Spectral Evolution of Hercules X-1 over Main High State from RXTE/PCA Observations

We test different X-ray spectrum models to find the one that best represents the observed Rossi X-ray Timing Explore/Proportional Counter Array (RXTE /PCA) spectra of Her X-1 during Main High state (MH). We then apply this model to MH observations taken over the lifetime of RXTE. From the results, we obtain patterns in the spectral parameters vs. 35-day phase during MH. The precessing-disc occultation model explains the 35-day cycle by changes in observer view of the emission regions by the accretion disc 35-day precession. Qualitatively, we find that this model can describe the main spectral changes. However, several spectral parameters show detailed changes that the models have not addressed yet. These changes will likely require modifica-tions to the basic precessing-disc model for the 35-day cycle.


Introduction
Hercules X-1 (Her X-1) is a well-studied X-ray binary with 1.7-day orbital period and 35-day long-term period [1] [2]. The 35-day period is caused by the precessing accretion disc around the neutron star, which causes periodic obscuration of the X-ray emitting accretion column on the neutron star [3]. The 35-day cycle normally shows a few day variations in its period [2] [4], but sometimes disappears for a few to several months: these periods are called anomalous low states (ALS) [5].
There are different physical origins for the X-ray emission from the system Her X-1/HZ Her. The X-ray emission regions consist of the following components: 1) Direct X-ray beams (pencil and/or fan beam) originating in the accre-off of the illuminated far side of a thick inner disc edge. The inner ring is tilted and precessing over the 35-day cycle, so the central source is most clearly viewed twice per 35-day cycle: when the ring axis is pointing most toward us at the peak of the MH-state; and when the ring axis is pointing most away from us, at the peak of SH-state. Also as can be seen from Figure 9 of [8], the far inner edge of the disc will be gradually occulted by the near inner edge of the disc. Using these two emission regions (i.e. central source and reflected emission by the inner ring) and the occultation by the disc, [8] reproduced the contribution of both emission regions over the 35-day cycle.
In this work, we make an observational study of the spectral evolution of Hercules X-1 over the Main High state using archival data from the Rossi X-ray Timing Explorer/Proportional Counter Array (RXTE/PCA). In Section 2, we describe the observations and how we choose and test the spectrum model. In Section 3, we give the results of applying the spectrum model to the data. The main results are the spectrum parameter changes as a function of 35-day phase during MH. In Section 4, we discuss these results. The summary is given in Section 5.

