Broadening of the thermal component of the prompt GRB emission due to rapid temperature evolution
Priya Bharali, Sunder Sahayanathan, Ranjeev Misra, Kalyanee Boruah
aa r X i v : . [ a s t r o - ph . H E ] F e b Broadening of the thermal component of the prompt GRB emissiondue to rapid temperature evolution
Priya Bharali a,d , Sunder Sahayanathan b , Ranjeev Misra c , Kalyanee Boruah d a Girijananda Chowdhury Institute of Management and Technology, Guwahati, India b Bhabha Atomic Research Centre, Mumbai, India c IUCAA, Pune, India d Department of Physics, Gauhati University, Guwahati, India
Abstract
The observations of the prompt emission of gamma ray bursts (GRB) by GLAST Burst Monitor(GBM), on board
Fermi
Gamma-ray Space Telescope, suggest the presence of a significant thermalspectral component, whose origin is not well understood. Recently, it has been shown that forlong duration GRBs, the spectral width as defined as the logarithm of the ratio of the energiesat which the spectrum falls to half its peak value, lie in the range of 0.84-1.3 with a medianvalue of 1.07. Thus, while most of the GRB spectra are found to be too narrow to be explainedby synchrotron emission from an electron distribution, they are also significantly broader than ablackbody spectrum whose width should be 0.54. Here, we consider the possibility that an intrinsicthermal spectrum from a fire-ball like model, may be observed to be broadened if the systemundergoes a rapid temperature evolution. We construct a toy-model to show that for bursts withdurations in the range 5-70 seconds, the widths of their 1 second time-averaged spectra can be atthe most . . . Thus, while rapid temperature variation can broaden the detected spectral shape,the observed median value of ∼ . requires that there must be significant sub-photosphericemission and/or an anisotropic explosion to explain the broadening for most GRB spectra. Keywords: gamma rays: bursts - radiation mechanisms: thermal - relativity
Email addresses: [email protected] (Priya Bharali), [email protected] (Sunder Sahayanathan)
Preprint submitted to New Astronomy March 19, 2018 . Introduction
Gamma ray bursts (GRB) are the most luminous, transient phenomena happening in the uni-verse with luminosities of the order of − erg/s (Piran, 2004). The prompt phase of theGRB corresponds to the initial episodic event which lasts typically for a duration ranging froma fraction of a second to few tens of seconds. This is often followed by a fading emission, longafter the initial burst decayed, termed as “after glow”. Optical study of these after glow confirmedthe cosmological origin of GRBs (Costa et al., 1997; van Paradijs et al., 1997). The distributionof the burst duration of GRBs is bimodal with a minima falling at ∼ s suggesting the GRB mayplausibly arise from two different process (Kouveliotou et al., 1993). Accordingly they were clas-sified into two types, namely short bursts with duration < s and long bursts with duration > s. Further, the spectra of short bursts are typically harder than the long ones supporting differentorigin of these two classes (Bhat et al., 2016; Kouveliotou et al., 1996; Dezalay et al., 1996). Theprogenitor of the GRBs are not well understood; however, observational and theoretical advance-ments suggests short GRBs to be associated with the mergers of compact objects, e.g. neutronstar-neutron star merger or neutron star-black hole merger(Eichler et al., 1989; Paczynski, 1991;Rosswog, Piran, & Nakar, 2013), and the long ones are associated with the collapse of a massivestar onto a black hole(Woosley, 1993; MacFadyen & Woosley, 1999).Initial attempts to understand the GRB emission were focussed on thermal nature originatingfrom a catastrophic event involving collapse of a massive star or merging of two compact objects(Paczynski, 1986; Goodman, 1986). The huge energy released within a small volume implies,the optical depth of the initial medium to be very high and the interaction between the energeticphotons and particles will cause the medium to expand at relativistic speed, commonly referredas “fire-ball”. However, after sufficient expansion, the fire-ball approaches the photospheric stagewhere further decrease in density transits the matter to optically thin regime and the trapped ther-mal photons are released (Goodman, 1986; Burgess & Ryde, 2015). On the contrary, the time-averaged spectra of GRBs, observed by BATSE on board
