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The gamma function is a good approximation to the luminosity function of astrophysical objects, and a truncated gamma distribution would permit a more rigorous analysis. This paper examines the generalized gamma distribution (GG) and then introduces the scale and the new double truncation. The magnitude version of the truncated GG distribution with scale is adopted in order to fit the luminosity function (LF) for galaxies or quasars. The new truncated GG LF is applied to the five bands of SDSS galaxies, to the 2dF QSO Redshift Survey in the range of redshifts between 0.3 and 0.5, and to the COSMOS QSOs in the range of redshifts between 3.7 and 4.7. The average absolute magnitude versus redshifts for SDSS galaxies and QSOs of 2dF was modeled adopting a redshift dependence for the lower and upper absolute magnitude of the new truncated GG LF.

The generalized gamma distribution (GG) was introduced by [

∫ 0 ∞ f ( L ) d L = 1, (1)

where L is the luminosity. A data oriented LF, Ψ ( L ) , is obtained by adopting Ψ * which is the normalization to the number of galaxies in a volume of 1 Mpc^{3}

Ψ ( L ) = Ψ * f ( L ) , (2)

which means

∫ 0 ∞ Ψ ( L ) d L = Ψ * . (3)

The above line of research allows exploring the LF for galaxies in the framework of well studied PDFs. Some examples are represented by the mass-luminosity relationship, see [

This paper brings up the GG and introduces the scale in Section 2. The new double truncation for the GG and the GG with scale is introduced in Section 3. The derivation of the truncated GG LF is done in Section 4. Section 5 contains the application of the GG LF to galaxies and quasars as well the fit of the averaged absolute magnitude for QSOs as function of the redshift.

Let X be a random variable defined in [ 0, ∞ ] ; the GG (PDF), f ( x ) , is

f ( x ; a , b , c ) = c b a c x a − 1 e − b x c Γ ( a c ) , (4)

where

Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t , (5)

is the gamma function, with a > 0 , b > 0 and c > 0 . The above PDF can be obtained by setting the location parameter equal to zero in the four parameters GG as given by [

The GG family includes several subfamilies, including the exponential PDF when a = 1 and c = 1 , the gamma PDF when c = 1 and the Weibull PDF when b = 1 .

The distribution function (DF), F ( x ) , is

F ( x ; a , b , c ) = 1 − Γ ( a c , b x c ) ( Γ ( a c ) ) , (6)

where Γ ( a , z ) is the incomplete Gamma function, defined by

Γ ( a , z ) = ∫ z ∞ t a − 1 e − t d t , (7)

see [

μ ( a , b , c ) = b − c − 1 Γ ( 1 + a c ) Γ ( a c ) , (8)

the variance, σ 2 , is

σ 2 ( a , b , c ) = b − 2 c − 1 ( − ( Γ ( 1 + a c ) ) 2 + Γ ( a c ) Γ ( 2 + a c ) ) ( Γ ( a c ) ) 2 . (9)

The mode, M, is

M ( a , b , c ) = a − 1 b c c . (10)

The rth moment about the origin is, μ ′ r ( a , b , c ) , is

μ ′ r ( a , b , c ) = b − r c Γ ( a + r c ) Γ ( a c ) . (11)

The information entropy, H, is

H ( a , b , c ) = ln ( 1 c b c Γ ( a c ) ) + Ψ ( a c ) ( 1 c − a c ) + a c , (12)

where Ψ ( z ) is the digamma or Psi function defined as

Ψ ( z ) = Γ ′ ( z ) / Γ ( z ) , (13)

where R z > 0 , see [

In some applications it may be useful to have a scale, b, and therefore the GG PDF, f s ( x ) , is

which has DF,

The average value,

the variance,

and the mode,

Let X be a random variable defined in

which is

where

and as a consequence the truncated GG PDF is,

The average value,

where

The DF,

where

The truncated GG PDF with scale requires the evaluation of the following integral,

which is

The constant of integration,

and as a consequence the truncated GG PDF with scale is,

The average value,

where

In this section we present the Schechter LF, we derive the GG LF, we introduce the double truncation in the LF and we develop the adopted statistics.

The Schechter LF, introduced by [

here

The equivalent distribution in absolute magnitude is

where

We replace in the GG with scale, see Equation (14) x with L (the luminosity), b with ^{3}

The magnitude version is

where M is the absolute magnitude and

We replace in the truncated GG with scale, see Equation (35), x with L (the luminosity), b with

where

The four luminosities

where the indexes u and l are inverted in the transformation from luminosity to absolute magnitude and

The magnitude version of the truncated GG LF is

The averaged absolute magnitude is

The unknown parameters of the LF can be found through the Levenberg-Marquardt method (subroutine MRQMIN in [

where n is number of datapoints and the two indexes theo and astr stand for theoretical and astronomical, respectively. The residual sum of squares (RSS) is

where

where

where L is the likelihood function and k the number of free parameters in the model. We assume a Gaussian distribution for the errors and the likelihood

function can be derived by the

The Bayesian information criterion (BIC), see [

where L is the likelihood function, k the number of free parameters in the model and n the number of observations. The phrase “better fit” used in the following means that the three statistical indicators:

In this section we apply the truncated GG LF to the SDSS galaxies and to QSOs. The introduction of the redshift dependence for lower and upper absolute magnitude allows to model the average absolute magnitude versus redshift for QSOs.