Observations and Light-Curves
The NASA satellite Rossi X-ray Timing Explorer (RXTE) [11] was designed to facilitate the study of time variability of X-ray sources with moderate spectral resolution. We make use of the data collected by the Proportional Counter Array (PCA) on-board the RXTE. The PCA [12] [13] consists of five identical multi-anode large-area Proportional Counter Units (PCUs).
Her X-1/HZ Her was observed numerous times by RXTE between 1996 and 2005 and the data has been archived at High Energy Astrophysics Science Archive Research Center (HEASARC). The RXTE/PCA database on HZ Her/Her X-1 contains 1.6 Ms of observations. The data were acquired by the RXTE/PCA instrument in the period from February, 1996 to August, 2005 (MJD 50,114 to 53,584) as a result of 23 study proposals for observing the HZ Her/Her X-1 system.
We carried out spectral analysis of PCA standard-2 data of Her X-1 for studying the spectral evolution over the 35-day super-orbital cycle. Standard-2 mode has temporal resolution of 16 s and energy ranges 2 -60 keV in 129 channels.
The times were converted to Dynamical Barycentric Modified Julian Day Time (MJD TDB) using the HEASARC time conversion procedures.
Binary orbital phase was determined using the ephemeris of [14]. To find the 35-day phase we used the nearest last MJD turn-on time of the 35-day cycles and cycles lengths of [15]. To study the spectral evolution over normal 35-day cycles, we removed data that were taken when the system (Her X-1) was in anomalous low state periods. In our data there are two anomalous low state (MJD 51226.4 to 51756.9 and MJD 52950.6 to 53159.4).
Our purpose is studying spectral evolution over the 35-day cycle other than D. A. Leahy, M. H. Abdallah dips and eclipse, thus we removed all data that falls within the dips and eclipsing periods. Significant changes in the spectra occur during dips [16] (and references therein) and during eclipse periods [17]. These periods include times during which the accretion stream impacts the disc and cover our line of sight to the central source appearing as absorption dips in the light curve [16] resulting in variations of the spectra. Dips times were taken from [16]. Eclipsing periods were filtered out by removing data with orbital phases less than 0.07 or greater than 0.93. After removal, to make sure that no residual dips remain in our data, we calculated the Softness Ratio (SR) defined as band 1 countrate divided by band 3 countrate, where band 1 and band 3 are the count rates in the energy ranges 2 -4 keV and 9 -20 keV respectively. We then rejected data points during MH-state and SH-state with SR < 0.35. For low states, SR is normally below 0.35, so this rejection was not done for low states. This resulted in a total ALS-eclipsedips-free exposure of 432 ks. The observations cover parts of twenty different main high states.
The band b and band d light-curves were scaled by the number of active PCUs at any given time to yield count rate per PCU. These and SR are plotted vs. 35-day phase for the Her X-1 data in Figure 1.
We observe that the softness ratio is very different during the main high and short high states compared to the low states. As noted previously by [18], the low state data appear similar to the data in main high and short high states when Her X-1 is in a dip. The band 1 and band 3 light-curves, and the band-1/band-3 softness ratio are shown for the individual MH states in

Spectrum Data Processing
In the current work, we make use of the standard-2 PCA data which has a time resolution of 16-s and 129 energy channels that cover the entire energy range (2 -60 keV) of the PCA detector. We added the data from all three layers and columns of the available PCUs and response matrices were generated accordingly. For Her X-1 we used the faint background model throughout. To obtain better statistics we average over time, to obtain the same signal-to-noise ratio for all of the spectra. The data were averaged over time intervals between 400-s and 1360-s, with a mean value of 889-s.
We use the graphical user interface (GUI) tool "XDF" (XTE Data Finder) to create a catalogue of files for further analysis. The standard-2 catalogue and the filter file that contains housekeeping parameters and the most recent background model are used to model the background for the selected data using the FTOOLS task "runpcabackest". We used the FTOOLS task "maketime" to create a good time interval (".gti") file selected from the filter file. The criteria chosen for data screening are: 1) The elevation above the Earth must be greater than 10 degrees; 2) The pointing must be stable with 0.02 degrees as an upper limit of the "offset"; 3) All data collected during the passage of RXTE through the South Atlantic Anomaly (SAA) were discarded; and 4) We rejected all data that has "electronX" (X represents the PCU number, 0-to-4) larger than 0.1. The ".gti" file contains the time intervals satisfying the above criteria. The desired spectra are extracted from the standard-2 and background catalogue files in turn, using the FTOOLS task "saextrct". As we carried out spectral analysis on a large collection of data, a Perl script was written to execute "saextrct" task for all data.