CGRO , are found to be non-thermaland are well explained by a broken power-law function with smooth transition at the break fre-quency (Band function) (Band et al., 1993). To understand this non-thermal emission, internal2hock models were proposed where particles are accelerated at a shock front initiated by the colli-sion between shells of matter expelled by the initial catastrophic event (Kobayashi, Piran, & Sari,1997; Panaitescu & Meszaros, 1998; Daigne & Mochkovitch, 1998). The GRB emission is thenmodelled as the synchrotron emission from the relativistic electron population accelerated at theseshock fronts. Alternatively, Blinnikov, Kozyreva, & Panchenko (1999) showed that a non-thermalspectrum can also be imitated by the time integration of blackbody emission arising from an evolv-ing GRB shells.The observed photon spectral indices of many GRBs; however, confront the synchrotron emis-sion interpretation as the indices are steeper than the one allowed by this model(Crider et al.,1997; Preece et al., 1998; Ghirlanda, Celotti, & Ghisellini, 2003). In addition, the low energy con-version efficiency of internal shock models posed severe drawback (Ryde, 2004; Pe’er & Ryde,2016). These discrepancies forced the addition of a thermal component in the GRB spectra again,along with the non-thermal one (Guiriec & Fermi/GBM Collaboration, 2010; Guiriec et al., 2013;Zhang & Yan, 2011; Burgess et al., 2014; Axelsson et al., 2012). After the advent of
Fermi and
Swift , satellite based experiments operating at gamma ray and X-ray energies, the presence of ther-mal component in GRB spectrum became more evident(Basak & Rao, 2014; Rao et al., 2014).Recently, Axelsson & Borgonovo (2015) performed an elaborative spectral study of long andshort bursts observed by Fermi/GBM and CGRO/BATSE and compared their spectral widths. Thefull width half maximum ( W ) of the EF E representation of the spectra, with E being the pho-ton energy and F E the specific flux, was calculated during the flux maximum of each bursts byfitting the observed fluxes using a Band function. Interestingly, the width distribution peaked at ∼ . for long GRBs and ∼ . for short GRBs, which is much broader than the Planck func-tion ( W ≈ . ) but narrower than the synchrotron spectrum due to a Maxwellian distribution ofelectrons ( W = 1 . ) or a power-law electron distribution with index − ( W = 1 . ). A similarstudy was also carried out by Yu et al. (2015) who studied the spectral curvature at the peak of theGRB spectrum for 1113 bursts detected by the Fermi
GBM experiment. Again, they concludedthat most of the bursts are inconsistent with synchrotron emission models or a single temperatureblackbody emission. In case of short bursts, a detailed study of spectral broadening was per-formed by B´egu´e & Vereshchagin (2014) using approximate analytical solution for the relativistic3ydrodynamic equations.In the present work, we study the broadening of the photospheric thermal emission due to rela-tivistic effects under the fire-ball model of GRBs. We consider a scenario where the temperature ofthe fire-ball decreases rapidly with increase in radius and this causes the high latitude emission tobe relatively hotter than the on-axis emission for a distant observer. In addition, the time integratedspectrum will cause further broadening due to the evolution of the fire-ball within the integrationtime. Particularly, we investigate the maximum attainable width of this multi temperature black-body emission within the burst timescale typical for long GRBs. In the next section, we describethe model and the spectral properties. In §
3, we study the maximum attainable width under thepresent model for the case of long GRBs and summarize the work in §
2. Spectral Evolution of an Expanding Fire-ball
