Radial distribution of stars in globular clusters inferred from the Monte Carlo approach

U N I V E R S I T A T I S M A R I A E C U R I E - 6 . à 2 ' 2 : 6 . $

L U B L I N – 3 2 / 2 1 , $

92//;,; 6(&7,2$$$ 2014

5$',$/',675,%87,212)67$56

,1*/2%8/$5&/867(56,1)(55('

)5207+(0217(&$5/2$3352$&+

7RPDV]3LHĔNRV

*

DQG6WDQLVáDZ+DáDV

Institute of Physics, Maria Curie-6NáRGRZVND8QLYHUVLW\-031 Lublin, Poland *_{Corresponding author, e-mail: WRPDV]SLHQNRV#XPFVSO}

ABSTRACT

,Q WKLV SDSHU ZH WU\ WR UHFRQVWUXFW WKH VSDWLDO GLVWULEXWLRQ of stars in globular clusters *&V IURP KHXULVWLF VWDWLVWLFDO LGHDV 6XFK ' UDGLDO GLVWULEXWLRQV DUH LPSRUWDQW IRU XQGHUVWDQGLQJSK\VLFDOFRQGLWLRQVDFURVVWKHFOXVWHUV2XUPHWKRGLVEDVHGRQFRQYHUWLQJ spherically symmetrical functions such as exp(-r2_/s2_{), exp(-r/s), 1/(1 + r}2_/s2₎ 2 _{and 1/(1 +}

r2_/s2₎m_{, (s and m are parameters) WR'VWDUGLVWULEXWLRQVLQD *&VE\WKH0RQWH&DUOR}

PHWKRG%\FRPSDULQJWKHREWDLQHG'SURILOHVZLWKREVHUYDWLRQDORQHVZHGHPRQstrate that Gaussian or exponential distribution functions yield too short extensions of periph-HUDOSDUWVRIWKH*&VSURILOHV7KHEHVWFDQGLGDWHIRUILWWLQJ*&VSURILOHVKDVEHHQIRXQG WREHWKHJHQHUDOL]HG6FKXVWHUGHQVLW\ODZ&r2_/s2₎m_{, ZKHUH&LVWKHQRUPDOL]DWLRQ}

constant and s and m are adjustable SDUDPHWHUV 7KHVH SDUDPHWHUV GLVSOD\ D QRQOLQHDU FRUUHODWLRQZLWKs YDU\LQJIURPWRSFZKLOVW m is FORVHWR8VLQJWKLVODZWKH UDGLDWLRQWHPSHUDWXUHVDFURVV0DQG7XFDQHZHUHHVWLPDWHG

Keywords: globular clusters, stars, density, UDGLDOGLVWULEXWLRQ'DQG'0RQWH&DUOR

10 73,(ē.26$1'6+$à$6

,1752'8&7,21

6LQFHWKHFODVVLFDOLQYHVWLJDWLRQVE\+DUORZ Shapley (1885-ZKRLGHn-tified Cepheids and calculated real distances to the globular clusters (GCs) [1], these remDUNDEOHREMHFWVEHFRPHWKHPRVWLQWHQVLYHO\VWXGLHG LQWKH0LON\:D\ DQG LQ QHLJKERULQJ JDOD[LHV :LWK WKH ODXQFK RI WKH +XEEOH 6SDFH 7HOHVFRSH +67 LW ZDV ILQDOO\ SRVVLEOH WR UHVROYH LQGLYLGXDO VWDUV LQ WKHLU GHQVH FHQWUDO FRUHV ,Q DGGLWLRQ WR VWDUV ZKRse presence is expected by the canonical stellar HYROXWLRQWKHRU\VHYHUDOPRUHH[RWLFREMHFWVOLNHEOXHVWUXJJOHVWDUV;-ray bina-ULHVPLOOLVHFRQGSXOVDUVHWFKDYHEHHQLQGHQWLILHGLQ*DODFWLF*&VVRIDUVHH HJUHI>@

7KH*&VSOD\DNH\UROHLQ astrophysics, because they may be considered as ODUJH DVVHPEOLHV RI FRHYDO VWDUV ZLWK D FRPPRQ KLVWRU\ EXW GLIIHULQJ RQO\ LQ WKHLULQLWLDOPDVVHVDOWKRXJKJURZLQJHYLGHQFHIRUVRPHVSUHDGLQVWDUIRUPDWLRQ DJHVLVEHLQJFROOHFWHGVHHHJ3LRWWR>@7KHVSUHDGRIVWDUDJHVLVVXUHO\ PXFKVKRUWHUWKDQWKHDJHRIWKHFOXVWHUV,WLVXVHIXOKHUHWKDWVWDUVLQD*&PD\ EHWUHDWHGVWDWLVWLFDOO\ZLWKKLJKGHJUHHRIFRQILGHQFH0RUHRYHU, the number of *&VLQWKH0LON\:D\LVTXLWHODUJHFORVHWR7hey differ in mass, lumi-nosity, total number of stars, and their spatial densities as a function of distance IURPWKHFHQWHU

The most fundamental characteristics of the GC such as the total number of stars, N, and their radial distribution are still poorl\NQRZQGXHWRWKHLUH[WUHPe-O\ODUJHFHQWUDOGHQVLWLHVDQGVORZJUDGXDOWUDQVLWLRQRIWKHLUSHULSKHUDOVWRZDUGV WKH *DODFWLF EDFNJURXQG $ EHWWHU NQRZOHGJH RI WKHVH FKDUDFWHULVWLFV LV QHFHs-sary for a proper estimation of the physical conditions in central SDUWVRI*&V,W LVSDUWLFXODUO\LQWHUHVWLQJWRNQRZWRZKDWH[WHQt their central temperatures dif-fer from the present-GD\EDFNJURXQGUDGLDWLRQWHPSHUDWXUH.DQGZKDWLV WKHWHPSHUDWXUHJUDGLHQWDFURVVDFOXVWHU

