---
abstract: 'Using a simple off-axis jet model of GRBs, we can reproduce the observed unusual properties of the prompt emission of GRB980425, such as the extremely low isotropic equivalent $\gamma$-ray energy, the low peak energy, the high fluence ratio, and the long spectral lag when the jet with the standard energy of $\sim10^{51}$ ergs and the opening half-angle of $10\degr\lesssim\Delta\theta\lesssim30\degr$ is seen from the off-axis viewing angle $\theta_v\sim \Delta \theta +10\gamma^{-1}$, where $\gamma$ is a Lorentz factor of the jet. For our adopted fiducial parameters, if the jet that caused GRB980425 is viewed from the on-axis direction, the intrinsic peak energy $E_p(1+z)$ is $\sim$2.0–4.0 MeV, which corresponds to those of GRB990123 and GRB021004. Our model might be able to explain the other unusual properties of this event. We also discuss the connection of GRB980425 in our model with the X-ray flash, and the origin of a class of GRBs with small $E_\gamma$ such as GRB030329.'
author:
- Ryo Yamazaki
- Daisuke Yonetoku
- Takashi Nakamura
title: 'GRB980425 in the Off-Axis Jet Model of the Standard GRBs'
---

[ address=[Department of Physics, Kyoto University, Kyoto 606-8502, Japan]{} ]{}

[ address=[Department of Physics, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920-1192, Japan]{} ]{}

[ address=[Department of Physics, Kyoto University, Kyoto 606-8502, Japan]{} ]{}

INTRODUCTION
============

There are some GRBs that were thought to be associated with SNe [@dv03; @stanek]. GRB980425 / SN1998bw, located at $z=0.0085$ (36 Mpc), was the first event of such class [@ga98; @kul98; @pian00; @pian03]. It is important to investigate whether GRB980425 is similar to more or less typical long duration GRBs. However, GRB980425 showed unusual observational properties. The isotropic equivalent $\gamma$-ray energy is $E_{iso}\sim 6\times10^{47}$ ergs and the geometrically corrected energy is $E_\gamma=(\Delta\theta)^2E_{iso}/2\sim3
\times10^{46}$ergs$(\Delta\theta/0.3)^2$, where $\Delta\theta$ is the unknown jet opening half-angle. These energies are much smaller than the typical values of GRBs. The other properties of GRB980425 are also unusual; the large low-energy flux [@fro00a], the low variability [@frr00], the long spectral lag [@norris00], and the slowly decaying X-ray afterglow [@pian00; @pian03].

Previous works suggest that the above peculiar observed properties may be explained if the standard jet is seen from the off-axis viewing angle (e.g.[@in01; @na01]). Following this scenario, the relativistic beaming effect reduces $E_{iso}$ and hence $E_\gamma$. In this paper, in order to explain all of the observed properties of GRB980425, we reconsider the prompt emission of this event using our simple jet model [@yin02; @yin03a; @yin03b; @yyn03].

SPECTRAL ANALYSIS OF GRB980425 USING BATSE DATA {#sec:analysis}
===============================================

We argue the time-averaged observed spectral properties of GRB980425. Using the BATSE data of GRB980425, we analyze the spectrum within the time of FWHM of the peak flux in the light curve of BATSE channel2. We fit the observed spectrum with the Band function. The best-fit values are $\alpha=-1.0\pm0.3$, $\beta=-2.1\pm0.1$, and $E_p=54.6\pm20.9$keV, which are consistent with those derived by the previous works [@fro00a; @ga98]. This spectral property is similar to one of the recently identified class of the X-ray flash (XRF) [@He01a; @ki02]. The observed fluence of the entire emission is $S$(20–2000keV) $=(4.0\pm0.74)\times10^{-6}$ergcm$^{-2}$, thus we find $E_{iso}=(6.4\pm1.2)\times10^{47}$ergs. The fluence ratio is $R_s=S$(20–50keV)/$S$(50–320keV) $=0.34\pm 0.036$.

