Finite-mass isothermal spheres and the structure of globular clusters

Transcription

1 Mon. Not. R. Astron. Soc. 280, (1996) Finite-mass isothermal sheres and the structure of globular clusters Jes Madsen* nstitute of Physics and Astronomy, University of Aarhus, DK-8000 Aarhus C, Denmark Acceted 1996 January 5. Received 1995 October 26; in original form 1995 March 17 ABSTRACT Utilizing the recently derived extension of the Maxwell-Boltzmann distribution to low occuation numbers by Simons, this investigation discusses the structure of stellar dynamical isothermal sheres. The resulting models, which resemble King models, constitute a new sequence of hysically well-motivated sherical equilibrium configurations of finite mass, and give nice fits to re-core collase globular cluster data. Key words: celestial mechanics, stellar dynamics - globular clusters: general - galaxies: ellitical and lenticular, cd - galaxies: fundamental arameters. 1 NTRODUCTON Globular clusters show much fine structure related to mass segregation, velocity anisotroies, core collase, etc., but as a first aroximation their structure (at least rior to core collase) is surrisingly well described by simle King models based on truncated isothermal sheres (see Elson, Hut & nagaki 1987 for a review). Globular clusters are small enough for two-body encounters among the stars to laya significant role in their evolution, so Maxwell-Boltzmann statistics (isothermal sheres) might be exected to be relevant for them, as in the case of elastically scattering molecules in a gas. Ellitical galaxies are too large for two-body encounters among the stars to be imortant, but, if they formed through rocesses resembling violent relaxation, Maxwell-Boltzmann statistics could laya role there also (see Binney & Tremaine 1987 for an overview and references on these toics). t has long been recognized that a self-gravitating stellar dynamical system described by a Maxwell-Boltzmann distribution of the form f(e) ex ex ( - /3E), where E is the stellar energy, has infinite mass, and therefore can at most rovide an aroximate descrition of, e.g., globular clusters or ellitical galaxies. n the context of model building, this has been remedied in several ways in terms of introducing cut-offs in the distribution function - most notably the King models (King 1966) based on the lowered isothermal shere with fk(e) ex [ex( - /3E) - ex (- /3<1>0)], where <1>0 is a constant discussed below. King models fit the observational roerties of globular clusters very well, and to a lesser extent * jesm@dfi.aau.dk ellitical galaxies, but their ad hoc nature is slightly disturbing. Recently Simons (1994) derived an exact exression for Maxwell-Boltzmann statistics by a method that correctly accounts for discreteness effects, by allowing macro cells in hase sace to be occuied by integral numbers of articles only. The resulting distribution, which forms the basis for the stellar dynamical alications in the resent investigation, is given by where [b] is the value of b rounded down to the nearest integer, F and g are the number of articles and the number of available states with energy E, anda and /3 are constants to be determined from the conservation equations F=N, m FE=Eto" where Nand Etot denote total number of articles and total energy, and m is the article mass. n the resent investigation all articles are assumed to be of equal mass, and the word 'energy' is used as a short synonym for energy er unit mass, excet Eto" which is a true energy. t can be shown exlicitly (Simons 1994) that this distribution maximizes entroy under the imlicit assumtion of fixed energy levels. For self-gravitating systems no maximum entroy state exists, since the entroy can be increased by concentrating the core and letting some articles go into a diffuse halo. However, in the case of, e.g., globular clusters undergoing slow evolution dominated by two-article encounters (i.e. rior to more violent henomena such as core collase) the gravitational otential is almost time indeendent, and stars can be exected to adjust into a distribution close to that which maximizes the entroy for a fixed otential. This is the distribution given in equation (1). (1) 1996 RAS

