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In this paper e try to reconstruct the spatial distribution of stars in globular clusters (GCs) from heuristic statistical ideas Such 3 radial distributions are important for understanding physical conditions across the clusters ur method is based on con erting spherically symmetrical functions such as exp(-r2/s2), exp(-r/s), 1/(1 + r2/s2) 2 and 1/(1 + r2/s2)m, (s and m are parameters) to 2 star distributions in a GCs by the Monte Carlo method By comparing the obtained 2 profiles ith obser ational ones e demonstrate that Gaussian or exponential distribution functions yield too short extensions of peripheral parts of the GCs profiles The best candidate for fitting GCs profiles has been found to be the generali ed Schuster density la : C/ (1 + r2/s2)m, here C is the normali ation constant and s and m are adjustable parameters These parameters display a nonlinear correlation ith s arying from 0 1 to 10 pc, hilst m is close to 2 Using this la the radiation temperatures across M 13 and 47 Tucane ere estimated Keywords: globular clusters, stars, density, radial distribution, 2 and 3 , Monte Carlo, Schuster la , M 13, 47 Tucane 1. INTRODUCTION Since the classical in estigations by arlo Shapley (1885-1972), ho identified Cepheids and calculated real distances to the globular clusters (GCs) , these remarkable objects become the most intensi ely studied in the Milky ay and in neighboring galaxies. ith the launch of the ubble Space Telescope ( ST) it as finally possible to resol e indi idual stars in their dense central cores. In addition to stars hose presence is expected by the canonical stellar e olution theory, se eral more exotic objects like blue struggle stars, -ray binaries, millisecond pulsars, etc., ha e been indentified in Galactic GCs so far, see e.g. ref. . The GCs play a key role in astrophysics, because they may be considered as large assemblies of coe al stars ith a common history, but differing only in their initial masses, although gro ing e idence for some spread in star formation ages is being collected, see e.g. Piotto, 2010 . The spread of star ages is surely much shorter than the age of the clusters. It is useful here that stars in a GC may be treated statistically ith high degree of confidence. Moreo er, the number of GCs in the Milky ay is uite large, close to 160. They differ in mass, luminosity, total number of stars, and their spatial densities as a function of distance from the center. The most fundamental characteristics of the GC such as the total number of stars, N, and their radial distribution are still poorly kno n due to their extremely large central densities and slo gradual transition of their peripherals to ards the Galactic background. A better kno ledge of these characteristics is necessary for a proper estimation of the physical conditions in central parts of GCs. It is particularly interesting to kno to hat extent their central temperatures differ from the present-day background radiation temperature (2.73 ) and hat is the temperature gradient across a cluster. GCs are the oldest objects in the Milky ay galaxy, of the order of 1012 years, i.e. large in comparison to a characteristic time-scale o er hich stars lose memory of their initial orbital conditions. This is a so-called relaxation time, of the order of 107 years according to Chandrasekhar . Therefore, GCs are old enough to attain a dynamic e uilibrium and a stable symmetric radial distribution, pro ided that they ere neither significantly disturbed during the last pass through the Galactic disk, nor they collided ith other GCs. hile the GC-GC collisions are actually rare, it ouldn t be so ith the passage through the disk. The radial distribution of stars is crucial in determining the dynamic properties of a GC, ho e er, this topic is beyond the scope of this study. It is the purpose of this paper to present step-by-step reconstruction of the 3-dimentional radial distributions (3D) of stars in a GC, from the 2-dimentional distributions recorded by telescopes. Our approach is based on the Monte Carlo method hich is applied to arious trial functions assumed to be symmetric 3D distributions. The Monte Carlo method allo s a fast con ersion of the 3D to 2D distribution hich is then compared to that obser ed in the sky. 2. T EORETICAL CONSIDERATIONS e ill start the calculations from the assumption of a 3D Gaussian as a trial function for spatial distribution of stars in a GC, because the Gaussian distribution may be considered as a standard radial-symmetric function to