Acta Polytechnica Acta Polytechnica 53(1):5–10, 2013 © Czech Technical University in Prague, 2013 available online at http://ctn.cvut.cz/ap/ ON SOME STATISTICAL PROPERTIES OF GRBS WITH MEASURED REDSHIFTS HAVING PEAKS IN OPTICAL LIGHT CURVES Grigori i Beskina,∗, Giuseppe Grecob, Gor Oganesyanc, Sergey Karpova a Special Astrophysical Observatory of Russian Academy of Sciences b Astronomical Observatory of Bologna, INAF, Italy c South Federal University, Russia ∗ corresponding author: beskin@sao.ru Abstract. We studied the subset of optical light curves of gamma-ray bursts with measured redshifts and well-sampled R band data that have clearly detected peaks. Among 43 such events, 11 are prompt optical peaks (P), coincident with gamma-ray activity, 22 are purely afterglows (A), and 10 more carry the signatures of an underlying activity (A(U)). We studied pair correlations of their gamma-ray and optical parameters, e.g. total energetics, peak optical luminosities, and durations. The main outcome of our study is the detection of source frame correlations between both optical peak luminosity and total energy and the redshift for classes A and A(U), and the absence of such a correlation for class P events. This result seems to provide evidence of the cosmological evolution of a medium around the burst defining class A and A(U) energetics, and the absence of cosmological evolution of the internal properties of GRB engines. We also discuss some other prominent correlations. Keywords: gamma-ray bursts, statistical methods. 1. Introduction The very first time when optical afterglows of GRBs were observed, it was immediately obvious that these phenomena are no mere extinguishing of the energy that powered GRBs, but rather, that they represent a vast wealth of physical phenomena that are not yet fully understood. The aim of our work is to develop a statistical anal- ysis of the prominent optical peaks that characterize several GRBs in the early stage of their emission. These are important for a study of the properties of the interstellar medium over a wide range of distances, and of the physical mechanisms operating during the transition phases between internal/external shock. With this purpose in mind, we discuss the selection criteria and the classification scheme for the GRB sample with a well-measured optical peak. We study the correlations of pairs of parameters of these GRBs, and define possible directions for future work. We adopt the concordance ΛCDM cosmology with ΩM = 0.3, ΩΛ = 0.7, and H0 = 70 km s−1 Mpc−1. 2. Method and classification We gathered all available GRBs with measured red- shifts z and well sampled optical light curves that present a clear peak or peaks during its evolution (over the period from February 28, 1997 until February 28, 2011). Subsequently, the entire optical peak sample was divided into sub-groups that took into account the main characteristics of the contemporaneous emissions at higher energies. In particular, we were interested in comparing the rising phases of optical emission with simultaneous behaviour at gamma-ray wavelengths, as observed