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Astron. Astrophys. 361, 121-138 (2000)

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4. Discussion

4.1. The nature of 4U1915-05

The classification introduced by Hasinger and Van der Klis (1989) distinguishes two classes of LMXBs, the Z and the atoll sources. We list the main evidence suggesting that 4U1915-05 is an atoll source. First, 4U1915-05 has always been detected at X-ray luminosities below [FORMULA] ergs s-1, which is typical for atoll sources. The Z sources are roughly 10 times brighter. Second, the orbital period of 4U1915-05 (50 min) is among the shortest observed for LMXBs. Now, it is observed that Z sources have longer periods (more than 10 h) than atoll sources, consistent with the idea that their companions are giants or subgiants (Van der Klis 1995). Furthermore, almost all persistent dippers and bursters belong to the atoll class (Cyg X-2 and GX 17+2 are the only Z sources to display type I bursts (Kahn & Grindlay 1984; Smale 1998; Sztajno et al. 1986)). Finally, no Z track in the color-color diagram has been observed neither in the GINGA data (Yoshida 1992), nor in the RXTE data. Hence, this favors the idea that 4U1915-05 belongs to the atoll class, as suggested by Yoshida (1992).

4.2. The states of 4U1915-05

Identifying the different atoll states in 4U1915-05 is not straightforward. From the hardness-intensity diagrams, we can see that the emission, on average, varies along the observations from a HS to a LS that could correspond to banana and island states respectively. However, we do not observe clearly separated regions in the color-color diagram but rather a continuous banana shape. Furthermore, in its LS, 4U1915-05 does not display HFN which is the typical aperiodic variability of atoll sources in their island state; 4U1915-05 displays VLFN in its both states. This behaviour was observed in the GINGA data as well. Unlike Yoshida (1992) who associated the two regimes with island and banana states, we believe that, during our observations, the source was not in its island state; it probably remained on the banana branch and hence never reached the state which may be at even lower luminosity. Such a state may have been observed by BeppoSAX (Church et al. 1998) since extrapolating the spectral parameters reported by Church et al. (1998) into the 2-50 keV energy range gives a luminosity of [FORMULA] ergs s-1 while the luminosity derived from the RXTE observations in the same energy band ranges between [FORMULA] and [FORMULA] ergs s-1 (Bloser et al. 2000).

The properties and presence of HFQPOs strongly depend on source states. For atoll sources, HFQPOs seem to occur at intermediate inferred accretion rates: they are generally not observed in extreme island or upper banana states corresponding respectively to the lowest and highest accretion rates in a given source, as in e.g. 4U1608-52 (Méndez et al. 1999). Twin simultaneous HFQPOs have now been detected in all atoll sources showing kHz variability but Aql X-1 (Van der Klis 1999). They are always seen during banana states with the exception of 4U1728-34 (Ford & Van der Klis 1998; Strohmayer et al. 1996; Méndez & Van der Klis 1999) and 4U1735-44 (Ford et al. 1998b). This gives further support to the idea that 4U1915-05 was in a banana state during our observation. The highest intensity state where no HFQPO is detected (February and March observations) was therefore more likely the upper banana state. The lowest intensity state was likely the lower banana state, although no HFQPO is detected during May 18th to 22nd observations where the count rate is the lowest; maybe due to a lack of sensitivity (the RMS upper limits corresponding to these observations and quoted in Table 4 are not really constraining).

4.3. Tracking the accretion rate with Sa

It has been shown that the relation between HFQPO frequency and intensity could be much more complex than a roughly one to one correlation as for example in 4U1820-30 (Smale et al. 1997). Numerous branches are clearly visible in the HFQPO frequency versus intensity diagram made with an extensive set of data from 4U1608-52 and 4U1728-34 (Méndez et al. 1999; Méndez & Van der Klis 1999). Zhang et al. (1998) also showed that Aql X-1, observed in very distinct flux or count rate ranges, could display HFQPOs in the same frequency range. The absence of a simple correlation had previously been noticed in 4U1705-44 (Ford et al. 1998a). For 4U1608-52, the HFQPO frequency was reported to correlate well with the count rate but only on timescales of hours (Méndez et al. 1999). Thus, the relation between HFQPO frequency and intensity seems to differ from source to source; sometimes being complex. On the contrary, the position on color-color diagram yields a more universal and unique relation as in, e.g. 4U1608-52, 4U1728-34 (Méndez et al. 1999; Méndez & Van der Klis 1999). Our study of 4U1915-05 confirms these results. The HFQPO frequency has a simpler relation with the position Sa on the color-color than with the count rate (Figs. 9 and 10). The frequency is well correlated with Sa within its full range.