hich other distributions may be simply compared. The follo ing physical analogy is releant to the Gaussian distribution function. The diffusion phenomenon may con ert the initial distribution of any particle system to the Gaussian one, generally ith time-dependent standard de iation parameter, . or example, a droplet of ink immersed inside a large ater pool ill diffuse continuously, and ink density ill attain, due to the chaotic motion of ater molecules, a Gaussian distribution ith standard de iation increasing proportionally to the s uare root of time. o e er, hen diffusing particles attract each other, the dispersion parameter, , can finally achie e a constant alue, just alike in the case of stars distribution in a massi e GC. Nonetheless, a lo mass cluster ill suffer a loss of stars becoming gradually conerted to an open cluster, as e.g. M 67 . De iations of a real distribution from the spatial Gaussian distribution ill be considered later on. It is expected, ho e er, that such a de iation ill be a rather small correction only to the second and some hat larger to the fourth central statistical moment, because of rather high spherical symmetry of all the clusters obser ed in the Milky ay (see McMaster Uni ersity Catalog [7,8] for eccentricity parameter). Therefore, in the first approximation, the third statistical central moment is zero, and only significant moments remain the second ( ariance) and the fourth. Consider a reference frame (x, y, z) ith the origin located in the center of a cluster and the z-axis oriented out ards a remote obser er. The obser ed distribution of stars in the (x, y) plane being a small section of the celestial sphere is the projection of their radial 3D distribution. This projection can be obtained from the assumed normal distributions along the three axes. These distributions are defined by a common parameter , due to GC symmetry. So, the probability to find a star in the range bet een x and x + dx is gi en by the follo ing expression: (1) Similarly are defined dPy and dPz, hence the probability to find a star in an infinitesimal box of size dxdydz is: No that (2) e can replace the Cartesian coordinates by the spherical ones nothing = sin . In order to calculate the probability of a star position bet een spheres of radius r and r + dr, e ha e to integrate the transformed expression (2) o er the angular coordinates and : sin (3) The number of stars, dNr, bet een spheres of radius r and r + dr is: (4) As it is seen from the abo e formula dNr can be calculated from the total numbers of stars, N, in a considered cluster and its characteristic radius hich is defined by the standard de iation parameter . Substituting s for 2 , e can easily con ert e uation (4) to the follo ing e ui alent form: (4a) It should be noted at this point that for any spherically symmetric function f(r/s), here s is a characteristic distance parameter, the fraction of stars of the total number N dispersed bet een spheres of radius r and r + dr may by calculated in similar ay: here (5) = 1/[ ] is the normalization constant, and u = r/s. 1 s 2 e 2s x2 s2 x s 2 x2 1 2 s s x2 3 1 2 4s s -4 -3 -2 -1 3 x/ s FIG. 1. The probability density functions f(x) = dPx/dx considered in this study. In this paper e ill consider other spherically symmetric functions as candidates for spatial star distribution around a GC center. Therefore, instead of e uation (1) for f(x) = dPx/dx e ill consider a double exponential function, = exp(-|x|/s), and the next it ill be a s uared Cauchy distribution func- = 1/(1+x2/s2)2. The first function is also kno n as the Laplace distrition, bution, hereas the second belongs to the Pearson type VII family probability density functions. The rationale for using the double exponential function is that the physical conditions in a GC ith a massi e black hole resemble the electron-proton interaction in the hydrogen atom. The uantum mechanics exactly describes the probability distribution of an electron (radial density) in the lo est energy state by the double exponential function. This function has 4 times larger ariance, 2, and much larger fourth statistical moment, 4, than the Gaussian (see Table 1). On the other hand, the s uared Cauchy distribution function has a slightly larger ariance than the Gaussian, but the fourth statistical moment is infinite, therefore it may be a better candidate for describing a broad star distribution in GCs. Actually the s uared Cauchy function nicely resembles a Gaussian, except that it has a larger o erall dispersion. These normalized functions are sho n in ig. 1 