*3.2. Density Profile of the External Medium*

Following the established connection between long GRBs and the core-collapse of massive stars, the jet is expected to produce the afterglow while propagating in the wind of the star in its free-streaming phase. Afterglow radiation of long GRBs should then be produced in the interaction with a medium with a radial density profile *n* ∝ *R* −2 . However, several investigations have demonstrated that about half of the long GRBs are better explained if the blast-wave is assumed to run into a medium with constant density. We revise the evidence in support of the constant density medium and discuss the difficulties in reconciling these observations with expectations on the environment surrounding long GRB progenitors.

Long GRBs originate from the core-collapse of massive stars, most likely rapidly rotating, and with a possible evidence of a preference for low-metallicity. The most convincing

evidence in support of this paradigm is the association with type Ic supernovae and the proximity of GRBs to young star-forming regions. While the connection of long GRBs (or at least with the bulk of the population) with the core-collapse of massive stars is solid, the role of metallicity and rotation in the launch of the GRB jets, and the identification of the progenitor star, is still uncertain. The progenitor is usually identified with Wolf-Rayet stars, massive stars (*M* > 20 *M*) in the final stages of their evolution, characterized by powerful winds and a high mass loss rate [101]. The wind from the star is expected to interact and deeply modify the environment where the GRB explodes and leaves imprints on its afterglow emission.

More in detail, the interaction between the stellar wind and the ISM four concentric regions with different properties is expected to form. In the inner part (i.e., close to the star), the circumburst medium is permeated by the free-streaming wind, producing a density with radial profile *n* ∝ *R* −2 . The density is related to the mass loss rate *M*˙ and to the velocity *v<sup>w</sup>* of the free-streaming stellar wind by:

$$m(\mathbf{R}) = \frac{\dot{\mathbf{M}}}{4\pi \,\mathbf{R}^2 \, m\_p \, \upsilon\_w} \,. \tag{52}$$

A termination shock separates the unshocked from the shocked wind: the latter forms a hot bubble of thermalized wind material, with a nearly constant density profile, as the formation of pressure and density gradients is prevented by the high sound speed inside the bubble. The hot bubble, in its outer part, is enclosed by a shell of shocked ISM, surrounded by the unshocked ISM. The GRB jet is supposed to trill its way in this stratified medium [10].

To understand where most of the afterglow evolution occurs, we have to estimate the deceleration radius *R<sup>d</sup>* and the non-relativistic radius *RNR* (i.e., the radius where the blast-wave has decelerated to non-relativistic velocity) and compare them to the termination shock radius. For typical parameters (*M*˙ <sup>=</sup> <sup>10</sup>−5*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>10</sup><sup>3</sup> km s−<sup>1</sup> ), the fit to numerical models of Wolf-Rayet stars [102] give the following relation between the termination shock radius and the density of the unshocked ISM: *R<sup>T</sup>* = 10 *n* −1/2 *ISM* pc, where *nISM* is the density of the unshocked ISM. From the blast-wave dynamics, the deceleration and the non-relativistic radius are *<sup>R</sup><sup>d</sup>* <sup>=</sup> <sup>6</sup> <sup>×</sup> <sup>10</sup>−<sup>5</sup> *<sup>E</sup>*<sup>52</sup> *<sup>v</sup>w*,3/(*M*˙ <sup>−</sup><sup>5</sup> <sup>Γ</sup>0,2) pc and *<sup>R</sup><sup>d</sup>* <sup>=</sup> 0.6 *<sup>E</sup>*<sup>52</sup> *<sup>v</sup>w*,3/*M*˙ <sup>−</sup><sup>5</sup> pc, respectively. It is evident how the complete evolution of the afterglow radiation occurs well inside the free-streaming region.

In afterglow modeling of long GRBs it is then customary to assume a density profile described by Equation (52), where *M*˙ and *v<sup>w</sup>* are treated as unknown parameters (normalized to the typical values of a Wolf-Rayet star) combined in one single free model parameter *<sup>A</sup>*?: *<sup>n</sup>*(*R*) = <sup>3</sup> <sup>×</sup> <sup>10</sup>35*A*?*<sup>R</sup>* −2 . Despite this robust prediction, the modeling of afterglow observations shows that in a relevant fraction of cases, observations are better explained by adopting a circumburst medium with a constant density *n* = *n*0.

The fraction of this case varies depending on the method and on the selected sample, and is on average about 50% [103–106].

