# A Left and Right Truncated Schechter Luminosity Function for Quasars

## Abstract

**:**

## 1. Introduction

## 2. The Flat Cosmology

## 3. The Adopted LFs

#### 3.1. The Schechter LF

#### 3.2. The Truncated Schechter LF

#### 3.3. The Double Power Law

#### 3.4. The Pei Function

## 4. The Astrophysical Applications

#### 4.1. K-Correction

#### 4.2. The Sample of QSO

#### 4.3. The Luminosity Function for QSOs

#### 4.4. Evolutionary Effects

#### 4.5. The Photometric Maximum

## 5. Conclusions

## Conflicts of Interest

## References

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1. |

**Figure 1.**The absolute magnitude ${M}_{{B}_{bj}}$ computed with the nonlinear Equation (36) for 22,413 QSOs versus the redshift (green points). The lower theoretical curve (upper absolute magnitude) as represented by the nonlinear Equation (37) is the red thick line. The redshifts cover the range $[0,3]$.

**Figure 2.**The observed LF for QSOs is reported with the error bar evaluated as the square root of the LF (Poissonian distribution) when z$[0.3,0.5]$.

**Figure 3.**The observed LF for QSOs when z $[0.3,0.5]$ and M $[-24.45,-21.50]$ (empty stars), $[0.7,0.9]$ and M $[-26.49,-23.50]$ (full triangles), $[1.1,1.3]$ and M $[-27.59,-24.60]$ (empty crosses) and $[1.5,1.7]$ and M $[-28.43,-25.50]$ (stars of David).

**Figure 4.**The observed LF for QSOs, empty stars with error bar, and the fit by the truncated Schechter LF when z$[0.3,0.5]$ and M$[-24.93,-22]$.

**Figure 5.**The observed LF for QSOs, empty stars with error bars, and the fit by the double power LF when the redshifts cover the range $[0.3,0.5]$

**Figure 6.**The observed LF for QSOs, empty stars with error bar, and the fit by the Pei LF when the redshifts cover the range $[0.3,0.5]$.

**Figure 7.**Average observed absolute magnitude versus redshift for QSOs (red points), average theoretical absolute magnitude for truncated Schechter LF as given by Equation (30) (dot-dash-dot green line), theoretical curve for the empirical lowest absolute magnitude at a given redshift, see Equation (42) (full black line) and the theoretical curve for the highest absolute magnitude at a given redshift (dashed black line), see Equation (37), RSS = 1.212.

**Figure 8.**Average observed absolute magnitude versus redshift for QSOs (red points), average theoretical absolute magnitude for the double power LF as evaluated numerically (dot-dash-dot green line), theoretical curve for the empirical lowest absolute magnitude at a given redshift, see Equation (42) (full black line) and the theoretical curve for the highest absolute magnitude at a given redshift (dashed black line), see Equation (37), RSS = 1.138.

**Figure 9.**Average observed absolute magnitude versus redshift for QSOs (red points), average theoretical absolute magnitude for the Pei LF as evaluated numerically (dot-dash-dot green line), theoretical curve for the empirical lowest absolute magnitude at a given redshift, see Equation (42) (full black line) and the theoretical curve for the highest absolute magnitude at a given redshift (dashed black line), see Equation (37), RSS = 5.41.

**Figure 10.**The QSOs with $20.16\le m\le 20.85$ are organized in frequencies versus spectroscopic redshift, points with error bars. The redshifts cover the range $[0,3]$ and the histogram’s interval is 0.14. The maximum frequency of observed QSOs is at $z=1.478$ and the number of bins is 20. The full line is the theoretical curve generated by $\frac{dN}{d\Omega dzdm}\left(z\right)$ as given by the application of the truncated Schechter LF, which is Equation (52) with parameters as in Table 1 but ${M}^{*}=-22.5$. The theoretical maximum is at $z=1.491$.

**Table 1.**Parameters of the truncated Schechter LF in the range of redshifts $[0.3,0.5]$ when n = 10 and k = 5.

${\mathit{M}}_{\mathit{l}}$ | ${\mathit{M}}^{*}$ | ${\mathit{M}}_{\mathit{u}}$ | ${\mathit{\Psi}}^{*}$ | $\mathit{\alpha}$ | ${\mathit{\chi}}^{2}$ | ${\mathit{\chi}}_{\mathit{red}}^{2}$ | Q | AIC |
---|---|---|---|---|---|---|---|---|

–24.93 | –23.28 | –22.29 | 3.38 × ${10}^{-8}$ | –0.97 | 12.89 | 2.57 | 0.024 | 22.89 |

${\mathit{M}}^{*}$ | ${\mathit{\Psi}}^{*}$ | $\mathit{\alpha}$ | ${\mathit{\chi}}^{2}$ | ${\mathit{\chi}}_{\mathit{red}}^{2}\mathbf{B}$ | Q | AIC |
---|---|---|---|---|---|---|

–23.75 | 8.85 × ${10}^{-7}$ | –1.37 | –10.49 | 1.49 | 0.162 | 16049 |

**Table 3.**Parameters of the double power LF in the range of redshifts $[0.3,0.5]$ when n = 10 and k = 4.

${\mathit{M}}^{*}$ | ${\mathit{\varphi}}^{*}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{\chi}}^{2}$ | ${\mathit{\chi}}_{\mathit{red}}^{2}$ | Q | AIC |
---|---|---|---|---|---|---|---|

–23.82 | 5.44 × ${10}^{-7}$ | –3.57 | –1.48 | 9.44 | 1.57 | 0.15 | 17.44 |

${\mathit{M}}^{*}$ | ${\mathit{\varphi}}^{*}$ | $\mathit{\beta}$ | ${\mathit{\chi}}^{2}$ | ${\mathit{\chi}}_{\mathit{red}}^{2}$ | Q | AIC |
---|---|---|---|---|---|---|

–16.47 | 3.68 × ${10}^{-5}$ | 0.924 | 14.4 | 2.05 | 0.044 | 20.40 |

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Zaninetti, L.
A Left and Right Truncated Schechter Luminosity Function for Quasars. *Galaxies* **2017**, *5*, 25.
https://doi.org/10.3390/galaxies5020025

**AMA Style**

Zaninetti L.
A Left and Right Truncated Schechter Luminosity Function for Quasars. *Galaxies*. 2017; 5(2):25.
https://doi.org/10.3390/galaxies5020025

**Chicago/Turabian Style**

Zaninetti, Lorenzo.
2017. "A Left and Right Truncated Schechter Luminosity Function for Quasars" *Galaxies* 5, no. 2: 25.
https://doi.org/10.3390/galaxies5020025