and their statistical properties are collected in Table 1. All the functions listed in Table 1 ill be used belo as trial functions for their con erting to 2D radial densities. TABLE 1. Statistical properties of the normalized distribution functions f(x) considered in this study, 2 is the ariance and 4 is the 4-th statistical moment, hich are defined as 2 Function x 1 es s ] and 2 Name ], respecti ely, here x = r/s. Normal or Gaussian Double-exponential s 2 3 4 s 2 24s 4 1 e 2s x s 2s 2 s2 2 x2 1 2 s s x2 3 1 2 4s s C ( s, m) 1 S uared Cauchy Pearson type VII s2 2 , for m >2 x2 s2 Po er la or generalized Schuster la The s uared Cauchy distribution function is a slightly modified function = 1/(1+r2/s2)2.5, hich is kno n from archi al literature (Plummer 1911 and Dicke 1939) listed as refs [5, 6]. This function has been obtained as one of elementary functions found ithin the solutions of the Emden's polytropic gas sphere e uation = 0, (6) here is the gas density, r is radial distance, is the ratio of specifics heats of the gas, and b is a parameter. The abo e mentioned function 1/(1+r2/s2)2.5 is strictly rele ant for = 1.2 only, hereas atomic and molecular hydrogen has alue 1.67 and 1.40, respecti ely. ence the s uared Cauchy function has rather statistical rationale only, and it is not intimately related to the conditions of early gas nebula from hich the cluster as formed as it as proposed by Plummer. et more general e uation for radial distribution of stars in globular clusters is alike double Cauchy distribution, but ith po er treated as an adjustable parameter. This type of radial distribution is kno n as the po er la or generalized Schuster la and it as considered by i ko and Ninko ic  as a simple formula for replacement of the King's radial distribution in spherical stellar systems. 3. NUMERICAL CALCULATIONS In the next step e ha e to project the assumed 3D distributions onto the x, y plane, in order to compare the obtained 2D distributions ith that recorded by telescopes. or numerical con ersion of any 3D radial distribution to 2D e ill apply the Monte Carlo method. The algorithm de eloped for this purpose initially diides the space around the center of a GC into concentric spheres. The first sphere has radius r, hilst the radii of the subse uent spheres are increased by r. The number of stars Nr bet een t o neighboring spheres, indexed by n and n+1, is calculated from e uation (5) for r = rn + ½ r. or each star of the sub-set of Nr, the spherical coordinates r and are randomly dra n from the inter als (rn, rn+ r) and (0, 2 ), respecti ely. The coordinate as calculated from arcsin function, the alues of hich ere randomly dra n from the inter al (-1, 1). The described procedure creates a uniform star distribution ithin the each sphere. In the last step of the numerical procedure the Cartesian coordinates (x, y, z) of all the stars are calculated from the obtained (r, , ) coordinates. The projection of the stars onto the planar surface x,y is made by setting z = 0 for all the N stars. rom the obtained planar distribution of stars, a 2D radial density function is calculated (i.e. GC profile) hich is then compared to obser ations. e adjust the parameters C, s and m in order to obtain the best agreement of the plotted profile ith that taken from ref.  using as a criterion the lo est alue of rootmean-s uare de iation. The sum of stars dra n in the simulation at optimum distribution parameters is treated as the total number of stars, N. Normalized radial distribution functions of stars in 3D space, hich ere considered in this paper are listed in Table 2. TABLE 2. Normalized radial distribution functions applied in this study. Name Normal or Gaussian Double-exponential S uared Cauchy Pearson type VII Radial distribution function 4 r 2 rs e dr s3 1 r 2 rs e dr 2 s3 4 r2 r2 1 2 s3 s dr r2 r2 1 2 s3 s dr K 4. RESULTS AND DISCUSSION In ig. 2a e sho the 2D star distribution in the x,y plane generated for N = 7·104 stars distributed in 3D space according to the s uared Cauchy radial function. This figure sho s the simulated stars distribution in the M 13 (NGC 6205) globular cluster, the photo of hich is sho n in ig. 2b for comparison. A certain amount of eccentricity is seen in the photo of M 13. According to the catalog data in refs. [7,8] M 13 has an absolute magnitude - 8.55 M, core radius 0.62 arc min, and half-light radius 1.69 arc min, the eccentricity 1- b/a = 0.1, here a and b are axes of the ellipse o erlapping the cluster core. FIG. 2. a. The stars distribution in M13 cluster simulated by the Monte Carlo