We consider the thermal emission from the photosphere of GRB to be associated with the re-lease of radiation trapped in an initial optically thick and relativistically expanding ball of plasma(Fire-ball). We also assume the expansion to be associated with a rapid fall in temperature de-scribed by T ( r ) = T (cid:16) r r (cid:17) ψ (1)Here, T is the temperature when the fire-ball radius is r , T corresponds to the temperature atradius r and ψ is the temperature index describing the cooling. The burst is assumed to be in itscoasting phase and hence the expansion velocity will be approximately constant till the internalshock or other dissipative events occur (Vedrenne & Atteia, 2009). Further, if the expansion isadiabatic, conservation of entropy limits the value of ψ to be / during the coasting phase of theburst (Piran, Shemi, & Narayan, 1993; Pe’er, 2015).The relativistic expansion of the fire-ball and the light travel time effect will cause the emissionfrom a higher latitude to be hotter than the on axis emission for a distant observer. The flux atfrequency ν received by the observer located at a distance D will then be a modified blackbody4pectrum given by (Appendix Appendix A) f ν ( r ) = 4 πhc r D (1 − β ) ν Z β µ dµ (cid:26) exp (cid:20) hν Γ kT (1+ βµ ) (cid:16) − β − βµ (cid:17) ψ (cid:21) − (cid:27) ( µ − β )(1 − βµ ) (2)where, β is the expansion velocity in units of the speed of light c and Γ = (1 − β ) − / is thecorresponding Lorentz factor, r is the on axis radius of the fire-ball measured by the observer, µ is the cosine of the latitude, h is the Planck constant and k is the Boltzmann constant. If weconsider the photosphere to be spherical, then the relativistic beaming effects will cause the off-axis emission received by the observer to be limited within an angle / Γ subtended at the centreof the fire-ball. When the comoving plasma density varies as r − and for an energy independentphoton scattering cross section, the photosphere can be significantly different from a simple sphere(Abramowicz, Novikov, & Paczynski, 1991). However, within the opening angle / Γ along theline of sight of the observer, the surface of the photosphere can still be approximated as spherical(Pe’er, 2008). For simplicity, the photosphere emission beyond this angle is not considered in thepresent work. In Fig. 1, we show the instantaneous spectrum (normalized) for Γ = 500 , ψ = 2 / and T = 10 keV along with equivalent blackbody spectrum. For Γ ≫ , as in the case of GRBs,the emission cone will be narrower and the instantaneous spectrum observed will drift towards asimple blackbody, unless ψ is large enough to create significant off-axis temperature variation.Following Axelsson & Borgonovo (2015), if we define the width of the resultant spectrum as(Fig. 1) W = log (cid:18) ν ν (cid:19) (3)where, ν and ν are the photon frequencies at the full width at half maximum (FWHM) of the νf ν (unit: ergs/cm /s ) spectrum, then for ψ > , W will be larger than the one expected froma simple blackbody spectrum ( W ≈ . ). In addition, for Γ ≫ , W will depend mainly on ψ and in Fig. 2 we show its variation with respect to the latter. As ψ increases, W increases from ∼ . to a maximum of ≈ . corresponding to ψ ≈ . . Beyond this, the temperature gradientbecomes too large such that the on-axis emission fall below the FWHM of the hotter high latitudeemission, causing W to decrease with further increase in ψ . Under the adiabatic limit ( ψ = 2 / ),5
1 21.7 21.8 21.9 22 22.1 22.2 22.3 N o r m a li z ed F l u x ( e r g s / c m / s ) log Frequency (Hz) Figure 1: The instantaneous spectrum corresponding to Γ = 500 and ψ = 2/3 for T = 10 keV. The dashed line indicatesthe spectral width W measured at FWHM. The dot-dashed line represents equivalent blackbody spectrum. the maximum value of W that can be attained is ≈ . and beyond this range is shown as theshaded region in Fig. 2.A time-averaged spectrum will involve the temporal evolution of the fire-ball within the dura-tion and this will further broaden the spectrum. Considering the fire-ball expands from a radius r to r during an interval, the time-averaged spectrum will then be F , ( ν ) = R r /r ¯ f ν ( x ) dx − r /r (4)where, ¯ f ν ( x ) = f ν ( xr ) and the corresponding temperature T x = T x ψ with T the temperatureof the fire-ball at radius r . In Fig. 3, we show the dependence of W on the ratio r /r for thecase Γ ≫ and ψ = 2 / , . and . . If the time-averaged spectrum is obtained for a time step ∆ consecutively over the entire burst, then the radius of the fire-ball after n ∆ duration can beexpressed in terms of the initial radius of the burst ( r ) as r n = nβc ∆Γ + r pe c t r a l w i