2 1090 J. Madsen n the following sections equation (1) will be discussed in the context of stellar dynamics, fits of globular cluster data will be obtained, and the fundamental arameters and assumtions in the model will be described. 2 THE POSSON EQUATON FOR SMONS MAXWELL-BOLTZMANN STATSTCS To find the structure of a sherically symmetrical stellar system described by equation (1) one has to solve Poisson's equation in the form -- 1 d ( 2 r - d<l» = 41tG, r2 dr dr where the density is given by = J (E) d 3 v, and 1 denotes a mass er length cubed er velocity cubed; for stars of identical mass such a coarse-grained distribution is obtained from equation (1) by multilying with 1//g, where 1/ is the hasesace density of a microcell occuied by one star. Macrocells will be constructed so as to contain the same number of microcells, go t is customary to introduce a relative gravitational otential, 'P, related to the true otential <> via 'P == - <> + <1>0' and a relative energy e == - E + <1>0 = 'P -!v 2, with the constant <1>0 chosen so that > 0 for e > 0, and 1=0 otherwise. Then 'P satisfies Poisson's equation of the form V2'P = - 41tG with 'P -+ <1>0 for r -+ 00, or more exlicitly 2-1 d ( r2_ d'p) = -161t f' 2 G l(e)[2('p-e)j112 de, r dr dr o with (e) =!i [ex (Pe)]. (4) g This corresonds to <l>o=ln (Ag)/. ntroducing a dimensionless radius s ==r/a, where a == [P12g/(G1/)]12, and scaling otentials and energies in units of, one finally gets a dimensionless Poisson equation (2) (3) The dimensionless otential decreases monotonically and aroaches a negative constant, P<>o, for s The corresonding density [in units of 1//(g312)] decreases from a central value, *, and reaches 0 when P'P = 0 at the dimensionless tidal radius, St. This is the major difference relative to the 'traditional' isothermal shere, where the density only asymtotically aroaches zero. Thus the total mass, M = J 41tr2 dr, is finite. To illustrate the imortance of discreteness effects, consider first the simlified situation where the hysical gravitational otential as a function of radius, <>(r), is fixed a riori [with <>(r) -+0 for r-+ 00]. n this case one can secify a artitioning of (x, v) hase sace, including secification of energy levels, E, with equal numbers of available states, go One may choose the numbering of levels such that E is the lowest level, i.e. closest to <1>(0), etc. Energy level is occuied by F articles, where F=[Ag ex (- E)] ==[ex(pe)]] == [ex (P<>o) ex ( - E)] == [ex [P'P(O)] ex {P [<1>(0) - E]}], using the identities Ag == ex (P<>o) and e == <1>0 - E == 'P(O) + <1>(0) - E. The classical isothermal shere has (fractional) occuation of energy levels all the way to E-+O for (with articles reaching r -+ 00), so, to rescale the gravitational otential, one would normally use <1>0=0 and 'P(r) = - <> (r). For the true isothermal sheres of the resent investigation there is a finite number of articles, and levels are not fractionally occuied, so only levels with below some finite Pmax contribute, corresonding to E 5:. Em" < O. Choosing <1>0 = EPmox the rescaled otential 'P (r) == <1>0 - <> (r) now has the roerty that it is ositive for exactly that range of radii where articles with <1>(0) 5:.E 5:. EPmox ['P(O) ;;::: e;;::: 0] are able to move. The total mass M and hysical energy (not er unit mass) Etot are given by M==mN=m F=m [ex [P'P(O)] ex fp[<>(o)-e]}], (6) ~ ~ [S2 d(p'p)] S2 ds ds = -161t2 f~ [ex (Pe)] [2(P'P - e)]112 d(e) = -161t 2 C~ [:j+l) j[2(p'p - e)j112 d(e) + [~m" jmax[2(p'p - e)j112 d(e)} t2 jm" = --- (P'P _lnj)312. (5) 3 j~1 Herejmax=[ex (P'P)]. A solution of the dimensionless Poisson equation exists for each choice of scaled central otential, P'P (0), subject to the boundary