Synchrotron spectrum

The typical observed afterglow spectrum is composed by a set of power-law segments joined at some specific frequencies. These frequencies are known as the characteristics break frequencies and are derived based on the standard definition of the general synchrotron frequency

ν_syn(γ_e) =K_νγBγ_e²(1+z)⁻¹, K_ν = 3q_esinα

4πm_ec , (2.28)

for an electron with Lorentz factor γ_e. The radiated power per electron with energy γ_edue to synchrotron radiation is given by

P(γ_e) =K_Pγ²B²γ_e², K_P= 4σ_Tc

3 8π , (2.29)

with the maximum power emitted at a frequencyν(γe) P_m= P(γ_e)

ν(γ_e), (2.30)

and the peak fluxF_mof the observed synchrotron spectrum given by F_m= 1

4π·10⁵⁶N_e(r)P_md_L⁻²

28, (2.31)

whered_L is the luminosity distance andN_e(r)is the number of electrons in a volume with radius r.

The synchrotron spectrum is defined byF_mtogether with three characteristic break frequencies:

the cooling frequencyν_c=ν(γ_c), the injection frequencyν_m=ν (γ_m) and the self-absorption break frequency ν_sa. Based on the relative position of the break frequencies, different spectral regimes are defined. First, fast (ν_c<ν_m) and slow (ν_c>ν_m) cooling regimes. In the fast cooling regime the bulk of the electrons have energies aboveγ_c, being able to cool down fast and efficiently, i.e., within the dynamical time of the system. The time duration of the fast cooling regime depends on the CBM profile. A few hundred of seconds for an ISM density profile and, a few thousand of seconds for a stellar wind-like density profile. In the slow cooling regime most of the electrons have Lorentz factorγ_m. Therefore only a small fraction of the electron (γ_e>γ_c) is affected by the cooling effects.

Whenν_sais taken into account, there is a subdivision of both, the fast and slow cooling regimes.

Following Granot & Sari (2002) five spectral regimes can describe the whole evolution of the GRB afterglow. The shape of the five different spectral energy distribution regimes is given by

F_ν^[1]=Fm^[1]

2.2 Radiation processes

where the lower index refers to the spectrum number and, the labels from [A] to [H] refers to the specific power-law segment on the spectrum (Fig. 2.3). Expressions forF_m⁶in the optically thick region andν_saandν_acare discussed in more detail later in the chapter.

Cooling break

When the energy injection contribution is included, a temporal evolution ofν_cis modifies as ν_cinj∼ν_ct

(3k−4)(1−q) 2(4−k)

dz . (2.38)

Finally, whenY 1 the SSC component is included andν_cis of the form ν_c^IC ∼ η_IC⁻²ε⁻

without and with energy injection component, respectively.

Injection break

ν_m is given by ν(γ_m), corresponding to the breaks (2), (4) and (9) in Fig. 2.3. If p>2 then γm ∼γ, but if 1< p<2 this proportionality changes and so does the temporal evolution of the break frequencies and peak flux. Due to the lack of consensus on the definition of γ_M, I present

6Eq. 2.31 applies forF_min Eq. 2.32 and Eq. 2.36. For the remaining three spectra the peak flux is modified.

2.2 Radiation processes

Figure 2.3:Schematic representation of the observed synchrotron spectra at five different stages during the afterglow evolution. Each spectra is described by Eqs. 2.32, 2.33, 2.34, 2.35 and 2.36, from top to bottom, respectively. Taken from Granot & Sari (2002)

two approaches: DC⁷ (Dai & Cheng 2001) whereγ_Mis defined by Eq. 2.23 and, GS whereγ_mis assumed to keep the proportionality toγ (i.e., p∼2) and the equations presented in Granot & Sari (2002) for the break frequencies and the peak flux are still valid.νmis then given by

ν_m∼





 ε

1 2

Bε¯_e²E

1 2

52t

−3 2

d (1+z)¹² ,for p>2,

ε¯e^4(4−k)ε_B^4−kt_d^{p(k−3)+k−6}E₅₂^{4+k(p−2)−p}A∗^p−2

(1+Y)2(2−p)(4−k)(1+z)14+k(p−3)−5p

2(4−k)(p−1)¹

,for 1<p<2.

