Effect of Dust Extinction on Gamma-ray Burst Afterglows
aa r X i v : . [ a s t r o - ph . H E ] A ug Effect of Dust Extinction on Gamma-ray Burst Afterglows
Gu-Jing L¨u , Lang Shao , , Zhi-Ping Jin , and Da-Ming Wei [email protected](L.S.) ABSTRACT
In order to study the effect of dust extinction on the afterglow of gamma-raybursts (GRBs), we carry out numerical calculations with high precision based onrigorous Mie theory and latest optical properties of interstellar dust grains, andanalyze the different extinction curves produced by dust grains with differentphysical parameters. Our results indicate that the absolute extinction quantityis substantially determined by the medium density and metallicity. However,the shape of the extinction curve is mainly determined by the size distributionof the dust grains. If the dust grains aggregate to form larger ones, they willcause a flatter or grayer extinction curve with lower extinction quantity. On thecontrary, if the dust grains are disassociated to smaller ones due to some uncer-tain processes, they will cause a steeper extinction curve with larger amount ofextinction. These results might provide an important insight into understandingthe origin of the optically dark GRBs.
Subject headings: gamma-rays burst: general–interstellar medium: dust, extinc-tion
1. INTRODUCTION
Gamma-ray burst (GRB) is known as one of the most energetic stellar explosions in theuniverse. At present, the
Swift satellite (Gehrels et al. 2004), a NASA mission dedicatedto monitor this phenomenon, carries three instruments with separate wave bands: BurstAlert Telescope (BAT; ∼ −
150 keV), X-Ray Telescope (XRT; ∼ . −
10 keV) andUltraviolet/Optical Telescope (UVOT). BAT is able to catch about 100 GRBs per year,and XRT is able to follow them rapidly in the X-ray band and pinpoint their positions Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China Department of Physics, Hebei Normal University, Shijiazhuang 050016, China ∼
2. PHYSICAL MECHANISM OF DUST EXTINCTION
Considering a spherical dust grain with radius a and complex refractive index ˜ m , andbased on Mie theory (van de Hulst 1957), the extinction cross-section for the incident lightwith wavelength λ is σ ext = 2 πk Σ ∞ n =1 (2 n + 1)Re { a n + b n } , (1)the scattering cross-section is σ sca = 2 πk Σ ∞ n =1 (2 n + 1)( | a n | + | b n | ) , (2)and the absorption cross-section is σ abs = σ ext − σ sca , where k = 2 π/λ , and the scatteringcoefficients a n and b n are a n = ˜ mψ n ( ˜ mx ) ψ ′ n ( x ) − ψ n ( x ) ψ ′ n ( ˜ mx )˜ mψ n ( ˜ mx ) ξ ′ n ( x ) − ξ n ( x ) ψ ′ n ( ˜ mx ) , (3) b n = ψ n ( ˜ mx ) ψ ′ n ( x ) − ˜ mψ n ( x ) ψ ′ n ( ˜ mx ) ψ n ( ˜ mx ) ξ ′ n ( x ) − ˜ mξ n ( x ) ψ ′ n ( ˜ mx ) , (4) 3 –where x = ka = 2 πa/λ is the dimensionless size parameter, and ψ n ( x ) and ξ n ( x ) are Riccati-Bessel functions.Methods for numerical calculations based on Mie theory are mature now. Owing to therapid development of computer science, the calculation of the infinite sums can be performedon popular PCs at present, instead of using supercomputers in the past. Numerical analysesindicate that the infinite series summation in Equations (1)-(4) can be approximated bythe first N = x + 4 x / + 2 terms with a sufficiently high precision (Wiscombe 1980),where x is the dimensionless size parameter mentioned above. Currently, there are a fewpopular FORTRAN codes (Wiscombe 1980; Bohren & Huffman 1983), which can be veryefficient (for a single calculation with x ≈ , it only takes a couple of seconds on an IntelPC with a main frequency of 2.6 GHz), but is also numerically unstable and can be verytime-consuming for multi-wavelength calculations, due to the lack of real-time adjustmentof the