basic concepts to understand Gamma Ray Bursts

The waves

Direction of propagation  perturbation

ELETTROMAGNETIC WAVE

Continuum series of pulses originated from a variation of the electromagnetic field.

It is a perturbation of the electromagnetic field.

The redshift and the distance Measurement

redshift:

The light frequency is lower than the frequency was emitted.

This happens when the source is receding in the observer We look at the spectrum of an electromagnetic light emission of an object and we compare it with another nearer

z

  

OBSERVED EMITTED

 1 1 

z

  

OBSERVED EMITTED

SUN GALAXY

Gamma-Ray Bursts: The story begins

The Vela are American satellites that try to see if the URSS respects the Treats banning nuclear tests between USA and URSS in the early 60s

Brief, intense flashes of

g

-rays

They were too much long to be nuclear explosions and too much short to be a known phenomenon!

Klebesadel R.W., Strong I.B., Olson R., 1973, Astrophysical Journal, 182, L85 `Observations of Gamma-Ray Bursts of Cosmic Origin’

GRBs phenomenology

– – – – – –

Basic phenomenology Flashes of high energy photons in the sky (typical duration is few seconds).

Isotropic distribution in the sky Cosmological origin accepted (furthest GRBs observed z ~ 7 – billions of light-years).

Extremely energetic and short: the greatest amount of energy released in a short time (not considering the Big Bang).

Sometimes x-rays and optical radiation observed after days/months (afterglows), distinct from the main γ-ray events (the prompt emission).

Observed non thermal spectrum

The energetics of GRBs An individual GRB can release in a matter of seconds the same amount of energy that our Sun will radiate over its 10-billion-year lifetime

Isotropical distribution in the sky

Short vs Long GRBs

Short (hard) Long (soft)

Short GRBs -> T 90 <2 s Short GRBs -> T 90 <5 s

Kouveliotou et al., 1996, AIP Conf. Proc., 384, 42.

Paciesas et al., 1999, ApJS, 122, 465.

Donaghy et al., 2006, astro-ph/0605570.

Long GRBs -> T 90 >2 s Long GRBs -> T 90 >5 s Norris et Bonnell 2006 12

What is the T

90 •

Time interval in which the instrument reveal the 5% of the total counts and the 90%.

•

To the duration of this event it is associated the 90% of the emission

Progenitors for traditional model

core collapse of massive stars (M > 30 M sun ) long GRBs

Collapsar or Hypernova

(MacFadyen & Woosley 1999 Hjorth et al. 2003; Della Valle et al. 2003, Malesani et al. 2004, Pian et al. 2006)

GRB simultaneous with SN

compact object mergers (NS-NS, NS-BH)

short GRBs

Discriminants: host galaxies, location within host, duration, environment, redshift distribution, ...

Collapsar model

Woosley (1993) • Very massive star that collapses in a rapidly spinning BH. • Identification with SN explosion.

prompt emission

FRED (Fast Rise, Exponential Decay)

Pre-Swift vs Swift for the afterglows

Swift zmedio = 2.5!!!

Pre-Swift zmedio = 1.2

Typical lightcurve for BeppoSAX Typical lightcurve for Swift

Definition of the Flux and Energy

• • • The flux F is the energy carried by all rays passing through a given area dA .

dA normal to the direction of the given ray all rays passing through dA whose direction is within a solid angle dΩ of the given ray • • • E=I ν *dA*dt*dΩ*d ν Iv= is the brightness or specific intensity dF ν =E/(dA*dt*dν) • dF=

F

  

I

 cos 

d

 For some arbitrary orientation n

Gamma-ray Burst Real-time Sky Map

http://grb.sonoma.edu/

• • • • • • • •

Burst List

Burst ID GRB 090301A Date 2009/03/01 Time 06:55:55 Mission Swift RA 22:32 Dec 26:38 brief Burst Description

This burst had a complex multipeak structure and a duration of ~50 seconds. Due to observing constraints Swift cannot slew to this position until after April 15. No XRT or UVOT was available as a result.

