On the High Energy Emission of the Short GRB 090510
Long-lived high-energy () emission, a common feature of most Fermi-LAT detected gamma-ray burst, is detected up to s in the short GRB 090510. We study the origin of this long-lived high-energy emission, using broad-band observations including X-ray and optical data. We confirm that the late MeV, X-ray and optical emission can be naturally explained via synchrotron emission from an adiabatic forward shock propagating into a homogeneous ambient medium with low number density. The Klein-Nishina effects are found to be significant, and effects due to jet spreading and magnetic field amplification in the shock appear to be required. Under the constraints from the low-energy observations, the adiabatic forward shock synchrotron emission is consistent with the later-time () high-energy emission, but falls below the early-time () high energy emission. Thus we argue that an extra high energy component is needed at early times. A standard reverse shock origin is found to be inconsistent with this extra component. Therefore, we attribute the early part of the high-energy emission () to the prompt component, and the long-lived high energy emission () to the adiabatic forward shock synchrotron afterglow radiation. This avoids the requirement for an extremely high initial Lorentz factor.
Gamma-ray bursts (GRBs) are the most luminous explosions in the universe. Their basic scenario based on the emission from extremely relativistic outflows with bulk Lorentz factors and isotropic energies of has been tested, although many questions remain open. The Fermi satellite has advanced our knowledge of GRBs significantly, while raising some new puzzles. During its first of operation as of July 27th, 2010, Fermi has observed 19 GRBs with photons detected in the LAT (Large Area Telescope) instrument. These observations reveal three new properties (Granot et al., 2010; Abdo et al., 2009d, c, 2010, b, a; Ackermann et al., 2010a, b; Fermi-LAT & Fermi-GBM collaborations, 2011): i) A delayed high energy emission, e.g., in GRB 080916C, GRB 081024B, GRB 090510, GRB 090902B and GRB 090926A111The first photons arrive later than the first lower energy photons detected by GBM (Gamma-ray Burst Monitor).. ii) A temporally extended high energy emission: at least 10 of the first 19 Fermi LAT GRBs have long-lived high energy emission, lasting much longer than the burst duration in the sub-MeV band (which declines very rapidly); in 4 out of 10 GRBs, the long-lived LAT light curves have a relatively steeper slope, for example, for GRB 080916C, for GRB 090510, for GRB 090902B, for GRB 090926A according to Zhang et al. (2010). iii) A deviation from a pure Band spectral function, showing an extra component in GRB 090510, GRB 090902B, GRB 090926A. These are among the brightest bursts, but the observations are compatible with the hypothesis of having such a component also in the other, less bright bursts, where it is harder to detect (Granot et al., 2010).
Among these 19 Fermi LAT GRBs, GRB 090510 is a short, hard burst, with a duration (De Pasquale et al., 2010; Ukwatta et al., 2009), located at a redshift (McBreen et al., 2010; Rau et al., 2009). It has been detected by Fermi (Guiriec et al., 2009; Ackermann et al., 2010b), AGILE (Longo et al., 2009), Swift (Hoversten et al., 2009), Konus-Wind (Golenetskii et al., 2009) and Suzaku (Ohmori et al., 2009). Thus, a large amount of high-quality broadband information is available on this burst, including optical, X-ray, MeV and GeV emission. The Swift BAT instrument triggered on GRB 090510 at UT, May 10th, 2009 (Hoversten et al., 2009), while the GBM instrument onboard Fermi triggered on at UT, May 10th, 2009 (Abdo et al., 2009b). Thus, there is a deviation between the two trigger times, which is . Hereafter we adopt the BAT trigger time as a natural start time for computing the afterglow evolution, this being the onset of the main burst.
The high energy emission of GRB 090510 has all three of the new features we summarized above: i) the bulk of the photons above 30 MeV arrive later than those below (Abdo et al., 2009b); ii) the high energy emission above shows a simple power law decay lasting with a temporal decay index (De Pasquale et al., 2010) 222Here we use the convention .; iii) the time-integrated spectrum from to is best fit by a Band function and a power-law spectrum (Abdo et al., 2009b); the extra power-law component photon index of can fit the data well up to the highest-energy () photon (Abdo et al., 2009b).
The XRT observations alone give a spectral index (Hoversten et al., 2009), while a detailed analysis of the temporal XRT emission combined with the LAT emission indicates a spectral index ranging from to (De Pasquale et al., 2010), and a temporal decay index before a break time , which subsequently steepens to . The optical emission initially rises with a temporal index , and after a break time it decays with a temporal index (De Pasquale et al., 2010).
