Standard and Truncated Luminosity Functions for stars in the Gaia Era

The luminosity function (LF) for stars is here fitted by a Schechter function and by a Gamma probability density function. The dependence of the number of stars on the distance, both in the low and high luminosity regions, requires the inclusion of a lower and upper boundary in the Schechter and Gamma LFs. Three astrophysical applications for stars are provided: deduction of the parameters at low distances, behavior of the average absolute magnitude with distance, and the location of the photometric maximum as a function of the selected flux. The use of the truncated LFs allows to model the Malmquist bias.


Introduction
The stellar luminosity function (LF) is the relative numbers of stars of different luminosities in a standard volume of space ,usually a cubic parsec. The determination of the LF for stars is complicated at a local level by the presence of five classes for the stars, as given by the MK system, and by the mass-luminosity relation. The presence of the Malmquist bias, after [1,2,3], for an introduction, see section 3.6 in [4] or the historical section 2 in [5], modifies the distribution in absolute magnitude as a function of the distance and therefore complicates the modeling of the LF for stars.
The LFs for stars started to be fitted by a Gaussian probability density function (PDF) in absolute magnitude, see [6]. In order to deal with the boundaries, a double truncated Gaussian in absolute magnitude has been considered, see [7]. The astronomical derivation of the LF takes account of a standard volume with a radius of ≈ 20 pc. As an example [8] has derived the first local LF for stars in a spherical volume having radius of 22 pc and more recently [9] has measured the volume luminosity density and surface luminosity density generated by the Galactic disc, using accurate data on the local luminosity function and the vertical structure of the disc. A new sample of stars, representative of the solar neighborhood LF, has been constructed from the Hipparcos (HIP) catalogue and the Fifth Catalogue of Nearby Stars, see [10].
From the previous analysis, the following questions can be raised.
• Is it possible to model the LF for stars with the Schechter function and the Gamma LF?
• Is it possible to model the absolute magnitude-distance plane with the truncated Schechter function or the truncated Gamma LF?
• Is it possible to model the observational maximum in the number of stars and the average number of stars versus distance at a given flux?

The Gaia Catalog
A great number of stars with mean apparent magnitude in the G-band, flux, f , expressed in electron-charge per second (e-/s) and parallax, ≈ two million, are available at the Gaia Data Release 1 (Gaia DR1) astrometric catalogs, see [11,12], with data at http://vizier.u-strasbg.fr/viz-bin/VizieR and specific Table I/337/tgasptyc. The above catalog gives stellar parallax, G-band flux, G-band magnitude, Tycho-2 or HIP BT magnitude and Tycho-2 or HIP VT magnitude. As pointed out by [13] there is an average offset of −0.25 ± 0.05 mas in the Gaia parallaxes and therefore we increased by 0.25 the parallax. According to Gaia DR1, the luminosity as deduced from the flux will be expressed in Gaia units, namely, e − /s pc 2 . The G magnitude, see [14] is where zp is the photometric zero derived as in [15], we found numerically zp = 25.52. The distribution of all Gaia DR1 sources in the sky is illustrated in Figure 1.
The observational Hertzsprung-Russell (H-R) diagram in M G as obtained by the Gaia DR1 parallaxes versus (B-V) , evaluated as BT-VT, is presented in Figure 2 and in a contour density version in Figure 3, see also figure 1 in [16].  The distance modulus is where m g is the apparent magnitude in the G-band, M g is the absolute magnitude in the G-band and d is the distance in pc. Isolating M G in the above equation we obtain the theoretical curve for the upper observable absolute magnitude once the maximum apparent magnitude in the g-band, m lim , is inserted, i.e. m G =12.71. Figure 4 presents the absolute magnitude as function of the distance as well the upper theoretical curve in magnitude.
The completeness of the sample can be evaluated by the following relationship for the absolute magnitude M g = − −m lim ln (10) + 5 ln (d) − 5 ln (10) ln (10) .  On inserting in the above formula m lim =12.71 we obtain a numerical relationship between selected absolute magnitude and numerical relationship over which the sample is complete, see Figure 5.
In the case here considered the absolute magnitude covers the range [3, 12 mag] and therefore we deal with a complete sample.

Standard LFs
Here we introduce an algorithm to build the LF, the statistical tests adopted, as well as the Schechter and Gamma LFs. The derived parameter for the local LF will be applied in Section 5.1 according to the general principle that the LF is equal everywhere but the upper observable absolute magnitude decreases with distance.

The astronomical LF
A LF for stars is built according to the following points (i) A standard distance is chosen, i.e. 20 pc, (ii) The GAIA's stars are selected according to the following ranges of existence: We organize an histogram with bins large 1 mag (iv) We divide the obtained frequencies by the involved volume, (v) We do not apply the 1/V a method because our sample is complete at 20 pc, (vi) The error of the LF is evaluated as the square root of the frequencies divided by the involved volume.
The LF for Gaia's stars is reported in Figure 6 together the LF main sequence in the V band as extracted from Table 2, column 9, in [10].

