A statistical study on the low-frequency quasi-periodic oscillation amplitude spectrum and amplitude in GRS 1915+105 Free

Abstract

A statistical study was made on both the energy dependence of the low-frequency quasi-periodic oscillation (QPO) amplitude (LFQPO amplitude spectrum) and the LFQPO amplitude from all the RXTE observations of GRS 1915+105. Based on the two-branch correlation of the LFQPO frequency and the hardness ratio, the observations that were suitable for evaluating the LFQPO amplitude spectrum were divided into two groups. According to a comparison between the radio and X-ray emissions, we deduced that the jets during the two groups of observations are very different. A negative correlation between the LFQPO frequency and the radio flux was found for one group. The LFQPO amplitude spectrum was fitted by a power law with an exponential cutoff in order to describe it quantitatively. It reveals that as the LFQPO frequency increases, the power law hardens. Furthermore, the cutoff energy first decreases, and then smoothly levels off. The fit also shows that the LFQPO amplitude spectra of the two groups are essentially the same, suggesting that the LFQPO does not originate from the jet. The LFQPO amplitude spectra are hard, indicating a possible origin of the LFQPO in the corona. As the LFQPO frequency increases, the LFQPO amplitude first increases and then decreases. The effects of the low-pass filter and the jet on the LFQPO amplitude are discussed.

INTRODUCTION

The black hole binary system (BHB) GRS 1915+105 was discovered in 1992 with WATCH onboard GRANAT (Castro-Tirado, Brandt & Lund 1992). It is at a distance of ∼11 kpc (e.g. Fender et al. 1999; Zdziarski et al. 2005), and comprises a spinning black hole (Zhang, Cui & Chen 1997; McClintock et al. 2006) with mass 14 ± 4 M_⊙, and a K–M III giant star with mass 0.8 ± 0.5 M_⊙ as the donor (Harlaftis & Greiner 2004; Greiner et al. 2001b). The orbital separation of the binary components is about 108 ± 4 R_⊙, and the orbital period is 33.5 ± 1.5 d (Greiner, Cuby & McCaughrean 2001a). GRS 1915+105 was the first microquasar to be found and produces superluminal radio jets (Mirabel & Rodríguez 1994; Fender et al. 1999).

It displays various X-ray light curves and complex timing phenomena. Based on the appearances of the light curves and colour-colour diagrams, the behaviours of GRS 1915+105 are classified into more than 10 categories (Belloni et al. 2000; Klein-Wolt et al. 2002; Hannikainen et al. 2005). These categories can be reduced to transitions among three basic states, namely states A, B and C. Three types of quasi-periodic oscillations (QPOs) with different QPO frequency bands have been observed in GRS 1915+105 (e.g. Morgan, Remillard & Greiner 1997; Chen, Swank & Taam 1997; Strohmayer 2001; Belloni, Méndez & Sánchez-Fernández 2001; Belloni et al. 2006). The low-frequency (∼0.5–10 Hz) QPO (LFQPO) is the type most commonly observed. Considerable effort has been put into exploring the origin of the LFQPO of GRS 1915+105. It has been shown that the LFQPO frequency is positively correlated with the fluxes of the thermal and power-law components as well as with the total flux (e.g. Chen et al. 1997; Markwardt, Swank & Taam 1999; Muno, Morgan & Remillard 1999; Trudolyubov, Churazov & Gilfanov 1999; Reig et al. 2000; Tomsick & Kaaret 2001; Muno et al. 2001). Muno et al. (1999) and Rodriguez et al. (2002b) reported that as the LFQPO frequency increases, the temperature of the inner accretion disc increases, and the disc radius decreases. These results indicate that the LFQPO is related to both the accretion disc and the region where the hard component is produced, for example the corona. It should be noted, however, that most of these results are dependent on spectral models, but the origin of the hard spectral component is still a matter of debate (e.g. Muno, Morgan & Remillard 1999; Rau & Greiner 2003; Vadawale et al. 2003; Zdziarski et al. 2005; Titarchuk & Seifina 2009; Van Oers et al. 2010; Neilsen, Remillard & Lee 2011).

