ROSSI X-RAY TIMING EXPLORER OBSERVATIONS OF THE LOW-MASS X-RAY BINARY 4U 1608–522 IN THE UPPER-BANANA STATE

In comparison with the previous work by Mitsuda et al. (1984), we first examine relatively slow (≳ 10³ s) spectral variations along the upper-banana state of 4U 1608–522. For this purpose, we selected four representative spectra, hereafter Spec A through Spec D, obtained at the most luminous (several times 10³⁷ erg s⁻¹) end (i.e., the right side) of the HIDs. Three of them (Spec A, Spec B, and Spec C) come from the same vertical line, while the other is located on a different line. Their observation IDs, average intensities, and soft/hard colors are summarized in Table 1, while their locations on the hard HID are indicated in Figure 1. These spectra and their ratios to their average spectrum are shown in Figure 2.

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In Figure 2, the brightest (Spec A) and the second brightest (Spec B) spectra of the four are selected from the same vertical line in the HID. By comparing them, it is clear that the intensity in the hard energy band changes significantly, while that in the soft band is kept constant. Moreover, in energies above ∼15 keV, their ratios saturate as a function of energy. As originally proposed by Mitsuda et al. (1984), these results can be explained when the spectra consist of a variable hard component that dominates in ≳ 15 keV energy band and a stable soft component, and the normalization (but not the shape) of the former changes between the two spectra. Indeed, the difference spectrum between them, shown in Figure 3 as Spec 1, is represented successfully by a BB model the temperature of which is kT_BB ∼ 2.5 keV. The obtained best-fit parameters are summarized in Table 2. Although the derived absorption column density, ∼3 × 10²³ cm⁻², is significantly higher than that reported previously (0.8 × 10²² cm⁻²; Grindlay & Liller 1978; Rutledge et al. 1999), the difference may be attributed to slight changes in the assumed stable soft component (as assessed later in this subsection), and/or those in kT_BB. These results reconfirm the achievements with Tenma (Mitsuda et al. 1984) and Ginga (Makishima et al. 1989).

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From the above results, we consider that the harder part of the spectrum is carried by a "hard component" which is approximated by a kT_BB ∼ 2.5 keV BB. Its variation (in normalization rather than in kT_BB; Makishima et al. 1989) causes significant changes in the hard color, but does not affect the intensity very much because the total source counts are dominated by softer photons. This explains the formation of the vertical stripes in the hard HID of Figure 1. This hard component is most naturally attributed to optically thick emission from the NS surface because the measured temperature agrees with the local Eddington temperature of a 1.4 M_☉ NS, or ∼2.0 keV.

Contrary to the above case, the other two spectra, the faintest spectrum (Spec D) and the second faintest one (Spec C) which lie on a pair of stripes adjacent to each other, differ mainly in the soft energy band, with the hard energy end kept almost constant. This suggests that the source variation in this case is carried by a soft spectral component which is presumably identified as the stable component suggested by Spec A and Spec B. To constrain its spectral shape, we derived the difference spectrum (denoted Spec 2) between Spec C and Spec D, and show the results in Figure 4. Clearly, this difference spectrum is much softer than the previous one (Spec 1), in agreement with the inference derived from the spectral ratios. This Spec 2 can be reproduced by either an MCD model (kT_in ∼ 1.7 keV) or a BB (kT_BB ∼ 1.3 keV), but as shown in Table 2, the former is more successful and gives a more reasonable absorption. We hence consider that the "soft component" is approximated by an MCD model with kT_in ∼ 1.7 keV.

