# Analysis of the Angular Dependence of Time Delay in Gravitational Lensing

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## Abstract

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## 1. Introduction

## 2. The Model

- The space is divided into two regions: the first, far from L, where $\mathsf{\Phi}\approx 0$, and the second, close to L, where $\mathsf{\Phi}\ne 0$.
- We approximate the curve ${Q}_{i}{P}_{i}$ by an arc of a circle centered in L. We want to point out that this is not a necessary condition: the arc of a circle is a good choice to represent ${Q}_{i}{P}_{i}$, but it is not the only one possible.
- The universe is spatially flat, compatibly with observations [23].

## 3. Smoothness Condition

## 4. Small Angles Limit

## 5. A Constraint on the Lens Gravitational Potential

## 6. Determination of the Lens Mass for Central Potential

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**S = source. L = lens. E = observer. ${b}_{i}$ = impact parameters. $S{Q}_{i}{P}_{i}E$ = approximated paths of the photons from S to E, indexed by $i=1,2$.

Lens | ${\mathit{z}}_{\mathit{L}}$ | ${\mathit{z}}_{\mathit{S}}$ | ${\mathit{\theta}}_{1}$ (${10}^{-6}$ rad) | ${\mathit{\theta}}_{2}$ (${10}^{-6}$ rad) | ${\mathit{\theta}}_{\mathit{E}}$$({10}^{-6}$ rad) |
---|---|---|---|---|---|

QJ0158 − 4325 | 0.317 | 1.29 | $3.95\pm 0.07$ | $1.99\pm 0.07$ | $2.8$ |

J1004 + 1229 | 0.95 | 2.65 | $6.156\pm 0.034$ | $1.309\pm 0.037$ | $4.02$ |

HE1104 − 1805 | 0.73 | 2.32 | $10.216\pm 0.021$ | $5.269\pm 0.015$ | $6.8$ |

SDSS1226 − 0006 | 0.52 | 1.12 | $3.992\pm 0.021$ | $2.120\pm 0.021$ | $2.76$ |

HE2149 − 2745 | 0.5 | 2.03 | $6.563\pm 0.027$ | $1.670\pm 0.031$ | $4.1$ |

Lens | ${\mathit{M}}_{\mathbf{std}}$ (${\mathit{M}}_{\odot}$) | ${\mathit{M}}_{\mathbf{our}}$ (${\mathit{M}}_{\odot}$) | $100\xb7({\mathit{M}}_{\mathbf{std}}-{\mathit{M}}_{\mathbf{our}})/{\mathit{M}}_{\mathbf{std}}$ |
---|---|---|---|

QJ0158 − 4325 | $5.80\times {10}^{10}$ | $5.81\times {10}^{10}$ | $-0.17$ |

J1004 + 1229 | $2.97\times {10}^{11}$ | $1.48\times {10}^{11}$ | 50.13 |

HE1104 − 1805 | $6.81\times {10}^{11}$ | $7.93\times {10}^{11}$ | $-16.41$ |

SDSS1226 − 0006 | $1.14\times {10}^{11}$ | $1.26\times {10}^{11}$ | $-11.11$ |

HE2149 − 2745 | $1.75\times {10}^{11}$ | $1.14\times {10}^{11}$ | 34.82 |

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**MDPI and ACS Style**

Alchera, N.; Bonici, M.; Cardinale, R.; Domi, A.; Maggiore, N.; Righi, C.; Tosi, S.
Analysis of the Angular Dependence of Time Delay in Gravitational Lensing. *Symmetry* **2018**, *10*, 246.
https://doi.org/10.3390/sym10070246

**AMA Style**

Alchera N, Bonici M, Cardinale R, Domi A, Maggiore N, Righi C, Tosi S.
Analysis of the Angular Dependence of Time Delay in Gravitational Lensing. *Symmetry*. 2018; 10(7):246.
https://doi.org/10.3390/sym10070246

**Chicago/Turabian Style**

Alchera, Nicola, Marco Bonici, Roberta Cardinale, Alba Domi, Nicola Maggiore, Chiara Righi, and Silvano Tosi.
2018. "Analysis of the Angular Dependence of Time Delay in Gravitational Lensing" *Symmetry* 10, no. 7: 246.
https://doi.org/10.3390/sym10070246