by space-borne telescopes. This operational classification leads to the division of our sample into three main sub-groups. • P: optical peaks arise during the main phases of the prompt gamma-ray emission; we will refer to these as prompt optical emission. • A(U): there is still some residual underlying gamma-ray activity simultaneously with the for- mation of the optical peak. • A: no significant gamma-ray activity is seen simul- taneously with the onset of the optical peak • P?: events that may not be unambiguously clas- sified either as A(U) or as P. Three of the P? objects fall into the A(U)-populated regions of the correlation plots. We also use P-3 notation to refer to all prompt events except P? events. The observed optical peak flux Fopt was obtained using the calibration of [1] Fopt = 1568 · ( 2.15 · 10−9 · 10−0.4mag ) erg cm−2 s−1 (1) and was then corrected for galactic extinction (based on the map of [2]) and for the brightness of the host galaxy (if this value is available). For the bursts whose the spectral index β and the host galaxy extinction Ahost are not yet available, we assumed β = 0.75 and 5 http://ctn.cvut.cz/ap/ G. Beskin, G. Greco, G. Oganesyan, S. Karpov Acta Polytechnica the mean value of Ahost measured in the correspond- ing ranges of the redshifts. Namely, the Ahost data collected in the golden sample presented in [3] were divided into five redshift ranges and for each distance interval the corresponding average value of Av was obtained. The isotropic equivalent luminosity Lopt of the op- tical peak is related to the peak optical flux Fopt by Lopt = 4πκopt(z)D2l (z)Fopt, (2) where Dl(z) is the luminosity distance for the standard cosmological model and κopt(z) is the cosmological κ- correction that takes into account the transformation of the R passband to the proper GRB rest frame: κopt = ( νR11+z∫ νR0 1+z ν−βdν )/( νR1∫ νR0 ν−βdν ) = 1 (1 + z)1−β , (3) Here, νR0 and νR1 are the frequency boundaries of the R band and β is the power-law index of the optical spectrum Fν ∝ ν−β. The optical fluence Sopt was determined by numer- ical integration of the afterglow light curve over the interval from the earliest observation to the latest observation with a power-law interpolation of the flux F(t) in the segments between the observational points. Since only a part of the optical afterglow light curve may be recorded in practice, this quantity is actually only a lower limit for the fluence. The isotropic equivalent of the total optical energy in the R band Eopt in the rest frame of the source was determined from the optical fluence Sopt using the relation: Eopt = 4πκopt(z)D2l (z)Sopt 1 + z . (4) Below is a list of quantities we use to characterize the GRBs. The duration of the burst until the emission of 90 % of its energy, topt; the delay of the maximal optical flux after the gamma-ray trigger, tpeak; the width of the optical peak on the 10 % from the maximum, twidth. These quantities are converted to the comoving frame by dividing by (1 + z), in the same way as the total fluences in the optical and gamma ranges (Sγ and Sopt) are converted to the