X-ray spectral variations of Circinus X-1 observed with NICER throughout an entire orbital cycle

We start with the stable phase. The continuum spectral shape shows only a gradual decrease in flux. This is shown in snapshot 91’s data in Figure 3 (f). The continuum emission can be modeled with a simple disk black body component (Mitsuda et al., 1984; Makishima et al., 1986) with two fitting parameters (the normalization and the innermost disk temperature $T_{\mathrm{in}}$), partially covered by a neutral cloud with two parameters (the covering fraction $f^{\mathrm{(cov)}}$ and the H-equivalent column density $N_{\mathrm{H}}^{\mathrm{(cov)}}$) and interstellar absorption with a fixed column density of 2.0$\times 10^{22}$ cm^-2. Schulz et al. (2008) showed that there can be several types of covering matter with different degrees of ionization along the line-of-sight. Our data did not allow us to distinguish these different types, thus we only assumed neutral matter. We further added the Gaussian model to account for the Fe K fluorescence line at 6.4 keV.

We next investigate the dip phase. The spectral shape changed in the middle of the dip phase. During the period of the first ten data points (hereafter “the early dip” period), the spectral shape exhibits one peak, which we represent with snapshot 34 (Fig. 3a). In contrast, during the period of the remaining six data points (hereafter ”the late dip” period), the shape exhibits two peaks, which we represent with snapshot 46 (Fig. 3b). Despite the difference, both can be described with different parameters of the same model. The two peaks are, in fact, reproduced by uncovered and covered components of the partially covered disk black body emission. The spectra in this phase are also characterized by numerous emission lines, most notably in highly ionized Fe, but also in lighter metals such as Mg, Si, and S. We fit them with Gaussian lines likely originating from a photo-ionized plasma. The plasma emission should accompany continuum emission in the soft band. In fact, we did observe excess emission in the softest end of the spectrum, which we phenomenologically represented with a power-law model.

There are significant flux variations in the flaring phase (Fig. 2). We chose three snapshots representing high, medium, and low flux (snapshot 58, 50, and 65), respectively). The flux varies by a factor of $>$5, but the spectral shape does vary as much. We thus applied the same model (partially covered disk black body emission) and we were able to model the spectra through variations in the covering fraction and column density of the partial covering component. Several Gaussian lines were added, both positive and negative in flux, to account for the emission and absorption features, respectively.

To summarize the result in § 3.2.1–§ 3.2.3, despite the variety of the spectral shapes, the spectra of the six selected snapshots were described well with the same approach; the disk black body emission partially covered by neutral cloud and fully covered by the ISM photo-electric absorption. Some minor differences remain, such as the soft excess emission required for the late dip phase (Fig. 3b) and the Gaussian lines observed either positive (emission) or negative (absorption) in different phases.

We thus constructed a common model to encompass these minor differences. The soft excess was required only in the dip phase, but we included this component in all phases with the parameters fixed to the parameter values derived from the dip phase. This does not influence the fitting in other phases, as the soft excess is overwhelmed by the uncovered disk black body emission. We also added Gaussian lines at the energies of Ly$\alpha$ and He$\alpha$ transitions of H-like and He-like ions of Mg, Si, S, and Fe plus Fe I K$\alpha$ fluorescence. Here, the He$\alpha$ line refers to all unresolved lines of (1s)(2p) or (1s)(2s) $\rightarrow$ (1s)² transitions, while the Ly$\alpha$ line refers to (2p) $\rightarrow$ (1s) transitions. We allowed both positive and negative values in the fit to account for both emission and absorption. No significant Doppler shift or broadening was observed in any of these lines, thus we fixed the line energy shift and width to 0. The best-fit models shown in Figure 3 are the results of applying this common model.

In the previous work of Cir X-1, the X-ray continuum spectra were often explained by a combination of multiple spectral components of a black body, disk black body, and Comptonized emission (e.g., Shirey et al. 1999; Iaria et al. 2005; D’Aì et al. 2012). We argue that a single disk black body component is sufficient to explain the variety of spectra for the present data set. The partial coverage of the disk black body emission alone can make two continuum peaks for the covered and uncovered parts. The addition of a black body component was not required in all spectra throughout the orbit. We note that the actual emission would be a mixture of emissions from the neutron star surface, the boundary layer, and the accretion disk, but a single disk black body component is sufficient to explain the data.

The phase-dependent change of the spectral model parameters for the continuum components is given in green symbols in Figure 4. The free parameters are the normalization and $T_{\mathrm{in}}$ for the disk black body emission and $f^{\mathrm{(cov)}}$ and $N_{\mathrm{H}}^{\mathrm{(cov)}}$ for the partial covering absorption. To mitigate a degeneracy in the parameters space, firstly we fitted the data only above 4 keV, obtained the best-fit $T_{\mathrm{in}}$ values. We then fitted using all the available energy range with the fixed $T_{\mathrm{in}}$ values.