As model-independent approaches, it is useful to study the LFQPO frequency–LFQPO amplitude relation, the energy dependence of the LFQPO frequency (LFQPO frequency spectrum), and the energy dependence of the LFQPO amplitude (LFQPO amplitude spectrum) for GRS 1915+105. The LFQPO amplitude refers to the LFQPO fractional rms amplitude, which is measured by using a Lorentzian fit to the power spectrum (see Section 2 for details). It was found that the LFQPO amplitude is inversely correlated with the LFQPO frequency (e.g. Muno et al. 1999; Trudolyubov et al. 1999; Reig et al. 2000). Qu et al. (2010) studied the LFQPO frequency spectrum of GRS 1915+105 and found that as the centroid frequency of the LFQPO increases, the relation between LFQPO frequency and photon energy evolves from a negative correlation to a positive one. Three additional combined patterns of the negative correlation and the positive one have been discovered (Yan et al. 2012). Furthermore, as photon energy increases, the LFQPO amplitude increases and then flattens in some cases (e.g. Tomsick & Kaaret 2001; Rodriguez et al. 2002a, 2004; Zdziarski et al. 2005; Sobolewska & Życki 2006), indicating a possible association between the LFQPO and the corona (e.g. Morgan et al. 1997; Ingram & Done 2012).

There is no statistical study, however, in which all the RXTE observations of GRS 1915+105 on both the LFQPO amplitude spectrum and the LFQPO amplitude are utilized. In order to reveal more details of the LFQPO phenomenology and to investigate the origin of the LFQPO, in this study we analysed all the RXTE/PCA data of GRS 1915+105 and found that as the LFQPO frequency increases, the LFQPO amplitude spectrum becomes harder, and the LFQPO frequency–amplitude relation evolves smoothly from a positive correlation to a negative one. A negative correlation between the LFQPO frequency and the radio flux is also found. The observations and data reduction methods are described in Section 2, the results are presented in Section 3, and a discussion and the conclusions are given in Section 4.

OBSERVATIONS AND DATA REDUCTION

We searched the LFQPOs from all the RXTE/PCA observations of GRS 1915+105. Only some observations are suitable for evaluating the LFQPO amplitude spectrum. The LFQPO frequency sometimes varies obviously during an observation. In order to obtain credible results, this kind of observation is split into several time intervals during which the LFQPO frequency is relatively stable. A total of 168 observation intervals during which the X-ray emission is relatively hard and steady were obtained (Table 1) .

It is a common technique to combine the timing analysis with the spectral analysis. In view of the debate over the spectral model, we investigate the relation between the hardness ratio (HR) and the LFQPO frequency as an approximate spectral analysis. The HR is defined as the ratio of the count rate in 7–60 keV to that in 2–7 keV. The corresponding PCA absolute channel intervals of the two energy bands in PCA gain epochs 3, 4 and 5 are 19–255 and 0–18, 16–255 and 0–15, and 17–255 and 0–16, respectively. The count rate is obtained by extracting the background-subtracted PCA Standard-2 light curve using the heasoft version 6.7 package.

In order to investigate the LFQPO amplitude spectrum, the light curves are extracted from the binned and event mode data. Good time intervals are defined as follows: a satellite elevation over the Earth limb >10° and an offset pointing <0| $_{.}^{\circ}$|02. In order to acquire the details of the LFQPO amplitude spectrum with high enough confidence, only the binned mode data with energy channel number ≥4 and time resolution ≤8 ms are selected. The light curves are extracted with a time resolution of 8 ms in PCA energy bands defined in Table S1 in the Supporting Information. The power density spectra (PDSs) are computed with a 64-s sampling duration, with the normalization of Miyamoto et al. (1992), which gives the periodogram in units of (rms/mean)² Hz⁻¹. The Poisson noise is also corrected (e.g. van der Klis 1989; Vaughan et al. 2003). Following Belloni, Psaltis & van der Klis (2002), we fit the PDS with a model that includes several Lorentzians to represent the LFQPOs, the continuum and other broad features, respectively. The uncorrected LFQPO amplitude is defined as A_raw (per cent rms) = |$100 \times \sqrt{WN\pi /2}$|⁠, where W is the full width at half-maximum (FWHM) of the Lorentzian that represents the LFQPO, and N is the Miyamoto normalization of the Lorentzian. The LFQPO amplitude is further corrected for background (Berger & van der Klis 1994; Rodriguez & Varnière 2011). The errors are derived by varying the parameters until Δχ² = 1 at the 1σ level.

