# Analytical Constraints on the Radius and Bulk Lorentz Factor in the Lepto-Hadronic One-Zone Model of BL Lacs

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of Constraints

#### 2.1. Constraint from Variability Timescale

**In order to avoid**temporal integrations over different portions of the blob, the timescale for variation in the radiation, ${t}_{\mathrm{var}}^{\prime}$, should be longer than the light crossing time ${t}_{\mathrm{lc}}^{\prime}\sim {R}^{\prime}/c$ [44], i.e.,

#### 2.2. Constraint from SSC Luminosity

#### 2.3. Constraint from Optical Depth

#### 2.4. Constraint from the Hadronic Process

## 3. Application to TXS 0506+056

## 4. Application to PKS 0735+178

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**One situation of constraints for $\Gamma $ and ${R}^{\prime}$, where $q=2.0$ and ${L}_{\mathrm{p}}^{*}={10}^{3}{L}_{\mathrm{Edd}}$. The highlighted region shows the available parameter space. The arrows indicate the allowed parameter space for diverse constraint methods. The green dot-dashed line is from Equation (2); the purple dot-dashed line is from Equation (3); the yellow dot-dashed line is from Equation (11); the blue and red lines are all from Equation (18), where the neutrino fluxes of the red dashed line and the red dotted line are ${F}_{\nu}\times 0.3\%$ (Equation (21)) and ${F}_{\nu}$ (Equation (20)), respectively. The allowed parameter values, in this case, are 5.9–299.9 for $\Gamma $ and $1.0\times {10}^{-4}$–$3.6\times {10}^{-2}$ pc for ${R}^{\prime}$.

**Figure 2.**(

**Left panel**) Different constraints when q is fixed as 2.0, while ${L}_{\mathrm{p}}^{*}=10{L}_{\mathrm{Edd}},{10}^{2}{L}_{\mathrm{Edd}}$ and ${10}^{3}{L}_{\mathrm{Edd}}$, respectively. The arrows indicate the allowed parameter space for diverse constraint methods as the same as in Figure 1. Constraints of time variability, SSC and opacity will not vary with q and ${L}_{\mathrm{p}}^{*}$; hence, we only demonstrate the constraints given by neutrinos with $0.3\%$ of the observed flaring neutrino flux (dashed, solid and dot-dashed line in red) for different ${L}_{\mathrm{p}}^{*}$ values, while ignoring the constraints of gamma rays from the photomeson process and neutrinos with $100\%$ of the observed flaring neutrino flux, since they have no effects on the parameter space in a selected condition. There are allowed parameter spaces in the condition of ${L}_{\mathrm{p}}^{*}={10}^{2}{L}_{\mathrm{Edd}}$ (highlighted region in green) and ${L}_{\mathrm{p}}^{*}={10}^{3}{L}_{\mathrm{Edd}}$ (highlighted region in yellow and green both), suggesting an ultra-high proton injection luminosity is required. (

**The right panel**) is zoomed in to visually distinguish the green region.

**Figure 3.**A similar demonstration to Figure 2, except ${\mathit{L}}_{\mathbf{p}}^{\mathbf{*}}\mathbf{=}{\mathbf{10}}^{\mathbf{2}}{\mathit{L}}_{\mathbf{Edd}}$ is fixed while q is varied. There are allowed parameter spaces in the condition of $\mathit{q}\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{0}$ (highlighted region in green) and $\mathit{q}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{8}$ (highlighted region in green and yellow in both).

Parameter | Symbol | TXS 0506+056 | PKS 0735+178 |
---|---|---|---|

Syn-radiation peak flux | ${F}_{\mathrm{syn},\mathrm{p}}$ | $4\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ | $3.9\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ |

Gamma ray peak flux | ${F}_{\gamma ,p}$ | $5\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ | $8.6\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ |

Neutrino flux | ${F}_{\nu}$ | $5.4\times {10}^{-10}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ | $9.4\times {10}^{-11}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ |

Critical photon flux | ${F}_{\mathrm{X},0}$ | $8\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ | $9.3\times {10}^{-13}\phantom{\rule{0.166667em}{0ex}}\mathrm{erg}/\mathrm{s}/{\mathrm{cm}}^{2}$ |

Critical photon energy | ${E}_{\mathrm{X},0}$ | $4.0\times {10}^{3}$ eV | $5.8\times {10}^{3}$ eV |

Maximum photon energy | ${E}_{\mathrm{max}}$ | $5\times {10}^{11}$ eV | $3.4\times {10}^{9}$ eV |

Low energy peak frequency | ${\nu}_{\mathrm{l},\mathrm{p}}$ | $1\times {10}^{15}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ | $1.3\times {10}^{15}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ |

High energy peak frequency | ${\nu}_{\mathrm{h},\mathrm{p}}$ | $5\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ | $4.5\times {10}^{22}\phantom{\rule{0.166667em}{0ex}}\mathrm{Hz}$ |

Redshift | z | $0.3365$ | $0.65$ |

Spectral index ${}^{1}$ | $\alpha $ | $-0.48\phantom{\rule{0.166667em}{0ex}}\mathrm{or}\phantom{\rule{0.166667em}{0ex}}0.31$ | $0.44$ |

