Gravitational wave memory of gamma-ray burst jets

Gamma-ray bursts (GRBs) are now considered as relativistic jets. We analyze the gravitational waves from the acceleration stage of the GRB jets. We show that (i) the point mass approximation is not appropriate if the opening half-angle of the jet is larger than the inverse of the Lorentz factor of the jet; (ii) the gravitational waveform has many step function like jumps, and (iii) the practical DECIGO and BBO may detect such an event if the GRBs occur in the Local Group. We found that the light curve of GRBs and the gravitational waveform are anticorrelated so that the detection of the gravitational wave is indispensable to determine the structure of GRB jets.

Figure 1

The amplitude of the gravitational wave memory for the GRBs jet in Eq. (7) is shown as a function of the viewing angle ${\theta }_v$ by solid lines. We adopt the Lorentz factor $\gamma =100$. The opening half-angle of the jet is $\Delta \theta =0.01,0.1$ and $0.2$ from top to bottom. We can see that the amplitude is suppressed in the forward direction ${\theta }_v\lesssim \Delta \theta$. The amplitude for the point mass in Eq. (2) is also shown by the dashed line. Clearly the point mass approximation is good when the opening half-angle is smaller than the inverse of the Lorentz factor $\Delta \theta <{\gamma }^{-1}$ or the viewing angle is larger than the opening half-angle ${\theta }_v\gtrsim \Delta \theta$. The amplitude is normalized by $h_0=\gamma m{\beta }^2/r\sim 2.7\times {10}^{-23}(\gamma m/{10}^{51}\mathrm{e}\mathrm{r}\mathrm{g})(r/1\mathrm{M}\mathrm{p}\mathrm{c}{)}^{-1}$.

Figure 2

The angular distribution of $N_{\mathrm{t}\mathrm{o}\mathrm{t}}=350$ subjets confined in the whole GRB jet in our simulation. The whole jet has the opening half-angle of $\Delta {\theta }_{\mathrm{t}\mathrm{o}\mathrm{t}}=0.2\mathrm{r}\mathrm{a}\mathrm{d}$, while the subjets have $\Delta {\theta }_{\mathrm{s}\mathrm{u}\mathrm{b}}=0.02\mathrm{r}\mathrm{a}\mathrm{d}$. The axes and the angular size of subjets are represented by crosses and the dotted circles, respectively. A represents the center of the whole jet and is hidden by the lines of subjets. (Also refer to Yamazaki, Ioka and Nakamura 27.)

Figure 3

The observed X-ray and gamma-ray photon number flux from the multiple subjets, corresponding the cases A (the upper left), “B.1”(the upper right), “B.2”(the lower left) and C (the lower right) in Fig. 2. Here the flux in $30-400\mathrm{k}\mathrm{e}\mathrm{V}$ band for each case is shown. (Also refer to Yamazaki, Ioka and Nakamura 27.)

Figure 4

The waveform of the gravitational wave memory from the GRB is shown. There are three timescales that determine the temporal structure of the waveform, the total duration $\Delta T\sim 30\mathrm{s}$, the separation between successive jets $\Delta T_{\mathrm{s}\mathrm{e}\mathrm{p}}\sim \Delta T/N_{\mathrm{t}\mathrm{o}\mathrm{t}}=0.086\mathrm{s}$ and the rise time of the amplitude due to a single jet $\Delta T_{\mathrm{a}\mathrm{c}\mathrm{c}}\sim {10}^{-3}\mathrm{s}$ (although we neglect the rise time here). The final amplitude is reached after the total duration $\Delta T$. The normalization, $h_0$ is the same one in Fig. 1.

Figure 5

The characteristic amplitude $h_c$ of the gravitational wave memory is plotted as a function of frequency for the whole GRB jets. The solid line represents $h_c$ for the multiple step waveform (case “B.2”) with the total energy $2\gamma m=3\times {10}^{51}\mathrm{e}\mathrm{r}\mathrm{g}$, the duration $\Delta T=30\mathrm{s}\mathrm{e}\mathrm{c}$ and the distance to the source $r=1\mathrm{M}\mathrm{p}\mathrm{c}$. The dashed line represents $h_c$ for the slope waveform with the duration $t_m=30\sec $. Here the amplitude $\Delta h_m$ in Eq. (11) is chosen so that both amplitudes coincide in the low-frequency limit.

Figure 6

The noise amplitude $h_n$ of LIGO II, LISA and DECIGO/BBO is also plotted by the dotted line. The characteristic amplitude for the gravitational wave memory of GRBs is also plotted by the bold solid line. The upper bold solid line corresponds to a GRB with the total energy $2\gamma m=3\times {10}^{51}\mathrm{e}\mathrm{r}\mathrm{g}$, the duration $\Delta T=30\sec $ at the $10\mathrm{k}\mathrm{p}\mathrm{c}$, and the lower one to a GRB with $2\gamma m=3\times {10}^{51}\mathrm{e}\mathrm{r}\mathrm{g}$ , $\Delta T=10\sec $ at the $1\mathrm{M}\mathrm{p}\mathrm{c}$. The characteristic amplitude for the slope waveform with $\Delta h_m=4m\gamma {\beta }^2/r$ is used for the low-frequency region, $f\lesssim 1/\Delta T$. The amplitude at $f\gtrsim 1/\Delta T$ (plotted by the bold dashed line) depends on the structure of the GRB jet. The SNR can be obtained by Eq. (13).