The GRB Afterglow Modeling Project (AMP):

Statistics and Absorption and Extinction Models

Adam S. TrotterUNC-Chapel Hill

PhD Oral Prelim Presentation30 January 2009

Advisor: Prof. Daniel E. Reichart

AMP: The GRB Afterglow Modeling Project

• AMP will fit statistically self-consistent models of emission, extinction and absorption, as functions of frequency and time, to all available optical, IR and UV data for every GRB afterglow since 1997.• Will proceed chronologically, burst-by-burst, rougly divided into BeppoSAX, Swift and Fermi satellite eras, and published as an ongoing series in ApJ.• Before we can begin modeling bursts, we must establish a solid statistical foundation, and a complete model of every potential source of line-of-sight extinction and absorption.• We must also test this model first on a hand-selected set of GRB afterglows with good observational coverage that are known to exhibit particularly prominent absorption and extinction signatures.

An “Instrumentation Thesis”Forge a Tool: A new statistical formalism for fitting model curves to two-dimensional data sets with measurement errors in both dimensions, and with scatter greater than can be attributed to measurement errors alone.

Build an Instrument: A complete model of absorption and extinction for extragalactic point sources, including: dust extinction and atomic and molecular hydrogen absorption in the host galaxy; Ly forest/Gunn-Peterson trough; and dust extinction in the Milky Way.

Conduct the Tests: Model fits to IR-Optical-UV photometric observations of a selected set of seven GRB afterglows that exhibit various signatures of the model and/or signs of time-dependent extinction and absorption in the circumburst medium.

Forge a Tool: A new statistical formalism for fitting model curves to two-dimensional data sets with measurement errors in both dimensions, and with scatter greater than can be attributed to measurement errors alone.Work 100% complete, to be submitted to ApJ this spring as AMP I.

So, how do we compute pn?

yxyxc

c

dxdyyyGxxGxyyyxp

xy

,mod ),,(),,())((),(

Gaussian 2D with )( curve ofn Convolutio on Distributi Model

points data and model ofy probabilitJoint

1tot

N

nnpp

N

yx

ynnxnnnn ydxdyyGxxGyxpp,

mod ),,(),,(),(

The General Statistical Problem: Given a set of points (xn,yn) with measurement errors (xn,yn),how well does the curve yc(x) and sample variance (x,y) fit the data?

xn

yn

x

y

yc(x)

yc(x)

xn

yn

(xn , yn)

It can be shown that the joint probability pn

of these two 2D distributions is equivalent to...

yx

yxc dxdyyyGxxGxyyyxp,

mod ),,(),,())((),(

yx

ynnxnnnn ydxdyyGxxGyxpp,

mod ),,(),,(),(

x

y

yc(x)

xn

yn

(xn , yn)

...a 2D convolution of a single 2D Gaussian with a delta function curve:

2222

,

and where

),),((),,())((

ynyynxnxxn

yxynncnxnnncn dxdyyxyGxxGxyyp

But...the result depends on the choice of convolution integration variables.

Also...the convolution integrals are not analytic unless yc(x) is a straight line.

yc(x)

xn

(xtn , ytn)

yn

(xn , yn)

If yc(x) varies slowly over (xn, yn), we can approximate it as a line ytn(x) tangent to the curve and the convolved error ellipse, with slope mtn= tantn

tn

ytn(x)

?dxdy

?else something

Now, we must choose integration variables for the2D convolution integral

yx

ynntnnxnnntnn dxdyyxyGxxGxyyp,

),),((),,())((

yc(x)

xn

yn

(xn , yn)

ytn(x)

?||dzdz

Gaussian. D2 then through integratiopath linear 1D ofelement

where

2

1exp

1

: toreduces always integraly probabilit thechoice, heWhatever t2

222222

dz

dx

dz

m

xxmyy

mp

tnxntnyn

tnntntnn

xntnyn

n

. uses (2005) AgostiniD' dxdz

.1 uses (2001)Reichart 222 dxmdydxdsdz tn

Both D05 and R01 work in some cases, and fail in others...

A new dz is needed.

Linear Fit to Two Points, xn = yn

x

y

D05

R01

Linear Fit to Two Points, xn = yn

x

yx

y

Linear Fit to Two Points, xn = yn

x

y

D05

R01

Linear Fit to Two Points, xn = yn

x

yx

y

Linear Fit to Two Points, xn = yn

x

y

D05

R01

mxy

myx

mxy= myx

R01 is invertible

D05 is not

Linear Fit to Two Points, xn << yn

x

y

D05R01

Linear Fit to Two Points, xn << yn

x

yx

y

Linear Fit to Two Points, xn << yn

x

y

D05

R01

Linear Fit to Two Points, xn << yn

x

yx

y

Linear Fit to Two Points, xn << yn

x

y

D05R01

mxy= myx

mxy

myx

Again, R01 is invertible...though, in this case, it gives thewrong fit.

D05 gives the correct fit for y vs. x,but not for x vs. y, and is stillnot invertible.

Summary of D05 and R01 Statistics: 2 Point Linear Fits

D05

dz = dx

R01

dz = dx/cosAlways Invertible?