In the stable phase, the parameters change gradually. Some spectra do not require the partial covering material, thus we fixed both $N_{\mathrm{H}}^{\mathrm{(cov)}}$ and $f^{\mathrm{(cov)}}$ to null. In the flaring phase, we found that the fitting results widely vary, in particular, for normalization. As this parameter represents the entire disk luminosity, it is inconceivable that it changes as rapidly as the revisit time of $\sim$4 hours. We thus constrained its variation with a low-pass-filtered trend as shown with red symbols in Figure 4. The rapid variation is now attributed mostly to the changes in the partial coverage.

The trend of the normalization was further extrapolated backward to the late dip phase, which yielded acceptable fits. However, this did not work for the early dip phase. This is because of the different appearance of the spectral shape as exemplified in the spectra from snapshot 34 and 46 (Fig. 3). In the late dip, we observe both the partially covered and uncovered disk black body emission. In the early dip, we only observe the uncovered component, as the covered component is attenuated too much to be visible in the NICER energy band. The degeneracy between the following two cases cannot be broken in the fitting: (a) a large value of the partial covering column or (b) a small value of the disk normalization with a small covering column. The solution (b) yields fitted results with a large scatter in the disk black body parameters ($T_{\mathrm{in}}$ and normalization), which is unlikely. We thus opted for (a) by constraining the partial covering column to be $>10^{24}$ cm^-2 for the best-fit parameter. For this reason We include large error estimates in the early dip phase in Figure 4. The NICER data do not allow us to track the trend of the disk black body parameters during the early dip phase.

The phase-dependent change of the line normalization parameter is given in Figure 5 separately for Mg, Si, S, and Fe. Different colors are used for the H-like and He-like ions.

For Mg, Si, and S, emission lines were detected in almost all the snapshots mainly during the dip phase. During the dip phase, their normalization decreases, particularly, in the late dip phase.

For Fe, a more stringent constraint was obtained, revealing changes between the emission and absorption features. In the dip phase, the line appears as emission at the beginning for the early dip, but the line normalization decreases in the late dip similar to the other elements. In the end, the line feature changed from emission to absorption, which continues for the flaring phase and reverts back to emission in the stable phase.

We use the emission or absorption line features to derive ionization parameters. We calculate the intensity ratio between the Ly$\alpha$ and He$\alpha$ lines when both of them are detected. The emission line pair was detected for Mg in the dip phase and for Fe throughout orbit, while the absorption line pair was detected for Fe from the end of the dip phase to the flaring phase.

We employ the xstar code (Kallman & Bautista, 2001) to numerically estimate the expected line ratio of the pairs. The code solves the radiative transfer equation for the non-local thermal equilibrium condition for the charge and level populations in the photo-ionized plasma by balancing radiative cooling and heating, ionization and recombination, and excitation and de-excitation. It uses the two-stream approximation solver in the inward and outward directions. The energy input (incident emission) is given and the outputs (transmitted emission and diffuse emission inward and outward) are calculated.

Because the xstar code is only for one-dimensional geometry, we approximated the photo-ionized plasma as a plane-parallel slab. The slab is static, uniform, and optically thin for the electron scattering with hydrogen density of $n_{\mathrm{H}}=3\times 10^{14}$ cm^-3 and column density of $N_{\mathrm{H}}=1\times 10^{18}$ cm^-2. The incident emission has the shape of the disk black body with representative parameters over the stable phase.

For the emission lines, they are mainly produced as a result of the recombination cascades of e.g., Fe²⁶⁺ and Fe²⁵⁺ for Fe. The Ly$\alpha$ to He$\alpha$ ratio is thus very sensitive to the ionization parameter $\xi=L/(n_{\mathrm{e}}r^{2})$, in which $n_{\mathrm{e}}$ is the electron density and $r$ is the distance from the incident source to the slab. We took into account the effect of radiative excitation by the incident emission, which is known to alter the line strength when a large fraction of the incident emission is outside of the beam (Kinkhabwala et al., 2002) as in the dip phase. The expected line ratio was calculated as a function of $\log{\xi}$ (erg s^-1 cm) over 0–5 in increments of 0.1 and a $\xi$ value was selected that matches the observed ratio.

For the absorption lines, we followed the same procedure but evaluated the column density of Fe²⁵⁺ and Fe²⁴⁺ through the slab that is responsible for the absorption lines respectively of Ly$\alpha$ and He$\alpha$ by absorbing the incident emission at each transition energy. These column densities are related to the observed equivalent width through the curve of growth. We selected the best $\log{\xi}$ value to reproduce the observed ratio between the two lines.

The result is shown in Figure 7. The $\xi$ values using the Mg, Si, S emission, Fe emission and Fe absorption pair are, respectively, $\sim 1.9\pm 0.02$, $\sim 2.1\pm 0.04$, $\sim 2.5\pm 0.02$, $\sim 3.4\pm 0.04$ and $\sim 3.7\pm 0.08$ for the means and standard deviations. They differ to some extent, which indicates that the actual photo-ionized plasma has a stratified ionization structure of varying $\xi$ as suggested in Schulz et al. (2008). However, they are stable across different phases. This suggests that the photo-ionized plasma remains mostly unchanged throughout the orbital cycle. The $\xi$ values using the Fe emission and absorption features match well, indicating that the differences are created by changes in the viewing geometry and/or the incident X-ray flux, not by changes in the photoionized plasma itself.

