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Astron. Astrophys. 345, L32-L34 (1999)

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2. Gamma-ray pulsations from the Crab

We chose to analyze the energy dependence of pulse arrival times from the Crab pulsar as it has the largest ratio of distance to pulse period of the bright gamma-ray pulsars, thus maximizing the constraints which can be placed. Also, the pulses from the Crab are well aligned in time from radio waves, through optical and x-ray emission, to gamma-rays. Thus, it is likely that the photons of different energies are produced nearly simultaneously.

We used data from the Energetic Gamma-Ray Experiment Telescope (EGRET) (Thompson et al. 1993) of the Compton Gamma-Ray Observatory (CGRO). We extracted gamma-ray photon event lists from the CGRO public archive for observations pointed within [FORMULA] of the Crab and then, using the program pulsar (version 3.2, available from the CGRO Science Support Center) selected events lying within the energy dependent 68% point spread function of EGRET and calculated the phase of each photon relative to the radio ephemeris of the Crab (Arzoumanian et al. 1992). The pulse period of the Crab changed from 33.39 ms to 33.49 ms over the course of these observations. The radio timing must be corrected for the variable dispersion along the line of sight to the Crab. The accuracy of this correction is estimated to be 0.2 ms (Nice 1998), consistent with previous estimates of the accuracy of the dispersion correction for the Crab (Gullahorn et al. 1977).

Pulse phase histograms for several energy bands are shown in Fig. 1. The main pulse peak, near phase 0.0, is the most appropriate feature for timing. The main peak is similar across the energy range from 70 MeV to 2 GeV (Fierro 1995). The peak width is about 0.05 in phase, and appears somewhat narrower at high energies. There is no obvious shift of the peak centroid with energy.

[FIGURE] Fig. 1. Crab pulsar phase histograms for various [FORMULA]-ray energy bands. Zero phase is set by the radio ephemeris.

To study the energy dependence of the speed of light, we measured the main peak pulse arrival time in each energy band. We did this in two ways. First, we calculated the average arrival time for photons in the main peak. We found the average time for each energy band using photons with phases between -0.0464 and 0.0336, an interval centered on the mean arrival time for all photons used in this analysis. Second, we parameterized the pulse arrival times by fitting a Lorentzian to the pulse profile, within the same phase range specified above, for each energy band. Before fitting, a constant rate equal to the average rate between phases -0.4 and -0.2 was subtracted. The resultant was then fit with a Lorentzian using a gradient-expansion algorithm to compute a non-linear least squares fit. The fits were all acceptable with [FORMULA] in the range 2.9 to 7.6 for 5 degrees of freedom.

Fig. 2 shows the pulse arrival times calculated via both methods. The errors in Fig. 2 correspond to [FORMULA] (68% confidence). The energy of each point is the median photon energy for each energy band. For the highest energy band, the median energy is substantially lower than the average, 2.9 GeV versus 5.0 GeV. The zero pulse phase is set by the radio ephemeris. The pulse arrival time for all photons used in this analysis is shown as a dashed line and differs by 0.21 ms from the radio zero phase. This is within the error in the radio dispersion correction (Nice 1998). We note that errors in the radio zero phase can broaden the gamma-ray peak, but will not induce an energy dependent shift in the gamma-ray pulse arrival time. The accuracy of the pulse arrival time determination for the Lorentzian fit is 0.07 ms ([FORMULA] or 95% confidence for a single parameter of interest) in the 100-200 MeV band and 0.21 ms (95% confidence) in the highest energy band. The accuracy in the highest energy band is limited mainly by statistics.

[FIGURE] Fig. 2. Pulse arrival time versus energy for the Crab. The diamonds indicate the average arrival time for photons within the main pulse for each energy band. The squares indicate the centroid of a Lorentzian fit to the pulse profile for each energy band. The energies plotted are the median energy for each band; the diamonds are shifted slightly in energy for clarity. The dashed line is the centroid of a Lorentzian fit to the pulse profile for all energies above 70 MeV.

It is apparent from the figure that there is no statistically significant variation in pulse arrival time with energy. To place an upper bound on any energy dependence in the speed of light, we compare the arrival time for photons with energies above 2 GeV (median energy 2.93 GeV) to that for the 70-100 MeV band (median energy 82.8 MeV). The 95% confidence upper limit on the difference of the arrival times is 0.35 ms. Adopting a distance to the Crab of 2.2 kpc (Zombeck 1990), this leads to a lower limit on the energy scale of quantum gravity effects on the speed of light of [FORMULA] (95% confidence). This limit lies below the range of interest, but within an order of magnitude of some predictions in the context of string theory (Witten 1996).

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999