d t h ( W ) Temperature index ( ψ )2/3 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0 0.5 1 1.5 2 2.5 3 Figure 2: Variation of the instantaneous spectral width W with respect to ψ The ratio of the radii falling on the beginning and the end of ∆ will then be r n r n +1 = 1 − n + 1 + ξ (5)where, ξ = r βc ∆Γ . Since the ratio of the radii continuously evolves during a burst, the width of thetime-averaged spectrum will additionally depend on the initial burst radius r and the time segment n , along with Γ and ψ . From equation (5), the minimum value of the ratio of radii attained willbe . corresponding to ξ = 0 and n = 1 , which approaches to as n increases. Hence, W willeventually attains a constant value with negligible change soon after the explosion. The light curveof the burst can be obtained by integrating equation (4) over the frequency range of interest. Theduration of the burst ( τ ) can then be obtained by clipping the light curve at 5% of the total fluxin the beginning and the end of the burst.
3. Spectral width of Long Bursts
To study the maximum attainable width of long GRBs under the expanding fire-ball scenario,we obtain one second time-averaged light curve integrated over the energies 8 keV to 40 MeV,consistent with the observations (Axelsson & Borgonovo, 2015). In Fig. 4, we show the evolution7 S pe c t r a l w i d t h ( W ) r /r ψ = 2/3 ψ = 0.8 ψ = 1.0 Figure 3: Variation of the time-averaged spectral width W with respect to the ratio of initial and final radius r /r corresponding to the time segment, for ψ = 2 / , . and . . of the normalized flux, time-averaged spectral width W and temperature corresponding to Γ = 500 and T = 10 keV. The parameters ψ and ξ are chosen to be ( / , . × − ), ( . , . × − )and ( . , . × − ) such that τ ≈ s, the typical duration of long GRBs. We find that thechoice of ψ significantly varies the temporal profile, with the burst peaking earlier for smaller ψ values. For a given ψ , the burst peak time ( t peak ) can be elongated by increasing ξ ; nevertheless,this will also increase the τ from the desired value.The spectral width W can be as large as ∼ . , during the initial phase of the burst; however,it rapidly decreases to a nearly constant value due to increase in ratio of radii r n r n +1 . Maintaining aconstant τ , the width of the spectrum during the flux peak ( W p ) increases with ψ and, in principle,can be adjusted to obtain the desired value ≈ . . For example, choosing ξ = 0 . , Γ = 600 and ψ = 2 . , we obtain W p ≈ while τ ≈ s; however, such choices will cause the burst to peakmuch earlier ( t peak < s), inconsistent with the observations. Reduction of ψ to the adiabaticlimiting value / reduces W p and this should be associated with a decrease in initial radius r inaddition, to maintain τ at the desired value. 8 N o r m a li z ed F l u x ( e r g s / c m / s ) W Log T e m pe r a t u r e ( k e V ) Figure 4: The evolution of the normalized flux (top), time-averaged spectral width W (middle) and tempera-ture(bottom) corresponding to
Γ = 500 and T = 10 keV for ψ = 2 / (bold), . (dashed) and . (dotted) W p and τ with respect to ψ and Γ . In Fig. 5a and 5b, we showthe variation of W p , τ and t peak with ψ for Γ = 200 (solid), (dashed) and (dotted). Thevalue of ξ is chosen to be − and T is fixed at keV. For a given Γ , W p increases with ψ and sharply beyond ψ ∼ . . However, this is also associated with a significant decline in τ and t peak (gray lines). In Fig. 5c and 5d, we again show the variation of W p , τ and t peak withrespect to Γ for ψ = 0 . (solid), . (dashed) and . (dotted). Here, larger Γ is associated withdelayed t peak and hence narrower W p (see Fig. 4). Hence, requirement of a broader W p demandsa larger ψ and a smaller Γ which on the other hand shortens the burst duration as well as deviatelargely from the adiabatic limit. This enforces a limit on maximum attainable width under theassumed fire-ball scenario and significantly hampers to achieve the width ( . ) demanded by theobservations. Through the present study, we found that with an optimal choice of parametersand ψ = 2 / , maintaining τ and t peak at a reasonable values corresponding to long GRBs, onecan only attain W p . . (e.g. W p ≈ . with τ ≈ s and t peak ≈ s for Γ = 100 and ξ = 1 . × − and ψ = 2 / ).
