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Search for Pulsed TeV Gamma-ray Emission from the Crab Pulsar

We present the results of a search for pulsed TeV emission from the Crab pulsar using the Whipple Observatory's 10 m gamma-ray telescope. The direction of the Crab pulsar was observed for a total of 73.4 hours between 1994 November and 1997 March. During this period the Whipple 10 m telescope was operated at its lowest energy threshold to date. Spectral analysis techniques were applied to search for the presence of a gamma-ray signal from the Crab pulsar over the energy band 250 GeV to 4 TeV. We do not see any evidence of the 33 ms pulsations present in other energy bands from the Crab pulsar. The 99.9% confidence level upper limit for pulsed emission above 250 GeV is derived to be 4.8x10^-12 cm^-2 s^-1 or<3% of the steady flux from the Crab Nebula. These results imply a sharp cut-off of the power-law spectrum seen by the EGRET instrument on the Compton Gamma-Ray Observatory. If the cut-off is exponential, it must begin at 60 GeV or lower to accommodate these upper limits.


Introduction
The Crab pulsar/Nebula system is one of the most intensely studied astrophysical sources with measurements throughout the electromagnetic spectrum from the radio to the TeV energy band. In most regions of the spectrum, the characteristic 33 ms pulsations of the pulsar are clearly visible. The pulse profile is unique amongst known pulsars in that it is aligned from radio to gamma-ray energies. The study of the pulsed emission in different energy ranges is of considerable importance to understanding the underlying emission mechanisms (e.g., Eikenberry & Fazio 1997). The EGRET instrument on the Compton Gamma-Ray Observatory (CGRO) has shown that there is pulsed gamma-ray emission from the pulsar up to at least 10 GeV (Ramanamurthy et al. 1995). Current imaging atmospheric Cherenkov telescopes have firmly established the Crab Nebula as a steady source of gamma rays from 300 GeV to 50 TeV (Hillas et al. 1998;Tanimori et al. 1998). However, these observations have not detected any significant modulation of this TeV signal at the period of the pulsar. In contrast to these reports, other groups have reported TeV emission modulated at the 33 ms period of the Crab pulsar. Some of these reports have been associated with episodic activity (Gibson et al. 1982;Bhat et al. 1986;Acharya et al. 1992). A persistent pulsed signal from the Crab pulsar was reported by the Durham group (Dowthwaite et al. 1984). However this has not been confirmed by more sensitive observations which show that less than 5% of the total very high energy (VHE) flux is pulsed (Weekes et al. 1989;Reynolds et al. 1993;Goret et al. 1993). At ultra-high energies, the CASA-MIA experiment does not find any statistically significant evidence for pulsed gamma-ray emission at the Crab pulsar period, on an interval of one day or longer, based on the analysis of data recorded during the interval 1990 March to 1995 October (Borione et al. 1997).
Pulsed emission from the Crab pulsar at IR energies and above is generally believed to originate in the magnetosphere of the system far from the stellar surface. In each of the two models which address the pulsed gamma-ray emission in detail, the outer gap model (Cheng, Ho & Ruderman 1986;Romani 1996) and the polar cap model (Daugherty & Harding 1982), the high energy flux arises from curvature radiation of pairs as they propagate along the open field lines of the magnetosphere. The specific details of the pulse shapes in different pulsars are explained by the line of sight geometry of the observer relative to the spin and magnetic axes of the rotating neutron star in these models. The energy at which the pulsed flux begins to cut-off and the detailed spectral shape of the cut-off can help to distinguish between the two models. Given the detection of pulsations out to 10 GeV by EGRET (Ramanamurthy et al. 1995) and the restrictive upper limits above 300 GeV (Weekes et al. 1989;Reynolds et al. 1993;Goret et al. 1993), the cut-off necessarily resides in the ∼100 GeV energy range. This is our primary motivation for this deep search for pulsations from the Crab in the 100 GeV range. The outer gap model by Romani 1996 also includes TeV emission via the synchrotronself-Compton mechanism which produces a peak spectral energy density above 1 TeV. Such a mechanism could in principle explain the detection of pulsed emission by the Durham group, which operates at an energy threshold of 1 TeV, and still be consistent with the upper limits reported at lower energies. For this reason we have applied spectral analysis techniques to search for a gamma-ray Crab pulsar signal over the energy band 250 GeV to 4 TeV.