Spectra

Non thermal spectra N(E)

E α E break E β

E

Epeak =(α +2) Ebreak α ~-1 β ~ -2 Ebreak ~ 100 keV - MeV The phenomenological Band law hold in a wide energy interval 2

keV

-100

MeV

GRB spectrum evolves with time within single bursts

E peak F ~ E

a

F ~ E

b featureless continuum power-laws - peak in  F  Hard to soft evolution Time [sec]

Jet effect Log(F)



>> 1/

  

1/

 Jet break

Log(t)

Jet half opening angle

X-ray Flashes and X-ray Rich Bursts

•

XRFs prompt emission spectrum peaks at energies tipically one order of magnitude lower than those of GRBs.

XRFs empirically defined by a greater fluence in the X-ray band (2-30keV) than in the γ-ray band (30 400keV).

•

XRR are an intermediate class between XRFs and GRBs

Why GRBs are so studied for the correlations?

GRBs are extremely energetic events and are expected to be visible out to

z ~ 15-20

(Lamb & Reichart, 2000, ApJ, 536, 1), which is further than that obtainable by quasars (zmax ~ 6). GRB z ~ 6.7 (Tagliaferri et al. 2005) Potential use of GRBs to derive an extended z Hubble diagram.

Peak energy – Isotropic energy Correlation

9+2 BeppoSAX GRBs

+ 21 GRBs (Batse, Hete-II, Integral)

E iso

1- cos



jet

Why is the Ghirlanda relation, Eg  (Epeak) 1.5, different from the Amati relation, Eiso  Epeak 0.5 ?

Because of the correction of the beaming angle

A completely empirical correlation between prompt (E E iso ) and afterglow properties (t break ) p , (Liang & Zhang 2005) Model dependent: uniform jet + wind density Model dependent: uniform jet + homogeneous density Through simple algebra it can be verified that the model dependent correlations are consistent with the empirical correlation! (Nava et al. 2006)

… still not convinced ? …

A new correlation between L iso , E p , T 0.45

Good fit Consistent with other corr ONLY PROMPT EMISSION PROPERTIES Firmani et al. 2006

The study of prompt vs afterglow A further step to build L X –Ta relation A lot of kinetic energy should remain to power the afterglow Prompt SAX X-ray afterglow light curve

E

afterglow

< E

prompt

s

=0.48

Flux vs observed time

Clustering of the optical luminosities Luminosity vs rest frame time

s

=0.28

GRB – Afterglow – Temporal Properties

GRB multiwavelength emission Panaitescu & Kumar

L X Eg correlate in optical and in X?

No corr.

The X-ray luminosities are more widely used for testing correlations

We also choosed X-ray luminosity for our analysis

Why we are searching a new correlation?

• to find a relation involving an observable property to standardize GRBs • in the same way as the Phillips law with SNeIa

Why we study the L-Ta correlation?

• GRBs possible cosmological distance estimators since they are observed up z = 8.26, much larger than SNeIa (z = 1.77) • but GRBs seem to be everything but the standard candles • with their energetics spanning over seven orders of magnitude • as an attempt to solve this problem we have probed the relation L-Ta that tries to standardize GRBs:

Dainotti, Cardone and Capozziello,

Mon. Not. R. Astron. Soc. 391, L79–L83 (2008) • the presentation describes a new analysis of the extended GRB afterglow sample INAF, Bologna, Italy 10 January 2011

Focusing on L 2

• L 2 =

at b

log

L

 log

a



b

log

t

Linearization provides a visual evidence of the claimed model and it gives the quantities as logarithms ready to compute the distance moduli Linear fits are used to find parameters also of other models which can be linearized through a suitable transformation of the variables.



x x

 

y y

Non-linear least-squares (NLLS) Marquardt-Levenberg algorithm, .

a

and

b

computed by the fit.

Values for -1.17 <

b

< -1.91

Time rescaled to restframe

The t break of the lightcurve is highly variable 10 3

L



t start b

The Spearman coefficient of correlation is 0.75

The correlation is new because it involves only the afterglow quantities

How can we improve it?

Increasing the statistic of GRBs observed by the same instruments to see if there is a selection effect depending on the instruments and improving the statistical method We used Ta and Fa values computed through Willingale et al. 2007 of the afterglow and the D’Agostini method as statistical method .

SWIFT

a

a t

 (Tc, Fc) is the transition between the exponential and the power law

T c t c

a

c

α c the time constant of the exponential decay, T c /α c • tc marks the initial time rise and the time of maximum flux occurring at

t



T c

a

t c c

In most cases t a =T p .

a

No case in which the two componets were sufficiently separated such that this time could be fitted as a free parameter.

