Main

The radiation mechanism of γ-ray bursts (GMBs), the most luminous explosions in the Universe, remains unidentified after 45 years since their discovery in the late 1960s. A typical GRB prompt emission spectrum is a smoothly connected broken power law called the Band function2, whose typical low- and high-energy photon spectral indices (in the convention of dN/dEγ Eα or Eβ) are α −1 and β −2.2. Synchrotron radiation of electrons accelerated in relativistic shocks has been suggested as the leading mechanism6,7. However, for nominal parameters, the magnetic field strength in the GRB emission region is strong enough that the electrons are in the fast-cooling regime; that is, the cooling timescale tc is shorter than the dynamical time scale tdyn. In this regime, it has been believed that the photon index should be −1.5 (corresponding to a spectral density distribution Fν ν−1/2; ref. 1). As a result, a fast-cooling synchrotron mechanism has been disfavoured8. Proposed solutions include introducing a spatially decaying magnetic field behind the internal shock front9,10,11, or slow heating from a turbulent downstream region of the shock12. Applying a synchrotron model to directly fit the GRB data has recently been carried out13,14. However, the electron cooling is not tracked to calculate a time-dependent photon spectrum in their modelling.

This well-known index α = −1.5 can be derived from a simple argument. Let us consider a continuity equation of electrons in energy space , where dNe/dγe is the instantaneous electron spectrum of the system at the epoch t, and Q(γe, t) is the source function above a minimum injection Lorentz factor γm of the electrons. For synchrotron radiation, the electron energy loss rate is

where σT, me and c are Thomson cross-section, electron mass and speed of light, respectively, and B is the strength of magnetic fields in the emission region. For fast cooling, electrons are cooled rapidly to an energy γc(t) (cooling energy) below the injection energy γm at time t. In the regime γc < γe < γm, one has Q(γe, t) = 0. Also consider a steady-state system (/∂t = 0), then one immediately gets dNe/dγe γe−2; that is, the electron spectral index is . The specific intensity of the synchrotron spectrum would have a spectral index15 (with the convention Fν νs). The photon spectral index (defined as dNγ /dEγ Eγα, where Eγ is the photon energy, and Nγ is the photon number flux) would then be α = −(1 + s) = −1.5.

The above argument relies on a crucial assumption of a steady state, which is achieved when a constant B is invoked. However, in a rapidly expanding source such as a GRB, the magnetic field strength in the emission region cannot be preserved as a constant. In the rest frame of a conical jet, flux conservation indicates that16 the radial (poloidal) magnetic field component decreases as Br r−2, whereas the transverse (toroidal) magnetic field component decreases as Bt r−1. As a result, at a large radius from the central engine where γ-rays are radiated, one has a toroidal-dominated magnetic field with B r−1. Here, r is the distance from the central engine. Considering other effects (for example, magnetic dissipation, non-conical geometry), the decay law may be described by a more general form

We investigate a generic problem of electron fast cooling in a decreasing magnetic field delineated by equation (1), and study the synchrotron emission spectrum. To interpret the GRB prompt emission spectra, we adopt some parameters that are relevant for GRBs. To be more generic, our calculation does not specify a particular energy dissipation mechanism or particle acceleration mechanism, and hence, can apply to a variety of GRB prompt emission models such as internal shocks17,18 and internal collision-induced magnetic reconnection and turbulence19. We consider a toy box that contains electrons and a co-moving magnetic field B′, which moves relativistically towards the observer with a bulk Lorentz factor Γ. The relativistic electrons are accelerated into a power-law distribution Q(γe, t′) = Q0(t′)(γe/γm(t′))p (for γe > γm(t′)) of a slope p and continuously injected into the box at an injection rate where t′ is the time measured in the co-moving fluid frame. Here Rinj(t′)δt′ gives the number of electrons injected into the box during the time interval t′ and t′ + δt′.

Electrons undergo both radiative and adiabatic cooling. In the rest frame that is co-moving with the relativistic ejecta, the evolution of the Lorentz factor γe of an electron can be described by20 (noting pressure p is ne4/3 in an adiabatically expanding relativistic fluid)

For a conically expanding toy box, we take the co-moving electron number density ne r−2, which gives d ln ne = −2d ln r. We divide the injection function Q(γe, t′) into small divisions in time space t′ and also in energy space γe, and numerically follow cooling of each group of electrons (between [t′, t′ + δt′] and [γe, γe + δγe]) individually using equation (2). We then find the instantaneous global electron spectrum dNe/e of the system at any epoch.

