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Statistical Re-examination of Reported Emission Lines in the X-ray Afterglow of GRB 011211

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Statistical Re-examination of Reported Emission Lines in the X-ray Afterglow of GRB 011211Robert E.Rutledge 1and Masao Sako 1,2ABSTRACT A 0.2-12keV spectrum obtained with the XMM-Newton EPIC/pn instrument of GRB 011211,taken in the ?rst 5ksec of a 27ksec observation,was found by Reeves et al.(2002;R02)to contain emission lines which were interpreted to be from Mg XI ,Si XIV ,S XVI ,Ar XVIII ,and Ca XX ,at a lower-redshift (z obs =1.88)than the host galaxy (z host =2.14).We examine the spectrum independently,and ?nd that the claimed lines would not be discovered in a blind search.Speci?cally,Monte Carlo simulations show that the signi?cance of reported features,individually,are such that they would be observed in 10%of featureless spectra with the same signal-to-noise.Imposing a model in which the two brightest lines would be Si XIV and S XVI K αemission velocity shifted to between z =1.88–2.40,such features would be found in between ～1.3-1.7%of observed featureless spectra (that is,with 98.3-98.7%con?dence).When we account for the number of trials implicit in a search of ?ve energy spectra (as were examined by R02),and permit a wider z -phase space search (z =2.14±1.0),the detection con?dence of the two line complex decreases to 77-82%.We ?nd the detection signi?cances to be insu?cient to justify the claim of detection and the model put forth to explain them.Kαline complexes are also found at z =1.2and z =2.75of signi?cance equal to or greater than that at z =1.88.Thus,if one adopts the z =1.88complex as signi?cant,one must

also adopt the other two complexes to be signi?cant.The interpretation of these data in the context of the model proposed by R02is therefore degenerate,and cannot be resolved by these data alone.Our conclusions are in con?ict with those of R02,because our statistical signi?cances account for the multiple trials required –but not accounted for by R02–in a blind search for emission features across a range of energies.In addition,we describe a practical challenge to the reliability of Monte Carlo ?χ2tests,as employed by R02.

Subject headings:gamma rays:bursts—gamma rays:observations

1.Introduction

It was recently reported that the X-ray afterglow of gamma-ray burst GRB011211, as observed with XMM-Newton EPIC/pn,contained spectral emission lines(Reeves

et al.2002a,hereafter,R02)–the?rst report of multiple X-ray emission lines from a GRB. These lines,at0.45,0.70,0.89,1.21,and1.44keV were interpreted to be from He-like Mg XI(rest energy1.35keV)and H-like Si XIV(2.0keV),S XVI(2.62keV),Ar XVIII (3.32keV)and Ca XX(4.10keV),redshifted to z=1.88.The di?erence between this and the known redshift of the host galaxy z host=2.14was modeled as due to supernova ejecta traveling at v=25800±1200km s?1,which had originated during a supernova4days prior to when the GRB jet illuminated it,producing the afterglow(a more detailed analysis by the same authors was completed after this paper was in its initial form;Reeves et al.2002b, R02b hereafter

The statistical signi?cance of the individual lines was not reported in R02;it was stated that joint analysis of the lines taken together produced an improvement in theχ2value which,by an F-test,yielded a signi?cance level of99.7%.In addition,it was found that Monte Carlo(MC)simulations were unable to produce the the same improvement inχ2 found between the best-?t power-law model and the best-?t?ve emission-line model more than0.02%of the time.Speci?cally,it was found that the best-?tχ2value for a power-law model was improved by?tting to a model of a MEKAL plasma with emission lines at rest energies corresponding to unresolved Mg XI,Si XIV,S XVI,Ar XVIII,and Ca XX redshifted to z=1.88in only0.02%of the simulated spectra(a99.98%con?dence detection).

The implications of the model discussed by R02–a delay between a supernova and a GRB on a timescale of days,the formation of a thin shell of supernova ejecta,

an apparent under-abundance of Fe relative to the detected nuclei–provide severe constraints on gamma-ray burst emission models.In addition,as demonstrated by R02, the future detection of multiple emission lines can provide extremely strong constraints on the production mechanisms,due to the inherent required out?ow velocity and emission timescales which can be derived from them,not to mention the implied association with supernovae.Similar spectra observed with greater S/N in the future would greatly aid in unravelling the emission mechanisms and geometry of gamma-ray bursts.Therefore,it is of wide theoretical(https://www.doczj.com/doc/8814539858.html,zzati et al.2002;Kumar&Narayan2002)and observational

interest to further interpret the observed X-ray spectrum of this GRB011211,in hopes of determining what more could be learned from future,more precise observations.