Now, currently admitted models for HFQPOs, which are beat frequency models (BFM), involve a bright spot at a specified radius of the Keplerian disk, e.g. the sonic radius in the "sonic-point model" (Miller et al. 1998c) to produce the upper HFQPOs. As the accretion rate increases, the disk inner radius moves inwards until the innermost stable orbit is reached. Then, the HFQPO frequency is supposed to correlate with the accretion rate up to a saturation. Thus, the observed complex relations between HFQPO frequency and count rate show that the count rate is not as good as Sa to track the accretion rate. This confirms the studies carried out with EXOSAT data which revealed the position on color-color diagrams as a better indicator of accretion rate than intensity (Hasinger & Van der Klis 1989; Van der Klis et al. 1990; Hasinger et al. 1990).

However, the position on the color-color diagram may not be the only or the best indicator of the accretion rate or (and) of the timing behaviour. The properties of the HFQPO RMS amplitude found in 4U1915-05 (Figs. 9 and 10) may be new evidence for this. Indeed, in the sonic-point model, the HFQPO amplitude is expected to decrease as the accretion rate increases: the optical depth and electron density increase near the neutron star, and the oscillations generated there are more attenuated during their propagation (Miller et al. 1998c). Therefore, one would expect the HFQPO RMS to be anticorrelated with the accretion rate. However, for 4U1915-05, the HFQPOs RMS does not seem to be anticorrelated with Sa but rather may be anticorrelated with the count rate (Figs. 10 and 9). Thus, the timing properties do not probably depend only on the position Sa. Furthermore, it is not clear why the HFQPOs RMS amplitude and frequency would be driven by different parameters.

Moreover, we have shown that 4U1915-05 can display different timing behaviours (presence or absence of HFQPOs) during observations overlapping in the color-color diagram. The same situation occurs in other sources, e.g. 4U1636-53 (Méndez 1998).

One explanation may be that these timing behaviours are in fact the same but seem different in appearance because of instrumental limitations. For example, a QPO may be present in two observations overlapping in the color-color diagram but not detected in one of them because one of the detectors has been switched off leading to a loss of sensitivity (see e.g. the RMS upper limits in Table 4).

A second explanation may be that within the timespan for which Sa is computed, the source might have moved in and out the Sa span, washing out temporal signals.

Another explanation is that the source has actually intrinsically different timing behaviours. In this case, this is evidence that the position in the color-color diagram is not the best indicator of the timing behaviour of a source. Then two different conclusions may be considered. First, assuming that the timing behaviour is governed by the accretion rate, then Sa is not a good indicator of the accretion rate. This quantity may be better represented by a combination of parameters or by other parameters than Sa. Second, assuming that Sa is a good indicator of the accretion rate, then the timing behaviour is not governed only by the accretion rate.

