Issue 
A&A
Volume 673, May 2023



Article Number  A20  
Number of page(s)  11  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202245414  
Published online  26 April 2023 
An analytic derivation of the empirical correlations of gammaray bursts
^{1}
School of Astronomy and Space Science, Nanjing University, Nanjing 210023, PR China
email: carpedieminreality@163.com; hyf@nju.edu.cn
^{2}
Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing, PR China
^{3}
Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, PR China
^{4}
School of Cyber Science and Engineering, Qufu Normal University, Qufu 273165, PR China
^{5}
College of Physics and Engineering, Qufu Normal University, Qufu 273165, PR China
Received:
9
November
2022
Accepted:
18
March
2023
Empirical correlations between various key parameters have been extensively explored ever since the discovery of gammaray bursts (GRBs) and have been widely used as standard candles to probe the Universe. The Amati relation and the Yonetoku relation are two good examples that enjoyed special attention. The former reflects the connection between the peak photon energy (E_{p}) and the isotropic γray energy release (E_{iso}), while the latter links E_{p} with the isotropic peak luminosity (L_{p}), both in the form of a powerlaw function. Most GRBs are found to follow these correlations well, but a theoretical interpretation is still lacking. Some obvious outliers may be offaxis GRBs and may follow correlations that are different from those at the onaxis. Here we present a simple analytical derivation for the Amati relation and the Yonetoku relation in the framework of the standard fireball model, the correctness of which is then confirmed by numerical simulations. The offaxis Amati relation and Yonetoku relation are also derived. They differ markedly from the corresponding onaxis relation. Our results reveal the intrinsic physics behind the radiation processes of GRBs, and they highlight the importance of the viewing angle in the empirical correlations of GRBs.
Key words: radiation mechanisms: nonthermal / methods: numerical / gamma rays: general / stars: neutron
© The Authors 2023
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1. Introduction
After about five decades of research, some insights into gammaray bursts (GRBs) have been obtained. Generally speaking, there are two different phases in GRBs: the prompt emission, and the afterglow. The afterglow can be relatively well interpreted by the external shock model (Mészáros & Rees 1997; Sari et al. 1998; Huang et al. 2006; Geng et al. 2016; Lazzati et al. 2018; Xu et al. 2022). However, the radiation process of the prompt emission is still under debate. Different models have been proposed to explain the complicated prompt emission, such as the internal shock model (Rees & Meszaros 1994; Kobayashi et al. 1997; Bošnjak et al. 2009), the dissipative photosphere model (Rees & Mészáros 2005; Pe’er & Ryde 2011), and the internalcollisioninduced magnetic reconnection and turbulence (ICMART) model (Zhang & Yan 2011; Zhang & Zhang 2014). The most commonly discussed model is the internal shock model, which is naturally expected for a highly variable central engine. In this model, the collision and merger of shells create relativistic shocks to accelerate particles. Then the accelerated particles will cause the observed GRB prompt emission.
The spectrum of GRB prompt emission has traditionally been described by the empirical Band function (Band et al. 1993). It has been suggested that synchrotron radiation may be responsible for the nonthermal component (Rees & Meszaros 1994; Sari et al. 1998). However, for a standard fastcooling spectrum, the theoretical lowenergy spectral index is too soft (Sari et al. 1998). Several studies have shown that this problem can be mitigated when a decaying magnetic field and the detailed cooling process are considered (Uhm & Zhang 2014; Zhang et al. 2016; Geng et al. 2018). Some authors argued that the lowenergy index of the empirical Band function may be misleading (Burgess et al. 2020). They proposed that the synchrotron emission mechanism can interpret most spectra and can satisfactorily fit the observational data (Zhang et al. 2020).
In addition to the spectrum, the diversity of GRB energy is another mystery. For a typical long GRB (duration longer than 2 s), the isotropic energy is around 10^{52} − 10^{54} erg. However, there also exist some lowluminosity GRBs (LLGRBs) with energies down to 10^{48} erg (GRB 980425) for long GRBs and 10^{46} erg (GRB 170817A) for short GRBs (duration shorter than 2 s). Some authors claimed that LLGRBs may form a distinct population of GRBs (Liang et al. 2007). However, a later study reported no clear separation between LLGRBs and standard highluminosity GRBs in a larger GRB sample (Sun et al. 2015). On the other hand, LLGRBs can naturally be explained by offaxis jets. Due to the relativistic beaming effect, a GRB will become dimmer when the viewing angle (θ_{obs}) is larger than the jet opening angle (θ_{jet}) (Granot et al. 2002; Huang et al. 2002; Yamazaki et al. 2003a). The interesting short GRB 170817A clearly proves that at least some LLGRBs are viewed offaxis (Granot et al. 2017).