Most of the models aimed at explaining the long-lived high energy emission of GRB 090510 have favored the view that the high energy photons arise from the afterglow emission, being generated via synchrotron emission in the external forward shock (e.g., Kumar & Barniol Duran, 2009, 2010; Corsi et al., 2010; De Pasquale et al., 2010; Gao et al., 2009; Ghirlanda et al., 2010; Ghisellini et al., 2010; Wang et al., 2010; Razzaque, 2010) . This explanation is fairly natural, since an external forward shock model can account for, at least in its gross features, not only for the observed delay of the photons, which corresponds to the deceleration time-scale of the relativistic ejecta, but also for the long lasting emission, which can be attributed to the power-law decay of the synchrotron external forward shock emission (Kumar & Barniol Duran, 2009, 2010). However, various of the above cited authors use somewhat different readings of the publicly available spectral and temporal slope data, which lead them to favor different explanations for the rapid decay of the long-lived high energy emission, falling into five different classes of models as follows.
One set of models interprets the LAT emission as synchrotron emission of electrons accelerated in a standard adiabatic ISM forward shock (Kumar & Barniol Duran, 2009, 2010; Corsi et al., 2010; De Pasquale et al., 2010). These authors argue that they can explain the LAT, X-ray, and optical data with plausible parameter values (and we revisit these arguments below). However, these authors did not perform a complete enough study to confirm whether the whole LAT data including the first second can be explained by this type of models. Kumar & Barniol Duran (2009, 2010) fit the late LAT data as , but do not give much significance to the early-time LAT data. Corsi et al. (2010) fit the whole LAT data as , which seems to explain the early-time LAT data but exceeds the upper limits at late times (). De Pasquale et al. (2010) suggest a steeper decay slope of the high energy emission (evolving as ) for the whole LAT data by taking a larger electron distribution index , but they don’t take into account in sufficient depth the early-time LAT data (), which is necessary in order to conclude what is the origin of the entire LAT emission (see more discussion in Section 6).
Other models, e.g., Neamus (2010) attribute the high-energy photons to synchrotron-self-Compton scattering (SSC) from an adiabatic forward shock propagating into a wind-like medium; this, however, requires an extremely small magnetic energy fraction . Another, different adiabatic forward shock model analyses in greater detail the Klein-Nishina (KN) effects on the high energy inverse Compton process (Wang et al., 2010). For some reasonable parameters, the KN effect, as it weakens in time, results in the synchrotron high energy emission being increasingly suppressed by the SSC cooling, which steepens the synchrotron high energy emission decay slope by a factor as large as . A fourth model views the high-energy emission as decaying proportional to , which is interpreted as being caused by synchrotron emission of electrons accelerated in a forward shock in the radiative, rather then adiabatic, regime. For this, the electron population must be significantly enriched, which is attributed to pair production between back-scattering photons and prompt outward-going photons (Ghisellini et al., 2010). This model explains the high energy emission without considering the constraints from lower energy, e.g., GBM band, XRT band and UVOT band emission, and the pair formation becomes inefficient at shock radii larger than , while for this burst (see §4). A fifth type of model for the afterglow of this burst (and others) is a hadronic model (Razzaque, 2010), which explains the high energy emission as proton synchrotron emission, while attributing the low energy emission to electron synchrotron emission from a forward external shock. This requires a large total kinetic energy 333Similarly large energies are required for hadronic models of the prompt emission of this burst (Asano et al., 2009)., with a low radiation efficiency and an extremely small fraction of electron energy .
In this article, we re-examine the first set of models (the standard adiabatic forward shock model) in significantly greater detail than hitherto. We present detailed arguments indicating that it is most likely that the forward shock synchrotron emission can only explain the LAT emission from sec. This conclusion disfavors the external shock origin of the early-time LAT emission (Ghisellini et al., 2010; Kumar & Barniol Duran, 2009), but supports the suggestions that it is related to the prompt emission (Corsi et al., 2010; De Pasquale et al., 2010). In 2, we examine the XRT and UVOT observations, and set up a model of the long-lived emission based on synchrotron emission from electrons accelerated in a forward shock in a uniform ambient environment, including these X-ray and optical/UV observations. We discuss the impact of the Klein-Nishina effects on the high energy emission, under the constraints imposed by the lower energy observations, which are found to be significant for suppressing the SSC cooling. In we use a semi-analytical model to calculate the development of the dynamical quantities of the forward shock across the deceleration time and into the self-similar phase, and use this to calculate the radiation properties of the long-lived high energy emission produced by synchrotron emission. We find a reasonable set of parameters which can explain most of the late afterglow, except for the six earliest LAT data points in the light curve. In we check several possibilities for the origin of this early high energy emission at times . In we discuss the possibility of the line-of-sight prompt emission as the origin of the early-time high-energy emission. In we discuss our conclusions concerning the most probable origin of the high energy emission from the short GRB 090510.
