# A Revised Limit of the Lorentz Factors of GRBs with Two Emitting Regions

###### Abstract

Fermi observations of GeV emission from GRBs have suggested that the Lorentz factor of some GRBs is around a thousand or even higher. At the same time the same Fermi observations have shown an extended GeV emission indicating that this higher energy emission might be a part of the afterglow and it does not come from the same region as the lower energy prompt emission. If this interpretation is correct then we may have to reconsider the opacity limits on the Lorentz factor which are based on a one-zone model. We describe here a two-zone model in which the GeV photons are emitted in a larger radius than the MeV photons and we calculate the optical depth for pair creation of a GeV photon passing the lower energy photons shell. We find that, as expected, the new two-zone limits on the Lorentz factor are significantly lower. When applied to Fermi bursts the corresponding limits are lower by a factor of five compared to the one-zone model. It is possible that both the MeV and GeV regions have relatively modest Lorentz factors () , which is significantly softer then one zone limit .

## 1 Introduction

Highly relativistic motion, essential to overcome the Compactness problem (Ruderman, 1975), is a basic ingredient of all GRB models. The value of the bulk Lorentz factor, , of the relativistic outflow is of outmost interest. It is essential for understanding the nature of the inner engine, the outflow and its acceleration and collimation mechanisms, the conditions at the emitting regions and the radiation mechanism. So far the most robust method to estimate was using the Compactness. The high energy photons set an upper limit on the optical depth for pair production (Fenimore, Epstein & Ho, 1993; Piran, 1995; Woods & Loeb, 1995; Lithwick & Sari, 2001). The observations of GeV photons from several bursts enabled the Fermi team to set very high ( lower limits on for those GRBs (Abdo et al., 2009a, b; Ackermann et al., 2010). While other methods to estimate or limit depend on various assumptions (see e.g. Zou & Piran (2010)), the compactness limit seems to be independent of any model assumptions and hence it is considered to be the most robust one.

However,
recent Fermi observations have also shown that the onset of higher energy ( MeV to a few GeV - denoted hereafter GeV)
emission lags after the onset of the lower energy ( keV to a few MeV - denoted hereafter MeV) prompt
emission (Abdo et al., 2009a, b; Ackermann et al., 2010; Ghisellini et al., 2010). Fermi confirmed earlier EGRET results that
the GeV emission also lasts longer than the MeV emission (Hurley et al., 1994; Fishman & Meegan, 1995; González et al., 2003).
These two facts suggest the possibility of a different origin for the
MeV and the GeV emission. This would be the case, for example, if
the GeV emission arises from an external shock afterglow (Kumar & Barniol Duran 2009, 2010; Ghisellini et al. 2010; Piran & Nakar 2010; Gao et al. 2009), from multi-zone internal shocks (Xue, Fan & Wei, 2008; Aoi et al., 2009; Zhao, Li & Bai, 2010) or if the MeV emission is
the quasi-thermal radiation of the baryonic outflow while the GeV emission is
mainly from the subsequent internal shocks^{1}^{1}1The synchrotron radiation of electrons accelerated by such internal shocks may peak at eV energies while the inverse Compton radiation can give rise to a significant GeV component. In such a scenario, the
GeV emission and energetic ultraviolet/optical flare are tightly correlated..

The compactness limits on the Lorentz factor are based, however, on an implicit assumption that the MeV and the GeV photon arise from the same region. We show here that the relaxation of this assumption reduces significantly the estimated lower limit on (see also Aoi et al., 2009; Li, 2010; Zhao, Li & Bai, 2010). The existence of two regimes leads to a rich variety of possibilities. We consider in the following the most natural configuration, which is also consistent with the temporal delay of the GeV emission, the MeV emission is produced at lower radii (say via internal shocks) and the GeV emission is produced at a larger radius (say via external shocks) - see Fig. 1. Other geometrical options that we don’t consider are that the GeV emission is emitted at a lower radius than the MeV emission or that the MeV and the GeV emission are produced at different angular regions. Our estimates don’t depend on the origin of the emission (e.g. internal or external shock) but just on the overall geometry of the system. We calculate the optical depth of a GeV photon passing through the MeV photons shell and we obtain new compactness limits on .

## 2 The Model

Consider two emitting regions denoted by , for MeV and , for GeV respectively (see Fig. 1). The MeV (GeV) emission region has a radius () (for simplicity we consider emission from thin shells) with , a Lorentz factor () and an angular width (). We assume that the MeV and GeV jets are aligned and both are pointing toward us. In this configuration a GeV photon passes through a shell of MeV photons on its way to the observer (see Fig. 2).
Our goal is to estimate the optical depth for pair production of a GeV and an MeV photons shell.
The width of the MeV photons shell is^{2}^{2}2Zhao, Li & Bai (2010) consider erroneously an MeV shell width of (see their Eq. (10)).:

(1) |

where is the observed duration of the MeV pulse and is the redshift of the burst. A GeV photon, emitted along the axis at and is immersed in MeV photons until it leaves the MeV shell at (see Fig. 2):

(2) |

for , while for .