In order to study the LFQPO frequency–amplitude relation, the light curves are extracted from the binned mode data in PCA absolute channel 0–35 (∼2–13 keV) and the event mode data in channel 36–255 (∼13–60 keV) with a time resolution of 8 ms. With the asynchronous rows deleted from the FITS files, the binned and event mode light curves in the same observation interval are added together to obtain a light curve that is used to measure the LFQPO frequency and amplitude.

RESULTS

LFQPO frequency–hardness ratio relationship

Fig. 1 shows the relationship between the LFQPO frequency and the HR. It can be seen that the points in this figure form two obviously separated branches. In order to clearly describe and analyse the results, we refer to the lower branch as ‘Branch 1’ (filled circles) and to the upper branch as ‘Branch 2’ (crosses), and divide the observation intervals into two groups corresponding to the two branches, respectively. Branch 1 is in the range ∼0.4–8 Hz, and Branch 2 is in the range ∼2–5.5 Hz. For Branch 1, as the LFQPO frequency increases, the HR first decreases, then smoothly levels off, and then increases slightly. For Branch 2, the HR decreases monotonically.

Spectral states of the Branch 1 and Branch 2 observations

Muno et al. (2001) investigated the radio and X-ray properties of GRS 1915+105 when its X-ray emission was hard and steady, and defined three spectral states/conditions. The energy spectra of the Branch 1 and Branch 2 observations are different based on the two-branch correlation of the LFQPO frequency and the HR. Thus, it is useful to identify the spectral states of the Branch 1 and Branch 2 observations. In order to clarify the states of the two groups of observations, we plot the RXTE/ASM count rate and the radio flux from the Ryle Telescope at 15.2 GHz as functions of time, and show the times of the observations analysed in this work (Figs 2a and b). The values of the radio flux were obtained from Muno et al. (2001). At first glance, the Branch 1 observations are in the time intervals (B1s in Fig. 2a) during which GRS 1915+105 produces the brightest radio emissions, and the Branch 2 observations are in the time intervals (B2s in Fig. 2a) when GRS 1915+105 produces fainter radio emissions.

The LFQPO frequency and amplitude as functions of time are also presented. The behaviour of the LFQPO frequency is similar to the behaviour of the count rate (Fig. 2c). The LFQPO amplitude is, however, a non-monotonic function of the count rate (Fig. 2d).

In order to show the relationship between the radio emission and the LFQPO more clearly, we need to bin the observations into time intervals. The time intervals of the radio fluxes presented in Fig. 2 are about 1 d. We therefore select 1 d as the bin size. Fig. 3 shows the radio flux as a function of time, and the relationship between the radio flux and the LFQPO frequency. Obviously, most of the radio fluxes corresponding to Branch 1 observations are larger than 30 mJy, and all except one flux corresponding to Branch 2 observations are lower than 40 mJy. As shown in Fig. 3(b), for Branch 1, the LFQPO frequency is negatively correlated with the radio flux. For Branch 2, it has no obvious correlation with the radio flux. The points of Branch 1 in Fig. 3(b) are fitted using least squares. The slope of the best-fitting line is −0.047 ± 0.015 Hz mJy⁻¹, and the adjusted R² is 0.62.

The Branch 2 point whose radio flux is about 90 mJy is located at some distance from the main Branch 2 group. We therefore checked it and found that its RXTE observation time was a bit earlier than its Ryle observation time, and that the radio flux on the days around this observation time was ∼20–30 mJy. It is thus possible that the radio flux corresponding to the RXTE observation is not actually so high, and the outlying Branch 2 point might actually be located within the main group. This is just speculation, however, owing to the lack of data. We also checked all the other radio fluxes and found that they are close to those on either side of them. These fluxes thus seem to be more reliable, although some uncertainty still exists. The states of the two group observations will be discussed in Section 4.

