The truncated version of the two-parameter Sujatha distribution is analysed. In particular, its probability density function and distribution function are obtained. The results are applied to the initial mass function for stars, to the luminosity function for galaxies, to the number of galaxies as a function of the redshift and to the average absolute magnitude of a galaxy as a function of its redshift.
The one-parameter Sujatha distribution was introduced by [
1) Is it possible to introduce the effect of truncation for the Sujatha distribution?
2) Is the Sujatha distribution useful to model the initial mass function (IMF) for the stars?
3) Is it possible to model the luminosity function for galaxies (LF) with the Sujatha distribution?
This paper reviews the Sujatha distribution in Section 2, introduces the scale in Section 3, evaluates the main statistical parameters of left- and right-truncated Sujatha distributions with scale in Section 4, derives the luminosity function (LF) for galaxies in Section 5, derives the truncated LF for galaxies in 6 and applies the results to same astrophysical samples, see Section 7.
Let X be a random variable with values in [ 0, ∞ ] . Then, the one-parameter Sujatha probability density function (PDF), f ( x ) , is
f ( x ; b ) = b 3 ( x 2 + x + 1 ) e − b x b 2 + b + 2 , (1)
where b is the shape parameter, see [
The distribution function (DF), F ( x ) , is
F ( x ; b ) = ( − 2 + ( − x 2 − x − 1 ) b 2 + ( − 2 x − 1 ) b ) e − b x + b 2 + b + 2 b 2 + b + 2 . (2)
Its average value or mean, μ , is
μ ( b ) = b 2 + 2 b + 6 b ( b 2 + b + 2 ) , (3)
its variance, σ 2 , is
σ 2 ( b ) = b 4 + 4 b 3 + 18 b 2 + 12 b + 12 b 2 ( b 2 + b + 2 ) 2 , (4)
its skewness is
s k e w n e s s ( b ) = 2 b 6 + 12 b 5 + 72 b 4 + 88 b 3 + 108 b 2 + 72 b + 48 ( b 4 + 4 b 3 + 18 b 2 + 12 b + 12 ) 3 2 , (5)
and its kurtosis is
k u r t o s i s ( b ) = 9 b 8 + 72 b 7 + 516 b 6 + 1128 b 5 + 2208 b 4 + 2592 b 3 + 2736 b 2 + 1440 b + 720 ( b 4 + 4 b 3 + 18 b 2 + 12 b + 12 ) 2 . (6)
Its kth moment about the origin, μ ′ k , is
μ ′ k ( b ) = b − k ( k + 2 ) ! + b 1 − k ( 1 + k ) ! + b − k + 2 k ! b 2 + b + 2 . (7)
Its mode, M o d e ( b ) , is
M o d e ( b ) = − b + 2 + − 3 b 2 + 4 2 b . (8)
An approximate expression for its median, M e d ( b ) , can be obtained by a Taylor expansion of order 2 for the DF around the average value
M e d ( b ) = − ( b 2 + b + 2 ) 4 e b 2 + 2 b + 6 b 2 + b + 2 + 4 b 8 + 20 b 7 + 84 b 6 + 220 b 5 + 494 b 4 + 804 b 3 + 1104 b 2 + 984 b + 704 2 b ( b 2 + b + 2 ) ( b 2 + 3 b + 3 ) ( b 4 + 6 b 2 + 12 ) . (9)
As an example, the above approximate median at b = 1 has a percentage error of 1.11%. The moment generating function, M X , is
M X ( t ; b ) = ( b 2 − 2 b t + t 2 + b − t + 2 ) b 3 ( b 2 + b + 2 ) ( b − t ) 3 . (10)
A random generation of the Sujatha variate X can be obtained by solving the nonlinear equation
F ( x ; b ) = R , (11)
where R is the unit rectangular variate and F is given by Equation (2). The parameter b can be derived from the sample mean, x ¯ , by solving the cubic Equation (3)