2 Forward shock model
2.1 Constraints from low energy emission of GRB 090510
The afterglow emission of GRB is generally well explained by synchrotron emission from electrons accelerated by the shock produced during a spherical relativistic shell colliding with an external medium. From the spectral index and the light curve slope (De Pasquale et al., 2010), the closure relation for the X-ray afterglow ranges from to , suggesting a slow-cooling ISM external forward shock model with 444Hereafter we use the subscripts or superscripts and to represent the quantities of the forward-shocked and reverse-shocked regions, respectively, and we use the convention in cgs units throughout the paper., which implies that the decay index of the X-ray light curve before the jet break is and the X-ray spectral index is (Sari et al., 1998).555 The closure relation indicates the slow/fast-cooling ISM/wind external forward shock model with and the spectral index , but this implies , which is not favored by numerical simulations of shock acceleration (e.g., Achterberg et al. (2001)) or by observational data of general GRB afterglows (Freedman & Waxman, 2001). From the spectral index we can get a constraint on which is . In the external shock model, the break seen in X-ray light curve can be explained as a jet break. We assume that the jet expands sideways (Sari et al. 1999). In this case the X-ray light curve slope steepens gradually from to , so that we have , which is reduced to by taking . Combining the above two constraints we have . Then the X-ray decay slope before the break should be , which can be acceptable if we take into account the observed fluctuation of the X-ray flux, as the light curves we will show below.
For the rising portion of the optical light curve before the break time, it is natural to assume that the optical band is below , which induces an optical light curve slope of (Sari et al., 1998; Kumar & Barniol Duran, 2010). The predicted spectral slope is consistent with the observations within the large error bars (De Pasquale et al., 2010). Thus, one can try to explain both the X-ray emission and the optical emission before the break time with synchrotron emission of electrons accelerated in an adiabatic external forward shock with the assumptions and . Then, assuming that the optical band is in the regime of after the break time, the slope of the optical post-break light curve is the same as that of the X-ray light curve.
In the analytical calculations of this section, we assume that the self-similar phase conditions have been established, and for simplicity we neglect the structure of the shock wave, considering a spherical shock with a total isotropic energy and a Lorentz factor . At late times, the adiabatic dynamical evolution of the spherical shock is in the Blandford Mckee self-similar phase, where is constant and the scaling law of the shock wave is (Blandford & McKee, 1976). The shock propagates a distance during the small observing time (Sari, 1997), and integrating this and using the scaling law, one obtains .
According to Sari et al. (1998), the cooling and minimum Lorentz factors of the electrons in the forward shock depend on the total isotropic kinetic energy , the number density of the external environment , and the fraction of the electron energy and magnetic field energy and , which can be expressed as
and , is the power-law index of the electron energy distribution, is the Compton parameter of the electrons with Lorentz factor , and is Thomson cross section. The cooling and minimum frequencies of electrons are
respectively. The peak flux density of the forward shock synchrotron emission is
where the total number of the electrons that the forward shock swept up is , and is the luminosity distance.
We get two constraints from the UVOT and XRT data as follows:
(i) The optical flux density is about at (De Pasquale et al., 2010), which indicates that
(ii) The X-ray flux density is about at , which indicates
where and and we can naturally get from the condition .
If the shocks involve a magnetic field amplification factor (in addition to shock compression), the upstream magnetic field strength would be
Thus, the upstream magnetic field strength could be G (e.g., Kumar & Barniol Duran, 2010), apparently compatible with shock-compression of the typical magnetic field in the interstellar medium, with no need of magnetic field amplification. However, as we discuss in §6, the external density deduced here is much below the average interstellar value, and likely so would be the external magnetic field, so that additional field amplification may be needed.
respectively. The above equations show that until , and under the constraint , which is easy to satisfy, where as discussed in §2.2. Thus, they are consistent with our previous assumptions. The radiative efficiency in the assumed slow cooling regime is (Sari & Esin, 2001)
which is much less than unity, consistent with our previous assumption of an adiabatic forward shock model.
