# Particle-in-Cell Simulations of Astrophysical Relativistic Jets

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## Abstract

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Contents | ||

1. Introduction......................................................................................................................................................................... | 2 | |

1.1. Astrophysical Jets...................................................................................................................................................... | 2 | |

1.2. The TRISTAN Code................................................................................................................................................... | 2 | |

1.3. Particle-in-Cell Approach and Plasma Instabilities............................................................................................... | 3 | |

2. Microscopic and Macroscopic Processes in Plasma Jets................................................................................................ | 5 | |

3. PIC Simulations................................................................................................................................................................... | 6 | |

3.1. Unmagnetized Jets..................................................................................................................................................... | 6 | |

3.1.1. Self-Consistent Synthetic Spectra from Shocks........................................................................................... | 6 | |

3.1.2. Shear Velocity Simulations with the Slab Model and Cylindrical Jets..................................................... | 8 | |

3.1.3. Global Simulations of Unmagnetized Relativistic Jets.............................................................................. | 9 | |

3.2. Magnetized Jets......................................................................................................................................................... | 12 | |

3.2.1. Topology of Relativistic Helical Jets............................................................................................................ | 12 | |

3.2.2. Global Simulations with Helical Jets and Large Radii.............................................................................. | 14 | |

3.2.3. Global Jet Simulations with a Toroidal Magnetic Field and a New Injection Scheme........................... | 18 | |

4. Summary.............................................................................................................................................................................. | 22 | |

References................................................................................................................................................................................ | 23 |

## 1. Introduction

#### 1.1. Astrophysical Jets

#### 1.2. The TRISTAN Code

#### 1.3. Particle-in-Cell Approach and Plasma Instabilities

## 2. Microscopic and Macroscopic Processes in Plasma Jets

## 3. PIC Simulations

#### 3.1. Unmagnetized Jets

#### 3.1.1. Self-Consistent Synthetic Spectra from Shocks

#### 3.1.2. Shear Velocity Simulations with the Slab Model and Cylindrical Jets

#### 3.1.3. Global Simulations of Unmagnetized Relativistic Jets

- 1.
- Jet electrons are collimated by strong toroidal magnetic fields generated by MI;
- 2.
- Electrons are perpendicularly accelerated along with the jet collimation;
- 3.
- The toroidal magnetic field polarity switches from clockwise to counterclockwise about halfway down the jet.

- 1.
- Jet electrons and positrons mix with the ambient plasma;
- 2.
- Magnetic fields around current filaments generated by a combination of kKHI, MI, and Weibel instability merge and generate density fluctuations;
- 3.
- A larger jet radius is required to properly simulate the ${e}^{\pm}$ jet case, since the jet and ambient particles mix strongly.

#### 3.2. Magnetized Jets

#### 3.2.1. Topology of Relativistic Helical Jets

#### 3.2.2. Global Simulations with Helical Jets and Large Radii

#### 3.2.3. Global Jet Simulations with a Toroidal Magnetic Field and a New Injection Scheme

- 1.
- How does a toroidal magnetic field affect the growth of kKHI, MI, and WI within the jet and in the jet–ambient plasma boundary?
- 2.
- How do jets composed of electrons and positrons and jets composed of electrons and protons evolve in the presence of a large-scale toroidal magnetic field?
- 3.
- How and where are particles accelerated in jets with different plasma compositions?

## 4. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Panel (

**a**) shows synthetic spectra for jets with Lorentz factors of $\gamma =10,20,50,100,$ and 300 and cold (thin lines) or warm (thick lines) jet electrons. Panel (

**b**) shows modeled Fermi spectra in $v{F}_{v}$ units at early (

**a**) to late (

**e**) times. Straight red lines indicate the slope of $v{F}_{v}=1$. Figure adapted from [68].

**Figure 2.**Panel (

**a**) shows the 3D PIC simulation setup. Panels (

**b**,

**c**) show the magnetic field component ${B}_{y}>$ 0 (red) and ${B}_{y}<0$ (blue) plotted in the x-z plane (jet flow indicated by large arrows) at the center of the simulation box, $y=100\Delta $ at $t=300\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$, (

