CHANDRA, KECK, AND VLA OBSERVATIONS OF THE CRAB NEBULA DURING THE 2011-APRIL GAMMA-RAY FLARE

The Crab Nebula, the relic of a stellar explosion recorded by Chinese astronomers in 1054, has a special place in the history of astronomy. It is our most frequently observed laboratory for high-energy astrophysics. Located at a distance of ≈2 kpc, the system is energized by a pulsar of spin-down luminosity L_plsr ≈ 5 × 10³⁸ erg s⁻¹ and current spin period P ≈ 34 ms. The history and general properties of the system are nicely summarized in the review by Hester (2008). Optical and X-ray images (Hester et al. 1995, 2002; Weisskopf et al. 2000) of the inner nebula show features such as an inner ring, toroidal structure, knots, and two opposing jets originating from the pulsar—these latter presumably aligned with its rotation axis and proper motion vector (Caraveo & Mignani 1999; Ng & Romani 2007; Kaplan et al. 2008 and references therein). The "inner-ring," prominent in X-rays, is commonly accepted as being the termination shock produced by the relativistic wind of particles accelerated by the pulsar. Many of the optical and X-ray features brighten and fade and/or move over weeks or months (e.g., Hester et al. 1995, 2002).

The quiescent or average spectral energy distribution (SED) of the Crab Nebula has a characteristic two-humped form (see, e.g., Figure 8; Atoyan & Aharonian 1996; Bucciantini et al. 2011 and references therein). The synchrotron spectrum extends from ≈30 MHz to ≈1.2 × 10²² Hz (500 MeV). Most of the power is radiated by ≈ TeV electrons in the near UV ≈10 eV with an associated luminosity of ≈1.3 × 10³⁸ erg s⁻¹ (Hester 2008). This roughly matches the loss of rotational energy by the pulsar, which releases its energy electromagnetically, generating a current ≈200 TA and inducing an electro-motive force (EMF) ≈50 PV. However, the nebula is currently varying on a few-year timescale (Wilson-Hodge et al. 2011). At higher energies, Compton scattering has a luminosity of ≈10³⁶ erg s⁻¹, peaking around 60 GeV (Albert et al. 2008) and measured up to ≈80 TeV (e.g., Aharonian et al. 2004; Abdo et al. 2010).

Since 2007, the AGILE and Fermi satellites have detected several γ-ray flares from the Crab Nebula (Tavani et al. 2011; Abdo et al. 2011; Striani et al. 2011a; Buehler et al. 2012) in the 0.1–1 GeV range. The most dramatic flares exhibit variability on timescales as short as a few hours, although it is unclear whether they are distinct events or just the largest variations from a stationary power spectrum of fluctuations. Prior to the 2011-April event, the only Crab γ-ray flare covered by a multi-wavelength observing program was the 2010-September flare, which triggered observations in radio, optical (using both ground-based telescopes and Hubble Space Telescope; HST), and X-ray bands. Despite the γ-ray brightness of the flares, there has been no evidence for correlated variations in radio (Lobanov et al. 2011; this paper), near infrared (Kanbach et al. 2010; this paper), optical (Caraveo et al. 2010), or X-ray bands (Evangelista et al. 2010; Shaposhnikov et al. 2010; Tennant et al. 2010, 2011; Ferrigno et al. 2010; Horns et al. 2010; Cusumano et al. 2011; Tavani et al. 2011; Striani et al. 2011b; this paper).

Here we focus on the Fermi-LAT results for the 2011-April flare (Buehler et al. 2012), which allow us to assess the source behavior in detail. The source doubled its γ-ray flux within eight hours and reached a peak flux 30 times its average. The isotropic luminosity increased to ≈2 × 10³⁷ erg s⁻¹ in ≈10 hr and the spectrum peaked at ≈400 MeV. Table 1 gives the γ-ray power law photon spectral index, the integrated photon flux above 100 MeV, and the photon spectral flux at 100 MeV, as measured during the 10 ks time intervals when X-ray data (Section 2) were taken.