The GRB empirical relations can help us understand the physical nature of GRBs. The most famous relation is the socalled Amati relation (Amati et al. 2002). This relation connects the isotropic energy (E_{iso}) and the peak photon energy (E_{p}). The index of the Amati relation (E_{p} − E_{iso}) is about 0.5 (Amati 2006; Nava et al. 2012; Demianski et al. 2017; Minaev & Pozanenko 2020). However, it was found that this relation is not followed by some LLGRBs. These outliers of the Amati relation appear to follow a flatter track on the E_{p} − E_{iso} plane (Farinelli et al. 2021). Previous studies showed that the offaxis effect may play a role in this phenomenon (RamirezRuiz et al. 2005; Dado & Dar 2012).
Analytical derivation of the onaxis and offaxis Amati relation indices has been attempted in several researches (Granot et al. 2002; Eichler & Levinson 2004; RamirezRuiz et al. 2005; Dado & Dar 2012). Eichler & Levinson (2004) derived the index of the Amati relation as 0.5 by considering a uniform, axisymmetric jet with a hole cut out of it (i.e., a ringshaped fireball). Dado & Dar (2012) argued that the Amati relation should have an index of 1/2 ± 1/6 in the framework of the socalled cannonball model. Here, we show that the standard fireball model will also lead to an index of 0.5 for the onaxis Amati relation. For the offaxis Amati relation, previous studies showed that the index should be 1/3 based on the effect of the viewing angle alone (Granot et al. 2002; RamirezRuiz et al. 2005). However, the effect of the Lorentz factor was usually ignored or not fully considered for the offaxis cases. Here we argue that the Lorentz factor is equally responsible for determining the index of the offaxis Amati relation. We suggest that this index should be between 1/4 and 4/13 after fully considering both the effect of the viewing angle and the Lorentz factor.
Researchers have also tried to reproduce the Amati relation by means of numerical calculations (Yamazaki et al. 2004; Kocevski 2012; Mochkovitch & Nava 2015). Recently, Farinelli et al. (2021) used an empirical comovingframe spectrum (a spectrum described by a smoothly broken powerlaw function) to simulate the prompt emission. Their radiation flux was calculated by averaging over the pulse duration. They found that the Amati relation should be for onaxis and for offaxis cases. Here, we consider the detailed synchrotron spectrum instead of an empirical one in our study, so that we can gain more insights into the detailed physics. Furthermore, the isotropic peak luminosity L_{p} can be precisely calculated in our model. As a result, in addition to deriving the Amati relation, we can also examine the Yonetoku relation between E_{p} and L_{p} (Yonetoku et al. 2004), which was previously discussed only for the onaxis cases (Zhang et al. 2009; Ito et al. 2019).
Our paper is organized as follows. In Sect. 2 we briefly describe our model. Then, in Sect. 3, we present an analytic derivation for the quantities of E_{p}, E_{iso}, and L_{p} for both onaxis and offaxis cases. The relations between these parameters are also derived. Numerical results are presented in Sect. 4. Next, in Sect. 5, Monte Carlo simulations are performed to confirm the theoretically derived correlations. The theoretical results are compared with the observational data in Sect. 6. Conclusions and discussion are presented in Sect. 7.
2. Jet model
We mainly focus on long GRBs. We used a simple tophat jet model similar to that of Farinelli et al. (2021). For simplicity, the radiation was assumed to only last for a very short time interval (δt) in the local burst frame. In other words, photons are emitted from the shell at almost the same time. Hence the photon arrival time is largely decided by the curvature effect. A photon emitted at a radius r with a polar angle θ will reach the observer at
where c is the speed of light, and z is the redshift of the source. θ_{1} refers to the angle from which the photons first arrive. We have θ_{1} = 0 for an onaxis jet (θ_{jet} ≥ θ_{obs}), and θ_{1} = θ_{obs} − θ_{jet} for an offaxis jet (θ_{jet} < θ_{obs}). The emission is produced through internal shocks, which can naturally dissipate the kinetic energy of a baryonic fireball (Zhang 2018). As a result, we have r ∼ γ^{2}d, where γ is the bulk Lorentz factor of the shell, and d is the initial separation between the clumps ejected by the central engine (Kobayashi et al. 1997). A schematic illustration of our model is presented in Fig. 1.
Fig. 1. Geometry of our jet model. The upper panel shows the case for an onaxis jet, where θ_{jet} ≥ θ_{obs}. The lower panel shows an offaxis jet with θ_{jet} < θ_{obs}. 
Usually, the light curve of a single GRB pulse is composed of a fast rising phase and an exponential decay phase (Li & Zhang 2021). Complicated processes may be involved in the prompt GRB phase, such as the hydrodynamics of the outflow and the cooling of electrons (Zhang et al. 2007). Analytical solutions will be too difficult to derive if these ingredients are included. Especially in the context of the internal shock model, our assumption of a small δt implies that the pulse is simply produced by the collision of two very thin shells. In realistic cases, a reverse shock may form and propagate backward during the collision. The pulse profile could then be affected by the hydrodynamics of the flow, not only by the curvature of the emitting shell. Depending on the initial distribution of the Lorentz factor, the effect can remain moderate, however, and the twoshell model we considered is approximately valid.