In Sec.2,we describe the observation,and perform a basic spectral analysis using continuum models.In Sec.3,we compare Monte Carlo(MC)realizations of acceptable continuum models with the GRB spectrum,and?nd that the reported features would

be produced in～10%of the continuum model spectra,due only to Poisson noise.In Sec.4,we adopt the model that the two apparently most signi?cant lines are Kαlines of Si XIV and S XVI;we perform a blind search for features of the same signi?cance in MC realizations of continuum spectra,and?nd that they would be reported from～1.2-2.6%of such spectra,again,due only to Poisson noise.We describe in Sec.5a practical challenge to the reliability of the MC?χ2analysis produced by R02.We conclude in Sec.6that the lines are not individually signi?cant in the absence of an imposed model,and are only marginally signi?cant when the adopted model is imposed.

These conclusions con?ict with those of R02.R02derived the model(that is,the observed line energies,or redshift)from the data,and then applied statistics for detection as if the energies were known prior to examining the data(that is,single-trial statistics). This is not appropriate when the model line energies are derived directly from the X-ray data,and not from an a priori model–one derived without examination of the X-ray data (for example:line energies of multiple features with redshifts of the host galaxy).We adopt statistics appropriate to a blind-search for these features,across a range of energies or redshifts(multi-trial statistics).This accounts for the diminished signi?cance we?nd for the features.We further discuss the reasons for this con?ict and conclude in§6.

2.Observation and Observed Spectrum

We analyzed the identical source and background XMM-Newton/EPIC-pn(Str¨u der et al.2001)spectrum as used by R02(their Fig.2),which was kindly made available to us by the authors in electronic form(J.Reeves,https://www.doczj.com/doc/8814539858.html,m.).We used the same response matrix(epn sdY9

implemented to maximize the signal-to-noise associated with the reported emission features at the reported energies.We?rst binned data with energy bins centered at the?ve best-?t line energies found by R02,with bin-sizes approximately equal to the FWHM EPIC/pn energy response at each energy(respectively:62eV,66eV,68eV,72eV and75eV;see Eq.1).The remaining data were binned with60eV or greater(<1keV),and70eV or greater(>1keV).Between0.2and5keV,each bin has>15counts(although they were not binned on this basis),for whichχ2?tting is valid.

We?t an absorbed photon power-law spectrum to the data,shown in Fig.1.The model spectrum was statistically acceptable(model parameters are listed in Table1, along with the obtainedχ2values).We also?t the model with a thermal bremsstrahlung spectrum(wabs*bremss),and derived an acceptable best?t.Finally,we found best-?t model parameters for the values of the power-law photon slope(models2and3)and

kT bremss(models5and6)at the90%con?dence limits of the best-?t,which will be used in MC simulations in§3.

3.Individual Emission Line Signi?cances

We?rst determine which of the reported lines are individually statistically signi?cant, when one is searching for emission lines at a priori known energies.We compared the observed spectrum with MC simulations of six featureless spectra–the three absorbed power-law and three absorbed thermal bremsstrahlung which are models1-6in Table1.

We used a“matched?lter”,convolving the observed pulse-invariant(PI)counts spectrum with a Gaussian energy response,with the energy resolution response of the detector.The matched?lter approach maximizes the signal-to-noise ratio as a function of energy of unresolved lines in the X-ray PI spectrum.

Based on Figure18in XMM-Newton v1.1Users’Handbook(Dahlem1999),we modelled the photon energy redistribution as a Gaussian response,with FWHM:

F W HM(E)=57+13(E/1keV)?0.29(E/1keV)2eV(1) This approximation was derived from the line in this?gure.The EPIC/pn energy resolution has been demonstrated to be stable over9months of in-?ight calibration(Str¨u der et al.2001).We expect that this analysis(and that of R02,since that work is based on the same energy response matrices)is valid as long as the energy resolution is within20%of this approximation(corresponding to3of～15PI channels at0.75keV).