precision, especially when x is very large (for X-rays and large dust grains, x & ).For the evaluation of absorption and scattering cross-sections over a broad bandpass and awide size range of dust grains, multiple analytical approximations are usually adopted forinterpolation (Wiscombe 1980). In order to study the properties of X-ray scattering andabsorption by large dust grains, we make an extensive use of the latest MieSold code in theadvanced language Mathematica which can make real-time adjustment of the precision witha sacrifice of the speed (for a single calculation with x ≈ , it takes about 1 h on an IntelPC with a main frequency of 2.6 GHz). Nevertheless, the precision is greatly improved byself-adapting calculations over a much larger parameter space .There are a variety of substances in the interstellar medium (ISM). The compositionand optical properties of most dust grains can not be obtained directly by experiment orobservation. They are mainly measured jointly by laboratory experiments, theoretical mod-elings and astronomical observations. At present, silicate and graphite are known as thetwo most important ingredients in ISM (Draine 2003b). Their optical properties have beensystematically studied by Draine and his colleagues and the latest results on their complexrefractive indices ˜ m have been summarized in Figure 1 (Draine & Lee 1984; Laor & Draine1993; Li & Draine 2001; Draine 2003a). The optical properties of graphite are highlyanisotropic, and the value of ˜ m is dependent on the angle included between the directionof electric field and the crystal axis. The “1 / − /
3” approximation is usually adopted inmost evaluations, i.e., graphite is assumed as a mixture of two types of isotropic substances.Among them, 1 / / Zimmer C, Aragon S R, Mie Scattering and Absorption from Bubbles and Spheres, Mathematica Journal,to be submitted. m ), the imaginary part of ˜ m ). The edgeabsorptions of silicate are quite abundant, including the multiple edge absorptions from Mg,Fe, Si and O, while graphite has only a K edge absorption between 282 and 310 eV (Draine2003a).Based on the above-mentioned optical properties of silicate and graphite, we can obtaintheir absorption and scattering cross-sections as functions of the grain size a and the energy ofthe incident light E by precise evaluations according to Mie theory. As shown in Figure 2, theresults given by the MieSolid code (Rigorous Mie) are very consistent with those previouslycombined results based on multiple analytical approximations (Mie, Rayleigh-Gans [RG]and Geometric Optics [GO]) and have higher spectral resolution with better performanceon the edge absorptions. The only flaw is that it is unstable in the ultraviolet and softX-ray band when the grain size is larger than 1 µ m, which shall be tackled in the future codedebugging. In this work, we adopt the existing approximative results for the unstable regionvia interpolation which will have negligible effect on the final results, since large grains inmost standard dust models are deficient.Our results (as in Figure 2) indicate that, when the typical grain size is small ( a < . µ m) absorption dominates the extinction with most of the incident energy transformedinto ambient thermal energy. In this case, the scattering will be relatively weak, and it hasbeen overlooked in previous works. On the contrary, when the typical grain size is large( a > . µ m), the scattering dominates the extinction, especially the X-ray scattering willplay an important role. As the grain is getting larger, the scattering is more effective. Theseresults might have crucial implications for the studies of X-ray scattering in GRB afterglows(Shao & Dai 2007; Shao et al. 2008).