We first consider four models with different decay indices b in equation (1). The ‘normalization’ parameter of magnetic field decay law is taken as B0′ = 30 G at r0 = 1015 cm, and a constant injection rate Rinj = 1047 s−1 is adopted. Model [a] takes the unphysical parameter b = 0, that is, a constant co-moving magnetic field B′ = B0′ = 30 G, to be compared with other models. It implies that there should be no change in the volume of the box. Thus, for this model we drop the adiabatic cooling term from equation (2). As shown in column 1 of Figs 1 and 2, this model gives the familiar electron spectrum dNe/dγe γe−2 below γm, and the well-known photon spectrum Fν ν−1/2 in the fast-cooling regime. One can see that the standard fast-cooling spectrum is reproduced for a steady-state system with a constant B′ and Rinj. Model [b] takes b = 1.0 in equation (1). This is the case of free conical expansion with flux conservation (no significant magnetic dissipation). As shown in column 2 of Figs 1 and 2, the electron spectrum and the photon spectrum both harden with time. At 1.0 s after injecting the first group of electrons, the global electron energy spectral index deviates significantly from the nominal value, and hardens to around . The corresponding photon spectrum is nearly flat (Fν ν0), which corresponds to a photon index α −1, the typical low-energy photon index observed in most GRBs (refs 3, 4). In columns 3 and 4 of Figs 1 and 2, we present models [c] and [d], for which steeper decay indices b = 1.2 and b = 1.5 are adopted, respectively. They may correspond to the cases when significant magnetic dissipation occurs during the course of synchrotron radiation. As shown in Fig. 2, both models also give spectra that are consistent with the observations.

Figure 1: The co-moving-frame fast-cooling electron energy spectrum evolution as a function of time.
figure 1

The injected electrons have a power-law distribution Q(γe, t′) = Q0(t′)(γe/γm)p above a minimum injection Lorentz factor γm = 105, with a power-law index p = 2.8 forγe > γm. We take a co-moving magnetic field B′ = B0′(r/r0)b with B0′ = 30 G and r0 = 1015 cm. Four models are investigated: b = 0 ([a]); b = 1 ([b]); b = 1.2 ([c]); and b = 1.5 ([d]). For all of the models, a constant injection rate is used, with both Q0 andγm as constants. The electron injection into the box begins at r = 1014 cm. The ejecta is assumed to be moving towards the observer with a Lorentz factor Γ = 300, and the burst is assumed at a cosmological redshift z = 1. For each model, the instantaneous electron spectra at four different epochs since the beginning of electron acceleration are calculated. The four epochs in the observer’s frame are: 0.1 s (black), 0.3 s (blue), 1.0 s (red) and 3.0 s (green). For each epoch, the sharp cutoff at low energies corresponds to the ‘cooling energy’ of the system, which is defined by the strength of the magnetic fields and the age of the electrons. Given the same field at r = 1015 cm, the B′ field is stronger at earlier epochs, so that electrons undergo more significant cooling initially. One can see that the cooling energy is systematically lower than that of the constant B′ case (model [a]) as b steepens (models [b], [c] and [d]). The lower panel of each model shows the local electron spectral index as a function of electron energy γe. For model [a] (constant magnetic field), the electron spectrum shows the well-known broken power law, with the spectral indices p + 1 and 2 above and below the injection energy, respectively. For other models, even though the index above γm remains the same, the index below γm is much harder. At later epochs (for example, 3 s spectra), the index approaches an asymptotic value a = (6b − 4)/(6b − 1).

