1 Hz FLARING IN THE ACCRETING MILLISECOND PULSAR NGC 6440 X-2: DISK TRAPPING AND ACCRETION CYCLES

3.0.1. Energy Dependence, Coherence, and Time Lags

We detect a clear increase of the 1 Hz modulation rms amplitude at high energies, with an rms amplitude that grows linearly from 40% up to a peak of ≃ 60% rms in the 16–28 keV energy band (Figure 4). The intrinsic coherence between two broad energy bands (2.47–7.35 keV and 7.76–28.67 keV) is ≈1 at frequencies between 0.5 and 1.5 Hz, and drops substantially outside this frequency range. Above 3 Hz and below 0.1 Hz, the statistics become poor and the data points in the intrinsic coherence function have large error bars (Figure 5). We use the cross-spectra to inspect for the presence of time lags between the soft and hard energy bands. The time lags are defined as

$$\begin{equation} \tau (\nu) = \frac{\Phi (\nu)}{2\pi \nu }= \frac{{\rm arg}\left[\langle \langle C(\nu) \rangle \rangle \right]}{2\pi \nu }, \end{equation} \tag{ 7 }$$

where Φ(ν) is the Fourier phase lag between two identical Fourier frequencies (ν) measured between the soft and hard energy bands. No time lags are detected between the two time series, with typical upper limits on the time lags between a specific Fourier frequency of less than 5 ms.

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We detect significant pulsations at a frequency of 205.892 Hz in all seven ObsIDs. The frequency shows a Doppler modulation consistent with the orbital solution reported in Altamirano et al. (2010), although we do not attempt to refine the solution with a coherent timing analysis. The pulsations are detected in coincidence with the presence of the 1 Hz modulation.

Our analysis of the outbursts of NGC 6440 X-2 shows that the 1 Hz modulation is present with high amplitude in at least 6 out of 10 outbursts and that it is a recurrent phenomenon. The lack of the 1 Hz modulation in the remaining four outbursts is unconstrained due to the poor photon flux or time resolution limits. The timing and energy properties of the outburst strongly suggest that the modulation is produced by variations in the accretion rate onto the star.

Coherent pulsations at 206 Hz are clearly detected during all outbursts, demonstrating that the channeled accretion flow persists when the 1 Hz modulation is seen. A complete study of the periodic variability of the pulsations will be presented elsewhere.

The energy spectrum (Heinke et al. 2010) and energy-timing information (this work) also suggests a large-amplitude variation in accretion rate onto the star at 1 s intervals. The spectral energy distribution across all outbursts can be fit with a power law with an index between Γ ≈ 1.7 and 3.4 (Heinke et al. 2010), which is most likely produced by inverse Compton scattering of thermal seed photons, both from hot accretion-shocked gas on the stellar surface and hot optically thin gas in the inner accretion flow.

The soft and high energy bands are strongly coherent between 0.5–1.5 Hz, consistent with an overall change in luminosity produced by a changing accretion rate onto the star, where the variability timescale is long enough for the optically thin corona above the neutron star and thermal emission from the neutron star surface to reach quasi-equilibrium (Haardt & Maraschi 1991). The drop in coherence above this frequency may then be a reflection of the intrinsic timescale for density or temperature fluctuations in the hot corona.

We also measure a steep increase (by about 20%) of the 1 Hz modulation rms amplitude with energy, so that the flux at higher energies varies more than the flux at low energies. Assuming that the overall emission is dominated by the accretion shock right above the neutron star's surface (see, e.g., Gierliński & Poutanen 2005), the increased rms amplitude at higher energies naturally results from a changing accretion rate.

Changes in the density and temperature in an optically thin corona above the neutron star will lead to a variable slope of the power-law index in the spectrum. (The slope is set by the Compton y parameter, a function of the gas's temperature and optical depth, where higher temperatures and higher densities both produce harder spectra). A fluctuation in accretion rate will thus change not only the overall flux but also the hardness of the inverse Compton spectrum, so that flux at higher energies is more variable than at lower energies, as seen in both NGC 6440 X-2 and SAX J1808.4−3658.

