LOW-FREQUENCY QPOS AND POSSIBLE CHANGES IN THE ACCRETION GEOMETRY DURING THE OUTBURSTS OF AQUILA X−1

Figure 1 shows the long-term X-ray light curve of Aquila X-1. We converted the RXTE/ASM 2–12 keV flux in units of Crab by dividing the ASM 2–12 keV count rate by the mean Crab count rate of 75.6 cts s⁻¹. Similarly, for the later period when RXTE stopped operations, we obtained the source flux in 2–10 keV by the combining publicly available 2–4 keV and the 4–10 keV MAXI data.

Figure 2 shows the X-ray light curve and the X-ray hardness ratio (HR) seen with the pointed observations of RXTE/PCA. We extracted source light curves from PCA standard products¹ which provide data with a time resolution of 16 s from the STD 2 PCA data. The light curves in the PCA standard products have five energy bands, namely, 2–9, 2–4, 4–9, 9–20, and 20–60 keV. We identified the time intervals associated with Type-I bursts following the screening method mentioned in Klein-Wolt & van der Klis (2008) and excluded these time intervals. Then, we calculated the mean count rates per proportional counter unit (PCU) in the five energy bands and the HR, which was defined as the count rate ratio between 9–20 keV and 2–9 keV in each observation.

We made use of those PCA data with a time resolution better than 0.25 ms and combined all of the available data in different energy bands whenever possible. We extracted the data and rebinned the light curves into 1/4096 s resolution and then generated the Fourier power spectrum for every 128 s segment. In the analysis of each power spectrum, we took the average power above 1800 Hz to estimate the white noise level, which was then subtracted. We fit each power spectrum with a model composed of several Lorentzians in which the zero-centered Lorentzian represents the band-limited noise components and narrow Lorentzians represents the QPOs following the previous approach (Belloni et al. 2002). A power-law component was included to fit very low-frequency noises when necessary. These model fits were performed using a custom package using the MPFIT package.² The best-fit model was achieved by minimizing the ${{\chi }^{2}}$, and the uncertainty was obtained from the distribution of ${{\chi }^{2}}$. Throughout this paper, the error quoted for the power spectral fit represents the 1σ uncertainty.

In order to estimate the X-ray flux of the source, we fit the 3–20 keV Standard 2 PCA energy spectra of the observations when the source was in the hard state during the rising phase of each outburst. The X-ray spectral model we used consisted of four components, namely, a power-law component representing the emission from the hot accretion flow, a blackbody component for the emission from the NS surface, a Gaussian line component with its central energy fixed at 6.5 keV representing a broad Iron emission line, and the absorption by Galactic neutral hydrogen. We fixed the column density of the neutral hydrogen to $3.4\times {{10}^{21}}\;{\rm atoms}\;{\rm c}{{{\rm m}}^{-2}}$, according to Dickey & Lockman (1990). The 2–20 keV flux was then obtained from the spectral fits in XSPEC 12.8.2.

As shown in Figure 1, the combined RXTE/ASM and MAXI 2–10 keV light curve of Aql X−1 indicates 16 outbursts in total over a period of 16 years; RXTE observations covered the hard X-ray state during the rising phase of seven outbursts which occurred in 1999, 2000, 2001, 2004, 2009, 2010, and 2011, respectively, as indicated by the triangles in the plot.