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Since the above result supports the Eastern model, the four spectra must be reproduced by a linear combination (though with different weights) of the soft component expressed by a kT_in ∼ 1.7 keV MCD, and the hard component identifiable with a kT_BB ∼ 2.5 keV BB. Then, adding a narrow Gaussian component to represent an Fe–K emission line at 6–7 keV, we fitted the four spectra with this composite model, denoted MCD+BB+Gau model. The results are shown in Figure 5 and Table 3; there, r_in is the innermost radius of the accretion disk for an assumed inclination angle i = 0°, and r_BB is that of the BB model assuming a spherical emission region. Thus, all the four spectra have been fitted successfully by the model, and the obtained values of kT_in ∼ 1.8 keV and kT_BB ∼ 2.7 keV indeed agree with those obtained by the two difference spectra. These two temperatures are also consistent with those found in previous works that employed the Eastern model (Mitsuda et al. 1984; Makishima et al. 1989).

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Can we explain these results with alternative modeling? For example, a model proposed by White et al. (1988), often called the "Western" model, assumed that Thomson opacity dominates free–free opacity in the accretion disk and the emission is significantly modified by electron scattering, while the emission from the NS surface is hardly modified and is observed as a simple BB emission. Then, an unsaturated Comptonized emission is expected to dominate all over the energy band, while a soft BB model is added in the intermediate range. This model can actually explain the shape of Spec 2 in terms of variations of the soft BB component. However, Spec 1 is very difficult to explain with this model because its shape is different from those of either component of the model; we would need extreme fine tunings among the parameters of the BB and Comptonized components.

Returning to the Eastern model, a simple simulation was performed to examine the increased absorption issue in Figure 3. Employing the best-fit model to Spec A, we created a fake spectrum to be called Spec A'. Yet another spectrum, named Spec B', was created, in which the BB normalization was reduced to 67% of that of Spec A', and the MCD normalization was set 5% higher than that in Spec A'. These are meant to emulate Spec B within the allowed fit errors in Table 2. Then, the difference between Spec A' and Spec B' was indeed fitted successfully by a single BB model with nearly the same BB temperature as assumed in the input, but with the absorption increased to ∼3 × 10²³ cm⁻².

The Eastern model allows us to assign clear physical meanings to absolute values of the physical parameters obtained by the MCD and BB components. For that purpose, however, we need to convert the measured color temperature kT_X (X = "in" or "BB") to the effective temperature kT^eff_X, using the so-called hardening factor κ as

$$\begin{equation} kT^{\rm eff}_{\rm X} = \kappa ^{-1} kT_{\rm X}. \end{equation} \tag{ 1 }$$

Then, the true radius r^eff_X is estimated from the measured value r_X as

$$\begin{equation} r^{\rm eff}_{\rm X} = \kappa ^{2} r_{\rm X}. \end{equation} \tag{ 2 }$$

The value of κ is numerically estimated as 1.4–1.6 for Type I bursts (Ebisuzaki et al. 1984) and 1.7–2.0 for accretion disks (Shimura & Takahara 1995).

If we adopt κ = 1.7, the observed soft-component parameters of kT_in ∼ 1.8 keV and r_in ∼ 4 km yields kT^eff_in ∼ 1.1 keV and r^eff_in ∼ 12 km. Thus, the estimated true radius becomes larger than the NS radius of 10 km. Moreover, the value of r^eff_in is close to the radius of last stable orbit in terms of general relativity, 3R_s = 12.4 km, where R_s ≡ 2GM/c² is the Schwarzschild radius, G is the gravitational constant, M is the NS mass, and c is the light speed. As to the BB component, kT_BB ∼ 2.7 keV and r_BB ∼ 1.2 km yields kT^eff_BB ∼ 1.8 keV and r^eff_BB ∼ 2.7 km, assuming κ ∼ 1.5 (London et al. 1986; Ebisuzaki 1987). The estimated kT^eff_BB is close to the local Eddington temperature (2.0 keV) of a 1.4 M_☉ NS, suggesting that the BB emission arises from a region on the NS surface. In addition, r^eff_BB is smaller than 10 km, when assuming isotropic emission from a spherical source. Therefore, as previously suggested by Mitsuda et al. (1984), the BB component can be regarded as being emitted from an equatorial zone of the NS, where the accretion disk contacts the surface.