4. Summary & Discussion
The FWHM spectral width of the Band spectrum used to fit the time resolved spectrum of longGRBs is observed to be ≈ . . This is larger than the width of the simple blackbody spectrum de-fined by a Planck’s function and smaller than the synchrotron spectrum from a Maxwellian/power-law distribution of electrons (Axelsson & Borgonovo, 2015). To understand the maximum attain-able width under a simple fire-ball interpretation of GRB, we consider a scenario where the GRBemission to be a multi temperature blackbody distribution arising from a rapidly cooling and rel-ativistically expanding fire-ball of hot thermal plasma. The dynamics of the fire-ball will causethe instantaneous spectrum to be broader than the simple blackbody spectrum with the emissionfrom the higher latitudes hotter than the on-axis emission due to light travel time and relativisticeffects. In addition, the time averaging of the emission will incorporate considerable evolution ofthe fire-ball and this will further broaden the emission. However, under this scenario, we are onlyable to obtain a maximum spectral width of ≈ . , while maintaining the burst duration to be10 -0.4 0.6 1.6 2.6 0.55 0.57 0.59 0.61 0.63 0.55 0.57 0.59 0.61 0.63 ψ Log τ ( s ) Log t pea k ( s ) Γ W p (a)(b) (c)(d) Figure 5: Left (a) and (b): Variation of the spectral width during the burst peak, W p (a), and τ (b) with respect to ψ for Γ = 200 (solid), (dashed) and (dotted). Right (c) and (d): Variation of the spectral width during the burstpeak, W p (c), and τ (d) with respect to Γ for ψ = 0 . (solid), . (dashed) and . (dotted). Gray lines in (b) and(d) indicate the burst peak time t peak . / Γ of an uniformly expanding fire-ball) which canfurther broaden the observed spectrum. References
Abramowicz M. A., Novikov I. D., Paczynski B., 1991, ApJ, 369, 175Axelsson M., et al., 2012, ApJ, 757, L31Axelsson M., Borgonovo L., 2015, MNRAS, 447, 3150Band D., et al., 1993, ApJ, 413, 281Basak R., Rao A. R., 2014, MNRAS, 442, 419B´egu´e D., Vereshchagin G. V., 2014, MNRAS, 439, 924Bhat P. N., et al., 2016, ApJS, 223, 28Beloborodov A. M., 2010, MNRAS, 407, 1033Blinnikov S. I., Kozyreva A. V., Panchenko I. E., 1999, Astronomy Reports, 43, 739Burgess J. M., et al., 2014, ApJ, 784, L43Burgess J. M., Ryde F., 2015, MNRAS, 447, 3087Costa E., et al., 1997, Nature, 387, 783Crider A., et al., 1997, ApJ, 479, L39Daigne F., Mochkovitch R., 1998, MNRAS, 296, 275Dezalay J. P., et al., 1996, ApJ, 471, L27Eichler D., Livio M., Piran T., Schramm D. N., 1989, Nature, 340, 126Ghirlanda G., Celotti A., Ghisellini G., 2003, A&A, 406, 879Goodman