No thermal component was detected in the 49.1 ks Chandra observation during the 2009 July 28 outburst (Heinke et al. 2010) or any other (less constraining) observation. This is in contrast to SAX J1808.4−3658, in which a thermal component is frequently seen in outbursts, and is attributed to a hot spot on the surface of the neutron star (Poutanen & Gierliński 2003; Gierliński & Poutanen 2005; Patruno et al. 2009a; Papitto et al. 2009; Kajava et al. 2011). In SAX J1808.4−3658,  the thermal component is observed to lag the high energy one by ∼0.125 ms (Cui et al. 1998; Gierliński et al. 2002), which Gierliński & Poutanen (2005) attribute to strong Doppler beaming of the blackbody emission compared to the more diffuse inverse Compton emission. The lack of thermal emission and low pulsation fraction may indicate that the main accretion hot spot in NGC 6440 X-2 is beamed out of our line of sight, while the upper limits on phase lags between hard and soft energy bands (of the order of 5 ms) are not particularly constraining.

The instability studied in Spruit & Taam (1993) and D'Angelo & Spruit (2010, 2012) appears when the inner edge of the disk moves into a marginally dead state, with r_m ≃ r_co. This condition requires a low mean accretion rate so that the inner edge of the disk moves temporarily outside the corotation radius. NGC 6440 X-2, which has one of the lowest outburst luminosities of all known AMXPs, supports this picture. Its typical X-ray outburst luminosity is ≈10³⁶ erg s⁻¹ (Heinke et al. 2010), corresponding to a mass accretion rate of ≈2 × 10⁻¹⁰ M_☉ yr⁻¹ for a canonical neutron star mass of 1.4 M_☉ and radius of 10 km.

The disk truncation radius r_m is set by the magnetic field, and is subject to considerable uncertainty due to the uncertainty in exactly how the disk and field interact. We follow Spruit & Taam (1993) and D'Angelo & Spruit (2010) and define the inner disk radius as where the field is strong enough to force the gas to corotate with the star:

$$\begin{eqnarray} r_{\rm m} &\equiv & 47.7\, {\rm km} \left(\frac{0.1}{B_\phi /B_*}\right)^{1/5}\left(\frac{B_*}{4\times 10^8\,{\rm G}}\right)^{2/5}\left(\frac{M_*}{1.4\,M_\odot }\right)^{-1/10}\nonumber\\ \nonumber &&\left(\frac{R_*}{10\,{\rm km}}\right)^{6/5}\left(\frac{\dot{M}}{2\times 10^{-10}\,M_{\odot } \,{\rm yr}^{-1}}\right)^{-1/5}\left(\frac{P_*}{4.9\,{\rm ms}}\right)^{3/10},\nonumber\\ \end{eqnarray} \tag{ 8 }$$

where the subscript "*" identifies quantities referred to the neutron star and B_ϕ/B_* is the fractional size of the toroidal field induced by the relative rotation between the magnetic field and the disk. For B_ϕ/B_* > 1, the field lines will rapidly inflate and sever the connection between the disk and the star (e.g., Aly 1985; Lovelace et al. 1995), but the exact average value is uncertain. Assuming B_ϕ/B_* ∼ 0.1–1 implies an uncertainty of up to ∼60% on the value of B_* that will put r_m ≃ r_c. The inferred magnetic field for NGC 6440 X-2 (assuming B_ϕ/B_* = 0.1 and r_m = r_co for $\dot{M} \sim 10^{-10}\,M_{\odot }\,{\rm yr}^{-1}$) will be 4 × 10⁸ G, well in line with most AMXPs (see, e.g., Patruno & Watts 2013 for a review).

The timescale of the 1 Hz modulation also suggests an instability in the inner accretion disk. This timescale is much longer than the spin frequency of the star (206 Hz) or the dynamical timescale of the inner disk (similar when r_m ∼ r_co). It is also likely much longer than the timescale on which the field lines are twisted open and reconnect to the disk, although this is currently somewhat uncertain (e.g., Romanova et al. 2009).