Spectrum Model Testing
For the purpose of testing different spectral models, energy spectra were extracted with a fixed exposure times of 256 s. For the spectral analysis we used XSPEC. The first three energy channels (<2.5 keV) were ignored because of uncertain background modelling, while channels 59 -129 (>30 keV) were ignored because of poor counting statistics.
In the 2.5 -30 keV energy range there are three spectral models commonly used: 1) a power-law times exponential cutoff model [19]; 2) Fermi-Dirac cutoff (FDCO) model [20]; and 3) Negative-Positive EXponential cutoff (NPEX) model [21]. Reference [22] examined the fits with these models, on data obtained for 10 accreting pulsars by PCA/HEXTE instruments on-board of RXTE. They found that the spectral parameters, in both FDCO and NPEX models, show a high degree of correlations. This correlation between spectral parameters makes the search for a best fit difficult and results in larger uncertainties in the obtained values for the parameters. The power-law times exponential cutoff model did not show such high correlations between spectral parameters. Thus following [22], we use the power-law times exponential cutoff model (powerlaw and highecut in XSPEC) for fitting the spectra. In functional form, the model is given as: where F is the photon number flux, E is the energy, is the photon index, E cut is the cutoff energy, and E fold is the folding energy.
The discontinuity at the cutoff energy in this model was smoothed by multiplying it by a Gaussian-shaped absorption function (gabs in XSPEC) at the cutoff energy with width sigma_{g} and depth tau_{g} fixed at 1.5 keV and 0.35, respectively. The gabs function in XSPEC is given by the equation: A Gaussian line in emission for the Fe K α line was also added to the continuum model for fitting the spectra.
This model can be applied to the spectra in three ways: 1) without low energy absorption-we call this the NA model (No-Absorption model); 2) with low energy absorption, called the TA model (Total-Absorption model); 3) partial covering absorption, in which the continuum consists of an unabsorbed continuum component plus an absorbed component, referred to it as PC model (Partial-Covering-model). We have tried all three alternatives in fitting the spectra.
In the TA-model we used for absorption phabs model in XSPEC, while in PC-model we used pcfabs in XSPEC for partial covering absorption. The spectral fits were done in XSPEC by tcl scripts.
For each of the above models (i.e. NA, TA, and PC models, each with N parameters) we fitted all the spectra with all N parameters free. Example fits to four observed spectra are shown in Figure 5. Then the fits were redone by fixing one parameter. This was done for each of the spectral parameters. We compared the χ 2 for fits with a particular parameter fixed compared χ 2 for fits with that same parameter free. The parameter with the lowest increase in χ 2 is the one that shows least variation in the spectra, and was then chosen to be fixed. The procedure was repeated with the remaining N − 1 spectral parameters to find the second parameter that caused least increase in χ 2 . Then it was repeated for the remaining N-2 spectral parameters, and repeated until there were no free parameters.
This procedure resulted in fixing the parameters in the following order of causing lowest increase in χ 2 to higher (but still small) increases in χ 2 : 1) constant Iron line width (labelled W); 2) constant Iron line width and folding energy (labelled WF); 3) constant line width, folding energy and iron line energy (WFL); and 4) constant width, folding, line, and cutoff energies (WFLC). Fixing further parameters beyond four resulted in large increases in χ 2 , thus showing real variations between spectra. Thus, it is not reasonable to fix the remaining parameters and they were left free. The same test procedure was applied to NA, TA and PC models and resulted in the same ordering of parameters that could be fixed.
Next we compared the χ 2 values between the three (NA, TA and PC) models with the aim of determining which is most reasonable to use to represent the spectra of Her X-1. Figure 6 shows the difference between χ 2 values of NA-model and PC-model fits (χ 2 (NA model) − χ 2 (PC model)) as a function of 35-day phase during Main High state (35-day phase 0 to 0.27). The NA-model has two more degrees of freedom than PC-model, thus data points in these plots above the threshold of 4.61 for the increase in χ 2 (red line in Figure 6) show worse fits, at more than 90% significance, with the NA-model compared to the PC model. In the spectral fits starting from the models with all parameters free to the one with the largest number of fixed parameters (i.e. free, W, WF, WFL, and WFLC fits), the percentage of spectra that show better fits with a significance level of 90% for PC-model over the NA-model are 70%, 68%, 74%, 75%, and 88% of the spectra respectively.
A similar comparison was made between the TA-model and PC-model. The difference between χ 2 values of TA-model and PC-model have the same shape as for NA-model compared to PC-model shown in Figure 6, except that the χ 2 differences are typically half of the amount found for the NA-model compared to PC-model fits. There is 1-extra degree of freedom in TA-model than in the PC-model, thus the 90% significance level threshold value for the difference in χ 2 is 2.71. Starting from models with all parameters free to the model with largest number of fixed parameters (WFLC), the percentage of data points with difference in χ 2 larger than 2.71 are 53%, 40%, 45%, 49%, and 70% respectively.
We have also tried to fit the spectra with a partial covering absorption by partially-ionized material (zxipcf in XSPEC). The spectral fits using partial covering with a partially-ionized absorber is referred to as IPC-model (Ionized Partial Covering). The partial covering with cold absorber (PC-model) shows better fits Journal of High Energy Physics, Gravitation and Cosmology From the above test fits we find that the spectra of Her X-1 in the 2.5 -30 keV energy range are best described by a two-component power-law with high energy cutoff continuum (unabsorbed and absorbed by a cold absorber) model. This is consistent with previous results, where the two component (absorbed and unabsorbed) was found to be needed throughout the 35-day cycle [23] [24] [25]. Thus we proceeded for the final spectral analysis using the PC-model.