MODEL OF PROMPT EMISSION OF GRB980425
=====================================

We use a simple jet model of prompt emission of GRBs, where an instantaneous emission of infinitesimally thin shell is adopted [@in01; @yin02; @yin03a; @yin03b; @yyn03]. See @yyn03 for details. We fix model parameters as $\alpha_B=-1$, $\beta_B=-2.1$, $\gamma\nu'_0=2600\,{\rm keV}$, and $\gamma=100$. Normalization of emitted luminosity is determined so that $E_\gamma$ be observationally preferred value of $1.15\times10^{51\pm0.35}(h/0.7)^{-2}$ergs [@bloom03] when we see the jet from the on-axis viewing angle $\theta_v=0$. Our calculations show that on-axis intrinsic peak energy becomes $E_p^{(\theta_v=0)}(1+z)\sim1.54\gamma\nu'_0\sim 4.0$ MeV, in order to reproduce the observed quantities of GRB980425. Indeed, there are some GRBs with higher intrinsic $E_p$; for example, $E_p(1+z)\sim2.0$MeV for GRB990123 and 3.6MeV for GRB021004 [@amati02; @bar03].

The left panel of Figure.1 shows $E_{iso}$ as a function of the viewing angle $\theta_v$. When $\theta_v\lesssim\Delta\theta$, $E_{iso}$ is constant, while for $\theta_v\gtrsim\Delta\theta$, $E_{iso}$ is considerably smaller than the typical value of $\sim10^{51-53}$ergs because of the relativistic beaming effect. We next calculated $E_p$ and $R_s$ for the set of $\Delta\theta$ and $\theta_v^\ast$ that reproduces the observed $E_{iso}$ of GRB980425. For our parameters, $\Delta\theta$ should be between $\sim$18$\degr$ and $\sim$31$\degr$, and then $\theta_v^\ast$ ranges between $\sim$24$\degr$ and $\sim$35$\degr$ in order to reproduce the observation results. Thorough discussions on the right panel of Figure1 is found in [@yyn03].

![ (Left panel): the isotropic equivalent $\gamma$-ray energy $E_{iso}$ is shown as a function of the viewing angle $\theta_v$ for a fixed jet opening half-angle $\Delta\theta$. The source is located at $z=0.0085$. The values of $\Delta\theta$ are shown in parentheses. Solid lines correspond to the case of $\gamma\nu'_0=2600$ keV, while dotted lines $\gamma\nu'_0=1300$ keV. Horizontal dashed line represents the observed value of GRB980425. (Right panel): the upper panel shows $\theta_v^\ast$ for which $E_{iso}$ is the observed value of GRB980425, while the middle and the lower panels represent the fluence ratio $R_s^\ast=R_s^{(\theta_v=\theta_v^\ast)}$ and the peak energy $E_p^\ast=E_p^{(\theta_v=\theta_v^\ast)}$, respectively. Solid lines correspond to the fiducial case. The dotted lines represent regions where $E_{iso}$ becomes $(6.4\pm1.2)\times10^{47}$ ergs when $\E_\gamma$ is in 1 $\sigma$ and 5 $\sigma$ level around the fiducial value, respectively. The dot-dashed line in the upper panel represents $\theta_v^\ast=\Delta\theta$. Horizontal dashed lines in the middle and the lower panels represent the observational bounds. ](nakamura_fig1 "fig:"){width="45.00000%" height=".27\textheight"} ![ (Left panel): the isotropic equivalent $\gamma$-ray energy $E_{iso}$ is shown as a function of the viewing angle $\theta_v$ for a fixed jet opening half-angle $\Delta\theta$. The source is located at $z=0.0085$. The values of $\Delta\theta$ are shown in parentheses. Solid lines correspond to the case of $\gamma\nu'_0=2600$ keV, while dotted lines $\gamma\nu'_0=1300$ keV. Horizontal dashed line represents the observed value of GRB980425. (Right panel): the upper panel shows $\theta_v^\ast$ for which $E_{iso}$ is the observed value of GRB980425, while the middle and the lower panels represent the fluence ratio $R_s^\ast=R_s^{(\theta_v=\theta_v^\ast)}$ and the peak energy $E_p^\ast=E_p^{(\theta_v=\theta_v^\ast)}$, respectively. Solid lines correspond to the fiducial case. The dotted lines represent regions where $E_{iso}$ becomes $(6.4\pm1.2)\times10^{47}$ ergs when $\E_\gamma$ is in 1 $\sigma$ and 5 $\sigma$ level around the fiducial value, respectively. The dot-dashed line in the upper panel represents $\theta_v^\ast=\Delta\theta$. Horizontal dashed lines in the middle and the lower panels represent the observational bounds. ](nakamura_fig2 "fig:"){width="40.00000%" height=".3\textheight"}

