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
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
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
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
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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