We performed a convolution between the raw PI spectrum(that is,number of counts vs.PI bin)and the gaussian energy response function,as a function of energy:

C(E i)=j(E i+3σ(E i))

j(E i?3σ(E i))I(j)12πσ(E i)exp?1σ(E i) 2δE j(2)

where N is the number of PI bins,and we sum across PI bins which are within±3σ(E i)

of E i.I(j)is the raw PI spectrum,which contains both source and background counts,

and j=1,2,..,N is the PI bin number.The centroid(average)energies and energy widths

(?E j)of the PI bins were taken from the EBOUNDS extension of the response matrix,where

i is the PI bin number andσ(E)=F W HM(E)/2.35.We do not correct the PI spectrum

for the detector area;however the detector area does not change dramatically across the

FWHM of the lines.If the area did change dramatically across the FWHM of a line,and

a statistical excess were observed in the area-corrected PI spectrum but not in the raw PI

spectrum,then such an excess could well be be due to calibration uncertainties.

The resulting C(E i)is shown in Fig.2a.By visual inspection,there are indeed features

in the spectrum near energies where the reported lines occur.

To determine if these features are signi?cant,we produced MC spectra of models

1-6(see§2).The MC realizations of the raw PI spectra were performed as follows.We

simulated the spectral models1-6in XSPEC,using the same response matrix as above,

so that the resulting PI spectra(without Poisson noise added)were convolved as the

observed spectrum through the telescope and detector response.The simulated PI spectra

N(E)each had a total of>9×108counts in PI bins between0.2-3keV.We then produced

integrated spectra I(E)= E0.2keV N(E)dE/ 3keV0.2keV N(E)dE,so that I(0.2keV)=0and I(3keV)=1(the integrated normalized model is used for the MC simulation as described

below).These constitute our six acceptable featureless spectral models;we will compare

the data with results from all six,as a?rm conclusion that emission lines are present should

be independent of the underlying broad-band model assumed.

We implemented a background spectral model,to simulate the～10%of the counts

due to background.Taking background from a di?erent part of the detector,we?nd that

it can be parameterized by a broken photon power-law(bknpower),withα1=2.4at low

energies,break energy1.35keV,andα2=0.44at high energies,between0.20-7.3keV(there

is a strong background line at8keV).In fact,there are statistically signi?cant deviations

from this pure continuum model between0.55-0.6keV;we ignore these deviations in our

background model.In our MC simulation the e?ect of ignoring what would appear to be

a line in the observed spectrum is conservative,in the sense that by ignoring its presence

in the background model,we could detect as“signi?cant”a line in the0.55-0.6keV range which is in fact produced by instrument background.

We simulated spectra between0.2and3keV,in which there were560counts in the observed spectrum,of which we estimate～66±1.2counts are due to background.We drew, for each MC realization,a number of background counts which is random poisson deviate (using poidev,Press et al.1995)with an average of66counts,with the remaining(of560) counts from the source.To produce a simulated spectrum,we generate a random uniform deviate r between0and1,and we place a count in the PI bin in which I(E)=r.

To produce our con?dence limits to C(E),we produced1667MC realizations each of models1-6for a total of10002realizations.We set the99%and99.9%con?dence limits at the100th and10th greatest values,respectively,of C(E)of all such realizations.This insures that the conclusions are not dependent upon the assumed featureless spectral model.

The results of this MC simulation are shown in Fig.2a.Two of the reported features (near0.7keV and0.85keV;Si XIV and S XVI)have single-energy-trial probabilities of >99%con?dence in comparison with the featureless spectral models(the claimed S XVI line peaks just below the99.9%con?dence limit;we will treat it as having met99.9% con?dence,while the reader may regard this as an upper-limit).The remaining three lines are not signi?cant in comparison with single-energy-trial probability of99%con?dence.

In Fig.2,we also show C(E)for single MC realizations of the best-?t power-law spectrum(model1),which also contain apparent features.The bumps in the single simulated spectra appear because in any spectrum which contains Poisson noise,the counts will not be distributed uniformly in energy,but will be clustered in energy simply due to counting statistics.

3.1.Multi-Energy-Trial(Blind Search)Probabilities

Since it was necessary to perform a blind-search for emission line features in the GRB spectrum–as the redshifted line energies were not known a priori,but were measured from the data–it is necessary to estimate the chance probability that the reported features are produced from a featureless spectrum during a blind search for such features.

We produced10000MC realizations for each of models1-6as described in the previous section.We compared the C(E)of these between0.4and1.5keV against the single-energy-trial99%and99.9%con?dence limits we found in the previous section,for the models1-6individually.