method, hile b is a photo of this GC for comparison, source: http:// .osser atoriomtm.it 1 0.5 log(N(r)/(arcsec^2)) 0 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 log(r/arcsec) N total = 50000, s = 50 arcsec Gaussian Exponential Squared Cauchy M13 observed (Miocchi at al. 2013) 0.5 1 1.5 2 2.5 3 3.5 FIG. 3. The comparison of 2D distribution of stars in modeled M 13 cluster using 3 different trial functions for 3D radial distribution ha ing identical characteristic size parameter, s (the disagreement ith the outermost 3 points of the M13 profile is due to the nearly constant 2D density superimposed profile of the Galactic stellar background). Each function as normalized for the total number of stars N = 50,000. The obtained distributions are compared ith the obser ed distribution by Miocchi et al. . It is seen that using the s uared Cauchy function ill lead to a better agreement for assumed larger number of stars and size parameter. 1 0.5 0 log(N(r)/(arcsec^2)) -0.5 -1 -1.5 -2 -2.5 -3 -3.5 log(r/arcsec) Squared Cauchy, N total=86000, s = 70 arcsec M13 observed (Miocchi at al. 2013) FIG. 4. The comparison of the con erted 3D distribution, hich is normalized s uared Cauchy function 1/(1+r2/s2)2 ith the obser ed 2D profile  for assumed larger number of stars and optimally adjusted s alue. The obtained 2D distribution fully agrees ith the obser ed profile of M 13 cluster. ig. 3 sho s the profiles of the projected distributions of stars into x,y plane for N = 5·104 stars ith the same size parameter, s, of the follo ing 3D radial distributions: (i) Gaussian, (ii) double exponential, and (iii) s uared Cauchy. All these functions ere normalized by an appropriate multiplier C to obtain the same total number of stars (N = 5·104) and all of them ha e identical dimensional parameter s = 50 arcsec. s (parsec) FIG. 5. The empirical relationship bet een the size parameter s in parsecs of a GC and the po er factor m determining the slope of the obser ed profile. It is seen that the larger the core ith respect to the o erall system size, the smaller the radial extent of the outer en elope region and ice ersa. K Although the obtained plots resemble a real- orld obser ed star distribution in M 13, hich is plotted as green line in ig. 3 using data from recent study by Miocchi et al. , neither of them fits ell to the obser ed distribution. The best fit is obtained ith the s uared Cauchy distribution, here by arying its s parameter e can finally achie e excellent agreement ith the obser ed distribution, as sho n in ig. 4. TABLE 3. Results of numerical simulation of 3D star distributions in GCs for those star counting profiles ere a ailable (Miocchi et al. ). The distance as taken from  hereas C, m, and s are parameters of formula (9) ere found by the Monte Carlo method as optimal. The total number of stars, N, is calculated from the fitted 3D distribution by counting the stars dra n in the simulation. NGC 104 1851 1904 2419 5024 5139 5272 5466 5824 5904 6121 6205 6229 6254 6266 6341 6626 6809 6864 Distance [kpc] 4.5 12.1 12.9 82.6 17.9 5.2 10.2 16 32.1 7.5 2.2 7.1 30.5 4.4 6.8 8.3 5.5 5.4 20.9 C 0.21 0.0025 0.019 0.16 0.07 0.26 0.17 0.1 0.012 0.28 0.04 1.75 0.054 0.12 1.4 0.025 0.01 0.18 0.021 N 147100 4400 6400 11700 17100 104100 20900 4500 1900 35000 13300 89400 3900 8300 156000 7300 5400 13300 6700 m 1.5 1.4 1.7 2 1.8 1.5 1.8 2.2 1.7 1.8 1.5 2.2 2 2 2.1 1.7 1.55 2.3 1.7 s [arcsec] 30 4 11 25 26 200 29 100 5 35 55 75 12 60 55 18 12 150 6 s [pc] 0.65 0.23 0.69 10.01 2.26 5.04 1.43 7.76 0.78 1.27 0.59 2.58 1.77 1.28 1.81 0.72 0.32 3.93 0.61 Although the proposed star distribution in GCs (i.e. s uared Cauchy) is not directly related to the dynamics of the system, it seems to be not far from those based on mechanical principles . Actually the s uared Cauchy radial function as considered by us to be more appropriate than Cauchy distribution function hich has infinite ariance or standard de iation, hereas the s uared Cau- chy function has a finite standard de iation. Through its larger dispersion in comparison to Gaussian or exponential function it appears to be most appropriate of 3D star distribution in M 13 ( igs 3 and 4). o e er, often the best fit to the obser ed profiles leads to the po er la function or Schuster density la [10-12], here the po er m aries from 1.4 to 2.3 as it is sho n in ig. 5. Studying a sample of Milky ay GCs for hich star counting profiles ha e been published recently , e ha e noticed an interesting non-linear correlation bet een parameters s and m ( ig. 5). In this ay by using the Monte Carlo