No Yes

Reduces to

1D 2?

Yes if xn = 0

No if yn = 0

No

Fitted Slope Biased low unless

xn = 0

Biased unless

xn = yn

xn = yn xn << yn yn << xn

y

x

R01

D05

Circular Gaussian Random Cloud of Points

y

x y

x

R01

D05

Circular Gaussian Random Cloud of Points

y

x

D05myx

D05mxy

R01mxy= myx

Circular Gaussian Random Cloud of Points

D05

R01

p( cosNStrongly biased towardshorizontal fits

p( constNo direction is preferredover another

Fitting to an Ensemble of Gaussian Random Clouds

tnntntnntnxntnyntn

tntn

tn

tnn

xxmyym

dx

dzp

n

and

where

2

1exp

1

:point data andon distributi model ofy probabilitjoint theRecall

222

2

models.linear for "point test two" thepasses statistic The 3.

and ;0or 0 when 1D the toreduces statistic The 2.

;invertible is statistic The 1.

:such that ofation parametriz a find want toWe

2

ynxn

tndx

dz

.cos

11 R01,For 1.factor theD05,For 2

tntn

tntn

mdx

dz

dx

dz

A New Statistic: TRF09

A New Statistic: TRF09

., ellipseerror intrinsic theto

curve theofpoint tangent theand , connecting

segment thelar toperpendicu line the toparallel be to

definingby satisfied are conditions theseall that found have We

ynxn

cnn xyyx

dz

tn

tn

yc(x)

xn

yn

(xn , yn)

dz

A New Statistic: TRF09

., ellipseerror intrinsic theto

curve theofpoint tangent theand , connecting

segment thelar toperpendicu line the toparallel be to

definingby satisfied are conditions theseall that found have We

ynxn

cnn xyyx

dz

tn

tn

xn

yntn

xn

yntn

tnm

m

dx

dz

cos

cos4

2

22

tn

tn

yc(x)

xn

yn

(xn , yn)

dz

A New Statistic: TRF09

., ellipseerror intrinsic theto

curve theofpoint tangent theand , connecting

segment thelar toperpendicu line the toparallel be to

definingby satisfied are conditions theseall that found have We

ynxn

cnn xyyx

dz

tn

tn

xn

yntn

xn

yntn

tnm

m

dx

dz

cos

cos4

2

22

.modellinear for Statistic R011, If 2

tn

tnynxn m

dx

dzσ

.in errors with datafor Statistic D1/05D1,0 If 2 ydx

dzσ

tnxn

.in errors with datafor Statistic D1,0 If 2 xmdx

dzσ tn

tnyn

.spoint test two thepasses and ,invertiblely analytical is TRF09

tn

tn

yc(x)

xn

yn

(xn , yn)

yc(x)

xn

yn

(xn , yn)

tn

dzD05

TRF09

R01

ytn(x)

Build an Instrument: A complete model of absorption and extinction for extragalactic point sources, including: dust extinction and atomic and molecular hydrogen absorption in the host galaxy; Ly forest/Gunn-Peterson trough; and dust extinction in the Milky Way.Model 90% complete, to be submitted to ApJ as AMP II, after testing on a selected sample of GRB afterglows.

Piran, T. Nature 422, 268-269.

Anatomy of GRB EmissionBurst

r ~ 1012-13 cmtobs < seconds

Afterglowr ~ 1017-18 cm

tobs ~ minutes - days

Synchrotron Emission from Forward Shock:Typically Power Laws in Frequency and Time

GRB 010222Stanek et al. 2001, ApJ 563, 592.

CircumburstMedium

Host Galaxy

Ly Forest

Milky Way

Modified DustExcited H2

Jet

GRB

Host DustDamped Ly Lyman limit

MW Dust

Sources of Line-of-Sight Absorption and Extinction

IGM

GP Trough

Parameters & Priors

• The values of some model parameters are known in advance, but with some degree of uncertainty.• If you hold a parameter fixed at a value that later measurements show to be highly improbable, you risk overstating your confidence and drawing radically wrong conclusions from your model fits.• Better to let that parameter be free, but weighted by the prior probability distribution of its value (often Gaussian, but can take any form).• If your model chooses a very unlikely value of the parameter, the fitness is penalized.• As better measurements come available, your adjust your priors, and redo your fits. • The majority of parameters in our model for absorption and extinction are constrained by priors.• Some are priors on the value of a particular parameter in the standard absorption/extinction models (e.g., Milky Way RV). • Others are priors on parameters that describe model distributions fit to correlations of one parameter with another (e.g., if a parameter is linearly correlated with another, the priors are on the slope and intercept of the fitted line).

Historical Example: The Hubble Constant

Sandage 1976: 55±5

GRB Host Galaxy: •Prior on zGRB from spectral observations {1}

Assume total absorption blueward of Lyman limit in GRB rest frame•Dust Extinction (redshifted IR-UV: CCM + FM models):

Free Parameters: AV, c2, c4 [3]Priors on: x0, , c1(c2), RV(c2), c3 /2(c2) from fits to MW, SMC, LMC stellar measurements (Gordon et al. 2003, Valencic et al. 2004) {20}May fit separately to extinction in circumburst medium (could change with time) and outer host galaxy (constant).