**b**) for the electron–proton case and (

**c**) for the electron–positron case, both with ${y}_{core}=15$. The smaller arrows indicate the magnetic field direction in the plane. Panels (

**b**,

**c**) cover one-fifth of the simulation system length in the x-direction. The maximum and minimum magnetic field strength is ${B}_{y}\approx \pm 0.367$ (

**b**,

**c**) $\pm 0.173$. Figure adapted from [69].

**Figure 3.**Isocontour plots of the ${J}_{x}$ magnitude with magnetic field lines (one-fifth of the jet size) (

**a**) for an ${e}^{-}-{p}^{+}$ and (

**b**) for an ${e}^{\pm}$ jet at simulation time $t=300{\Omega}_{pe}^{-1}$. The 3D images are snipped perpendicularly and along the jet in order to view its interior. Figure adapted from [69].

**Figure 4.**Midplane slices of the electron density for jet Lorentz factor ${\gamma}_{jt}=5$ at a simulation time $t=500{\omega}_{pe}^{1}$. The jet is injected at $x/\Delta =100$ and propagates to the right. The jet front is located at $x/\Delta =600$. The upper panel (

**a**) shows the electron density structure for the mass ratio ${m}_{i}/{m}_{e}=1836$ and the lower panel (

**b**) for the electron–positron case. Figure adapted from [69].

**Figure 5.**Global jet simulations for ${e}^{-}-{p}^{+}$ (

**a**) and ${e}^{\pm}$ (

**b**) at time $t=500{\omega}_{pe}^{-1}$. Figure adapted from [69].

**Figure 6.**Schematic simulation setups: (

**a**) injection scheme for shock simulations where jets are injected at $x=25\Delta $ in the y-z plane, (

**b**) cylindrical injection scheme for shear flow simulations where jets are initially placed along the entire length of the x-axis at the center of the y-z plane, and (

**c**) global jet injection scheme where the jet is injected at $x=100\Delta $ with a jet radius ${r}_{jt}=100\Delta $ at the center of the y-z plane (not scaled). Figure adapted from [17].

**Figure 7.**Electron density with magnetic field arrows in the plane at time $t=1700{\omega}_{pe}^{1}$ for the ${e}^{-}-{p}^{+}$ jet (

**a**,

**b**) and the ${e}^{\pm}$ jet (

**c**,

**d**). Panels (

**a**,

**c**) show electron density in the x-z plane at $y=500\Delta $, and Panels (

**b**,

**d**) show electron density in the y-z plane at $x=1200\Delta $, where development is in the nonlinear stage. Color bars: (

**a**) 0–143.3, (

**b**) 4.58–37.92, (

**c**) 0–119.4, and (

**d**) 0–100.6. Figure adapted from [17].

**Figure 8.**Panel (

**a**) shows the schematic simulation setup of a global jet. The jet is injected at $x=100\Delta $ with a jet radius of ${r}_{jt}$ at the center of the ($y,z$) plane (not scaled). Panel (

**b**) shows the helical magnetic fields ${B}_{x}$ (black), ${B}_{\varphi}$ (red) for pitch profiles $\alpha =0.7$ (dashed), 1.0 (solid), and 2.0 (dotted) with damping functions outside the jet with $b=800.0$. The jet boundary is located at ${r}_{jt}=120\Delta $. Figure adapted from [23].

**Figure 9.**Upper panels: (

**a**) the y-component of the magnetic field, ${B}_{y}$, with the x-z electric field depicted by arrows, and (

**b**) the x-component of the electron current density, ${J}_{x}$, with the x-z magnetic field depicted by arrows, both in the x-z plane at $t=1000{\omega}_{pe}^{-1}$. The lower panels show the total magnetic field strength in the y-z plane at $x/\Delta =700$ (

**c**) and $x/\Delta =835$ (

**d**). The arrows indicate the magnetic field (${B}_{y},{B}_{z}$). Figure adapted from [20].

**Figure 10.**Magnetic field vectors within a cubic section of the simulation grid ($820<x/\Delta <1120,231<y/\Delta ,z/\Delta <531$) at time $t=900{\omega}_{pe}^{-1}$ (

**a**) and at $t=1000{\omega}_{pe}^{-1}$ (

**b**). To illustrate the magnetic field inside the jet, the plots show the rear half of the jet with a cut in the x-z plane ($381<y/\Delta <531$). Figure adapted from [20].