From the above spectral analysis, it has been confirmed that the Eastern (MCD+BB+Gau) model successfully reproduces the selected four spectra in the upper-banana state, and yields physically reasonable interpretations. Given these, we applied the MCD+BB+Gau model to the 95 spectra in the upper-banana state, and obtained χ²/dof of <1.4 for all of them. Therefore, we regard the Eastern model as applicable to all the present data sets.

Figure 6 summarizes the relation among the unabsorbed disk bolometric luminosity L_disk, the unabsorbed BB bolometric luminosity for isotropic emission L_BB, and their sum L_tot ≡ L_disk + L_BB, as well as the temperatures and radii. Here, L_disk is calculated assuming a face-on disk (i.e., the inclination angle i = 0°) as

$$\begin{equation} L_{\rm disk} = 2 \times \int ^{\infty }_{r_{\rm in}} 2\pi {r}\sigma T(r)^{4} dr = 4\pi r_{\rm in}^{2} \sigma T_{\rm in}^{4}, \end{equation} \tag{ 3 }$$

where the first factor of 2 means emission from the two sides of the disk, σ is the Stefan–Boltzmann constant, and T(r) is the disk temperature at the radius r.

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In Figures 6(a), both L_disk and L_BB are seen to increase as L_tot increases from 1 × 10³⁷ to 4 × 10³⁷ erg s⁻¹. However, a closer inspection reveals that L_disk varies more steeply than L_BB. Actually, the data behavior in Figure 6(a) can be approximated as

$$\begin{equation} L_{\rm disk} \propto L_{\rm tot}^{1.1},\quad L_{\rm BB} \propto L_{\rm tot}^{0.7},\quad L_{\rm BB} \propto L_{\rm disk}^{0.6}. \end{equation} \tag{ 4 }$$

As a result, the ratio between L_BB and L_disk decreases from 0.6 to 0.4 as L_tot increases by a factor of four, in agreement with previous reports (Gilfanov et al. 2003; Revnivtsev & Gilfanov 2006). Similarly, Figure 6(b) shows the luminosity dependences of the temperature and radius parameters, derived in the same way as in Section 3.1, and presented without applying any of the correction factors mentioned in Section 3.2. The relations are approximated as

$$\begin{eqnarray} &&kT_{\rm in} \propto L_{\rm tot}^{0.19},\quad r_{\rm in} \propto L_{\rm tot}^{0.12},\quad kT_{\rm BB} \propto L_{\rm tot}^{0.10},\quad r_{\rm BB} \propto L_{\rm tot}^{0.11}.\nonumber\\ \end{eqnarray} \tag{ 5 }$$

Instead of L_tot, we may utilize the mass accretion rate $\dot{M}$, which can be estimated in the following way. Generally, the accreting matter releases energy at a rate of ${\it GM}\dot{M}/r_{\rm in}$, as it flows through the disk down to the innermost radius r_in. According to the picture of the standard accretion disk, a half of this luminosity is radiated away in the accretion disk as $L_{\rm disk} = \frac{1}{2} \frac{{\it GM}\dot{M}}{r_{\rm in}}$, and the other half is stored in the Keplerian kinetic energy. Therefore, the mass accretion rate is estimated as

$$\begin{equation} \dot{M} = \frac{2 L_{\rm disk} r_{\rm in}}{GM} \propto L_{\rm disk} r_{\rm in} \propto r_{\rm in}^{3} T_{\rm in}^{4}. \end{equation} \tag{ 6 }$$

This relation remains valid even if r_in varies, as long as the disk stays in the standard state. Figure 7 shows the same physical parameters as presented in Figure 6, but this time, as a function of $\dot{M}$ estimated via Equation (6), where the unit of $\dot{M}$ is arbitrary.