Figure 2: The synchrotron emission flux-density (Fν) spectra of electrons with energy distribution presented in Fig. 1.
figure 2

The full synchrotron spectrum of each electron15 is taken into account. The observed spectra are calculated by considering the Lorentz blueshift and cosmological redshift. Whereas the constant B′ case (model [a]) gives rise to the familiar Fν ν−1/2 spectrum, the decaying B′ cases (models [b], [c] and [d]) all give rise to a much harder spectrum below the injection break νm. For the spectra in the seconds timescale (1 s, red; 3 s, green), the low-energy spectral index is nearly flat, consistent with the typical observed photon index −1. Lower panels show local spectral slopes as a function of observed frequency. The energy peak Ep corresponds to the transition break towards the p/2 index. Thus, a clear hard-to-soft evolution of Ep is predicted, which is consistent with the data of most broad pulses observed in γ-ray bursts (ref. 22).

To understand the physical origin of such an effect, in Fig. 3 we decompose the tobs = 1.0 s instantaneous electron spectrum into the contributions of 10 injection time slices, each lasting for 0.1 s. For the constant B′ case (Fig. 3a), one can see that as the electrons age, they tend to distribute more narrowly in logarithmic energy space, so that the electron number per energy bin increases. This is because in the fast-cooling regime, as time elapses, the original electrons with a wide range of energy distribution tend to cool down to a narrower range of energy distribution defined by the ages of the electrons in the group, which are very close to each other at late epochs. Above γm, the electron energy density distribution remains unchanged with time, because it is always determined by the same injection rate and cooling rate.

Figure 3: Decomposition of electron spectrum at 1 s in the observer’s frame.
figure 3

To see the contributions of electrons injected at different epochs, the electrons are grouped into 10 slices in injection time, each with a duration of 0.1 s. The contributions of each group to the instantaneous electron spectrum at 1 s are marked in different colours. Older groups are cooled down further towards lower energies, so from left to right, curves with different colours denote the electron energy distribution of the electron groups injected from progressively later epochs, with a 0.1 s time step. Dashed curves are the summed total of all electrons.

The cases of B′ decay show a more complicated behaviour. The distribution of each group of electrons still shrinks as the group ages. However, because at early epochs the magnetic field was stronger, it had a stronger cooling effect so that for the same injection time duration (0.1 s), initially it had a wider spread in energy at a given age (which can be seen by comparing the 0.1 s electron spectrum for models [a] and [b] in Fig. 1). The later injected electrons are cooled in a weaker B′ field, so that their initial spread is narrower. After the same shrinking effect due to cooling pile-up, the groups injected in earlier time slices have a wider electron distribution than the constant B′ case. Also the electron spectrum above the injection energy, although possessing the same spectral index, has a normalization increasing with time owing to progressively less cooling in a progressively weaker magnetic field. These complicated effects all work in the direction to harden the spectral index, as seen in Fig. 3b, c, d. For a steeper B′-decay index (for example, b = 1.2 and b = 1.5), the late-time injection occurs in an even weaker magnetic field, so that slow cooling is possible. This results in the accumulation of electrons around the minimum injection energy γm, so that a sharper break in the electron energy distribution is achieved.

The model predicts that the low-energy spectrum below the injection frequency νm is curved, owing to the complicated cooling effect as delineated in Fig. 3. Most GRB detectors have a narrow band pass so that below the peak energy (typically a few hundred kiloelectronvolts), there are at most 2 decades in energy. Nonetheless, in the detector band pass, the observed spectra are usually fitted by a Band function, with the low-energy spectral index α −1. In most situations, time-resolved spectral analyses are carried out with a time bin in seconds4. This is the typical timescale of the slow variability component in most GRB light curves21. We therefore focus on the 1 s and 3 s model spectra. We truncate these spectra in a narrow band (5 keV–5 MeV) and compare them to the empirical Band function fits (Fig. 4). One can observe that most of our model spectra are consistent with the Band function with the correct low-energy spectral indices.

Figure 4: A comparison of our 1 s and 3 s model spectra (solid) with the empirical Band function fits (dashed) for all four models in a narrower band pass from 5 keV to 5 MeV.
figure 4

The energy spectra (νFν) are presented to show clear peak energy (Ep) in the spectra. It is seen that the model spectra can mimic the Band function spectra well. The plotted Band function parameters are the following: model [a]: α = −1.5, β = −2.3, E0 = 1,800 keV for both 1 s and 3 s; model [b]: α = −1.22, β = −2.26, E0 = 490 keV for 1 s, and α = −1.17, β = −2.26, E0 = 220 keV for 3 s; model [c]: α = −1.16, β = −2.25,E0 = 400 keV for 1 s, and α = −1.12, β = −2.19, E0 = 160 keV for 3 s; model [d]: α = −1.1, β = −2.21, E0 = 320 keV for 1 s, and α = −1.05, β = −2.09, E0 = 90 keV for 3 s.

