An Efficient Numerical Simulation for Gamma Ray Burst Energy Profiles

October 6, 2025

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Gamma-Ray Bursts (GRBs) are one of the most energetic events observed in the universe that can outshine both stars and galaxies alike. GRBs are composed of high-energy photons in the gamma-ray range of the electromagnetic spectrum and propagate at relativistic speeds through the interstellar medium. Due to their extreme speeds, scientists consider the sheer amount of energy that launch the ejecta at speeds nearing the speed of light, or relativistic speeds. These high levels of energy are expended through a beam, which is referred to as the jet of a GRB; these jets are theorized to drive the ejecta at velocities approaching the speed of light. In this work, we observe the dynamics of these jets through a devised and numerically simulated model that dictates energy levels with respect to time and simulate the emission using a Python program for a visual analysis of the model. More specifically, we analyze the jet energy emission (in erg/s) and energy per solid angle analysis for three example GRBs and their properties. We then discuss the outcomes and implications of the results of the simulation, as well as conclusions regarding the experiment. Through our simulation, we show that the jet dynamics produce a theorized “plateau” in energy emission, directly being caused where the GRB turns non-relativistic, transitioning into slower speeds relative to the speed of light. We open source our code to support further research and experimentation, available at https://github.com/Om-Kasar/GRB-Jet-Energy-Model.

Keywords: Gamma-Ray Bursts, Energy Distributions, GRB Modeling, Numerical Simulation, Jet Dynamics

1. Introduction

Gamma Ray Bursts (GRBs) are concentrated beams of photons that occur during especially large cosmic events. GRBs are related to many fundamental areas of astrophysics, including the properties of the surrounding medium and host galaxy, gravitational wave analysis, and the study of cosmic rays¹. After a GRB’s initial emission, it proceeds to travel in the ambient medium at a speed nearing the speed of light, and slows down as a result of a great amount of accumulated mass and energy emission throughout its lifespan in the medium. As a GRB slows down and progresses through the interstellar medium, it slowly transitions to the non-relativistic stage, the stage in which the GRB stops advancing the outside medium at speeds nearing the speed of light. As the GRB experiences a transition to this regime, the dynamics fundamentally change², resulting in new dependencies for the central engine, most notably a change from dependence on the initial angle to its observed luminosity profile. For instance, the GRB dynamics transition from a dependency on the emission angle to the total energy emitted of a GRB as it transitions from becoming relativistic to non-relativistic.

As a result of their chaotic outburst being closely associated with cosmic cataclysms³, scientists hypothesize that the main engine causing the gamma rays, accumulated mass, and other GRB contents (referred to as the ejecta), to move at such speeds is the concept of a jet, a heavily concentrated beam of plasma⁴ that occurs at the start of its launch and guides the GRB ejecta through the interstellar environment. The main causes of the jet is by either the merging of a binary system or, more commonly, the collapse of especially massive stars known as collapsars⁵. This hypothesis fails to account for specific dynamics of the jet, such as its energy levels and angular distributions, and remains fairly elusive despite a multitude of studies being done on the topic. Due to our limited understanding of energy emission⁶, this work attempts to model an approximation of the energy emission of a given GRB jet throughout its initial prompt emission and its lifetime in the interstellar medium.

The development of multiple technologies, such as the Fermi/LAT telescope⁷, has enabled the use of complex GRB afterglow detections to collect vital data of numerous instances of GRB explosions. These technologies directly facilitate many works in the understanding the fundamentals of GRB dynamics. In this work we thank the Swift/BAT GRB catalog in providing the data for conceptual analysis this work⁸, as well as the ClassiPyGRB Python library to integrate the catalog with approaches to data analysis⁹.

In this work, we begin by explaining the energy emission model in Section 2, including detailed explanation of each component of the model. In Section 3, we simulate the model using a C++ program with a Python interface and compare it to similar models from other works. We then explain the implications and potential flaws that could be improved upon in future studies in Section 4 and draw conclusions in Section 5 based on our results.

2. Energy Emission Model

In developing this model, we aim to create an accurate and computationally efficient method to track the energy levels of a given GRB jet in a consistent manner. The model leverages the relationship between the initial ejection energy and the decrease in energy levels as the GRB travels through space. For the general model, it’s vital to recognize the use and meaning of each ratio, and the magnitude at which it affects the energy of the GRB jet.