GCs are the oldest objects in the MiON\ :D\ JDOD[\ RI WKH RUGHU RI 12

\HDUVLHODUJHLQFRPSDULVRQWRDFKDUDFWHULVWLFWLPH-VFDOHRYHUZKLFKVWDUVORVH memory of their initial orbital FRQGLWLRQVThis is a so-called relaxation time, of the order of 107 _{\HDUVDFFRUGLQJWR&KDQGUDVHNKDU>@7KHUHIRUH, GCs are old}

enough to attain DG\QDPLFHTXLOLEULXPDQGDVWDEOHV\PPHWULFUDGLDOGLVWULEu-WLRQSURYLGHGWKDWWKH\ZHUH neither significantly disturbed during the last pass WKURXJKWKH*DODFWLFGLVNQRUWKH\FROOLGHGZLWKRWKHU*&V:KLOHthe GC-GC FROOLVLRQVDUHDFWXDOO\UDUHLWZRXOGQWEHVRZLWKWKHSDVVDJHWKURXJKWKHGLVN

The radial distribution of stars is crucial in determining the dynamic proper-WLHVRID*&KRZHYHU, this topic is beyond the scope of WKLVVWXG\ It is the pur-pose of this paper to present step-by-step reconstruction of the 3-dimentional radial distributioQV'RIVWDUVLQD*&, from the 2-dimentional distributions UHFRUGHG E\ WHOHVFRSHV 2XU DSSURDFK LV EDVHG RQ WKH 0RQWH &DUOR PHWKRG

ZKLFKLVDSSOLHGWRYDULRXV WULDOIXQFWLRQVDVVXPHGWREHV\PPHWULF'GLVWULEu-WLRQV7KH 0RQWH&DUORPHWKRGDOORZV a IDVWFRQYHUVLRQRIWKH'WR'GLVWUi-EXWLRQZKLFKLVWKHQFRPSDUHGWRWKDWREVHUYHGiQWKHVN\

7+(25(7,&$/&216,'(5$7,216

:HZLOOVWDUWWKHFDOFXODWLRQVIURPWKHDVVXPSWLRQRID'*DXVVLDQDVDWUi-al function for spati:HZLOOVWDUWWKHFDOFXODWLRQVIURPWKHDVVXPSWLRQRID'*DXVVLDQDVDWUi-al distribution of stars in a GC, because the Gaussian distri-bution may be considered as a standard radial-V\PPHWULFIXQFWLRQWRZKLFKRWKHU GLVWULEXWLRQVPD\EHVLPSO\FRPSDUHG7KHIROORZLQJ physical analogy is rele-YDQWWRWKH*DXVVLDQGLVWULEXWLRQIXQFWLRQ

The diffusion phenomenon PD\FRQYHUWWKH initial distribution of any parti-cle system WRWKH*DXVVLDQRQHJHQHUDOO\ZLWKWLPH-GHSHQGHQWVWDQGDUGGHYLa-tion parameter, ı)RUH[DPSOHDGURSOHWRILQNLPPHUVHGLQVLGHDODUJHZDWHU SRRO ZLOO GLIIXVH FRQWLQXRXVO\ DQG LQN GHQVLW\ ZLOO DWWDLQ GXH WR WKH FKDRWLF PRWLRQRIZDWHUPROHFXOHVD*DXVVLDQGLVWULEXWLRQZLWKVWDQGDUGGHYLDWLRQLn-FUHDVLQJ SURSRUWLRQDOO\ WR WKH VTXDUH URRW RI WLPH +RZHYHU Zhen diffusing particles attract each other, the dispersion parameter, ı FDQ ILQDOO\ DFKLHYH D FRQVWDQWYDOXHMXVWDOLNHLQWKHFDVHRIVWDUVGLVWULEXWLRQLQDPDVVLYH*&1RQe-WKHOHVV D ORZ PDVV FOXVWHU ZLOO VXIIHU DORVV RI VWDUVEHFRPLQJ JUDGXDOO\ FRn-YHUWHGWRDQRSHQFOXVWHUDVHJ0>@

'HYLDWLRQVRIDUHDOGLVWULEXWLRQIURPWKHVSDWLDO*DXVVLDQGLVWULEXWLRQZLOO EHFRQVLGHUHGODWHURQ,WLVH[SHFWHGKRZHYHUWKDWVXFKDGHYLDWLRQZLOOEH a rather small correction only to the second and someZKDW ODUJHU WR WKH IRXUWK central statistical moment, because of rather high spherical symmetry of all the FOXVWHUVREVHUYHGLQWKH0LON\:D\VHH0F0DVWHU8QLYHUVLW\&DWDORJ>7,8] for HFFHQWULFLW\SDUDPHWHU7KHUHIRUHLQWKHILUVWDSSUR[LPDWLRQWKHWKLrd statistical central moment is zero, and only significant moments remain the second YDUi-DQFHDQGWKHIRXUWK