Notification as to the level of flaring prompted us to trigger pre-approved Target of Opportunity observations with Chandra and with the NRAO¹⁸ Karl G. Jansky Very Large Array (VLA). We were also fortunate to obtain a Keck image in the near infrared, albeit not under ideal conditions. Figure 1 shows the Fermi-LAT γ-ray counting rate as a function of time and also indicates the times of the Chandra, Keck (Section 3), and some of the VLA (Section 4) observations.

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With the back-illuminated ACIS S3 CCD on the Chandra X-Ray Observatory approximately centered on the Crab pulsar, we obtained five observations (Table 1) during and somewhat after the 2011 April γ-ray flare. For these observations, the spacecraft dithered with an amplitude set to 1''. Although standard processing typically produces an aspect solution better than 05, even this small uncertainty can introduce noticeable shifts when comparing different data sets. Thus, we re-registered images for our analysis using the read-out streak and the pulsar as guides. As each of these five images was placed at approximately the same CCD location, spatial non-uniformity in the ACIS response (e.g., due to contamination) does not introduce spurious temporal variability.

Owing to the Crab's high flux, the ACIS observations employed a special mode with 0.2 s frame time, which limits the CCD read-out to a 300 × 300 ACIS-pixel (≈150'' × 150'') subarray. Although each observation lasted about 10 ks, telemetry saturation reduced the effective integration time to approximately 1200 s per observation. Despite the short frame time of the special ACIS mode, regions of high surface brightness suffer somewhat from pile-up effects. We consider only data in the range 0.5–8.0 keV because of severe interstellar absorption at low energies and declining flux at high energies. Using these data, we then search for X-ray variations approximately contemporaneous with the 2011-April γ-ray flare.

2.1. X-Ray Image Analysis

Figure 2 shows an image of the number of counts per ACIS pixel, summed over the five observations. For each observation, we re-binned a 120 × 120 ACIS-pixel image centered on the pulsar into a 60 × 60 array of 2 × 2 ACIS pixels. Each of these I = 3600 "analysis pixels" is sufficiently large (about 1 arcsec²) to enclose most of the Chandra point-spread function anywhere in the field of view. Note that we also performed an analysis using a circular bin of radius 1 ACIS pixel on an oversampled grid of cadence 0.1 × ACIS pixel—i.e., a spherical top hat smoothing of the events. As each method gave similar results, we here report the results for the "analysis pixel" binning, for which each pixel is statistically independent.

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For each analysis pixel i, we calculate the mean count rate r_i averaged over the J = 5 observations, weighted¹⁹ by the respective (counting-rate) statistical error σ_ij for each analysis pixel and observation. For evaluating statistical significance of temporal variations over the J = 5 observations, we compute²⁰ χ²_i along with a derived significance measure S_i.

For purposes of discussion, we also compute the appropriately weighted²¹ statistical error σ_i and sample standard deviation s_i for each pixel i. As properly weighted, χ²_i = (J − 1) s_i²/σ²_i. While we have rigorously calculated r_i, χ²_i, σ_i, and s_i for each pixel i using appropriate weightings, we note that the weightings are nearly uniform as the effective duration of the each of the J = 5 observations was nearly the same—about 1200 s.

2.2. Variability of the X-Ray Images

The (counting) statistical error σ_i is the primary noise term and thus governs the sensitivity for detecting temporal variations at the analysis-pixel (square-arcsec) scale. Figure 3 shows the image and the corresponding histogram of the distribution of σ_i, which ranges from 0.0025 to 0.024 ct s⁻¹ per analysis pixel. Based upon the χ² probability distribution and the number of "tries" (I = 3600 independent analysis pixels), a 99% confidence detection would require a $\chi _{i, 99\%}^{2} > 31.2$ on (J − 1) = 4 degrees of freedom. This corresponds to a sample standard deviation $s_{i, 99\%} > 2.80\, \sigma _{i}$, which ranges from 0.0071 to 0.068 ct s⁻¹ over the field.