3. EXTINCTION OF GRB AFTERGLOWS
The size distribution of the dust grains around GRBs (Mathis et al. 1977) can beassumed obey a power law between ( a min , a max ) given byd N i d a ( a ) = A i × N H a β ( a min ≤ a ≤ a max ) , (5)where d N i / d a are the column densities per unit radius of silicate ( i = 1) and graphite( i = 2), respectively, A and A are the coefficients that quantify their absolute columndensities, N H is the column density of hydrogen atoms, β is the dimensionless power-lawindex. Accordingly, the dust grain mass per unit hydrogen mass (Laor & Draine 1993), i.e.,the equivalent metallicity is given by 5 – - - - - Re H m Ž L m Ž - H m Ž L Silicate
Radio IR Optical UV X - ray - - - m Ž Re H m Ž L m Ž - H m Ž L Graphite H Parallel L - - - - - E (cid:144) eV - - - - Re H m Ž L m Ž - H m Ž L Graphite H Perpendicular L Fig. 1.— Complex refractive indices ˜ m of spherical silicate and graphite as functions of theenergy of an incident photon. The real part and imaginary parts of the complex refractiveindices represented by black and gray solid lines, respectively, and the values of | ˜ m − | thatis frequently used in literatures are represented by dotted lines. 6 – - - - - Rigorous MieMie + RG + GO Silicate a = Μ m10 - - - - a = Μ m10 - - - - Q = Σ Π a a = Μ m10 - - - - absorption scattering a = Μ m10 - - E (cid:144) eV10 - - - - Radio IR Optical UV X - ray a = Μ m 10 - - - - Rigorous MieMie + RG + GO Graphite a = Μ m10 - - - - a = Μ m10 - - - - Q = Σ Π a a = Μ m10 - - - - absorption scattering a = Μ m10 - - E (cid:144) eV10 - - - - Radio IR Optical UV X - ray a = Μ m Fig. 2.— Variations of absorption and scattering cross sections of spherical silicate andgraphite with frequency. 7 – f d = 4 π m H a β +4max β + 4 " − (cid:18) a min a max (cid:19) β +4 i A i ρ i , (6)where ρ = 3 . / cm and ρ = 2 . / cm are the mass densities of silicate and graphite,respectively, m H is the mass of hydrogen atom. Here f d , A and A are not completelyindependent. Thereafter f d and A /A will be considered as two independent parameters.Meanwhile, the extinction optical depth can be given by τ ( λ ) = Z X i σ i ext ( λ ) d N i d a d a , (7)and the extinction magnitude is A ( λ ) = 1 . τ ( λ ).The GRB afterglows that are emitted by the shock-accelerated electrons in the rela-tivistic outflow usually exhibit a power-law spectrum from the optical to X-ray band, whichhereafter is assumed to be F ν ∝ ν − (as shown by the gray solid line in Figure 3; Shao et al.(2010)). The column density of hydrogen atoms is the principal quantity that dominatesthe extinction from ultraviolet to soft X-ray band. As shown in Figure 3, from top to bot-tom, the solid, dotted, short-dashed and long-dashed lines represent that the values of N H are 10 cm − , 10 cm − , 10 . cm − and 10 cm − , respectively. All the other physicalparameters have the typical vales in ISM, where the metallicity f d is 0.01, the ratio of sili-cate and graphite A /A is 1 and the parameters for grain size distribution are β = − . a min = 0 . µ m and a max = 0 . µ m. In general, the column density of hydrogen atoms N H determines the absolute amount of extinction. In some dense regions of the surroundingmedium N H could be very high (usually N H > cm − ), the optical to soft X-ray emissionsfrom GRB afterglows would be severely attenuated.Besides the column density of hydrogen atoms, many other factors will also affect theextinction curve (including the absolute amount of extinction and the profile of the extinctioncurve). Herein we mainly consider some key physical quantities: the ratio of silicate andgraphite A /A , the metallicity f d , parameters for dust grain size distribution β and a max .The impact of A /A on the extinction curve is shown in Figure 4, where A /A = 0 .
6, 1 .