In Table2we list the fraction of the10000MC spectra in which C(E)in at least one PI bin reaches a single-energy-trial probability of99%or99.9%con?dence.These fractions are78-79%and14-17%,respectively;if?nding a single-energy-trial99%(Si XIV) and99.9%(S XVI)feature were statistically independent,then the probability of observing both a99%and99.9%single-energy-trial“line”in a single spectrum,such as we?nd in the present spectrum of GRB011211,is≈10%.

Therefore,in a blind-search of the EPIC/pn spectrum for emission features,we would expect to?nd features which have single-energy-trial signi?cance equal or greater to those observed in one of approximately ten observed featureless spectra.

4.Line Complex Signi?cance As a Function of Redshift

In this section,we examine if the reported lines,taken together,implicate Kαemission features from the particular redshift of z=1.88as reported by R02.We do so by summing the C(E i/(1+z)),using the values of the rest energies of the reported lines:

χ(z)=N lines

i N j I(j)12πσ(E j)exp?1σ(E j) 2(3)

(4)

where j denotes the PI bin number,E j is the centroid energy of the j th PI bin,i denotes the index[1,5]of the?ve lines reported detected;E i denotes the rest energies of the

?ve lines identi?ed by R02,which were1.35(Mg XI),2.00(Si XIV),2.62(S XVI),3.32 (Ar XVIII),and4.10keV(Ca XX).We examined the range of0

The average value ofχwill systematically increase with increasing z as the lines are shifted to lower energies,where the intensity is higher in the power-law spectrum and the detector e?ective area is larger and so there are a greater number of counts.To examine if any particular maximum inχ(z)is signi?cant,we performed this convolution for10000 MC realizations using the simulated spectral models1-6,taking the100th and10th highest

values,as described in the previous section,to produce the99%and99.9%con?dence limits respectively.

The results of the calculation using all5reported lines,as well as the99%and99.9% MC con?dence limits,are in Fig.3.The value ofχ(z)is in excess of the99%MC con?dence limit at z=[1.86?1.98]and z=[2.62?2.865],and in excess of the99.9%MC con?dence limit at z=2.76.

We also performed this convolution and Monte-Carlo simulation using what appear to be the most signi?cant two lines from Fig.2of R02(Si XIV and S XVI),the results of which are also shown in Fig.3.The value ofχ(z)is in excess of the99%MC con?dence limit at z=[1.155?1.275]and z=[1.80?2.01],and in excess of the99.9%con?dence limit at z=[1.86?1.95].

4.1.Multi-Redshift-Trial(Blind Search)Probabilities

What fraction of featureless spectra,with the same number of source and background counts as the observed spectrum,would produce values ofχ(z)of comparable signi?cance to the excess inχ(z=1.88)from the observed spectrum?If one examinesχ(z)only at z=1.88,the answer is<1%,which is the single-z-trial probability.However,the reported z=1.88is di?erent from the known redshift of the host galaxy z=2.14;it is therefore unlikely that z=1.88was the only redshift which would be considered consistent with an a priori model by R02.The pertinent statistical question to ask,then,is what is the fraction of featureless X-ray spectra,examined for a redshifted pair of S XVI and Si XIV lines,would produce a value ofχ(z)comparable to the single-z-trial signi?cance observed,allowing for a blind-search at values of z between1.88and2.40(a range of equal magnitude redshift and blueshift from the host galaxy)?

To address this,we simulated10000MC spectra of each of spectral models1-6,and foundχ(z)in the same way as for the observed spectrum in the previous section.We used only the2-line model,as this gave the apparently most signi?cant result near z=1.88. We comparedχ(z)with the99%and99.9%MC limits,found in the previous section,and noted when these were exceeded in at least one z bin for the99%con?dence limit,and in at least seven consecutive z bins for the99.9%con?dence limit between z=1.88and z=2.40.We require seven consecutive z bins as this is the number of z bins inχ(z)we?nd in excess of the single-trial99.9%con?dence limit near z=1.88.(We require only1bin for the99%con?dence limit to satisfy a minimal“detection”requirement;whereas we require seven bins for the99.9%con?dence limit,since this was what was actually observed near

z=1.88,and we wish to evaluate the likelihood of producing the observedχ(z)excess).

