TIMING AND FLUX EVOLUTION OF THE GALACTIC CENTER MAGNETAR SGR J1745−2900

For timing purposes, for each NuSTAR observation, we extracted photons in a 1' radius around the nominal source position using the NuSTAR Data Analysis Software Data Analysis Software version 1.2.0, along with HEASOFT version 6.13. The data were reduced to the solar system barycenter assuming the Chandra position reported by Rea et al. (2013) and the DE200 planetary ephemeris. We also filtered the data to extract only photons with energies 3–10 keV, as this generally provided an optimal signal-to-noise strength on the pulse. Average pulse times-of-arrival (TOAs) were extracted from the event lists by first folding the X-ray time series at the nominal pulse period and then aligning the resulting profile with a high signal-to-noise ratio template in the Fourier domain taking into account six Fourier harmonics, although our results are not strongly dependent on this choice. We have observed the pulse profile to be largely stable long-term, apart from the slow disappearance of the first peak seen in Figure 1 of Mori et al. (2013); see our Figure 1. This gives us confidence in the reliability of the TOA extraction method as the primary and third peaks have remained unchanged. The resulting TOAs were then fed into the tempo software package¹⁵ for further analysis. Note that we have verified that NuSTAR's clock is sufficiently stable (i.e., reliable in absolute timing to well under 10 ms on comparable timescales to those considered here) that it contributes negligibly to the uncertainties in the TOAs.

Zoom In Zoom Out Reset image size

We further supplemented the NuSTAR TOAs with timing data extracted from Swift XRT (Burrows et al. 2005) Windowed Timing (WT) observations of the source. Photon-counting (PC) mode data could not be included as they had insufficient timing resolution. To produce the Swift TOAs, Level 1 data products were obtained from the HEASARC Swift archive and reduced using the xrtpipeline standard reduction in HEASOFT version 6.13. We extracted photons in a 47'' radius around the nominal source position, and reduced the resulting event times to the solar system barycenter. The Swift data were also filtered to include only photons from 3–10 keV. The resulting events were then subjected to the same TOA extraction analysis as for the NuSTAR data. The details of the Swift observations are presented in Table 1. Note that one WT-mode observation (ObsID 00032811004 on MJD 56429) was omitted from the analysis due to poor statistics.

The ephemeris reported by Mori et al. (2013) provided an excellent fit to the initial NuSTAR data, and we further refine it here with subsequent observations. Our best-fit parameters for this first ephemeris are presented in Table 2. However, TOAs added from observations made on MJDs 56439 and 56457 deviated significantly by ∼0.1 from the prediction of an ephemeris fit using all earlier data. This deviation alone is not large enough to rule out extrapolation of this ephemeris to those epochs, as their phase deviation could be mostly fit out using a large second frequency derivative. By MJD 56480, however, this first ephemeris clearly described the phase data poorly, precluding proper phase counting, even with a second derivative.

Table 2. Phase-Coherent Timing Ephemerides for SGR J1745−2900

Parameter	Value
First Ephemeris
MJD range	56408–56450
Epoch	56415.42
Frequency ν (Hz)	0.2657067288(20)
Frequency derivative $\dot{\nu }$ (Hz)	−4.32(9) × 10−13
Second derivative $\ddot{\nu }$ (Hz s−1)	−8(1) × 10−20
Period P ≡ 1/ν (s)	3.763547895(29)
Period derivative, $\dot{P}$	6.12(12) × 10−12
Second derivative, $\ddot{P}$ (s−1)	1.15(15) × 10−18
rms residual (ms)	38
χ2/dof/pa	113/86/0.03
Second Ephemeris
MJD range	56457–56519
Epoch	56513.00
Frequency ν (Hz)	0.265700350(9)
Frequency derivative $\dot{\nu }$ (Hz)	−9.77(10) × 10−13
Second derivative $\ddot{\nu }$ (Hz s−1)	−2.7(4) × 10−20
Period P ≡ 1/ν (s)	3.76363824(13)
Period derivative, $\dot{P}$	1.385(15) × 10−11
Second derivative, $\ddot{P}$ (s−1)	3.9(6) × 10−19
rms residual (ms)	51
χ2/dof/pa	52/41/0.12

Note. ^aχ², degrees of freedom, probability of chance occurrence.