Consider a reference frame (x, y, z) ZLWKWKHRULJLQORFDWHGLQWKHFHQWHURID cluster and the z-axis oriented outZDUGVDUHPRWHREVHUYHU7KHREVHUYHGGLVWUi-bution of stars in the (x, y) plane being a small section of the celestial sphere is WKH SURMHFWLRQ RI WKHLU UDGLDO ' GLVWULEXWLRQ This projection can be obtained IURPWKHDVVXPHGQRUPDOGLVWULEXWLRQVDORQJWKHWKUHHD[HV7KHVHGLVWULbutions are defined by a common parameter ı, due to GC syPPHWU\6R the probability WRILQGDVWDULQWKHUDQJHEHWZHHQx and x + dx LVJLYHQE\WKHIROORZLQJH[SUHs-sion:

݀ܲ

௫

=

_ξଶగఙଵ

݁

ି

ೣమ

12 73,(ē.26$1'6+$à$6

Similarly are defined dPy and dPz, hence the probability to find a star in an

infinitesimal box of size dxdydz is:

݀ܲ = ݀ܲ

_௫

݀ܲ

_௬

݀ܲ

_௭

= ቀ

ଵ ξଶగఙ

ቁ

ଷ

݁

ିೣమశ೤మశ೥మమ഑మ

݀ݔ݀ݕ݀ݖ

₍₂₎ 1RZZHFDQUHSODFHWKH&DUWHVLDQFRRUGLQDWHVE\WKHVSKHULFDORQHVQRWKLQJ that

ݔ

ଶ

_{+ ݕ}

ଶ

_{+ ݖ}

ଶ

_{= ݎ}

ଶ

݀ݔ݀ݕ݀ݖ

՜ ݀ݎ ή ݎ݀ߠ ή ݎ sin ߠ ݀߮

,QRUGHUWRFDOFXODWHWKHSUREDELOLW\RIDVWDUSRVLWLRQEHWZHHQVSKHUHVRIUa-dius r and r + drZHKDYHWRLQWHJUDWHWKHWUDQVIRUPHGH[SUHVVLRQRYHUWKH angular coordinates ij and ș:

݀ܲ

௥

=

ௗே_ேೝ

= ׬ ݀߮

_଴ଶగ

׬ sin ߠ ݀ߠ

_଴గ

ή ቀ

_ξଶగఙଵ

ቁ

ଷ

݁

ିమ഑మೝమ

݀ݎ

(3)

The number of stars, dNrEHWZHHQVSKHUHVRIUDGLXVr and r + dr is:

݀ܰ

_௥

= ܰට

ଶ_గ௥_ఙమௗ௥_య

݁

ିమ഑మೝమ



(4)

$V LW LV VHHQ IURP WKH DERYH IRUPXOD dNr can be calculated from the total

numbers of stars, N, in a considered cluster and its characteristic radius ZKLFKLV GHILQHGE\WKHVWDQGDUGGHYLDWLRQSDUDPHWHUı6XEVWLWXWLQJs for

ξ2

ıZHFDQ HDVLO\FRQYHUWHTXDWLRQWRWKHIROORZLQJHTXLYDOHQWIRUP ௗேೝ ே

=

_S

4

௥మௗ௥ ௦య

݁

ିೝమ_ೞమ

_

(4a) It should be noted at this point that for any spherically symmetric function

f(r/s)ZKHUHs is a characteristic distance parameter, the fraction of stars of the

total number N dispersed EHWZHHQVSKHUHVRIUDGLXVr and r + dr may by calcu-ODWHGLQVLPLODUZay: ௗேೝ ே

= ܥ

௥మௗ௥ ௦య

݂ ቀ

௥ ௦

ቁ

, (5) ZKHUH

ܥ = 1/[׬ ݑ

ஶ ଶ

_݂(ݑ)݀ݑ]

FIG. 1. The probability density functions f(x) = dPx/dx FRQVLGHUHGLQWKLVVWXG\

,QWKLVSDSHUZHZLOOFRQVLGHURWKHUspherically symmetric functions as can-GLGDWHV IRU VSDWLDO VWDU GLVWULEXWLRQ DURXQG D *& FHQWHU Therefore, instead of HTXDWLRQ  IRU f(x) = dPx/dx ZH ZLOO FRQVLGHU D GRXEOH H[SRQHQWLDO IXQFWLRQ

݂ ቀ

௫_௦

ቁ

= exp(-|

x|/sDQGWKHQH[WLWZLOOEHDVTXDUHG&DXFK\GLVWULEXWLRQIXQc-tion,

݂ ቀ

௫_௦

ቁ =

1/(1+x2_/s2₎2_{ 7KHILUVWIXQFWLRQLVDOVRNQRZQDVWKH/DSODFH}

distri-EXWLRQZKHUHDVWKHVHFRQGEHORQJs to the Pearson type VII family probability GHQVLW\IXQFWLRQV