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In the case of a standard accretion disk, L_disk and kT_in are expected to be proportional to $\dot{M}$ and $\dot{M}^{0.25}$, respectively (e.g., Makishima et al. 1986; Ebisawa et al. 1993; Tanaka & Shibazaki 1996), as long as r_in is kept constant. Indeed, in the range of $\dot{M} \lesssim 300$ (corresponding to L_tot ≲ 1.5 × 10³⁷ erg s⁻¹), Figure 7 reveals tight scalings as

$$\begin{eqnarray} &&L_{\rm disk} \propto \dot{M}^{0.92 \pm 0.07},\quad kT_{\rm in} \propto \dot{M}^{0.21 \pm 0.05},\quad r_{\rm in} \propto \dot{M}^{0.05 \pm 0.07},\nonumber\\ \end{eqnarray} \tag{ 7 }$$

which agree, within errors, with the predictions for standard disks. This result is consistent with that reported by Lin et al. (2007).

As the accretion rate $\dot{M}$ increases to ≳ 300 in Figure 7, the disk parameters start deviating from the scalings of Equation (7), and follow new relations approximated as

$$\begin{equation} L_{\rm disk} \propto \dot{M}^{0.80 \pm 0.03},\quad kT_{\rm in} \propto \dot{M}^{0.11 \pm 0.02},\quad r_{\rm in} \propto \dot{M}^{0.18 \pm 0.03}. \end{equation} \tag{ 8 }$$

Thus, the inner disk radius r_in apparently starts "retreating," with increasing fluctuations, but the increase in r_in is compensated by the flattering in kT_in. As a result, L_disk starts to weakly saturate, instead of increasing in proportion to $\dot{M}$.

Turning to the BB parameters, we find kT_BB rather stable, with a weak positive dependence on $\dot{M}$. On the contrary, r_BB exhibits a large scatter, which causes similar scatter in L_BB. The $\dot{M}$ dependence of these BB parameters can be approximated as

$$\begin{equation} L_{\rm BB} \propto \dot{M}^{0.5},\quad kT_{\rm BB} \propto \dot{M}^{0.07 \pm 0.01},\quad r_{\rm BB} \propto \dot{M}^{0.13}. \end{equation} \tag{ 9 }$$

This BB behavior is the same as previously reported (Mitsuda et al. 1984; Gilfanov et al. 2003; Revnivtsev & Gilfanov 2006). We also find that kT_BB is always higher than kT_in, and r_BB is smaller than r_in. These relations are consistent with the basic assumptions of the Eastern model; the BB emission comes from the NS surface, while the MCD component from the surrounding accretion disk.

We have so far described the energy spectra and their variations in terms of the four quantities, kT_in, r_in, kT_BB, and r_BB. However, it is not yet clear how many of them can be regarded as independent variables (with the rest depending on them). In other words, we need to know how many degrees of freedom are involved in the observed spectral variations in the upper-banana state. This can be done by applying "fractal dimension analysis" (e.g., Matsumoto et al. 2005) to the four model parameters.

For this purpose, let us define a four-dimensional vector space spanned by the four variables, Y₁ = kT_in, Y₂ = r_in, Y₃ = kT_BB, and Y₄ = r_BB, wherein each data set is represented by a vector

$$\vec{V(i)}=\left\lbrace Y_1(i), Y_2(i), Y_3(i), Y_4(i) \right\rbrace,$$

with i = 1, 2, ... 95 denoting the data number. Let us also define the average vector $\langle \vec{V} \rangle = \Sigma _{i=1}^{95} \vec{V(i)}/95 \equiv \left\lbrace \langle Y_1 \rangle, \langle Y_2 \rangle, \langle Y_3 \rangle, \langle Y_4 \rangle \right\rbrace$, and the zero-mean vectors, $\vec{v(i)} = \vec{V(i)} - \langle \vec{V} \rangle \equiv \left\lbrace y_1(i), y_2(i), y_3(i), y_4(i) \right\rbrace$ with y_k(i) ≡ Y_k(i) − 〈Y_k〉. Then, a "distance" D(i) of the vector $\vec{v(i)}$, measured from the origin, is calculated as

