the magnetosphere will be quasi-spherical. In this geometry, a small fraction of the flow is able to reach the NS despite the centrifugal barrier present in the propeller regime (Menou et al. 1999). Whether material is expelled in a strong or weak outflow, or else accumulates outside r_co (e.g., a "dead" disk), depends upon details of the disk–magnetosphere interaction (Spruit & Taam 1993; D'Angelo & Spruit 2010, 2012; Lii et al. 2014). However, observationally, Bernardini et al. (2013) have shown that a strong propeller outflow can likely be ruled out in Cen X-4.

Our observations support the conclusion of Bernardini et al. (2013) that low-level accretion is occurring during X-ray quiescence in Cen X-4, indicating that a small fraction of the accretion flow must eventually reach the NS. However, it remains unclear what combination of the above mechanisms ultimately controls what that fraction is.

After Cen X-4, the next brightest well-studied quiescent NS/LMXB transient is Aql X-1. Unlike Cen X-4, Aql X-1 has a relatively short recurrence timescale of 1–2 yr. There have been several recent studies of its quiescent emission (Cackett et al. 2011; Coti Zelati et al. 2014; Sakurai et al. 2014), all observing a soft thermal component and hard power-law component with no cutoff. Sakurai et al. (2014) argue that, for their 2007 Suzaku observations (L/L_E = [3–9] × 10⁻⁵ M_1.9 for an assumed distance of 5.2 kpc), the most appropriate model for the hard component is Comptonization in either an optically thin (τ_es ≃ 0.3), very hot (kT_e > 100 keV) corona or an optically thick (τ_es > 3), somewhat cooler (kT_e ∼ 50 keV) corona. They do not find the same inconsistency between their measured τ_es and radially uniform accretion that we found in Equation (6). This is a consequence of their $\dot{M}$ being higher and their τ_es being lower than in our Cen X-4 observation. However, we noted in Section 4.1 that τ_es and kT_e vary inversely, and the high kT_e values (and corresponding cutoff energies) that Sakurai et al. (2014) fit lie above their observation bandpass. The Suzaku data are thus unable to rule out the presence of a cutoff below 50–100 keV (but still above their bandpass); this would introduce the same difficulties for a Comptonization model that we found in Cen X-4, although it would be somewhat mitigated by the higher $\dot{M}$. We note that our bremsstrahlung model could explain the hard component in Aql X-1 for kT_e ≳ 30 keV.

It is interesting to also compare the behavior of Cen X-4 with systems that have been observed to transition between LMXB and radio pulsar states during X-ray quiescence. The theoretical expectation is that such transitions are controlled by the location of the NS magnetospheric boundary (Stella et al. 1994). We can compare r_m to both r_co and the light-cylinder radius:

$$\begin{equation} r_{\rm lc} = \frac{cP}{2\pi } = 144\,P_{\rm 3\,ms}\, {\rm km} \,. \end{equation} \tag{ 56 }$$

For sufficiently low $\dot{M}$, we have r_co < r_m < r_lc, and the system will be in the propeller regime, with accretion onto the NS (mostly) shut off (Illarionov & Sunyaev 1975; Ustyugova et al. 2006). For even lower $\dot{M}$, we will have r_co < r_lc < r_m. In this case, the radio pulsar mechanism can turn on, with the radiation pressure of a radio pulsar wind clearing the magnetosphere and an intrabinary shock giving rise to synchrotron X-ray emission (Stella et al. 1994; Campana et al. 1998a; Burderi et al. 2001). The $\dot{M}$ implied for transition to a millisecond radio pulsar state corresponds to L_x ≲ 10³³ erg s, where the exact value depends on details of the system and the disk–magnetosphere interaction.

Indirect evidence for such transitions in NS/LMXBs during X-ray quiescence was previously reported in Aql X-1 (L_x = 6 × 10³² erg s⁻¹; Campana et al. 1998b) and SAX J1808.4−3658 (L_x = 5 × 10³¹ Burderi et al. 2003; Campana et al. 2004a; Deloye et al. 2008). However, more direct evidence has been reported more recently in at least three systems. The 1.7 ms radio pulsar PSR J1023+0038 (hereafter J1023) has made two transitions between the LMXB and radio pulsar states. It is now understood to have been in an LMXB state during 2000–2001, with direct optical evidence for the presence of an accretion disk (Wang et al. 2009). However, a state transition then occurred, with subsequent observations establishing the absence of an accretion disk during 2002–2013 (Thorstensen & Armstrong 2005) as well as the presence of a millisecond eclipsing radio pulsar during 2007–2013 (Archibald et al. 2009), along with a low X-ray luminosity associated with intrabinary synchrotron shock emission (Archibald et al. 2010; Bogdanov et al. 2011). A second state transition was observed more recently, with the radio pulsar turning off and an accretion disk reemerging (Stappers et al. 2014; Patruno et al. 2014).