rest frame energies Eiso and Eopt, also taking into account the kappa- correction. The shape of the optical light curves is characterized by the power-law indices of the rise before the peak and decay after the peak, αr and αd. The peak optical and gamma light curve values are characterized by the fluxes Fγ and Fopt, and also by the rest frame isotropic equivalent luminosities Liso and Lopt. Finally, the gamma-ray spectrum is characterized by the photon index α . To measure the relations between different parame- ters we use unweighted Pearson correlation coefficient R, computed for the logarithms of all quantities. The significances of these estimates are computed using Student’ t-distribution for the quantity T = R √ n−2√ 1−R2 , where n is the sample size. 3. Results Table 1 lists all the pair correlations for various classes of optical companions with correlation coefficients greater than 0.5 and significances better than 1 %. We found a strong correlation between the peak op- tical luminosity and the redshift for all object classes except for the prompt class (see the lower right panel of Figure 1). There ls a less obvious, but still sig- nificant, correlation of the optical energetics and the redshift. This may provide evidence for the cosmo- logical evolution of interstellar medium parameters in regions around the progenitors of long gamma-ray bursts. We detected significant connections between ener- gies and peak luminosities, both in the gamma-ray range and in optical range, for all classes of objects (see the upper row of panels in Figure 1), and also between the optical energetics and the peak luminosi- ties and energetics in gamma-rays. It is worth noting that in the latter cases the characteristics of these connections are similar in all classes. This suggests similar jet opening angles in gamma-ray and optical emission regions. In Table 2, we show the parameters of linear fits of quantities whose correlation coefficients are shown in bold in Table 1. A detailed discussion of all detected correlations will be performed in the future. 4. Conclusions Monitoring of robotic telescopes has helped us to understand some characteristics of optical emissions; from the region of peak formation, the gamma-ray emission can proceed in coincidence with the optical luminosity (P), can be extinguished (A), or can carry the remnants of an underlying activity (A(U)). In the case of A or A(U), the peak events that arise when prompt gamma-ray emission is completely extin- guished appear to carry the signatures of the circum- burst environment at different cosmological epochs. In the P type, the optical luminosity of the peak – coin- ciding with a major activity of the prompt gamma-ray emission – seems to deviate from any cosmological evolution trend. This fact provides independent con- firmation that the prompt physical mechanisms are independent of their location in the Universe. How- ever, the mechanisms of the prompt optical emission are still unclear; with the exception of a few rare cases, the morphology of these peaks is complex and poorly re-sampled. 