Observation and Analysis Techniques
The VHE observations reported in this paper utilize the atmospheric Cherenkov technique (Cawley & Weekes 1995) and the 10 m optical reflector located at the Whipple Observatory on Mt. Hopkins in southern Arizona (elevation 2.3 km) (Cawley et al. 1990). A camera, consisting of photomultiplier tubes (PMTs) mounted in the focal plane of the reflector, detects the Cherenkov radiation produced by gamma-ray and cosmic-ray air showers from which an image of the Cherenkov light can be reconstructed. For most of the observations reported here, the camera consisted of 109 PMTs (each viewing a circular field of 0. • 259 radius) with a total field of view of 3 • in diameter. In 1996 December, 42 additional PMTs were added to the camera, increasing the field of view to 3. • 3.
We characterize each Cherenkov image using a moment analysis (Reynolds et al. 1993). The roughly elliptical shape of the image is described by the length and width parameters and its location and orientation within the field of view are given by the distance and α parameters, respectively. We also determine the two highest signals recorded by the PMTs (max1, max2) and the amount of light in the image (size). These parameters are defined in Table 1 and are depicted in Figure 1. Gamma-ray events give rise to more compact shower images than background hadronic showers and are preferentially oriented towards the putative source position in the image plane. By making use of these differences, a gamma-ray signal can be extracted from the large background of hadronic showers.

Selection Methods
The standard gamma-ray selection method utilized by the Whipple Collaboration is the Supercuts criteria (see Table 2; cf., Reynolds et al. 1993;Catanese et al. 1996). These criteria were optimized on contemporaneous Crab Nebula data to give the best sensitivity to point sources. In an effort to remove the background of events triggered by single muons and night sky fluctuations, Supercuts incorporates pre-selection cuts on the size and on max1 and max2. While the introduction of a pre-selection is desirable from the point of view of optimizing overall sensitivity, it automatically rejects many showers below ∼ 400 GeV. In the context of a search for pulsed emission from the Crab pulsar, which must have a low energy cut-off to accommodate existing upper limits, this is clearly undesirable. Accordingly, a modified set of cuts (Table 3; cf., Moriarty et al. 1997), developed to provide optimal sensitivity in the ∼ 200 GeV to ∼ 400 GeV region and referred to hereafter as Smallcuts, was used for the events which failed the Supercuts pre-selection criteria. The most notable difference between Smallcuts and Supercuts is the introduction of a cut on the length/size of an image. Such a cut is effective at discriminating partial arcs of Cherenkov light rings arising from single muons, which become the predominant background at lower energies. These images tend to be long compared to their intensity and so may be rejected on the basis of the length/size ratio. When a combination of Supercuts and Smallcuts is used, Monte Carlo simulations indicate that this analysis results in an energy threshold of ∼ 250 GeV. This threshold is the energy at which the differential rate from a source with a spectral index equal to that of the steady Crab Nebula reaches its peak. The collection area as a function of gamma-ray energy is depicted in Figure 2 and results in an effective collection area of 2.7 × 10 8 cm 2 . Details of the methods used to estimate the energy threshold and effective area are given elsewhere (Mohanty et al. 1998).
The data from 1997 were analyzed with slightly modified cuts (see Tables 2,3) which were re-optimized after an upgrade to the Whipple camera which increased the field of view. The greatest effect of the larger field of view was that images appeared longer and at a greater distance from the center of the field of view due to less image truncation than caused by the smaller camera.
Supercuts was optimized to give the best point source sensitivity but in doing so it rejects many of the larger gamma-ray events. Another selection process, known as Extended Supercuts (Table 4; cf., Mohanty et al. 1998), was utilized to facilitate a search for pulsed emission over the energy band 250 GeV to 4 TeV. This method is quite similar to Supercuts but scales the various cuts with the shower size and retains approximately 95% of gamma-ray events compared to approximately 50% of gamma-ray events passed by the Supercuts criteria. By applying a lower bound on the size of an image, the energy threshold of the analysis increases. Figure 2 depicts the collection area as a function of gamma-ray energy as derived by Monte Carlo simulations for a lower bound on the size of an image of 500, 1000, 2000 and 5000 digital counts. These cuts impose energy thresholds of 0.6, 1.0, 2.0 and 4.0 TeV respectively.