Assume, for simplicity, that the emitted flux of the MeV photons is constant over time then the number density of the MeV photons is:

(3) |

where is the isotropic equivalent MeV luminosity, is the observed peak energy and is the radius. The spectrum of the MeV photons is described by the Band function:

(4) |

where, in view of the two region model, the MeV component has an upper limit to the energy: . Fortunately is insensitive to the exact value of .

At The MeV photons are beamed with an angular width along the radial direction outwards. Since the MeV photons travel almost radially outwards. Relative to the GeV photons, that move along the axis, the angular width of the MeV photons is of order and it decreases with . This leads to two angular regimes: Along the axis (for ), the MeV photons have very small angles relative to the GeV photons. This leads to a very small optical depth along the axis. For the angular spread of the MeV photons can be neglected and the typical angle between the MeV and the GeV photons is simply .

More generally, the angle between an MeV photon emitted radially outwards at and a GeV photon emitted parallel to the axis at is:

(5) |

where is the distance the GeV photon travels before it collides with the MeV photon and is the angle of a given MeV photon relative to the axis parallel to the GeV photon axis.

A GeV photon, with an (observed) energy , and an MeV photon, with an (observed) energy , can produce a pair if , where:

(6) |

The cross section behaves like (e.g. Jauch & Rohlich, 1980):

(7) |

where

(8) |

and is the Thompson cross section.

We can estimate now the overall optical depth for the GeV photon:

(9) |

The effective optical depth for an observed photon is averaged over all angles:

(10) |

where is the Doppler factor, is the bulk velocity and is the photon index of the GeV emission. It is interesting to note that Eq. (9) is independent of . The overall dependence of on arises from Eq. (10) where determines (via ) the effective width of the integration.

## 3 A Simplified model

We can simplify the above model and obtain an almost analytic formula by making a few approximations. We show later that the full numerical results indeed agree with these formulae. First, we approximate the MeV spectrum in the relevant energy range using a single power law: . Second, we approximate the cross section (Eq. (7)) as (This form allows us for an analytic integration. The numerical factor is chosen by comparison of the analytic approximate results with the full numerical solution.). Third, we divide the analysis to two regimes: For all the MeV photons move radially and the collision angle is simply . For the scatter in the directions of the MeV photons is important and we approximate the collision angle as .

Define and . For large angles the collision angle between the MeV and GeV photons is . The minimal energy of the MeV photon for pair production is:

(11) |

The number density of the MeV photons:

(12) |

where . Collecting the above expressions:

(13) | |||||

where and .

We use to express in terms of and , where is the time when the GeV photon is observed. Generally with for an external shock and () for an internal shock. As we see later the result is not very sensitive to . Realizing that the GeV photons are mainly coming from , we can invert now this equation to obtain a rough estimate of the minimal Lorentz factor. The solution will be consistent if namely if . To do so we assume that the factor in the square brackets of Eq. (13) is order of unity. We obtain:

(14) | |||||

where the notation is used and . For , the overall coefficient is 34. Note the weak (with a power to ) dependence of on .

For the typical collision angle of the MeV photons is . The optical depth is similar to Eq. (13), with replacing and :

(15) | |||||

Interestingly, this formula depends on and it is independent of . For typical values, , , GeV, MeV, erg, cm, cm, we obtain , which provides only a very week constraint on the bulk Lorentz factor of the MeV region. If we set , this expression reduces to the one zone case, where the relevant collisions occur at in the observer’s frame. Taking , where s is the typical duration of -ray pulse, , consistent with Lithwick & Sari (2001).

## 4 Results

Fig. 3 depicts the dependence of on for a set of typical parameters: , s and cm, GeV, MeV, erg, and s. is always chosen to be 2. for these parameters (with , ) for . Note that numerical experiments reveal that to obtain the average effective optical depth using Eq. (10) we have to integrate up to . The results in this figure depict both the full (Eq. (9)) and the simplified (Eq. (13)) calculations. The later are depicted by the two thicker lines which are almost superposed on the corresponding numerical results, showing consistency of the analytical solution with the full numerical one. As the angle increases, the small angle approximation () breaks down and the approximate solution slightly diverges from the numerical one. However, the deviation is small and it usually takes place in a regime that is not critical for the overall optical depth.

One can clearly see different segments of power law dependence of the optical depth on . For a given GeV photon determines the colliding angle and hence . As most photons are in the lowest energy and the cross section is largest near , the optical depth is dominated by the low energy photons at and the photon index can be simply taken the one at . For small angle, (see Eq. (6)) and the effective spectral slope is . For large angles and the effective spectral slope is . Correspondingly, lines with the same coincide at small angles while lines with the same conicide at large angles. These results suggest that the single power law approximation for the spectrum is useful. determines the relevant spectral, or .

A second transition takes place when approaches unity, namely the width of the interaction region is small compared to . According to Eq. (13), for and for . The transition between the two takes place as . For some parameters the two transitions may coincide to one and we have chosen so that the two transitions are clearly seen.

