1. The Investigation of the Luminosity Function for Sample 6168 Galaxy Clusters
W. Godłowski1 J. Popiela1 & P. Flin2
Hot topics in Modern Cosmology
May 2018 Cargèse
1. Institute of Physics, Uniwersytet Opolski,
2. Institute of Physics, Jan Kochanowski University,

2. Members of research group
dr Katarzyna Bajan UP (Kraków)
dr Monika Biernacka Instytut Fizyki UJK (Kielce)
dr hab. Piotr Flin, retired prof. UJK, Instytut Fizyki UJK – (Kielce)
dr hab. Włodzimierz Godłowski Instytut Fizyki UO (Opole)
mgr Paulina Pajowska (Opole)
dr hab. Elena Panko Nikolaev State University (Ukraina)
mgr Joanna Popiela Instytut Fizyki UO (Opole)
Prof. dr hab. Włodzimierz Stefanowicz IF UO (Opole)
Prof. Zong Hong Zhu (Chiny)

3. (Fairall et al. 1988)

4. Structures in the Universe

6. Method of the computation autocorrelation function
What do we have to do to determine the autocorrelation function?
From definition – difficult, but it is possible to find estimators (Peacock)
We generate the theoretical catalogue as the real (the same area, luminosity function, selection effects, systematic errors).
Computation of the correlation function requires a comparison of the number of the galaxy pairs - separated on (r,r+dr) (or (q, q+dq) for angular correlation function) in real and theoretical catalogue
In the case of two-dimensional it is basically enough but: If we want to make a detailed interpretation of either If we examine three-dimensional distributions: THIS IS THE KEY DETERMINATION OF LUMINOSITY FUNCTION

7. Luminosity Function
Schechter (1976) was the first to study the mass function and then the brightness function for galaxies
The luminosity function is an important quantity for analysis of large scale structure statistics, interpretation of galaxy counts ( Lin & Kirshner 1996).
The problem of the analysis of luminosity function for galaxy cluster, (contrary to the analysis of luminosity function for individual optical galaxies and radio galaxies) is rather negligent till now

8. Luminosity Function II
We have decided to construct and study the luminosity function of galaxy clusters.
This was performed counting brightness of galaxies belonging to 6168 galaxy clusters in PF Catalogue (ten galaxies in the magnitude range (m3; m3 +3), structure was finding involving the Voronoi tessellation, taking data for galaxies from MRSS.
The obtained luminosity function is significantly different than that obtained both for individual optical and radio galaxies (Machalski & Godłowski 2000).
The implications of this result for theories of galaxy formation are discussed as well.

9. Luminosity of the clusters
Investigation was based on counting brightness of galaxies belonging to individual galaxy clusters
Astronomical objects with brightest I0 =  10-6 Lx has magnitude 0 (it is because reproduce historical ancientt value)
Now magnitude of object is given by formulae:
m = –2.5 log(I/I0) = –2.5 log I log I0
It mean that:
m1 – m2 = –2.5 log (I1/I2) = 2.5 log (I2/I1)
Now we are able to compute total luminosity of cluster Itot = S Ii
As well as total magnitude of the cluster mtot
Absolute Magnitude Mtot is magnitude if object would be on the distance 10 pc
M =m - 5 log D + K(z) + 5

10. Data
Catalogue of Galaxy Clusters Panko-Flin (PF) (2006) - on the base of MRSS Catalogue (Munster))

11. Advantages and Disadvantages of PF Catalogue
First so rich catalogue of galaxy clusters - complete till 18.3m, clusters
Obtained on the base of MRSS catalogue, every object in the catalogue is a galaxy
No radial velocities - no possibility of removing background objects
Distances extrapolated from 10th brightest galaxy in clusters – no errors from peculiar motions of galaxies

12. Obtaining Luminosity Function
It is a lot of methods of estimation of the luminosity function
1. Direct counting (Condon 1989)
V_max is the maximum volume in which astronomical object(i.e. galaxy, radiogalaxy, cluster) with respective absolute magnitude M would appear in catalog – in our case galaxy have m>18.3m
2. Fitting Schechter Function :
3. Maximum likelihood method:

13. Essence of Methods SWML

14. Luminosity Function (optical) –SWML method
Schechter Function (STY Method)

15. Sechter Luminosity Function
Schechter luminosity function
The Schechter luminosity function provides a parametric description of the space density of galaxies as a function of their luminosity. The form of the function is
where , and is a characteristic galaxy luminosity where the power-law form of the function cuts off. The parameter has units of number density and provides the normalization. The galaxy luminosity function may have different parameters for different populations and environments; it is not a universal function. One measurement from field galaxies is
It is often more convenient to rewrite the Schechter function in terms of magnitudes, rather than luminosities. In this case, the Schechter function becomes:
Note that because the magnitude system is logarithmic, the power law has logarithmic slope
This is why a Schechter function with is said to be flat.

