We present preliminary data fit results of synthetic light curves computed
from numerical afterglow blast wave simulations. Our technique uses Markov
chain Monte Carlo (MCMC) in a new data analysis tool, ScaleFit. Scaling
relations in both the hydrodynamics and radiation equations allow light curves
to be parameterized by a small set of scale-invariant characteristic
quantities. These quantities have been calculated and tabulated from high
resolution two-dimensional hydrodynamic simulations. Producing a light curve
from the characteristics takes only a millisecond, allowing for the use of MCMC
data fitting techniques which can require millions of iterations. ScaleFit is a
portable, lightweight, python package which performs this analysis on afterglow
light curves. Using the set of Swift-XRT light curves from 2011 & 2012 with
known redshifts, we find ScaleFit can measure the jet opening angle, observer
angle, and spectral index of most afterglows. Globally we find gamma-ray burst
afterglows tend to be observed off axis, at a significant fraction of the jet
opening angle.Comment: 8 pages, 3 figures. 7th Huntsville Gamma-Ray Burst Symposium, GRB
  2013: paper #30 in eConf Proceedings C130414

MacFadyen, Andrew

Ryan, Geoffrey

van Eerten, Hendrik

English

arXiv

We present preliminary data fit results of synthetic light curves com-puted from numerical afterglow blast wave simulations. Our technique uses Markov chain Monte Carlo (MCMC) in a new data analysis tool, ScaleFit. Scaling relations in both the hydrodynamics and radiation equations allow light curves to be parameterized by a small set of scale-invariant characteristic quantities. These quantities have been calculated and tabulated from high resolution two-dimensional hydrodynamic sim-ulations. Producing a light curve from the characteristics takes only a millisecond, allowing for the use of MCMC data fitting techniques which can require millions of iterations. ScaleFit is a portable, lightweight, python package which performs this analysis on afterglow light curves. Using the set of Swift-XRT light curves from 2011 &amp; 2012 with known redshifts, we find ScaleFit can measure the jet opening angle, observer angle, and spectral index of most afterglows. Globally we find gamma-ray burst afterglows tend to be observed off axis, at a significant fraction of the jet opening angle

Geoffrey Ryan

Hendrik Van Eerten

Andrew Macfadyen

CiteSeerX

Fitting Afterglows With Multi-Dimensional Simulations

We present preliminary data fit results of synthetic light curves computed from numerical afterglow blast wave simulations. Our technique uses Markov chain Monte Carlo (MCMC) in a new data analysis tool, ScaleFit. Scaling relations in both the hydrodynamics and radiation equations allow light curves to be parameterized by a small set of scaleinvariant characteristic quantities. These quantities have been calculated and tabulated from high resolution two-dimensional hydrodynamic simulations. Producing a light curve from the characteristics takes only a millisecond, allowing for the use of MCMC data fitting techniques which can require millions of iterations. ScaleFit is a portable, lightweight, python package which performs this analysis on afterglow light curves. Using the set of Swift-XRT light curves from 2011 &amp; 2012 with known redshifts, we find ScaleFit can measure the jet opening angle, observer angle, and spectral index of most afterglows. Globally we find gamma-ray burst afterglows tend to be observed off axis, at a significant fraction of the jet opening angle

Ryan, G.

van Eerten, H.

MacFadyen, A.