condition P'P'(O) =0 (no force at the centre). and Etot=m EF=m E[ex [P'P(O)] ex {P[<>(O) -E]}]. (7) f P were fixed, then a system with fixed total mass M could be realized for given article mass, m, with that choice of 'P(O) which obeys equation (6). Clearly, lowering m (and thereby increasing the number of articles, N) corresonds to icking a higher value of 'P(O) to increase the sum. This is equivalent to choosing the (negative) value <1>0 closer to 0, allowing articles to occuy states with higher 'quantum number'. Thus, a decrease in article mass is directly related to an increase in the scaled otential, P'P, for fixed total mass. As one sees from the figures, solutions with higher dimensionless central otential P'P(O) (corresonding, as just argued, to more articles for fixed total mass, hysical otential and hase-sace artitioning) are,

3 as exected, closer to the standard isothermal shere solution. That is, discreteness effects are less consicuous for large N. n reality, would not be fixed in this rocedure since the solution must also obey the equation for E,o" but since (for dimensional reasons) E'o,=M-lh[P'P(O)], where h is a slowly varying function of P'P(O), does not change much if M and E,o' are fixed. For the self-consistent case where the otential <1>(r) is given by Poisson's equation rather than fixed a riori, the icture is more comlicated. However, <1>(r) is not strongly varying with N for fixed M and E,o" so qualitatively the argument above remains valid - discreteness effects are more ronounced for low N, and for fixed M and E,o, (and artitioning of hase sace) a change in m (N) corresonds to a change in 'shae arameter' P'P(O) for the isothermal sheres. 3 COMPARNG WTH OBSERVATONS As exlained above, the new Maxwell-Boltzmann statistics in a natural way account for finite-mass systems. However, many new arameters aear (A, g' '1, P), so it is not obvious that much has been gained in terms of constructing useful stellar dynamical models. t will now be demonstrated how these arameters are related to observable roerties of, e.g., globular clusters. Fig. 1 shows the density rofiles for several choices of P'P(O). Densities are scaled in units of the central density, and radii in units of the King radius, Finite-mass isothermal sheres 1091 J tPG(0) = 41t* a. Also shown in Fig. 1 is the standard isothermal shere, which the resent models aroach for P'P(O) -> 00, and the corresonding King models. The similarities are striking - not surrisingly, since King models also have an energy cutoff at E = <1>0 and an element of rounding down {fkoc[ex (e) -1] rather than [ex (e)]}. Equally similar are the root-mean-square velocities at a given satial radius in Fig. 2. {For the full Simons Maxwell-Boltzmann statistics one can show that The velocities shown in Fig. 2 are.jv 2 (r)p3}. Fig. 3 shows the so-called concentration arameter defined as c =:= loglo (r,lro). The value of c is tyically 0.3 units lower than the corresonding King model with the same P'P(O). King models fit globular cluster data quite well for 0.75 < c < 1.75 and ellitical galaxies moderately well for c > 2.2 (Mihalas & Binney 1981; Binney & Tremaine 1987). Similar limits are reasonable for the isothermal sheres. This corresonds to 5 ::s; P'P(O) ::s; 9 and P'P(O) ~ 11 for the isothermal sheres. Each. choice of dimensionless central otential, P'P(O), leads to values for the dimensionless central density, * =:= (0)gP3/21'1, the dimensionless total mass, M*=:=MGla, (8) (9)! > ~-L~~~~~~~~~~'~~~iW~-M~ r/ro Figure 1. Density rofiles of isothermal sheres with central otential l:jl(o) =3, 6, 9,12 (solid curves, from lower to uer) and 00 (dotted curve; equivalent to the classical, infinite-mass isothermal shere). For comarison, the corresonding King models are shown for the same choices of l:jl(o) (dashed curves) r/ro Figure 2. One-dimensional root-mean-square velocity disersion rofiles for the same models as deicted in Fig. 1. VRMS =.Jv 2 (r) P/3.