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The PCA 2–20 keV count rate and the HR during the seven outbursts are shown in Figure 2. The peak count rate ranged from ∼200 (the 2001 outburst) to ∼1800 (the 2011 outburst) ${\rm counts}\;{{{\rm s}}^{-1}}\;{\rm PC}{{{\rm U}}^{-1}}$, spanning nearly one order of magnitude in the RXTE observations. For the 1999, 2000, 2004, 2010, and 2011 outbursts, the hard-to-soft state transitions are clearly shown by the abrupt decrease of the HR from ∼1.0 to ∼0.2. For the 2001 outburst, the hard-to-soft state transition was identified in Yu & Dolence (2007) based on the HEXTE(15–250 keV)/PCA(2–9 keV) HR. We therefore used the previous identification. The time of the hard-to-soft state transition corresponds to observation 60054-02-04-00 on MJD 52095. It is also worth noting that the 2001 outburst was peculiar since it was not a fast rise with an exponential decay outburst as were the others. In addition, the hard-to-soft state transition in the 2001 outburst was also peculiar since it occurred when the luminosity declined (Yu & Dolence 2007). For the 2009 outburst, we identified the time corresponding to the hard-to-soft state transition based on source timing properties. This is based on the fact that for other atoll sources, such as 4U 1608−52, the state transition was accompanied by the transition of the properties of the power spectra (see for example, Figure 7 of van Straaten et al. 2003). In Aql X−1, we saw such a transition in the power spectra from observation 94076-01-04-04 taken on MJD 55152.2 to observation 94076-01-04-05 taken on MJD 55153.1, and hence we conclude that the source left the hard state between these two observations.

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Figure 3 shows the evolution of the power spectra in the rising phase of the seven outbursts during which the LFQPOs and the band-limited noise components were observed. The most obvious component below 100 Hz is the LFQPO component for which the frequency evolved gradually from several Hz to tens of Hz. The LFQPO component also corresponds to the maxima in the $\nu {{P}_{\nu }}$ versus frequency plot below 100 Hz. We found that the LFQPOs had the following properties.

• 1.  
  In all except the 2001 outburst, the frequency increased with time before the state transition. This indicates that the characteristic timescale corresponding to the LFQPO decreased with time during the rising phase of the outbursts.
• 2.  
  In all except the 2001 outburst, the frequency appeared to increase more rapidly just before the hard-to-soft state transition than during the earlier stages. This is best illustrated in the 1999 outburst. Notice that the power spectra are plotted in logarithmic scale and the power spectra are shifted by multiplying a factor proportional to the time separation between the observations.

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To quantitatively analyze the frequency evolution of the LFQPO and other noise components, we fit the power spectra with multi-Lorentzians following a previous approach (Belloni et al. 2002). By doing so, the power spectra can be empirically decomposed into several Lorentzian components (e.g., Belloni et al. 2002; van Straaten et al. 2002, 2003; Klein-Wolt & van der Klis 2008), allowing us to study the frequency correlations among the characteristic frequencies. Some examples of our model fits can be seen in Figure 4 where we show the power spectra with the best-fit models corresponding to three observations performed during the 1999 outburst. The component with the lowest frequency can be identified as the noise component, L_b, as defined in previous studies. Two distinct components, a narrow component at a lower frequency and a prominent broader component with a higher frequency, were found in the frequency range from several Hz to tens of Hz. The broader, stronger components are the LFQPOs. In more than half of the observations, the narrow feature is not required in the power spectral fit. Those components with higher frequencies are identified as L_l and L_u, following the conventions used in Belloni et al. (2002).

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It has been found that the characteristic frequencies of the broad noise and narrow QPO components are correlated (e.g., Psaltis et al. 1999; Wijnands & van der Klis 1999; Belloni et al. 2002). To compare with the results obtained from other sources and to study how these characteristic frequencies are correlated during single outbursts in the RXTE era, we plotted the LFQPO characteristic frequency ${{\nu }_{{\rm LF}}}$ versus break frequency ${{\nu }_{b}}$, along with the data shown in the WK99 relation (Wijnands & van der Klis 1999) and the BPK02 relation (Belloni et al. 2002) in Figure 5. The correlation of ${{\nu }_{{\rm LF}}}$ and ${{\nu }_{b}}$ in Aql X−1 is consistent with the correlation in other sources, as expected.