OPUS

        Citation for published version:Ryan, G, van Eerten, H & MacFadyen, A 2013, 'Fitting afterglows with multi-dimensional simulations' Paperpresented at 7th Huntsville Gamma-Ray Burst Symposium, GRB 2013, Tennessee, USA United States,14/04/13 - 18/04/13, .Publication date:2013Document VersionPeer reviewed versionLink to publicationUniversity of BathGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 13. May. 2019Article 30 in eConf C1304143July 25, 2013Fitting Afterglows With Multi-Dimensional SimulationsGeoffrey Ryan, Hendrik van Eerten, Andrew MacFadyenCenter for Cosmology and Particle Physics, Department of PhysicsNew York University, New York, NY 10003We present preliminary data fit results of synthetic light curves com-puted from numerical afterglow blast wave simulations. Our techniqueuses Markov chain Monte Carlo (MCMC) in a new data analysis tool,ScaleFit. Scaling relations in both the hydrodynamics and radiationequations allow light curves to be parameterized by a small set of scale-invariant characteristic quantities. These quantities have been calculatedand tabulated from high resolution two-dimensional hydrodynamic sim-ulations. Producing a light curve from the characteristics takes only amillisecond, allowing for the use of MCMC data fitting techniques whichcan require millions of iterations. ScaleFit is a portable, lightweight,python package which performs this analysis on afterglow light curves.Using the set of Swift-XRT light curves from 2011 & 2012 with knownredshifts, we find ScaleFit can measure the jet opening angle, observerangle, and spectral index of most afterglows. Globally we find gamma-rayburst afterglows tend to be observed off axis, at a significant fraction ofthe jet opening angle.PRESENTED ATHuntsville Gamma Ray Burst SymposiumNashville TN, USA, April 14-18, 2013arXiv:1307.6334v1  [astro-ph.HE]  24 Jul 20131 IntroductionGamma-ray burst (GRB) afterglows present a rich opportunity for the study of GRBsthemselves and their extragalactic environments. The complex nature of afterglowemission requires the use of numerical simulations for accurate construction of lightcurves from basic physical parameters. However, since state-of-the-art simulationsrequire days to produce a light curve, it is challenging to use the most accurate sim-ulations in a live data analysis situation. The BoxFit package made use of scalinginvariance between explosion energies and circumburst medium densities in the hy-drodynamic equations to speed the process, tabulating the results of simulations soonly radiative transfer need be performed at run time [1]. ScaleFit extends thiswork, using scale invariance in synchrotron spectra to generate light curves directlyfrom a precomputed table [2]. These light curves have the accuracy of advanced nu-meric simulations, but can be generated in milliseconds, opening new afterglow dataanalysis possibilities.ScaleFit is a python package which implements this scaling procedure to performMarkov-chain Monte Carlo (MCMC) data-fitting on GRB afterglow light curves. Itoutputs central values and uncertainties in all fit parameters as well as a list of samplesapproximating the full posterior probability distribution function (PDF) of the fit.As a first run we have performed fits for all Swift-XRT afterglows in 2011 and 2012with known redshifts. We find ScaleFit can constrain values for θ0, θobs, and p forseveral bursts, while other parameters are relatively unconstrained by the single-bandfit. Our preliminary results indicate most afterglows are observed significantly off-axiswith p ≈ 2.1.2 ScaleFitWe model an afterglow as synchrotron radiation produced from a collimated rela-tivistic blast wave propagating through the GRB circumburst medium. The observedradiation has a synchrotron spectrum parameterized as series of power laws with apeak flux Fpeak and break frequencies νm and νc [3, 4, 1, 2].For now we ignore self-absorption effects as the corresponding frequency νa lieswell-below the Swift x-ray band currently under consideration. Each of these spec-tral parameters (Fpeak, νm, and νc) vary with time and depend on qualities of theblastwave, its environment, and its distance/orientation from the observer. We pa-rameterize this dependence through the redshift z, luminosity distance dL, isotropic-equivalent energy Eiso, the circumburst medium density n0, jet half-opening angleθ0, observer angle θobs, spectral index p, electron energy fraction e, magnetic energyfraction B, and fraction of accelerated particles ξN . We assume a global cooling timeand homogenous circumburst medium. The dependence of the synchrotron spectrum1Parameter Bounds Parameter Boundsz [0.0, 10.0] log10(dL/1028cm) [−5.0, 5.0]log10(Eiso/1053erg) [−7.0, 3.0] log10(n0/1cm−3) [−5.0, 5.0]θ0 [0.045, 0.5] θobs/θ0 [0.0, 1.0]p [2.0, 3.0] log10 e [-5.0, 0.0]log10 B [-5.0, 0.0] log10 ξN [-5.0, 0.0]Table 1: Default priors for all parameters in ScaleFit. For parameters which varyover several orders of magnitude the fit is performed on the logarithm of the parameterinstead of the parameter itself.on these parameters is given by simple scaling relations [2].After scaling, all the dynamic behaviour of the spectral light curve Fν(tobs) isenclosed in characteristic quantities fpeak, fm, and fc which only depend on θ0 andθobs. A series of high resolution, two