Electrons accelerated by internal shocks should be in the fastcooling regime. They will follow a distribution of (Geng et al. 2018)
with
where is the Lorentz factor of electrons, and N_{e} is their total number. The spectral index p usually ranges from 2–3 (Huang et al. 2000). and are the minimum and maximum electron Lorentz factor, respectively. is less than the initial Lorentz factor of the injected electrons () in the fastcooling regime (Burgess et al. 2020), while can be calculated approximately as (Dai & Lu 1999; Huang et al. 2000), where B′ is the magnetic field strength in the comoving frame.
In the comoving frame, the synchrotron radiation power of these electrons at frequency ν′ is (Rybicki & Lightman 1979)
where e is the electron charge, and m_{e} is the electron mass. and , where and K_{5/3}(k) is the Bessel function.
In the local burst frame, the angular distribution of the radiation power is (Rybicki & Lightman 1979)
where 𝒟 = 1/[γ(1 − β cos θ)] is the Doppler factor, and ν′=(1 + z)ν_{obs}/𝒟. The radiation is assumed to be isotropic in the comoving frame.
Due to the different lighttraveling time, photons emitted at different angles (θ) will be received by the observer at different times. This is called the curvature effect. Summing up the contribution from the whole jet, we can obtain the observed γray light curve. We first consider a time interval of Δt in the local burst frame. It corresponds to a thin ring of [θ, θ + Δθ]. When the electrons are uniformly distributed in the shell, the number of electrons in the ring is
where f_{b} = 1 − cos θ_{jet} is the beaming factor, N_{tot} is the total number of electrons in the shell, and ϕ(θ) is the annular angle of the ring. N_{tot} can be derived from the isotropicequivalent mass of the shell m_{sh} as N_{tot} = 2πf_{b}m_{sh}/m_{p}, where m_{p} is the mass of the proton. The annular angle ϕ(θ) takes the form of
Since the radiation process only lasts for a short interval of δt, the energy emitted into one unit solid angle is
This energy corresponds to a duration of Δt = r sin θΔθ/c in the local burster frame. The luminosity is then L_{ν} = ΔE_{ν}/Δt. For an observer at distance D_{L}, the observed flux at t_{obs}(θ) should be , that is,
The total isotropic energy can be calculated as
where t_{end} = r(cos θ_{1} − cos(θ_{jet} + θ_{obs}))(1 + z)/c is the end time of the pulse in the observer frame. The isotropic peak luminosity is
where θ_{p} is the angle corresponding to the peak time of the light curve. The peak photon energy E_{p} can be derived from the timeintegrated spectrum. In the next section, we present an analytic derivation for the prompt γray emission, paying special attention to the three parameters E_{iso}, L_{p}, and E_{p}.
3. Analytic derivation of E_{p}, E_{iso}, and L_{p}
We first consider the onaxis cases. From Eqs. (1) and (10), we can derive E_{iso} as
where θ_{2} = θ_{jet} + θ_{obs}. Further combining Eq. (4), we have
We note that C_{1} in Eq. (3) is largely dependent on and . In this study, a simple case of is assumed, which naturally leads to . Then with Eqs. (2), (3), and (6), we can further obtain
For an onaxis observer, most of the observed photons are emitted by electrons within a small angle around the line of sight. According to Eq. (7), it is safe for us to simply take ϕ(θ) as 2π. Changing the integral order in Eq. (14), we obtain
In the last step above, the integral of has been simplified as a constant (k_{0}). Equation (15) can be further reduced as
where k_{1} and k_{2} are integration constants. When θ ≪ 1 and β ∼ 1, we have 1/(1 − β cos θ)∼2γ^{2}/(θ^{2}γ^{2} + 1). The integral in Eq. (16) can therefore be approximated as
Finally, combining Eqs. (16) and (17), it is easy to obtain .
Now we derive the peak luminosity of L_{p}. Since the main difference between Eqs. (10) and (11) is the integral of time, we can first consider the luminosity corresponding to a particular angle θ as
The flux peaks at θ_{p} ∼ 0, thus the peak luminosity is .
The peak photon energy in the observer frame can be derived by considering the standard synchrotron emission mechanism,
Therefore, the peak photon energy in the comoving frame is .
Next, we consider the offaxis cases. Again, we first focus on L_{p}. For an offaxis jet, the main difference is that ϕ(θ) can no longer be taken as 2π. Instead, it should be calculated as . Equation (18) now becomes
Here θ ∈ (θ_{obs} − θ_{jet}, θ_{obs} + θ_{jet}). We define . Then it is obvious that R(θ)∈(0, 1), which leads to the approximation . We note that R(θ) can be simplified as
When , we have . Combining Eqs. (20) and (21), we obtain
In this case, L_{p} will peak at . We note that the condition of leads to . Under this condition, L_{p} can be further written as
The last factor is . Its effect is not important compared with (θ_{obs} − θ_{jet})^{−8.5}, and therefore, we ignore this factor for simplicity. Finally, we obtain .