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The evolution of the characteristic frequency of the LFQPOs in seven outbursts is shown in Figure 6. First, we examined the maximum LFQPO frequency which the source can reach in the hard states; this frequency may correspond to a certain radius or timescale in the accretion flow when the state transition occurs. As seen from Figure 6, the maximum LFQPO frequency varies among the outbursts, from $6\pm 0\;{\rm Hz}$ in the 2000 outburst to $38\pm 12\;{\rm Hz}$ in the 2001 outburst, although the uncertainty for the latter is quite large. In the 2009 outburst, the maximum frequency was $30\pm 5\;{\rm Hz}$. We took the weighted average of the maximum LFQPO frequency of the 1999 and the 2001 outburst, i.e., $\sim 31\;{\rm Hz}$, as the estimate of the maximum LFQPO frequency seen in Aql X−1 in the following discussion.

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The evolution pattern of ${{\nu }_{{\rm LF}}}$ was quite similar among the outbursts. In the following analysis, for the 2004 outburst, we concentrated on the last four observations in the hard state before the hard-to-soft transition, since earlier observations corresponded to a special flare in the light curve, as can be seen from Figure 2. For the peculiar 2001 outburst, the frequency was around 10 Hz at the beginning, then increased rapidly to around 30 Hz at about 6 days before the state transition, and then decreased to around 24 Hz at about just 1 day before the transition, and finally turned to around 40 Hz at the time of the state transition. For all of the other outbursts, the frequency of the LFQPO increased consistently with time during the rising phase of the outbursts.

In our model-independent approach, we estimated the rate of change of the LFQPO frequency following $\dot{\nu }=({{\nu }_{2}}-{{\nu }_{1}})/({{t}_{2}}-{{t}_{1}})$, where ν is the LFQPO frequency, t is the time, and the subscripts 1 and 2 denote two consecutive observations. Then, we calculated the fractional rate of change of the frequency $2\dot{\nu }/({{\nu }_{1}}+{{\nu }_{2}})$, corresponding to the time at $({{t}_{2}}+{{t}_{1}})/2$. Figure 7 shows the time evolution of the fractional rate of change. The fractional rate of change increased from 10% day⁻¹ at the beginning of the outburst, to ∼100% day⁻¹ just before the state transition.

The pattern of the increase of the LFQPO frequency deviates from a single exponential pattern, which would follow a straight line if plotted in logarithmic scales, and the fractional rate of change would be constant with time. We tried two phenomenological models to fit the pattern. The first model is an exponential quadratic polynomial, i.e., $\nu ={{e}^{a{{t}^{2}}+bt+c}}$. This function is continuous and the fractional rate of change increases with time when the parameter a is positive. The second model is of a broken exponential function, i.e., the frequency first increases with a constant e-folding timescale t₁, then afterwards the frequency increases with another constant e-folding timescale t₂. We fit the models to the data of the 1999 outburst and the 2009 outburst, which have the most data points (which is also more than the number of model parameters). The dashed and dotted lines show the best-fit broken exponential and exponential quadratic polynomial models, respectively. Note that the time at zero corresponds to the time when the last observation of the source in the hard state was taken.

For the 2009 outburst, ${{\chi }^{2}}$ was 8.76 with 6 degrees of freedom (dof) and 34.25 with 7 dof for the broken exponential function and exponential quadratic polynomial function, respectively, indicating that the former model describes the pattern of frequency evolution better. The result from the best-fit broken exponential model showed that the frequency increased with an e-fold timescale of 5.13 ± 0.17 and 1.31 ± 0.17 days before and after the moment at 2.80 ± 0.22 days before the state transition, respectively. For the 1999 outburst, the model fit with an exponential quadratic polynomial provided a good fit with ${{\chi }^{2}}=1.89$ and 2 dof, while the fit with the broken exponential model yielded a ${{\chi }^{2}}=0.11$ with 1 dof, indicating that we lacked the statistics to constrain the later model. The trend in both 2009 outbursts suggests that the frequency increased with time at different speed in the early and late stages of the hard state during the rising phase of an outburst, further supporting the idea that the transition is associated with non-stationary accretion during these outbursts.