2.1 Model Assumptions and Concepts

We begin by providing preliminary explanation of the fundamentals of our energy analysis. As the jet expands from the central engine, it is expected to collect a large portion of the mass from the external medium at which it is launched in combination with its initial mass gained from the central engine¹⁰. As the jet steadily sweeps up more mass, the GRB is expected to rapidly decelerate as a result of inertia. Because of this, one of the main factors that directly contributes to the jet’s deceleration is the accumulation of mass in the external medium, thus causing a reduction¹¹ in its energy emission¹². We define the following values for the mass swept up per solid angle of the GRB according to Wang¹³:

(1) $\begin{equation*}M_{sw} = \int \Gamma \rho_{sw} , r^2 , dr\end{equation*}$

where $\rho_{sw} = 4 \rho_{0} \Gamma$ and $rho_{0} propto r^{-k}$ ( $k$ is the power-law index), in accordance with the shock jump condition as seen by Wang¹⁴ and Uhm¹⁵. The radial integrated value as a function of time will be evaluated in combination with total ejecta mass, defined as the radial integrated value plus the initial mass per solid angleof the GRB. This gives us the following:

(2) $\begin{equation*}M_{ej} = M_{sw} + \frac{M_{0}}{\Omega_{0}}\end{equation*}$

where $M{0}$ is the initial mass of the GRB and $\ohm_{0}$ is the initial solid angle of the GRB. In addition, the initial mass of the GRB will be calculated with the following relationship mentioned by Sari¹⁶:

(3) $\begin{equation*}E_{0} = M_{0} \Gamma_0 c^2\end{equation*}$

in which $E_{0}$ is the initial ejection energy of the GRB.

Furthermore, we assume an energy model in which the energy level dependencies change after the GRB advances through the medium at times $t \geq t_{NR}$ , or the times at which the GRB becomes non-relativistic. $t_{NR}$ is defined as the following according to vanEerten’s conjecture on GRB propagation

(4) $\begin{equation*}t_{NR} = 1100 \left( \frac{E_{\text{iso},0}}{10^{53} n_{0}} \right)^{1/3} \text{days}\end{equation*}$

where the $E_{\text{iso},0}$ is the initial isotropic equivalent energy of the GRB, and $n_{0}$ is the external medium density in which the GRB is launched. We note that the expression for $t_{NR}$ is in days.

The model utilizes a piecewise approximation of the Lorentz factor during a GRB’s propagation, where the Lorentz factor is constant until the threshold t = $t_{dec}$ , or the time of rapid deceleration after the GRB’s initial outburst, as suggested by Figure 1. We define the approximation to be the following expression in accordance to Uhm’s findings¹⁷ on GRB afterglow speeds:

(5) $\begin{equation*}\Gamma \approx\begin{cases}\Gamma_{0}, & 0 \leq t \leq t_{dec} \\[6pt]\Gamma_{0} \cdot \left( \frac{t_{dec}}{t} \right), & t > t_{dec}\end{cases}\end{equation*}$

Figure 1: A GRB afterglow analysis model, detailing the approximate Lorentz factor with respect to radial displacement from the central engine. As shown, the Lorentz factor becomes constant until the r = 10¹⁷ cm threshold, suggesting a constant Lorentz factor approximation until some drop-off, which may be defined as some time nearing t = t_dec. Adapted from Uhm¹⁵.

We utilize this model in order to suggest the same speed throughout the interval [0.01, t_dec] and then a sudden drop in velocity after the t_dec threshold, to align with Uhm’s findings¹⁷ on decreasing Lorentz factors in afterglow propagation.

Lastly, the jet energy levels before t_NR depend on the Lorentz factor Γ and ejection angle θ of the GRB. After the t = t_NR threshold, the GRB becomes dependent on the radius r from the central engine as well as the total ejection energy emission from luminosity E_ej². These parameters will be represented as decreasing ratios, since the energy decreases as the GRB progresses through the medium.

2.2 Jet Energy Model

Following these assumptions, we now derive the approximate jet energy model. We delve into the possible energy emission, logic behind each component, and its implications on jet energy emission as a whole.

