5.2.1 2D Fitting Technique
5.3 Energy Spectrum
Key information concerning the acceleration mechanisms at work in the binary system of PSR B1259−63 is contained in the energy spectrum of the detected VHE γ-ray emission. The differential energy spectrum, or differential flux of the sourceF(E) is defined as the number of particles from the source detected per unit area, time, and energy. Thus, the spectrum can be expressed as:
wheretlive is the live time,Nγ is the measured number of excess events, andAeff is the effective collection area of the instrument. The latter quantity cannot be measured and must be deter-mined using simulatedγ-rays.
This section first describes the determination of the effective collection area of the detector.
Subsequently, the method to reconstruct an energy spectrum is introduced and tested with data from the Crab Nebula, and finally the spectra for the PSR B1259−63 data set are given.
5.3.1 E ff ective Area and Energy Threshold for γ-Rays
The effective area represents that area, within which the detector could detect a flux ofγ-rays as-suming a detection efficiency of 100%. This quantity depends on aspects of the shower physics and the detector and cannot be derived analytically but rather must be obtained from simula-tions. Considering a simulated point source ofγ-rays (see Sec. 3.3), the effective area is given by:
Aeff(E,Θ, φ, θcam)= 2π
∞
Z
0
P(E,Θ, φ, θcam,R)R dR, (5.4) whereRis the distance from the shower core to the centre of the array, andPis the probability of a γ-ray shower to be detected and pass the event selection. P reflects the efficiency of the array to collect the Cherenkov emission of a shower and therefore depends on
. E, the shower energy, proportional to the intensity of the Cherenkov emission,
. Θ, the zenith angle, determining the distance between detector and shower maximum, . φ, the azimuth of the telescope pointing and therefore the orientation of the shower with
respect to the Earth’s magnetic field,
. θcam, the distance from the shower direction to the camera centre (offset), reflecting the vari-ation of the system acceptance.
At a large distance R = R0, the corresponding detection probability becomes negligibly small and only showers with core locations within a circle of this radiusR0need to be considered.
For a sample of simulated γ-rays, the probability P is simply the ratio of the number of simulatedγ-rays to those triggering the detector, and Eq. (5.4) simplifies to:
Aeff(∆E,Θ, φ, θcam)=2π πR20 n(∆E,Θ, φ, θcam) N(∆E,Θ, φ, θcam),
representing the effective area for an energy interval∆E, withNas the total number of simulated γ-rays, andnthe number of events remaining after the detector simulation and selection cuts.
Figure 5.11 shows the calculated effective area after selection cuts as a function of the true and reconstructed shower energy (left and right, respectively) for typical parameters regarding the observations of PSR B1259−63. The area steeply rises at low energies towards a turning point and then increases slowly until it reaches its maximum at several TeV and declines towards higher energies. At low energiesAeff is limited by the amount of Cherenkov light collected with the detector while at high energies, the Cherenkov images become too large to fit into the camera and are rejected by the cut on the local distance of an image.
The difference between the effective area versus the true and reconstructed energy arises from the bias in the energy reconstruction (c.f. Fig. 4.21). In order to compensate for this bias, for the energy spectrum determination method the effective area was used as a function of the reconstructed energy. Note, that this correction depends on the spectrum of simulated γ-rays (see next section).
Figure 5.12 illustrates the dependence ofAeff on the zenith angle (left) and the camera offset (right). For all possible combinations of Θ = 0,30,40,45,50,55◦, φ = 0,180◦, and θcam = 0,0.5, . . . ,2.5◦,Aeff(∆E) was computed and fitted with an analytical function in order to reduce uncertainties resulting from the finite binning and statistical fluctuations.
[TeV]
Effective Area vs. Reconstructed Energy
Figure 5.11: Effective areas Aeff for simulatedγ-rays forΘ = 40◦,Θ =180◦, andθcam =0.5◦(points) and the corresponding fitted function (solid line). Left: Effective area as a function of simulated (true) energy.Right:Effective area as a function of reconstructed shower energy.
0
Effective Area for Different Zenith AnglesΘ
[TeV]
Effective Area for Different Offsets
Figure 5.12: Effective areas Aefffor simulatedγ-rays for different zenith angles and offsets as a function of the true energy. Left:Aeff for a range of zenith angle values. With increasing zenith angle, the energy threshold rises, and the collection area increases for energies above the threshold.Right:Aefffor several camera offsets. While the area does not change significantly for offsets< 1◦, the collection efficiency drops dramatically towards the camera edge.