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Here we do not display the analogous image and histogram for the sample standard deviation s_i, which ranges from 0.0014 to 0.048 ct s⁻¹ per analysis pixel. Instead, Figure 4 shows the image and histogram of the distribution of a calculated (Section 2.1) significance measure S_i, related to χ²_i = (J − 1) s_i²/σ²_i. The statistically most significant variation has χ²_i = 23.5 on ν = (J − 1) = 4 degrees of freedom giving S_i = 6.9. Such a fluctuation is expected statistically in at least 1 of 3600 pixels in 31% of realizations. Table 2 gives the sample standard deviation and 99%-confidence upper limit to the count-rate variation, for the analysis pixel with the statistically most significant X-ray variation. While we detect no variations statistically significant at 99%confidence, it is curious that the three most significant variations occur at locations on the inner ring.

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Table 2. X-Ray Results at 1 keV for the Analysis Pixel with the Most Significant Variation

Quantity	Unit	Sample stdev (s)			99%-Upper Limit
Rate	ct s−1	0.0480			0.0554
Γx		$\frac{2}{3}$	1	2	$\frac{2}{3}$	1	2
NE	10−4 ph (cm2 s keV)−1	0.48	0.61	1.18	0.55	0.70	1.36
FE	10−13 erg (cm2 s keV)−1	0.76	0.97	1.89	0.88	1.12	2.18
ELE	1032 erg s−1	0.37	0.47	0.91	0.42	0.54	1.05
Γxγ		1.18	1.20	1.26	1.20	1.22	1.27

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Note that if a feature, such as one of the knots, does not change in intensity but moves from one analysis pixel to another, then our χ²_i test would detect this as a variation. We know that features in the inner ring of the Nebula do move and expect to detect some variability due to this motion. However, as our five observations span only 14 days and 1'' corresponds to 11.5 light days, only relativistic motion would be detectable. Other effects, such as changes in the roll angle of the read-out streak, can also lead to spurious variability. Indeed, this may play a role for the analysis pixel with the most significant variation, which lies adjacent to the average read-out streak (Figure 2).

2.3. Limits to the X-Ray Flux

Thus far, we have described the X-ray data for each analysis pixel in units of ACIS count rate. Neglecting for the moment pile-up effects, the photon spectral flux (or other related radiation quantity) is proportional to the count rate for an assumed spectral shape. Consequently, any change in count rate corresponds to a proportionate change in photon spectral flux (for an assumed spectral shape). Using the Chandra PIMMS²² for the ACIS-S detector and an absorption column N_H = 3.1 × 10²¹ cm⁻², we determine (ignoring pile-up) this constant of proportionality for an X-ray power-law photon index Γ_x = (2/3), 1, and 2: At E_x = 1 keV, N_E(E_x)/r = 0.99, 1.26, and 2.46 × 10⁻³ ph (cm² s keV)⁻¹ per ct s⁻¹, respectively. Correcting for pile-up has little effect in low-count-rate regions, but would raise these flux upper limits by ≈10% or so for high-count-rate regions.

Table 2 calculates the photon spectral flux N_E(E_x), the energy spectral flux F_E(E_x), and the indicative (isotropic) luminosity EL_E(E_x) = 4πD²EF_E(E_x) at D = 2 kpc, corresponding to the sample standard deviation and 99%-confidence upper limit for the count-rate variation in the analysis pixel with the most significant X-ray variation. Figure 5 displays an image of the energy spectral flux F_E(E_x) at E_x = 1 keV for Γ_x = 1, based upon the sample standard deviation s_i of the count rate in each analysis pixel.

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2.4. Constraints on the X-Ray to γ-Ray Spectral Index

We now compare the X-ray data with the γ-ray data to quantify the implications of our lack of detection of time variations in the X-ray data. Our approach compares a variability measure for the X-ray (1 keV) photon spectral flux ΔN_E(E_x) in each analysis pixel with the analogous variability measure for the γ-ray (100 MeV) photon spectral flux ΔN_E(E_γ). In particular, we calculate the sample standard deviation of the γ-ray spectral flux at 100 MeV, using power-law fits to the five Fermi-LAT measurements that were simultaneous with the five Chandra observations. Table 1 lists the five Chandra ObsIDs, their dates, along with the γ-ray photon index Γ_γ, integrated photon flux N(>100 MeV), and photon spectral flux N_E(100 MeV). For the five Fermi-LAT observations, the mean and sample standard deviation of the photon spectral flux at 100 MeV are 1.21 × 10⁻¹⁰ and 5.77 × 10⁻¹¹ ph (cm² s keV)⁻¹, respectively.