LFQPO amplitude spectrum

For each interval listed in Table 1, we have drawn a diagram to show the LFQPO amplitude spectrum. Although these spectra have various shapes, they evolve with the LFQPO frequency. Fig. 4 shows several representative spectra of the Branch 1 observations. When the LFQPO frequency is very low, the amplitude increases slightly with energy (Fig. 4a). As the LFQPO frequency increases, the amplitude in the higher energy band gradually increases (Figs 4b and c), and then the amplitudes in both the higher and lower energy bands gradually decrease (Fig. 4d and e). For the LFQPOs with higher frequency, however, the amplitude in the higher energy band is relatively high (Fig. 4e). Fig. 5 shows representative spectra of the Branch 2 observations. When the LFQPO frequency is low, the amplitude spectrum is steep (Fig. 5a). As the LFQPO frequency increases, the amplitude in the higher energy band decreases (Fig. 5b), and then the amplitudes in both the higher and lower energy bands decrease (Figs 5c and d).

In order to describe the LFQPO amplitude spectrum quantitatively, we fitted the spectrum by a power law with an exponential cutoff, A(E) = KE^−αexp (−E/E_c), where α is the power-law index and E_c is the e-folding energy of exponential roll-off. Fig. 6 shows α and E_c as functions of the LFQPO frequency. Clearly, the two groups of points are essentially identical, indicating that there is not much difference between the two groups of observations in the LFQPO amplitude spectrum. Thus, the two groups of points will later be fitted as a whole. As the LFQPO frequency increases from ∼0.4 to ∼8 Hz, α decreases from ∼−0.4 to ∼−1.1. (Fig. 6a). The points are fitted using least squares. The slope of the best-fitting line is −0.087 ± 0.012 Hz⁻¹, and the adjusted R² is 0.53. Fig. 6(b) presents the LFQPO frequency dependence of E_c. At lower LFQPO frequencies, the dependence starts at an E_c of ∼80 keV and follows a negative correlation until a certain LFQPO frequency, where it levels off. The points are fitted with the function E(f) = A − D Bln {exp [(f_tr − f)/D] + 1} (function (1) in Shaposhnikov & Titarchuk 2007). The best-fitting values are A = 25.1 ± 2.8 keV, B = −21.6 ± 8.0 Hz⁻¹, D = 0.3 ± 0.5 Hz and f_tr = 2.84 ± 0.55 Hz. The errors for the best-fitting parameters are standard deviations.

LFQPO frequency–amplitude relationship

The relationship between the LFQPO frequency and amplitude is shown in Fig. 7. For Branch 1, as the LFQPO frequency increases, the LFQPO amplitude increases from ∼7 to ∼13 per cent at f < 2 Hz, and then decreases from ∼13 to ∼2 per cent at f > 2 Hz. For Branch 2, as the LFQPO frequency increases, the LFQPO amplitude decreases monotonically from ∼16 to ∼5 per cent.

The LFQPO absolute amplitude is estimated by multiplying the LFQPO amplitude by the corresponding count rate (see e.g. Méndez et al. 1997; Gilfanov, Revnivtsev & Molkov 2003; Zdziarski et al. 2005). For Branch 1, as the LFQPO frequency increases, the LFQPO amplitude increases at f < 2 Hz and then decreases, similar to the behaviour of the LFQPO amplitude. For Branch 2, the points are widely scattered (Fig. 8).

DISCUSSION AND CONCLUSIONS

LFQPOs have been detected in many BHBs (see e.g. van der Klis 2004; McClintock et al. 2006; Remillard & McClintock 2006). Their frequencies and amplitudes are usually correlated with spectral parameters of both the thermal and power-law components (e.g. Chen et al. 1997; Markwardt, Swank & Taam 1999; Muno, Morgan & Remillard 1999; Trudolyubov et al. 1999; Reig et al. 2000; Sobczak et al. 2000; Revnivtsev, Trudolyubov & Borozdin 2000; Tomsick & Kaaret 2001; Muno et al. 2001; Vignarca et al. 2003). However, neither the QPO mechanism (e.g. Stella & Vietri 1998; Stella, Vietri & Morsink 1999; Wagoner 1999; Tagger & Pellat 1999; Chakrabarti & Manickam 2000; Nobili et al. 2000; Titarchuk & Osherovich 2000; Psaltis & Norman 2000; Ingram, Done & Fragile 2009) nor the origin of the power-law component (see e.g. Done, Gierliński & Kubota 2007) is very clear. Thus, we have adopted a model-independent strategy to study the phenomenon of the LFQPOs.