We are unable to see the rise of the afterglow component because the prompt component always dominates at early times and t a could be much less than T p for most GRBs

.

The phenomenological formula

• 1)f_c(t) = f_p(t) + f_a(t) • 2)f_p(t)=

p e

( a

p



t

a

p

/

T p

)

e

( 

t p

/

t

) • fa(t)=

F a



t T a

 a

a

exp( 

t a t

) For t

t



T a

Willingale at al. 2007 Negligible if ta=0 and in that case we return to the simple case of power law decay

General treatment

3) L bol = 4 πDL 2 (z) P bol

P bolo



P

 / 1  10000

keV keV

 / 1

E

 1 

z z

(

E

)

dE E

max 

E

 (

E

min

E

)

dE

P bolo is the bolometric flux, while P is the peak flux, E min -E max is the energy range in which P occurs

We compute the X-ray luminosities at the time Ta so that we have to set f(t)=f (Ta)=fa(Ta) Since the contribution on the prompt component is typically smaller than the 5%, Much lower than the statistical uncertainty on Fa(Ta).

Neglecting Fp(Ta) we reduce the error on Fx(Ta) without introducing any bias. 3) LX(Ta) = 4 πDL 2 (z) FX(Ta)=aTa b

F X



f

(

t

) 

E E

max/  min/ 1 

z E

 ( 1 

z E

)

dE E

max

E



E

 min (

E

)

dE

E_min, E_max = (0.3, 10) keV set by the instrument bandpass

• • Due to the limited energy range, the GRB spectrum may be described by a simple power law  (

E

) 

E

b • • • • • β(t) β p for the prompt phase β pd for the prompt decay β a for the plateau observed at the time T a β ad for the afterglow at t > T a We estimate β a because we compute Fx(Ta)

Afterglow LT correlation - canonical vs irregular light curves

Dainotti et al. ApJL, 722, L 215 (2010)

L X

(

T a

) 

T a L X

(

T a

) = the source rest frame isotropic X-ray luminosity

T a

= the transition time separating the afterglow plateau phase and the power-law decay phase

L X

(

T a

)  4 

D L

2

F X

(

T a

) 

a

*

T a b F X



f

(

t

) 

E E

max/(  1  min/( 1 

z E



z

) ) (

E

)

dE E

max 

E

 (

E

min

E

)

dE D L



c H

0 ( 1 

z

) 0 

z

(E min , E max ) = (0.3, 10) keV - the instrument energy band

dz

' 

M

( 1 

z

' ) 3  ( 1  

M

) INAF, Bologna, Italy 10 January 2011

Data and methodology

 Sample : afterglows detected by

Swift

from January 2005 up to March 2009  Redshifts : from the Greiner's web page http://www.mpe.mpg.de/jcg/grb.html

.

 Spectrum for each GRB was computed using the Evan's web page http://www.swift.ac.uk/xrt curves in the filter time Ta ± σ Ta For some GRBs in the sample the error bars are so large that determination of the observables (Lx, Ta ) is not reliable. We therefore study effects of excluding such cases from the analysis (for details see Dainotti et al. 2011 to appear in ApJ ). To study the low error subsamples we use the respective formally define

the error parameter

logarithmic errors bars to

u

 ( s 2

Lx

 s 2

Ta

) 1 / 2 INAF, Bologna, Italy 10 January 2011

L

*

X



T

*

a

D’Agostini method (

D’Agostini 2005

) L*x(Ta) vs T*a distribution for the sample of 62 long afterglows errors measurements on both x and y INAF, Bologna, Italy 10 January 2011

The computations errors

• the parameters of interest are given with their 90%confidence ranges. • Following Willingale (priv. comm.), we have assumed independent Gaussian • errors and obtained 1 sigma uncertainties by roughly dividing by 1.65 the 90% errors.