Outside the band pass, our model predicts an asymptotic value of the low-energy electron energy spectral index of , which is 2/5 for b = 1. This is seen in the numerical results of the models (lower panels in Fig. 1), and can be derived analytically (Methods). According to the simple relationship , one gets sa = −3/(12b − 2), which is −0.3 for b = 1 (or Fν ν0.3). In reality, owing to the contribution of the 1/3 segment of the individual electron spectrum, which becomes significant when approaches 1/3 from above, the asymptotic photon spectrum limit is softened. In this case, s is about −0.2. This corresponds to a photon index of −0.8, which is much harder than the nominal value −1.5.

Besides the decay index b as discussed above, the value of low-energy photon index α also depends on several other factors: the ‘normalization’ parameter at r0 = 1015 cm, the time history of electron injection, and the bulk Lorentz factor Γ. To see how different parameters affect the predicted α values, we have carried out more calculations by varying these parameters (Methods).

This model predicts a hard-to-soft evolution of the peak energy Ep during a broad pulse. This is consistent with the observational trends of a large fraction of GRBs (ref. 22). According to Figs 1 and 2, the electron spectrum also tends to harden with time, as does the α value. This model therefore predicts that for a broad pulse in a GRB, during the very early epochs, the α value would harden with time. If the α value of a GRB is already very hard from the very beginning, then the above-mentioned α evolution is no longer significant, even though the electron spectrum continues to harden with time. This is because the contributions from the 1/3 spectral segment for individual electrons become more important.

Two caveats to apply this model to interpret GRB prompt emission should be noted. First, observations showed that a growing sample of GRBs have a quasi-thermal component superposed on the Band component23,24,25. Whereas the Band component is probably of a synchrotron origin13,26,27, the quasi-thermal component is widely interpreted as emission from the GRB photosphere28,29,30,31, the relative strength of which with respect to the synchrotron component depends on the composition of the GRB ejecta, and could be dominant if the ejecta is a matter-dominated fireball. As the hardest α value we get is about −0.8, an observed α harder than this value would be evidence of a dominant photosphere component28. Second, some requirements on the parameters are needed to account for the GRB data. The observed high-energy spectral index β requires a relatively large, yet reasonable, value of the electron injection index p (for example, >2.5). More importantly, to interpret the observed α distribution peaking at α −1 in our model, one demands a relatively high γm 105 and low (10–100) G. A plausible scenario to satisfy these parameter constraints may be magnetic dissipation models that invoke a large dissipation radius, such as the internal collision-induced magnetic reconnection and turbulence (ICMART) model19. Owing to the large emission radius R 1015 cm, this model allows seconds-duration broad pulses as fundamental radiation units, during which particles are continuously accelerated. Owing to a moderately high magnetization parameter σ in the emission region, the minimum injected electron Lorentz factor γm 105 can be achieved, because a small amount of electrons share a similar amount of dissipated energy. One potential difficulty is that there is a preferred range of (10–100 G) for α to fall into the observed distribution. The magnetization parameter

is required to be in the range of 2.7 × 10−5 − 2.7 × 10−3 for Γ = 300, R = 1015 cm and L = 1052 erg s−1, which is relatively low. Within the ICMART scenario, the electrons probably radiate in the outflow region of a reconnection layer, in which magnetic fields are largely dissipated. One therefore expects a relatively low (and hence, low σ) as compared with the undissipated regions in the outflow. Nonetheless, detailed studies of magnetic reconnection and particle acceleration processes are needed to address whether the range demanded by the model could be achieved.

The new physics in the moderately fast-cooling regime discussed in this paper would find applications in many other astrophysical systems invoking jets and explosions, such as active galactic nuclei, galactic ‘micro-quasars’ in X-ray binaries and jets from tidal disruption of stars by supermassive black holes. Within the GRB context, it also finds application in the afterglow phase where electrons never enter a deep fast-cooling regime. Further investigations of this physical process in other astrophysical environment are called for.