Firstly, we propose the model as a sudden decrease in kinetic and luminous energies shortly after a given GRB’s observation, namely at t_dec, in which energy levels suddenly drop. Furthermore, a multitude of afterglow simulations and observations noted a sudden plateau in afterglow observation as a result of the theorized “energy reservoir”¹⁸ as the GRB propagates through the medium (observed in Figure 2). We encapsulate these observations through a decreasing reciprocal function, where the value asymptotically approaches some arbitrary horizontal asymptote (say, 10⁴⁵ erg) and produces a ”plateau” in overall jet ejection energy. Therefore, in the relativistic stage, we define the expression and scale factor for energy emission (denoted as dE_j,R/dt and α, respectively) after our analysis as:

(6) $\begin{equation*}\frac{dE_{j,R}}{dt} = E_{0} \cdot \left[ t \left( 1 + \left( \frac{t}{t_{dec}} \right)^{\alpha(t)} \right) \right]^{-1}\end{equation*}$

To detail these dependencies in our model, we define $\alpha$ as:

(7) $\begin{equation*}\alpha = \left[\left( \frac{M_{ej}}{M_{sw}} \right)^{0.25}\left( \frac{\Gamma}{\Gamma_{0}} \right)^{0.5}\left( \frac{\theta_{0}}{\theta} \right)^{0.5}\right]^{0.5}\end{equation*}$

We include the ratio Mej/Msw, since it elaborates on the dependence of the swept-up mass pattern of the GRB jet, which is thought to cause the deceleration of the GRB with the accumulated mass during its propagation as a result of inertia and conservation of momentum¹⁹. We set this dependency as a ratio of the total ejecta mass and the swept-up mass to analyze the factor of change in the GRB’s mass profile to suggest the change in GRB dynamics with respect to the mass.

Figure 2: A representation of the flux density (mJy) from a proposed GRB afterglow model. Represents a theorized “plateau” in observed flux over the interval [10³, 10⁴]. Adapted from Uhm¹⁵.

The ratio Γ/Γ₀ reflects the dependence on both the Lorentz factor and the initial Lorentz factor on the theoretical kinetic energy emissions of the GRB jet²⁰. The Lorentz factor Γ is expected to experience a rapid decrease as it progresses through the medium because the mass is swept by the GRB jet, causing a decrease in the kinetic energy of the ejecta in the interstellar medium.

The component θ₀/θ of our model represents a relationship in which the jet energy directly correlates with the ejection angle compared to its initial launch angle, θ₀, before the time t = t_NR, according to vanEerten². The desired output will be similar to a decreasing pattern for the typical GRB explosion. Since θ(t) is inversely proportional to θ₀, this gives us a decreasing pattern in the observed emission angle that is thought to further decelerate the concentrated ejecta in the medium.

After analyzing the jet energy emission in the relativistic stage, we now present the non- relativistic dynamics of GRB jets. To extend upon the aforementioned plateau, we utilize an exponential decrease in energy in order to elongate the observed plateau and propose a decrease in energy beyond the asymptote in times t_NR and beyond. The expression and scale factor, denoted as dE_j,NR/dt and β, respectively, can be defined as the following:

(8) $\begin{equation*}\frac{dE_{j,NR}}{dt} = \left \frac{dE_{j,R}}{dt} (t_{NR}) \cdot \exp \left( -\beta (t - t_{NR}) \right)\end{equation*}$

in which $\beta$ is:

(9) $\begin{equation*}\beta = \left( \frac{r_{NR}}{r} \cdot \frac{E_{0}}{E_{ej}} \cdot \left( \frac{M_{ej}}{M_{sw}} \right)^{0.25} \right)\end{equation*}$

$E_{ej}$ is defined as the integral of a given light curve of a GRB, given as the following⁴’²¹:

(10) $\begin{equation*}E_{ej}(t) = \frac{\theta(t)^{2}}{2} \int_{0.01}^{t} L_{v}(x) , dx\end{equation*}$

where $L_{v}(t)$ is the light curve of the GRB and $E_{iso}$ is the isotropic equivalent energy of the light curve. We employ this dependency as a result of the patterns from many Swift/BAT GRB light curves and their tendency to portray a decreasing pattern of luminous energy emission shortly after their violent outburst. After further analysis of GRB light curves from the Swift/BAT catalog, we point out that, as $t rightarrow infty$ , the ejection energy has negligible changes as a result of the continued sharp decrease of its energy levels²² from t = t_dec and t = t_NR thresholds. Therefore, for simplicity, we define the following limit for an approximate expression for E_ej(t) in the analyzed computational model to encapsulate the luminous energies of the GRB:

(11) $\begin{equation*}\lim_{t \to \infty} E_{ej}(t) \approx E_{0} + 1.5 \times 10^{50}\end{equation*}$

We note that the function is directly proportional to $dE_{j,R}/{dt}(t_{NR})$ in order to ensure a continuous function over the interval (0, t]. This condition is necessary in order to derive the energy per solid angle parametric function, discussed later on in this paper.