4 1996MNRAS M 1092 J. Madsen 4 3 """",,,,,,,,,,,,,,,,,,,,",,,,, o~~~~~~~~~~~~~~~~~ o ~+(O) Figure 3. The concentration arameter c =loglo(r.lro) as a function of the dimensionless central otential for isothermal sheres (solid curve) and King models (dashed curve). the dimensionless tidal radius, s,=r.ja, and the dimensionless, central line-of-sight velocity disersion, 2 S' ['P(s) _lnj]5/2 ds o J a*2= a2= 5 S' ['P(s) _lnj]3/2 ds o J Exlicitly exressing observable quantities in terms of '1 [in units of Mo c-3 (an S-)-3], P [in units of (an S-)-2], and g' one gets (O)=P*'1g- 1 P-3/2Mo c- 3, (10) M=3.546 X 103M*-3/4'1-/2g~/2 r,=15.25s,1/4'1-/2g~ c, M o' (11) (12) and a2=a*2-1 (km S-)2. (13) Notice that '1 and g always occur in the combination '1/g [g is further couled toa via <l>o=ln(ag)/, but this can be considered as a consistency equation for A]. A tyical model with P'P(O) = 8, which gives a good fit to the surface brightness structure of globula>: clusters, has s,=0.153, * =4.677 x 10 M* =0.1290, a*2= For M=6 x 10 5 Mo and a2=(7 an S-)2 this gives -=(7.126 km S-)2, '1/g =2.10 x 10-4 Mo c- 3 (an S-)- corresonding to a tidal radius of 60.3 c and a central density of 3.6 x 103 Mo c-3. Choosing '1 to be higher than the maximal (central) hase-sace density of the system ['1 > 10 Mo c-3 (an S-)-3] means that g < 5 X 10 4 Thus, statistical mechanics gives reasonable results for a globular cluster of 6 x 1051-Mo stars if one divides hase sace into microcells smaller than ~ 0.1 c3 (an S-)3 and into macrocells containing of the order of 10 4 or fewer microcells. 4 DSCUSSON AND CONCLUSONS t has been demonstrated in this investigation that a sherical stellar dynamical equilibrium configuration described by Maxwell-Boltzmann statistics, contrary to the 'standard' isothermal shere, has finite mass. This is because hasesace cells can be occuied by an integral number of stars only, rather than the continuous distribution imlicit in the classical Maxwell-Boltzmann distribution. The comlete Simons-Maxwell-Boltzmann distribution (Simons 1994) exlicitly takes this into account, and in stellar dynamical alications this leads to finite mass because one does not integrate fractionally occuied cells over infinite sace. A new class of hysically well-motivated models for globular clusters were introduced based on the new distribution function. They resemble King models because of some common features such as an energy cut-off and a lowering of the classical Maxwellian, but in the King models these features are not included in a comletely correct manner [King models still allow fractional occuation of hase-sace cells, and the lowered isothermal shere does not maximize entroy, which equation (1) does for fixed gravitational otential, total energy and mass, resembling the situation for globular clusters rior to core collase]. The new models resented here fit the overall roerties of globular clusters rior to core collase well, for aroximately the same choice of central otential as for King models. They have constant 'temerature' (P-), but, in contrast to the infinite-mass isothermal shere, this does not reflect itself in constant velocity disersion (cf. Fig. 2). For large central otentials the models get closer to the standard isothermal shere, and the velocity disersion gets closer to. For very low central otential {jrnax = [ex (P'P)] = 1} the models go over into n = 3/2 olytroes, but this is far below the range of otentials giving reasonable fits to globular clusters. Any use of distribution functions in stellar dynamics involves an (often imlicit) assumtion about the 'graininess' of hase sace and a choice of unit cell. Contrary to quantum mechanics, where Planck's contant defines a fundamental cell size, there is no natural choice in stellar dynamics. n fact there are even several 'schools' arguing for a choice related either to the observational resolution (interreting entroy as a measure of information content), or to the tyical hase-sace size of elements which are not disruted by interactions over the lifetime of the system studied. The arameters extracted from fits to observations for the resent model include measures of the fundamental hase-sace density, '1, and the number of microcells in a macrocell, g, but only in the combination '1/g [which can also be interreted as the mass of a fundamental article (star) divided by the volume of a macrocell]. This could indicate that there is a hysically motivated relation between the central otential (or more generally the structure of the cluster) and the scale on which mixing in hase sace takes lace. As discussed above, values giving fits to 1996 RAS, MNRAS 280,

5 globular cluster data seem reasonable, and are not very deendent on the choice of model, as long as /3'1'(0) is not too small. ACKNOWLEDGMENTS thank Jens Hjorth for useful discussions. This work was suorted by the Theoretical Astrohysics Center under the Danish National Research Foundation. REFERENCES Finite-mass isothermal sheres 1093 Binney J., Tremaine S., 1987, Galactic Dynamics. Princeton Univ. Press, Princeton Elson R., Hut P., nagaki S., 1987, ARA&A, 25, 565 King. R., 1966, AJ, 71, 64 Mihalas D., Binney J. J., 1981, Galactic Astronomy, 2nd edn. Freeman, San Francisco Simons S., 1994, Am. J. Phys., 62, 515