b = ( 3 3 7 x ¯ 4 + 28 x ¯ 3 + 171 x ¯ 2 + 32 x ¯ + 5 x ¯ + 8 x ¯ 3 + 66 x ¯ 2 + 6 x ¯ + 1 ) 1 3 3 x ¯ − ( x ¯ − 1 ) ( 5 x ¯ + 1 ) 3 x ¯ ( 3 3 7 x ¯ 4 + 28 x ¯ 3 + 171 x ¯ 2 + 32 x ¯ + 5 x ¯ + 8 x ¯ 3 + 66 x ¯ 2 + 6 x ¯ + 1 ) 1 3 − x ¯ − 1 3 x ¯ . (12)
The two-parameter Sujatha PDF, f s ( x ) , introduces the scale, s, in the Sujatha PDF
f s ( x ; b , s ) = b 3 ( x 2 s 2 + x s + 1 ) e − b x s ( b 2 + b + 2 ) s , (13)
which has the DF
F s ( x ; b , s ) = ( ( − s 2 − s x − x 2 ) b 2 − s ( s + 2 x ) b − 2 s 2 ) e − b x s + s 2 ( b 2 + b + 2 ) s 2 ( b 2 + b + 2 ) . (14)
We now determine some quantities, among others, its average value is
μ s ( b , s ) = μ ( b ) × s , (15)
its variance is
σ s 2 ( b , s ) = σ 2 ( b ) × s 2 , (16)
its median is at
M e d s ( b , s ) = M e d ( b ) × s , (17)
and its mode is
M o d e s ( b , s ) = M o d e ( b ) × s . (18)
The two parameters b and s can be derived by the numerical solution of the two following equations, which arise from the maximum likelihood estimator (MLE),
3 n b − n ( 2 b + 1 ) b 2 + b + 2 − ∑ i = 1 n x i s = 0 , (19a)
− 3 n s + ( ∑ i = 1 n b x i 3 + b s x i 2 + s 2 ( b + 1 ) x i + 2 s 3 s 2 + x i s + x i 2 ) s 2 = 0, (19b)
where the x i are the elements of the experimental sample with i varying between 1 and n.
Let X be a random variable with values in [ x l , x u ] . The truncated two-parameter Sujatha PDF, f T ( x ) , is
f T ( x ; b , s , x l , x u ) = b 3 ( s 2 + s x + x 2 ) e − b x s A , (20)
where
A = ( ( s 2 ( b 2 + b + 2 ) + b x l ( b + 2 ) s + b 2 x l 2 ) e − x l b s − e − x u b s ( s 2 ( b 2 + b + 2 ) + b x u ( b + 2 ) s + b 2 x u 2 ) ) s . (21)
The truncated DF, F T , is
F T ( x ; b , s , x l , x u ) = ( ( s 2 + s x l + x l 2 ) b 2 + s ( s + 2 x l ) b + 2 s 2 ) e − x l b s − ( ( s 2 + s x + x 2 ) b 2 + s ( s + 2 x ) b + 2 s 2 ) e − b x s B , (22)
where
B = ( ( s 2 + s x l + x l 2 ) b 2 + s ( s + 2 x l ) b + 2 s 2 ) e − x l b s − ( ( s 2 + s x u + x u 2 ) b 2 + s ( s + 2 x u ) b + 2 s 2 ) e − x u b s . (23)
Its average value, μ T , is
μ T ( b , s , x l , x u ) = N D , (24)
where
N = ( x l ( s 2 + s x l + x l 2 ) b 3 + s ( s 2 + 2 s x l + 3 x l 2 ) b 2 + 2 s 2 ( s + 3 x l ) b + 6 s 3 ) e − x l b s − ( x u ( s 2 + s x u + x u 2 ) b 3 + s ( s 2 + 2 s x u + 3 x u 2 ) b 2 + 2 s 2 ( s + 3 x u ) b + 6 s 3 ) e − x u b s (25)
and
D = b ( ( ( s 2 + s x l + x l 2 ) b 2 + s ( s + 2 x l ) b + 2 s 2 ) e − x l b s − e − x u b s ( ( s 2 + s x u + x u 2 ) b 2 + s ( s + 2 x u ) b + 2 s 2 ) ) . (26)
The first two parameters, x l and x u , of the truncated Sujatha distribution can be obtained from the empirical data by evaluating the minimum and maximum elements of the sample. Consider a sample X = x 1 , x 2 , ⋯ , x n and let x ( 1 ) ≥ x ( 2 ) ≥ ⋯ ≥ x ( n ) denote their order statistics, so that x ( 1 ) = max ( x 1 , x 2 , ⋯ , x n ) , x ( n ) = min ( x 1 , x 2 , ⋯ , x n ) . The first two parameters x l and x u are
x l = x ( n ) , x u = x ( 1 ) . (27)
The remaining two parameters, b and s, are obtained by maximizing the MLE but the results are complicated and therefore we omit them.