(2.41)

7I derive independently all the equations for the regime where 1 < p < 2. The formalism is described in detail along the chapter and is similar to GS withγ_Mintroduced. SSC and energy injection components in this regime are analysed in the fast and slow cooling regime. The euqationss were derived for a general density profile with slope k.

2.2 Radiation processes The temporal evolution ofν_mwhen the energy injection phase is ongoing is given by

νm_inj ∼

Finally, only when 1< p<2γ_Mdepends on the mechanisms responsible for the electron cooling and, therefore SSC radiation becomes important. The general expression for ν_mwith SSC effect included for 1<p<2, without and with energy injection contribution included, is given by

ν_m^IC ∼ η

The emission region can be thin or thick during the afterglow emission. The optical depth (τ) to electron scattering is only important if the mission region is optically thick. When τ >1 there is an important change in the observed flux density and in the evolution of νsaand νac. In this optically thick case, i.e.,νsa>ν, the location in the system of the emitting electrons is important.

In the optically thin emission all the electron will escape the system regardless their distribution (homogeneous or inhomogeneous), while in the optically thick part of the spectrum they will not.

Following Rybicki & Lightman (1979) and Panaitescu & Kumar (2000), an expression for the optical thickness in terms ofν, ν_m, ν_cis derived by equating the synchrotron emission (optically thin) and the blackbody emission (optically thick). τ is given by

τ_ν = cooling regimes, respectively. τ_pis define by

τ_p(γ_p) =τ_genγ_p⁻⁵, with τ_gen=5 q_e(3−k)⁻¹n r B⁻¹. (2.46) There are two scenarios forτ_p. The fast cooling regime whereτ_pis evaluated atγ_cas

τ_c(γ_c) =τ_gen·γ_c⁻⁵, (2.47)

and the slow cooling regime whereτpis evaluated atγmas

τm(γ_m) =τgenγ_m⁻⁵. (2.48)

In the optically thick emission region two layers are identified. A thick layer of electrons that have cooled down to γ_m/2 and, a thin layer of uncooled electrons, right behind the shell. The transition between the two layers is observed as a break in the afterglow spectrum. At this break, two power-law segments are joined. A segment corresponding to a standard blackbody spectrum (F_ν ∝ν²) and a segment with a blackbody spectrum that has an effective temperature that depends on the frequency (F_ν ∝ν^11/18). The transition between the absorption due to uncooled electrons

2.2 Radiation processes

to the absorption due to cooled electrons occurs at ν_ac. This break is difficult to observe and is usually not included in the afterglow analysis. However, νacadd an important contribution to the flux density at low frequencies, with its main relevance in the analysis of early radio observations (Granot et al. 2000). Settingτ_ν =1 in Eq. 2.45 an expression forν_sacan be derived. The specific expressions forνsaandνacare given for each one of the five spectral regimes are given by

νsa=

In order to include the energy injection or the SSC component the proper expression forν_c,ν_m,τ_c andτ_mhave to be used, as well as the cases for p>2 or 1<p<2.

Peak Flux

The peak flux,F_m, can be written in terms of the maximum powerP_maxas in Eq. 2.31 (for breaks 2 and 11 in Fig. 2.3). The peal flux is not affected by the especific mechanism for the electron cool-ing, but it is affected by a prolonged energy injection. This energy injection component modified theF_mtermporal evolution as

F_{min j} ∼ F_mt

(q−1)(3k−8) 2(4−k)

dz . (2.51)

When the frequency at which the peak flux occurs is in the self-absorbed region, the absorption effects affectF_m. The correction factor for the spectra (2), (3) and (4) where this absorption effects are important are given by

The flux of the afterglow is described by F ∼ν^−βt^−α. For a specific model and synchrotron spectrum, there is a unique set of relations betweenα andβ that constrained the cooling regime, the circumburst environment, the jet geometry and the electron energy distribution indexp(Rees &

Meszaros 1994; Wijers et al. 1997; Sari et al. 1998; Dai & Cheng 2001; Zhang & Mészáros. 2004), this set of relations is called "closure relations". Here I present the closure relations for the standard model in the case of a deceleration blast wave for an ISM or wind-like density profiles, p>2 and 1<p<2, including energy injection relations forp>2 and a jet break for both cases: a spreading phase and a non-spreading phase. The table is taken from (Racusin et al. 2009). For details on

Im Dokument Testing the standard GRB afterglow model with the snapshot method using multi-epoch multi-wavelength data  (Seite 24-28)