approach e ha e confirmed a great significance of po er-la distribution function. Though the po er-la is considered in literature as ad hoc fitting function , in most cases it better fits to the obser ation data than King and ilson models . The major eakness of this function o er the King model is that it is not dynamically self-consistent in the sense that it produces a dynamical e uilibrium. o e er, for the purposes of this study the po er-la radial distribution is fully sufficient , because e do not consider star elocities, but their spatial distribution only. 5. RADIATION TEMPERATURE ACROSS GCS e can no use the Monte Carlo approach to estimate the radiation temperature across a GC. Let us assume for this purpose that each star of a GC produces the same amount of electromagnetic radiation flux of 1366 /m (solar constant) at the distance of one astronomical unit. According to this simplified assumption the radiation flux density from a star at distance ri from a fixed point in the free space of GC can be calculated, using formula: 1366 /m 2 2 ri / 1 AU at this point is N i i (7) , here N is total The total irradiation flux density number of stars in the considered GC. The total flux density of electromagnetic radiation determines the temperature T of black body, hich fully absorbs this radiation. The relation bet een and T is described by the StefanBoltzmann la = T4, (8) here in formula (8) is the StefanBoltzmann constant. Using the abo e t o e uations, e can calculate approximately the radiation temperature in the space K inside a modeled GC (by the Monte Carlo method) as a function of distance from its center. T o examples of such temperature profiles are sho n in ig. 6. 12 T (K) 10 r_min (parsec) 8 5 T (K) 6 4 3 4 2 2 1 0 0 10 20 30 r (parsec) 40 50 60 0 distance to the nearest star (parsec) 6 8 7 FIG. 6. Radiation temperatures (abo e background of 2.7 K) as a function of distance from the center of modeled M13 and 47 Tucane clusters (black lines). The spikes in black lines are due to proximity to the nearest star, the distances of hich are plotted as gray lines. 6. CONCLUSIONS A critical discussion of the calculations presented abo e leads to a conclusion that 3D radial density of stars is ell described by t o-parameters function kno n as the po er-la distribution or generalized Schuster density la : r2 f (r ) C 1 2 s (9) here C is the normalization constant, s is the size parameter and m is related to the obser ed slope of the star density profile. ith this function e ha e calculated present-day radial temperature distribution in the free space inside t o GCs: M 13 and 47 Tucane. The last one, being one of the largest Milky ay cluster, has the central radiation temperature of 16 K abo e the present-day Uni erse background temperature (2.7 K). Though temperatures across GCs are meaningless in the astrophysical modeling of stars e olution, ho e er e suppose that the temperature gradient plays a great role of a mop hich cleans the acuum inside the GCs. Thanks to its action and perhaps some gas accretion by hite d arfs, e ha e an ideal insight into the interiors of GCs by the ST. Recent density determination of ionized gas (probably the dominant component of the intra-cluster medium) by radio-astronomical obser ations of 15 pulsars in 47 Tucane yields 0.067±0.015 cm 3 only . This is about 100 times the free electron density of the interstellar medium in the icinity of this GC. Such a lo density is undetectable by other methods. ACKNO LEDGEMENTS The authors ish to express their gratitude to Dr. Paolo Miocchi from the Department of Physics and Astronomy, Uni ersity of Bologna, Italy, for comments on the manuscript and help in access to recent literature. e are grateful to Dr. Tomasz Durakie icz from Los Alamos National Laboratory for corrections of English. RE ERENCES 1. 2. 3. Stru e O. and ebergs V. (1962) Astronomy of the 20th Century, Macmillan Co. erraro .R., Exotic Populations in Galactic Globular Clusters, in: The Impact of HST on European Astronomy, Astrophysics and Space Science Proceedings, Springer Science + Business Media B.V., 2010, p. 51. Piotto, G., 2010. Observational Evidence of Multiple Stellar Populations in Star Clusters, PKAS 25, 91. K Chandrasekhar S. (1960) Principles of Stellar Dynamics, Do er, Ne ork. Plummer .C. (1911). On the problem of distribution in globular stars clusters, Monthly Notes L I, 5, 460-470. Dicke R. . 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Annales UMCS, Physica – de Gruyter
Published: Mar 1, 2015
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