0, 1 . . f d = 0 . N H = 10 . cm − , β = − . a min = 0 . µ m and a max = 0 . µ m. As revealed in the figure, A /A mainly affectsthe extinction bump around 2175 ˚A, which has been known to be caused by small graphitegrains (Draine & Malhotra 1993). Therefore, as A /A increases, the extinction bump getsflatter. In general, the composition of dust grains has weak effect on the extinction curveand can not account for why we can observe evidently different extinction curves from GRBafterglows. 8 – N H = cm - N H = cm - N H = cm - N H = cm - - - - - - - - Ν (cid:144) Hz F l ux (cid:144) A r b it r a r y Fig. 3.— Extinction of GRB afterglow by circum-stellar dust grains with different columndensities of hydrogen nuclei N H . Gray straight line represents the intrinsic spectrum of theafterglow. Λ - (cid:144) Μ m - A H Λ L A (cid:144) A = A (cid:144) A = A (cid:144) A = A (cid:144) A = - - - - - - - Ν (cid:144) Hz F l ux (cid:144) A r b it r a r y Fig. 4.— Extinction of GRB afterglow by circum-stellar dust grains with different relativeabundances between silicate and graphite A /A . Left panel is the extinction curve, and rightpanel is the attenuated afterglow spectrum. Gray straight line in the right panel representsthe intrinsic spectrum of the afterglow. 9 –The metallicity f d has a great effect on the extinction curve. Being similar to the columndensity of hydrogen atom N H , which dominates the absolute amount of extinction, largermetallicity causes stronger extinction. As shown in Figure 5, f d = 0 . . .
007 and0 .
01 are represented by the solid, dotted, short-dashed and long-dashed lines, respectively.The other parameters also have their respective typical values, i.e., A /A = 1, N H =10 . cm − , β = − . a min = 0 . µ m and a max = 0 . µ m. The ambient environmentaround a GRB is very complicated. There might be a high metallicity if the explosionoccurs in the latter phase of the massive progenitor star which is an ideal place for the dustformation. This might be the leading cause of the severe extinction and the optically darkGRBs.Obviously, as the computing results indicate, the power-law index of the dust grain sizedistribution β mainly determines the profile of the extinction curve. The chief reason is thatdust grains with different sizes have different contributions to the extinction at different pho-ton frequencies. This is governed by the physics of dust scattering, which is weakly affectedby the ingredients of dust grains. As shown in Figure 6, β = − . − . − . − . f d = 0 . A /A = 1, N H = 10 . cm − , a min = 0 . µ m and a max = 0 . µ m. As the size of dust grains increases, the number ofsmall dust grains decreases and the extinction bump around 2175 ˚A also becomes less ev-ident. Another interesting feature is that β barely affects the optical extinction A V . Thismay explain why we can usually observe different extinction curves from GRB afterglows,but A V is barely correlated with N H (Schady et al. 2007). Our computing results indicatethat difference of size distribution of dust grains might be the internal cause. Λ - (cid:144) Μ m - A H Λ L f d = f d = f d = f d = - - - - - - - Ν (cid:144) Hz F l ux (cid:144) A r b it r a r y Fig. 5.— Extinction of GRB afterglow by circum-stellar dust grains with different metallici-ties. Left panel is the extinction curve, and right panel is the attenuated afterglow spectrum.Gray straight line in the right panel represents the intrinsic spectrum of the afterglow.The upper limit for the size distribution of dust grains a max is also an important pa- 10 –rameter that determines the absolute amount of extinction. Being different from the above-mentioned column density of hydrogen atom N H and the metallicity f d , as a max increases, theabsolute amount of extinction decreases. As shown in Figure 7, the relations a max = 10 − . ,1, 10 . and 10 µ m are represented by the solid, dotted, short-dashed and long-dashed lines,respectively. The other parameters still have the typical values, i.e., Λ - (cid:144) Μ m - A H Λ L Β = -
Β = -
Β = -
Β = - - - - - - - - Ν (cid:144) Hz F l ux (cid:144) A r b it r a r y Fig. 6.— Extinction of GRB afterglow by circum-stellar dust grains with different indicesof size distribution β . Left panel is the extinction curve, and right panel is the attenuatedafterglow spectrum. Gray straight line in the right panel represents the intrinsic spectrumof the afterglow. f d = 0 . A /A = 1, N H = 10 . cm − , β = − . a min = 0 . µ m. Thisphenomenon is due to an underlying assumption in our calculations that the total mass ofthe dust grains is conserved. Larger dust grains are formed by the aggregation of smaller ones.As the number of larger dust grains increase, the total number density of the dust grains willnaturally decrease. Our computing results indicate that the absolute amount of extinctionwill remarkably decrease, and as the size of dust grain increases, the extinction curve willbecome flatter, causing gray extinction (Stratta et al. 2004; Li et al. 2008). Therefore, with β and a max both varying, we would expect that the dust grains with typically larger sizeswill cause weak extinction and have a flatter extinction curve, i.e., causing gray extinction.This explains why most observed optically bright afterglows exhibit flat extinction curves(Stratta et al. 2004). On the contrary, the dust grains with typically smaller sizes wouldcause more severe extinction, i.e., causing optically dark bursts, and have remarkably steeperextinction curves. Therefore, numerous computing results indicate that the discrepancy andevolution of the sizes of dust grains can have very crucial effects on the extinction curves ofGRB afterglows. 11 – Λ - (cid:144) Μ m - A H Λ L a max = Μ m a max = Μ m a max = Μ m a max = Μ m10 - - - - - - - Ν (cid:144) Hz F l ux (cid:144) A r b it r a r y Fig. 7.— Extinction of GRB afterglow by circum stellar dust grains with different parametersof size distribution a max . Left panel is the extinction curve, and right panel is the attenuatedafterglow spectrum. Gray straight line in the right panel represents the intrinsic spectrumof the afterglow.