In the dip phase, it should be noted that the Fe line pairs changed from emission to absorption in the middle at $\phi=0.95$ before this phase ends (Fig. 5). Since the emission from the photo-ionized plasma is considered stable over orbit, this change should be created by an increase in the fraction of the incident emission through the photo-ionized plasma, which contributes to the absorption lines. This is presumably made by the increase of both the normalization of the disk black body emission and the decrease of the partial covering fraction in the line-of-sight as shown in Figure 4(d) and depicted in Figure 6.

The size of the system can be estimated by assuming the electron density $n_{\mathrm{e}}$. The NICER data are incapable of constraining this value, so we used the estimate of the upper limit ($10^{15}$ cm^-3) given by the Chandra grating spectroscopy of the triplet lines of He-like ions (Schulz et al., 2020).

We estimate the size of the photo-ionized plasma. The inner radius $r_{\mathrm{in}}=\sqrt{L_{\mathrm{X}}/(\xi n_{\mathrm{e}})}\sim 10^{10}$ cm for the best-fit $\log\xi$ parameter for Fe emission and the observed disk luminosity of $L_{\mathrm{X}}$. The $r_{\mathrm{in}}$ value is a lower limit as we used an upper limit for $n_{\mathrm{e}}$.

The outer radius ($r_{\mathrm{out}}$) is constrained by the observed the maximum total amount of Fe absorption column by

in which $N_{\mathrm{Fe}i+}$ is the observed absorption column of the Feⁱ⁺ ions ($i\in\left\{24,25\right\}$), $f_{\mathrm{Fe}i+}$ is the charge fraction of the ion among all Fe, $A_{\mathrm{Fe}}$ is the assumed solar Fe abundance relative to H, and $n_{\mathrm{H}}$ is the H density, which we approximate as $n_{\mathrm{H}}=n_{\mathrm{e}}$. The Fe column is derived from the observed equivalent width through the curve of growth. The charge fraction and the curve of growth are derived from an XSTAR simulation, in which the column density of the slab $\log{N_{\rm{H}}}$ cm^-2 was changed 15–25 in increments of 1 for the ionization degree of $\log{\xi}=3.5$ erg s^-1 cm. We used snapshot 50 and 65 (Fig. 3 c and e, respectively) where the Fe absorption features were strong. For both H-like ($i=25$) and He-like ($i=24$) ions, the result is in the range of $r_{\mathrm{out}}-r_{\mathrm{in}}=10^{9-10}$ cm. The photo-ionized plasma has some structures for different $\xi$ values for different elements (§ 3.3.2). The part contributing to the Fe features is relatively thin with $r_{\mathrm{out}}/r_{\mathrm{in}}=1.1-2$.

This still yields strong emission lines from the plasma. XSTAR calculates that Fe XXV He$\alpha$ will be 6–120 $\times 10^{36}$ erg s^-1 for $N_{\rm{H}}=10^{24-25}$ cm^-2, which is much larger than the observed value of $1.7\times 10^{35}$ erg s^-1. This is the case in the dip phase when there is no significant absorption features imprinted in the disk black body emission that cancels the emission. Therefore, we consider that the volume filling factor of the photo-ionized plasma is small in comparison to the xstar assumption of a spherical medium.

We further compare the inner and outer radius of the photo-ionized plasma ($r_{\mathrm{in}}$ and $r_{\mathrm{out}}$) with those of the accretion disk ($R_{\mathrm{in}}$ and $R_{\mathrm{out}}$). In our spectral analysis, the normalization of the disk black body emission is related to $R_{\mathrm{in}}$ as $(R_{\mathrm{in}}/0.94)^{2}\cos{\theta}$, in which $\theta$ is the viewing angle. For a complete edge-on $\theta=0$ and the normalization of 100 (the largest in the trend in Fig. 4b when a large fraction of the disk is considered to be exposed), we obtain $R_{\mathrm{in}}\sim$10⁶ cm. The $R_{\mathrm{in}}$ value is close to a typical value of a neutron star radius since the accretion disk is thought to stretch inward almost close to the neutron star surface (e.g., Frank et al. 2002).

We cannot constrain $R_{\mathrm{out}}$ from the present data. At least, it must be smaller than the effective Roche lobe radius of the binary of $\sim 10^{10}$ cm (Begelman et al., 1983). If we assume that a typical value for low mass X-ray binaries of $R_{\mathrm{out}}\sim 10^{9-10}$ cm (Jimenez‐Garate et al., 2002) is applicable to Cir X-1, $r_{\mathrm{in}}\gtrsim R_{\mathrm{out}}$, suggesting that the photo-ionized plasma is located outside of the accretion disk.