In Figure 8, we show how the LFQPO frequency evolved with the 2–20 keV flux. During the rising phase of an outburst, the frequency generally increased with the X-ray flux, except for the peculiar 2001 outburst. The frequency–flux relation for each outburst deviates from a straight line on a log–log plot, showing that it does not follow a simple power-law relation. The power-law model was only acceptable for the 2000 outburst, with ${{\chi }^{2}}=0.59$ and 1 dof, while for other outbursts the model was rejected. The model parameters in the fits with the power-law model are tabulated in Table 1. We fit the data with a broken power-law model instead. Such a model consists of four free parameters: the normalization, the break frequency, and the power-law indices before and after the break. We fit the model to the data corresponding to the 1999 and 2009 outbursts, which have more than four data points.

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Table 1.  Power-law Model Fit to the Frequency vs. Flux Relation

Year	α	N	${{\chi }^{2}}$	dof
1999	2.45 ± 0.08	0.47 ± 0.05	36.80	3
2000	0.88 ± 0.12	0.27 ± 0.07	3.48	2
2004	2.50 ± 0.22	1.04 ± 0.23	98.50	2
2009	2.20 ± 0.09	2.23 ± 0.13	39.36	7
2010	1.81 ± 0.12	0.79 ± 0.11	0.59	1
2011	1.26 ± 0.12	0.45 ± 0.05	4.89	1

Note. The parameters of best-fit power-law model of frequency versus 2–20 keV flux: ${{\nu }_{10}}=Nf_{3}^{\alpha }$, where ${{\nu }_{10}}={{\nu }_{{\rm LF}}}/10\;{\rm Hz}$ and f₃ is flux in $3\times {{10}^{-9}}\;{\rm erg}\;{{{\rm s}}^{-1}}\;{\rm c}{{{\rm m}}^{-2}}$.

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For the 1999 outburst, the broken power-law model yielded a ${{\chi }^{2}}=1.36$ with 1 dof while the power-law model gave a ${{\chi }^{2}}=36.80$ with 3 dof, showing a preference for the broken power-law model. In that model, the break occurred at $\sim 2\;{\rm Hz}$. The index before the break was not well constrained, providing 0.64 ± 0.43, while the index after the break was well constrained as 2.89 ± 0.12. Fitting with the broken power-law model yielded ${{\chi }^{2}}=12.35$ with 5 dof, while the power-law model fit gave ${{\chi }^{2}}=39.36$ with 7 dof. The break frequency occurred at $\sim 7\;{\rm Hz}$. The power-law indices were 1.34 ± 0.22 and 3.58 ± 0.34 before and after the break, respectively. The best-fit power-law and broken power-law models are plotted as dashed and dotted lines, respectively, in Figure 8.

Among the different outbursts, the power spectra show that the source can reach significantly different characteristic QPO frequencies at similar X-ray fluxes. For instance, as shown in Figure 8, at a source flux of $2\times {{10}^{-9}}\;{\rm erg}\;{\rm c}{{{\rm m}}^{-2}}\;{{{\rm s}}^{-1}}$, the frequency of the LFQPO was around 2 Hz for the 1999 and 2011 outbursts, but around 30 Hz for the 2001 outburst. It is interesting to note that for a certain characteristic frequency, the X-ray flux is usually higher in the outbursts in the years 1999, 2000, and 2011, which were associated with higher state transition fluxes, while the X-ray flux corresponding to the same QPO frequency was lower in the outbursts in the years 2001, 2004, 2009, and 2010, which were associated with lower state transition fluxes. There is another anti-correlation between the maximum LFQPO frequency and the state transition flux (or the outburst peak X-ray luminosity) with a Spearman correlation coefficient of −0.96 at a significance level of $4.5\times {{10}^{-4}}$. This is shown in the upper right corner of Figure 8. Actually, the maximum frequency of the LFQPO was reached during the 2001 and 2009 outbursts, and was associated with the lowest state transition fluxes.