Based upon the sample standard deviation (s_i) of photon spectral flux at E_x = 1 keV for each X-ray analysis pixel and the measured standard deviation (5.77 × 10⁻¹¹ ph (cm² s keV)⁻¹) at E_γ = 100 MeV, we constrain the effective X-ray to γ-ray photon index of the flaring component: Γ_xγ ≡ −log [ΔN_E(E_γ)/ΔN_E(E_x)]/log [E_γ/E_x]. Figure 6 shows the image and corresponding histogram of the distribution of upper limits to Γ_xγ based upon the sample standard deviation of the X-ray measurements and assuming Γ_x = 1.

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In that the γ-ray variations are statistically significant and the X-ray variations are not, we compute 99%-confidence upper limits to Γ_xγ (Table 2, last row). Note that the upper limits to Γ_xγ are marginally consistent with the low-energy extrapolation of the γ-ray spectrum (Γ_γ = 1.27 ± 0.12) of the flaring component (Buehler et al. 2012).

2.5. Variability within an X-Ray Image

The extreme saturation of the pulsar in the X-ray images means that we cannot easily study the central 2'' in X-rays. However, this region does contain a nebular structure of particular interest: the "inner knot" whose peak is 065 southeast of the pulsar at position angle 118° east from north (Hester 2008). This structure, an oval shape extending ≈075, is well measured in HST and ground-based near-IR images. Given its relatively red spectrum (energy spectral index α_ν = −1.3 ± 0.1 versus α_ν = 0.27 ± 0.03 for the pulsar; Sandberg & Sollerman 2009), it is one of the near-IR brightest structures in the Nebula. Sandberg & Sollerman (2009) note that the knot varies by a factor of two; we confirm typical variability of 20%–30% in archival HST images. Komissarov & Lyutikov (2011) have proposed that this structure represents radiation from an oblique termination shock in the pulsar wind nebula. In this picture, the Earth line-of-sight is tangent to the flow at the inner knot position, and thus the intensity experiences substantial Doppler boosting for synchrotron emission in the mildly relativistic post-shock flow. Indeed, in relativistic MHD simulations they find that this bright spot is highly variable and can dominate the γ-ray synchrotron emission. Alternatively, the knot could be a time varying standing shock in the polar jet flow itself, a flow known to be highly variable from HST imaging (Hester et al. 1995, 2002; Hester 2008).

It is thus of interest to check the status of the knot during the 2011 April γ-ray flare. Unlike the sequence of multiwavelength observations performed after the 2010 September flare, it was impossible to trigger an allocated HST Target of Opportunity observation owing to solar constraints in April. We were able to obtain a Keck Near Infrared Camera (NIRC2) K' exposure (Figure 7, left image) on MJD 55667.250, almost precisely at the peak of the γ-ray flux and 2.5 hr before the ACIS image ObsID 13152 (Figure 1). Unfortunately, the observations occurred during twilight and only one 20 × 4 s integration without dithering was obtained. Under these conditions the adaptive optics (AO) loop did not close, leaving an undithered image with native 046 FWHM seeing. This frame was dark subtracted and an approximate background was removed using an immediately subsequent image. Despite the modest image quality, the inner knot was well detected. After subtracting the pulsar with a scaled image of the comparably bright companion star 4'' northeast, we measured the knot flux and position. We find a magnitude K' = 15.60 ± 0.03 and an offset 064 ± 004 from the pulsar. For comparison, we measured a high-quality NIRC2 K' image (Figure 7, right image) obtained 2005 November 10. Here the knot is K' = 15.94 ± 0.02 at offset 058 ± 002. We also note that Sandberg & Sollerman (2009) measured K_s = 15.80 ± 0.03 on 2003 October 18. We conclude that the knot was in a relatively bright state during the flare (≈35% brighter than in 2005), but well within the normal range of flux (and position) variation. Thus, there is no dramatic change in the inner knot in the near-IR band. We use the amplitude of the measured variation as an upper limit to any variation in the inner knot associated with the γ-ray flare (Figure 8).

