The GRB Afterglow Modeling Project (AMP): Statistics and Absorption and Extinction Models

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The GRB Afterglow Modeling Project
(AMP)Statistics and Absorption and Extinction
Models

• Adam S. Trotter
• UNC-Chapel Hill
• PhD Oral Prelim Presentation
• 30 January 2009
• Advisor Prof. Daniel E. Reichart

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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.

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An Instrumentation Thesis
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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.
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The General Statistical Problem Given a set of
points (xn,yn) with measurement errors
(sxn,syn), how well does the curve yc(x) and
sample variance (sx,sy) fit the data?
yc(x)
So, how do we compute pn?
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sy
sx
syn
sxn
(xn , yn)
yc(x)
It can be shown that the joint probability pn of
these two 2D distributions is equivalent to...
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...a 2D convolution of a single 2D Gaussian with
a delta function curve
Syn
Sxn
(xn , yn)
yc(x)
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.
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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
(xtn , ytn)
Syn
qtn
Sxn
(xn , yn)
yc(x)
ytn(x)
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Now, we must choose integration variables for
the 2D convolution integral
Syn
Sxn
(xn , yn)
yc(x)
ytn(x)
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Both D05 and R01 work in some cases, and fail in
others... A new dz is needed.
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Linear Fit to Two Points, sxn syn
y
R01
D05
x
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Linear Fit to Two Points, sxn syn
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Linear Fit to Two Points, sxn syn
x
R01
D05
y
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Linear Fit to Two Points, sxn syn
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Linear Fit to Two Points, sxn syn
y
R01
myx
mxy myx
D05
mxy
R01 is invertible
D05 is not
x
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Linear Fit to Two Points, sxn ltlt syn
y
R01
D05
x
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Linear Fit to Two Points, sxn ltlt syn
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Linear Fit to Two Points, sxn ltlt syn
x
R01
D05
y
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Linear Fit to Two Points, sxn ltlt syn
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Linear Fit to Two Points, sxn ltlt syn
y
R01
mxy myx
D05
myx
mxy
Again, R01 is invertible... though, in this case,
it gives the wrong fit.
D05 gives the correct fit for y vs. x, but not
for x vs. y, and is still not invertible.
x
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Summary of D05 and R01 Statistics 2 Point Linear
Fits
D05 dz dx R01 dz dx/cosq
Always Invertible? No Yes
Reduces to 1D c2? Yes if sxn 0 No if syn 0 No
Fitted Slope Biased low unless sxn 0 Biased unless sxn syn
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Circular Gaussian Random Cloud of Points
R01
D05
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Circular Gaussian Random Cloud of Points
R01
D05
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Circular Gaussian Random Cloud of Points
D05 myx
R01 mxy myx
D05 mxy
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Fitting to an Ensemble of Gaussian Random Clouds
D05
p(q ) µ cosNq Strongly biased towards horizontal
fits
R01
p(q ) const No direction is preferred over
another
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A New Statistic TRF09
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A New Statistic TRF09
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dz
jtn
syn
qtn
sxn
(xn , yn)
yc(x)
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A New Statistic TRF09
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dz
jtn
syn
qtn
sxn
(xn , yn)
yc(x)
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A New Statistic TRF09
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jtn
syn
qtn
sxn
(xn , yn)
yc(x)
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dz
D05
TRF09
R01
Syn
qtn
Sxn
(xn , yn)
yc(x)
ytn(x)
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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 Lya 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.
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Anatomy of GRB Emission
Burst r 1012-13 cm tobs lt seconds
Afterglow r 1017-18 cm tobs minutes - days
Piran, T. Nature 422, 268-269.
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Synchrotron Emission from Forward
Shock Typically Power Laws in Frequency and Time
GRB 010222 Stanek et al. 2001, ApJ 563, 592.
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Sources of Line-of-Sight Absorption and Extinction
Circumburst Medium
Host Galaxy
Milky Way
IGM
Jet
GRB
Lya Forest
Modified Dust Excited H2
Host Dust Damped Lya Lyman limit
MW Dust
GP Trough
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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).

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Historical Example The Hubble Constant
Sandage 1976 555
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Extinction/Absorption Model Parameters Priors

• 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, g, c1(c2), RV(c2), c3 /g 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 Lya 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
• 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 1
• Prior on E(B-V)MW from Schlegel et al. (1998)
  1

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Optical Spectrum Provides Redshift Prior
GRB 050904 z 6.2950.002 Totani et al. 2006
PASJ 58, 485498.
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IR-UV Dust Extinction Model Cardelli, Clayton
Mathis (1988), Fitzpatrick Massa (1988)
FM Model
CCM Model
UV Bump Height
slope c2
c1
-RV -AV / E(B-V)
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c1 vs. c2 Linear Model Fit to 441 MW, LMC and SMC
stars
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UV Extinction in Typical MW Dust c2 1, RV 3
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Extinction in Young SFR c2 0, E(B-V) small,
RV large
Stellar Winds
Gray Dust
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Extinction in Evolved SFR c2 large, E(B-V)
large, RV small
SNe Shocks
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RV vs. c2 Smoothly-broken linear model Fit to 441
MW, LMC and SMC stars
Orion
SMC
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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...

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Bump Height vs. c2 Smoothly-broken linear model
Fit to 441 MW, LMC and SMC stars
SMC
Orion
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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
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Lya Forest Absorption Priors Transmission vs.
zabs in 64 QSO Spectra
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Typical GRB Absorption/Extinction Model Spectra
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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.
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A Hand-Selected Sample of GRB Afterglows
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 dust extinction model and compare to
  old modeling results
• GRB 030115, 050408 (Nysewander 2006, PhD Thesis)

• Test the Gunn-Peterson Trough and Lya 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, Lya 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
  Lya 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.)

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Example GRB 050904 Evidence 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.