4. CONCLUSION
In this work, in order to study the effect of dust extinction on GRB afterglows, we carryout numerical calculations based on dust physics and explore the effects of various dustparameters on the extinction curves. We find that the medium density and the metallicitydetermine the absolute amount of extinction, and the parameters for the size distribution ofdust grains β and a max determine the profile of the extinction curve. When β is larger or a max is larger, i.e., larger grains are more excessive, the extinction curve will be flatter with weakextinction. On the contrary, when β is smaller or a max is smaller, i.e., smaller grains are moreexcessive, the extinction curve will be steeper with severe extinction, most likely causingthe optical dark bursts. This may also explain why most bright afterglows tend to haveflatter extinction curves. Therefore, the massive stellar birth of the GRB and its complexprogenitor environment, should be the major cause of origin of the optical dark bursts anddiverse optical afterglows. Observing and analyzing the extinction of afterglows would alsobe important to the studies of the GRB progenitors and their explosion mechanisms.This work made use of the tabulated data of interstellar dust provided by B. T. Drainefrom Princeton University. We are grateful to S. Aragon from San Francisco State Universityfor discussion on the MieSolid code and M. A. Caprio from University of Notre Dame forproviding the updated LevelScheme package and enthusiastic technical support. 12 – REFERENCES
Bohren, C. F., & Huffman, D. R. 1983, Absorption and Scattering of Light by Small Particles.New York: John Wiley & Sons, Inc., 475-482Chen, S.-L., Li, A., & Wei, D.-M. 2006, ApJ, 647, L13Draine, B. T. 2003a, ApJ, 598, 1026Draine, B. T. 2003b, ARA&A, 41, 241Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89Draine, B. T., & Malhotra, S. 1993, ApJ, 414, 632Gehrels, N., et al. 2004, ApJ, 611, 1005Laor, A., & Draine, B. T. 1993, ApJ, 402, 441Li, A., & Draine, B. T. 2001, ApJ, 554, 778Li, Y., Li, A., & Wei, D.-M. 2008, ApJ, 678, 1136Mathis, J. S., Rumpl, W., & Nordsieck, K. H. 1977, ApJ, 215, 425Schady, P., et al. 2007, MNRAS, 377, 273Shao, L., & Dai, Z.-G. 2007, ApJ, 660, 1319Shao, L., Dai, Z.-G., & Mirabal, N. 2008, ApJ, 675, 507Shao, L., Fan, Y.-Z., & Wei, D.-M. 2010, ApJ, 719, L172Stratta, G., et al. 2004, ApJ, 608, 846Stratta, G., et al. 2005, A&A, 441, 83van de Hulst. 1957, Light Scattering by Small Particle. New York: John Wiley & Sons, Inc.,114-130van der Horst, A. J., et al. 2009, ApJ, 699, 1087Wiscombe, W. J. 1980, ApOpt, 19, 1505