J., 1986, ApJ, 308, L47 uiriec S., Fermi/GBM Collaboration, 2010, Bulletin of the American Astronomical Society, 42, 654Guiriec S., et al., 2013, ApJ, 770, 32Kobayashi S., Piran T., Sari R., 1997, ApJ, 490, 92Kouveliotou C., et al., 1993, ApJ, 413, L101Kouveliotou C., et al., 1996, AIPC, 384, 42Lundman C., Pe’er A., Ryde F., 2013, MNRAS, 428, 2430MacFadyen A. I., Woosley S. E., 1999, ApJ, 524, 262Paczynski B., 1986, ApJ, 308, L43Paczynski B., 1991, A&A, 41, 257Panaitescu A., Meszaros P., 1998, ApJ, 526, 707Pe’er A., 2008, ApJ, 682, 463Pe’er A., 2015, AdAst, 2015, 907321Pe’er A., Ryde, F., 2016, arXiv:1603.05058Piran T., Shemi A., Narayan R., 1993, MNRAS, 263, 861Piran T., 2004, RvMP, 76, 1143Preece R. D., et al., 1998, ApJ, 506, L23Rao A. R., et al., 2014, RAA, 14, 35-46Rosswog S., Piran T., Nakar E., 2013, MNRAS, 430, 2585Ryde F., 2004, ApJ, 614, 827van Paradijs J., et al., 1997, Nature, 386, 686Vedrenne G., Atteia J.-L., 2009, Gamma-Ray Bursts: The Brightest Explosions in the Universe, Springer PraxisBooks. Springer, Berlin, HeidelbergWoosley S. E., 1993, ApJ, 405, 273Yu H.-F., et al., 2015, A&A, 583, A129Zhang B., Yan H., 2011, ApJ, 726, 90 Appendix A. Instantaneous thermal spectrum from the expanding fire-ball
Let r be the instantaneous radius of the fire-ball. For an observer located at a distance D, theemission from r will coincide the emission from higher latitude ( θ ) at an earlier radius r θ due tolight travel time effects (Fig. A.6). If the expansion velocity is βc ( c being the speed of light), then r θ = r (cid:20) (1 − β )(1 − βcosθ ) (cid:21) (A.1)13 Equal time surfacer r θ r - r θ cos θ D Figure A.6: Schematic representation indicating the light travel time effects on the instantaneous spectrum seen by adistant observer.
The net flux at frequency ν emitted from the surface of the fire-ball will then be F ν = 2 πr Γ D (1 − β ) Z β I ν ′ (cid:18) βµ − βµ (cid:19) ( µ − β ) µ dµ (A.2)Here, µ = cosθ , I ν ′ is the specific intensity measured in the rest frame of the ejecta and ν ′ = ν [Γ(1 + βµ )] − . For thermal emission, I ν ′ can be replaced by the Planck function and we get F ν = 4 πhc r D (1 − β ) ν Z β µ dµ n exp h hν Γ kT θ (1+ βµ ) i − o ( µ − β )(1 − βµ ) (A.3)where, T θ is the temperature corresponding to radius r θ , h is the Planck constant and k is theBoltzmann constant. Using equation (1) and (A.1), T θ can be expressed as T θ = T (cid:18) − βµ − β (cid:19) ψ (A.4)and equation (A.3) can be written as f ν ( r ) = 4 πhc r D (1 − β ) ν Z β µ dµ (cid:26) exp (cid:20) hν Γ kT (1+ βµ ) (cid:16) − β − βµ (cid:17) ψ (cid:21) − (cid:27) ( µ − β )(1 − βµ )3