DISCUSSION {#sec:dis}
==========

We have found that when the jet of opening half-angle of $\Delta\theta\sim10$–30$\degr$ is seen from the off-axis viewing angle of $\theta_v\sim\Delta\theta+6\degr$, observed quantities can be well explained. Observed low variability can be explained since only subjets at the edge of the cone contribute to the observed quantities [@yin02]. If the time unit parameter $r_0/c\beta \gamma^2$ is about 3sec, which is in the reasonable parameter range, the spectral-lag of GRB980425 can be also explained.

Our result might be able to explain the slowly decaying X-ray afterglow of GRB980425. If we assume the density profile of ambient matter as $n=n_0 (r/r_{ext})^{-2}$ with $n_0r_{ext}^2=4\times10^{17}$cm$^{-2}$, the break in the afterglow light curve should occur at $t_b = 3.1\times10^2$days$E_{51}(\Theta/0.4\,{\rm rad})^2$, where $\Theta$ is defined by $\Theta^2=(\Delta\theta)^2+\theta_v^2$, and $E$ is the total energy in the collimated jet [@na99; @na01]. Since our calculation suggests $\Theta$ should range between 0.4 and 0.67rad, $t_b$ is consistent with the observation [@na01]. Up to the break time, one can estimate the flux in the X-ray band as $F$(2–10keV) $\propto t^{-0.2}$, where we assume $\theta_v\gg\Delta\theta$ and the spectral index of accelerated electrons as $p=2.2$ [@na99; @na01]. This result is also consistent with the observation [@pian00; @pian03]. Furthermore, the adopted value of $n_0r_{ext}^2$ corresponds to the mass loss rate of the progenitor star $\dot{M}=1.3\times10^{-6}M_\odot$yr$^{-1}$$(v_{\rm W}/10^3\,
{\rm km}\,{\rm s}^{-1})$, which might be able to explain the radio data (see [@wa03]).

The observed quantities of small $E_p$ and large fluence ratio $R_s$ are the typical values of the XRF [@fro00a; @He01a; @ki02]. The operational definition of the [*BeppoSAX*]{}-XRF is a fast X-ray transient with duration less than $\sim10^3$ s which is detected by WFCs and not detected by the GRBM. If the distance to the source of GRB980425 were larger than $\sim90$ Mpc, the observed flux in the $\gamma$-ray band would have been less than the limiting sensitivity of GRBM, so that the event would have been detected as an XRF. We might be able to explain the origin of a class with low $E_\gamma$ such as GRB980326 and GRB981226 [@bloom03], and GRB030329 whose $E_\gamma$ is about $\sim 5 \times 10^{49}$ergs if the jet break time of $\sim0.48$ days is assumed [@tamagawa; @vanderspek]. Let us consider the jet seen from a viewing angle $\theta_v\sim\Delta\theta+\gamma_i^{-1}$, where $\gamma_i$ is the Lorentz factor of a prompt $\gamma$-ray emitting shell. Due to the relativistic beaming effect, observed $E_\gamma$ of such a jet becomes an order of magnitude smaller than the standard energy. At the same time, the observed peak energy $E_p$ is small because of the relativistic Doppler effect. In fact, the observed $E_p$ of the above three bursts are less than $\sim$70keV. In our model the fraction of low-$E_\gamma$ GRBs becomes $2/(\gamma_i\Delta\theta)\sim 0.1$ since the mean value of $\Delta \theta \sim 0.2$, while a few of them are observed in $\sim$30 samples [@bloom03]. In later phase, the Lorentz factor of afterglow emitting shock $\gamma_f$ is smaller than $\gamma_i$, so that $\theta_v<\Delta\theta+\gamma_f^{-1}$. Then, the observed properties of afterglow may be similar to the on-axis case $\theta_v\ll\Delta\theta$; hence the observational estimation of the jet break time and the jet opening angle remains the same.

This work was supported in part by Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology, No.05008 (R.Y.), No.14047212 (T.N.), and No.14204024 (T.N.).

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