The disk instability studied by Spruit & Taam (1993) and D'Angelo & Spruit (2010, 2012) causes density fluctuations in the inner parts of the disk, which generally evolve on viscous timescales. The instability frequency roughly scales with the viscous timescale around the inner edge of the disk. This timescale is given by

$$\begin{eqnarray} T_{\rm visc} & \sim & 80 \left(\frac{0.1}{\alpha }\right)^{-4/5} \left[\frac{\dot{M}}{2\times 10^{-10}\,{M_\odot \,{\rm yr}^{-1}}}\right]^{-3/10}\nonumber \\ && \times \left[\frac{M_*}{1.4\, M_\odot }\right]^{1/4} \left[\frac{r}{50 \,{\rm km}}\right]^{5/4}\,{\rm s}, \end{eqnarray} \tag{ 9 }$$

where α ∼ 0.1 is the canonical α-disk parameter for geometrically thin and optically thick disks (Siuniaev & Shakura 1977) and r ∼ r_m. For NGC 6440 X-2, T_visc ∼ 80 s, and the modulation at 1 Hz is well in the instability range seen by D'Angelo & Spruit (2010, 2012). In Section 4.4, we discuss this further and place tighter physical constraints on the disk–magnetosphere interaction based on the instability properties.

As well as being weak with short durations, the outbursts seen in NGC 6440 X-2 have the shortest recurrence time of any known AMXP, with 10 outbursts seen at roughly monthly intervals between 2009 and 2011. Similar behavior was seen in another AMXP, IGR J00291+5934, which in 2008 underwent two small outbursts (lasting around 10 days each) separated by a month in quiescence (Patruno 2010; Hartman et al. 2011). This AMXP also showed a QPO at 0.5 Hz (Hartman et al. 2011), although this QPO is much more typical (with substantially lower amplitude of (13 ± 2)%, and a standard Lorentzian shape) than those seen in NGC 6440 X-2 and SAX J1808.4−3658. There also may be a connection between the appearance of the instability in SAX J1808.4−3658  and the "reflaring" portion of the light curve. The "reflaring" refers to large luminosity variations (up to three orders of magnitude) on timescales of three to five days in the decay tail of SAX J1808.4−3658  (see, e.g., Figure 1 in Patruno et al. 2009b), suggesting large-scale changes in the accretion rate as the source approaches quiescence.

The short variation timescale of the reflares and outbursts is incompatible with the standard ionization instability model believed to power normal outbursts (e.g., Lasota 2001) and suggests that a substantial amount of gas remains in the inner regions of the accretion disk (close to r_co) when the accretion rate decreases, exactly as is expected for a dead disk. As was first noted by D'Angelo & Spruit (2012), the unusual frequent outbursts and strong 1 Hz modulation are both well explained by a disk that never moves into a strong propeller phase at very low $\dot{M}$, but instead stores mass as a "dead disk," which becomes unstable at higher $\dot{M}$.

The density structure of a dead disk, with its low temperature and a substantial amount of stored mass in the disk inner regions, is considerably different from the standard accretion disk and could strongly alter the ionization instability cycle. Although detailed calculations of how the ionization instability would be triggered in a dead disk are beyond the scope of the current work, the additional stored mass during quiescence could lower the density threshold for which newly accumulated mass (from the star or the outermost regions of the accretion disk) can trigger a new outburst. Such outbursts would be expected to have both shorter recurrence times and weaker outbursts, since less additional mass is accumulated between outbursts.

A further prediction of the dead disk picture is that spin-down should be enhanced during the quiescent phase, since the disk inner edge remains close to r_co  and continues to spin down the star. Note, however, that the disk–field coupling parameters we infer for this source (in Section 4.4) imply a narrow connected region between the star and the disk, which will also limit the spin-down from this source.