The fraction of MC featureless spectra which contained at least1z bin between

z=1.88and z=2.40in excess of the single-z-trial MC probability of99%are given in Table3.Because the observed spectrum gaveχ(z)>99.9%in seven consecutive z bins, we also used this as our criterion to count“hits”in the>99.9%con?dence comparison, also shown in Table3.Of10000MC spectra,between20-22%produced“hits”for the single-z-trial99%con?dence limit,and1.5-1.9%produced“hits”for the the single-z-trial 99.9%con?dence limit.

We note that when we search the range z=2.14±1.0(instead of±0.26)the percentage of featureless spectra which have seven consecutive z bins withχ(z)greater than the99.9% limit is3.8-5.0%.However,it is unclear if R02would have attached equal signi?cance to a detection at z=1.14as one at z=1.88,as no limits on excess line emission as a function of assumed redshift are given,and the redshift phase-space examined by R02was not given. We therefore rely on our search of the smaller phase-space;while this may underestimate the number of”e?ective trials”used by R02,it nonetheless serves as the probability of producing the claimed excess line emission due a statistical?uctuation within theδz=0.26 observed.If the redshift space examined by R02were1.14-3.14(δz=1.0),then the probability of?nding an excess equal or greater than that observed would be3.8-5.0%.If the full redshift space of0-5was in fact examined by R02then the probability of a false detection is>3.8-5.0%.

5.A Practical Challenge with Monte Carlo?χ2Tests for Multi-Parameter

Models

A MC?χ2test as employed by R02is not fundamentally?awed as is the analytic?χ2 test(that is,the F-test)for the application of spectral emission line discovery.In the F-test, the reference?χ2distribution was derived under the assumption that the null hypothesis lies on the border of the acceptable parameter space(Protassov et al.2002),which is not true in a search for emission lines;however,this assumption is not made in the MC?χ2 test.Thus,the simulated?χ2distribution can,in principle,provide a reliable reference distribution with which the value of?χ2from application to real data can be compared to determine the false positive rate.

However,as we show below,the MC?χ2test as employed by R02(and described more fully by R02b)su?ers from a practical problem which makes it an inferior approach to the one we have applied.Speci?cally,to apply the?χ2statistic using the MC approach,

one must assuredly?nd the global minimumχ2for the applied model for every single MC realization;the description of the analysis performed by R02(and R02b)does not assure that this has occurred.

In the case ofχ2minimization through local mapping of theχ2surface,as in the modi?ed Levenberg-Marquart method employed in XSPEC(Arnaud1996;modi?ed from the CURFIT algorithm as described by Bevington1969;see also Press et al.1995),one ?nds the vector in multi-parameter space along this surface which provides the most negative derivative,follows along this vector a short way,and iterates,until one reaches a point where there are no negative derivatives in any direction along theχ2surface(that is,when one has reached a minimum point.This approach su?ers from the well known problem of local minima,where the true global minimum can lie at a completely di?erent set of parameter values(see,for example Press et al.1995,p.394).On simpleχ2surfaces, where the second partial derivatives of theχ2surface are everywhere small–certainly in the case of the2-parameters power-law spectrum–it is rare that local minima di?erent from the global minimum are found.However,on complexχ2surfaces(those which contain large second partial derivatives ofχ2)–as will be the case when?tting a six parameter model of three emission lines of speci?ed rest energy with variable?uxes and redshift plus a power-law(slope and normalization)–local minima are common;subsequently, this approach is unsuited to the unassisted discovery(by computer alone,without human intervention)of the globalχ2minimum.It is,for example,common occurrence when using a multi-component(of more than,say,three)parameters in XSPEC that some?nal human assistance is required to?nd the global minimum,since almost always it is a local minimum which is found unassistedly by the computer;the quantitative di?erence inχ2 between the computer-discovered local minimum and the true global minimum will depend on the complexity of theχ2surface.In general,χ2surfaces become more complex with the addition of more model parameters,and the discrepancy will be greater when there are greater covariances between model parameters(such as can be expected between?ux for line1vs.line2,or vs.the continuum,or for each of the lines and the continuum,or for the relative?ux for the lines and the spectral slope;and so on).The only certain means to overcome this known de?ciency is to evaluateχ2on a parameter grid with resolution in each parameter dimension much smaller than ranges where theχ2value changes by1.