The rationale for using the double exponential function is that the physical FRQGLWLRQVLQD*&ZLWKDPDVVLYHEODFNKROHUHVHPEOHWKHHOHFWURQ-proton inter-DFWLRQ LQ WKH K\GURJHQ DWRP 7KH TXDQWXP PHFKDQLFV H[DFWO\ GHVFULEHV WKH SUREDELOLW\GLVWULEXWLRQRIDQHOHFWURQUDGLDOGHQVLW\LQWKHORZHVWHQHUJ\VWDWH E\WKHGRXEOHH[SRQHQWLDOIXQFWLRQ7KLVIXQFWLRQKDVWLPHVODUJHUYDULDQFHı2_,

and much larger fourth statistical moment, ȝ4WKDQWKH*DXVVLDQVHH7DEOH

2QWKHRWKHUKDQGWKHVTXDUHG&DXFK\GLVWULEXWLRQIXQFWLRQ has a slightly larger YDULDQFHWKDQWKH*DXVVLDQEXWWKHIRXUWKVWDWLVWLFDOPRPHQWLVLQILQLWHWKHUHIRUH LWPD\EHDEHWWHUFDQGLGDWHIRUGHVFULELQJDEURDGVWDUGLVWULEXWLRQLQ*&V$c-WXDOO\WKHVTXDUHG&DXFK\ IXQFWLRQQLFHO\UHVHPEOHVD*DXVVLDQH[FHpt that it KDVDODUJHURYHUDOOGLVSHUVLRQ7KHVHQRUPDOL]HG IXQFWLRQVDUHVKRZQLQ)LJ and their statistical properties are collected LQ7DEOH$OOWKH functions listed in 7DEOH  ZLOO EH XVHG EHORZ DV WULDO IXQFWLRQVIRUWKHLU FRQYHUWLQJ WR ' UDGLDl GHQVLWLHV -4 -3 -2 -1 0 0 1 2 3 4         x/s 2 2 1 x s e s S  2 2 x s e s  2 2 2 2 1 x s s S  § _ · ¨ ¸ © ¹  2 2 3 ₁ 4 x s s  § _ · ¨ ¸ © ¹

14 73,(ē.26$1'6+$à$6

TABLE 1. Statistical properties of the normalized distribution functions f(x) considered

in this study, ı2 _{LVWKHYDULDQFHDQGȝ}

4 is the 4-WKVWDWLVWLFDOPRPHQWZKLFKDUHGHILQHGDV 2

׬ ݔ

_଴ஶ ଶ

݂(ݔ)݀ݔ]

and 2

׬ ݔ

_଴ஶ ସ

݂(ݔ)݀ݔ]

UHVSHFWLYHO\ZKHUHx = r/s Function Name ı2 ȝ4 2 2 1 e x s s S  Normal or Gaussian 2 2 s 3 4 2s 1 e 2 x s s  'RXEOH-exponential _2s2 _24s4 2 2 2 2 1 x s s S  § _ · ¨ ¸ © ¹ 6TXDUHG&DXFK\ 2 s ’  2 2 3 1 4 x s s  § _ · ¨ ¸

© ¹ Pearson type VII

2 2 s ’ 2 2 ( , ) 1 m x C s m s  § · ˜ _{¨ ¸} © ¹

3RZHUODZRUJHQHUDl-ized 6FKXVWHUODZ ’IRUm >2 ’

The VTXDUHG &DXFK\ GLVWULEXWLRQ IXQFWLRQ is a slightly modified function

݂ ቀ

௥_௦

ቁ =

1/(1+r2_/s2₎_{ZKLFKLVNQRZQIURPDUFKLYDOOLWHUDWXUH3OXPPHU}

DQG'LFNHOLVWHGDVUHIV >@7KLVIXQFWLRQKDVEHHQREWDLQHGDVRQHRI HOHPHQWDU\IXQFWLRQVIRXQGZLWKLQWKHVROXWLRQVRI the Emden’s polytropic gas VSKHUHHTXDWLRQ ௗ ௗ௥

ቀ

௥మ ఘ ௗఘം ௗ௥

ቁ + ܾ

ଶ

ߩݎ

ଶ

= 0

, 

ZKHUHȡ is the gas density, r is radial distance, Ȗ is the ratio of specifics heats of the gas, and b LVDSDUDPHWHU7KHDERYHPHQWLRQHGIXQFWLRQr2/s2) is strictly UHOHYDQW for Ȗ RQO\ZKHUHDVDWRPLFDQGPROHFXODUK\GURJHQKDVȖ YDOXH DQG  UHVSHFWLYHO\ +HQFH WKH VTXDUHG &DXFK\ IXQFWLRQ KDV UDWKHU VWDWLVWLFDO rationale only, and it is not intimately related to the conditions of early gas nebula IURPZKLFKWKHFOXVWHUZDVIRUPHGDVLWZDVSURSRVHGE\3OXPPHU

<HWPRUHJHQHUDOHTXDWLRQIRUUDGLDOGLVWULEXWLRQRIVWDUVLQJOREXODUFOXVWHUV LVDOLNHGRXEOH&DXFK\GLVWULEXWLRQEXWZLWKSRZHU treated as an adjustable pa-UDPHWHU7KLVW\SHRIUDGLDOGLVWULEXWLRQLVNQRZQDV the ³SRZHU ODZ´RUJHQHUDl-L]HG6FKXVWHUODZDQGLWZDV considered by äLYNRYDQG1LQNRYLF>@DVDVLm-ple formula for replacement of the King’s radial distribution in spherical stellar V\VWHPV

3180(5,&$/&$/&8/$7,2NS

,QWKHQH[WVWHSZHKDYHWRSURMHFWWKHDVVXPHG'GLVWULEXWLRQVRQWRWKH x, y SODQHLQRUGHUWRFRPSDUHWKHREWDLQHG'GLVWULEXWLRQVZLWKWKDWUHFRUGHG by WHOHVFRSHV )RUQXPHULFDOFRQYHUVLRQRIDQ\'UDGLDOGLVWULEXWLRQWR'ZHZLOODSSO\ WKH0RQWH&DUORPHWKRG7KHDOJRULWKPGHYHORSHGIRUWKLVSXUSRVHLQLWLDOO\Gi-YLGHV WKH VSDFH DURXQG WKH FHQWHU RI D *& LQWR FRQFHQWULF VSKHUHV 7KH ILUVW sphere has radius ǻrZKLlVWWKHUDGLLRIWKHVXEVHTXHQWVSKHUHVDUHLQFUHDVHGE\