In order to check whether the LAT emission predicted by synchrotron emission from an adiabatic forward shock can explain the LAT observations we need to study two situations.
(i) Under the condition , we yield at , the average synchrotron flux density from the forward shock in the LAT band ( to ) is
which is independent of the electron energy . In addition, the slope of the LAT light curve is also constrained, as , the same as that of the X-ray light curve. For , at is satisfied if , then the predicted LAT flux is at s, consistent with the observational data within the error bars at around s. The corresponding slope of the predicted LAT light curve is around , which can explain the late-time data of the LAT observation. If we take a small electron index, for example, , we have at if . Then the predicted LAT flux can be calculated by equation (17), approximated as at s, which is almost one order of magnitude smaller than the observed LAT flux. What’s more, the slope of the predicted LAT light curve is about , which is too shallow to explain the late-time LAT observation. Thus, in the case at s, we can exclude the model.
(ii) If the electron energy satisfies the condition , the lower end of the LAT band, MeV, is above the frequency , and the predicted LAT flux at can be calculated as
Inserting the condition into equation (18), we see that . For , the predicted LAT flux at s is , which is almost one order of magnitude lower than the observed LAT flux. Thus in the case at s, we can exclude the model.
For , to be consistent with the observed LAT flux at s, the condition is required, constraining the electron energy to a very small value. Here we adopt fairly standard values of , similar to those observed in long GRBs, since these are determined by collisionless shock physics processes on microphysical scales, which should be independent of the global properties of GRBs. Then the constrained total kinetic energy could be . Since the isotropic energy at energy band during is (Ackermann et al., 2010b), the radiative efficiency of the prompt emission is in the range . Moreover, we constrain and by inserting the constraint into equations (9) and (10). The constrained is at , and when . Since the expected slope of the LAT light curve is for in the case, similar to that expected from the model in the case, this can also explain the slope of the late-time LAT light curve. Even though decreases by a factor of by taking rather than taking , it doesn’t change the optical break time by much, and we may not rule out the model in the case.
Thus, both the model in the case and the model in the case can explain the late-time LAT light curve, leading to the same conclusion, since the predicted LAT light curves have the similar slope. For presentation purposes, hereafter, we discuss the model in the case.
2.2 Impact of Klein-Nishina effects on the constraints
Wang et al. (2010) have studied the Klein-Nishina (KN) effects on high-energy gamma-ray emission in the early afterglow, and find that at early times the KN suppression on the IC scattering cross section for the electrons that produce the high-energy emission is usually strong, and therefore their inverse-Compton losses are small, with a Compton parameter of less than a few for a wide range of parameter space. This leads to a relatively bright synchrotron afterglow emission at high energies at early times. However, as the KN effects weaken with time, the inverse-Compton losses increase and the synchrotron high energy emission is increasingly suppressed, which leads to a more rapid decaying synchrotron emission. This provides a potential mechanism for the steep decay of the high-energy gamma-ray emission seen in some Fermi LAT GRBs.
The Compton parameter for electrons with Lorentz factor is defined as the ratio of the synchrotron self-inverse Compton (SSC) to the synchrotron emissivity, i.e.
When the KN suppression on the scattering cross section is negligible, is a constant for the slow-cooling case (Sari & Esin, 2001). However, for high energy electrons with a significant KN effect, is no longer a constant and this affects the electron radiative cooling function, as well as the continuity equation of the electron distribution. The self-consistent electron distribution is given by
for the slow-cooling case (Nakar et al., 2009; Wang et al., 2010), where is a constant. The high energy synchrotron photons with energy are produced by electrons with Lorentz factor which typically have . Thus, the number density of electrons of is
where is the number density of electrons of when only the synchrotron cooling is considered (Sari et al., 1998). Therefore, the number density of electrons with Lorentz factor is a factor of lower than that in the case where only the synchrotron cooling is considered. Thus the synchrotron luminosity is correspondingly reduced by the same factor. We have , in the slow cooling regime, as long as (Wang et al., 2010). In that case, if the synchrotron luminosity is suppressed by the factor which is in proportion to , i.e. the light curve decay of the high energy synchrotron emission could be steepened by a factor at most. Meanwhile, the distribution of electrons which are in the region is not affected by , due to equation 20, thus the lower energy synchrotron emission decay slope is normal.