4.4. Low and high frequency quasi-periodic oscillations

LFQPOs between a few and [FORMULA] 80 Hz have now been observed in several atoll sources (see Table 1). In 4U0614+09 (Ford 1997), 4U1608-52 (Yu et al. 1997), 4U1702-43 (Markwardt et al. 1999), 4U1728-34 (Strohmayer et al. 1996; Ford & Van der Klis 1998), KS1731-260 (Wijnands & Van der Klis 1997) and 4U1735-444 (Wijnands et al. 1998c), such LFQPOs are detected simultaneously with HFQPOs. In the case of 4U0614+09, 4U1728-34 and 4U1702-43, the LFQPO frequency is observed to increase with the HFQPO frequency and with the intensity of the source. First, this indicates that the LFQPOs detected in atoll sources are comparable to the horizontal branch QPOs (HBOs) detected in Z sources, suggesting that similar physical processes are at work in both kind of LMXBs. Second, this indicates that LF and HFQPOs are produced either by related mechanisms or by the same mechanism seen under different aspects or occuring at different locations in the disk. Our study of 4U1915-05 confirms both these conclusions. For the first time, we show a LFQPO in an atoll source varying in frequency and RMS amplitude as a function of the position on the color-color diagram (Fig. 10) the same way as HBOs in Z sources. We note however that in 4U1915-05, the LFQPOs are not strictly correlated with the source intensity, which is a characteristic of HBOs and also of the atoll sources mentioned above showing LFQPOs and HFQPOs. 4U1915-05's behaviour is thus different in this respect. This may be explained by the fact that in 4U1915-05, the intensity is not correlated with the position in the color-color diagram over the entire range where LFQPOs are detected. On the contrary, the intensity is strictly correlated with the position in the color-color diagram in horizontal branches (Van der Klis 1995) and probably in some portions of the track of atoll sources. In 4U1915-05, we clearly show (Figs. 10 and 9) that Sa, and not the count rate, is the key parameter of the quasi-periodic variability at low frequencies. Thus, the HF and LFQPOs frequencies depend the same way upon Sa.

Psaltis et al. (1999a) have shown that the frequency of the QPOs detected between 0.1 and 100 Hz in numerous Z, atoll or black hole binaries, simultaneously with HFQPOs followed one of a small number of correlations with the lower HFQPO frequency. For 4U1915-05, we could detect simultaneously the LFQPO and the lower HFQPO only during three segments. On the diagram derived by Psaltis et al. (Psaltis et al. 1999a, see their Fig. 2), these points would fall in the region where the two main correlations merge. Hence, our RXTE observation does not help to conclude about the unified picture suggested for QPOs.

The fact that HBOs and HFQPOs have now been observed simultaneously in Z sources, as in Cyg X-2 (Wijnands et al. 1998a) calls into question the models proposed for QPOs. Indeed, the magnetospheric BFM was first proposed to explain the HBOs in Z sources (Alpar & Shaham 1985). Later, the same model was suggested to explain the HFQPOs (Strohmayer et al. 1996) observed both in atoll and Z sources. But simultaneous LFQPOs and HFQPOs are difficult to explain with the same model. This requires that the disk is still present inside the radius where the magnetosphere couples the disk and channels the matter (Miller et al. 1998c). In this case, HBOs and HFQPOs can be explained both with the magnetospheric BFM. Psaltis et al. (1999b) have shown that the correlations observed between HBO and HFQPO frequencies in Z sources were indeed consistent with this model. However, this result has been shown so far for Z sources only. The striking similarities between the LFQPOs detected in 4U1915-05 and HBOs suggest that both phenomena could be interpreted within a same model. But the magnetospheric BFM may not be easily extrapolated to atoll sources because of their lower inferred magnetic fields and accretion rates.

Another possible interpretation comes from Stella and Vietri (1998) who proposed that the LFQPOs observed around 15-50 Hz in several atoll and Z (horizontal branch) sources could result from the precession of the innermost disk regions. In the first version of this model, the LFQPO frequency is the nodal precession ([FORMULA]) of slightly titled orbits in these regions. This precession frequency (relativistic -Lense-Thirring- and classical contributions) depends on the equation of state of the neutron star but also on its spin frequency and on the Keplerian frequency of the innermost accretion disk region. In the framework of BFMs, these latter frequencies are inferred for LMXBs displaying twin HFQPOs, from the frequency separation of the peaks and the frequency of the upper peak respectively. The precession frequency is predicted to vary approximately as the square of the innermost Keplerian frequency. For 4U1915-05, the LFQPO frequency varies as [FORMULA] where [FORMULA] is the inferred Keplerian frequency (see Fig. 13). This is not far from the expected quadratic dependence within the errors. However, the observed LFQPO frequency is [FORMULA] 2 times greater than the expected precession frequency drawn as a dashed line on Fig. 13. We assumed a neutron star spin frequency of 348 Hz which is the mean peak separation between the twin HFQPOs detected in 4U1915-05. Furthermore, we used the same parameters as Stella and Vietri (1998) for the neutron star: mass [FORMULA], ratio [FORMULA] where [FORMULA] is the moment of inertia in units of 1045 g cm2. With these values, Stella and Vietri (1998) found that the precession frequency matched the LFQPO frequency for three different atoll sources (4U1728-34, 4U0614-09 and KS1731-260). Since the neutron star parameters used yield precession frequencies close to the maximum values that can be reached keeping the parameters in the ranges allowed in classical neutron star models, it is unlikely that the Lense-Thiring model can be pushed up to match the LFQPOs in 4U1915-05. The precession frequencies are actually lower than the observed LFQPOs frequencies for any values of M and [FORMULA] within their reasonable respective ranges. This would also be the case if the spin frequency were lower than the one inferred from the twin peaks separation (348 Hz). On the opposite, the precession frequencies would roughly match the LFQPOs if the spin frequency were twice that value ([FORMULA] 700 Hz). With a spin frequency of 348 Hz, matching the observed LFQPO frequencies with precession frequencies would require a ratio [FORMULA] of [FORMULA] 3.5 which is too large. Indeed, this ratio is inferred to be in the range [FORMULA] 0.5-2 for realistic rotating neutron star models (Stella & Vietri 1998; Markovic & Lamb 1998; Kalogera & Psaltis 2000; Miller et al. 1998a).