The relationship $r_{NR}/r(t)$ signifies the dependency of the radial distance from the central engine after the GRB is in the non-relativistic stage, which is thought to be an indicator of many aspects of GRB energy levels²³. As the increase in radius from its entity at which the GRB becomes non-relativistic (denoted as r_NR, or r(t_NR)) becomes more significant, the ratio suggests a desired decreasing relationship in radius as a result²⁴.

The dependency E₀/E_ej(t) demonstrates how the observed ejection energy is the dominant factor in the overall dynamics of the jet after passing the t_NR threshold²⁵. Since the approximate value for overall ejection energy heavily decreases after the main energy emission, we approximate the observed ejection energy independent to the initial energy emitted as 1.5·10⁵⁰ for a standard GRB.

Finally, we consider the mass profile of our model due to the inertial effect mentioned earlier. We believe that this mass component will continue to create an impact on GRB energy profiling because of a noticeable accumulation of mass even after the t_NR threshold.

Composing each component of the equation yields the approximate derivation of the jet energy model (denoted as dE_j/dt) with respect to time. We use a piecewise structure for our model to denote the desired plateau behavior observed in Figure 2, seen as:

(12) $\begin{equation*}\frac{dE_{j}}{dt} =\begin{cases}\frac{dE_{j,R}}{dt}(t), & 0 \leq t \leq t_{NR} \\[8pt]\frac{dE_{j,NR}}{dt}(t), & t > t_{NR}\end{cases}\end{equation*}$

The energy of the jet model in the relativistic and non-relativistic time ranges can then be represented as the following:

(13) $\begin{equation*}\frac{dE_{j}}{dt}(t) =\begin{cases}E_{0} \left[ t \left( 1 + \left( \frac{t}{t_{dec}} \right)^{\alpha(t)} \right) \right]^{-1},& 0 \leq t \leq t_{NR} \\[10pt]\frac{dE_{j,R}}{dt} (t_{NR}) \cdot \exp \left( -\beta (t - t_{NR}) \right),& t \geq t_{NR}\end{cases}\end{equation*}$

3.1. Experimentation

We now discuss the experiments used for energy analysis, including the simulation setup as well as our results from numerical simulations. We elaborate on the general experimentation process and possible implications of the numerical simulation in this section as a whole.

3.1 Simulation Setup

For the numerical simulation, our computer’s specifications involve an RTX 4060 GPU, an 12^th generation Intel CPU, and 16 gigabytes of RAM. We use Python in order to provide a wrapper for the C++ system to interface with the user. We store constants and experiment configurations into a .yaml file for calculations through the Parameters.yaml file and the dynamically constructed CalculatedParameters.yaml file, where the values are created as a result of dependencies located in the Parameters.yaml file. Detailed instructions can be found at the GitHub repository. We then compare 3 sample GRBs’ (narrow, medial, and wide GRBs, respectively) properties and jet expansions. We discuss their implications (refer to Table 1 and Table 2), and demonstrate their modeled energy emissions in comparison to one another.

Next, we use C++ in our program for general computational efficiency for each function. After the .yaml files have defined the experimental constants and parameters, we reference the files’ parameters in a structure function in the source code, and define them globally for experimentation in the main.cpp file. We then utilize the pybind11 package to compile the C++ program into a .pyd file in order interface with the C++ module in Python. All the ratios involved, including the mass, Lorentz factor, and ejection angle ratios, are included in the package for individual testing if needed.

Parameter	GRB1	GRB2	GRB3	GRB4	GRB5	GRB6	GRB7
t_dec(s)	10	12	15	25	50	70	100
E₀(erg)	5.00 * 10⁵¹	9.00 * 10⁵¹	2.5 * 10⁵²	10⁵³/4π	8.00 * 10⁵²	10⁵³	2.00 * 10⁵³
A (cm)	10⁵	3.5 * 10⁵	7.5 * 10⁵	4.5 * 10⁶	6 * 10⁶	8 * 10⁶	10⁷
k	2.2	2.0	1.7	1.3	1.27	1.22	1.2
n₀(cm^-3)	0.01	0.35	0.75	1.0	1.15	1.35	1.5
θ₀ (rad)	0.05	0.2	0.45	0.7	0.72	0.77	0.8

Table 1: Experiment Configurations for each GRB [Narrow (1, 2), Medial (3, 4), and Long (5, 6, 7)] (Parameters.yaml files)

We evaluate the model by displaying the numerical simulation results through the matplotlib library. We investigate the shapes at which the GRB graphs $d E_j / d t$ change with respect to time for the three GRBs. In addition, we also perform energy per solid angle analysis in order to compare the data with other energy per solid angle models by by parameterizing $\theta$ and $d E_j / d \Omega$ for analysis using a similar process.