6 vol. 53 no. 1/2013 On Some Statistical Properties of GRBs Acknowledgements This work was supported by Bologna University Progetti Pluriennali 2003, by grants of CRDF (No. RP1-2394-MO- 02), RFBR (No. 04-02-17555, 06-02-08313, 09-02-12053 and 12-02-00743), INTAS (04-78-7366), by the Presid- ium of the Russian Academy of Sciences Program and by a grant of the President of the Russian Federation for the support of young Russian scientists. S.K. has also been supported by a grant from the Dynasty foundation. G.B. thanks Landau Network-Cenro Volta and the Cariplo Foundation for fellowship and Brera Observatory for hos- pitality. G.G. also gratefully acknowledges support from Foundatione CARISBO. References [1] Fukugita, M., Shimasaku, K. and Ichikawa, T.: Galaxy Colors in Various Photometric Band Systems. PASP, 107, 1995, 945. [2] Schlegel, D.J., Finkbeiner, D.P. and Davis, M.: Maps of Dust Infrared Emission for Use in Estimation of Reddening and Cosmic Microwave Background Radiation Foregrounds. ApJ, 500, 1998, 525. [3] Kann, D.A., Klose, S., Zhang, B. et al.: The Afterglows of Swift-era Gamma-ray Bursts. I. Comparing pre-Swift and Swift-era Long/Soft (Type II) GRB Optical Afterglows. ApJ, 720, 2010, 1513–1558. lo g T p ea k lo g T w id th lo g T op t lo g L op t lo g E op t α r α d lo g T 90 lo g L is o lo g E is o α 1 0.23 −0.00 0.01 0.11 0.19 −0.55 0.43 0.19 0.47 0.57 0.12 2 −0.20 −0.26 −0.06 0.82 0.68 0.16 −0.31 0.19 0.73 0.82 0.21 3 z −0.25 −0.64 0.31 0.74 0.54 −0.33 −0.25 0.02 0.46 0.31 0.44 4 −0.03 −0.21 0.25 0.59 0.57 −0.18 −0.06 0.16 0.59 0.60 0.15 5 −0.31 −0.44 0.06 0.82 0.67 0.07 −0.34 0.22 0.72 0.73 0.31 6 0.72 0.22 −0.64 −0.18 0.03 0.10 0.22 0.31 0.22 0.46 −0.15 7 −0.25 −0.35 0.12 0.83 0.69 −0.17 −0.20 0.20 0.74 0.75 0.30 1 0.54 −0.07 −0.77 −0.55 0.12 0.03 0.35 −0.51 −0.36 −0.62 2 0.65 0.10 −0.21 −0.02 −0.02 −0.11 0.46 −0.39 −0.16 −0.13 3 lo g T p ea k 0.74 −0.50 −0.37 0.07 −0.20 0.24 −0.67 −0.12 −0.25 0.30 4 0.76 0.38 −0.39 0.07 −0.19 0.27 0.02 −0.43 −0.31 −0.37 5 0.72 −0.12 −0.35 −0.09 −0.10 0.23 0.10 −0.46 −0.25 −0.15 6 0.64 −0.72 −0.76 −0.54 −0.64 0.20 0.40 −0.36 −0.21 −0.59 7 0.74 −0.03 −0.32 0.01 −0.10 0.20 0.06 −0.40 −0.25 −0.21 1 −0.40 −0.62 −0.78 −0.14 −0.08 −0.52 −0.10 −0.32 −0.05 2 0.03 −0.10 0.03 0.04 −0.08 0.23 −0.20 −0.13 −0.09 3 lo g T w id th −0.51 −0.57 −0.27 −0.24 0.59 −0.58 −0.56 −0.21 −0.12 4 0.22 −0.36 −0.02 −0.20 0.36 −0.16 −0.30 −0.27 −0.29 5 −0.20 −0.34 −0.12 −0.09 0.42 −0.09 −0.35 −0.21 −0.18 6 −0.76 −0.70 −0.81 −0.74 0.10 −0.44 −0.25 −0.52 −0.26 7 −0.11 −0.31 −0.02 −0.13 0.38 −0.13 −0.26 −0.17 −0.21 1 −0.06 0.08 0.19 0.10 0.03 −0.33 −0.27 −0.27 2 −0.07 0.17 −0.25 0.07 −0.05 −0.27 −0.18 0.04 3 lo g T op t 0.32 0.34 0.30 −0.51 0.48 0.33 0.49 0.11 4 −0.04 0.39 −0.19 0.12 −0.06 −0.19 −0.19 −0.28 5 0.07 0.22 −0.09 −0.22 0.08 −0.11 0.00 0.10 6 0.47 0.49 0.34 −0.29 0.22 −0.16 0.08 0.27 7 0.10 0.28 −0.18 −0.16 0.05 −0.04 0.03 0.06 (continued on the next page) 7 G. Beskin, G. Greco, G. Oganesyan, S. Karpov Acta Polytechnica (cont.) lo g T p ea k lo g T w id th lo g T op t lo g L op t lo g E op t α r α d lo g T 90 lo g L is o lo g E is o α 1 0.88 −0.03 0.24 −0.28 0.72 0.74 0.79 2 0.88 0.36 −0.55 0.09 0.77 0.79 0.33 3 lo g L op t 0.76 −0.36 0.09 −0.02 0.69 0.58 0.70 4 0.77 0.12 −0.12 0.10 0.76 0.75 0.51 5 0.85 0.19 −0.30 0.15 0.78 0.76 0.47 6 0.87 0.83 0.14 −0.42 0.74 0.74 0.87 7 0.83 −0.04 −0.19 0.15 0.76 0.76 0.45 1 0.12 0.30 0.10 0.64 0.76 0.47 2 0.22 −0.59 0.29 0.66 0.73 0.54 3 lo g E op t −0.40 0.12 −0.01 0.83 0.69 0.69 4 −0.01 0.02 0.17 0.62 0.61 0.35 5 0.12 −0.30 0.27 0.70 0.74 0.58 6 0.81 0.13 0.00 0.75 0.86 0.54 7 −0.15 −0.16 0.23 0.67 0.70 0.48 1 −0.25 −0.04 −0.32 −0.12 −0.29 2 −0.60 −0.02 0.16 0.20 0.08 3 α r −0.72 0.27 −0.28 −0.20 −0.26 4 −0.45 0.06 −0.04 0.07 0.05 5 −0.56 0.06 0.12 0.13 0.02 6 0.18 −0.01 0.67 0.84 0.79 7 −0.60 0.06 −0.17 −0.09 −0.01 1 −0.04 0.17 0.34 −0.04 2 −0.15 −0.20 −0.23 −0.29 3 α d −0.45 0.03 0.18 0.05 4 −0.28 −0.10 −0.00 −0.23 5 −0.33 −0.22 −0.13 −0.18 6 −0.31 −0.05 0.28 −0.17 7 −0.31 −0.09 −0.04 −0.16 1 −0.40 −0.13 −0.65 2 0.14 0.45 0.28 3 lo g T 90 0.48 0.21 −0.24 4 0.16 0.37 0.14 5 0.25 0.44 0.18 6 −0.48 −0.08 −0.70 7 0.20 0.40 0.17 1 0.89 0.68 2 0.89 0.41 3 lo g L is o 0.70 0.44 4 0.88 0.47 5 0.87 0.44 6 0.82 0.51 7 0.88 0.44 1 0.63 2 0.45 3 lo g E is o 0.56 4 0.53 5 0.50 6 0.45 7 0.51 Table 1. Pair correlation coefficients for rest frame parameters. Those shown in bold have correlation coefficients greater than 0.5 and significance better than 1%. 1: P, 2: A, 3: A(U), 4: P+A+A(U), 5: A(U)+A, 6: P-3, 7: [A(U)+3]+A 8 vol. 53 no. 1/2013 On Some Statistical Properties of GRBs 50,0 50,5 51,0 51,5 52,0 52,5 53,0 53,5 50,5 51,0 51,5 52,0 52,5 53,0 53,5 54,0 54,5 A A(U) P P? Fit Linear A+A(U)+P Is ot ro ph ic e ne rg y of g am m a- ra y bu rs t, er g Isotrophic peak gamma luminosity, erg/s 50,0 50,5 51,0 51,5 52,0 52,5 53,0 53,5 54,0 54,5 46 47 48 49 50 51 52 53 A A(U) P P? Fit Linear A Fit Linear P-3 Is ot ro ph ic o pt ic al e ne rg y, e rg Isotrophic energy of gamma-ray burst, erg 50 51 52 53 46 47 48 49 50 51 52 53 A A(U) P P? Fit A Fit A(U) Is ot ro ph ic o pt ic al e ne rg y, e rg Peak isotrophic gamma luminosity, erg/s 45 46 47 48 49 50 46 47 48 49 50 51 52 53 A A(U) Prompt P? Fit Linear P Fit Linear A Is ot ro ph ic o pt ic al e ne rg y, e rg Peak optical luminosity, erg/s 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 46 47 48 49 50 51 52 53 A A(U) Prompt P? Fit Linear A+A(U)+P? Is ot ro ph ic o pt ic al e ne rg y, e rg log(z+1) 50,5 51,0 51,5 52,0 52,5 53,0 53,5 54,0 54,5 44 45 46 47 48 49 50 A A(U) Prompt P? Fit Linear A+A(U) P ea k op tic al lu m in os ity , e rg /s lsotrophic energy of gamma-ray burst, erg 49,5 50,0 50,5 51,0 51,5 52,0 52,5 53,0 53,5 54,0 44 45 46 47 48 49 50 A A(U) Prompt P? Fit Linear A P ea l o pt ic al lu m in os ity , e rg /s Peak gamma luminosity, erg/s 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 44 45 46 47 48 49 50 A A(U) Prompt P? Fit A+A(U)+P? P ea k op tic al lu m in os ity , e rg /s log(1+z) Figure 1. Scatter plots of pair correlations between various parameters. 9 G. Beskin, G. Greco, G. Oganesyan, S. Karpov Acta Polytechnica C or re la ti on T yp e a b C or re la ti on T yp e a b E is o − L is o P 1. 58 ± 5. 60 0. 99 ± 0. 