In order to assign an effective area to a single event, Aeff(E,Θ, φ, θcam) is derived by eval-uating the fit function and interpolating linearly in cosΘ and θcam. Since the dependence on the azimuth angle φ is rather weak and most of the observation time of PSR B1259−63 and the Crab Nebula was spent close to their culmination, the azimuth corresponding to that of the target culmination was used.
Figure 5.13: Expected differential rate of γ-rays for a Crab-like spectrum using the effective area curve appro-priate for PSR B1259−63 observations.
The energy threshold Eth is de-fined as the maximum of the expected differential rate of excess events
dR
dE = Aeff dNγ
dE
with dNγ/dE ∝ E−2.62, as measured for the Crab Nebula by the HEGRA collaboration [2004]. Figure 5.13 shows this differential rate for the ef-fective area curve for the mean zenith angle and camera offset of the obser-vations of PSR B1259−63. The re-sulting threshold ofE1259th ≈ 380 GeV represents the mean threshold energy for this data set.
5.3.2 Energy Spectrum Determination
Using Eq. (5.4), a source spectrum can be derived from the number of excess events Nγ and the effective area Aeff in an energy interval dE. Since the energy of the excess events is not a directly measured quantity in the sense that the sample of events from the signal region also contains background events, the flux of the on and offevents must be calculated separately and the difference yields the flux of signal events.
The considered energy range was divided into finite energy bins∆E, and for each observa-tion run an energy threshold Esafe was determined, defined as the lowest energy, for which the energy bias
|Etrue−Ereco| Etrue
(hΘi,hθcami),
dependent on the mean zenith angle and offset of the run, is below 10%. Esafe is typically 15–30% higher than the energy thresholdEth.
The flux for each energy bin was determined by using the on and offevents obtained with the 7-background model. For each energy bin∆E, the flux is given by
F(∆E)= 1
as the effective area for the particular event. Only events for which∆E lies aboveEsafe for the particular run were considered and thus the normalisation αand live timetlive can be different for a given energy bin.
In order to obtain an appropriate bin coverage within the energy range, the initial finely binned distribution was re-binned according to the signal strength. For each bin, a spectral point (E,F(E))∆E was assigned, with E as the mean energy of the energy distribution within the bin and F(E) = F(∆E)/∆E. All points in the spectrum were required to have a maximum relative error∆F/F =1, and a flux above zero.
The resulting energy spectrum was fitted with a power law F(E)= dN/dE =F0
E 1 TeV
−Γ
. (5.6)
Since the reconstructed spectrum depends on the spectral index of the simulatedγ-rays used for the determination of the effective area,Aeffwas recalculated using the same sample of simulated γ-rays, but weighting the distribution in energy such that it resembles a power law with an index matchingΓobtained from the fit to the spectrum above. Subsequently, the spectrum was redetermined with the modified effective areaA0eff in a second iteration. A third iteration always yielded a result which did not change noticeably compared to the second iteration.
Validation with Data from the Crab Nebula
The method for the determination of the energy spectrum was validated with data from the Crab Nebula, since its spectrum has been measured by nearly all 1st generation IACT instruments [e.g. in Hillas et al., 1998, HEGRA collaboration, 2004] and thus the results can be compared with these measurements.
The observations of the Crab Nebula were performed in late 2003 with wobble offsets of
±0.5◦ and±1◦ Dec and yielded a data set of roughly 4 hours live time after quality selection cuts (for a list of runs see Appendix B). The energy threshold of the observations is similar to that of the PSR B1259−63 data at≈400 GeV.
Although the γ-ray selection cuts were optimised for faint sources of 0.1 Crab, the cuts were not changed for the spectral analysis of the Crab data in order to test the validity of the analysis. The reconstructed spectrum should not depend on the event selection used. Figure 5.14 shows the differential energy spectrum for the Crab Nebula data set. The spectrum is well fit (χ2/ndf = 16.3/18) by a power law with a photon indexΓ = 2.63±0.05statand normalisation F0 = (3.25± 0.13stat)× 10−11cm−2s−1TeV−1. This result is in reasonable agreement with the results obtained by:
. Hillas et al. [1998]
Γ =2.49±0.06stat±0.04sys,
F0 = (3.20±0.17stat±0.60sys)×10−12cm−2s−1TeV−1, . HEGRA collaboration [2004]
Γ =2.62±0.02stat±0.05sys,
F0 = (2.83±0.17stat±0.60sys)×10−11cm−2s−1TeV−1.
Energy [TeV]
1 10
−1−2−1
dN/dE [TeV cm s ]
10−15 10−14 10−13 10−12 10−11 10−10 10−9
/ ndf
χ2 16.3 / 18
Prob 0.573
Norm 3.25e−11 ± 1.27e−12 Index −2.63 ± 0.0494
−11 −2 −1