The critical photon energy above which the scattering with electrons of energy just enters the KN scattering regime is defined as , i.e.
which is much smaller than under the condition . In this case the synchrotron-self Compton scattering is strongly suppressed due to KN effects, and
From the above equation, we find that under the condition , which is easy to be satisfied before . Therefore, since , the distribution of electrons which contribute to the high-energy synchrotron photons doesn’t change, according to equation (20), which means that the decay slope of synchrotron high energy emission can not be affected by .
3 Forward Shock Synchrotron Emission Evolution
We adopt the following parameters for the calculations. From equation (17), we see that the flux density in the LAT range is independent of the total isotropic kinetic energy and the electron energy fraction at , under the constraint . Thus, we can fix the total energy of the afterglow as , indicating a radiation efficiency as . Even if we choose other values of the total kinetic energy under the constraint , our conclusion will not change. Then the fraction of electron energy is constrained as . We adopt fairly standard values of . From the equations (9) and (10), we can get the values of , corresponding to an external density range . These densities range from much less than the intergalactic medium (IGM), up to a low density inter-cluster medium (ICM) or possibly galactic halo baryon density. The critical Lorentz factor which is the boundary between the thin and thick shell cases (Kobayashi et al., 2007) is , where the duration of GRB 090510 is . For an initial Lorentz factor which is not too large, it is reasonable to assume that the initial Lorentz factor is smaller than the critical Lorentz factor, which means the shell has given the ambient medium an energy comparable to its initial energy at the deceleration time , indicating the thin shell case. Then the initial Lorentz factor of the forward shock can be expressed as a function of the deceleration time, which is . If, for example, we take an initial Lorentz factor as suggested by previous Fermi analyses of , and a not too small number density of , we would obtain a deceleration time of .
To calculate the dynamics of the evolution of the blast wave including the transition from the quasi-free expansion, through the deceleration and into the self-similar phase, we use the relativistic hydrodynamics equations for the evolution of the shock radius , the mass swept up by the shock, the opening half-angle of jet and the Lorentz factor of the shock (e.g., Huang et al., 2000b) (see Appendix A), and solve these equations numerically. The solution of these equations provides the dynamical quantities which we use to calculate the radiation spectrum and the light curves discussed in the following. We take into account also the jet evolution as it goes through the jet break and starts to expand sideways. The half-angle of the jet evolves as , with a spreading velocity in the comoving frame approximated by the sound speed (Rhoads, 1997, 1999; Huang et al., 2000a). After the inverse Lorentz factor becomes larger than the initial jet angle, the spreading of the jet speeds up the shock deceleration significantly, which leads to a steeper light curve.
In figures 1 and 2 we present model light curves in the LAT, XRT and UVOT bands for two choices of parameters. The forward shock light curves (solid black lines) are shown, for each density choice, for two different Lorentz factors and jet opening angles. A smaller opening angle is chosen for a larger Lorentz factor in order to get a certain jet break time.
In figure 1, we show a solution with a density of , which would correspond to a sub-average density IGM environment. Under the assumptions before the optical break time and , the steep decay of the latter parts of the X-ray and optical light curves can be explained using such a forward shock synchrotron emission going through a jet spreading phase, as shown in figure 1 and 2. Note, however, that, as in other analyses too, the late-time optical data are challenging to fit. This leads to a model forward shock light curve and spectral fit to the X-ray and optical observations satisfying all the constraints of §2.1, for a set of parameters , , , , and for , or for . The collimation corrected energies are and erg.
In figure 2 we show the forward shock solutions for a more moderate density of , corresponding to a galactic halo or interarm medium, which satisfy the low energy constraints as well as the afterglow epoch LAT, XRT and UVOT data points. The parameters are , , and erg, for a choice of and , and also for a choice of and . The collimation corrected energies are and erg, which is a typical energy for a short GRB. Despite the larger in this fit, the forward shock is still in the adiabatic regime according to equation (16).
In Figures 1 and 2 the LAT energy band light curves predicted by the forward shock synchrotron model shown in black solid lines (thicker for the larger Lorentz factor and smaller opening angle choice, thinner for the smaller Lorentz factor and larger angle choice). These appear to explain well the late time LAT emission, i.e. after 3 seconds, either with the larger or smaller angle/Lorentz factor, no matter how small the deceleration time is, but they fall below the first six LAT data points. This difference between the early time observed LAT data and the synchrotron forward shock emission suggests that there may be another radiation component (whose contribution can be represented by the gray dotted lines in Figures 1 and 2, corresponding to the high latitude emission of the prompt emission with variability timescale ), in addition to the forward synchrotron contribution (shown as the black solid lines), at least for its decaying portion. The gray solid lines are the sum of the steep decay component and the forward synchrotron emission with a larger Lorentz factor (the first, lower data point could be part of either a rising portion or a variable portion of the gray dotted component). Thus, it is possible that the early-time high-energy emission is not from the afterglow forward shock synchrotron emission, instead having a different origin. In the next section we discuss possible origins for the first six data points in the LAT band around the deceleration time.