In this work, we found that the maximum frequency of the LFQPOs in Aql X−1 observed with the RXTE was ∼31 Hz. In a similar study of the BH transient GX 339−4, we found that the maximum frequency of the LFQPOs was around 5 Hz, which is about six times lower than that of Aql X−1. Özel et al. (2012) showed that the mass of the NSs in the recycled pulsars has a mean value of $1.48\;{{M}_{\odot }}$ with a dispersion of $0.2\;{{M}_{\odot }}$. We know that the BH binary GX 339−4 has a mass function of $5.8\;{{M}_{\odot }}$ (Hynes et al. 2003), so we do not know the exact mass ratio. If we take its mass as a typical value of 10 solar masses in BH X-ray binaries, then the mass ratio between the compact stars in the two systems seems consistent with the ratio between the maximum frequencies of the LFQPOs. This suggests that the maximum frequency of the LFQPOs may be an indicator of the mass of the compact star. By observing the maximum frequency of the LFQPOs during outbursts, we may be able to determine empirically the nature of the compact star in new transient LMXBs.

Using the analyzed observations, we can investigate whether or not it is possible to trace the evolution of the truncation radius using the LFQPO based on current disk truncation models and to constrain these models using observations. It has been believed that the frequency of one of the twin kHz QPOs might be associated with the Keplerian frequency at the inner disk edge (e.g., Miller et al. 1998; Stella & Vietri 1998; Osherovich & Titarchuk 1999). Therefore, we can probably associate the frequency of the LFQPOs with the inner disk edge (or the truncation radius in some popular models) based on the empirical frequency correlation between the kHz QPOs and LFQPOs. The empirical correlations found in the RXTE data suggest that for the lower kHz QPO at frequency ${{\nu }_{1}}$, the frequency of the LFQPO is ${{\nu }_{{\rm LF}}}\approx (42\pm 3\;{\rm Hz}){{({{\nu }_{1}}/500\;Hz)}^{0.95\pm 0.16}}$ (Psaltis et al. 1999). The empirical relation between the lower and upper kHz QPOs is ${{\nu }_{1}}=(724\pm 3){{\left( {{\nu }_{2}}/1000\;{\rm Hz} \right)}^{1.9\pm 0.1}}\;{\rm Hz}$ (Psaltis et al. 1998). This yields the relation between the LFQPO and upper kHz QPOs as ${{\nu }_{{\rm LF}}}\propto \nu _{2}^{1.85\pm 0.21}$.

In the magnetospheric model (Ghosh et al. 1977; Ghosh & Lamb 1979), the inner edge of the disk is where the ram pressure is balanced by the magnetic pressure. Ghosh & Lamb (1979) studied the structure of a geometrically thin disk around a magnetized NS and found that the disk is truncated at ${{r}_{0}}\approx C{{r}_{{\rm A}}}$, where r_A is the Alfvén radius, i.e., ${{r}_{{\rm A}}}={{\mu }^{4/7}}{{(2GM)}^{-1/7}}{{\dot{M}}^{-2/7}}$, where $\mu =B{{R}^{3}}$ is the magnetic dipole moment and R is the NS radius, and C is a parameter that does not much affect the configuration. The magnetospheric model therefore predicts that the truncation radius is ${{r}_{{\rm tr}}}\propto {{\mu }^{4/7}}{{\dot{M}}^{-2/7}}$.