In the model presented in D'Angelo & Spruit (2010, 2012), the instability develops in two different regimes, depending mainly on the mean mass accretion rate through the disk. The more-studied regime (called RI in D'Angelo & Spruit 2012) occurs at very low accretion rates, with $\dot{M} \approx 10^{-4}\hbox{--}0.1 \dot{M}_{\rm co}$ (where $\dot{M}_{\rm co}$ is the accretion rate that sets r_co = r_m). It is characterized by long periods of quiescence while mass accumulates in the inner regions of the disk followed by brief accretion outbursts. Indeed, the authors found that the period of the instability could span several orders of magnitude, from around 0.1 to 10⁴ T_visc (where T_visc is the viscous timescale at r_m, Equation (9)), depending chiefly on the time needed to fill up the reservoir and tip the source into an accretion burst.

The instability outburst shape and duration also depends quite sensitively on the details of the disk–field interaction, which D'Angelo & Spruit (2010, 2012) parameterized as two length scales: the width of the coupled region between the field and the disk, Δr, and Δr₂, the transition length around r_co  between the accreting and non-accreting solutions (see Figures 6 and 7). Both of these length scales are assumed to be small: Δr/r_co, Δr₂/r_co ≪ 1. In the RI instability region, D'Angelo & Spruit (2010, 2012) found that the instability occurs for a broad range of Δr/r_m (see Figures 3 and 4 in D'Angelo & Spruit 2012) but only when Δr₂/r_m is very small (of order of 0.01).

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However, in D'Angelo & Spruit (2012) the authors identified a second instability region, which they referred to as RII. Here, the mean accretion rate is $0.1\hbox{--}10\times \dot{M}_{\rm co}$—much higher than RI—and accretion continues throughout the cycle, so that the amplitude of the instability is much smaller. The instability timescale is also much shorter than for RI, with periods around 0.01–0.1 T_visc, and has a more sinusoidal, regular shape, and a smaller amplitude, although the outburst length is still typically shorter than the build-up time (see, e.g., Figure 5 of D'Angelo & Spruit 2012). The instability also appears for different disk–field interaction parameters: here Δr/r_m is very small (≈0.05), while Δr₂/r_m has a much broader range of instability.

In NGC 6440 X-2, the corotation radius is located at approximately 50 km from the neutron star center, but the surface magnetic field is still unknown, so that it is not possible to estimate the location of the magnetospheric radius from the luminosity alone. However, if we assume a typical AMXP magnetic field strength of ≈(1–5) × 10⁸ G, the observed mass accretion rate is of the same order of magnitude or lower than $\dot{M}_{\rm co}$, suggesting that we are observing the RII instability. In SAX J1808.4−3658  (with B ≈ 10⁸ G; see Hartman et al. 2008; Patruno et al. 2012), the 1 Hz modulation was seen at fluxes corresponding to $\dot{M}/\dot{M}_{\rm co}\approx 1$, suggesting that source is also unstable in the RII region. In this regime, the instability only appears over a relatively narrow range of $\dot{M}$, offering a plausible explanation for why the instability is only observed in the decay tail of the outburst.

One obvious point of interest is why the 1 Hz modulation central frequency is nearly the same in both SAX J1808.4−3658  and NGC 6440 X-2, given that the system parameters are quite different. Despite the comparable X-ray flux detected in both AMXPs when the 1 Hz modulation is observed, the corresponding mass accretion rate of SAX J1808.4−3658  is about an order of magnitude lower than in NGC 6440 X-2 because the first source lies at about 3.5 kpc (Galloway & Cumming 2006) whereas the latter is at 8.5 kpc (Ortolani et al. 1994). If we compare the fundamental centroid frequency of the 1 Hz modulation observed in all observations of NGC 6440 X-2 with those of SAX J1808.4−3658  then it is, on average, systematically lower in NGC 6440 X-2 by only a factor of ≈2. Without knowing the magnetic field in either source, it is not easy to constrain exactly where the disk is truncated (or, to invert the problem, the actual ratio $\dot{M}/\dot{M}_{\rm co}$), but if we assume that the relevant timescale is the viscous timescale at r_co  then this will be ∼80 s for NGC 6440 X-2 versus ∼40 s for SAX J1808.4−3658, so the model predicts roughly the same ν for both sources, as observed. An independent constraint on B_* would allow a much more secure determination of $\dot{M}/\dot{M}_{\rm co}$ and also Δr₂ and Δr.

