Confirmed width-Eiso and width-Liso relations in GRB: comparison with the Amati and Yonetoku relations

In this paper, we select a sample including 141 BEST time-integrated F spectra and 145 BEST peak flux P spectra observed by the Konus-Wind with known redshift to recheck the connection between the spectral width and $E_{iso}$ as well as $L_{iso}$. We define six types of absolute spectral widths. It is found that all of the rest-frame absolute spectral widths are strongly positive correlated with $E_{iso}$ as well as $L_{iso}$ for the long burst for both the F and P spectra. All of the short bursts are the outliers for width-$E_{iso}$ relation and most of the short bursts are consistent with the long bursts for the width-$L_{iso}$ relation for both F and P spectra. Moreover, all of the location energy, $E_{2}$ and $E_{1}$, corresponding to various spectral widths are also positive correlated with $E_{iso}$ as well as $L_{iso}$. We compare all of the relations with the Amati and Yonetoku relations and find the width-$E_{iso}$ and width-$L_{iso}$ relations when the widths are at about 90\% maximum of the $EF_{E}$ spectra almost overlap with Amati relation and Yonetoku relation, respectively. The correlations of $E_{2}-E_{iso}$, $E_{1}-E_{iso}$ and $E_{2}-L_{iso}$, $E_{1}-L_{iso}$ when the location energies are at 99\% maximum of the $EF_{E}$ spectra are very close to the Amati and Yonetoku relations, respectively. Therefore, we confirm the existence of tight width-$E_{iso}$ and width-$L_{iso}$ relations for long bursts. We further show that the spectral shape is indeed related to $E_{iso}$ and $L_{iso}$. The Amati and Yonetoku relations are not necessarily the best relationships to relate the energy to the $E_{iso}$ and $L_{iso}$. They may be the special cases of the width-$E_{iso}$ and width-$L_{iso}$ relations or the energy-$E_{iso}$ and energy-$L_{iso}$ relations.

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