In all of these observations, J1023 has remained in X-ray quiescence in the sense that a high-luminosity (L_x ≳ 10³⁶ erg s⁻¹) transient X-ray outburst was not observed. However, two distinct substates are evident: a faint X-ray–quiescent state (L_x ∼ 10³² erg s⁻¹) during which radio pulsations are seen, and a bright X-ray–quiescent state (L_x ∼ 10³³ erg s⁻¹) during which the radio pulsar is off. This suggests that r_m has moved outside the light cylinder in the fainter state. In both substates, the 0.3–10 keV X-ray emission has a power-law spectrum with little or no thermal component, with Γ = 1.3 in the faint state (Archibald et al. 2010) and Γ = 1.69 in the bright state (Patruno et al. 2014). NuSTAR observations show that these power-law spectra remain unbroken up to at least 79 keV (Tendulkar et al. 2014). During faint X-ray quiescence, the X-ray flux shows high-amplitude modulation at the orbital period (Archibald et al. 2010; Tendulkar et al. 2014), similar to what is seen in the X-ray emission from some (but not all) eclipsing millisecond radio pulsars (the so-called "black widow" and "redback" systems; Roberts 2013; Roberts et al. 2014). During bright X-ray quiescence, the X-ray flux shows strong, rapid flickering, with the intensity varying by an order of magnitude on timescales of <100 s (Patruno et al. 2014; Tendulkar et al. 2014).

A second object in which two LMXB/radio pulsar state transitions were seen is the M28 globular cluster source PSR J1824−2452I (hereafter M28I), a 3.9 ms radio pulsar. This radio pulsar underwent a bright transient X-ray outburst (L_x ∼ 10³⁶ erg s⁻¹) in 2013 March, during which accretion-powered millisecond pulsations and a thermonuclear X-ray burst were observed, establishing the system as an NS/LMXB. The system returned to X-ray quiescence within a month, at which point radio pulsations were again detected (Papitto et al. 2013). These observations demonstrate that LMXB/radio pulsar state transitions can occur on timescales as short as days. During X-ray quiescence, rapid (≲500 s) intensity variations of nearly an order of magnitude are again seen [(0.6–4) × 10³³ erg s⁻¹], with no change in the 0.3–10 keV X-ray spectrum: an absorbed power law with Γ = 1.2 and no detectable thermal component (Linares et al. 2014). Remarkably, this rapid variability seems to toggle between two stable flux levels; Linares et al. (2014) suggest that this represents fast transitions between synchrotron shock emission and magnetospheric accretion. No orbital variability of the X-ray flux is reported in M28I.

The third object in which an LMXB/radio pulsar transition was seen is XSS J12270−4859 (hereafter J12270), a faint hard X-ray source associated with a relatively bright Fermi γ-ray source. During 2003–2012, the source was in a quiescent LMXB state with an absorbed power-law X-ray spectrum with Γ = 1.7, no evidence for a thermal spectral component, a 0.1–10 keV luminosity of L_x = 2 × 10³³ D_kpc erg s⁻¹, highly variable X-ray flaring, and multiwavelength evidence for the presence of an accretion disk (de Martino et al. 2010, 2013). In late 2012, the source made a transition to a lower (6 × 10³¹ D_kpc erg s⁻¹) luminosity state with a power-law X-ray spectrum with Γ = 1.2, a thermal fraction <9%, and a large-amplitude orbital modulation of the X-ray flux (Bassa et al. 2014; Bogdanov et al. 2014). After this transition, 1.69 ms radio pulsations were also detected (Roy et al. 2014).

Cen X-4 has a luminosity comparable to J1023, M28I, and J12270, and so it is presumably at least close to the regime where LMXB/radio transitions could occur. The rapid X-ray variability we observe (see Section 3.1) is quite different from the large-amplitude orbital modulation seen in the low-quiescent state of J1023, but it is qualitatively similar to (although somewhat weaker than) the flickering seen in the high-quiescent states of J1023 and J12270 as well as the M28I quiescent variability. On the other hand, unlike those three sources, the X-ray spectrum of Cen X-4 has a substantial thermal fraction (≃60%). This may indicate that more of the accretion flow reaches the surface of Cen X-4 than in the other systems. If we apply our Cen X-4 Compton-cooled bremsstrahlung model to the hard X-ray emission in J1023, M28I, and J12270 during their radio quiet or X-ray quiescent states, then we would expect a high (50–100 keV) electron temperature and an unbroken 2–10 keV power-law X-ray spectrum, consistent with what was observed.

Of course, synchrotron shock emission can also produce an unbroken power law in the X-ray band, and we expect this mechanism to dominate in the radio pulsar state. At sufficiently low luminosity, Cen X-4 should transition into a radio pulsar state; we suggest that this may be what occurs at $L_{\rm th}\lesssim 10^{32}\,D_{\rm kpc}^2$ erg s⁻¹ in Figure 8 and that the jump in Γ might reflect a sharp transition from high-temperature bremsstrahlung emission to synchrotron shock emission. This is consistent with the suggestion by Jonker et al. (2004) that the power-law component in quiescent NS/LMXBs arises from accretion at higher $\dot{M}$ and from some different, nonaccretion mechanism (e.g., synchrotron shock emission) at lower $\dot{M}$; they used this to explain how the fractional power-law contribution to the quiescent luminosity varies with $\dot{M}$ in quiescent NS/LMXBs. It would be interesting to search for millisecond radio pulsations from Cen X-4 when its X-ray luminosity next drops to ≲ 10³² D_kpc erg s⁻¹.