Important remark

• The presence of the luminosity distance in the equation 1) DL(z) = (c/H 0 ) dL(z) dL(z)= (1+z)* 0 

z dz

' 

M

( 1 

z

' ) 3  ( 1  

M

) constrain us to adopt a cosmological model to compute Lx(Ta) (ΛCDM) • with (Ω M, h)=(0.291, 0.697)

The program to compute dl

• • • • ΩM=0.291

h0=0.697*100 c=300000 Mpc=3.08*10^(24) DL

z

: 1

z

NIntegrate 1 Sqrt 1 t ^3 1 , t , 0,

z

name

ReadList " directoryname grb name.txt

", Table String, 1

z name

Part

ReadList " name

, All, 1

directoryname grb z.txt

", Table Number, 1 z Part z , All, 1 For i Print 1, i name 107, i i , " , z ", z i , " DL ", Mpc ^2 c h0 DL z i ^ 2

• • • • • • • • • • • The error computations The D’Agostini Method L and Ta measurement errors :σ L , σ Ta statistical uncertainties on log(L), log(Ta) : (σL)/L *(1/ln(10) , (σTa)/Ta *(1/ln(10) respectively. These errors may be comparable so that it is not possible to decide what is the independent variable to be used in the usual χ 2 fitting analysis.

Moreover, the relation L = a Ta b may be affected by an intrinsic scatter σint of unknown nature that has to be taken into account. to determine the parameters (a, b, σint) a Bayesian approach D’agostini 05 thus maximizing the likelihood function

L

(a, b, σint) = exp (-L(a, b, σint).

Error on Lx(Ta)

• error[L_X(T_a)] = L_X(Ta) * {(DeltaF/F)^2 + [log(1+z)]^2 * DeltaBetaa}^{1/2} • (DeltaF/F)^2 = (TpErr/Tp)^2 + (TaErr * Tp/Ta^2)^2 + (FaErr/Fa)^2 (FaErr/Fa)^2 = [ln(10)]^2 * [(logTaErr/logTa)^2 + (logFaTaErr/logFata)^2]

The likelihood

• whose maximization is performed in the two parameter space (b, σ int ) since a may be estimated analytically • so that we will not consider it anymore as a fit parameter.

• (a, b, σ int ) = (48.54, -0.74, 0.43)

The goodness of the fit

Defining the best fit residuals as

δ = y obs – y fit , < δ>=-0.08

δ rms = 0.52

δ does not correlate with the other parameters of the fitted flux Sperman correlation coefficient r =-0.23 between and δ and z favours no significative evolution of the Lx - Ta relation with the redshift ( in the exercise you will do the same but simply beetwen Lx-z and Ta-z)

The comparison between the statistical methods

• • the best fit obtained through a Levemberg Marquardt algorithm with 1.5 σ outliers rejection (a, b) = (48.58, -0.79) in good agreement with the below maximum likelihood estimator results are independent on the fitting method (a, b, σ int ) = (48.54, -0.74, 0.43). • • since the Bayesian approach is better motivated and also allows for an intrinsic scatter, we hereafter elige this as our preferred technique.

(but you will use in the excercise for semplicity the Levemberg Marquardt algorithm)

Best fit curves

solid lines D’Agostini method dashed lines Levemberg-Marquardt estimator

Advantages

• • • • • Two parameters correlation A small scatter compared to the other correlation Well defined quantities involved Lx(Ta) and Ta A good sample No evolution with redshift

Disavantage

• Lx(Ta) is not an observable!

HOW TO FIND THE LIGHTCURVES AND SPECTRA

http://www.swift.ac.uk/burst_analyser/00148225/#BAT http://www.swift.ac.uk/xrt_spectra/00100585/ http://www.swift.ac.uk/xrt_curves/00100585/ The temporal decays parameters corresponding to a certain time region and the spectral decay ones are in the following papers arXiv.org

> astro-ph > arXiv:0812.3662v1

arXiv.org

> astro-ph > arXiv:0812.4780v1 arXiv.org

> astro-ph > arXiv:0704.0128v2

• • • • • • • •

Exercise

1) dowload the name of the GRBs with firm redshifts From http://www.oa.uj.edu.pl/M.Dainotti/GRB2010/ GRBupdate_se lescted_z.xls

2) Download all the lightcurves indicated in the file with firm redshift go to http://www.swift.ac.uk/burst_analyser/ In the all range XRT and BAT lightcurve 3)Install the EDA package on the PC 4)take off from the lightcurves the flares 5)find the parameters Tp,to,Fp, from the BAT 6)find using Tp the parameters Ta,Fa, necessary to compute Lx(Ta) 7) compute the spectral index beta_afterlow 8) compute Lx(Ta) for the GRBs in the table