ǻr7KHQXPEHURIVWDUVǻNr EHWZHHQWZRQHLJKERULQJVSKHUHVLQGH[HGE\QDQG

QLVFDOFXODWHGIURPHTXDWLRQIRUr = rn + ½ ǻr)RUHDFKVWDURIWKHVXE-set

of ǻNr, the spherical coordinates r and ij DUHUDQGRPO\GUDZQ IURPWKHLQWHUYDOV

(rn, rn+ ǻr) and (0, 2ʌ UHVSHFWLYHO\ 7he coordinate ș ZDV FDOFXODWHG IURP

arcsin(șIXQFWLRQWKHYDOXHVRIZKLFKZHUHUDQGRPO\GUDZQIURPWKHLQWHUYDO (-  7KH GHVFULEHG SURFHGXUH FUHDWHV D XQLIRUP VWDU GLVWULEXWLRQ ZLWKLQ WKH HDFKVSKHUH

In the last step of the numerical procedure the Cartesian coordinates (x, y, z) of all the stars are calculated from the obtained (r,ij,șFRRUGLQDWHV7KHSURMHc-tion of the stars onto the planar surface x,y is made by setting z = 0 for all the N VWDUV)URPWKHREWDLQHGSODQDUGLVWULEXWLRQRIVWDUVD'UDGLDOGHQVLW\IXQFWLRQ LVFDOFXODWHGLH*&SURILOHZKLFKLVWKHQFRPSDUHGWRREVHUYDWLRQV:HDGMXVW the parameters C, s and m in order to obtain the best agreement of the plotted SURILOHZLWKWKDWWDNHQIURPUHI>@XVLQJDVDFULWHULRQWKHORZHVWYDOXHRIUoot-mean-VTXDUH GHYLDWLRQ 7KH VXP RI VWDUV GUDZQ LQ WKH VLPXODWLRQ DW RSWLPXP distribution parameters is treated as the total number of stars, N

1RUPDOL]HG UDGLDO GLVWULEXWLRQ IXQFWLRQV RI VWDUV LQ ' VSDFH ZKLFK ZHUH FRQVLGHUHGLQWKLVSDSHUDUHOLVWHGLQ7DEOH

TABLE 2. 1RUPDOL]HGUDGLDOGLVWULEXWLRQIXQFWLRQVDSSOLHGLQWKLVVWXG\

Name Radial distribution function

Normal or Gaussian 4 S 2 2 2 3e r s r dr s  'RXEOH-exponential 1 2 2 3e r s r dr s  6TXDUHG&DXFK\ 4 S 2 2 2 3 1 2 r r dr s s  § _ · ¨ ¸ © ¹

Pearson type VII 3

 2 2 3 1 2 r r dr s s  § _ · ¨ ¸ © ¹

 73,(ē.26$1'6+$à$6

45(68/76$1'',6&866,21

,Q)LJa ZHVKRZWKH'VWDUGLVWULEXWLRQLQWKHx,y plane generated for N = 7·104 _{VWDUVGLVWULEXWHGLQ'VSDFHDFFRUGLQJto}

WKHVTXDUHG&DXFK\UDGLDOIXQc-WLRQ7KLVILJXUHVKRZVWKHVLPXODWHGVWDUVGLVWULEXWLRQLQWKH01*& JOREXODUFOXVWHUWKHSKRWRRIZKLFKLVVKRZQLQ)LJb IRUFRPSDULVRQ$FHr-tain amount of eccentricity is seen in the photo of 0$FFRUGLQJWRWKHFDWa-ORJGDWDLQUHIV>@0KDVDQDEVROXWHPDJQLWXGH- M_{, core radius }

arc min, and half-OLJKWUDGLXVDUFPLQWKHHFFHQWULFLW\- b/a ZKHUHa and b DUHD[HVRIWKHHOOLSVHRYHUODSSLQJWKHFOXVWHUFRUH

a b

FIG. 2. a. The stars distribution in M13 cluster simulated by the Monte Carlo method,

ZKLOHb LVDSKRWRRIWKLV*&IRUFRPSDULVRQVRXUFHKWWSZZZRVVHUYDWRULRPWPLW

FIG. 3. 7KH FRPSDULVRQ RI ' GLVWULEXWLRQ RI VWDUV LQ PRGHOHG 0  FOXVWHU XVLQJ 

GLIIHUHQW WULDO IXQFWLRQV IRU ' UDGLDO GLVWULEXWLRQ KDYLQJ LGHQWLFDO FKDUDFWHULVWLF VL]H parameter, s WKHGLVDJUHHPHQWZLWKWKHRXWHUPRVWSRLQWVRIWKH0SURILOHLVGXHWR the QHDUO\FRQVWDQW'GHQVLW\VXSHULPSRVHGSURILOHRIWKH*DODFWLFVWHOODUEDFNJURXQG (DFK IXQFWLRQ ZDV QRUPDOL]HG IRU WKH WRWDO QXPEHU RI VWDUV N   7KH REWDLQHG GLVWULEXWLRQVDUHFRPSDUHGZLWKWKHREVHUYHGGLVWULEXWLRQE\0LRFFKLHWDO>@It is seen WKDWXVLQJWKHVTXDUHG&DXFK\IXQFWLRQZLOOOHDGWRDEHWWHUDJUHHPHQWIRUDVVXPHGODUJHU QXPEHURIVWDUVDQGVL]HSDUDPHWHU 0 0.5 1 1.5 2 2.5 3 3.5 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 N total = 50000, s = 50 arcsec Gaussian Exponential Squared Cauchy