The Sujatha luminosity function (LF) can be derived from Equation (13):
Ψ ( L ; b , L * , Ψ * ) d L = Ψ * b 3 ( L 2 + L L * + L * 2 ) e − b L L * ( b 2 + b + 2 ) L * 3 , (28)
where L is the luminosity, L * is the characteristic luminosity and Ψ * is a normalization, i.e. the number of galaxies per cubic Mpc. The mean luminosity per unit volume, j, is
j = ∫ 0 ∞ L Ψ ( L ) d l = L * ( b 2 + 2 b + 6 ) Y * b ( b 2 + b + 2 ) , (29)
and the average luminosity, 〈 L 〉 , is
〈 L 〉 = L * ( b 2 + 2 b + 6 ) b ( b 2 + b + 2 ) . (30)
The version in absolute magnitude is
Ψ ( M ; b , M * , Ψ * ) d M = 0.4 b 3 Y * ln ( 10 ) e − b 10 − 0.4 M + 0.4 M * ( 10 − 0.4 M + 0.4 M * + 10 − 1.2 M + 1.2 M * + 10 − 0.8 M + 0.8 M * ) b 2 + b + 2 , (31)
where M * is the characteristic magnitude. We now introduce the parameter h, which is H 0 / 100 , where H 0 is the Hubble constant. The scaling with h is M * − 5 log 10 h and Ψ * h 3 [ Mpc − 3 ] .
We start with the truncated Sujatha PDF with scale as given by Equation (20)
Ψ ( L ; b , L * , Ψ * , L l , L u ) d L = Ψ * b 3 ( L 2 + L L * + L * 2 ) e − b L L * L * F i s t a r E d L , (32)
where L is the luminosity, L * is the characteristic luminosity, L l is the lower boundary in luminosity, L u is the upper boundary in luminosity, Ψ * is a normalization factor and
E = L * 2 ( L * 2 e − L l b L * b 2 − L * 2 e − L u b L * b 2 + L * e − L l b L * b 2 L l − L * e − L u b L * b 2 L u + e − L l b L * b 2 L l 2 − e − L u b L * b 2 L u 2 + L * 2 e − L l b L * b − L * 2 e − L u b L * b + 2 L * e − L l b L * b L l − 2 L * e − L u b L * b L u + 2 L * 2 e − L l b L * − 2 L * 2 e − L u b L * ) . (33)
The magnitude version is
Ψ ( M ; b , M * , Ψ * , M l , M u ) d L = F G , (34)
where
F = − 0.4 b 3 ln ( 10 ) e − 1.0 b 10 − 2 M 5 + 2 M * 5 ( 10 0.4 M l + 0.4 M u + 0.4 M * − 1.2 M + 10 0.4 M l + 0.4 M u − 0.8 M + 10 0.4 M l + 0.4 M u − 0.4 M * − 0.4 M ) , (35)
where
G = e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * b 2 + e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * b + 2.0 e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * + e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 − 0.4 M l + 0.4 M u b 2 + e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 − 0.4 M * + 0.4 M u b 2 + 2.0 e − 1.0 10 − 0.4 M l + 2 M * 5 b 10 − 0.4 M * + 0.4 M u b − e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * b 2 − e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * b − 2.0 e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 0.4 M l + 0.4 M u − 0.8 M * − e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 0.4 M l − 0.4 M u b 2 − e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 − 0.4 M * + 0.4 M l b 2 − 2.0 e − 1.0 10 − 0.4 M u + 2 M * 5 b 10 − 0.4 M * + 0.4 M l b , (36)