Thus,while the global minimum will likely be found from unassisted discovery for the power-law spectral model,it is more likely that only a local minimum will be found for the six-parameter model when the search for this minimum is not human-assisted.This will underestimate the value of?χ2for that realization;over the entire ensemble of MC realizations,there are then fewer false positives,and the signi?cance of the?χ2in support of the presence of lines will be overstated.In addition to the problem of local minima,

the local minimum found will be dependent upon initial parameter values(that is,the algorithm is path-dependent);it therefore does not lend itself to duplication by di?erent groups.Also,the spectral?tting is non-analytic,such that the magnitude of a possible discrepancy cannot be evaluated a priori.

XSPEC–which R02states was used for the MC simulation–does perform theχ2 minimization approach.We suggest that it is unlikely that human-assisted spectral?tting–as is common practice when attempting to?nd the globalχ2minimum for a single spectrum in XSPEC–was performed for all10,000MC spectra by R02,as we found ourselves was necessary for the single spectral?t to the real data,as this would be an impractically long task.

We are unable to attempt to duplicate the result of R02,because performing assisted spectral?tting on10,000MC spectra is impractical,and in any case,we know of no

de?ciency in our present approach.Our approach,in contrast,is analytic and not

path-dependent and,therefore,more robust.

6.Discussion and Conclusions

We have attempted to con?rm the observational statistical signi?cance of emission lines in the X-ray afterglow of GRB011211.In a blind-search for individual emission lines between0.4and1.5keV,features of signi?cance equal to those observed will be found in one in ten featureless spectra.Thus,the reported features can be said to be detected with 90%con?dence in a model-independent way.

Also,a blind-search for the two-line complex(Si XIV and S XVI)at any redshift between the reported value(z=1.88)and a blueshift of equal magnitude from the host galaxy(z=2.40)would?nd such features with equal signi?cance to that observed in

1of60featureless spectra(1.3-1.7%of the time,depending on the intrinsic spectrum). Thus,the features as reported can be said to be detected with98.3-98.7%con?dence,in a model-dependent interpretation,where we search for two features due to Si XIV and S XVI Kαredshifted to some value of z in the range z=2.14±0.26.

The di?erence between the present statistics and those of R02are due to the di?erent statistical arguments used to establish the existence of the emission lines.While R02relies on single-trial statistics,we?nd the model used by R02(Kαlines,at a redshift di?erent from that of the host galaxy)was derived directly from the data,which therefore requires a statistical analysis appropriate to a blind search.By expanding the searched phase-space, and taking into account the multiple trials of a blind search,the con?dence in the detection

drops from the99.98%of R02to,the98.7%(best case)we?nd here.Moreover,R02did

not estimate the individual signi?cances of the lines as we do here;thus we?nd that such

“lines”would appear in between15-78%of observed featureless spectra for a single-trial

signi?cance comparable to that of the reported Si XIV or S XVI lines.

The analysis of these data has otherwise recently been called into question.Borozdin

&Trudolyubov(2002)have shown that there is a background line associated with the

EPIC/pn detector edge during the observation,which would have been included in the GRB spectrum from the?rst5ksec,when the source was near the detector edge,but not afterwards,after the source had been moved away from the detector edge.In our

own analysis,we cannot con?rm this result unless we adopt non-standard event selection

criteria,which di?er from the ones used by R02.R02removed events near the CCD

detector edge(FLAG==0)and selected only single and double events(PATTERN<=4)(J.

Reeves,https://www.doczj.com/doc/8814539858.html,m.).These selections result in a smooth,featureless background spectrum

with with no bright line-like feature near～0.7keV(as seen in Fig.4f of Borozdin&

Trudolyubov2002)as well as a reduction of the count rate by a factor of～>2in the range E=0.2?3keV(see Fig.4).Therefore,we are not able to con?rm the applicability of

(Borozdin&Trudolyubov2002)to the analysis of R02.

An alternative approach to the one we have taken is employed using XSPEC,in which

one?ts a featureless spectrum to the data,and then a spectrum which includes emission

lines,to determine if the change inχ2νis signi?cant,as according to an F-test;this is the

approach taken by R02.However,this approach for the detection of emission or absorption

lines is formally incorrect,and gives false statistical results(Protassov et al.2002)

particularly so when the true continuum is not well constrained,as in the present case.We

therefore prefer our approach of applying a matched energy response?lter for line detection

at arbitrary energies,and to compare this with application of the matched?lter to MC

realizations of featureless spectra.It is a trivial statistical exercise to demonstrate that

matched?ltering maximizes the signal-to-noise ratio(and thus detectability)for detection

of in?nitely narrow emission lines.