Now we continue to calculate E_{iso}. Since most of the γray energy is released in the decay stage of the light curve, we only need to integrate the energy over a θ ranging from θ_{p} to θ_{obs} + θ_{jet}, that is,
Here the value of I(θ_{obs}, θ_{jet}, θ) will not change significantly in the range from θ_{p} to θ_{obs} + θ_{jet}. Hence it can be taken as a constant for simplicity. Since , we obtain the result as .
Finally, we calculate E_{p}. Similar to Eq. (19), the peak energy in the rest frame is
We see that .
To summarize, in the onaxis cases, we have
while in the offaxis cases (θ_{obs} − θ_{jet} > 1/γ), we obtain
The equations derived above are very simple but intriguing. There are eight input parameters in our model: γ, B′, , θ_{jet}, θ_{obs}, m_{sh}c^{2}, d, and the electron spectral index p. Seven of them are closely related to the prompt emission parameters of E_{p}, E_{iso}, and L_{p}, while p is largely irrelevant. Of all the input parameters, γ plays the most important role in affecting the observed spectral peak energies and fluxes for onaxis cases (Ghirlanda et al. 2018). Usually, γ may vary in a wide range for different GRBs, from lower than 100 to more than 1000. In contrast, and m_{sh} only vary in much narrower ranges. Taking and m_{sh} approximately as constants, from Eq. (26), we can easily derive the onaxis Amati relation as and the onaxis Yonetoku relation as . In both relations, the powerlaw indices are 0.5, which is largely consistent with observations (Nava et al. 2012; Demianski et al. 2017).
In the offaxis cases, previous studies mainly focused on the effect of the variation of the viewing angle among different GRBs. Consequently, the powerlaw index of the offaxis Amati relation is derived as 1/3 based on the sharpedge homogeneous jet geometry (Granot et al. 2002; RamirezRuiz et al. 2005). Here, from our Eq. (27), we see that E_{iso} and L_{p} sensitively depend on both the viewing angle and the Lorentz factor. This indicates that both γ and θ_{obs} are important parameters that will affect the slope of the Amati relation and the Yonetoku relation. In Eq. (27), if only the variation of the viewing angle is considered (i.e., the indices of (θ_{obs} − θ_{jet}) in Eq. (27) are taken into account, but the item of γ is omitted), an index of 4/13 will be derived for the offaxis Amati relation, which is very close to the previous result of 1/3. When the combined effect of varying γ and θ_{obs} is included, the offaxis Amati relation is derived as , and the corresponding offaxis Yonetoku relation is . The powerlaw indices become smaller than in the onaxis cases.
4. Numerical results
Some approximations have been made in deriving Eqs. (26) and (27). In this section, we carry out numerical simulations to confirm whether the above analytical derivations are correct. For convenience, a set of standard values were taken for the eight input parameters in our simulations, as shown in Table 1. When studying the effect of one particular parameter, we only changed this parameter, but fixed all other parameters at the standard values.
Standard values assumed for the eight input parameters.
To examine the correctness of our analytical derivations, we chose one input parameter as a variable, and standard values were taken for all other input parameters so that the dependence of E_{p}, E_{iso}, and L_{p} on that particular parameter can be illustrated.
We first consider the effect of the bulk Lorentz factor γ for both onaxis and offaxis cases. Here we let γ vary between 50 and 1000. Figure 2 shows the timeintegrated spectrum obtained for different γ. The peak photon energy is correspondingly marked with a black star on the curve.
Fig. 2. Timeintegrated spectrum for different γ values (marked in the figure). Panel a: Spectrum of onaxis cases. Panel b: Spectrum of offaxis cases. The black star on each curve marks the position of the peak energy (E_{p}). 
For onaxis cases (θ_{jet} ≥ θ_{obs}), the numerical results are plotted in the left panel of Fig. 2. When the value of γ increases, the peak photon energy and the flux increase simultaneously. However, a different trend is found for the offaxis cases (θ_{jet} < θ_{obs}), as shown in the right panel of Fig. 2. Both the flux and E_{p} decrease as γ increases. The flux is significantly lower than that of the onaxis cases. The results are consistent with those of Farinelli et al. (2021).
Figure 2 clearly shows that both E_{p} and the flux are positively correlated with γ in the onaxis cases, but they are negatively correlated with γ in the offaxis cases. Next, we wish to determine the precise relation between the prompt emission parameters and the input parameters. Hence, we plot E_{p}, E_{iso}, and L_{p} as functions of γ, B′, , θ_{obs} − θ_{jet}, and p in Fig. 3.
Fig. 3. Numerical results showing the dependence of the prompt emission features on the input parameters. The calculated prompt emission parameters (E_{p}, E_{iso}, and L_{p}) are plotted against the input parameters (γ, B′, , θ_{obs} − θ_{jet}, and p) for both the onaxis cases (with cross symbols) and the offaxis cases (with circle symbols). For comparison, the powerlaw indices derived from our analytical solutions are marked correspondingly in each panel. 