In the following, we derive the energy per solid angle function, with respect to time. The equations for energy per solid angle (denoted as $d E_j / d \Omega(t)$ or $epsilon(t)$ ) are given by the following:

(14) $\begin{equation*}\frac{d E_j}{d \Omega}=\frac{d E_j}{d t} \cdot \frac{d t}{d \Omega}\end{equation*}$

According to Salafia⁴, energy per solid angle equates to approximately the initial luminous energy emitted before the relativistic threshold. Therefore, dt/dΩ must then be defined as:

(15) $\begin{equation*}\frac{d t}{d \Omega}=\frac{E_0}{d E_{j, R} / d t}=t \left(1+\left(\frac{t}{t_{d e c}}\right)^{\alpha(t)} \right)\end{equation*}$

Using this identity, we can derive the first component of the energy per solid angle model, yielding:

Parameter	GRB1	GRB2	GRB3	GRB4	GRB5	GRB6	GRB7
E_{iso, 0} (erg)	8.00 * 10⁵³	4.45 * 10⁵³	2.47 * 10⁵³	3.25 * 10⁵²	3.09 * 10⁵³	3.37 * 10⁵³	6.25 * 10⁵³
Г₀	793.5	474.2	343.0	212.0	212.9	186.0	173.4
M₀(g)	1.40 * 10²⁷	2.11 * 10²⁸	8.11 * 10²⁸	4.18 * 10²⁸	4.18 * 10²⁹	6.00 * 10²⁹	1.28 * 10³⁰
t_NR (s)	8.28 * 10⁶	2.51 * 10⁶	1.33 * 10⁶	6.13 * 10⁶	1.23 * 10⁶	1.21 * 10⁶	1.44 * 10⁶
r_NR(cm)	1.37 * 10²⁹	1.27 * 10²⁸	2.64 * 10¹⁷	7.52 * 10³¹	3.07 * 10³²	2.93 * 10³²	2.06 * 10²⁹

Table 2: Dynamically Created Configurations for each GRB (CalculatedParameters.yaml files)

(16) $\begin{equation*}\epsilon_1(t)=E_0 \cdot \left[t \left(1+ \left(\frac{t}{t_{d e c}} \right)^{\alpha(t)} \right) \right]^{-1} \cdot \frac{d t}{d \Omega}= E_0\end{equation*}$

We can now define energy per solid angle as the following parametric by dividing both parts of the jet energy emission function, giving us:

(17) $\begin{equation*}\epsilon(t)= \begin{cases}E_0, & 0 \leq t \leq t_{N R} \\[6pt]E_0 \cdot \exp \left(-\beta(t) \cdot \left(t-t_{N R} \right) \right) \cdot \frac{d t}{d \Omega}, & t \geq t_{N R}\end{cases}\end{equation*}$

3.2 Results

In this section, we now discuss the operation of the computational model, its results, and its implications on jet energy emission as a whole.

For the model, we first input each requested value into the Parameters.yaml file. The experiment configurations are seen in Table 1. We then run the parameters.py file in order dynamically create the CalculatedParameters.py file, which can be seen in Table 2. These experiment configurations will then be processed through the “burstConfigs” alias in the experimental Python files in order to output a numerically simulated graph through the matplotlib Python library.

As seen in Figure 3, the jet emission model somewhat agrees with our hypothetical image of GRB jets. We include the plateau after the t = t_NR threshold in order to align with our afterglow analysis. Each simulated GRB represents some form of the theorized plateau in energy emission, as suggested by Figure 1. In addition, as hypothesized, the fundamental dynamics of each GRB jet changes after passing the t = t_NR threshold for each GRB.

Figure 3: A numerically simulated graph for dE_j/dt using the approximate jet energy emission model. Burst configuration parameters for each simulated GRB are given in Table 1 and Table 2. Dotted lines represent times in which t = t_NR for the narrow, medial, and wide GRBs, denoted with a 1-7 as such with corresponding colors for their respective GRBs. Model error estimates include an approximate margins of ±10³ for each analyzed data point in our curves.

However, we also observe sharp turns in the model, as shown in the results from GRB1, GRB2, and GRB3. The energy emission model includes these sharp turn as likely as a result of sudden conversion of other forms of energy (e.g. luminosity) or a sudden interaction with a dense medium²⁶. Being a narrow GRB with less mass, the jet could have accumulated more mass and rapidly decelerated through an especially dense medium in order for the plateau phase of its emission much earlier than the other GRBs. However, this hypothesis fails to account for the extended length of the plateau for GRB1; the jet is expected to accumulate more mass as it propagates through the medium, causing the jet to move slower and exert more energy with a surefire decrease in ejection energy rather than a standard plateau.

