Adam S. Trotter UNC-Chapel Hill, Dept. of Physics & Astronomy PhD Final Oral Examination

The Gamma-Ray Burst Afterglow Modeling Project (AMP):

Foundational Statistics and Absorption & Extinction Models

Adam S. TrotterUNC-Chapel Hill, Dept. of Physics & Astronomy

PhD Final Oral Examination30 June 2011

Advisor: Prof. Daniel E. Reichart

AMP: The GRB Afterglow Modeling Project

Model, in a statistically sound and self-consistent way, every GRB afterglow observed since the first detection in 1997, using all available radio, infrared, optical, ultraviolet and X-ray data.

Can get physical information about GRBs… Eiso, εe, εB, p, jet geometry

Can get physical information about GRB environments…n(r)rk, AV, NH, Extinction Curves,

Dust/Gas Modification

An “Instrumentation Thesis”Forge a Tool: Statistic

A new statistical technique for fitting models to 2D data with uncertainties in both dimensions

Build an Instrument: ModelGRB emissionMW extinctionSource-frame extinction & absorptionIGM absorption

Test it Out: Fit the FirstGRB 090313 z = 3.375IR/optical/X-ray dataTests all aspects of model

Forge a Tool: The TRF StatisticA new statistical formalism for fitting model

distributions to 2D data sets with intrinsic uncertainty (error bars) in both dimensions, and with extrinsic uncertainty (slop) greater than can

be attributed to measurement errors alone

So, how do we compute pn?

The General Statistical Problem: Given a set of points (xn,yn) with measurement errors (sxn,syn),how well does the model distribution fit the data?

sxn

syn

sx

sy

yc(x)

yx

yxc dxdyyyGxxGxyyyxp,

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

Model Distribution = Curve yc(x) convolved with 2D Gaussian

1

N

nnpp

yx

ynnxnnnn ydxdyyGxxGyxpp,

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

Joint Probability of Model Distribution and Data

yc(x)

sxn

syn

(xn , yn)

It can be shown that the joint probability pn

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

yx

yxc dxdyyyGxxGxyyyxp,

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

yx

ynnxnnnn ydxdyyGxxGyxpp,

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

sx

sy

yc(x)

Sxn

Syn

(xn , yn)

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

2222

,

and where

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

ynyynxnxxn

yxynncnxnnncn dxdyyxyGxxGxyyp

ssss

SS

SS

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)

Sxn

(xtn , ytn)

Syn

(xn , yn)

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

qtn

ytn(x)

?dxdy

?else something

Now, we must choose integration variables for the2D convolution integral

SSyx

ynntnnxnnntnn dxdyyxyGxxGxyyp,

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

yc(x)

Sxn

Syn

(xn , yn)

ytn(x)

?||dudu

Gaussian. D2 then through integratiopath linear 1D ofelement where

21exp1

: toreduces always integraly probabilit thechoice, heWhatever t2

222222

SS

SS

du

m

xxmyydxdu

mp

xntnyn

tnntntnn

tnxntnyn

n

. uses (D05) AgostiniD' dxdu

.1 uses (R01)Reichart 222 dxmdydxdsdu tn

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

A new du is needed.

directionin that ellipse of radius 1 andpoint tangent todistance radial where

21exp1

:point data andon distributi model ofy probabilitjoint The2

222

s

S

S

SS

tn

tn

tn

tn

tnxntnyn

n dxdu

mp

n

A New Statistic: TRF

S

SS

2

222 21exp1

tn

tn

tnxntnyn

n dxdu

mp

Starting to look like 2…same for all statistics

directionin that ellipse of radius 1 andpoint tangent todistance radial where

21exp1

:point data andon distributi model ofy probabilitjoint The2

222

s

S

S

SS

tn

tn

tn

tn

tnxntnyn

n dxdu

mp

n

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

;invertible is statistic The 1.