Periodic Analysis
The arrival times of the Cherenkov events were registered by a GPS clock with an absolute resolution of 250 µs. An oscillator, calibrated by GPS second marks (relative resolution of 100 ns), was used to interpolate to a resolution of 0.1 µs. After an oscillator calibration was applied, all arrival times were transformed to the solar system barycenter by utilizing the JPL DE200 ephemeris as described by Standish (1982). As the acceleration of the pulsar relative to the solar system barycenter is negligible, the only additional correction factor is due to the gravitational redshift. The conversion of the coordinated universal time (UTC) as measured at the telescope, to the solar system barycenter arrival time (TDB), is given by The international atomic time (TAI) differs from UTC time by an integral number of leap seconds. The terrestrial dynamical time (TDT) is used as a timescale of ephemerides for observations from the Earth's surface and differs from TAI by 32.184 s. The correction to the Earth's surface requires the telescope's geocentric coordinates and a model of the Earth's motion. The final correction applied, ∆ REL , accounts for the variation of the gravitational potential around the Earth's orbit.
The corrected times were folded to produce the phases, φ j , of the events modulo the pulse period according to where ν,ν are the frequency and first frequency derivative at the epoch of observation t 0 . For each source run the valid frequency parameters were derived from the J2000 ephemeris obtained from Jodrell Bank where the Crab pulsar is monitored on a monthly basis.
To check the Whipple Observatory timing systems an optical observation of the Crab pulsar was undertaken on the nights of 1996 December 2 (UT), 1996 December 18 (UT) and 1997 March 11 (UT), using the 10 m reflector with a photometer at its focus (Srinivasan et al. 1997). The signal from the photometer was recorded by the data acquisition electronics and timing system of the telescope thereby providing a direct test of the instrument's timing characteristics. The phase analysis of the event arrival times, depicted in Figure 3, yielded a clear detection of the optical signal from the Crab pulsar in phase with the radio pulse. This demonstrates the validity of the timing, data acquisition and barycentering software in the presence of a pulsed signal.

Observations and Results
The position of the Crab pulsar was observed between 1995 January and 1997 March. The traditional mode of observing potential periodic sources with the Whipple Observatory gamma-ray telescope is to track the putative source location continuously for runs of 28 minute duration. After filtering runs for bad weather and instrumental problems, the data set consists of 159 runs for a total source observing time of 73.4 hrs. The radio position (J2000) of the Crab pulsar (α = 05 h 34 m 31.949 s , δ= 22 • 00 ′ 52.057 ′′ ) was assumed for the subsequent timing analysis.
The numbers of events passing the selection criteria described above are given in Table 5. The phases of these events, shown in Figure 4, are used for periodic analysis. We find no evidence of pulsed emission at the radio period. To calculate upper limits for pulsed emission we have used the pulse profile seen at lower energies by EGRET. That is, we assume emission occurs within the phase ranges of both the main pulse, phase 0.94-0.04, and the intrapulse, phase 0.32-0.43 (Fierro et al. 1998). The number of events with phases within these intervals constitutes the number of candidate pulsed events, N on . N of f , an estimate of the numbers of background events, is obtained by multiplying the number of events with phases outside these pulse intervals by the ratio of ranges spanned by the pulse and non-pulse regions. The results are given in Table 5. The statistical significance of the excess is calculated using the maximum likelihood method of Li & Ma (1983). The 99.9% confidence level upper limits calculated using the method of Helene (1983) are given in Table 6.
Several reports of pulsed emission from the Crab pulsar at very high energies claim to have seen evidence of episodic emission on time scales of several minutes. For this reason we have performed a run-by-run search for periodic emission from the Crab pulsar based on the above pulse profile. The statistical significance of excess events for each observation and the corresponding distribution of significance for the lowest and middle energy ranges are given in Figure 5. In each energy band the distribution of significance is consist with the statistical expectation for zero excess.