M13 observed (Miocchi at al. 2013)

log(r/arcsec) lo g(N (r )/( arc sec ^2) )

FIG. 4. 7KHFRPSDULVRQRIWKHFRQYHUWHG'GLVWULEXWLRQZKLFKLVQRUPDOL]HGVTXDUHG

Cauchy function 1/(1+r2_/s2₎2 _{ZLWKWKHREVHUYHG'SURILOH>@IRUDVVXPHGODUJHUQXPEHU} of stars and optimally adjusted s YDOXHThe obWDLQHG'GLVWULEXWLRQ IXOO\DJUHHV ZLWK WKHREVHUYHGSURILOHRI0FOXVWHU

)LJVKRZVWKHSURILOHVRIWKHSURMHFWHGGLVWULEXWLRQVRIVWDUVLQWRx,y plane for N = 5·104 _{VWDUVZLWKWKHVDPHVL]HSDUDPHWHUs, RIWKHIROORZLQJ'UDGLDOGLVWULEXWLRQV}

(i) Gaussian, (ii) double exponential, and (iii) VTXDUHG&DXFK\$OOWKHVHIXQFWLRQV ZHUHQRUPDOL]HGE\DQDSSURSULDWHPXOWLSOLHUC to obtain the same total number of stars (N = 5·104_{DQGDOORIWKHPKDYHLGHQWLFDOGLPHQVLRQDOSDUDPHWHUs DUFVHF}

0 2 4 6 8 10 1.4 1.6 1.8 2.0 2.2 2.4 m s (parsec)

FIG. 5. 7KHHPSLULFDOUHODWLRQVKLSEHWZHHQWKHVL]HSDUDPHWHUs in parsecs of a GC and

WKHSRZHUIDFWRUm GHWHUPLQLQJWKHVORSHRIWKHREVHUYHGSURILOHIt is seen that the larger WKHFRUHZLWKUHVSHFWWRWKHRYHUDOOV\VWHPVL]Hthe smaller the radial extent of the outer HQYHORSHUHJLRQDQGYLFH YHUVD 0 0 1 1.5 2 2.5 3 3.5 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1

Squared Cauchy, N total=86000, s = 70 arcsec M13 observed (Miocchi at al. 2013)

log(r/arcsec) log( N (r) /( ar cs ec ^2 ))

18 73,(ē.26$1'6+$à$6

Although the obtained plots resemble a real-ZRUOGREVHUYHGVWDUGLVWULEXWLRQ LQ0ZKLFKLVSORWWHGDVJUHHQOLQHLQ)LJXVLQJGDWDIURPUHFHQWVWXG\E\ 0LRFFKLHWDO>@QHLWKHURIWKHPILWVZHOOWRWKHREVHUYHGGLVWULEXWLRQ7KHEHVW ILWLVREWDLQHGZLWKWKHVTXDUHG&DXFK\GLVWULEXWLRQZKHUHE\YDU\LQJLWVs pa- UDPHWHUZHFDQILQDOO\DFKLHYHH[FHOOHQWDJUHHPHQWZLWKWKHREVHUYHGGLVWULEu-WLRQDVVKRZQLQ)LJ TABLE 3. 5HVXOWVRIQXPHULFDOVLPXODWLRQRI'VWDUGLVWULEXWLRQVLQ*&VIRUWKRVHVWDU

FRXQWLQJ SURILOHV ZHUH DYDLODEOH 0LRFFKL HW DO >@ 7KH GLVWDQFH ZDV WDNHQ IURP >@ ZKHUHDVC, m, and s are parameters of formula (9) ZHUHIRXQGE\WKH0RQWH&DUORPHWh-od as RSWLPDO7KHWRWDOQXPEHURIVWDUVN, is calculated IURPWKHILWWHG'GLVWULEXWLRQ E\FRXQWLQJWKHVWDUVGUDZQLQWKHVLPXODWLRQ NGC 'LVWDQFH _>NSF@ C N m s [arcsec] s [pc] 104   147100  30  1851   4400  4  1904     11  2419   11700 2 25  5024   17100    5139   104100  200  5272   20900  29     4500  100  5824   1900  5  5904   35000  35     13300  55     89400  75     3900 2 12     8300 2        55     7300  18     5400  12     13300  150         $OWKRXJKWKHSURSRVHGVWDUGLVWULEXWLRQLQ*&VLHVTXDUHG&DXFK\LVQRW directly related to the dynamics of the system, it seems to be not far from those EDVHGRQPHFKDQLFDOSULQFLSOHV>@Actually the VTXDUHG&DXFK\UDGLDOIXQc- WLRQZDVFRQVLGHUHGE\XVWREHPRUHDSSURSULDWHWKDQ&DXFK\GLVWULEXWLRQIXQc-

WLRQZKLFKKDVLQILQLWHYDULDQFHRUVWDQGDUGGHYLDWLRQZKHUHDVWKHVTXDUHG&Du-FK\ IXQFWLRQ KDV D ILQLWH VWDQGDUG GHYLDWLRQ 7KURXJK LWV ODUJHU GLVSHUVion in comparison to Gaussian or exponential function it appears to be most appropri-DWHRI'VWDUGLVWULEXWLRQLQ0)LJVDQG