Generally, the simulation results are well consistent with our analytical results of Eqs. (26) and (27). Especially E_{p}, E_{iso}, and L_{p} have a positive dependence on the Lorentz factor γ in the onaxis cases, but they have a negative dependence on γ in the offaxis cases. It is also interesting to note that the prompt emission parameters are nearly independent of θ_{obs} − θ_{jet} in the onaxis cases, since the line of sight is always within the jet cone. However, in the offaxis cases, the dependence of the prompt emission parameters on θ_{obs} − θ_{jet} can be described as a twophase behavior. When θ_{obs} − θ_{jet} ≪ 1/γ, the prompt emission parameters are independent of θ_{obs} − θ_{jet}. When θ_{obs} − θ_{jet} > 1/γ, however, they decrease sharply with the increase of θ_{obs} − θ_{jet}, as indicated in Eq. (27). Finally, the last column of Fig. 3 shows that the parameter p has little impact on the prompt emission parameters.
5. Monte Carlo simulations
In this section, we perform Monte Carlo simulations to generate a large number of mock GRBs to further test the existence of the Amati relation and the Yonetoku relation. For this purpose, we performed Monte Carlo simulations as follows. First, a group of eight input parameters is generated randomly, assuming that each parameter follows a particular distribution. This group of parameters defines a mock GRB. Second, our model is applied to numerically calculate the corresponding values of E_{p}, E_{iso}, and L_{p} for this mock GRB. The above two steps are repeated until we obtain a large sample of mock GRBs. Finally, the mock bursts are plotted on the E_{p} − E_{iso} plane and the E_{p} − L_{p} plane, and a bestfit correlation is obtained for them.
For convenience, we designate the distribution function of a particular parameter a as D(a). Here a refers to the input parameters (i.e., γ, B′, , θ_{jet}, θ_{obs}, p, m_{sh}c^{2}, and d). Three kinds of distributions are adopted in this study: (i) A normal Gaussian distribution, which is noted as N(μ, σ), where μ and σ refer to the mean value and standard deviation, respectively. (ii) A uniform distribution, which is noted as U(min,max), where min and max define the range of the parameter. Finally, (iii) a lognormal distribution, noted as D(log(a)), which means that log(a) follows a normal Gaussian distribution of N(μ, σ). The detailed distribution for each of the input parameters adopted in our Monte Carlo simulations is described below.
The ranges of some input parameters, such as γ, θ_{jet}, and θ_{obs}, can be inferred from observations. For the bulk Lorentz factor, a distribution of D(log(γ)) = N(2.2, 0.8) was assumed, which is consistent with observational constraints (Liang et al. 2010; Ghirlanda et al. 2018). As for the angle parameters, we set θ_{jet} as a uniform distribution of D(θ_{jet}) = U(0.04, 0.2) for the on and offaxis cases (Frail et al. 2001; Wang et al. 2018). The distribution of θ_{obs} was taken as D(θ_{obs}) = U(0.0, 0.2) in the onaxis cases, together with the restriction of θ_{jet} ≥ θ_{obs}. On the other hand, in the offaxis cases, θ_{obs} was taken as D(θ_{obs}) = U(0.1, 0.5), together with θ_{jet} < θ_{obs}.
Other parameters cannot be directly measured from observations. Their distributions are then taken based on some theoretical assumptions. The isotropic ejecta mass is usually thought to range from 10^{−3} M_{⊙} to 10^{−1} M_{⊙} for a typical long GRB. Hence we assumed that m_{sh}c^{2} has a lognormal distribution. We set its mean value as 10^{−2} M_{⊙} c^{2} and its 2σ range as 10^{−3} M_{⊙} c^{2} to 10^{−1} M_{⊙} c^{2}, that is, D(log(m_{sh}c^{2}/erg)) = N(52.2, 0.5). The distribution of d was taken as D(log(d/cm)) = N(10, 0.2), which, combined with the above distribution of γ, gives the range of the internal shock radius as r ∼ γ^{2}d ∼ 10^{13} − 10^{16} cm. It is largely consistent with the theoretical expectation of 10^{11} − 10^{17} cm (Zhang 2018).
Some parameters (B′, , p) concern the microphysics of relativistic shocks. The comoving magnetic field strength (B′) is hard to estimate from observations (Zhang & Yan 2011; Burgess et al. 2020). Here, a lognormal distribution with a mean value of 10 G and a standard deviation of 0.4 G was assumed for it, that is, D(log(B′/G)) = N(1, 0.4). The mean value of was taken as 10^{5}, with a distribution of (Burgess et al. 2020). For the parameter p, we simply assumed a normal distribution of D(p) = N(2.5, 0.4).
With the above distribution functions of the input parameters, two samples of mock GRBs were generated through Monte Carlo simulations. One sample included 200 onaxis events, and the other sample included 200 offaxis cases. E_{p}, E_{iso}, and L_{p} were calculated for each mock event.