:such that ofation parametriz a find want toWe

2 SS

ynxn

tndxdu

.1 R01,For 1.factor theD05,For 2tn

tntn

mdxdu

dxdu

A New Statistic: TRF

A New Statistic: TRF

., ellipseerror intrinsic theto

curve theofpoint tangent theand , connectingsegment thelar toperpendicu line the toparallel be to

definingby satisfied are conditions theseAll

ynxn

cnn xyyxdu

SS

D05

TRF

R01

A New Statistic: TRF

., ellipseerror intrinsic theto

curve theofpoint tangent theand , connectingsegment thelar toperpendicu line the toparallel be to

definingby satisfied are conditions theseAll

ynxn

cnn xyyxdu

SS

442

222

ynxntn

ynxntn

tn m

mdxdu

SS

SS

.in errors with datafor statistic like- D1/05D1,0 If 2 ydxdu

tnxn

S

.in errors with datafor statistic like- D1,0 If 2 xmdxdu

tntn

yn

S

Analytically Invertible:same fit to y vs. x as x vs. y

Reduces to 2 in 1D limits

S

S

2

2TRF

21exp1

tn

tn

tnnp

2-like measured in direction of closest approach of curve to data

point…intuitive!

y

x

TRF

D05

Circular Gaussian Random Cloud of Points

y

x y

x

TRF

D05

Circular Gaussian Random Cloud of Points

y

x

D051/myx

D05mxy0

TRFmxy= 1/myx

Circular Gaussian Random Cloud of Points

D05

TRF

p(q cosNqStrongly biased towardshorizontal fits

p(q constNo direction is preferredover another

Expected Fits to an Ensemble of Gaussian Random Clouds

Actual Fits to Ensemble of 1000 Gaussian Random Clouds

But…TRF is not Scalable

S

SSSS

2

2

4242

222

21exp

tn

tn

ynxntn

ynxntnn sm

mp

s cannot be factored out of total joint probability

Best-fit curve depends on choice of s

Distribution of slop into x- and y-dimensionsdepends on s

TRF at s0 = D05

TRF at s = Inverted D05

TRF at intermediate s

Slop-Dominated Linear Fit

(This is what Excel would give)

TRF at smax

Inverted D05

TRF at intermediate s

Linear Fit with Slop and Error Bars

TRF at smin

D05

smax is scale where fittedslop σy0

smin is scale where fittedslop σx0

D05 limited to inversion or non-inversionTRF can fit to a continuum of scales

Range of Physical Scales 0 as Error Bars Dominate Over Slop

Pearson Correlation CoefficientR2

xy = myxmxy

Useless for Invertible StatisticR2

xy 1TRF Scale-Based Correlation Coefficient

used to find “optimum scale” s0

22maxmin

2'TRF 24

tan),( maxmin

xyss RssR

qq

qsmin

qsmax

qs0

2TRFmax0

2'TRF0min

2'TRF ),(),( RssRssR

TRF can be generalized to non-linear fits…

smin

smax

s0

…And to asymmetric intrinsic and extrinsic uncertainties

(See Appendix A, B…)

Build an Instrument: Models

GRB emissionMW Dust Extinction

Source-Frame Dust ExtinctionSource-Frame Lyα Absorption

IGM Lya forest/Gunn-Peterson Trough

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

See, e.g., Meszaros & Rees, 1997; Sari et al., 1998; Piran, 1999; Chevalier & Li, 1999; Granot et al., 2000; Meszaros, 2002.

log

N(E

)

log EEm

p < -2

CircumburstMedium Host Galaxy

Lya Forest

Milky Way

Modified Dustand Gas

Jet

GRB

Host Dustand Gas

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 in Source Frame: Free Parameters: AV, c2, c4 [3] Priors on: c1(c2), RV(c2), BH(c2), x0, g from fits to MW, SMC, LMC stellar

measurements (Gordon et al. 2003, Valencic et al. 2004) {22}

• Damped Lya Absorber: Free Parameter: NH [1]

Lya 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 {3}• Prior on: E(B-V)MW from Schlegel et al. (1998) {1}