$$\begin{equation} D(i) = \left[ \sum _{k=1}^{4}\left\lbrace y_k(i)/\sigma _k \right\rbrace ^2 \right]^{1/2} (i = 1, 2, \ldots, 95), \end{equation} \tag{ 10 }$$

where σ_k = [∑⁹⁵_{i = 1}y_k(i)²/94]^1/2 is the standard deviation in y_k(i) around the origin (or in Y_k(i) around its mean 〈Y_k〉). Finally, we calculate the number N(< D) of those data points the distance D(i) of which is less than a given value D.

Figure 8 shows the normalized data point number N(< D)/95 from Equation (10) as a function of D, over the range of N/95 = 0.2–0.8 (or N = 23–76). If the variations are controlled by n(1 ⩽ n ⩽ 4) independent parameters, we expect the vectors $\vec{v(1)}, \vec{v(1)}, \ldots \vec{v(95)}$ to form an n-dimensional subspace in the vector space, so that N(< D) should increase as ∝Dⁿ. Indeed, Figure 8 reveals a tight power-law relation as

$$\begin{equation} N \propto D^{1.8 \pm 0.1}. \end{equation} \tag{ 11 }$$

Since this result is consistent with n = 2, we infer that the spectral behavior in the upper-banana state of 4U 1608–522 has effectively two degrees of freedom.

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Of the two independent variables describing the spectral variability (Section 3.4), one is obviously the total luminosity L_tot, or nearly equivalently, the mass accretion rate $\dot{M}$. Actually in Figure 6 (or Figure 7), the four quantities are all observed to depend primarily on L_tot (or $\dot{M}$). However, they show significant scatters around the L_tot-dependent correlation, so that none of them can be regarded as a single-valued function of L_tot (or $\dot{M}$). This is consistent with the presence of the second degree of freedom revealed in Section 3.4. In order to identify what is causing this extra freedom, we remove the L_tot dependence from the behavior of the four model parameters, and study the residual variations.

To eliminate the major L_tot dependence from the physical parameters, we calculated their de-trended counterparts Z'(i) (Z = L_disk, L_BB, kT_in, r_in, kT_BB, and r_BB), employing Equations (4) and (5), as

$$\begin{eqnarray} &&L^{\prime }_{\rm disk}(i) \propto L_{\rm disk}(i)/L_{\rm tot}(i)^{1.1}, L^{\prime }_{\rm BB}(i) \propto L_{\rm BB}(i)/L_{\rm tot}(i)^{0.7},\nonumber\\ \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} &&kT^{\prime }_{\rm in}(i) \propto kT_{\rm in}(i)/L_{\rm tot}(i)^{0.19}, r^{\prime }_{\rm in}(i) \propto r_{\rm in}(i)/L_{\rm tot}(i)^{0.12},\nonumber\\ \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &&kT^{\prime }_{\rm BB}(i) \propto kT_{\rm BB}(i)/L_{\rm tot}(i)^{0.10}, {\rm {and}}\; r^{\prime }_{\rm BB}(i) \propto r_{\rm BB}(i)/L_{\rm tot}(i)^{0.11}.\nonumber\\ \end{eqnarray} \tag{ 14 }$$

(As the de-trending parameter, we chose L_tot rather than $\dot{M}$ because the latter is less directly estimated from the data.) Absolute values of the de-trended parameters are not meaningful, and we hereafter consider their relative values only. Since r'_in shows the largest variation among them, we plot in Figure 9 the de-trended parameters as a function of r'_in. Table 4 also summarizes those of Spec A to D in Section 3.1. From the behavior of the de-trended luminosities, the data points can be divided into two branches; one has almost constant luminosities as r'_in varies, while the other is characterized by significant r'_in-dependent variations both in L'_disk and L'_BB. Hereafter, the two branches are denoted as "constant-luminosity branch (CLB)" and "variable-luminosity branch (VLB)," respectively. The two branches may connect at r'_in ∼ 0.85, rather than behaving independently from each other.