Alternatively, we believe that a more plausible explanation for these sharp turns involves the energy reservoir²⁷; we believe that these turns result in the narrow GRB’s tendency to transition towards their energy reservoir at an increasingly sudden rate during its propagation, which is especially emphasized in Kumar’s study²⁸. We attribute this theory to a more physically significant explanation of these sharp turns because many studies have alluded to energy-reservoir transitions in narrow and medial GRB’s, which are yet to be answered as of now, as seen in Frail’s²⁹ and Panaitescu’s³⁰ works.

A nearly instantaneous change in the jet’s internal energy reservoir could also explain the relatively abrupt plateau of GRB1 and GRB2. Through potential interactions with the reverse shock during its growth or other internal processes, the GRBs may have instantly dissipated some of its energy in the process of its expansion³¹. This hypothesis could also explain the broad, constant emission of the medial GRBs’ graphs, suggesting a negligible difference in energy emission. This could also apply to the sharp turn seen at GRB5, GRB6, and GRB7, but at a much shorter scale throughout its propagation in the medium overall.

We also recognize that, in Figure 3, the GRBs experience an accelerated linear decrease after its initial ejection energy at times nearing t = 10¹ s. This rapid decrease shortly after each GRBs’ emission agrees with the conventional understanding of GRBs, in which GRB energy emission decreases from 1-100 seconds after the initial burst emission in the prompt phase¹⁹.

Lastly, we discuss the end behavior of each GRB after its plateau phase in Figure 3. We see that these graphs align with conventional understanding of GRBs; the graphs reveal a constant energy per solid angle profile until it reaches the θ = θ_NRthreshold. We observe an especially heavy decrease over the timescale ∼ t >> t_NR in order to account for the increased ejecta density and mass accumulation over the large time period³². In combination with the general spreading of the GRB as a result of forward shock expansion, this conjecture agrees with the behavior of many GRBs analyzed in previous studies³³^,³⁴.

We now elaborate on the extent of the energy per solid angle analysis. As seen in Figure 4, the energy per solid angle appears to be constant, with a immediate, exponential drop off at θ values around ∼ ∆θ = 10² − 10³. This behavior somewhat aligns with the standard understanding of GRBs. As each GRB traverses the ambient medium, the GRBs are thought to retain its’ conical shape before transitioning to the non-relativistic regime, as well as a minimal change in ejection angle from the jet axis³⁵. Therefore, we justify the constant energy per solid angle at the start of the energy emission.

Figure 4: A numerically simulated graph for dE_j/dΩ(t) (ϵ(t)) using the approximate energy per solid angle model. The parametric equation for ϵ(t) is given in Eq. 17 Simulated burst configurations are given in Tables 1 and 2. Dotted lines represent times in which θ = θ_NR for the narrow, medial, and wide GRBs, denoted as a number from 1-7 as such with corresponding colors in the legend. Error predictions include values nearing an average of ±10⁴ for each data point in each energy per solid angle curve.

However, this hypothesis does not account for a crucial factor in the profile of energy per solid angle, being the excessive elongation of the energy per solid angle ratio during the interval ∆θ < θ_NR. This could be due to the constant relationship occurring as a result of minimal changes in angle from the jet axis before the θ = θ_NR threshold. Therefore, we propose that the model has a substantial decrease in energy per solid angle only after a noticeable change in θ in jet dynamics from its initial angle³⁶, namely at the θ_NR threshold.

The energy per solid angle drop-off is accurate to the average illustration of GRB energy emission as a whole. The steep drop-off appears to be relevant to the general ejection energy after the θ = θ_NR threshold. The simulated model suggests that, as each GRB becomes non-relativistic, the energy per solid angle function experiences a great decrease in energy in proportion to the general energy profile dE_j/dt³⁷. This conclusion agrees with the general idea of the model, in the sense that a rapid decrease in ejecta energy occurs mainly after the GRB becomes non-relativistic³⁸, thus yielding accurate models to the jet emission patterns of GRBs.

4. Discussion

4.1 Parameter Sensitivity Analysis and Observational Uncertainties

We now conduct a brief parameter sensitivity analysis for our model to analyze the effects of each parameter usage. For our sensitivity analysis, we run multiple instances of our program with GRB4 configurations from Tables 1 and 2 and vary the values of the deceleration time t_dec, initial energy E₀, density constant A, and initial angle θ₀ to observe the changes in our jet energy curve. We then assess what these changes suggest about the GRB’s propagation in the interstellar medium.