Discussion
Data taken with the Whipple Observatory's 10 m gamma-ray telescope have been used to search for pulsations from the Crab pulsar above 250 GeV. We find no evidence of pulsed emission at the radio period and upper limits on the integral flux have been given.
To model the pulsed gamma-ray spectrum, a function of the form was used, where E is the photon energy, γ is the photon spectral index and E o is the cut-off energy. The source spectrum in the EGRET energy range is well fitted by a power law with a photon spectral index of −2.15 ± 0.04 (Nolan et al. 1993). The pulsed upper limit above 250 GeV reported here is ∼ 3 orders of magnitude below the flux predicted by the EGRET power law. Equation 3 was used to extrapolate the EGRET spectrum to higher energies constrained by the TeV upper limit reported here and indicates a cut-off energy E o ≤ 60 GeV for pulsed emission (see Figure 6).
As indicated in § 2.1, the energy threshold of the technique is derived assuming a source with a spectral index equal to that of the steady Crab Nebula. With the above model, this assumption is invalid. If we assume a source spectrum as given by Equation 3 and define energy threshold and effective collection area as stated in § 2.1 we simultaneously solve for an energy threshold of 180 GeV and energy cut-off of 60 GeV. The derived cut-off energy is the same as that obtained assuming a Crab Nebula spectrum, and indicates the robustness of defining the energy threshold of the technique in this way.
The sharpness of the spectral cut-off of the emission models depicted in Figure 6 provides a good discriminant. The status of current observations and the derived cut-off given above indicates that the cut-off must lie in the 10-60 GeV range. However, the upper limits reported here are well above the flux predicted by the polar cap and outer gap models and offer no discrimination between them. In contrast, the outer gap model of Romani (1996) predicts TeV emission via the synchrotron-self-Compton mechanism. The flux produced via this mechanism is dependent on the density and spectrum of primary electrons and positrons in the gap, as well as the density of local soft photon fields. The predicted pulsed TeV flux for a young gamma-ray pulsar is somewhat less than 1% of the pulsed GeV flux. The results reported here derive an upper limit to this fraction of < 0.07%. Fig. 1.-Images of Cherenkov light produced by gamma-ray and cosmic-ray induced air showers are parameterized using a moment analysis to describe the image shape and orientation in the image plane. Fig. 2.-The collection areas for Supercuts combined with Smallcuts (SC+SM) and Extended Supercuts (ESC) combined with a lower bound on image size, which selects only the higher energy events.  The χ 2 probability that each distribution is consistent with it's mean is given in each panel. We find no evidence of pulsed emission at the radio period.   Nolan et al. 1993. The thin solid line is the polar cap model fit to the EGRET data (Harding 1999). The dotted line is the outer gap model for the Vela pulsar (scaled to match the EGRET Crab pulsar flux at peak intensity) and is included to indicate the shape of the cut-off this model predicts (Romani 1999). The dashed line represents the power-law fit to the EGRET data (Nolan et al. 1993). The dot-dashed line represents Equation 3 with a cut-off energy E o = 60 GeV. The upper limits for pulsed emission presented in this paper are represented by the open squares. The thick solid curve depicts the model of unpulsed GeV -TeV emission from the Crab Nebula (Hillas et al. 1998). Table 1. Definition of the image parameters, which are used to characterize the image shape and orientation (see Figure 1).
parameter definition max1: largest signal recorded by the PMTs max2: second largest signal recorded by the PMTs size: sum of all signals recorded width: the root mean square (RMS) spread of light along the minor axis of the image; a measure of the lateral development of the shower. length: the RMS spread of light along the major axis of the image; a measure of the vertical development of the shower. distance: the distance from the centroid of the image to the center of the field of view. α: the angle between the major axis of the image and a line joining the centroid of the image to the center of the field of view.