Results of Spectrum Modelling
From these test fits the final model chosen is the PC model. We extracted energy spectra from standard-2 data of RXTE/PCA in 129 channels with larger exposure times to 1) reduce the number of spectra and 2) obtain better statistics for each spectrum. To obtain nearly constant signal-to-noise ratios for all spectra, we split the data into time intervals between 400-s and 1360-s with a mean value of 889-s. This resulted in a total of 609-spectra covering 20 MH-state. The extracted spectra include events detected in all layers and columns of the active PCUs combined together. Similarly, all layers and columns of the active PCUs are combined for background files using the faint background model. The corresponding response matrices were created accordingly.
We fit the spectra in the energy range 2.5 -30 keV with the PC-model consisting of an unabsorbed continuum component and an absorbed component. Both continuum components have the same spectral parameters except for their normalizations. A multiplicative Gaussian line in absorption (gabs in XSPEC), was added to the model to smooth the continuum at the cutoff-energy. The energy of the gabs function was tied to the cutoff energy, while its width and depth were fixed at 1.5 keV and 0.35 respectively. Another Gaussian line (gauss in XSPEC) in emission for the Fe K α line was added to the continuum model. Test fits showed that allowing a free iron line width and folding energy did not improve the fits, thus we fixed the line width and folding energy to 0.6 and 7.8 respectively.
The value of folding energy obtained here is lower than that obtained by [22] of 10.8 +0.2/−0.3 and the value obtained by [26] of 10.22 +/− 0.07. Reference [22] studied the spectra of Her X-1 during the peak of the MH-state in 5 hours of observations on MJD 50,705, 35-day phase 0.02 (cycle number 18 in our data) and used data from PCA and HEXTE instruments (with energy ranges 3.5 -60 keV and 20 -70 keV, respectively). Reference [26] analyzed data from PCA and HEXTE covering the MH-state (MJD 52,595 to 52,604, 35-day phase 0.01 to 0.25 (cycle number 73 in our data).
There could be a number of reasons responsible for this difference in the folding energy: 1) references [26] and [22] used data from two instruments (PCA/HEXTE) over a larger energy range than was used in our analysis; 2) references [26] and [22] used only data collected by the top layer of the PCA instrument, while we added all three layer of active PCUs; or 3) reference [22] did not use any absorption, while [26] used a total absorption model. We used the same continuum model as theirs but with a partial covering for the absorption. Journal of High Energy Physics, Gravitation and Cosmology By revisiting our test fits with different representations for absorption (NA, TA, and PC models) we rule out that the difference in folding energy to be due to different representation for the low energy absorption. In our test fits we found that the mean values obtained for folding energies are 8 Figure 9). This is a feature that should be explained by any model for the 35-day cycle. Qualitatively it is telling us that the illumination of the fluorescing gas is different than the illumination by the source in the direction of the observer, thus is providing a constraint on the source, fluorescing gas and observer geometry.