The distribution of the simulated bursts on the E_{p} − E_{iso} and E_{p} − L_{p} planes is shown in Fig. 4. In the onaxis cases, the bestfit slope is 0.495 ± 0.015 for the E_{p} − E_{iso} correlation, and it is 0.511 ± 0.016 for the E_{p} − L_{p} correlation. These two indices are very close to our theoretical results of 0.5, proving credibility for our analytical derivations. The intrinsic scatters of both relations are about 0.21, indicating that the correlations are rather tight.
Fig. 4. Monte Carlo simulation results. (a,b) Distribution of simulated bursts on the E_{p} − E_{iso} plane and the E_{p} − L_{p} plane for the onaxis cases. (c,d) Distribution of simulated bursts on the E_{p} − E_{iso} plane and the E_{p} − L_{p} plane for offaxis cases. The solid lines show the bestfitting results for the simulated data points, and the dashed lines represent the 3σ confidence level. To conclude, for onaxis bursts, the bestfit Amati relation is and the bestfit Yonetoku relation is . For offaxis bursts, the bestfit Amati relation is , while the bestfit Yonetoku relation is . 
In the offaxis cases, the bestfit slope is 0.282 ± 0.006 for the E_{p} − E_{iso} correlation, which is consistent with our expected range of 1/4 ∼ 4/13 for the offaxis Amati relation. Similarly, on the E_{p} − L_{p} plane, the bestfit slope is 0.206 ± 0.006, which also agrees well with our theoretical range of 1/6 ∼ 4/17 for the offaxis Yonetoku relation. Another interesting point in Figs. 4c and d is that most of the simulated offaxis GRBs have an E_{iso} lower than 10^{48} erg and an L_{p} lower than 10^{46} erg s^{−1}. Only about 5% of our simulated offaxis GRBs have greater E_{iso} and L_{p}. We thus argue that offaxis GRBs are usually LLGRBs. It also indicates that the majority of currently observed normal GRBs should be onaxis GRBs. In Fig. 5, all the simulated onaxis and offaxis bursts are plotted in the same panel. Only a very small number of offaxis events are strong enough to be detected. Their distribution obviously deviates from that of the onaxis events on both the E_{p} − E_{iso} and E_{p} − L_{p} plane.
Fig. 5. Distribution of simulated offaxis bursts as compared with that of onaxis ones. a) All mock bursts on the E_{p} − E_{iso} plane. b) All mock bursts on the E_{p} − L_{p} plane. The dashdotted line and the dotted line correspond to the bestfit result for on and offaxis samples, respectively. The dashed lines represent the corresponding 3σ confidence level. Most of the offaxis bursts are too weak to appear in the two panels here. 
Using the mock bursts, we can also test the Ghirlanda relation (Ghirlanda et al. 2004), which links E_{p} with the collimationcorrected energy of E_{γ} = (1 − cos θ_{jet})E_{iso}. The distribution of the simulated bursts on the E_{p} − E_{γ} plane is shown in Fig. 6. The data points are best fit by a powerlaw function of for the onaxis case (Fig. 6a), with an intrinsic scatter of 0.263 ± 0.031. This agrees well with the recently updated Ghirlanda relation of for a sample of 55 observed GRBs (Wang et al. 2018). For the offaxis case (Fig. 6b), we find , with a larger scatter of 0.272 ± 0.029. This indicates that the Ghirlanda relation may be largely connected with the Amati relation, but compared with the latter, the slope of the Ghirlanda relation is slightly flatter, and the data points are also obviously more scattered. It is difficult task to compare the offaxis Ghirlanda relation with observations because it needs the precise determination of both the beaming angle and the viewing angle for a number of offaxis GRBs, which themselves are rather dim.
Fig. 6. Distribution of the mock GRBs on the E_{p} − E_{γ} plane for the onaxis case. Panel a: onaxis case. Panel b: offaxis case. The solid line shows the bestfit result for the simulated data points, and the dashed lines represent the 3σ confidence level. The onaxis Ghirlanda relation is , and the offaxis Ghirlanda relation reads . 
6. Comparison with observations
In Sect. 5 we illustrated the simulated E_{p} − E_{iso} and E_{p} − L_{p} relations for both onaxis and offaxis cases. We now proceed to compare our results with observational data.
A sample containing 172 long GRBs was used for this purpose, 162 of which were collected from the previous study by Demianski et al. (2017) and Nava et al. (2012). E_{p} and E_{iso} parameters are available for all these 162 events. The L_{p} parameter is available for only 45 GRBs (Nava et al. 2012). Additionally, we collected another 10 bursts that were argued to be possible outliers of the normal Amati relation in previous studies. It has been suggested that they might be offaxis GRBs (Yamazaki et al. 2003b; RamirezRuiz et al. 2005). The observational data of these 10 events are listed in Table 2.
Some possible outliers.
