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

1. 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

2. 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.

3. 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; Lya 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. An “Instrumentation Thesis”

4. 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.

5. syn sy sx sxn 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?

6. 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...

7. ...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 straightline.

8. If yc(x) varies slowly over (Sxn, Syn), we can approximate it as a line y¢tn(x) tangent to the curve and the convolved error ellipse, with slope m¢tn= tanq¢tn (x¢tn , y¢tn) Syn q¢tn Sxn (xn , yn) yc(x) y¢tn(x)

9. Now, we must choose integration variables for the 2D convolution integral Syn Sxn (xn , yn) yc(x) y¢tn(x)

10. Both D05 and R01 work in some cases, and fail in others... A new dz is needed.

11. Linear Fit to Two Points, sxn = syn y R01 D05 x

12. y y x x Linear Fit to Two Points, sxn = syn

13. Linear Fit to Two Points, sxn = syn x R01 D05 y

14. x x y y Linear Fit to Two Points, sxn = syn

15. Linear Fit to Two Points, sxn = syn y R01 myx mxy= myx D05 mxy R01 is invertible D05 is not x

16. Linear Fit to Two Points, sxn << syn y R01 D05 x

17. y y x x Linear Fit to Two Points, sxn << syn

18. Linear Fit to Two Points, sxn << syn x R01 D05 y

19. x x y y Linear Fit to Two Points, sxn << syn

20. Linear Fit to Two Points, sxn << 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

21. sxn = syn sxn << syn syn << sxn Summary of D05 and R01 Statistics: 2 Point Linear Fits

22. y x Circular Gaussian Random Cloud of Points R01 D05

23. x y y x Circular Gaussian Random Cloud of Points R01 D05

24. y x Circular Gaussian Random Cloud of Points D05 myx R01 mxy= myx D05 mxy

25. 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

26. A New Statistic: TRF09

27. A New Statistic: TRF09

28. dz jtn syn qtn sxn (xn , yn) yc(x)

29. A New Statistic: TRF09

30. dz jtn syn qtn sxn (xn , yn) yc(x)

31. A New Statistic: TRF09

32. jtn syn qtn sxn (xn , yn) yc(x)

33. dz D05 TRF09 R01 Syn q¢tn Sxn (xn , yn) yc(x) y¢tn(x)

34. 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.

35. Anatomy of GRB Emission Burst r ~ 1012-13 cm tobs < seconds Afterglow r ~ 1017-18 cm tobs ~ minutes - days Piran, T. Nature422, 268-269.

36. Synchrotron Emission from Forward Shock: Typically Power Laws in Frequency and Time GRB 010222 Stanek et al. 2001, ApJ563, 592.

37. Sources of Line-of-Sight Absorption and Extinction Circumburst Medium IGM Milky Way Host Galaxy Jet GRB Modified Dust Excited H2 Host Dust Damped Lya Lyman limit GP Trough Lya Forest MW Dust

38. 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).

39. Historical Example: The Hubble Constant Sandage 1976: 55±5

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

41. Optical Spectrum Provides Redshift Prior GRB 050904: z = 6.295±0.002 Totani et al. 2006 PASJ58, 485–498.

42. 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)

43. c1 vs. c2 Linear Model Fit to 441 MW, LMC and SMC stars

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

45. Extinction in Young SFR: c2 ~ 0, E(B-V)small, RV large Stellar Winds “Gray Dust”

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

47. RV vs. c2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars Orion SMC

48. 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...

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

50. 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