Statistical Tests
The merit function χ 2 is computed as where n is the number of bins for the LF of the stars and the two indices theo and astr stand for 'theoretical' and 'astronomical', respectively. The reduced merit function χ 2 red is evaluated by where N F = n − k is the number of degrees of freedom and k is the number of parameters. The goodness of the fit can be expressed by the probability Q, see equation 15.2.12 in [17], which involves the number of degrees of freedom and χ 2 . According to [17], the fit "may be acceptable" if Q ≥ 0.001. The Akaike information criterion (AIC), see [18], is defined by where L is the likelihood function and k is the number of free parameters in the model. We assume a Gaussian distribution for the errors and the likelihood function can be derived from the χ 2 statistic L ∝ exp(− χ 2 2 ) where χ 2 has been computed by Equation (5), see [19], [20]. Now the AIC becomes

The Schechter LF
Let L, the luminosity of a star, be defined in [0, ∞]. The Schechter LF of the stars, Φ, originally applied to the stars, see [21], is where α sets the slope for low values of L, L * is the characteristic luminosity, and Φ * represents the number of stars per pc 3 . The normalization is where is the Gamma function. The average luminosity, L , is An equivalent form in absolute magnitude of the Schechter LF is where M * is the characteristic magnitude. The resulting fitted curve is displayed in Figure 7 with parameters as in Table 1.
where Ψ * is the total number of stars per pc 3 , where L * > 0 is the scale and c > 0 is the shape, see formula (17.23) in [22]. The average luminosity is The change of parameter (c − 1) = α allows obtaining the same scaling as for the Schechter LF (9), for more details, see [23].
The resulting fitted curve is displayed in Figure 8 with parameters as in Table 2.

Truncated LFs
Here we derive the truncated version of the Schechter and Gamma LFs.
where Γ(a, z) is the incomplete Gamma function, defined by see [24]. The average value is with The four luminosities L, L l , L * and L u are connected with the absolute magnitudes More details can be found in [25]. The resulting fitted curve is displayed in Figure 9 with parameters as in Table 3.
where the constant k is Its expected value is More details on the truncated Gamma PDF can be found in [26,27,23].
The averaged absolute magnitude, M , is defined numerically as in Equation 26. The resulting fitted curve is displayed in Figure 10 with parameters as in Table  4.

Distance effects
We model the average absolute magnitude of the stars as a function of the distance, the photometric maximum in the number of stars for a given flux as a function of the distance, and the average distance of the stars for a given flux in the framework of the two truncated LFs here considered.

Averaged absolute magnitude
In order to model the influence of the distance d in pc on the LF, an empirical variable lower bound in absolute magnitude, M l , has been introduced, The upper bound, M u was already fixed by the nonlinear equation (3). A second distance correction is where M u (d) has been defined in Equation (3). Figure 11 compares the theoretical average absolute magnitudes for the truncated Schechter LF with the observed ones; the value of M * in Equation (33) minimizes the difference between the two curves. Conversely Figure 12 compares the theoretical average absolute magnitudes for the truncated Gamma LF with the observed ones; also here the value of M * obtained from Equation (33) minimizes the difference between the two curves.

The photometric maximum
The definition of the flux, f , is where r is the distance and L the luminosity of the star. The joint distribution in distance, r, and flux, f, for the number of stars is were the factor ( 1 4π ) converts the number density into density for solid angle and the Dirac delta function selects the required flux. We now apply the sifting properties of the delta function, see [28], to the case of the Schechter LF as given by formula 9 Figure 11. Average observed absolute G-magnitude versus distance for Gaia (green points), average theoretical absolute magnitude for truncated Schechter LF with α = −0.61 as given by Equation (26)    We now introduce the critical radius r crit Therefore the joint distribution in distance and flux becomes The above number of stars has a maximum at r = r max : and the average distance of the stars, r , is Figure 13 presents the number of stars observed in Gaia as a function of the distance for a given window in the flux, as well as the theoretical curve. Figure 14 presents the observed position of the maximum of the number of stars as a function of the flux.
In order to shift to more familiar variables Figure 15 reports the position of the above maximum as function of the apparent Gaia magnitude Figures 16 and 17 present the observed average value of the number of stars as a function of the flux and apparent magnitude.
In the case of the Gamma LF, the maximum in the number of stars is at and the average distance of the stars r , is    Position of the average distance of the stars as function of the apparent magnitude, G, (empty stars) and theoretical curve as given by Equation (39) (full line) for the Schechter LF. The parameters are the same of Figure 16. Case of the Schechter LF.

Conclusions
Standard LFs. The Schechter function and the Gamma PDF can model the LF for stars, see Tables 1 and 2  The theoretical and observed average distance of the stars are also functions of the selected flux, see Figure 16.
Topics not covered The treatment here adopted deals with an homogeneous distribution of stars and therefore the the vertical scale-heights are not covered, see [10].