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In the CLB, the de-trended BB luminosity stays nearly constant at L'_BB ∼ 0.8, so do the two BB parameters, kT'_BB ∼ 1.6 and r'_BB ∼ 0.9. Similarly, the de-trended disk luminosity remains at L'_disk ∼ 0.9. As a result, the CLB data points are distributed in Figure 6(a) along the major correlation trends. Nevertheless, r'_in varies by ∼ ± 20%, accompanied by a clear decrease in the de-trended disk temperature as kT'_in∝r_in^−0.5. (This scaling is a natural consequence of the constant L'_disk and the relation of L'_disk∝r'_in²T'_in⁴.) There are two possibilities to explain this r'_in behavior. One is real changes in r_in and the other is variations in the color hardening factor κ of the disk. Since L'_disk is kept constant in the CLB, the latter case may be more likely. Specifically, the observed behavior will be explained if some unspecified mechanisms (possibly related to vertical structure changes in the disk) caused κ to increase by ∼10% while somehow keeping the effective temperature of the disk and its true radius constant; the observed color temperature kT'_in would then increase by ∼10% according to Equation (1), and the apparent disk radius r'_in would decrease by ∼20% after Equation (2), thus reproducing the observation.

In the VLB, r'_in varies over a relatively small range (∼ ± 10%), whereas the two luminosities both vary significantly in an anti-correlated way; L'_disk decreases from 1.9 to 1.5 as ∝r'_in⁻¹, while L'_BB increases from 0.8 to 1.2 as ∝r'_in². This complementary behavior between the MCD and BB components appears in Figure 6 as the large fluctuations away from the major trend, where the data points reach a level of L_disk ∼ L_BB ∼ 0.5L_tot. (This behavior cannot be a fitting artifact caused by strong couplings between the two spectral components, since the nominal fitting errors are not particularly larger in the VLB and are generally smaller than/comparable to the size of the plotting symbols in Figure 6 and Figure 7.) Since kT'_BB is almost constant (or only slightly decreasing) like in the CLB, the increase of L'_BB is mainly caused by that of r'_BB from 0.9 to 1.2 as ∝r'_in¹. We again observe kT'_in to decrease like in the CLB, but with a different r'_in dependence which is approximated as ∝r'_in^−0.75. This is exactly the behavior of a standard accretion disk (Shakura & Sunyaev 1973) when its radius varies under a constant $\dot{M}$ because we then expect $\dot{M} \propto r_{\rm in}^3 T_{\rm in}^4$ (Equation (6)). This also explains the observed disk luminosity behavior, as L'_disk∝r²_inT_in⁴∝r'_in⁻¹. Obviously, the increase in the disk radius under a constant $\dot{M}$ predicts that a larger fraction of the overall gravitational energy release should be emitted by the BB component; in fact, along the full VLB in Figure 7(b), L'_disk is seen to decrease by ∼0.3 in the employed arbitrary unit, while L'_BB to increase by ∼0.3, thus conserving the total luminosity. In other words, the VLB is characterized by actual changes in the innermost disk radius.

Based on these correlation analyses among the de-trended spectral parameters, we conclude that the second independent variable (besides $\dot{M}$; Section 3.4) causing the variability in the upper-banana state can be ascribed to sporadic changes in r'_in (or in r_in). However, this r'_in variability is likely to be further subdivided into two different mechanisms under a constant $\dot{M}$: one is an apparent effect due to variations in κ of the disk and the other is real changes.

As shown in Figure 9 and Table 4, the L'_disk parameters of Spec B, C, and D locate on the CLB line, while that of Spec A locates on the VLB line. Therefore, we consider that the hard/soft-band differences of Spec A/B and C/D observed in Figure 2 are mainly attributed to the VLB and CLB variation, respectively. On the other hand, the de-trended parameters of Spec B and C are similar, and the variability between them observed over the entire 3–30 keV band is thought to be dominated by the $\dot{M}$ change.