Download table as:  ASCIITypeset image

We therefore initiated a series of closely spaced observations in order to reacquire phase lock. This resulted in a second phase-coherent ephemeris with significantly different spin-down rate, as shown in Table 2; the featureless residuals from this new ephemeris are shown in Figure 2. The backward extrapolation of this second ephemeris to MJDs 56457 and 56439 also showed a significant phase deviation, however, again a (new) second frequency derivative could reasonably be fit to remove the deviation. Backward extrapolation of the new ephemeris beyond those epochs resulted in significant phase wraps, as shown in Figure 3, even with this new second derivative.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The two coherent ephemerides are plotted in solid lines in Figure 4. There, the overlap region between MJDs 56439 and 56457 is shown and the difference between the two ephemerides is clear. To determine which ephemeris better fits the data in the overlap region, we fit local frequencies to the TOAs at those epochs, as the NuSTAR integration times were sufficiently long to allow this measurement. The resulting frequencies are plotted in solid circles, with tempo-reported 1σ uncertainties shown. Clearly, the frequency from the MJD 56439 observation is inconsistent with the second ephemeris, indicating those data are best described by the first ephemeris.

Zoom In Zoom Out Reset image size

In Figure 4, we plot in red the epochs of the three X-ray bursts reported by Swift/BAT (Kennea et al. 2013a, 2013b, 2013d). Interestingly, the second burst coincided within uncertainties to the epoch at which our two ephemerides converge; specifically, at the observed burst epoch (11:17:26 UT on 2013 June 7 or MJD 56450.47044), the extrapolated frequencies of the two ephemerides agree at the 1.9σ level. This suggests that the burst likely coincided with the change in spin-down rate, if indeed the change was abrupt. It further suggests that the burst event occurred with no frequency discontinuity, i.e., with no spin-up or spin-down glitch. We set a 3σ upper limit on the amplitude of such a glitch, assuming that it coincided with the BAT burst, of Δν/ν < 1.1 × 10⁻⁶. This upper limit is in the mid-range of observed fractional frequency changes in magnetar bursts (e.g., Kaspi et al. 2003; Dib et al. 2008, 2009). If, in fact, the burst was not coincident with the ephemeris change, then the latter accompanied a spin-up glitch if it preceded the burst, and a spin-down glitch if it followed. On the other hand, the change in ephemeris may have been gradual and with no frequency discontinuity; in that case, however, the burst epoch coinciding with the convergence of the two independently determined ephemerides would have to be merely luck.

2.2.1. Swift Observations

2.2.2. NuSTAR Observations

The analysis of the NuSTAR spectral data for SGR J1745−2900 required particular care because two nearby transient sources impacted the data at different times. These transients are CXOGC J174540.0−290005, 24'' from the magnetar (henceforth Transient 1; see ATELs 5095, 5074) and AX J1745.6−2901, 88'' away (henceforth Transient 2; see ATELs 5226, 1513). Both transients are low-mass X-ray binaries and contaminated the NuSTAR magnetar spectral data significantly; the contamination was so severe that we ignored the spectral data in the epochs when Transient 1 was bright (MJDs 56430, 56439), and processed with great caution when Transient 2 was bright (MJDs 56504, 56512, 56513, 56517). Stray light from an unrelated source also contaminated the background in module B for one observation (see Table 1). Data were processed using nupipeline and nuproducts from the public release with HEASOFT 6.14. The extraction region was selected as a circle of 30'' radius centered on the source position. Response matrices appropriate for each data set were also generated using the standard software. In our spectral fitting, for interstellar X-ray absorption modeling, we assumed Wilms et al. (2000) abundances and Verner et al. (1996) cross sections. We considered only the energy range 3–30 keV, as the source was not detected at higher energies in any observation.

The GC region is crowded and the background is spatially variable. Moreover, the unrelated transients affected the data significantly. As a result, extracting a separate background region for the magnetar was, in general, not feasible. In order to evaluate the robustness of our results, we used two different independent methods to analyze these data. The first method subtracted background spectra obtained from pre-outburst data, while the second modeled the background spectra along with the magnetar's. For all the observations under consideration, in the source extraction region around the magnetar and in the energy band we analyzed (3–30 keV), GC diffuse emission dominated while the internal and stray-light backgrounds were negligible. Both of our methods described below are appropriate when the background is dominant. As we show, the two methods largely agree, implying that our reported results are robust.

In Method 1, we selected one of the pre-outburst images of the field (ObsID 30001002003 taken on MJD 56413) as our background exposure. The integration time for this observation was much longer than in any of the magnetar exposures, so the uncertainties in fit parameters are dominated by source statistics. We have verified that the off-axis angle of the source (and pre-outburst background) was similar for all epochs so that variations in the ARF between source and background regions are small (under ∼5%).