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On 2011 April 14, we triggered a prompt radio follow-up program with the VLA. The VLA observations occurred in eight epochs starting April 15 and ending July 10. These observations detected the pulsar at two epochs, but found no other point sources in the field. Observations were predominantly in the range 4–8 GHz, with additional observations at 1.4 GHz (not reported; see below) and at 22 GHz (for some later epochs). Unless otherwise noted, all observations used two sub-bands, each with a 128 MHz bandwidth. In each run, observations of the target were bracketed with scans of a phase calibrator (J0559+2353, except where noted) and a flux calibrator (3C 147). The fields of view are limited to the primary beam response of the antennas, with full width at half-power of 9'/ν₅ with ν₅ ≡ ν/(5 GHz). Table 3 provides a summary of the observational parameters and results.

Our initial observations were obtained through a Fermi guest-investigator cycle-3 program (S3184) approved for four 1 hr runs in the L-band and C-band (≈1.4 and 5 GHz, respectively). The VLA was in its B-array configuration during these observations, resulting in images with angular resolution ≈1''/ν₅. We found that the L-band data for the Crab were highly confused due to the brightness and complexity of the steep-spectrum nebular emission in the first (April 15) and second (April 19) epochs. Consequently, we modified our strategy for subsequent observations. After the first epoch, we split the C-band observations into two widely spaced side-bands centered at 4.2 and 7.8 GHz, aiming better to constrain the spectrum of any detected source. We also began scheduling observations only at frequencies greater than 4 GHz after the second epoch. In these B-array data, our point-source limits at the lower frequency are ≈3–4 × larger than at the higher frequency, again due to the Crab Nebula's steep-spectrum radio emission.

After non-detection of any significant radio point-source emission down to ≈1–7 mJy (3σ) sensitivities in the initial three VLA observations 1–7 days after the γ-ray peak (Hays et al. 2011), we purposely delayed the fourth observation until 9 days after the previous observation to probe longer timescales. Only in this last observation (April 30) did we obtain a significant point-source detection, which was coincident with the Crab pulsar position, but only in the upper side-band centered at 7.8 GHz. The detection is a factor of five greater than the 3σ limit from nine days prior. The source was not detected at the lower frequency side-band (4.2 GHz) with a limit indicating a source with a flat radio spectrum.

Following the point-source detection on April 30, we became aware of VLA TEST observations of the Crab obtained on April 22 (program TDEM0007; PI: D. Frail). These data were obtained with wide bandwidth (16 × 128 MHz wide sidebands), so were more sensitive than those from our observing runs. The flux densities were scaled to 3C 147; J0534+1927 was utilized for phase calibration. The flat-spectrum radio source coincident with the Crab pulsar detected in our Apr-30 observation was confirmed in the Apr-22 data in two bands, but with a much lower (10 ×) flux. Also, the source spectrum was rather steep, with an energy spectral index α = 2.21 ± 0.34 (S_ν∝ν^−α) between 5 GHz and 8.6 GHz.

Following the radio detections of the pulsar, we requested further VLA monitoring of the Crab through Director's Discretionary Time (program 11A-268 = AC1052). In addition to the C-band observations, we obtained exposures in the K band (centered at 22.396 and 22.254 GHz) aiming to constrain further the spectrum of any detected radio source. Through this program, we obtained 2 hr runs on May 12-13 (while the VLA was in its hybrid BnA array) and on July 10-11 (in A array), and an additional 1 hr run on May 28, using one of the early (April 19, from program S3184) frequency setups. An angular resolution of ≈(03/ν₅) is typically achieved in A-array VLA observations. With the higher resolution, we obtained systematically 4× lower flux limits than in the lower resolution B-array data, presumably due to lesser contribution from the extended nebular emission. In none of these later epoch follow-up observations, did we detect a point source, to typical limits of 1–2 mJy at each of the three frequencies (see Table 3).