Time variation | ${t}_{\mathrm{var}}$ | one week | 5000 s |

Maximum Lorentz factor | ${\gamma}_{\mathrm{p},max}^{\prime}$ | ${10}^{6}$ | ${10}^{6}$ |

Minimum Lorentz factor | ${\gamma}_{\mathrm{p},min}^{\prime}$ | 1 | 1 |

Injection index | q | $1.8-2.2$ | $1.8-2.2$ |

Proton luminosity (AGN frame) | ${L}_{\mathrm{p}}^{*}$ | $10{L}_{\mathrm{Edd}}-{10}^{3}{L}_{\mathrm{Edd}}$ | $10{L}_{\mathrm{Edd}}-2\times {10}^{3}{L}_{\mathrm{Edd}}$ |

^{1}The spectral index ($\nu {F}_{\nu}\propto {\nu}^{\alpha}$) could be different below ${E}_{\mathrm{X},0}$ and above ${E}_{\mathrm{X},0}$. For TXS0506+056, two different indexes are involved, while for PKS 0735+178, only one is involved. See text for details.

**Table 2.**Available values for ${R}^{\prime}$ and $\Gamma $ under various conditions, where “—” means no allowable parameter space in such situations, for the case of TXS 0506+056.

Conditions | q = 1.8 | q = 2.0 | q = 2.2 | |||
---|---|---|---|---|---|---|

${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | ${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | ${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | |

${L}_{\mathrm{p}}^{*}=10{L}_{\mathrm{Edd}}$ | — | — | — | — | — | — |

${L}_{\mathrm{p}}^{*}=50{L}_{\mathrm{Edd}}$ | $2.2\times {10}^{-3}-6.1\times {10}^{-3}$ | $7.2-12.3$ | — | — | — | — |

${L}_{\mathrm{p}}^{*}={10}^{2}{L}_{\mathrm{Edd}}$ | $7.7\times {10}^{-4}-1.6\times {10}^{-2}$ | $6.3-37.3$ | $2.3\times {10}^{-3}-4.7\times {10}^{-3}$ | $7.5-10.5$ | — | — |

${L}_{\mathrm{p}}^{*}={10}^{3}{L}_{\mathrm{Edd}}$ | $2.7\times {10}^{-5}-5.0\times {10}^{-2}$ | $5.9-1023.4$ | $1.0\times {10}^{-4}-3.6\times {10}^{-2}$ | $5.9-299.9$ | $6.2\times {10}^{-4}-2.0\times {10}^{-2}$ | $6.1-44.9$ |

**Table 3.**Available values for ${R}^{\prime}$ and $\Gamma $ under various conditions, where “—” means no allowable parameter space in such situations, for the case of PKS 0735+178.

Conditions | q = 1.8 | q = 2.0 | q = 2.2 | |||
---|---|---|---|---|---|---|

${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | ${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | ${\mathit{R}}^{\prime}$[pc] | $\mathbf{\Gamma}$ | |

${L}_{\mathrm{p}}^{*}=10{L}_{\mathrm{Edd}}$ | — | — | — | — | — | — |

${L}_{\mathrm{p}}^{*}=200{L}_{\mathrm{Edd}}$ | $7.3\times {10}^{-4}-7.7\times {10}^{-4}$ | $26.0-27.1$ | — | — | — | — |

${L}_{\mathrm{p}}^{*}=500{L}_{\mathrm{Edd}}$ | $1.4\times {10}^{-4}-1.1\times {10}^{-3}$ | $26.0-138.7$ | $6.3\times {10}^{-4}-8.0\times {10}^{-4}$ | $26.0-31.7$ | — | — |

${L}_{\mathrm{p}}^{*}={10}^{3}{L}_{\mathrm{Edd}}$ | $4.3\times {10}^{-5}-1.4\times {10}^{-3}$ | $26.0-463.0$ | $1.9\times {10}^{-4}-1.0\times {10}^{-3}$ | $26.0-108.5$ | — | — |

${L}_{\mathrm{p}}^{*}=2\times {10}^{3}{L}_{\mathrm{Edd}}$ | $1.2\times {10}^{-5}-1.9\times {10}^{-3}$ | $26.0-1558.1$ | $5.0\times {10}^{-5}-1.4\times {10}^{-3}$ | $26.0-392.5$ | $4.5\times {10}^{-4}-8.0\times {10}^{-4}$ | $26.0-44.3$ |

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Ma, Z.-P.; Wang, K.
Analytical Constraints on the Radius and Bulk Lorentz Factor in the Lepto-Hadronic One-Zone Model of BL Lacs. *Universe* **2023**, *9*, 314.
https://doi.org/10.3390/universe9070314

**AMA Style**

Ma Z-P, Wang K.
Analytical Constraints on the Radius and Bulk Lorentz Factor in the Lepto-Hadronic One-Zone Model of BL Lacs. *Universe*. 2023; 9(7):314.
https://doi.org/10.3390/universe9070314

**Chicago/Turabian Style**

Ma, Zhi-Peng, and Kai Wang.
2023. "Analytical Constraints on the Radius and Bulk Lorentz Factor in the Lepto-Hadronic One-Zone Model of BL Lacs" *Universe* 9, no. 7: 314.
https://doi.org/10.3390/universe9070314