All the observed GRB samples are plotted on the E_{p} − E_{iso} and E_{p} − L_{p} planes in Fig. 7. Most of the GRBs are well consistent with our onaxis results, thus they both follow the Amati relation and the Yonetoku relation.
Fig. 7. Distribution of all the observed GRBs. Panel a: E_{p} vs. E_{iso}. Panel b: E_{p} vs. L_{p}. The dashdotted lines represent our theoretical results for onaxis bursts, and the thick dotted line shows our theoretical results for offaxis bursts. The 3σ confidence levels are marked correspondingly. The solid lines are the bestfit results for normal GRBs. 
On the E_{p} − E_{iso} plane (Fig. 7a), four of the ten possible outliers are located between our on and offaxis lines. It is thus difficult to judge whether these events are onaxis or offaxis GRBs simply from the Amati relation. However, the remaining six bursts can only be matched by our offaxis line. They should indeed be offaxis events. Figures 4c and d clearly show that most offaxis GRBs should be LLGRBs with E_{iso} < 10^{48} erg, with only a small portion falling in the range of 10^{48} − 10^{52} erg (see also Fig. 5a). This is easy to understand. The input parameters should take some extreme values for an offaxis GRB to be strong enough. Especially these offaxis events should generally have a low θ_{obs} − θ_{jet} value (see Eq. (27)), which means that they are still slightly offaxis. In Fig. 7a, only a small number of GRBs are outliers of the onaxis Amati relation. This low event rate is also consistent with our simulation results.
For the E_{p} − L_{p} correlation (Fig. 7b), we note that almost all the ten possible outliers clearly deviate from the onaxis Yonetoku relation. From combining the Amati relation and the Yonetoku relation, we therefore argue that the ambiguous sample should also be offaxis GRBs. Furthermore, Fig. 7 seems to indicate that the Yonetoku relation may be a better tool for distinguishing between on and offaxis GRBs.
7. Conclusions and discussion
The prompt emission of GRBs was studied for both on and offaxis cases. Especially the three prompt emission parameters E_{p}, E_{iso}, and L_{p} were considered. Their dependence on the input model parameters was obtained via both analytical derivations and numerical simulations, which are consistent with each other. We confirmed that E_{p}, E_{iso}, and L_{p} are independent of θ_{obs} and θ_{jet} as long as our line of sight is within the homogeneous jet cone (i.e., the onaxis cases). However, they strongly depend on the value of θ_{obs} − θ_{jet} when the jet is viewed offaxis. Additionally, their dependence on γ is very different for on and offaxis cases, as shown in Eqs. (26) and (27). Through Monte Carlo simulations, we found that the Amati relation is in onaxis cases. Correspondingly, the Yonetoku relation is . On the other hand, in offaxis cases, the Amati relation is and the Yonetoku relation is . The simulated samples were directly compared with the observational samples. We found that they are well consistent with each other in the slopes and intrinsic scatters. The E_{p} − E_{γ} relation was also tested and was found to be for onaxis GRBs and for offaxis GRBs.
We have focused on the empirical correlations of long GRBs in the above analysis. Here we present some discussion of short GRBs. Generally speaking, long GRBs usually contain many pulses (tens or even up to hundreds) in their light curves, while short GRBs contain much fewer pulses. For short GRBs, we can therefore perform a similar analysis, but only when the number of pulses considered is reduced significantly. In our model, the total energy release (E_{iso}) is proportional to the number of pulses, while the peak luminosity (L_{p}) and the spectral parameter E_{p} are nearly irrelevant. In other words, a short GRB will have a much smaller E_{iso}, but its L_{p} and E_{p} are largely unchanged. As a result, short GRBs will follow a parallel track on the E_{p} − E_{iso} plane as compared with long GRBs, the only difference is that their E_{iso} are significantly smaller. Alternatively speaking, the powerlaw index of the Amati relation should be the same for short GRBs and long GRBs, consistent with currently available observations (Zhang et al. 2009, 2018). As for the Yonetoku relation on the E_{p} − L_{p} plane, we argue that it should be almost identical for both long and short GRBs. However, the number of currently available short GRBs that is suitable for exploring the Yonetoku relation (i.e., for which E_{p} and L_{p} are measured) in detail is still too small.
A simple tophat jet geometry was adopted in this study. It should be noted that the GRB outflow might be a structured jet (Ioka & Nakamura 2018). A recent study suggested that E_{p} and the flux will not change much, regardless of the choice of a tophat jet or a nottoocomplicated structured jet (Farinelli et al. 2021). However, in some more complicated scenarios, structured jets may further have an angledependent energy density (Lamb et al. 2021). They may be choked jets or even in jetcocoon systems (Ioka & Nakamura 2018; Mooley et al. 2018; Troja et al. 2019). The empirical correlations of GRBs in these situations are beyond the scope of this study and could be further considered in the future.
