A Simple Method To Test For Energy-Dependent Dispersion In High Energy Light Curves Of Astrophysical Sources
aa r X i v : . [ a s t r o - ph . I M ] O c t A Simple Method To Test For Energy-DependentDispersion In High Energy Light Curves OfAstrophysical Sources
M. K. Daniel ∗ and U. Barres de Almeida † ∗ Department of Physics, University of Durham, Durham, DH1 3LE. U.K. † Max-Planck-Institut für Physik, D-80805, München, Deutschland.
Abstract.
We present a method of testing for the presence of energy dependent dispersion intransient features of a light curve. It is based on minimising the Kolmogorov distance betweentwo cumulative event distribution functions. The unbinned and non-parametric nature of the testmakes it particularly suitable for searches of statistically limited data sets and we also show thatit performs well in the presence of modest energy resolutions typical of gamma-ray observations( ∼ Keywords: time series analysis
PACS:
INTRODUCTION
Timing analysis algorithms with the capability of resolving energy dependent propertiescan be an important tool for probing the physical mechanisms leading to flux variability,such as particle acceleration and cooling [1], or the nature of a propagating medium [2].In the case of very high energy gamma-ray sources, where high energy processes canbe responsible for extreme and short-lived variability events, the observational data areoften limited by low photon statistics and non-negligible uncertainty in the reconstructedenergy of indivdual events. This makes unbinned methods, which act on the informationof the entire available sample, the natural and preferential choice of approach to thetemporal analysis of these event lists.
METHOD
If the low ( L ) and high ( H ) energy particles are generated in the same region then theymust be able to exist co-spatially. The act of acceleration, or cooling, or moving througha dispersive medium will act to separate the L and H populations relative to each other.An energy dependent correction factor ( t ) can be applied to the event arrival times ( t i ) d t i = − t E a i (1)where d t is the difference in arrival time with and without dispersion, E i is the energyof the event and a is the scale of the correction (1 for linear, 2 for quadratic, etc). Byycling through a range of correction factors we can determine the one ( t ∗ ) where theshape of the H light-curve best fits that of the L one, here we use the Kolmogorovdistance between the cumulative distribution function (CDF) of the event arrival times,as seen in figure 1. Time [arb.]0 100 200 300 400 500 600 R a t e [ a r b .] LHL+H a) profiles at source
Time [arb.]0 100 200 300 400 500 60000.10.20.30.40.50.60.70.80.91 b) CDFs at source
Time [arb.]0 100 200 300 400 500 600 R a t e [ a r b .] L HL+H c) profiles dispersed
Time [arb.]0 100 200 300 400 500 60000.10.20.30.40.50.60.70.80.91 d) CDFs dispersed k D FIGURE 1.
Cartoon of the effect of the energy dependent dispersion on the shape of the low (L) andhigh (H) energy profiles. The panels on the left show the shape of the lightcurve and the panels on the rightthe event CDFs. The top plots show are the intrinsic (at source) shape and the bottom after propagation.
Simulating 10,000 lightcurves shows the Kolmogorov test always has a well definedminimum, with the difference between the expected and best correction ( t − t ∗ ) well fitby a Gaussian 2. The RMS of the fit is dependent on the width of the light-curve, butrelatively insensitive to the rise and fall times or the number of events contained within,provided there are ≥
10 events in the H sample. It is also relatively insensitive to theenergy binning provided the E H ≥ E L .We quantify the sensitivity to the burst width by the term sensitivity factor, h definedas h = d t D t (2)here D t is the width of the transient feature in the light-curve. In figure 3 we simulated10,000 Gaussian burst profiles of 500 events each and a power-law spectral index of-2.5. A dispersion was introduced that varied from 5-200% of the burst width. Wesee, as expected, that the narrower a burst relative to the dispersion the better it canbe determined. Also plotted in figure 3 are the results of varying energy resolutions( | D E / E | ) from ideal (0%, 10% and 20%). There is a small systematic trend for thereconstructed lag to be underestimated as the energy resolution worsens, again this isto be expected, but this is very small in comparison to the overall statistical error in t ∗ showing the method is robust to the modest energy resolutions expected in ground basedgamma-ray astronomy. It is possible to overcome this systematic trend with appropriateMonte Carlo modelling or bootstrapping, if necessary. DISCUSSION & CONCLUSIONS
We have presented a simple method to perform an unbinned, non-parametric energydependent timing analysis of data with low statistics and moderate energy resolutions.Further details of the performance of the method can be found in [3], simulations ofcurrent generation VHE gamma-ray instrument observations of AGN show the methodto be comparable in sensitivity to the more sophisticated analyses which have to makegreater assumptions on the instrinsic source physics and instrument response functions.The placing of Planck scale limits on the linear term in Lorentz invariance violation dueto quantum gravity models could be achievable in observations by the next generationinstrument CTA [4].
REFERENCES
1. W. Bednarek & R. M. Wagner,
A&A , 679 (2008). ] −1 [ s TeV * t −200 −150 −100 −50 0 50 100 150 200 nu m b e r o f t i m es -150-100-50 0 50 100 150-150 -100 -50 0 50 100 150 t * [ s / T e V ] t [s/TeV]PKS 2155 M QG =M Pl t =4.26 s/TeVBF1BF2 BF3BF4 BF5 t = t * FIGURE 2.
Performance of the method for recovering dispersion. The left plot shows the error in thebest estimate is well fit by a Gaussian; the right plot shows the accuracy to which the estimated dispersionmatches the actual simulated dispersion (see text for details). . . . . . [h] sensitivity factor r e c on s t r u c t ed l ag [ a r b .] FIGURE 3.
Sensitivity h of the algorithm for 0% (open circle), 10% (open square) and 20% (opentriangle) energy resolution.2. G. Amelino-Camelia, et al. Nature , 525 (1998).3. U. Barres de Almeida & M. K. Daniel
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