With the t_dec parameter, we observe a surprising pattern after input values ranging from 10 − 100 s; the decrease in energy emission before the t_NR phase appears to alleviate the harsh transition into the plateau phase before the t_NR threshold with higher values of t_NR. A plausible hypothesis that supports this behavior could be as a result of the GRB’s noticeable dependencies after its ultra-relativistic state before the t_dec threshold, as seen in a previous study by Sultana³⁹.

We now analyze the changes in the jet energy curves through various changes in the density⁴⁰ proportionality constant A of the GRB ejecta, ranging from 10⁴ − 10⁸ cm. As the values of A increases, we observe a GRB energy curve that gives a smoother transition to the energy reservoir state, similar to that of t_dec. We believe a main driver for this trend is the increase in the mass ratio of our model as a result of an increased GRB ejecta density, causing our curves to have an increased exponential value for every t during the GRB’s propagation.

Finally, we now survey values for θ₀in radians from 0.1 − 0.85 and observe how our GRB4 energy curve changes. Based on our analysis, we notice surprising artifacts of our energy curves; as θ₀ increases in value, we note an elongated plateau phase as the GRB traverses the medium before the t_NR threshold, likely as a result of the increased angular ratio in our model. This shows us that GRB4’s behavior towards the theorized energy reservoir is heavily dependent on higher values of its initial emission angle, which agrees with Zhang’s⁴¹ conclusion in his GRB analysis.

4.2 Comparison with Existing Models and Observational Data

We compare our energy per solid angle model with three other models. In addition to the simulation of this GRB and its implications, we also compare and contrast GRB2’s configurations with our model and three models in order to see how our model fares in comparison to them for energy per solid angle analysis.

We discuss comparative data using a standard Gaussian model. The Gaussian model is given by the following equation, adapted from Salafia⁴:

(18) $\begin{equation*}\frac{dE}{d \Omega}(\theta) = E_{0} \cdot \exp \left( -\frac{\left( \frac{\theta}{\theta_{c}} \right)^{2}}{2} \right)\end{equation*}$

We mention that the values for $\theta$ is in degrees for the Gaussian model, and where $\theta_{c}$ = 3 and 4.

Along with this we compare our approximate energy per solid angle model to the GRB energy per solid angle collapsar model detailed by Lazzati⁴², which elaborates on the relationship dE/dΩ(θ) ∝ θ⁻², given by the following formula:

(19) $\begin{equation*}\frac{dE}{d \Omega}(\theta) = \frac{2\sqrt{3},L r_{*}}{\pi c} \theta^{-2},\qquad \theta_{j,br} < \theta < \theta_{0}\end{equation*}$

It is worth nothing that this model is also dependent on a degree measure, and applies if the values of $\theta$ from the jet axis is within the interval $\theta_{j,br}$ (= 1 in the model) and initial angle $\theta_{0}$ . For the comparative model, we also define r∗, or the radius of the star in the collapsar model, to be equal to 109 cm and L to be constant luminosity 1.5 · 1050 erg. With that being said, we now move on to the assessment of the numerically simulated model in comparison to other models.

As shown in Figure 5, the output is relatively consistent with the formulation of our model. Both curves represent a similar relationship with each other, with a plateau present at the start of its formation. However, in comparison to the Gaussian models, the energy per solid angle ratio of our model appears to drop off much earlier in the jet’s propagation. Though each curve represents a similar decrease in energy after its drop-off threshold, the elongated energy per solid angle proves to be a further discrepancy in the model. In fact, the standard Gaussian models appear to asymptotically approach the line $\theta = \theta_{NR}$ instead of having a fundamental change in energy emission.

Figure 5: A numerically simulated graph for GRB4’s dE_j/dΩ(θ) or (ϵ(t)) over a multitude of simulation methods. θ = θ_NR is labeled as a blue dotted line to represent the threshold value for θ as it passes the non-relativistic threshold. The purple and yellow curves represents the uniform jet model of energy per solid angle between different values of θ_c. The green line represents the collapsar model with a host star radius of r_∗ = 10⁹ cm and constant luminosity L ≈ 1.5 · 10⁵⁰ erg (Refer to Equation 19).

For the collapsar model (refer to Equation 19), the model dictates the energy decrease in our model with an approximate linear decrease in energy with respect to θ. The model manages to estimate the drop off pretty well in comparison to our model, and represents a somewhat accurate depiction of an energy per solid angle profile for GRB2.

