7.5.4. Combo Correlation

The Combo correlation is a hybrid correlation linking the prompt emission *E*<sup>p</sup> and the observable quantities determined from the X-ray afterglow light curve, i.e., among all the rest-frame 0.3–10 keV plateau luminosity *L*0, its rest-frame duration *τ*, and the late power-law decay index *α* [140]. For each GRB, *L*0, *τ*, and *α* can be obtained by fitting the rest-frame 0.3–10 keV flare-filtered afterglow luminosity light curves with the function *L*(*t*) = (1 + *t*/*τ*) *α* . <sup>22</sup> The general expression is much more complicated than previous ones and reads

$$\log\left(\frac{L\_0}{\text{erg/s}}\right) = k\_0 + k\_1 \log\left(\frac{E\_\text{p}}{\text{keV}}\right) - \log\left(\frac{\pi/s}{|1+a|}\right). \tag{42}$$

Here, the constants *k*<sup>0</sup> and *k*<sup>1</sup> need to be determined by means of the calibration procedure. Again, the correlation is characterized by an extra scatter *σ*k.

The Combo correlation can be explained by the external shock scenario (see Section 3). The correlation is the result of the synchrotron emission from the electrons accelerated in a relativistic shock (see Section 3.1.5). The shock propagates through the external CBM and interacts with the magnetic field of the the turbulent plasma. Hence, the relationship among *E*p, *L*0, and *τ* and the corresponding comoving quantities scale with the initial Lorentz factor of the bulk motion Γ0, whereas the intrinsic scatter is due to the uncertainties on the source spectral energy distribution [165].

Keeping in mind the hybrid nature and the Combo correlation, which is a combination of the Amati and the *E* X iso–*E*iso–*E*<sup>p</sup> correlations, the same biases and selection effects at work in the *E* X iso–*E*iso–*E*<sup>p</sup> affect the Combo correlation as well.