The NuSTAR spectra are plotted in Figure 5. We tested various spectral models for the magnetar, using XSEPC version 12.8.1. We binned the spectra to have a minimum of 20 counts per spectral bin and used χ² statistics in this method. Physically, we do not expect the column density N_H toward this source to vary on the relevant timescales, hence we fit jointly for N_H across all observations using the tbabs function in XSPEC. The best fit was obtained by the conventionally used empirical two-component model consisting of an absorbed blackbody plus power law, with overall reduced $\chi ^2_{\nu } = 1.03$ for 3683 degrees of freedom. Our best-fit fluxes and blackbody temperatures and radii (calculated assuming a distance of 8.0 kpc) are shown in Figure 6. Fits with just one spectral component were significantly worse. Although initially we allowed the blackbody and power-law model parameters to vary, we also tried fitting the models assuming a constant power-law index. The resulting fits, with constant and variable power-law index, are of similar quality, and yield similar results for the blackbody component. The best-fit value obtained for N_H, (13.5  ±  0.5) × 10²² cm⁻², was consistent with the values reported by Mori et al. (2013) and Rea et al. (2013). The best-fit value of the assumed constant fit power-law index was 1.43 ± 0.15. The best-fit kT values and effective radii are plotted in Figure 6. The χ² value for the overall fit was 6722 for 6425 degrees of freedom. We note that our need for a power-law component, even a constant value as a minimum, is consistent with the hint of high-energy excess reported by Rea et al. (2013) from Chandra data.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Given that Transient 2 is ∼10 times brighter than the magnetar and that their PSFs overlap, our method for handling the data affected by Transient 2 requires a special description. Transient 2 was extracted and reduced in a similar fashion to the magnetar itself, but was fit with an absorbed blackbody plus disk model, based on similar fits done to this source in the past (Hyodo et al. 2009). The contamination of transient counts within the magnetar extraction region was then estimated using the PSF file from NuSTAR's CALDB; we concluded that 0.035 of the transient's total flux fell within our magnetar extraction region in our energy range, which amounted to roughly half of the magnetar's flux in its aperture. We then fit the spectra of both objects jointly but using the transient's parameters and ignoring the magnetar's flux within the transient's extraction region. Allowing the contamination factor to vary gave similar results, and the best-fit contamination factor was close to the calculated one. Note that we did not detect the source above 10 keV in the last four NuSTAR observations, so for those epochs 99% confidence upper limits are presented in the 10–30 keV energy range. For all other observations, the detection significances in the 10–30 keV band ranged from 10.4σ (at the start of the observations) to 5.6σ (at the last observation for which there was a significant detection in this band).

In spectral analysis Method 2, we selected a background region for each observation that included all of the chips on which the source region falls, but excluded two bright features (Sgr A^* and the Sgr A-E knot), as observed in a mosaic of all observations of the field in which no transient was present, as well as stray light patterns from nearby bright sources. In individual observations, we found the selected background to always be subdominant compared to the source spectrum in our energy range. In this method, we used XSPEC v.12.8.0m, binned the spectra by a minimum of three counts per bin, and used the "lstat" fitting statistic.

We first fit a spectral model to the diffuse emission in the background regions. We jointly fit all the background regions, using two velocity-broadened APEC thermal plasma models (bapec in XSPEC) plus a power-law component, with photoelectric absorption. The velocity broadening of both plasma models was linked, as were all diffuse emission parameters for all observations, except the overall normalization, which was allowed to vary between modules A and B. We fit the diffuse emission in observations where the magnetar is absent, and found a reasonable model with featureless residuals throughout the fitted band. This demonstrated to us that our background model is reasonable as a phenomenological description sufficient for our purposes. For a more physically relevant consideration of the background spectrum in this energy band, see Krivonos et al. (2014).

We then fitted each magnetar observation individually using this independently determined diffuse emission model with variable parameters as background. For the magnetar's spectrum, we assumed a model consisting of a blackbody plus power law, photoelectrically absorbed. We used a Markov-Chain Monte Carlo method to explore the likelihood landscape of a joint fit to all the data. The absorption column of the magnetar's spectral model was linked for all observations but the normalization was allowed to vary between modules (though linked for each module between all observations), with all other parameters left free to vary between observations (but linked for both modules of the same observation). Fitted power-law indexes were poorly constrained. When linking the magnetar's power-law index across all observations, we found this to be slightly statistically disfavored but the impact on the kT values not significant. Data contaminated by Transient 2 are not reported for Method 2 as we could not find stable results for these.