dimensional numerical simulations have beenperformed to cover this parameter space using the adaptive-mesh-refinement rela-tivistic hydrodynamics code RAM [1, 5]. The results are collated into lookup tables foreach of the characteristic quantities. These tables are the core input to ScaleFit,which can then use the scaling relations to produce light curves for arbitrary valuesof Θ ≡ {z, dL, Eiso, n0, θ0, θobs, p, e, B, ξN}.The ScaleFit parameter set is ten dimensional and may be highly correlated insome parameters (e.g. between Eiso and n0, or e, B, and ξN) [6]. For these reasonswe use MCMC as the core data-fitting routine. For given data D, MCMC producesa set of samples which well approximate the posterior PDF p(Θ|D). These samplesmay then be used to find central values, standard uncertainties, or any other requiredstatistic.Our data take the form of light curves: D = {(ti, Fi, σFi)}. We take the likelihoodfunction p(D|Θ) to be a product of independent gaussians for each data point in thelight curve. This gives a likelihood of the standard χ2 form:p(D|Θ) ∝ exp(−12χ2), χ2 =∑i(Fi − Fmodel(ti; Θ)σFi)2(1)We take the prior p(Θ) to be flat within appropriate bounds for each parameter. Forparameters which may vary over several orders of magnitude, the fit is performed (andthe prior applied) in log-space. The bounds used in this analysis are summarized inTable 1.To perform the MCMC analysis ScaleFit uses the emcee package [7]. emceeis a free, open source, python-based package for performing MCMC. In particularScaleFit uses the EnsembleSampler, an implementation of the affine-invariant en-semble MCMC algorithm [8]. The performance of this algorithm is invariant under2affine transformations, providing efficient sampling of highly correlated parameters (acommon problem when using simple Metropolis-Hastings type samplers).3 DatasetWe performed fits on a sample of afterglow light curves made publicly available bythe Swift collaboration [9]. To reduce the dimensionality of the fits we chose to onlyexamine afterglows with known redshifts z and used a benchmark ΛCDM cosmology(Ωm = 0.27, H0 = 71 km s−1 Mpc−1) to calculate dL. To further reduce the dimen-sionality and remove degeneracy between the parameters we fix ξN = 1 for all fits inthis analysis [6].The ScaleFit afterglow model includes the effects of shock deceleration andspreading but does not include flares, energy injection, or other effects in the earlytime evolution of the light curve. As such, these parts of the Swift data must beidentified (with some confidence) and cut out so that fits are only attempted in theregime where the model applies. Assuming one of νc or νm lies below the observationband the shallowest power law slope obtainable by ScaleFit is −0.25, and the powerlaw slope monotonically decreases with time [4]. Hence we include only late-time,steepening, sections of the data with a power law slope smaller than −0.25. We useautomatic pre-analysis data published by Swift to make this determination [9]. Iffewer than ten data points remain after the cut, the afterglow is not fit or includedin the sample.There are 38 Swift-XRT afterglows with redshifts in 2011 and 2012. Of these, 33pass the cut on number of data points. The raw data was the count-rate light curvefor each burst made available by Swift. This was translated to a intrinsic flux lightcurve via the published counts-to-flux conversion factors which take into account hostextinction and galactic absorption. To properly fit the data, the fit was performedfor the ScaleFit specific flux integrated over the Swift-XRT observation band: 0.3 -10.0 keV.4 ResultsFor each afterglow in the sample ScaleFit was run with 1500 random walkers for2000 iterations, with a burn-in run of 500 iterations. This produces three million sam-ples of the posterior PDF for each afterglow, covering several auto-correlation times(typically ∼ 120 iterations). Figure 1 shows a corner plot of the fit for 110503A, withthe marginalized distributions of each parameter along the diagonal and covarianceplots in the off-diagonal locations.The fit for 110503A shows typical behaviour for a well-fit afterglow in our sample.The values of Eiso, n0, and e are effectively unconstrained but show an extremely38404log(n0)0.10.20.30.4θ 00.20.40.60.8θ obs/θ 02.12.22.32.4p642log(²e)10.0 7.55.02.5log(²B )0 2 4 6log(Eiso)10.07.55.02.5log(²B)8 4 0 4log(n0 )0.10.20.30.4θ00.20.40.60.8θobs/θ02.12.22.32.4p6 4 2log(²e )110503A102 103 104 105 106Observer Time tobs (s)10-1410-1310-1210-1110-1010-910-8Intrinsic Flux F in (0.3-10.0) keV (erg/cm^2 s)110503AFigure 1: Preliminary fit result for 110503A. The diagonals along the corner plotshow the marginalized probabilities for each parameter. The off-diagonal contourplots show the covariances between all pairs of parameters. The best-fit values (MAP,maximum posterior probability) are shown in blue. The best-fit light curve is shownagainst the data in the upper right.high degree of correlation with each other, as expected from the model. This is due toa degeneracy in the scaling relations when the afterglow remains in a single spectralregime. We expect this degeneracy will be removed by performing multi-band fits.The angles θ0 and θobs as well as the spectral index p are well constrained by the fit,4demonstrating the use of this procedure even in single-band