Overall, each model appears to somewhat represent the general consensus of our understanding of GRB energy per solid angle analysis. Despite a few discrepancies in their estimation, they are fairly consistent in drop-off and plateau in general ejection energy at large.

4.3 Model Inconsistencies and Potential Improvements

Though the model appears to be accurate with the general idea of GRB prompt emission, it has some limitations that arise from the assumptions made in our model. We elaborate on how this work could be improved upon in future studies regarding energy emission as well as additional refinements for the model in order to increase its accuracy in modeling jet energy emission and energy per solid angle analysis for GRBs.

The jet energy model includes a broad assumption for the total ejection energy released via luminosity (E_ej(t)). The model approximates the expression to a constant ∼E₀ + 1.5 · 10⁵⁰ erg for the sake of simplicity and computational efficiency. However, we caution the use of this ratio in further models to ensure a more accurate representation of the jet energy emission model, since the luminosity does not stay constant throughout its entire duration. We encourage the use of the integral version of the formula for future models in order to have a more accurate profile on the luminosity model as a whole.

In addition, the jet energy graph has many sharp turns, which could either imply a computational error or a different aspect about jet energy emission as a whole. We believe each sharp turn is the direct result of an issue with the computational model, so future models may try to remedy this.

For other GRBs, the plateau in jet energy emission does not necessarily need to be at least at 10⁴⁵ erg. Depending on the burst parameters, future models could calculate a way to calculate this asymptote and apply it accordingly.

We also point out some discrepancies in the energy per solid angle model. In order to improve this model, a shorter horizontal line representing the ϵ(t) plateau for any GRB could be the best fit in general energy per solid angle analysis, in accordance to the Gaussian jet energy model.

We also note that the limit used in order to derive the relativistic part ϵ(t) is somewhat broad in the overall energy emission of the GRB. The assumption using the limit is could lead to a multitude of inaccuracies seen in Figure 4. A prime example of such behavior possibly due to this limit approximation can be seen through subtle rise in energy per solid angle in each of the GRBs’ behaviors. We believe that this discrepancy is an oversight in the calculation for this model. Therefore, we caution further use of the identity used for dt/dΩ in further models. Instead, we suggest the use of an- other, alternative expression for dt/dΩ in order to generate an expression for energy per solid angle.

Based on our analysis, we also detail a few possible observational uncertainties that may occur during the GRB’s propagation. We note that the GRB may collide with something during its travels in the medium, which may result in an increased inertial effect and the unexpected change in energy emission for the GRB. In addition, the GRB may experience sudden internal shocks during its initial ultra-relativistic outflow stage⁴³ and disrupt energy emission patterns in our numerical model. Therefore, we illustrate the need to account for these patterns in energy emission in future works containing our numerical model.

5. Conclusion

In this paper, we discussed the theoretical jet dynamics and jet energy emission in a given GRB’s propagation. We developed a reciprocal function and exponential model for mapping out the general energy profile of a GRB’s lifespan and observation accordingly by graphing jet energy emission and energy per solid angle analysis through a numerically simulated model in C++ and Python. After discussing the simulation of the model, we construe a few main conclusions regarding the dynamics of GRBs. The results for the energy profiles for each GRB are likely dependent on the energy reservoir that a given GRB as it traverses the medium. The main cause of the plateau behavior in both jet ejection energy and energy per solid angle models alike point to the idea of an energy reservoir and emits a constant energy release, as seen in the simulated models. Even though there are some discrepancies in the conceptualization and calculation of the model, each curve is consistent with other models and concepts. The energy per solid angle analysis, though having a more elongated plateau in its emission, still accurately represented a drop-off similar to that of the discussed energy per solid angle models. As a result of the similar behavior in accordance to other models, we conclude that the jet energy emission appear to be proportional to the ratio of each of the burst parameters and their function counterparts, represented as decreasing ratios. We encourage future works on the topic to further advance the fundamental understanding of GRBs using these ratios if needed. In addition, the GRB jet has some change in dynamics over its energy profile in order to produce a plateau. In our work, we proposed that this threshold is t = t_NR and used that as a basis for all of our analysis. The formula used to calculate t = t_NR could be different as a result of elongated constant energy per solid angle emissions in Figure 4 and Figure 5.

This study was facilitated Swift/BAT GRB Catalog and the ClassiPyGRB Python library. I would like to thank to Annika Thomas, a researcher from Massachusetts Institute of Technology, for guiding me through the writing of this paper.

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An Efficient Numerical Simulation for Gamma Ray Burst Energy Profiles

1. Introduction