**Figure 11.**Phase space ($x-\gamma {V}_{x}$) distribution of jet electrons (red) and ambient electrons (blue) at $t=900{\omega}_{pe}^{-1}$ (

**top**) and $t=1000{\omega}_{pe}^{-1}$ (

**bottom**). The two vertical lines show the regions for which Figure 10 displays the 3D magnetic field vectors. Figure adapted from [20].

**Figure 12.**Two-dimensional maps of the Lorentz factor of the jet electrons at $y/\Delta =381$ for ${e}^{\pm}$ jets (left panels) and ${e}^{-}-{p}^{+}$ jets (right panels) with ${r}_{\mathrm{jet}}=100\Delta $ at time $t=1000\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$. Panels (

**a**,

**b**) show unmagnetized jets and Panels (

**c**,

**d**) the jets with the toroidal magnetic field. Black arrows show the in-plane magnetic field $({B}_{x},{B}_{z})$. Figure adapted from [15].

**Figure 13.**Color maps of the magnetic field amplitude ${B}_{\mathrm{y}}$ and arrows depicting the magnetic field components in the x-z plane, both at $t=600\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$ (upper panels) and $1000\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$ (lower panels), respectively. The jet is injected at $x=100\Delta $ in the middle of the y-z plane and propagates in the $+x$-direction. Panels (

**a**,

**c**) are for an ${e}^{\pm}$ plasma, while Panels (

**b**,

**d**) are for an ${e}^{-}-{p}^{+}$ composition. The peak amplitudes of ${B}_{\mathrm{y}}$ are (

**a**) $\pm 1.591$, (

**b**) $\pm 3.339$, (

**c**) $\pm 2.691$, and (

**d**) $\pm 5.673$. Figure adapted from [15].

**Figure 14.**Upper panel: x - $\gamma {v}_{\mathrm{x}}$ distribution of jet electrons (red), jet positrons (green), and ambient (blue) electrons at $t=1000\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$. Lower panel: color map of ${E}_{\mathrm{x}}$ in the x-z plane at $y/\Delta =381$, with arrows indicating ${B}_{x,z}$. The maximum and minimum are $\pm 0.817$. The dislocation of jet electrons and positrons generates the strips of the positive and negative ${E}_{\mathrm{x}}$. Figure adapted from [15].

**Figure 15.**Particle energy distributions of the jet (red) and ambient (blue) electrons in and around the ${e}^{\pm}$ jet (

**a**) and ${e}^{-}-{p}^{+}$ jet (

**b**) in the two regions $x/\Delta <600$ (dashed lines) and $x/\Delta >600$ (solid lines) at $t=1000\phantom{\rule{0.166667em}{0ex}}{\omega}_{\mathrm{pe}}^{-1}$. Figure adapted from [15].

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**MDPI and ACS Style**

Meli, A.; Nishikawa, K.-i.
Particle-in-Cell Simulations of Astrophysical Relativistic Jets. *Universe* **2021**, *7*, 450.
https://doi.org/10.3390/universe7110450

**AMA Style**

Meli A, Nishikawa K-i.
Particle-in-Cell Simulations of Astrophysical Relativistic Jets. *Universe*. 2021; 7(11):450.
https://doi.org/10.3390/universe7110450

**Chicago/Turabian Style**

Meli, Athina, and Ken-ichi Nishikawa.
2021. "Particle-in-Cell Simulations of Astrophysical Relativistic Jets" *Universe* 7, no. 11: 450.
https://doi.org/10.3390/universe7110450