Discussion
Based on the disc model by [8] and the pulse evolution model of [3], we can understand the spectra of Her X-1 during the MH-state as being superposition of three components: 1) the spectrum of the direct emission from the pulsar (fan and pencil beams); 2) the spectrum of the reflected X-ray radiation off of the irradiated inner disc ring; and 3) the spectrum of magnetospheric/coronal emission of the scattered direct radiation.
The variation of the relative importance of these components is responsible for the variation of the spectral parameters over the 35-day cycle. Scattered emission, by hot material either in the magnetosphere high above the accretion column or in the corona above and below the accretion disc, is very weak and its energy spectrum is similar to the direct emission. Thus this emission does not contribute significantly to spectral changes during MH-state and SH-state, and is only significant during the low-states.
On the other hand, as can be seen from [8], the reflected emission from the inner disc ring contributes a significant fraction of the observed emission during both high states. The inner disc ring is blocked by the thick-hot atmosphere of the outer disc edge during the early MH-state. The uncovering of the inner ring by this hot-atmosphere is apparent in the gradual increase of its intensity up to 35-day phase 0.15. By the end of MH-state the near side of the inner-disc edge gradually covers the central source and the inner disc ring. Variation of spectral parameters over the 35-day cycle during the MH-state can be explained naturally in this picture.
There are cycles that show deviation from the typical trend of the spectral parameters with 35-day phase (cycle numbers 53, and possibly 51, 90, and 91).
These cycles occur after an emergence of the system Her X-1 from an extended low state (i.e. Anomalous Low State "ALS"). There are two ALS that occurred in the middle of our observations, during the periods MJD 51255.9 to 51756.9 and 52945.5 to 53159.4 (or in cycle numbers 34 to 48 and 83 to 88). It is generally believed that the ALS results from changes in the state of the accretion disc [28].
Thus the most probable reason for such deviation from the typical trend of spectral evolution is the change in the geometry of the inner disc edge due to proximity of these cycles from ALS periods.

Cutoff Energy
In the 35-day pulse-evolution model of [3], the reversed fan beam originates at a height of 2 neutron star radii above the neutron star surface and is much broader in solid angle than the pencil beam (by a factor of about 5 -10). Both the scattered and reflected emission is dominated by the spectrum of the fan beam. The inner disc ring will be illuminated largely by the fan-beam, so that the backscattered radiation will mostly be that of the incident fan-emission on the inner disc-ring.
Reference [26] carried out a pulse-phase-resolved spectral analysis at 4-intervals over the MH-state (35-day phases 0.03, 0.10, 0. 15, 0.20). The cutoff energy variation with pulse phase is shown in Figure 10 of [26]. The cutoff energy shows a large variation with pulse phase, reaching a high value around the central hard peak (pencil beam) of E cut of 24 -25 keV and a lower value of E cut of 18 -22 keV around the leading/trailing shoulders (fan beam).
By considering the phase-resolved variation of the cutoff energy, we expect for early MH-state, when the direct emission is the dominant component, the phase-average spectra will show a high value of the cutoff energy. As the 35-day cycle progresses, the reflected and scattered emission (mostly reflected, and scattered emission of the fan beams) becomes more important and results in a decrease in the pulse-phase-average cutoff energy. This is seen in our results in Figure 11.

Photon Index
For an irradiated slab of cold gas, the incident photons are Compton scattered by free or bound electrons, photoelectrically absorbed and reprocessed into fluorescent line emission and/or destroyed by Auger de-excitation. Because of the decrease of the photo-absorption cross-section with the third power of photon energy, most of the incident soft X-rays are absorbed and a fraction is reprocessed into the fluorescent emission line. On the other hand, higher energy photons are mostly backscattered. Thus, reflected/backscattered radiation by material illuminated by a central X-ray source shows an excess of the continuum above 10 keV [29] [30]. This is a result of electron scattered X-rays, called "the Compton reflection continuum". One of the main features of the reflected spectrum is the flattening of the spectrum above 10 keV. Figure 1 from [29] shows an example of an incident power law and reflected energy spectra.
As mentioned previously, the evolution of the spectral photon index ( Figure   10 in Figure 10).