In the disk evaporation model, the heat generated in the corona formed above the disk is conducted down to the lower layer of the corona and radiated away. If the density of the lower layer is too low to efficiently radiate away the heat, then the thin disk would be heated and then evaporated, leading to a steady mass flow from the disk to the corona. In a stationary state, the outer disk would be truncated at a radius r_tr where the evaporation is balanced by the mass inflow from the outer disk. This truncation radius lies at ${{r}_{{\rm tr}}}\propto {{\dot{M}}^{-1/1.17}}$. Other colleagues further developed the model by including the effect of the viscosity and the accretion disk magnetic field (Taam et al. 2012; Liu & Taam 2013), which gives the truncation radius ${{r}_{{\rm tr}}}\propto {{\dot{M}}^{-0.886}}{{\alpha }^{0.07}}{{\beta }^{4.61}}$, where α is the viscous parameter and β is the ratio of gas pressure to total pressure. The truncation radius depends only weakly on the viscosity.

It is worth noting that the parallel-track phenomenon of the LFQPO shown in Figure 8, which is similar to that seen for the kHz QPOs, actually suggests that the mass accretion rate is not the only parameter determining the QPO frequency. However, let us consider the simplest case where the mass accretion rate is the only parameter in the accretion system. As we know, both disk truncation models predict that the truncation radius is a function of the mass accretion rate, i.e., ${{r}_{{\rm tr}}}=c{{\dot{M}}^{\delta }}$, where c is the normalization coefficient. Then, we obtain the relation between the LFQPO frequency and the mass accretion rate as ${{\nu }_{{\rm LF}}}\propto {{\dot{M}}^{1.43\delta }}={{\dot{M}}^{{{\epsilon }_{1}}}}$ if we take the lower kHz QPO as the orbital frequency at the inner disk edge and as ${{\nu }_{{\rm LF}}}\propto {{\dot{M}}^{2.71\delta }}={{\dot{M}}^{{{\epsilon }_{2}}}}$ if we take the upper kHz QPO frequency as the Keplerian frequency at the inner disk edge. For the former case, the index is 1.22 ± 0.20 in the evaporation model and 0.57 ± 0.10 in the magnetospheric model; meanwhile, in the latter case, which corresponds to the upper kHz QPO assumption, the index is 2.31 ± 0.26 in the evaporation model and 0.77 ± 0.10 in the magnetospheric model. In the above estimates, we assumed that the bolometric correction factor was constant during those observations and that the bolometric luminosity was proportional to the mass accretion rate (i.e., the radiation efficiency did not change a lot).

In Section 3.4, we showed that for the well-sampled 2009 outburst, the frequency versus the X-ray flux relation was best fit by a broken power-law model with indices of 1.34 ± 0.22 and 3.58 ± 0.34 before and after the trend break. Both of the values are larger than that predicted by the magnetospheric model. As for the disk evaporation model, we found that for the case corresponding to the lower kHz QPO assumption, during the early rising phase of the outburst, the observed index before the trend break is consistent with the model, while after the break the observed index is much larger than that predicted by the disk evaporation model. This might indicate that the disk evaporation model is consistent with the observed evolution of the inner disk edge at the early rising phase of those outbursts but not later on. For the case corresponding to the upper kHz QPO assumption, the disk evaporation model overestimates the index in the early stage of the outburst but underestimates the index in the later stage. This suggests that additional physics should at least be considered in the current disk evaporation model. We also realized that additional constraints can be placed on the evaporation model as well from the so-called "parallel-track" phenomenon of the LFQPO. In the improved evaporation model investigated in Taam et al. (2012), the truncation radius, r_tr, depends on the magnetic field in the accretion disk as ${{r}_{{\rm tr}}}\propto {{\dot{M}}^{-0.886}}{{\alpha }^{0.07}}{{\beta }^{4.61}}$. In order to explain the observations showing that a lower characteristic frequency can be observed under similar mass accretion rates in the outbursts with a higher state transition luminosity, the disk evaporation theory with the consideration of the magnetic field has to be consistent with the fact the brighter an outburst is, the lower the magnetic pressure in the accretion disk should be due to the fact that there is empirical correlation between the luminosity of the hard-to-soft transition and the peak luminosity of the outburst or the following soft state (Yu et al. 2004; Yu & Dolence 2007; Yu & Yan 2009).