Through the analysis of the RXTE spectra of the atoll source 4U 1608–522 in its upper-banana state, we have confirmed that both time-averaged and difference energy spectra can be reproduced successfully by the Eastern (MCD+BB+Gau) model. Considering the hardening factor, the effective temperature and radius of the MCD component were obtained as kT^eff_in ∼ 1.1 keV and r^eff_in ∼ 12 km, respectively. The radius is larger than the representative NS radius, 10 km, and is consistent with the last stable orbit, 3R_s = 12.4 km, allowed by general relativity. This result agrees with the picture of a standard accretion disk which is formed around an NS. The BB parameters were found as kT^eff_BB ∼ 1.8 keV and r^eff_BB ∼ 2.7 km, assuming isotropic emission from a spherical source. The temperature is close to the local Eddington temperature (2.0 keV) at the NS surface, and the radius is smaller than 10 km. Then, the BB component can be regarded as being emitted from an equatorial zone of the NS, as previously suggested by Mitsuda et al. (1984).

According to the picture of standard accretion disks, half of the released gravitational energy is radiated from the accretion disk (L_disk), and the other half is stored in the Keplerian kinetic energy and emitted (L_BB) when the matter settles onto the NS surface. Then, we expect L_BB to be proportional to L_disk, and hence the L_BB/L_disk ratio to be constant. Indeed, L_disk was found to increase almost linearly with the total luminosity L_tot (Figure 6(a)). However, the same figure reveals that L_BB increases less steeply, making the L_BB/L_disk ratio decrease from 0.6 to 0.4 as L_tot increases from 1 × 10³⁷ to 4 × 10³⁷ erg s⁻¹.

We may think of two possibilities to explain the observed relative decrease of L_BB. One is that the accreting matter reaches the NS surface and emits L_BB which is equivalent to the Keplerian energy (i.e., ∝L_disk), but we cannot observe all the emission because its wavelength shifts outside the PCA energy band (3–30 keV), or the geometrical angle of the emission moves from our line of sight. However, we have not observed any hint of extra emission in the softest or hardest energy ends of the PCA spectra. In addition, other sources, which are thought to have different inclination angles, are also reported to show the same behavior of the decreasing L_BB/L_disk ratio (Revnivtsev & Gilfanov 2006). Therefore, this possibility is unlikely.

The other possibility is that the accreting matter flows all the way through the disk down to its inner radius that is close to the NS surface, but a progressively larger fraction fails to accrete onto the NS surface. If the accretion-failed matter stayed around the NS away from the surface (e.g., an expansion of the boundary layer), it would accumulate to become optically thick, eventually producing detectable emission. This would lead to a decrease in kT_BB because the emission radius should increase. As shown in Figure 6 and Equation (5), however, this is not the case; the kT_BB value stays almost constant (or rather increases) as the total luminosity increases. Then, a fraction of the matter must be outflowing and escaping from the system without releasing its kinetic energy as emission. Although the observed L_tot ∼ 4 × 10³⁷ erg s⁻¹ is only ∼20% of the Eddington luminosity for a 1.4 M_☉ NS, the observed value of kT^eff_BB ∼ 1.8 keV is already close to the spherical Eddington temperature at 10 km (2.0 keV). Considering further the non-spherical geometry of the disk, and the disk radius which is possibly larger than 10 km, the putative outflow is likely to be driven by increased radiation pressure in the innermost disk region. The presence of such outflows is indeed suggested by detections of broad absorption features from LMXBs (Schulz & Brandt 2002; Ueda et al. 2004), black hole binaries (Kotani et al. 2000; Yamaoka et al. 2001; Kubota et al. 2007), and active galactic nuclei (Pounds et al. 2003a, 2003b).