fits. The multi-modalityin the p distribution is most likely due to ScaleFit trying to fit different spectralregimes to the data. This multi modal behaviour is reflected in the correlationsbetween p and the other parameters, particularly n0.Taking our dataset as a representative sample of GRB afterglows we can histogramthe central values of a parameter to determine the global distribution for that param-eter. Figures 2 and 3 shows a histogram of median values of θ0, θobs, and p over allbursts in our sample.0.0 0.1 0.2 0.3 0.4 0.5θ0024681012All BurstsWell constrained θ00.0 0.2 0.4 0.6 0.8 1.0θobs/θ0024681012All BurstsWell constrained θobsFigure 2: Distribution of median values for θ0 and θobs. Well constrained fits satisfyδθ0/θ0 < 0.5, where δθ0 is the half-width of the 68% confidence interval. The samecriterion is applied to θobs. These results are preliminary.2.0 2.2 2.4 2.6 2.8 3.0p024681012All BurstsWell constrained pFigure 3: Distribution of median values for p. Well constrained fits satisfy δp/p < 0.5,where δp is the half-width of the 68% confidence interval. All bursts in the samplepassed this cut, so the curves are identical. These results are preliminary.5Several fits resulted in very broad (i.e. flat) distributions for θ0 and θobs. Thesedistributions tend to have medians near the midpoint of their domain, hence thelarge number of bursts with 0.25 < θ0 < 0.3 or 0.4 < θobs/θ0 < 0.5. In order to cutaway these ill-constrained values we make a cut on the fractional uncertainty in eachparameter, requiring it to be less than 0.5. This cut certainly introduces unknownbiases into the resulting distribution, so the distribution over all bursts is plotted aswell. The value for p is determined to good accuracy by all bursts in the sample.The well-constrained distribution of θ0 is broad, although the all-burst distributionincludes a peak at θ0 ∼ 0.1. The signals from p and θobs are more clear. The pdistribution favours smaller values, with a large peak (including about 2/3 of burstsin the sample) at p ∼ 2.1. The observer angle shows a clear preference to be off-axis,peaking at θobs ∼ 0.7θ0.Assuming random orientations of GRBs in the sky, the distribution of θobs/θ0 isexpected to grow with θobs until the effects of jet spreading become significant. Ourresults corroborate this hypothesis. This result is striking, as the effect of off-axisobservation is not usually included in afterglow fitting. Off-axis observers see jet-breaks smeared out and completing at later times compared to on-axis observers [10].This affects energy estimates and the determination of opening angles.5 ConclusionScaleFit is a new data-fitting package for GRB afterglows, allowing high performanceMCMC routines to fit afterglow light curves to high resolution, two-dimensional hy-drodynamic simulations. As a first test of the package, we perform single-band fits onall Swift-XRT afterglows with known redshifts in 2011 and 2012. Eiso, n0, e, and Bare difficult to constrain with the single band fit and display a high degree of correla-tion. θ0, p, and the observer angle θobs are well constrained by the data in several fits,allowing for study of the global distributions of these parameters. In our preliminaryresults, we find the afterglow blastwave tend to have p ∼ 2.1, and be viewed off-axiswith θobs a significant fraction of θ0. ScaleFit will undergo a future public release.A more thorough analysis is underway and will appear in a future publication [11].ACKNOWLEDGEMENTSThis research was supported in part by NASA through grant NNX10AF62G issuedthrough the Astrophysics Theory Program and by the NSF through grant AST-1009863 and by the Chandra grant TM3-14005X. Resources supporting this work wereprovided by the NASA High-End Computing (HEC) Program through the NASA Ad-vanced Supercomputing (NAS) Division at Ames Research Center. The software used6in this work was in part developed by the DOE-supported ASCI/Alliance Center forAstrophysical Thermonuclear Flashes at the University of Chicago. We thank DavidBurrows, Binbin Zhang, Judith Racusin, David Hogg, and Daniel Foreman-Mackeyfor their helpful comments.References[1] H. J. van Eerten, A. J. van der Horst and A. I. MacFadyen, Astrophys. J. 749,44 (2012) [arXiv:1110.5089 [astro-ph.HE]].[2] H. J. van Eerten and A. I. MacFadyen, Astrophys. J. 747, L30 (2012)[arXiv:1111.3355 [astro-ph.HE]].[3] R. Sari, T. Piran and R. Narayan, Astrophys. J. 497, L17 (1998)[4] J. Granot and R. Sari, Astrophys. J. 568, 820 (2002) [astro-ph/0108027].[5] W. Zhang and A. I. MacFadyen, Astrophys. J. Suppl. 164 255 (2006) [astro-ph/0505481].[6] D. Eichler and E. Waxman, Astrophys. J. 627, 861 (2005) [astro-ph/0502070].[7] D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, (2012)arXiv:1202.3665 [astro-ph.IM].[8] J. Goodman and J. Weare, Comm. App. Math. Comp. Sci. 5, 65 (2010)[9] P. A. Evans, A. P. Beardmore, K. L. Page, J. P. Osborne, P. T. O’Brien, R. Will-ingale, R. L. C. Starling and D. N. Burrows et al., Mon. Not. R. Astron. Soc.397, 1177 (2009) arXiv:0812.3662 [astro-ph].[10] H. J. van Eerten, W. Zhang and A. I. MacFadyen, Astrophys. J. 722, 235 (2010)[11] G. Ryan, H. J. van Eerten, and A. I. MacFadyen, In Preparation (2013)7

Fitting Afterglows With Multi-Dimensional Simulations

Abstract

Similar works

Full text

Available Versions

CiteSeerX

OPUS