In estimating the model-dependent con?dence limit for the detection of the line complex

(98.7%),we accounted only for searching the redshift phase space between z=1.88and

z=2.40,symmetric about the host galaxy redshift–an extremely minimal requirement.

We did not account for the full redshift phase space searched by R02,as such was not

given in that reference;if the redshift phase space searched by R02covered z=1.14?3.14

(z=2.14±1.0),then the detection signi?cance of the two strongest lines(Si XIV and S XVI)

together decreases from98.3-98.7%to95-96.2%con?dence.Finally,we did not include

in this con?dence limit the number of trials implicit in searching?ve X-ray spectra for

emission lines,which was performed by R02for di?erent time periods(0-5ksec,5-10ksec, 10-15ksec,15-20ksec,and20-27ksec).If we presume the same search was made on all ?ve spectra,as seems a reasonable a priori search to perform,then the detection con?dence for the Si XIV and S XVI lines together decreases to0.955–0.9625=77-82%.We regard 98.7%to be a conservative(in the sense of permitting a higher signi?cance)upper-limit to the con?dence of detecting the Si XIV and S XVI lines together,while a more accurate accounting of the number of trials and phase-space searched by R02produces a77-82% con?dence limit.

We consider neither a90%con?dence detection in a model-independent interpretation, nor a98.3-98.7%con?dence detection in a model-dependent interpretation,to be su?cient to justify the detection claims and subsequent interpretation put forth by R02.The77-82% con?dence limit,which accounts for the wide z-phase space and number of spectra examined by R02,is well below any comfortable detection con?dence.If the z phase space actually searched by R02is larger,the number of implicit trials is greater,and our estimate of the con?dence level for the detected line complex would decrease.

Moreover,if one concludes that the marginal detection of the2lines(Si&S)near

z=1.88is signi?cant,then one must also conclude that the detection of all5lines near z=2.75is equally signi?cant.In addition,if one concludes that the marginal detection of the5lines near z=1.88is signi?cant,then one must also conclude that the detection of2 lines(Si&S)near z=1.2is equally signi?cant.

Therefore,one cannot conclude simply that a complex of Kαline emission is detected near z=1.88;these data permit alternate interpretations of such complexes near z=1.2 and z=2.75.As the statistical excesses are due to the same“features”in the observed spectrum,the interpretation of the statistical excess in the context of the model presented by R02is degenerate and cannot be resolved with these data alone.

Prospects for con?rmation of line features in GRBs are very good,considering that the X-ray spectral integration for GRB011211was begun11hours after the GRB was initially detected,and required1.4hrs of integration to obtain.Decreasing the reaction time would permit a longer integration,while the afterglow is brighter in the X-rays,and the marginal results found here may well be improved upon.

We are grateful to J.Reeves,who generously made his observed spectrum of the

?rst5ksec of the XMM-Newton observation of GRB011211available to us,that we might independently analyze it.We gratefully acknowledge useful conversations with A. MacFadyen,R.Blandford,and D.Fox.The authors are grateful to F.Harrison,F.Paerels and an anonymous referee for useful comments on the manuscript.MS was supported by

NASA through Chandra Postdoctoral Fellowship Award Number PF1-20016issued by the Chandra X-ray Observatory Center,which is operated by the Smithsonian Astrophysical Observatory for and behalf of NASA under contract NAS8-39073.

References

Arnaud,K.A.,1996,in G.Jacoby&J.Barnes(eds.),Astronomical Data Analysis Software and Systems V.,Vol.101,p.17,ASP Conf.Series

Bevington,P.R.,1969,Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill

Borozdin,K.N.&Trudolyubov,S.P.,2002,ApJ,submitted,astro-ph/0205208 Dahlem,M.,1999,XMM Users’Handbook,Issue1.1,distributed by the XMM-Newton Science Operations Center,Vilspa

Kumar,P.&Narayan,R.,2002,ApJ,submitted,astro-ph/0205488

Lazzati,D.,Ramirez-Ruiz,E.,&Rees,M.J.,2002,ApJ572,L57

Press,W.,Flannery,B.,Teukolsky,S.,&Vetterling,W.,1995,Numerical Recipies in C, Cambridge University Press