Assuming that the decrease in L_BB/L_disk from 0.6 to 0.4 is due to outflows, about (0.6 − 0.4)/0.6 ∼ 30% of the accreting matter is estimated to escape from the system at a typical luminosity of ∼4 × 10³⁷ erg s⁻¹. Toward lower luminosities, the ratio appears to converge to ∼0.6, rather than 1.0 which would be theoretically expected. This may be attributed to effects due, e.g., to the system inclination, general relativity, and NS rotation (e.g., Sunyaev & Shakura 1986). Detailed theoretical discussion on this point is beyond the scope of the present paper.

As indicated by Equation (8), the innermost disk radius r_in is observed to increase gradually when the mass accretion rate increases to ≳ 300 in Figure 7, or equivalently, when the total luminosity becomes ≳ 1.5 × 10³⁷ erg s⁻¹ (corresponding to ≳ 7% of the Eddington luminosity). This is unlikely to be an apparent effect caused, e.g., by changes in the hardening factor, since this quantity would increase as the luminosity increases (Gierliński & Done 2004; Davis et al. 2005), and would hence make r_in smaller. Therefore, the increase in r_in toward higher accretion rates is considered to represent a real retreat of the innermost disk radius. This interpretation may agree with the results by Popham & Sunyaev (2001), who numerically calculated the physical condition of the boundary layer and showed that the disk inner edge retreats back by the increased radiation pressure when the luminosity approaches the Eddington limit.

The result of the fractal dimension analysis shows that the source variations are controlled by two independent variables. One of them is obviously the mass accretion rate (or nearly equivalently the total luminosity), which produces the major trend in Figure 7 as discussed so far. The other variable is thought to be random changes in the inner disk radius, as represented by the behavior of r'_in in the de-trended analysis performed in Section 3.5. As discussed therein, the variations of r'_in may be caused by two different mechanisms, corresponding to the CLB and VLB. These two branches are considered to degenerate in the fractal dimension analysis and are difficult to identify as two independent degrees of freedom.

In the CLB, the change in r'_in seems to be apparent rather than real, and is considered to reflect fluctuations in the hardening factor of the disk. As a result, the BB parameters remain unchanged, while the MCD parameters vary as kT'_in∝r'_in^−0.5, keeping the disk luminosity constant (L'_disk∝r'_in⁰). Such variations in the hardening factor may in turn stem, e.g., from fluctuations of the disk height because this would affect the temperature gradient and/or radiative transfer in the vertical direction of the disk.

In the VLB, the change in r'_in is considered real (Section 3.5) because we observe a scaling of kT'_in∝r'_in^−0.75 which agrees with the behavior of a standard accretion disk under a constant $\dot{M}$ and a constant hardening factor. In this branch, the disk is considered to randomly retreat back from its "home position," which is located in Figure 9 around r'_in ∼ 0.85 where the CLB and VLB meet. When this takes place, the accreting matter located between the retreated and the home-position radii will fall onto the NS surface without radiating the disk emission, and the gravitational energy it acquires will be released solely as the radiation from the NS surface. Then, L_disk will decrease while L_BB will increase. This can naturally explain the anti-correlation between the MCD and BB luminosities observed in the VLB (Figure 9).

The results of our study can be summarized in the following four points.

• 1.  
  The physical picture provided by the Eastern model can consistently explain the behavior of 4U 1608–522 in the upper-banana state.
• 2.  
  When the mass accretion rate increases, some fraction of the matter flowing through the accretion disk starts outflowing, presumably due to an increased radiation pressure.
• 3.  
  Even at the same accretion rate, the spectral parameter changes randomly on timescales of hours to days, mainly reflecting changes in the innermost disk radius.
• 4.  
  The change in the disk inner radius is a mixture of real and apparent effects, the former due to sporadic retreat of the disk while the latter due to fluctuations in the color hardening factor.

Facility: RXTE (PCA) - Rossi X-ray Timing Explorer