4.1. Discussion of the Radio Data

Here we discuss possible explanations for the absence in non-γ-ray bands of variability that is obviously correlated with the γ-ray flare. In addition we present a conceptual model for the production of the γ-ray flares.

5.1. γ-Ray Emission

5.2. Associated Emission

Using the Chandra, Keck, and VLA Observatories, we acquired X-ray, near-IR, and radio images of the Crab Nebula, contemporaneous with the 2011 April γ-ray flare. We searched for variability in the X-ray data over two timescale ranges: First we tested for pointing-to-pointing variations amongst the five pointings, each with an effective exposure time ≈1200 s and a minimum separation of 0.6 days. Second, we tested for variations within each of the five observations. In neither case did we detect statistically significant X-ray variations; thus we can set only upper limits to any X-ray variations associated with the γ-ray flare. As the Chandra ACIS images suffer severe pile-up near the Crab pulsar, our search for variability in the X-ray images was not sensitive to variations within the central ≈15 or so.

Comparing the upper limits to X-ray variations with the Fermi-LAT-measured γ-ray variations, we set upper limits at 99% confidence to the effective X-ray–γ-ray photon power-law index Γ_xγ ⩽ 1.20 to ⩽1.27, dependent upon assumptions about the X-ray index Γ_x. As Fermi-LAT measures a γ-ray index Γ_γ = 1.27 ± 0.12 for the flaring component, it is statistically possible that the flaring component's spectrum extends as a simple power-law from γ-rays to X-rays. Further, we note that our upper limit to Γ_xγ is consistent with transparent synchrotron emission, whose photon index must be >(2/3).

Comparison of two Keck near-IR observations found that the inner knot (≈065 from the pulsar) was somewhat brighter than average during the γ-ray flare, but well within the normal range of brightness fluctuations typically observed. We used the measured (≈35%) change in the near-IR flux from this knot as an upper limit to near-IR variations associated with the γ-ray flare. We also performed a number of VLA observations searching for a point source appearing either at an unusual location and/or contemporaneous with the γ-ray flare. Other than the pulsar itself, no such source was detected.

Figure 8 shows the SED of the Crab Nebula over the observed electromagnetic spectrum. The plot also shows the SED of the 2011 April γ-ray flare and the various limits determined here on variable radio, near-infrared, and X-ray emission possibly associated with the γ-ray flare.

Finally, we reviewed and discussed potential implications of γ-ray flares and theoretical issues to be addressed. We concluded that, apart from lower-energy emission directly associated with the γ-ray flare itself, the dynamical aftermath of a major flare could alter the ambient emission—e.g., through compression of the magnetic field. The associated surface brightness change would likely be only several percent, assuming a total emission region of size ≈03, consistent with our upper limits.

Although no "smoking gun" has been identified, one should be encouraged that we have identified a number of regions in the X-ray images that are possible candidates. We have also established further Target of Opportunity observations with Chandra and HST that will be triggered at the onset of the next γ-ray flare. The X-ray observations will also probe the region very close to the pulsar using the Chandra High-Resolution Camera (HRC).

The work of M.C.W., S.L.O., and A.F.T. is supported by the Chandra Program. The Chandra data was obtained in response to a pre-approved target of opportunity request granted under Chandra Director's Discretionary Time. The Fermi LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'Études Spatiales in France. C.C.C., G.B.T., and J.D.L. thank Tim Hankins for useful discussions, and the NRAO scheduling committee for alerting us to the VLA TEST data and for their prompt consideration of our Director's Discretionary Time observations. C.C.C., G.B.T., and J.B.L. were supported in part by NASA through a Fermi cycle-3 guest investigator grant. In addition, work by C.C.C. at NRL is supported in part by NASA DPR S-15633-Y. Our analyses utilized software tools provided by the Chandra X-ray Center (CXC) in the application package CIAO and from the High-Energy Astrophysics Science Archive Research Center (HEASARC, operated by the NASA Goddard Space Flight Center, Greenbelt, MD, and by the Smithsonian Astrophysical Observatory, Cambridge, MA).

Facilities: CXO - Chandra X-ray Observatory satellite, Fermi - Fermi Gamma-Ray Space Telescope (formerly GLAST), VLA - Very Large Array