•Damped Ly Absorber:Prior on NH from X-ray or preferably optical spectral observations, if available {1}

•Ro-vibrationally Excited H2 Absorption: Use theoretical spectra of Draine (2000)Free Parameter: NH2

(could change in time) [1]

Ly Forest/Gunn-Peterson Trough: •Priors on T(zabs) from fits to QSO flux deficits (Songaila 2004, Fan et al. 2006) {6}

Dust Extinction in Milky Way (IR-Optical: CCM model):•Prior on: RV,MW {1}•Prior on: E(B-V)MW from Schlegel et al. (1998) {1}

Total: minimum [4] free parameters, {30} priors

Extinction/Absorption Model Parameters & Priors

Optical Spectrum Provides Redshift Prior

GRB 050904: z = 6.295Totani et al. 2006 PASJ 58, 485–498.

m1μ

x

CCM Model FM Model

IR-UV Dust Extinction ModelCardelli, Clayton & Mathis (1988), Fitzpatrick & Massa (1988)

UV BumpHeight slope = c2

1

)(

)(

VV

AAR

VB

E

VE

c1

-RV = -AV / E(B-V)

c1 vs. c2 Linear ModelFit to 441 MW, LMC and SMC stars

priors with parameters 4, onsDistributi Sample

tan)(

12

2221

cc

pccbcc

UV Extinction in Typical MW Dust: c2 ~ 1, RV ~ 3

Extinction in Young SFR: c2 ~ 0, E(B-V) small, RV large

Stellar Winds “Gray Dust”

Extinction in Evolved SFR: c2 large, E(B-V) large, RV small

SNe Shocks

RV vs. c2 Smoothly-broken linear modelFit to 441 MW, LMC and SMC stars

SMC

Orion

priors with parameters 6, onsDistributi Sample

ln)(

V2

22222

12211 tantan

2V

Rc

ccbccb pp

eecR

The UV Bump

• Thought to be due to absorption by graphitic dust grains• Shape is described by a Drude profile, which describes the absorption cross section of a forced-damped harmonic oscillator• The frequency of the bump, x0, and the bump width, , are not correlated with other extinction parameters, and are parameterized by Gaussian priors.• The bump height, c3 / , is correlated with c2, with weak bumps found in star-forming regions (young and old), and stronger bumps in the diffuse ISM...

Bump Height vs. c2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars

SMCOrion

priors with parameters 6, onsDistributi Sample

ln)(BH

BH

tantan2

2

22222

12211

c

ccbccb pp

eec

Ro-vibrationally Excited H2 Absorption Spectra

• Fit empirical stepwise linear model to theoretical spectra of Draine (2000) for log NH2

= 16, 18, 20 cm-2

• Linear interpolation/extrapolation gives spectrum for model parameter NH2

log NH2 = 16 cm-2

log NH2 = 18 cm-2

log NH2 = 20 cm-2

Ly Forest Absorption Priors Transmission vs. zabs in 64 QSO Spectra

Gunn-PetersonTrough

priors with parameters 6 , onsDistributi Sample

10ln

1)(log

log

)()( 21

Tz

zzdczzba eezT

Typical GRB Absorption/Extinction Model Spectra

Conduct the Tests: Model fits to IR-Optical-UV photometric observations of a selected set of seven GRB afterglows that exhibit various signatures of the model and/or signs of time-dependent extinction and absorption in the circumburst medium.Work to commence this spring, results to be published partly in AMP II, and partly in later, chronological AMP series.

A Hand-Selected Sample of GRB Afterglows

• Test the dust extinction model and compare to old modeling results:• GRB 030115, 050408 (Nysewander 2006, PhD Thesis)

All exhibit relatively simple emission spectra and light curves. Preference for bursts with data extending to the Lyman limit, and bursts with data obtained using UNC-affiliated instruments (PROMPT, SOAR).

• Test the Gunn-Peterson Trough and Ly Forest models with high-z bursts:• GRB 080913, z = 6.7, GP Trough• GRB 050904, z = 6.3, GP Trough• GRB 060927, z = 5.5, Ly Forest

• Model time-dependent dust extinction (New):• GRB 070125, shows evidence of color evolution, UVOT data available to Lyman limit, UNC collaboration (Updike et al. 2008, ApJ 685, 361.)

• Model (time-dependent?) molecular hydrogen absorption (New):• GRB 980329, unexplained 2 mag drop redward of Ly forest (Fruchter 1999, ApJ 512, 1.)• GRB 050904, evidence of possible early-time H2 that is later destroyed by jet (Haislip et al. 2006, Nature 440, 181.)

Example: GRB 050904Evidence for H2 Evolution?

• Could be due to dissociation of H2 by the jet...• Or, lateral spreading of the jet at late times, so that emission traverses circumburst medium where H2 was never ro-vibrationally excited by the more collimated burst.

Haislip et al. 2006, Nature 440, 181.