Based on a statistical study of both the LFQPO amplitude spectrum and the LFQPO amplitude of GRS 1915+105, we find that in the LFQPO frequency–HR diagram the points form two branches, which are designated as Branch 1 and Branch 2 (Fig. 1). This indicates that the energy spectra of the observations corresponding to the two branches are very different. Similar phenomena have been found by other authors. For instance, Belloni et al. (2000) showed that the χ state points follow two branches in the colour-colour diagram and used different spectral models for the observations located on the two branches. Rau & Greiner (2003) studied four years of RXTE observations of GRS 1915+105 during the χ state and revealed a two-branch correlation of the power-law slope and the power-law normalization. Their two branches correspond to our Branch 1 and Branch 2. Van Oers et al. (2010) analysed two observations that belong to Branch 1 and Branch 2, respectively, and found that their best-fitting model parameters were significantly different.

Identifying the states of the Branch 1 and Branch 2 observations is helpful for analysing the properties and origin of the LFQPO. Muno et al. (2001) investigated the radio and X-ray properties of GRS 1915+105 when the X-ray emission was hard and steady, and established that radio emission always accompanies the hard state of GRS 1915+105, but that the radio flux and the X-ray flux are not correlated. They defined ‘radio plateau conditions’ (the radio flux at 15.2 GHz is >20 mJy and the radio spectrum is optically thick with power law |$E^{-\alpha _{\rm r}}$|⁠, where α_r < 0.2), ‘radio steep conditions’ (the radio flux at 15.2 GHz is >20 mJy and the radio spectrum is optically thin with α_r > 0.2), and ‘radio faint conditions’ (the radio flux at 15.2 GHz is <20 mJy) for the hard-steady X-ray observations. The radio emission is generally believed to be synchrotron emission from ejected plasma in sporadic or continuous jets (e.g. Fender et al. 1995). For GRS 1915+105, the optically thick radio emission during plateau conditions has been resolved as a compact jet of relativistic electrons (Dhawan, Mirabel & Rodríguez 2000). The optically thin radio emission during steep conditions originates from material ejected from the central source (Mirabel & Rodríguez 1994; Fender et al. 1999; Dhawan et al. 2000). The radio-faint observations show some properties similar to a weak radio-plateau state. The radio-steep observations represent the transition into and out of radio-plateau conditions. As the LFQPO frequency decreases, the radio emission becomes brighter and optically thick. The source is in plateau conditions when the LFQPO frequency is lower than 2 Hz. By combining our results shown in Section 3.2 with the definitions and conclusions in the literature presented here, we deduce that for Branch 1, as the LFQPO frequency increases, the source evolves from radio-plateau conditions to radio-faint conditions via radio-steep conditions, and for Branch 2, the source is mainly in the radio-faint condition (Figs 2 and 3). It should be pointed out that the radio conditions of some observations cannot be identified owing to the lack of radio data. Nevertheless, the two branches are clearly separate and smoothly evolve in the LFQPO frequency–HR diagram, which reflects the smooth evolution of spectral state. Therefore, the lack of radio data has no effect on our identification of state.

Despite the significant difference between the spectral states of the Branch 1 and Branch 2 observations, and the fact that the radio emissions at a given LFQPO frequency are always stronger during Branch 1 observations than during Branch 2 observations (Fig. 3b), there is no essential difference between the LFQPO amplitude spectra of the two branches (Figs 4, 5 and 6). Thus, the LFQPO seems not to originate from the jet, as the jets of GRS 1915+105 during Branch 1 and Branch 2 observations are very different. The spectrum of the LFQPO amplitude is hard (Figs 4 and 5), suggesting that the LFQPO is related to the corona (e.g. Morgan et al. 1997; Ingram & Done 2012). Shaposhnikov & Titarchuk (2007) showed that as the LFQPO frequency increases, the spectral index of the X-ray spectrum, α_x, increases linearly and then smoothly levels off to become a constant. The α_x–LFQPO frequency relationship is fitted with function (1) in Shaposhnikov & Titarchuk (2007), and the obtained transition frequency is 2.23 ± 0.07 Hz. Coincidentally, we find that as the LFQPO frequency increases, E_c of the LFQPO amplitude spectrum also decreases and then smoothly levels off (Fig. 6b). The E_c–LFQPO frequency relationship is fitted with the same function, and the transition frequency is 2.84 ± 0.55 Hz. The similarity of the behaviours of the two correlations is another indication of the link between the LFQPO and the corona, although the details are as yet not very clear.