The results of our NuSTAR spectral analysis for the magnetar for both methods are summarized in Figure 6. In the top two panels of the figure, we show the flux evolution with time in two bands, 3–10 keV and 10–30 keV. Note the difference in scale between the two in the figure. The results are qualitatively the same for Methods 1 and 2: the soft band flux decreases monotonically, quasi-linearly, by a factor of ∼2 over 80 days, while the hard band flux shows greater variation—notably an increase at the fourth epoch 23 days post-outburst, and a greater decrease subsequently such that by 80 days post-outburst, it is a factor of ∼4 lower than in our initial observation. In panel (c), we plot a hardness ratio, defined as the ratio of the flux in the 10–30 keV band to that in the 3–10 keV band. Methods 1 and 2 show good qualitative and near-quantitative agreement: the source indeed hardens significantly at the fourth epoch. The third panel from the top shows the evolution of blackbody temperature kT. There is clear evidence for a decrease in kT with time, at least until 60 days post-outburst, in spite of the apparent increase in 10–30 keV flux at 23 days. The bottom panel shows blackbody radius evolution assuming a distance to the magnetar of 8.0 kpc. Here we find that our methods disagree somewhat at the fourth and fifth epochs; regardless, overall it is clear that the blackbody radius remained nearly constant overall, to at least within 30%–40%.

The Swift spectral results described in Section 2.2.1 are consistent with those of NuSTAR in the 3–10 keV band, as shown in Figure 6, although we note a normalization offset. This may be a result of cross-calibration uncertainties or due to imperfect background subtraction, as we found the normalization of the Swift fluxes depended significantly on the exact location of the selected background region.

The coincidence of the second BAT burst with the intersection of the first and second ephemerides, as shown graphically in Figure 4, is striking. If not due to chance, the spin-down rate of the magnetar changed abruptly at the burst epoch and there was no frequency discontinuity of any type, either a spin-up or spin-down glitch, at that epoch.

Regardless of exactly how the change in spin-down rate occurred, we have shown that its magnitude has increased. Comparing extrapolated values of $\dot{\nu }$ at the start of our observations (MJD 56408) with that at the end (MJD 56519), we show that its magnitude has changed by a factor of 2.60 ± 0.07, that is, has nearly tripled in less than four months. Moreover, the magnitude is continuing to increase.

Large torque variations have been seen ubiquitously in magnetars (e.g., Woods et al. 2002; Kaspi et al. 2003; Gavriil & Kaspi 2004; Camilo et al. 2007, 2008; Dib et al. 2012; Archibald et al. 2013; Dib & Kaspi 2014). These torque variations can be categorized in two main classes: (1) those following glitches, and (2) those unassociated with glitches. The former can be seen immediately after a spin-up or spin-down event and are generally similar to the recoveries seen post-glitch in many radio pulsars (e.g., Yuan et al. 2010; Espinoza et al. 2011; Yu et al. 2013). These recoveries are thought to be related to repinning of angular momentum vortices in the superfluid component of the stellar crust after a major unpinning event (Alpar et al. 1984, 2000, 1993), although in the context of magnetars, scenarios involving the magnetosphere have also been suggested (e.g., Parfrey et al. 2012, 2013; Lyutikov 2013). Thompson et al. (2000) discussed these possibilities in the context of torque variations comparable to that seen in SGR J1745−2900 but observed in SGR 1900+14 following its 1998 giant flare. However, since we find evidence for a glitch in SGR J1745−2900, glitch-specific models do not seem relevant here.

Thompson et al. (2000) proposed that torque variations unassociated with glitches could arise due to particle outflow. In this picture, the energy output observed in the giant flare of SGR 1900+14 was insufficient to explain the torque change; Thompson et al. (2000) required a radiation-hydrodynamical outflow of even higher energy output. As no giant flare has been observed from SGR J1745−2900, this model does not seem to apply here.

Magnetar torque variations have also been suggested to be purely magnetospheric in origin (e.g., Beloborodov 2009). In general, a burst signals a sudden re-arrangement of a part of the magnetosphere. If this part involves the open field lines, the spin-down torque must change. Note that the open magnetic flux is a tiny fraction ∼10⁻⁴ of the total magnetic flux of the star, so the spin-down torque is sensitive to the behavior of a tiny fraction of the magnetosphere. In the context of the scenario proposed by Beloborodov (2009), the persistent luminosity is produced by a much more energetic closed j-bundle which may or may not contain the open flux. If a burst affects the open flux but does not greatly affect the j-bundle, the torque can change while the X-ray luminosity does not, as is observed here, since near the second burst epoch where the torque seems to have begun to change, there is no feature in the source's flux evolution beyond the continued decay following the source's initial appearance in 2013 April. Indeed, we find it interesting that at the second burst epoch, no glitch was seen and no flux change was detected (apart from the brief burst itself and the continued decay following the source's initial appearance). It could be that in magnetar outbursts, part of the observed enhanced flux results from the interior of the star and originates from heat released in an internal glitch, whereas another component of the enhanced flux originates from purely magnetospheric processes. This could explain why, for example, in the 2002 outburst of magnetar 1E 2259+586, there were two clearly different timescales associated with the decay of the initial flux enhancement: one very short, on a one- to two-day timescale, and one lasting months (Woods et al. 2004; Zhu et al. 2008). Perhaps the short-term X-ray flux enhancement and decay is a result of transient glitch-related emission, whereas the much longer decay is due to magnetospheric untwisting and/or crustal cooling.