Methods

Asymptotic value of α.

The asymptotic low-energy spectral index can be derived analytically from equation (2). Assuming a constant Lorentz factor Γ (which is relevant for GRB prompt emission), one has r = ctΓ. We first solve a simpler equation by dropping the adiabatic term, that is

where

We then find the solution of electron Lorentz factor at any time tj′ (> ti′)

where γe(ti′) is the electron Lorentz factor at an initial time ti′. For b > 1/2, tj ti′, and γe(tj′) γe(ti′), this solution gives γe(tj′) ti2b−1. We then get

For a constant injection rate Rinj, we have δNe δti′. Thus, we have an asymptotic behaviour of the global electron spectrum as follows.

Now we consider the full equation (2) that includes the adiabatic term. For B′(r) = B0′ (r/r0)b and r = ctΓ, equation (2) can be written as

This equation has an analytic solution

where C is the integration constant of the differential equation, to be determined by the initial condition; γe(ti′) at time ti′. The electron’s Lorentz factor γe(tj′) at a later time tj′ is found to be

For b > 1/6, tj ti′ and tj2/3γe(tj′) ti2/3γe(ti′), this solution gives γe(tj′) tj−2/3ti(6b−1)/3. A variation in γe(tj′) results only from δti′ for the instantaneous (that is, at a fixed time tj′) global electron spectrum; δγe(tj′) tj−2/3ti(6b−4)/3δti′. This gives the asymptotic behaviour of the global electron spectrum

where we have again assumed a constant injection rate Rinj. Therefore, we have the asymptotic low-energy electron spectral index .

Dependence on other parameters.

The ‘normalization’ factor B0′ is essential in defining the strength of magnetic fields seen by an electron during the cooling process since injection. So far, we have adopted the value B0′ = 30 G. In the following, we calculate the cases for B0′ = 10,100,300 G for b = 1 and constant injection rate (models [e], [f] and [g], respectively). To compare with model [b] (B0′ = 30 G and b = 1), we keep Γ = 300 fixed and the product of γm2B0′ as a constant to ensure the same observed Ep. We then repeat the calculations as described in the main text and perform the Band function fits to the model spectra. The resulting Band function parameters are presented in Table 1. One can see that an α value ranging from −1 to − 1.5 is obtained. The general trend is that a lower B′ tends to give rise to a harder α value.

Table 1 Spectral parameters of models [e], [b], [f] and [g] (constant injection rate).

The light curves of GRBs show erratic variability, and can be decomposed as the superposition of many ‘pulses’. For a clean pulse, the decay phase of a pulse would be controlled by the high-latitude ‘curvature’ effect32. GRB light curves are typically composed of superposed pulses, so that the decay phase can be contaminated by the rising phase of an adjacent pulse. A test of the curvature effect model of GRB pulses suggested that only about 40% of GRB pulses satisfy the model constraints33. Nonetheless, because the curvature effect only introduces Ep evolution but does not apparently modify the α value, in general the observed α values would be mostly defined by the rising history of electron injection. An increase in the injection rate with time gives more weight to electrons that are injected later, which tend to harden the spectrum. We test how the injection history during the rising phase affects α. First, we introduce a linear increase of the injection rate for B0′ = 10,30,100,300 G, respectively (with b = 1) and name the models as [e1], [b1], [f1] and [g1], respectively. The fitted spectral parameters of these models are presented in Table 2. One can see that by introducing a rise of injection rate with time, the resulting α values are systematically harder. For the four models discussed, the α value ranges from −0.92 to −1.48.

Table 2 Spectral parameters of models [e1], [b1], [f1] and [g1] (linear increase of injection rate).

The rising phase may be steeper than a linear increase with time. We next test the effect of different rising profiles on α. We fix B0′ = 10 G to check how hard a spectrum one may get. Considering the injection rate Rinj(t′) tq, we calculate the cases for q = 0,1,2,3 (models [e], [e1], [e2] and [e3], respectively). Table 3 shows the spectral parameters of these models. One can see that α hardens as q increases (a more rapid increase). For these four models, the α value is in the range between −0.82 and −1.03.

Table 3 Spectral parameters of models [e], [e1], [e2] and [e3] (B0′ = 10 G).