Iron Line and Continuum Flux
In accreting X-ray pulsars (AXP), the Fe K α line emission is produced by the fluorescent reprocessing of the direct emission from the central source by the surrounding matter. This is indicated by the proportionality between the continuum flux above 7.1 keV and iron line flux [31]. Iron line emission produced by an illuminated slab in reflection is emitted by more highly ionized material near the illuminated face, and thus shows a higher energy fluorescent line. Lines produced in transmission through illuminated material are more likely to be produced far from the illuminated face (i.e. in cold material), and thus produce fluorescent lines of lower energy [32].

Summary and Conclusions
Her X-1/HZ Her was observed by the RXTE/PCA instrument for 20 different Main High (MH) states over the 19-year period 1996 to 2005. We have obtained the set of observations from the RXTE archive at HEASARC and analyzed the light-curves and spectra of Her X-1 with the goal of characterizing the Main High state of the 35-day cycle.
The low energy (2 -4 keV) and high energy (9 -20 keV) light-curves ( Figure  2 and Figure 3) show that: one cycle (number 73) has extensive coverage of 35-day phase of MH (35-day phase 0 to ~0.3); the other 19 cycles have low coverage; the light-curves are consistent with having nearly the same shape but with different peak intensity (see Figure 1 for all MH data overlaid); the softness ratio ( Figure 4) has a consistent shape for all cycles. There are sharp drops in softness ratio which indicate absorption dips [16]: these are not the subject of the current study, so were removed prior to light-curve or spectrum analysis.
To find a universal spectral model which is adequate to describe all MH spectra and allow comparison of spectral parameters with 35-day phase in MH, we carried out a large set of test fits. The model which we find is best to describe MH spectra is the power-law with an exponential cutoff. A fluorescent iron emission line is also required. Partial covering absorption gives a better description than uniform absorption (or no absorption) and we find a neutral absorber fits the spectra better than an ionized absorber. With this universal spectral model, all MH spectra for the 20 different MH states were fit using χ 2 minimization and the XSPEC software.
The resulting spectral parameters variation with 35-day phase was obtained for MH (Figures 7-13). Cycle 73 serves as a prototype for the 35-phase dependence of spectral parameters, but the other 19 cycles are generally consistent with cycle 73. Column density does not show a clear trend with 35-day phase. However, covering fraction has a minimum in late MH at 35-day phase 0.22. Continuum normalization peaks at 35-day phase 0.1, which is peak of MH state. Photon index declines from 1.1 at 35-day phase 0.05 to 0.85 at 35-day phase 0.22, with values near 1.0 for earlier than 0.05 or later than 0.22. Cutoff energy is constant at 20.5 keV for early MH (35-day phase 0 to 0.1) then declines to 18 keV at end of MH. The iron line energy has large errors, but shows a decrease, indicating a less-ionized fluorescing gas, from 35-day phase 0.05 to 0.22. Lastly, we calculated unabsorbed source fluxes for each spectrum and showed the ratio of 20 -30 keV source flux to iron line intensity in Figure 14. This shows a smooth trend of decreasing flux to iron line ratio, indicating more efficient iron line production as 35-day phase increases.
The above trends can be qualitatively described by the precessing-disc occultation model for the 35-day cycle in Her X-1 ([3] [7] [8] [9] [28]) with the most detailed model to date presented by [33]. One goal of the current work is to provide light-curves and spectral parameters vs. 35-day phase for the MH state, which can be compared quantitatively to models for the 35-day cycle.