Fig. 4 depicts as a function of the MeV luminosity . As one can expect (see Eq. (14)) increases with . The optical depth for s (thin solid line, ) is much larger than the other ones for which s, again in agreement with Eq.(14). If , increases with as a single power law. This is consistent with Eq.(14). For , the relationship breaks into two power law segments that are dominated by different parts of Band spectrum of the MeV photons. As the luminosity increases the Lorentz factor increases, and the dominant contribution to the optical depth arises from , with a lower effective , increases and hence for large values of the opacity is dominated by the high energy spectral slope, . Conversely, for low values of the opacity is dominated by . The transition takes place at .

GRB | z | (s) | (MeV) | (erg/s) | (GeV) | ref | |||||

080916C | 4.35 | 66 | 1.02 | 2.21 | 1.17 | 40 | 193(414) | 880 | 1 | ||

090510 | 0.903 | 0.5 | 0.48 | 3.09 | 5.1 | 3.4 | 0.5 | 150 (277) | 1200 | 2 | |

090902B | 1.822 | 30 | 0.61 | 3.87 | 0.8 | 33.4 | 82 | 120 (218) | 1000 | 3,4 | |

090926A | 2.1062 | 20 | 0.693 | 2.34 | 0.27 | 19.6 | 26 | 150 (318) | 1200 | 5 | |

Shown are the low energy spectral parameters , and as well as the luminosity, and the energy, | |||||||||||

and time of the highest energy GeV photon as well as the two zone, , and the single zone, ,limits. | |||||||||||

(1) Abdo et al. (2009a); (2) Ackermann et al. (2010); (3) Abdo et al. (2009b); (4) de Palma et al. (2009); | |||||||||||

(5) Swenson et al. (2010) | |||||||||||

in bold face is the value for and brackets is the value for |

The two zone limits, , for four Fermi bursts are shown in table 1 together with the single zone limits, , and some parameters of the bursts. While the single zone limits, , are of order 1000 and even larger, the two zone limits are around 200 (400), for . It should be stressed that in two of these bursts, GRB 080916c and GRB 090510, the highest energy GeV photon used to determine the single zone limit is coincident with a large MeV flux, while an 11 GeV photon is contemporaneous in GRB 090902b with an MeV spike. Still in both GRB 090902b and GRB 090926a GeV photons are observed after the end of the prompt MeV (). Thus, it is not clear whether the single zone of the two zone limit should be used.

If the GeV photons are from an external shock, another direct constraint on arises from the dynamics of external shock: (Sari & Piran, 1999). With observed GeV peak emission time of s (Ghisellini et al., 2010), erg and a circum-burst density , we obtain . This value is larger than the lower limits obtained from the two-zone compactness estimate. However, it is quite uncertain, in view of the uncertainty in the determination of .

## 5 Conclusions

Following various indications that the high energy (GeV) emission in GRBs is produced in a different region than the lower energy (MeV) emission we derived here revised compactness limits on the Lorentz factor of GRB outflows within a two-zone model. We considered the “natural” model in which the GeV emission is produced at larger radii than the MeV emission. This would arise, for example, if the MeV emission is produced by internal shocks and the GeV emission by the afterglow, as has been suggested recently by several authors (Kumar & Barniol Duran, 2009, 2010; Ghisellini et al., 2010). We calculated the optical depth for pair production by a GeV photon passing through the MeV photons shell. Our results reduce to the one-zone model when the emission region of the MeV and GeV coincide.

Collisions between the GeV and MeV photons occur in the two-zone model at larger radii than the prompt emission radius, the density of the MeV emission is smaller (than in the prompt emission regime) and the MeV photons are more collimated along the line of sight. Consequently, the optical depth is smaller compared to the one-zone case and the compactness constraint on the Lorentz factor becomes weaker. The new constraint that we find is only for the Lorentz factor of the GeV region. The constraint on the Lorentz factor of the MeV emitting region, arising from the optical depth of the GeV photon, is rather weak. The weak limit does not contradict to the neutrino driven jets(Aloy, Janka & Muller, 2000) nor to the magnetic driven jets(Komissarov et al., 2009; Tchekhovskoy, McKinney & Narayan, 2009).

For a canonical set of parameters, like , s, GeV, MeV, erg, and s, the constraint on the GeV region is . When we apply the two zone constraint to four Fermi bursts, we find minimal Lorentz factors of about 200-400, about one fifth to one half (depending on ) of the the one-zone limit which is in the order of 1000. We conclude that one should proceed with care when applying the one-zone limits to the Fermi data and unless we can verify that the GeV emission is indeed produced in the same region as the lower energy prompt emission we should consider the more relaxed two zone limits.

We thank Ehud Nakar and Uri Vool for helpful discussions and an anonymous referee for helpful comments. The research was supported by an ERC grant, the Israel center of excellence for High Energy Astrophysics, a special grant of Chinese Academy of Sciences, National basic research programme of China grant 2009CB824800 and the National Natural Science Foundation of China under the grant 10703002 and 11073057. TP thanks the Purple mountain observatory of Nanjing and Huazhong University of Science and Technology for hospitality while some of this research was done.

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