where M is the absolute magnitude, M * the characteristic magnitude, M l the lower boundary in magnitude and M u the upper boundary in magnitude. The mean theoretical absolute magnitude, 〈 M 〉 , is
〈 M 〉 = ∫ M l M u M × Ψ ( M ; b , M * , Ψ * , M l , M u ) d M ∫ M l M u Ψ ( M ; b , M * , Ψ * , M l , M u ) d M . (37)
The Photometric MaximumFollowing a procedure outlined in Section 5.4 of [
d N ( z ; b , Ψ * , z c r i t ) d Ω d z d f = 4 z 4 b 3 e − b z 2 z c r i t 2 c l 5 π ( z 4 + z 2 z c r i t 2 + z c r i t 4 ) ( b 2 + b + 2 ) H 0 5 L * z c r i t 4 , (38)
where the critical value of z, z c r i t , is
z c r i t 2 = H 0 2 L * 4 π f c 2 , (39)
where L * was defined in Section 5 and c is the speed of light. Recall that in the pseudo-Euclidean universe, the flux of radiation, f, expressed in units of L ⊙ Mpc 2 units, where L ⊙ represents the luminosity of the sun, is
f = L 4 π D 2 , (40)
where D is the distance of the galaxy expressed in Mpc, and
D = c l z H 0 . (41)
The number of galaxies as a function of the redshift has a maximum at
z = G 6 b ( 3 3 3 b 4 + 6 b 3 − 13 b 2 + 108 b + 368 b + 7 b 3 + 15 b 2 + 12 b + 128 ) 1 6 , (42)
where
G = 2 1 6 3 ( − 2 × 2 2 3 ( b − 4 ) ( 3 3 3 b 4 + 6 b 3 − 13 b 2 + 108 b + 368 b + 7 b 3 + 15 b 2 + 12 b + 128 ) 1 3 − 8 b 2 + 2 × 2 1 3 ( 3 3 3 b 4 + 6 b 3 − 13 b 2 + 108 b + 368 b + 7 b 3 + 15 b 2 + 12 b + 128 ) 2 3 + 4 b + 64 ) 1 2 z c r i t . (43)
The mean redshift as a function of z c r i t and b is
〈 z 〉 ( b , z c r i t ) = 32 z c r i t ( b 2 + 3 b + 12 ) 3 π b ( 4 b 2 + 10 b + 35 ) . (44)
The merit function χ 2 , the reduced merit function χ r e d 2 , the goodness of fit, Q, the Akaike information criterion (AIC), the maximum distance in the Kolmogorov-Smirnov test (K-S), D, and the significance level, P K S , in the K-S test were defined in Section 5.1 of [
We tested the truncated Sujatha distribution on four samples of stars: NGC 2362 (271 stars), the young cluster NGC 6611 (207 stars), the γ Velorum cluster (237 stars), and the young cluster Berkeley 59 (420 stars). (For more details, see Section 5.2 of [
The empirical PDF visualized through histograms as well as the theoretical PDF for NGC 6611 are reported in
The first test was performed on the u * band of SDSS as in [
A careful examination of
The second test is on the LF for quasi stellar objects (QSO) in the case 0.3 < z < 0.5 , see [
| Cluster | parameters | AIC | χ r e d 2 | Q | D | P K S | LN |
|---|---|---|---|---|---|---|---|
| NGC 2362 | b = 6.95 × 10 − 2 , s = 1.61 × 10 − 2 , x l = 0.12 , x u = 1.47 | 44.44 | 2.24 | 1.81 × 10−3 | 4.85 × 10−2 | 0.534 | N |
| NGC 6611 | b = 8.49 , s = 3.57 , x l = 0.019 , x u = 1.46 | 43.15 | 2.17 | 2.71 × 10−3 | 5.67 × 10−2 | 0.5 | Y |
| γ Velorum | b = 69297 , s = 16663 , x l = 0.158 , x u = 1.317 | 42.99 | 2.16 | 2.85 × 10−3 | 6.14 × 10−2 | 0.322 | Y |