For Branch 1, the LFQPO amplitude increases with frequency until ∼2 Hz; above this, it decreases. Negative correlations between LFQPO amplitude and frequency have been observed in GRS 1915+105 and other BHBs (e.g. Muno et al. 1999; Sobczak et al. 2000; McClintock et al. 2009; Heil, Vaughan & Uttley 2011). The aperiodic variability also shows a decrease in amplitude above ∼ a few hertz (e.g. Pottschmidt et al. 2003; Axelsson, Borgonovo & Larsson 2005; Done et al. 2007; Kalemci et al. 2003, 2006). These negative correlations are often attributed to a low-pass filter acting to suppress variability above ∼2–5 Hz (e.g. Done et al. 2007; Gierliński, Nikołajuk & Czerny 2008; Heil et al. 2011). On the other hand, at f ≲ 2 Hz, the decrease in the LFQPO amplitude coincides with the growth of the jet, indicating a possible correlation between them. If the jet emits X-rays, considering the decreases in both the LFQPO fractional and absolute amplitudes (Figs 7, 8 and 6b), the decrease of the LFQPO fractional amplitude might partly be attributable to the increase in the X-ray flux of the jet, which is independent of the LFQPO. Even if the jet does not emit X-rays, the decrease might be attributable to the weakening of the LFQPO itself owing to some sort of process, for example more accretion material/energy forms the jet but not the corona. Yan et al. (2013) also presented a decrease in the LFQPO amplitude that coincides with the possible production of a short-lived jet in GRS 1915+105. Thus our result tends to support the existence of a short-lived jet. Because the low-pass filter mainly suppresses variability above several hertz and the radio flux is inversely correlated with the LFQPO frequency, it might be that both the low-pass filter and the jet affect the LFQPO amplitude, and that the former plays a dominant role at f ≳ 2 Hz while the later plays a dominant role at f ≲ 2 Hz. The word ‘dominant’ here refers only to the comparison between the effects of the low-pass filter and the jet.

If the LFQPO does come from the corona, then the negative correlation between the LFQPO frequency and the radio flux indicates a strong correlation between the corona and the jet. In the context of the truncated disc model, this means that decreasing the disc truncation radius leads to a higher QPO frequency (e.g. Done et al. 2007; Ingram et al. 2009) and a weaker jet.

For Branch 2, it is interesting to note that the points of the LFQPO absolute amplitude are distributed sporadically, while the points of the LFQPO amplitude are distributed regularly. More intriguingly, the LFQPO amplitude of Branch 2 is roughly in line with that of Branch 1 in the LFQPO amplitude–frequency relationship, hinting that a common mechanism, for example a low-pass filter, might work.

In summary, we have made a statistical study of both the LFQPO amplitude spectrum and amplitude in GRS 1915+105. The observations are divided into two groups based on the appearance of the LFQPO frequency–HR diagram. The jets of GRS 1915+105 during the two groups of observations are very different. For one group, the LFQPO frequency is negatively correlated with the radio flux. We fitted the LFQPO amplitude spectrum by a power law with a cutoff, and found that as the LFQPO frequency increases, the spectrum becomes harder. In addition, there is no significant difference between the two groups of observations in the LFQPO amplitude spectrum, indicating that the LFQPO does not originate from the jet. The LFQPO amplitude spectrum is hard, suggesting that the LFQPO originates from the corona. As the LFQPO frequency increases, the LFQPO frequency–amplitude relationship evolves from a positive correlation to a negative one, which might be a result of the combined effect of the low-pass filter and the jet on the LFQPO amplitude.

We thank the anonymous referee for comments that greatly improved the quality of the manuscript. We thank Ms Qian Yang for her help with the English. This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center. This work is supported by CAS (KJCX2-YW-T09), 973 Program (2009CB824800), NSFC (11143013, 11173024, 11203063 and 11203064), and WLF of CAS (XBBS201123, LHXZ201201 and XBBS201121).

REFERENCES

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article:

Table 1. The definitions of RXTE/PCA energy bands for light curve extraction (Supplementary Data).

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