Regardless of the origin of the torque variations, they clearly will dominate over dynamical effects due to the orbit around Sgr A^*. Rea et al. (2013) have argued that variations in the spin-down rate as large as 10% due to acceleration in the field of Sgr A^* might one day be observed, however, the observations reported here demonstrate that the spin evolution of this magnetar, like those of many others, is inherently unstable and unlikely to permit such a measurement. The discovery of a rotation-powered pulsar, particularly a millisecond radio pulsar, in similar proximity to Sgr A^* would be far more useful for dynamical studies of the black hole environment.

We have shown that the 3–10 keV flux of SGR J1745−2900 has declined since the source's initial appearance, albeit rather slowly, only by a factor of ∼2 over the first 80 days since the discovery outburst. This is similar to what was found by Rea et al. (2013) on the basis of three Chandra observations. The flux decay on a timescale of ∼10⁷ s was predicted by Mori et al. (2013) based on the observed emission area of the blackbody component, A, and the untwisting magnetosphere model of Beloborodov (2009). Specifically, Mori et al. (2013) estimated a luminosity evolution timescale of $t_{{\rm ev}} \simeq 10^7 \mu _{32} \Phi _{10}^{-1} A_{11.5}$ s, where μ₃₂ is the magnetic moment in units of 10³² G cm³, $\Phi _{10}^{-1}$ is the electric voltage sustaining e^± discharge in the magnetosphere in units of 10¹⁰ V, and A_11.5 is in units of 10^11.5 cm². This predicted timescale is roughly consistent with the flux decay we report. The model also predicts that the hot spot should shrink approximately as $A\propto L_{{\rm BB}}^{1/2}$, where L_BB is the blackbody luminosity. When L_BB decreases by a factor of two, area A is expected to decrease by ∼40% and the blackbody radius by ∼20%. There may be a hint of such a radius decrease in the bottom panel of Figure 4, but given the contamination due to Transient 2, our observations cannot confirm this.

The blackbody temperature also decreased monotonically by ∼20% over the first 60 days post-outburst, although as is clear in Figure 5, there is a possible hint of an increase in the subsequent NuSTAR observation at 80 days. Due to the presence of Transient 2, we cannot verify unambiguously whether this trend continued, since our subsequent observations may be contaminated and the Swift XRT observations yield insufficient statistics to detect such a change. Chandra observations, given that telescope's superior angular resolution that should preclude the transient source contamination, may be able to address this question. Meanwhile, however, we note with interest the apparent hardening of the magnetar's flux in our fourth observation 23 days post-outburst (see Figure 5); this was unaccompanied by any significant frequency or torque change or by any observed burst. This is puzzling and could indicate a burst that went unseen by all-sky monitors just prior, or perhaps it could be due to a different source appearing within the NuSTAR PSF. Regardless, we have not observed the common magnetar flux/hardness correlation (e.g., Woods et al. 2004; Gotthelf & Halpern 2007; Zhu et al. 2008; Tam et al. 2008; Rea & Esposito 2011; Scholz & Kaspi 2011) in this source; this may be due to the relatively small range of fluxes yet observed, although we note no clear correlation was seen for SGR 1627−41 either, for a much larger flux range (An et al. 2012). Given the typical behavior of other magnetars, we expect the source spectrum to gradually soften as the flux continues to decline, although presently both flux and temperature show evidence for leveling off. We further note the relative stability of the inferred effective blackbody radius (Figure 4); any model to explain the flux decline will also have to account for a relatively stable emitting area. Some crustal cooling models predict an increase in emitting area as the initially localized internal heat spreads around the neutron-star surface (Pons & Rea 2012), while relaxation following a magnetospheric twist as discussed above should involve a decrease in the emitting region, the footpoint of the j-bundle (Beloborodov 2009). Continued observations of the source spectrum as the emission fades may help distinguish between these two possible processes in the star.