The RII instability appears only for very small values of Δr, of the order of 0.01–0.05 r_m. This suggests a very small coupling region between the field and the disk—of the order of 1 km—in both SAX J1808.4−3658  and NGC 6440 X-2. Given the uncertainties in determining the system's physical parameters, Δr₂ cannot be as well constrained, although at these accretion rates Δr₂∝ν⁻¹ (where ν is the instability frequency; see Figure 7 of D'Angelo & Spruit 2012), so a smaller value for Δr₂ (of about 1–10 km) would also seem to be favored.

D'Angelo & Spruit (2012) found that when the instability is triggered in the RII regime, the accretion torques are reduced compared with the standard disk solution. An implication of this is that if an AMXP is observed to show the 1 Hz modulation in a certain mass accretion rate range, then the observed accretion torques must be smaller than the prediction made with the approximation:

$$\begin{equation} N = \dot{M}\sqrt{G\,M_{*}\,r_{\rm m}}. \end{equation} \tag{ 10 }$$

It is possible to speculate therefore that the lack of strong positive accretion torque in SAX J1808.4−3658  (Hartman et al. 2008, 2009) might be partially caused by the source being almost always close to this marginally accreting state. The fact that SAX J1808.4−3658  is found with r_m close to r_co was also suggested by Haskell & Patruno (2011) based on coherent timing and quiescent thermal properties of the accreting neutron star.

The fact that the 1 Hz modulation can be fitted with a Gaussian and not with a Lorentzian suggests some intrinsic difference between the generation of the 1 Hz modulation and the higher frequency QPOs observed in NGC 6440 X-2 and SAX J1808.4−3658. It also suggests some remarkable difference between most of the other QPOs broadly observed in other low mass X-ray binaries and the 1 Hz mechanism.

If a Gaussian stochastic process controls the fluctuations in mass accretion rate around the magnetospheric radius, then the expected $\dot{M}$ profile instability will be a Gaussian. This in turn might explain why the 1 Hz modulation has a Gaussian shape and not a Lorentzian. It is conceivable indeed to think that, once the instability is triggered and the gas is accumulating around r_m ≈ r_co, the main timescale of the instability will be strictly periodic if none of the quantities in the system are changing (i.e., $\dot{M}$, Δr/r_m and Δr₂/r_m). However, if we let $\dot{M}$ fluctuate (in a Gaussian stochastic way) around its mean value, the periodicity of the instability will slightly change, thus broadening the periodicity across adjacent frequencies. It is then possible to understand why the shape of the 1 Hz modulation is Gaussian and why it becomes a broad feature in some observations whereas in others it is a more periodic and narrow feature: the stronger the fluctuations in $\dot{M}$ around the mean value, the broader the 1 Hz modulation will appear in the power spectra.

Patruno et al. (2009b) reported an extensive discussion of alternative models to the trapped-disk instability presented in Spruit & Taam (1993; and later developed in D'Angelo & Spruit 2010, 2012). It was shown that none of the alternative models can explain at the same time the observed features of the 1 Hz modulation in SAX J1808.4−3658. Identical consideration apply also to the 1 Hz modulation in NGC 6440 X-2, as all the observational features closely resemble those in SAX J1808.4−3658 and identical considerations apply. More recently, Romanova et al. (2013) carried three-dimensional MHD simulations with a dipole magnetic field misaligned from the neutron star's rotational axis, in which a warped disk appears. Perturbations of the disk in the vertical direction produce low-frequency bending waves, whose frequency is approximately equal to the Keplerian frequency $(\Omega =\sqrt{GM/r^3})$ of the gas in the disk. Since the inner disk is truncated at about ∼50 km, the Keplerian frequency there is of the order of 200 Hz, which is two to three orders of magnitude off the observed instability frequency. The warped disk predicts oscillations with frequency down to 0.1–1 Hz, but the region of the disk that oscillates has to be far away from the neutron star, at several thousand kilometers, where no X-ray radiation is produced and the inner accretion flow (at about 50 km) would still be modulated at the much higher Keplerian frequency there.