To place the termination shock at least inside the non-relativistic radius, one should invoke a very large density of the ISM, *n* & 10<sup>5</sup> cm−<sup>3</sup> , typical of dense cores of molecular clouds: *R<sup>T</sup>* = 0.03 (*nISM*/10<sup>5</sup> cm−<sup>3</sup> ) <sup>−</sup>1/2 pc. Density profiles for different ISM densities are shown in Figure 8, upper panel. Alternatively, one can try to variate the wind parameters. How the termination shock radius changes for different values of *M*˙ and *v<sup>w</sup>* is shown in the bottom panel of Figure 8. A very low mass loss rate *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−7*<sup>M</sup>*yr−<sup>1</sup> (which may find a justification in case of a low-metallicity star) is needed to bring the termination shock radius below 1 pc (for *nISM* = 10 cm−<sup>3</sup> ). With this low mass loss rate, the deceleration and non-relativistic radius increase (*R<sup>d</sup>* <sup>∼</sup> <sup>6</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> pc and *<sup>R</sup>NT* <sup>∼</sup> <sup>60</sup> pc), placing the termination shock still after the deceleration radius but well within the non-relativistic radius, allowing for part of the observed emission to develop into a constant density environment. By increasing the blast-wave energy, the deceleration radius can further approach *RT*. This suggests that it is more likely for a very energetic GRB to cross the termination shock at early times and then expand in a ISM-like medium, as compared to

a faint GRB. An indication of an average larger *E<sup>γ</sup>* in GRBs with a wind-like medium as compared to GRBs with a ISM-like medium has been found in [107], but is in contrast with results from the study performed by [106] on a larger sample.

**Figure 8.** Density profile produced by a star wind as a function of the distance from the central object. The *R* <sup>−</sup><sup>2</sup> profile characterizes the region where the wind freely streams, which is separated from the shocked wind (with nearly constant density) by the termination shock. **Top** panel: the impact of different values of the ISM density on the termination shock is shown. For all the curves, it is assumed *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−5*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>10</sup><sup>3</sup> km s−<sup>1</sup> , while the ISM density varies from 10 to 10<sup>4</sup> cm−<sup>3</sup> , with the termination shock moving to smaller distances when the density increases. **Bottom** panel: for a fixed ISM density *nISM* = 10 cm−<sup>3</sup> , the impact of different *M*˙ and *v<sup>w</sup>* is shown. Black-solid: *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−4*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>10</sup><sup>3</sup> km s−<sup>1</sup> ; red-solid: *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−6*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>10</sup><sup>3</sup> km s−<sup>1</sup> ; black-dotted: *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−5*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>500</sup> km s−<sup>1</sup> ; red-dotted: *<sup>M</sup>*˙ <sup>=</sup> <sup>10</sup>−5*<sup>M</sup>* yr−<sup>1</sup> and *<sup>v</sup><sup>w</sup>* <sup>=</sup> <sup>2</sup> <sup>×</sup> <sup>10</sup><sup>3</sup> km s−<sup>1</sup> . From [102]. ©AAS. Reproduced with permission.

The parameter space for which part of the afterglow emission can indeed be produced in the ISM-like density profile of the shocked wind is very limited, as it corresponds to the most energetic GRBs, low-metallicity progenitors and high-density ISM, or a combination of these factors [107]. These considerations on the diversity of *E<sup>k</sup>* , *M*˙ , *v<sup>w</sup>* and ISM density may not be sufficient to explain the results of the modeling (i.e., the preference for a ISMlike environment). The fraction of GRBs, which might have these peculiar parameters can hardly account for the large fraction of GRBs for which a wind-like profile is excluded by observations. The required conditions are too extreme to be verified in half of the population. However, it is not clear if this percentage has been overestimated by present studies. To quantify the inconsistency, the first step would be to perform a dedicated study of afterglow emission to assess the percentage of long GRB afterglows that are not consistent with a wind-like environment.

Methods based on closure relations may not be valid if the spectrum is modified by Compton scattering in the Klein–Nishina regime (see also [108]). Moreover, these are based on a simple approximation of the synchrotron spectrum into power-law segments, while the wide curvature of the real synchrotron spectra might lead to incorrect estimates of the value of *p* if the observed frequency is in the vicinity of a synchrotron break frequency. A full modeling is then necessary to really assess the fraction of long GRBs for which an *R* <sup>−</sup><sup>2</sup> density profile is excluded, and ultimately understand if the paradigm for the environment of GRBs should be drastically modified. Radio observations may be of great help, since the flux temporal behavior does not depend on *p* and is quite different in the

case of constant or wind-like density profile. Similarly, the detection of SSC radiation can help solve this ambiguity.