In our calculations, the prompt emission was assumed to be due to nonthermal radiation from internal shocks. Other mechanisms such as photosphere radiation and magnetic reconnection above the photosphere may also contribute to the observed flux of GRBs. Generally, there are two types of photosphere models: nondissipative photosphere models, and dissipative photosphere models. The former can produce a narrow quasiPlanckian component and could account for the thermal component observed in some GRBs (Pe’er 2008; Beloborodov 2011), although this component does not play a dominant role in most GRBs (Guiriec et al. 2011; Axelsson et al. 2012). In the dissipative photosphere models, various subphotosphere dissipation processes (Rees & Mészáros 2005), especially Comptonization, may affect the spectrum (Thompson 1994; Pe’er et al. 2006; Pe’er & Ryde 2011; Veres & Mészáros 2012), which may lead to a nonthermal spectrum. Some authors have even used this model to explain the Band spectrum of GRBs (Thompson 1994; Beloborodov 2013; Lundman et al. 2013). Reconnection above the photosphere has also been considered, especially in the view of the ICMART model, focusing on Poyntingfluxdominated outflows (Zhang & Yan 2011; Zhang & Zhang 2014). It is still unclear how commonly photosphere radiation and magnetic reconnection are involved in the prompt phase of GRBs. If these radiation mechanisms are included, the derivation will become much more complicated, but it deserves a trial in the future.
The Amati and Yonetoku relations are essential for us to understand the nature of GRBs. They can also help us probe the highredshift universe (Xu et al. 2021; Hu et al. 2021; Zhao & Xia 2022; Jia et al. 2022; Deng et al. 2023). Our study shows that they are due to the ultrarelativistic effect of highly collimated jets. To be more specific, they mainly result from the variation in the Lorentz factor in different events. It may provide useful insights for a better understanding of these empirical correlations.
Acknowledgments
We would like to thank the anonymous referee for helpful suggestions. This study is supported by the National Natural Science Foundation of China (Grant Nos. 12233002, 12041306, 12147103, U1938201, U2031118, 12273113, 11903019, 11833003), by the National Key R&D Program of China (2021YFA0718500), by National SKA Program of China No. 2020SKA0120300, and by the Youth Innovations and Talents Project of Shandong Provincial Colleges and Universities (Grant No. 201909118).
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All Tables
All Figures
Fig. 1. Geometry of our jet model. The upper panel shows the case for an onaxis jet, where θ_{jet} ≥ θ_{obs}. The lower panel shows an offaxis jet with θ_{jet} < θ_{obs}. 

In the text 
Fig. 2. Timeintegrated spectrum for different γ values (marked in the figure). Panel a: Spectrum of onaxis cases. Panel b: Spectrum of offaxis cases. The black star on each curve marks the position of the peak energy (E_{p}). 

In the text 
Fig. 3. Numerical results showing the dependence of the prompt emission features on the input parameters. The calculated prompt emission parameters (E_{p}, E_{iso}, and L_{p}) are plotted against the input parameters (γ, B′, , θ_{obs} − θ_{jet}, and p) for both the onaxis cases (with cross symbols) and the offaxis cases (with circle symbols). For comparison, the powerlaw indices derived from our analytical solutions are marked correspondingly in each panel. 

In the text 
Fig. 4. Monte Carlo simulation results. (a,b) Distribution of simulated bursts on the E_{p} − E_{iso} plane and the E_{p} − L_{p} plane for the onaxis cases. (c,d) Distribution of simulated bursts on the E_{p} − E_{iso} plane and the E_{p} − L_{p} plane for offaxis cases. The solid lines show the bestfitting results for the simulated data points, and the dashed lines represent the 3σ confidence level. To conclude, for onaxis bursts, the bestfit Amati relation is and the bestfit Yonetoku relation is . For offaxis bursts, the bestfit Amati relation is , while the bestfit Yonetoku relation is . 

In the text 
Fig. 5. Distribution of simulated offaxis bursts as compared with that of onaxis ones. a) All mock bursts on the E_{p} − E_{iso} plane. b) All mock bursts on the E_{p} − L_{p} plane. The dashdotted line and the dotted line correspond to the bestfit result for on and offaxis samples, respectively. The dashed lines represent the corresponding 3σ confidence level. Most of the offaxis bursts are too weak to appear in the two panels here. 

In the text 
Fig. 6. Distribution of the mock GRBs on the E_{p} − E_{γ} plane for the onaxis case. Panel a: onaxis case. Panel b: offaxis case. The solid line shows the bestfit result for the simulated data points, and the dashed lines represent the 3σ confidence level. The onaxis Ghirlanda relation is , and the offaxis Ghirlanda relation reads . 

In the text 
Fig. 7. Distribution of all the observed GRBs. Panel a: E_{p} vs. E_{iso}. Panel b: E_{p} vs. L_{p}. The dashdotted lines represent our theoretical results for onaxis bursts, and the thick dotted line shows our theoretical results for offaxis bursts. The 3σ confidence levels are marked correspondingly. The solid lines are the bestfit results for normal GRBs. 

In the text 
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