| Berkeley 59 | b = 3465 , s = 1040 , x l = 0.16 , x u = 2.24 | 78.12 | 4.11 | 8.96 × 10−9 | 0.139 | 1.45 × 10−7 | N |
| LF | parameters | χ r e d 2 |
|---|---|---|
| Schechter | M * = − 17.92 , α = − 0.9 , Φ * = 0.03 / Mpc 3 | 0.689 |
| Sujatha | M * = − 20.70 , b = 18.83 , Ψ * = 0.029 / Mpc 3 | 6.6 |
| M * | Ψ * | α | χ 2 | χ r e d 2 | Q | AIC |
|---|---|---|---|---|---|---|
| −23.75 | 8.85 × 10−7 | −1.37 | 10.491 | 1.49 | 0.162 | 16.49 |
| M * | Ψ * | b | χ 2 | χ r e d 2 | Q | AIC |
|---|---|---|---|---|---|---|
| −25.2 | 2.41 × 10−6 | 12.56 | 32.05 | 4.57 | 3.96 × 10−5 | 38.05 |
The absolute magnitude that can be observed, as a function of the limiting apparent magnitude, m L , is
M L = m L − 5 log 10 ( c z H 0 ) − 25 , (45)
where m L = 11.75 for the 2MRS catalog.
The theoretical average absolute magnitude of the truncated Sujatha LF, see Equation (37), can be compared with the observed average absolute magnitude of the 2MRS as a function of the redshift. To fit the data, we assumed the following empirical dependence on the redshift for the characteristic magnitude of the truncated Sujatha LF
M * = − 25.14 + 2.5 ( 1 − ( z − z min z max − z min ) 0.7 ) . (46)
This relationship models the decrease of the characteristic absolute magnitude as a function of the redshift and allows us to match the observational and theoretical data. The lower bound in absolute magnitude is given by the minimum magnitude of the selected bin, the upper bound is given by Equation (45), the characteristic magnitude varies according to Equation (46) and
Sujatha distribution with scale
We derived the PDF, the DF, the average value, the variance, the median, the mode, an expression to generate random numbers and the way to obtain the two parameters, b and s, by the MLE for the Sujatha distribution with scale.
Truncated Sujatha distribution
We derived the PDF, the DF, and the average value of the truncated Sujatha distribution. The application of this distribution to the IMF for stars gives better results than the lognormal distribution for two out of four samples, see
Sujatha luminosity function
We derived the Sujatha LF in the standard and the truncated case. The application to both the SDSS Galaxies and to the QSOs in the range of redshift [ 0.3,0.5 ] yields a lower reduced merit function compared to Schechter LF, see
Cosmological applications
The number of galaxies as a function of the redshift, the flux and the solid angle for the Sujatha LF in the pseudo-Euclidean universe presents a maximum which can be compared with the observed one for the 2MRS, see
The author declares no conflicts of interest regarding the publication of this paper.
Zaninetti, L. (2021) New Probability Distributions in Astrophysics: VI. The Truncated Sujatha Distribution. International Journal of Astronomy and Astrophysics, 11, 517-529. https://doi.org/10.4236/ijaa.2021.114028