11 E op t − L is o A 2. 09 ± 12 .0 2 0. 93 ± 0. 23 A 5. 04 ± 6. 78 0. 93 ± 0. 13 A (U ) -2 4. 50 ± 18 .6 3 1. 44 ± 0. 36 A + A (U )+ P 5. 09 ± 4. 31 0. 92 ± 0. 08 A + A (U )+ P 2. 88 ± 9. 55 0. 91 ± 0. 18 A + A (U ) 7. 34 ± 5. 67 0. 88 ± 0. 11 A + A (U ) -0 .3 9 ± 9. 49 0. 97 ± 0. 18 A + A (U )+ P ? 8. 20 ± 4. 91 0. 86 ± 0. 09 A + A (U )+ P ? 2. 80 ± 9. 22 0. 91 ± 0. 18 E op t − L op t P 11 .6 2 ± 7. 14 0. 80 ± 0. 15 E op t − E is o P -2 0. 74 ± 19 .6 9 1. 32 ± 0. 37 A 12 .9 1 ± 4. 57 0. 80 ± 0. 10 A -4 .2 0 ± 11 .4 0 1. 03 ± 0. 22 A (U ) 28 .6 3 ± 0. 43 0. 46 ± 0. 14 A + A (U )+ P -1 .2 8 ± 10 .3 1 0. 97 ± 0. 20 A + A (U )+ P 17 .4 5 ± 4. 89 0. 69 ± 0. 09 A + A (U ) -3 .0 3 ± 8. 91 1. 01 ± 0. 17 A + A (U ) 17 .7 6 ± 3. 65 0. 69 ± 0. 08 P -3 -4 0. 95 ± 21 .4 9 1. 70 ± 0. 41 P -3 9. 87 ± 8. 85 0. 83 ± 0. 19 A + A (U )+ P ? -1 .8 ± 9. 16 0. 98 ± 0. 17 A + A (U )+ P ? 17 .2 6 ± 3. 82 0. 70 ± 0. 08 E op t − (z + 1) A 47 .9 0 ± 0. 51 4. 26 ± 1. 03 L op t − (z + 1) A 43 .9 2 ± 0. 42 5. 41 ± 0. 95 A + A (U ) 48 .0 9 ± 0. 41 3. 89 ± 0. 78 A + A (U )+ P 45 .0 3 ± 0. 55 3. 66 ± 1. 17 A + A (U )+ P 47 .8 1 ± 0. 46 4. 13 ± 0. 93 A + A (U ) 44 .0 3 ± 0. 39 5. 31 ± 0. 79 A + A (U )+ P ? 47 .9 2 ± 0. 39 4. 11 ± 0. 75 A + A (U )+ P ? 44 .0 5 ± 0. 35 5. 32 ± 0. 73 E is o − (z + 1) A 50 .8 4 ± 0. 28 3. 49 ± 0. 58 L is o − (z + 1) A 50 .2 8 ± 0. 36 2. 86 ± 0. 77 A + A (U )+ P 51 .7 3 ± 0. 31 2. 17 ± 0. 61 A + A (U )+ P 50 .8 2 ± 0. 32 2. 11 ± 0. 66 A + A (U ) 51 .0 9 ± 0. 28 2. 98 ± 0. 52 A + A (U ) 50 .3 9 ± 0. 29 2. 66 ± 0. 56 A + A (U )+ P ? 51 .0 9 ± 0. 26 3. 01 ± 0. 49 A + A (U )+ P ? 50 .3 0 ± 0. 28 2. 82 ± 0. 55 E op t − T w id th P 53 .4 9 ± 1. 22 -3 .9 4 ± 1. 04 L op t − T p e a k P 52 .7 3 ± 2. 03 -3 .8 6 ± 1. 24 P -3 54 .2 8 ± 1. 61 -4 .4 5 ± 1. 32 L op t − E is o P -3 2. 52 ± 24 .0 3 1. 50 ± 0. 45 L op t − L is o A -2 7. 03 ± 11 .9 5 1. 43 ± 0. 23 A -3 2. 3 ± 13 .2 1. 50 ± 0. 25 A + A (U )+ P -2 9. 88 ± 10 .0 1 1. 49 ± 0. 19 A + A (U )+ P -3 0. 16 ± 11 .9 5 1. 46 ± 0. 23 A + A (U ) -3 7. 96 ± 9. 85 1. 65 ± 0. 19 A + A (U ) -4 4. 11 ± 11 .6 0 1. 73 ± 0. 22 A + A (U )+ P ? -3 2. 90 ± 9. 65 1. 55 ± 0. 19 A + A (U )+ P ? -4 1. 69 ± 11 .1 4 1. 69 ± 0. 21 T p ea k − T w id th A 1. 24 ± 0. 24 0. 51 ± 0. 13 E op t − (z + 1) A 47 .9 0 ± 0. 51 4. 26 ± 1. 03 A + A (U )+ P 1. 00 ± 0. 13 0. 59 ± 0. 08 A + A (U ) 48 .0 9 ± 0. 41 3. 89 ± 0. 78 A + A (U ) 1. 22 ± 0. 16 0. 50 ± 0. 09 A + A (U )+ P 47 .8 1 ± 0. 46 4. 13 ± 0. 93 A + A (U )+ P ? 1. 16 ± 0. 14 0. 53 ± 0. 08 A + A (U )+ P ? 47 .9 2 ± 0. 39 4. 11 ± 0. 75 *E op t − α d ec ay A 48 .8 9 ± 0. 36 -1 .0 1 ± 0. 32 *L op t − α d ec ay A 46 .1 1 ± 0. 46 -0 .8 8 ± 0. 35 Table 2. Pair correlations for various classes of optical companions with correlation coefficients greater than 0.5 and significances less than 1%. Four columns are linear regression (a + b ·x) coefficients, derived through unweighted least squares. The stars mark the log-linear correlations, in contrast to the log-log correlations used otherwise. 10 Acta Polytechnica 53(1):5--10, 2013 1 Introduction 2 Method and classification 3 Results 4 Conclusions Acknowledgements References