Protassov,R.,van Dyk,D.A.,Connors,A.,Kashyap,V.L.,&Siemiginowska,A.,2002, ApJ571,545

Reeves,J.N.,Watson,D.,Osborne,J.P.,Pounds,K.A.,O’Brien,P.T.,Short,A.D.T., Turner,M.J.L.,Watson,M.G.,Mason,K.O.,Ehle,M.,&Schartel,N.,2002a,Nature 416,512

Reeves,J.N.,Watson,D.,Osborne,J.P.,Pounds,K.A.,O’Brien,P.T.,Short,A.D.T., Turner,M.J.L.,Watson,M.G.,Mason,K.O.,Ehle,M.,&Schartel,N.,2002b,?a, submitted,astro-ph/0206480

Str¨u der,L.,Briel,U.,Dennerl,K.,Hartmann,R.,Kendziorra,E.,Meidinger,N., Pfe?ermann,E.,Reppin,C.,Aschenbach,B.,Bornemann,W.,Br¨a uninger,H.,Burkert, W.,Elender,M.,Freyberg,M.,Haberl,F.,Hartner,G.,Heuschmann,F.,Hippmann,H., Kastelic,E.,Kemmer,S.,Kettenring,G.,Kink,W.,Krause,N.,M¨u ller,S.,Oppitz,A., Pietsch,W.,Popp,M.,Predehl,P.,Read,A.,Stephan,K.H.,St¨o tter,D.,Tr¨u mper,J., Holl,P.,Kemmer,J.,Soltau,H.,St¨o tter,R.,Weber,U.,Weichert,U.,von Zanthier,C., Carathanassis,D.,Lutz,G.,Richter,R.H.,Solc,P.,B¨o ttcher,H.,Kuster,M.,Staubert, R.,Abbey,A.,Holland,A.,Turner,M.,Balasini,M.,Bignami,G.F.,La Palombara, N.,Villa,G.,Buttler,W.,Gianini,F.,Lain′e,R.,Lumb,D.,&Dhez,P.,2001,A&A 365,L18

Fig.1.—(Top Panel):X-ray spectrum from XMM-Newton EPIC/pn of GRB011211,with energy binning optimized to?nd deviations from the best-?t power-law spectrum(solid line) due to emission lines at the reported energies,with a best-?t absorbed power-law model. The data show no signi?cant excess counts at the reported line energies.(Bottom Panel):χ=(model-data)/σ,residuals between the best-?t continuum spectra and the data.The 5-10keV energy bin,while included in our spectral?ts,is not included in this?gure,to better show the0.2-5keV energy spectrum.

Fig. 2.—(Panel a):Solid line is C(E)(Eq.2)from the observed raw PI spectrum–the convolution between the raw spectrum and the EPIC/pn energy response.The broken lines are the max(C(E))for spectral models1-6,showing the99%and99.9%con?dence single-trial upper-limits.(panels b-f):The solid line is the same observed convolved spectrum as in Panel a.Dotted lines are?ve(in the?ve separate panels)randomly selected Monte Carlo spectra using Model1.Features of similar magnitude to those found in the observed spectra are apparent in each;these are due to the Poisson noise distribution(in energy)in a spectrum with a?nite number of detected counts.

Fig.3.—(Top panel)Figure of meritχ(z)using all?ve reported line energies(solid line), for0

Fig. 4.—The background spectrum near the region near the CCD chip edge using two di?erent event selection criteria.(a)no explicit selection of PATTERN and FLAG as adopted by Borozdin&Trudolyubov(2002)(circles)binned at a minimum of5counts per bin and(b) using only PATTERN<=4and FLAG=0events as adopted by R02(stars)with binnsizes identical to those of(a).

Figure4

Table1.Best-?t X-ray Spectral Parameters

N H Nχ2ν/dof Model(1022cm?2)α/kT bremss phot/keV/cm2/s at1keV(prob a)

Thermal Bremmstrahlung

(9.0±1.2)×10?5 1.38/17(0.13) (4)Best Fit0.03±0.01 1.5+0.4

?0.2

50.05±0.01(1.1)(12±1)×10?5 1.50/18(0.08)

)×10?5 1.44/18(0.10) 60.014±0.01(2.1)(7.0+0.7

?0.4

10.780.16

20.780.15

30.780.14

40.780.15

50.790.16

60.780.17

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