Similar conclusions have been reached for 4U1728-34 (Ford & Van der Klis 1998, from a larger set of data than the one used by Stella and Vietri (1998), 4U1735-44 (Wijnands et al. 1998c), 4U1702-42 (in this case, the required ratio [FORMULA] is only 2.3 which is closer of the accepted range than for other sources (Markwardt et al. (1999))) and for the Z sources GX 17+2, GX 5-1, GX 340+0, Cyg X-2 and Sco X-1 (Stella & Vietri 1998; Jonker et al. 1998; Psaltis et al. 1999b; Kalogera & Psaltis 2000). For the atoll source 4U1608-52, the precession frequency model does not seem to apply either: the spin frequency predicted from the Lense-Thirring model (Yu et al. 1997) is inconsistent with the one inferred from the frequency separation of the twin HFQPOs later observed (Méndez et al. 1998b,c). In summary, the LFQPOs roughly follow the dependence upon the Keplerian frequency expected within the Lense-Thirring model but occur at a frequency lower than the precession frequency for a growing group of binaries that now includes 4U1915-05. Thus, in order to explain the LFQPOs with the Lense-Thirring effect, further hypothesis are required (Markovic & Lamb 1998; Morsink & Stella 1999; Armitage & Natarajan 1999; Kalogera & Psaltis 2000; Schaab & Weigel 1999). Another possibility is that the LFQPOs correspond to the second harmonics of the precession frequency 2[FORMULA] instead of [FORMULA] as proposed in a second version of the Lense-Thirring model, maybe because of a modulation at twice the precession frequency generated at the two points where the inclined orbit of the blobs intersects the disk (Vietri & Stella 1998; Morsink & Stella 1999; Stella & Vietri 1998; Stella et al. 1999).

However, Psaltis et al. (1999b) and Schaab and Weigel (1999) noticed another discrepancy with the model: the LFQPO frequency does not follow the expected quadratic dependence on the Keplerian frequency when the latter frequency is greater than 850 Hz in Z sources. In addition, radiation forces in the disk could significantly affect the precession frequencies and thus possibly challenge the Lense-Thirring model as the explanation for the LFQPOs (Miller 1999). Thus, the LFQPOs origins and relations with HFQPOs remain unclear.

Another model proposes to interpret both the LF and HF QPOs within the same framework: the two-oscillator model (Osherovich & Titarchuk 1999b; Titarchuk & Osherovich 1999; Titarchuk et al. 1999). In this model, the lower kHz QPO is the Keplerian frequency ([FORMULA] [FORMULA] 700 Hz) at the outer edge of the boundary layer between the Keplerian disk and the neutron star. Blobs thrown out from this region into the magetosphere oscillate in both radial and perpendicular modes to produce an upper kHz QPO (at [FORMULA] [FORMULA] 1000 Hz) and a low frequency QPO (at [FORMULA] [FORMULA] 50 Hz) respectively. Furthermore, viscous and diffusive processes within the boundary layer produce respectively a low frequency QPO (at [FORMULA] [FORMULA] 30 Hz) and a break (at [FORMULA] [FORMULA] 8 Hz) in the PDS. In this model, the angle [FORMULA] between the rotational frequency [FORMULA] of the magnetosphere and the normal to the neutron star disk is not zero.

