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. 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.

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. So, how do we compute p n ? The General Statistical Problem: Given a set of points (x n,y n ) with measurement errors (  xn,  yn ), how well does the curve y c (x) and sample variance (  x,  y ) fit the data?  xn  yn xx yy yc(x)yc(x)

6. yc(x)yc(x)  xn  yn (x n, y n ) It can be shown that the joint probability p n of these two 2D distributions is equivalent to... xx yy

7. yc(x)yc(x)  xn  yn (x n, y n )...a 2D convolution of a single 2D Gaussian with a delta function curve: But...the result depends on the choice of convolution integration variables. Also...the convolution integrals are not analytic unless y c (x) is a straight line.

8. yc(x)yc(x)  xn (x tn, y tn )  yn (x n, y n ) If y c (x) varies slowly over (  xn,  yn ), we can approximate it as a line y tn (x) tangent to the curve and the convolved error ellipse, with slope m tn = tan  tn  tn y tn (x)

9. Now, we must choose integration variables for the 2D convolution integral yc(x)yc(x)  xn  yn (x n, y n ) 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,  xn =  yn x y D05 R01

12. Linear Fit to Two Points,  xn =  yn x y x y

13. x y D05 R01

14. Linear Fit to Two Points,  xn =  yn x y x y

15. x y D05 R01 m xy m yx m xy = m yx R01 is invertible D05 is not

16. Linear Fit to Two Points,  xn <<  yn x y D05 R01

17. Linear Fit to Two Points,  xn <<  yn x y x y

18. x y D05 R01

19. Linear Fit to Two Points,  xn <<  yn x y x y

20. x y D05 R01 m xy = m yx m xy m yx 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.

21. Summary of D05 and R01 Statistics: 2 Point Linear Fits D05 dz = dx R01 dz = dx/cos  Always Invertible? NoYes Reduces to 1D  2 ? Yes if  xn = 0 No if  yn = 0 No Fitted SlopeBiased low unless  xn = 0 Biased unless  xn =  yn  xn <<  yn  yn <<  xn

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

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

24. y x D05 m yx D05 m xy R01 m xy = m yx Circular Gaussian Random Cloud of Points

25. D05 R01 p(   cos N  Strongly biased towards horizontal fits p(   const No direction is preferred over another Fitting to an Ensemble of Gaussian Random Clouds

26. A New Statistic: TRF09

28.  tn  tn yc(x)yc(x)  xn  yn (x n, y n ) dz

29. A New Statistic: TRF09

30.  tn  tn yc(x)yc(x)  xn  yn (x n, y n ) dz

31. A New Statistic: TRF09

32.  tn  tn yc(x)yc(x)  xn  yn (x n, y n )

33. yc(x)yc(x)  xn  yn (x n, y n )  tn dz D05 TRF09 R01 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; 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.

35. Piran, T. Nature 422, 268-269. Anatomy of GRB Emission Burst r ~ 10 12-13 cm t obs < seconds Afterglow r ~ 10 17-18 cm t obs ~ minutes - days

36. Synchrotron Emission from Forward Shock: Typically Power Laws in Frequency and Time GRB 010222 Stanek et al. 2001, ApJ 563, 592.

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

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 R V ). 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. GRB Host Galaxy: Prior on z GRB 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: A V, c 2, c 4 [3]  Priors on: x 0, , c 1 (c 2 ), R V (c 2 ), c 3 /  2 (c 2 ) 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 N H from X-ray or preferably optical spectral observations, if available {1} Ro-vibrationally Excited H 2 Absorption: Use theoretical spectra of Draine (2000)  Free Parameter: N H 2 (could change in time) [1] Ly  Forest/Gunn-Peterson Trough: Priors on T(z abs ) from fits to QSO flux deficits (Songaila 2004, Fan et al. 2006) {6} Dust Extinction in Milky Way (IR-Optical: CCM model): Prior on: R V,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

41. Optical Spectrum Provides Redshift Prior GRB 050904: z = 6.295  Totani et al. 2006 PASJ 58, 485–498.

42. CCM Model FM Model IR-UV Dust Extinction Model Cardelli, Clayton & Mathis (1988), Fitzpatrick & Massa (1988) UV Bump Height slope = c 2 c1c1 -R V = -A V / E(B-V)

43. c 1 vs. c 2 Linear Model Fit to 441 MW, LMC and SMC stars

44. UV Extinction in Typical MW Dust: c 2 ~ 1, R V ~ 3

45. Extinction in Young SFR: c 2 ~ 0, E(B-V) small, R V large Stellar Winds“Gray Dust”

46. Extinction in Evolved SFR: c 2 large, E(B-V) large, R V small SNe Shocks

47. R V vs. c 2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars SMC Orion

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, x 0, and the bump width, , are not correlated with other extinction parameters, and are parameterized by Gaussian priors. The bump height, c 3 /  , is correlated with c 2, with weak bumps found in star- forming regions (young and old), and stronger bumps in the diffuse ISM...

49. Bump Height vs. c 2 Smoothly-broken linear model Fit to 441 MW, LMC and SMC stars SMC Orion

50. Ro-vibrationally Excited H 2 Absorption Spectra Fit empirical stepwise linear model to theoretical spectra of Draine (2000) for log N H 2 = 16, 18, 20 cm -2 Linear interpolation/extrapolation gives spectrum for model parameter N H 2 log N H 2 = 16 cm -2 log N H 2 = 18 cm -2 log N H 2 = 20 cm -2

51. Ly  Forest Absorption Priors Transmission vs. z abs in 64 QSO Spectra Gunn-Peterson Trough

52. Typical GRB Absorption/Extinction Model Spectra

53. 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.

54. 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 H 2 that is later destroyed by jet (Haislip et al. 2006, Nature 440, 181.)

55. Example: GRB 050904 Evidence for H 2 Evolution? Could be due to dissociation of H 2 by the jet... Or, lateral spreading of the jet at late times, so that emission traverses circumburst medium where H 2 was never ro-vibrationally excited by the more collimated burst. Haislip et al. 2006, Nature 440, 181.