16. Estimation of the models parameter – SWML method

17. For example for Las Campanas optical galaxies it is described by Schechter function with possible parameters: alpha =-0.7, (Lin et al., 1996) or alpha=-0:63, (Machalski & Godlowski, 2001). On should note however that for radiogalaxies the luminosity function has more complexform, the Saundres function (Saunders et al., 1990; Machalski & Godlowski, 2000). The reason is that to be classied as radiogalaxy the object must be observed (i.e.be enough bright) both in optical and radio wavelength.

18. Density function and corrections for luminosity function

20. Galaxy Clusters
Histogram - number of clusters with particular absolute magnitude (all and m3<15.3) If m3 is well indicator of the distances it should be Luminosity Function – but:
Histogram of absolute magnitude for m3 galaxies in clusters

21. Probability density function for luminosity function for galaxy clusters
a) All clusters in teh sample (left panel) and b) in the case m3<15.3 (right panel)
M* = -23:1, alpha = 4.26 !!! Luminosity Function for galaxy clusters obtained from Condon’s methods – actually Gauss not Schechter!!!
Schechter function

22. Analysis of luminosity function for galaxy cluster, (contrary to the analysis of luminosity function for individual optical galaxies and radio galaxies) is rather negligent till now
But:
E. De Filippis i in. (2011)

23. ML fit obtained using a Schechter plus a Log-normal components (solid line). The individual components are shown as dashed lines. Data were extracted within 0.5 R200. Confidence
levels for the free parameters of the Schechter function are also plotted. Gray circles and empty diamonds represent the composite LFs obtained using the Colless and GMA methods, respectively (E. De Filippis at al. 2011)

24. Why the results are different??
In the paper Wen i Han (2014) analyzed separately the brightest cluster galaxies (BCG) and non BCG galaxies

25. Composite luminosity functions of BCGs and the best-fit Schechter functions of non-BCG member galaxies (thin lines) for clusters in the three ranges of relaxation parameters (Wen & Han 2014)

26. 2. Usually brightness of galaxy clusters is dominated by the brightness member of the clusters
3. Moreover:
Taking into account definition of galaxy cluster they can not be to faint :
For example the PF Catalogue regard as the cluster the structure which contain at least ten galaxies in a magnitude range between m3 and m3+3, where m3 is the magnitude of the third brightest galaxy located in the considered structure region. The criterion of m3+3 is well known criterion to galaxy membership for the cluster if, as usually we have no information about radial velocities of particular galaxies.

27. Summary
In the work we examine both the problem of determining the lighting function for galaxies that are part of the clusters and the functions of lighting the galaxy clusters themselves. While the function of lighting for optical galaxies or radio galaxies was determined many times, the lack of such data on the scale of clusters. Our base catalog is the Panko-Flin Cluster of Galaxies catalog based on the MRSS Catalog. This is the first such numerous (6188 clusters) directory of galaxy clusters complete to 18.3m. The problem of determining the function of luminescence for galaxy clusters is important from the point of view of scenarios for the creation of large-scale structures as well as in studies on the autocorrelation function for galaxy clusters (where our preliminary results showed that the scale of grouping is in the order of 100 Mph-1). Preliminary results show that the luminosity function for galaxy clusters: It is dominated by the brightest galaxies in the cluster and is described rather as an ordinary Gaussian function rather than a classic Schechter function.

28. Conclusions
The obtained luminosity function is significantly different than that obtained both for optical and radio galaxies (Machalski & Godłowski 2000).
It looks like Gauss not Schechter function
Probable explanation – differences in scenarios of formation of galaxies and clusters

29. Problem „śmieci”
Rozkład galaktyk w obszarze gromady Abell 2556 (dane z przegladu Superkosmos)

30. Problems with theoretical catalogue

31. Theory of galaxy formation
In the commonly accepted LCDM model the Universe is deemed to be spatially flat, homogeneous and isotropic at appropriate scale, but:
The dimension of this scale is changing with the growth of our knowledge of the Universe
Galaxies are clustering – what with clustering of radio sources, what with clustering of galaxy clusters?

32. Luminosity function for radio galaxies section 3
Luminosity function for radio galaxies section 3.2 – Simple Condon’s method

33. Dwupunktowa trójwymiarowa funkcja autokorelacji (analogicznie definiujemy kątową funkcję autokorelacji)