+RZHYHURIWHQWKHEHVWILWWRWKHREVHUYHGSURILOHVOHDGVWRWKH³SRZHUODZ´ IXQFWLRQRU6FKXVWHUGHQVLW\ODZ>10-12], ZKHUH WKHSRZHUm YDULHVIURPWR DVLWLVVKRZQLQ)LJ6WXG\LQJDVDPSOHRI0LON\:D\*&VIRUZKLFKVWDU FRXQWLQJSURILOHVKDYHEHHQSXEOLVKHGUHFHQWO\>@ZHKDYHQRWLFHGDQLQWHUHVt-ing non-OLQHDUFRUUHODWLRQEHWZHHQSDUDPHWHUVs and m )LJ 

,QWKLVZD\E\XVLQJWKH0RQWH&DUORDSSURDFKZHKDYHFRQILUPHGDJUHDW VLJQLILFDQFH RI SRZHU-ODZ GLVWULEXWLRQ IXQFWLRQ7KRXJK WKH SRZHU-ODZ LV FRn-sidered in literature as ad hoc fitting function [13], in most cases it better fits to WKHREVHUYDWLRQdata than King DQG:LOVRQPRGHOV>@7KHPDMRUZHDNQHVVRI WKLVIXQFWLRQRYHUWKH.LQJPRGHOLVWKDWLWLVQRWG\QDPLFDOO\VHOI-consistent in the sense that it produces a dynamLFDOHTXLOLEULXP+RZHYHU for the purposes of WKLVVWXG\WKHSRZHU-ODZUDGLDO GLVWULEXWLRQLVIXOO\VXIILFLHQWEHFDXVHZHGRQRW FRQVLGHUVWDUYHORFLWLHVEXWWKHLUVSDWLDOGLVWULEXWLRQRQO\

55$',$7,217(03(5$785($&5266*&S

:HFDQQRZXVHWKH0RQWH&DUORDSSURDFKWRHVWLPDWHWKHUDGLDWLRQWHPSHr-DWXUHDFURVVD*&

Let us assume for this purpose that each star of a GC produces the same DPRXQW RI HOHFWURPDJQHWLF UDGLDWLRQIOX[ RI  :Pð VRODU FRQVWDQW DWWKH GLVWDQFH RI RQH DVWURQRPLFDO XQLW $FFRUGLQJ WRWKLVVLPSOLILHG DVVXPSWLRQWKH radiation flux density from a star at distance ri from a fixed point in the free

space of GC can be calculated, using formula:





2 2 :P /1 AU i i ĭ r (7)

The total irradiation flux density Ɏ at this point is

¦

N

i i

ĭ  ZKHUH N is total

QXPEHURIVWDUVLQWKHFRQVLGHUHG*&7KHWRWDOIOX[GHQVLW\Ɏ of electromagnetic radiation determines the temperature T RI EODFN ERG\ ZKLFK IXOO\ DEVRUEV WKLV UDGLDWLRQ7KHUHODWLRQEHWZHHQɎ and T is described by the Stefan–%ROW]PDQQODZ

Ɏ = ıT4_{, (8)}

ZKHUH ı in formula (8) is the Stefan–%ROW]PDQQ FRQVWDQW 8VLQJ WKH DERYH WZR HTXDWLRQVZHFDQFDOFXODWHDSSUR[LPDWHO\WKHUDGLDWLRQWHPSHUDWXUHLQWKHVSDFH

20 73,(ē.26$1'6+$à$6

inside a modeled GC (by the Monte Carlo method) as a function of distance from its FHQWHU7ZRH[DPSOHVRIVXFKWHPSHUDWXUHSURILOHVDUHVKRZQLQ)LJ

FIG. 6. 5DGLDWLRQWHPSHUDWXUHVDERYHEDFNJURXQGRI .DVD IXQFWLRQRIGLVWDQFH

from the center of PRGHOHG 0 DQG  7XFDQH FOXVWHUV EODFN OLQHV 7KH VSLNHV LQ EODFNOLQHVDUHGXHWR SUR[LPLW\WRWKHQHDUHVWVWDUWKHGLVWDQFHVRIZKich are plotted as JUD\OLQHV 0 10 20 30 40 50 60 0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 8 T (K) r_min (parsec) r (parsec) T (K ) di stance to t h e nea rest sta r (p arsec )

 &21&/86,21S

$FULWLFDOGLVFXVVLRQRIWKHFDOFXODWLRQVSUHVHQWHGDERYHOHDGVWRDFRQFOu-VLRQWKDW'UDGLDOGHQVLW\RIVWDUVLVZHOOGHVFULEHGE\WZR-parameters function NQRZQDVWKHSRZHU-ODZGLVWULEXWLRQRUJHQHUDOL]HG6FKXVWHUGHQVLW\ODZ 2 2 ( ) 1 m r f r C s  § _ · ¨ ¸ © ¹ , (9)

ZKHUH&LVWKHQRUPDOL]DWLRQFRQVWDQWs is the size parameter and m is related to WKHREVHUYHGVORSHRIWKHVWDUGHQVLW\SURILOH