Total: [4] free parameters, {33} priors

Extinction/Absorption Model Parameters & Priors

m1μx

CCM Model FM Model

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

UV BumpHeight slope = c2

1

)(

)(

VV

AAR

VB

EV

E

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

pccbccss

q

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

eecRss

qq

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, g , are not correlated with

other extinction parameters, and are parameterized by Gaussian priors.• The bump height, c3 / g 2 , 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

ss

qq

c

ccbccb pp

eec

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

Gunn-PetersonTrough

priors with parameters 6

)()1(

ln))(lnln(2/1

abs0)lnln(

)(tan)(tan 222111

zz

eezT

T

zzbzzb

a

qq

ss

Typical GRB Absorption/Extinction Model Spectra

Test it Out: Fit the FirstGRB 090313, z = 3.375

Fit Models to NIR/optical/X-ray ObservationsLyα Forest and Lyman Limit in Optical

UV Extinction in NIR/OpticalObtain Dust Extinction Curve in High-z SFR

…Possibly Modified by GRBGalapagos-Enabled Science: Parameter Linking

…Rebrightening: Intrinsic or Extrinsic?

UV Bump

Lyα

Lyα Forest

Lyman Limit

GRB 090313: z = 3.375

= p/2 = -1.12

Cooling break mostly below the NIR/opticalCannot distinguish between

ISM (k = 0) and Stellar Wind (k = -2) Models

c

k = 0 k = -2

GRB 090313 Light Curve: Intrinsic Rebrightening?

Jet Break

Slow Rise

α = p = -2.24

α = p/2 = -1.12

logF 0.3

Galapagos-Enabled Science:Nested Models

Through parameter linking, can obtainrelative likelihood of rebrightening due to intrinsic or extrinsic causes.

Either case is statistically equally likelyfor this burst, but…

If rebrightening is due to variable extinctionAV and c2 … a lot, to account for NIR data

Opposite of what we expect if widening jetilluminates unmodified dust at late times

If it’s due to variable intrinsic emission, We can estimate changes necessary inmicro- and macro-physical quantities:

Eiso, e, B, and n

…dramatic changes in e, B not likely.

logF ~ 0.3 Factor of ~2 increase in Eiso (Energy Injection)

or Factor of ~7 increase in n (Density Variation)

If we can measure a , m , c , and Fm

We can compute values of (not just changes in)

Eiso, e, B, and n (or A*)

Requires radio – X-ray data at early and late times.

GRB-Modified Dust?

Fitted Extinction Curve: AV 0.3, c2 2.2, c4 0.3

Fitted NH < 61021 cm-2 (3σ) -- due to neutral hydrogen

XRT NH = 31022 cm-2 -- due to metals (and typical of GMCs)

Either old, SMC-like star forming region, or modified (fragmented) dust along LOS

Either higher than solar metallicity (not likely at z = 3.375), or hydrogen is ionized at > 80% level

Suggests gas (and dust) is local to GRB

Fragmentation of dust by GRB emission results in higher c2, c4

By fitting extinction curves to dozens of GRBs over a wide range of redshifts, AMP will probe evolution

(and modification) of dust and gas over the history of the universe.

By modeling with parameter linking, we can determine relative likelihood of nested models, and measure values of

and changes in intrinsic and environmental physical parameters of GRBs

We don’t know yet what else we’ll find…

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Thanks To

Brad BarlowMatt Bayliss

Summers BrennanTodd BorosonGerald Cecil

Art ChampagneChris ClemensRebecca EggerAndrew Foster

Nicholas FinkelsteinJason Freitas

Alyssa GoodmanLeonid GurvitsJosh Hailslip

Fabian HeitschGina HodgesKevin Ivarsen

Kannan JagannathanJohn KolenaHarlan Lane

Aaron LaCluyzéHelen Lineberger

Kitty MatkinsScott MitchellJustin MooreJim Moran

Melissa NysewanderApurva OzaFrank Philip

Richard PillardMichael Reilly

Jim RoseRachel RosenBrian ShumanMark Schubel

Eric SpeckhardDon Smith

Jana StyblovaRob TrotterElise Weaver

Special Thanks toDan Reichart

Dedicated in Honor ofMy Parents

Al & Gay Trotter

and of My Grandmothers

Ethel Trotter & Eleanor Chappell

and in Loving Memory of My Grandfathers

Frank Trotter & Robert Chappell

and of My Great-GrandfatherRobert Greeson Fitzgerald

UNC – Chapel Hill, Class of 1913

ParaVicente Rosario

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