7.4.3. Amati or *E* <sup>p</sup>–*E*iso Correlation

Likely, the most used and investigated relation is represented by the so-called *E*p–*E*iso or Amati correlation [151] that can be recast here by

$$\log\left(\frac{E\_{\rm p}}{\rm keV}\right) = a\_0 + a\_1 \left[\log\left(\frac{E\_{\rm iso}}{\rm erg}\right) - 52\right].\tag{39}$$

Here, we have two free constants, namely *a*<sup>0</sup> and *a*1, that represent the calibration constants to determine once the relation is somehow calibrated. A possible limitation of the *E*p–*E*iso correlation is due to the extra source of variability *σ*a. This is thought as a direct consequence of hidden variables that contributes to the overall calibration, albeit we cannot directly observe them [152].

A possible explanation for the *E*p–*E*iso correlation considers the thermal radiation emitted when the GRB jet drills through the core of the progenitor star (see Section 2.4.1), responsible for the thermal peak in the spectrum, and the Compton scattering of this radiation by relativistic electrons outside the photosphere (see Section 3.1.5 and Ref. [153], for details).

There are claims that the Amati correlation is caused by some selection effect of observations, rather than being an intrinsic property of GRBs [149,154]. However, there is a general consensus on the fact that the correlation is real [155–157], though detector sensitivity affects the correlations and a weak fluence dependence may be larger than the statistical uncertainty and contributes to the dispersion of the correlation [158,159].

7.4.4. Ghirlanda or *E* <sup>p</sup>–*E<sup>γ</sup>* Correlation

From theoretical and observational arguments in favor of the jetted nature of GRBs [36], the radiated GRB energy can be corrected by means of the collimation factor *f* = 1 − cos *θ*, leading to *E<sup>γ</sup>* = *f E*iso. In particular, the jet opening angle *θ* is evaluated at the characteristic time *t*<sup>b</sup> for specific assumptions on the circumburst medium that can be assumed to be homogeneous [28]. The functional form adopted here for the Ghirlanda relation reads

$$\log\left(\frac{E\_\mathrm{p}}{\mathrm{keV}}\right) = b\_0 + b\_1 \left[\log\left(\frac{E\_\gamma}{\mathrm{erg}}\right) - 50\right] \tag{40}$$

in which, as usual, *b*<sup>0</sup> and *b*<sup>1</sup> are the two free constants, fixed by means of calibration. The extra scatter, *σ*b, behooves us to better constrain the relation itself.

The Ghirlanda correlation shares with the *E*p–*E*iso one a similar physical interpretation (see Section 3.1.5 and Ref. [153], for details). This correlation also takes into account the jet correction in the computation of the GRB energy output (see Section 2.4).

The Ghirlanda correlation is linked to the Amati one and, thus, criticisms/analyses against/in favor of being an intrinsic property of GRBs. In Ref. [160], it was shown that as many as 33% of the BATSE bursts would not be consistent with the Ghirlanda relation, but these results depended upon the assumed distribution for the jet's correction factor *f* [161]. This fact limits the increase in the correlation sample. For details, see Ref. [130]. Likewise for the Amati correlation, the Ghirlanda one is statistically real but strongly affected by the thresholds of GRB detectors [159].