:LWKWKLVIXQFWLRQZHKDYHFDOFXODWHGSUHVHQW-day radial temperature distri-EXWLRQLQWKHIUHHVSDFHLQVLGHWZR*&V0DQG7XFDQH The last one, be-LQJRQHRIWKHODUJHVW0LON\:D\FOXVWHUKDVWKHFHQWUDOUDGLDWLRQWHPSHUDWXUHRI a.DERYHWKHSUHVHQW-GD\8QLYHUVHEDFNJURXQGWHPSHUDWXUH.7KRXJK temperatures across GCs are meaningless in the astrophysical modeling of stars HYROXWLRQKRZHYHUZHVXSSRVHWKDWWKHWHPSHUDWXUHJUDGLHQWSOD\VDJUHDWUROH RID³PRS´ZKLFKFOHDQVWKHYDFXXPLQVLGHWKH*&V7KDQNVWRLWs action and SHUKDSVVRPHJDVDFFUHWLRQE\ZKLWHGZDUIVZHKDYHDQLGHDOLQVLJKWLQWRWKH LQWHULRUVRI*&VE\WKH+675HFHQWGHQVLW\GHWHUPLQDWLRQRILRQL]HG gas (prob-ably the dominant component of the intra-cluster medium) by radio-astronomical REVHUYDWions of 15 pulsars in 47 Tucane yields ±FP-3 _{RQO\>@7KLV}

is about 100 times the free electron density of the interstellar medium in the YLFLQLW\RIWKLV*&6XFKDORZGHQVLW\LVXQGHWHFWDEOHE\RWKHUPHWKRGV

$&.12:/('*(0(176

7KH DXWKRUV ZLVK WR H[SUHVV WKHLU JUDWLWXGH WR 'U 3DROR 0LRFFKL IURP the 'HSDUWPHQW RI3K\VLFV DQG $VWURQRP\ 8QLYHUVLW\ RI %RORJQD ,WDO\ IRU FRm-PHQWVRQWKHPDQXVFULSWDQGKHOSLQDFFHVVWRUHFHQWOLWHUDWXUH:HDUHJUDWHIXO WR 'U7RPDV] 'XUDNLHZLF] Irom Los Alamos National Laboratory for correc-WLRQVRI(QJOLVK

5()(5(1&(6

1 SWUXYH2DQG=HEHUJV9 () Astronomy of the 20th _{Century0DFPLOODQ&R}

2 )HUUDUR)5Exotic Populations in Galactic Globular Clusters, in: The Impact

of HST on European Astronomy, Astrophysics and Space Science Proceedings,

6SULQJHU6FLHQFH%XVLQHVV0HGLD%9S

3 3LRWWR*Observational Evidence of Multiple Stellar Populations in Star

22 73,(ē.26$1'6+$à$6

4 &KDQGUDVHNKDU6 () Principles of Stellar Dynamics'RYHU1HZ<RUN 5 3OXPPHU+&(1911)On the problem of distribution in globular stars clusters,

0RQWKO\1RWHV/;;,, 5, -

 'LFNH5+ (1939) The radial distribution in globular clusters$VWURQRPLFDO-

1111, 108-

7 Catalogue of Milky Way Globular Cluster Parameters, KWWSZZZSK\VLFV PFPDVWHUFD*OREXODUKWPO

8 Catalog of Parameters for Milky Way Globular Clusters: The Database

Compiled E\:LOOLDP(+DUULV0F0DVWHU8QLYHUVLW\7KLVUHYLVLRQ'HFHPEHU

2010 KWWSSK\VZZZSK\VLFVPFPDVWHUFDaKDUULVPZJFGDW

9 Miocchi 3, Lanzoni % )HUUDUR ) 5 'DOHVVDQGUR (, Vesperini (, 3DVTXDWR 0, Beccari *, Pallanca &, and Sanna 1 (2013)Star count density profiles and structural parameters of 26 galactic globular clusters, The

Astrophysical Journal, 774, SS

10 1LQNRYLü 6 (1998) On the generalized Schuster density law, Serbian Astronomical Journal 158, 15-

11 äLYNRY 9 DQG 1LQNRYLF 6 (1998) On the generalized Schuster density law

and King’s formula , Serbian Astronomical Journal 158, 7-

12 9RQ/RKPDQQ: () Dichtegesetze und mittlere Sterngeschwindigkeiten in

Sternhaufen=$VWURSK\V60, 43-

13 McLaughlin 'HDQ ( DQG YDQ GHU 0DUHO 5RHODQG 3, Resolved massive star

clusters in the Milky Way and its satellites: brightness profiles and a catalog of fundamental parameters, The Astrophysical Journal Supplement Series 161,

304–'HFHPEHU 2005

14 Elson R $:)DOO60 DQG)UHHPDQK &(1987) The structure of young

star clusters in the Large Magellanic Cloud, The Astrophysical Journal 323,

54-

15 'LHGHULN .UXLMVVHQ - 0, 0LHVNH S The mass-to-light ratios of Galactic

Globular Clusters 7R DSSHDU LQ WKH SURFHHGLQJV RI *DOD[\ :DUV 6WHOODU

3RSXODWLRQV DQG 6WDU )RUPDWLRQ LQ ,QWHUDFWLQJ *DOD[LHV 7HQQHVVHH -XO\ 2009), DU;LY

 )UHLUH 3&, Kramer 0, Lyne $*, Camilo ), Manchester 51, '¶$PLFR 1 (2001) Detection of ionized gas in the globular cluster 47 Tucanae, The Astrophysical Journal 557, L105-/

17 .LQJ , 5 () The structure of star clusters. III. Some simple dynamical

models, Astronomical Journal 71, -

18 Naim 6*ULY ( (2012) Examining the M67 classification as an open cluster, IJAA 2-