4 Other Possible Components around the Deceleration Time
Besides the forward shock synchrotron emission, around the deceleration time there are several other possible emission components which could contribute to the early LAT observations. Their importance can be estimated using approximate values for the characteristic quantities at the deceleration time. At the deceleration time, the Blandford-McKee self-similar solution is not yet applicable. For this we can take the usual value of the energy as and as the radius at the deceleration time (Sari et al., 1998), from which we obtain the initial Lorentz factor of the shock and the deceleration radius, which are
Before we examine the other possible components around the deceleration time, we confirm that the forward shock synchrotron emission cannot explain the observation at early times in more details. We can then estimate the approximate high energy emission at the deceleration time under the previous constraints provided by the low energy observations. Here we take the Lorentz factor and radius at the deceleration time as and , which are close to the values indicated by the numerical evolution of the previous section. Inserting the above and into the first expressions of equations (1) (6), and meanwhile adopting equations (9) and (10), the characteristic frequencies and peak flux density of the forward shock synchrotron emission at the deceleration time are then
Ghisellini et al. (2010) suggests the radiative forward shock model to explain the steep temporal decay of high energy emission, which requires to get the fast cooling case. However, the radiative model cannot explain the shallower decay at later time () (as seen in Figure 7 in Ghisellini et al. (2010)), which agrees better with an adiabatic model in slow cooling case at that time. According to the constrained characteristic frequencies, the fast cooling case at the deceleration time requires that
which is inconsistent with their assumption of a very high energy fraction of electrons . This is because Ghisellini et al. (2010) assume the late-time emission to be also in the fast cooling case, and do not take into account the constraints from the low energy emission.
This is almost one order of magnititude below the observed LAT flux density, for the case when the deceleration time is smaller than (this is seen also in Figures 1 and 2). The synchrotron forward shock emission cannot explain the early-time high-energy emission (the first six LAT data points near the deceleration time), although it can explain very well the late LAT results (beyond the sixth LAT data point, when the afterglow can be considered as established), which are consistent with the numerical results in §3.
Thus, we find that there has to be another component, contributing to the early high energy emission. In the rest of this section we consider several such possibilities, such as a reverse shock synchrotron component, a reverse shock SSC component or a cross IC component(forward/reverse shock synchrotron photons scattered by electrons from reverse-shocked/forward-shocked region), a high latitude (curvature) component of the prompt high energy emission.
4.1 Reverse Shock Synchrotron Emission
We consider the thin shell case as is assumed in §3, the flux peaks at the crossing time of the reverse shock with Lorentz factor , and the ratio of the comoving number densities of the forward-shocked to that of the reverse-shocked regions is given by (Kobayashi et al., 2007). The internal energy densities and the bulk Lorentz factors of the two regions are equal with each other (Zhang et al., 2003). Consequently, we have that
characterized by the ratios , .
The minimum and cooling frequencies of the reverse shock synchrotron emission at the crossing time under the low energy constraints are
According to §2.2, we have around the deceleration time, thus the synchrotron spectrum does not change and the ratio of the reverse shock synchrotron flux to forward shock synchrotron flux can be calculated as
The ratio could be as high as since the total kinetic energy can be as high as by considering a reasonable radiation efficiency . Considering equation (30), the flux density from reverse shock synchrotron emission cannot be as high as to explain the observations at early times. Thus, the reverse shock synchrotron emission is unlikely to contribute to the early-time LAT observation.
4.2 The SSC and EIC Emission
Besides the reverse synchrotron emission, we consider the other four possible IC processes, including the synchrotron self-Compton (SSC) processes in forward and reverse shocks, and two combined-IC processes (i.e. scattering of reverse-shock synchrotron photons on electrons accelerated in forward shocks and froward-shock synchrotron photons on electrons accelerated in reverse shocks).
The optical depth of inverse Compton scattering in the reverse shock in the Thomson regime is (Wang et al., 2001). The optical depth to inverse Compton scattering in the forward shock in the Thomson regime is . Taking the peak flux of synchrotron emission at the crossing time of the reverse shock and forward shock, which are