The various observed QPOs of Sco X-1, 4U1728-34 and 4U1702-42 have been succesfully identified in the framework of the two-oscillator model (Titarchuk et al. 1999; Osherovich & Titarchuk 1999a). The angle [FORMULA] derived is [FORMULA], [FORMULA] and [FORMULA] respectively for each source. Furthermore, the predicted variation of [FORMULA] as [FORMULA] has been checked for number of atoll and Z sources (Titarchuk et al. 1999).

For 4U1915-05, the identification of the observed QPOs is not straightforward. 4U1915-05 does not show any break in its PDS. Only one LFQPO is detected in the frequency range where QPOs are predicted to be present at [FORMULA], [FORMULA] and 2[FORMULA].

We tried however to test the two-oscillator model. [FORMULA], [FORMULA] and [FORMULA] follow the relation [FORMULA] (see e.g., Eq. 2 in Osherovich and Titarchuk (1999a)), so that the rotational frequency can be derived when twin HFQPOs are detected simultaneously and interpreted as [FORMULA] and [FORMULA].

The bottom panel of Fig. 14 shows the inferred rotational frequency as a function of the lower HFQPO frequency, for the four segments where twin HFQPOs are detected above 550 Hz (see Table 1). Theoretically, [FORMULA] depends on the magnetic structure of the neutron star's magnetosphere. In the case of 4U1915-05, there are not enough simultaneous detections of [FORMULA] and [FORMULA] to reconstruct the [FORMULA] profile. We thus use the approximation [FORMULA], as was done also for 4U1702-42 (Osherovich & Titarchuk 1999a). As shown in Fig. 14 (bottom panel), the rotational frequency is indeed consistent with being constant with [FORMULA] Hz. We note that the pair of QPOs at 224 and 514 Hz would correspond to a rotational frequency of [FORMULA] Hz inconsistent with the previous value and not considered here.

[FIGURE] Fig. 14. Inferred rotational frequency (bottom panel) and angle [FORMULA] (top panel) as a function of the lower HFQPO frequency within the two-oscillator model. The dashed lines represent the mean values. The four points in the bottom panel are inferred from the 4 detections of twin HFQPOs above 550 Hz. The three points in the top panel correspond to the three cases where a LFQPO is simultaneously detected.

Knowing [FORMULA], the angle [FORMULA] can be derived when [FORMULA] and [FORMULA] are measured simultaneously (see e.g. Eq. 5 in Osherovich and Titarchuk (1999a)). The top panel of Fig. 14 shows the inferred angle [FORMULA] for 4U1915-05, assuming that the LFQPO frequency is [FORMULA]. [FORMULA] is consistent with being constant with a mean value [FORMULA]. It is in the range of values found for the other sources, but the error is particularly large.

We note however that the relation between [FORMULA], [FORMULA] and [FORMULA] written above requires [FORMULA], which becomes [FORMULA] Hz in the case of 4U1915-05. Now, according to Fig. 10 (left) showing the HFQPO frequency as a function of Sa, when only one HFQPO is detected, it seems to be the upper HFQPO. If we assume that it is actually the case and that its frequency is [FORMULA], then the observed frequency range of [FORMULA] would begin at [FORMULA] 500 Hz. This would be inconsistent with the condition [FORMULA] Hz derived according to the assumptions considered above.

4.5. Implications for the neutron star in 4U1915-05

Observations of HFQPOs allow to derive constraints on the neutron star present in the system. In the framework of beat frequency models, the frequency difference between twin HFQPOs is interpreted as the spin frequency of the neutron star. For 4U1915-05, the mean frequency separation is 348 Hz for four segments of observations. This is consistent with values reported in other LMXBs and would imply a neutron star rotating with a 2.8 ms period. However, a frequency separation of 290 Hz is detected for a pair of QPOs at 514 and 224 Hz. The spin frequency detection at 348 Hz needs to be firmly confirmed.

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Online publication: September 5, 2000
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