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The GRB Afterglows Flowchart^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Hydrodynamic Evolution

#### 2.1. Hydrodynamic Models

#### 2.2. The Flowchart of the Hydrodynamic Evolution

- Start with the initial parameters for all necessary physical quantities, such as those of the fireball and the external environment.
- Use the substitution $x={log}_{10}$ (R/cm) (logarithmic scale), which is suitable for large scales, dealing with large distances and long periods of time, such as the distance traveled.
- Specify the state of the fireball, such as whether it is radiated, constant or variable (depending on the effectiveness of the radiation). We will also choose a model with a minimum Lorentz factor.
- Choose the hydrodynamic model, then use the finite differences method as a numerical tool to obtain approximate solutions of the Lorentz factor differential equations as a function of the mass m of the surrounding medium that is swept up by the fireball. This method appears to be the simplest one. Furthermore, using this method in our code provides results that converge to the Sedov solution [14], and to the analytical solutions in case of an expanded and constant radiation.

## 3. Radiation Parte

#### 3.1. Synchrotron Radiation and Self-Absorption

#### 3.2. The Flowchart of the Radiation Emission

- To observe the nature of the energy emitted by the afterglows as a function of time with the light curves, we introduce the frequency in the observations, then create a DO loop for various time values to calculate the spectral intensity ${F}_{\nu}\left(t\right)$. In this loop, we call the subroutines of the relativistic transformation for each distance R.
- For the spectra, the opposite is carried out; that is, we set the time then open a DO loop to evolve the frequency, and call the subroutine of the relativistic transformation at every distance R.
- For the third calculations, we make changes to the time and frequency with two DO loops and use the condition IF to save the frequency that gives the maximum of $\nu F\left(\nu \right)$.

## 4. Numerical Results and Discussion

- Figure 3 shows that, in the adiabatic expansion case, the deceleration of the Lorentz factor is slower compared to that of a radiative regime that generates a faster deceleration due to radiation. Moreover, we can observe three sections of the deceleration, corresponding to:
- 1.
- The ultra-relativistic phase.
- 2.
- The relativistic phase.
- 3.
- The non-relativistic phase.

- Figure 4 displays the evolution of the radiative efficiency of the fireball as a function of the distance R, showing that the radiation in Haung’s models is more effective than that of Feng.
- Figure 5 shows the ratio between the absorption coefficient ${\alpha}_{{\nu}^{\prime}}$ for a radio frequency ${\nu}_{obs}=3\xb7{10}^{8}\mathrm{H}\mathrm{z}$ and an UV one ${\lambda}_{obs}^{-1}=500\mathrm{c}{\mathrm{m}}^{-1}$. Note that this is more important at low frequencies. This result is confirmed in Figure 6, where the spectra of GRB afterglow consist of an increased absorption at low frequencies compared to higher frequencies.
- Figure 7 shows that the majority of the radiation during the GRB-afterglow emissions starts by the hard gamma to the radio bands. Therefore, the detection of the prompt emission of GRBs overlaps with the early afterglows
- Figure 8 shows a good concordance between the GRB170202 data supporting the proposed model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Evolution of absorption coeffcient ${\alpha}_{{\nu}^{\prime}}$ for radio frequency and UV frequency (for Feng model).

**Figure 7.**Frequency ${\nu}_{{\left(\nu F\right)}_{max}}$ corresponding to the maximum emission in terms of $\nu {F}_{\nu}$ as a function of the distance R (for Feng model).

**Figure 8.**Comparison of calculated afterglow light curves (our code) to the observed data using the XRT / Swift satellite, considering of the integrated fluence, ${S}_{B}$ (in erg.s${}^{-1}$.cm${}^{-2}$ units) in the X-ray band (E = 0.2–10 keV).

Models | Internal Energy | Lorentz Factor Evolution | |
---|---|---|---|

Chaing [7] | $U=\int (\Gamma -1)dm$ | $dU=(\Gamma -1)dm$ | $\frac{d\Gamma}{dm}=-\frac{{\Gamma}^{2}-1}{M}$ |

Huang [8] | $U=(\Gamma -1)m{c}^{2}$ | $dU=(\Gamma -1)dm{c}^{2}+m{c}^{2}d\Gamma $ | $\frac{d\Gamma}{dm}=-\frac{{\Gamma}^{2}-1}{{M}_{0}+\epsilon m+2(1-\epsilon )\Gamma m}$ |

Feng [9] | $U=\int (1-\epsilon )d{U}_{ex}$ | $dU=(1-\epsilon )d{U}_{ex}\backslash \backslash d{U}_{ex}=(\Gamma -1)dm{c}^{2}+m{c}^{2}d\Gamma $ | $\frac{d\Gamma}{dm}=-\frac{{\Gamma}^{2}-1}{{M}_{0}+m+U/{c}^{2}+(1-\epsilon )\Gamma m}$ |

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**MDPI and ACS Style**

Zouaoui, E.; Mebarki, N.
The GRB Afterglows Flowchart. *Phys. Sci. Forum* **2023**, *7*, 51.
https://doi.org/10.3390/ECU2023-14045

**AMA Style**

Zouaoui E, Mebarki N.
The GRB Afterglows Flowchart. *Physical Sciences Forum*. 2023; 7(1):51.
https://doi.org/10.3390/ECU2023-14045

**Chicago/Turabian Style**

Zouaoui, Esma, and Noureddine Mebarki.
2023. "The GRB Afterglows Flowchart" *Physical Sciences Forum* 7, no. 1: 51.
https://doi.org/10.3390/ECU2023-14045