In order to perform a test we selected the data of the Sloan Digital Sky Survey (SDSS) which has five bands

Parameter | u^{*} | g^{*} | r^{*} | i^{*} | z^{*} |
---|---|---|---|---|---|

M_{l} − 5log10h | −20.65 | −22.09 | −22.94 | −23.42 | −23.73 |

M_{u} − 5log10h | −15.78 | −16.32 | −16.30 | −17.21 | −17.48 |

M^{*} − 5log10h | −17.34 | −19.45 | −20.28 | −20.29 | −20.77 |

Ψ^{*} [h^{3} Mpc^{−}^{3}] | 0.042 | 0.043 | 0.052 | 0.038 | 0.042 |

c | 0.473 | 0.078 | 0.015 | 0.247 | 0.10 |

a | 0.842 | 1.02 | 0.942 | 0.839 | 0.866 |

χ^{2} | 283.17 | 747.58 | 2185 | 1867 | 2916 |

0.591 | 1.256 | 3.261 | 2.648 | 3.961 | |

AIC k = 4 | 291.17 | 755.58 | 2193 | 1874 | 2923 |

BIC k = 4 | 307.89 | 773.16 | 2211 | 1893 | 2941 |

χ^{2} Schechter | 330.73 | 753.3 | 2260 | 2282 | 3245 |

0.689 | 1.263 | 3.368 | 3.232 | 4.403 |

The Schechter function, the new four parameters function as represented by formula (49) and the data are reported in Figures 1-5, where bands

_{T}. The range of existence in the truncated case is finite rather than infinite and therefore the luminosity density is always smaller than in the standard case.

For our first example, we selected the catalog of the 2dF QSO Redshift Survey (2QZ), which contains 22431 redshifts of QSOs with

In the second example we explored the faint LF for quasars in the range of redshifts

The first application is about galaxies: we processed the SDSS Photometric Catalogue DR 12, see [

parameter | u^{*} | g^{*} | r^{*} | i^{*} | z^{*} |
---|---|---|---|---|---|

4.35 | 2.81 | 2.58 | 3.19 | 3.99 | |

1.38 | 1.18 | 1.57 | 1.88 | 2.47 |

parameter | value |
---|---|

M_{l} − 5log10h | −24.93 |

M_{u} − 5log10h | −22.29 |

M^{*} − 5log10h | −22.48 |

Ψ^{*} [h^{3} Mpc^{−}^{3}] | 1.09 × 10^{−}^{6} |

c | 0.013 |

a | 0.652 |

χ^{2} | 10.17 |
---|---|

1.69 | |

AIC k = 4 | 18.17 |

BIC k = 4 | 19.38 |

χ^{2} Schechter | 10.49 |

1.49 |

parameter | value |
---|---|

M_{l} − 5log10h | −25.86 |

M_{u} − 5log10h | −22.56 |

M^{*} − 5log10h | −20.07 |

Ψ^{*} [h^{3} Mpc^{−}^{3}] | 8.57 × 10^{−}^{7} |

c | −4.38 |

a | 0.17 |

χ^{2} | 3.46 |

0.86 | |

AIC k = 4 | 11.46 |

BIC k = 4 | 11.78 |

χ^{2} Schechter | 5.82 |

1.16 |

The second application is about the QSO and we used the framework of the flat cosmology in order to find the absolute magnitude relative to the catalog of the 2dF QSO Redshift Survey (2QZ), exactly as [^{−1}Mpc^{−1}, and

where G is the Newtonian gravitational constant and

The above equation represents a useful theoretical reference. Another, more empirical, way explores numerically the maximum and the minimum in absolute magnitude functions of the redshift for the sample 2QZ. In order to fix the numbers we fitted the upper absolute magnitude with the third degree polynomial

with

with

Now different combinations of curves can be used.

Truncated GG: We derived an expression for the left and right truncated GG PDF in terms of the Whittaker M function, see Equation (22), its DF, see Equation (28), and its average value, see Equation (23).

Truncated LF: The truncated LF for galaxies or QSO is derived both in the luminosity form, see Equation (46), and in the magnitude form, see Equation (49). In all the astrophysical examples here analyzed which are the five bands of SDSS galaxies, see

Average Magnitude versus redshift: The averaged absolute magnitude of the SDSS galaxies and QSOs belonging to the catalog 2QZ as functions of the redshift are reasonably fitted by the averaged absolute magnitude of the truncated GG LF, see Equation (52). In order to perform the fit we provided for

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2019) New Probability Distributions in Astrophysics: I. The Truncated Generalized Gamma. International Journal of Astronomy and Astrophysics, 9, 393-410. https://doi.org/10.4236/ijaa.2019.94027

The Whittaker M function,

The regularized hypergeometric function,

where

The generalized hypergeometric series is denoted by

The relationship

allows to express the Whittaker M function in terms of the generalized hypergeometric function,