Modelling individual globular clusters An Update Douglas Heggie University of Edinburgh d.c.heggie@ed.ac.uk
Outline 1. Introduction: two kinds of model two kinds of star cluster 2. Monte Carlo models M4 NGC 6397 (in progress)
Two kinds of dynamical models 1. Static models: build a model of the cluster now King's model and its variants Models based on Jeans' equations Schwarzschild models 2. Dynamical evolutionary models Fokker Planck models Monte Carlo models Fluid models N body models
Monte Carlo code: the basic idea Assume spherical symmetry. Orbit of star characterised by energy, E, and angular momentum, J. The code repeatedly alters E, J to mimic the effects of gravitational encounters, using theory of relaxation. Additional processes: binaries, tidal cutoff, single and binary star evolution.
Two kinds of surface brightness profiles Collapsed core profile Rising central surface brightness profile No definable core radius 17% of Galactic GC (+6%?) King profile Flat central surface brightness profile Well defined core radius 77% of Galactic GC Source: Trager et al (1995)
What determines the core radius? 1. Primordial 2. Mass segregation (primordial/dynamical), core collapse 3. Stellar evolution (As bright stars evolve and fade, we see the core radius of the less segregated stars of lower mass) 4. Primordial binaries An energy source which can stop core collapse... but it is a non renewable energy source 5. Stellar mass black holes (like 2) 6. Intermediate mass black holes
Two nearby examples M4 and NGC 6397 M4 (NOAO)
M4 NGC 6397 Surface brightness profile King type profile Collapsed core profile Distance from sun (kpc) Distance to Galactic Centre (kpc) Log Mass (M ) 1.72 5.9 4.8 2.6 6.0 5.0 Half light radius (pc) Tidal radius (pc) Binary fraction 2.2 16.01.02 2.2 12.01.02
Monte Carlo models of M4 Fixed initial parameters: Binary parameters (period, mass ratio, mass function), as in Hurley et al, 2005, A complete N body model of the old open cluster M67, MNRAS, 363, 293 Kroupa double power law IMF Plummer model Z = 0.002 Free initial parameters: Mass M Tidal and half mass radii r, r t h Binary fraction fb Slope of lower mass function (Kroupa: = 1.3)
Comparison with Observational data on M4 By trial and error the initial conditions t = 0: M = 3.61 105, fb = 0.06, = 0.7, rt = 35pc, rh = 0.58pc gave present day values t = 12Gyr: M = 5.07 104, fb = 0.047, = 0.11, rt = 18.2pc, rh = 3.23pc and a satisfactory fit to the following observational data Surface brightness profile (Trager et al, 1995, AJ, 109, 218) Radial velocity dispersion (Peterson et al, 1995, ApJ, 443, 124) V Luminosity function (Richer et al, 2004, AJ,127, 2771) at two radii
The core of the M4 model According to this model, M4 is a core collapse cluster with a normal core The core is sustained (we think) by binary burning
NGC 6397 : a very preliminary model
Comparison of the two cluster models M4 Mass (M ) t=0 12Gyr 3.61 105 5.07 104 Binary fraction 0.06 0.047 0.7 0.11 Tidal radius 35pc 18.2pc Half mass radius 0.58pc 3.23pc Time of core collapse 8.3Gyr [Fe/H] 1.20 NGC 6397 t=0 12Gyr 4.26 105 7.5 104 0.06 0.043 0.7 0.08 40pc 22.4pc 0.40pc 3.40pc 7.8Gyr 1.95 These models are rather similar at 12Gyr. Why is one profile collapsed and the other uncollapsed?
Why are the profiles so different? 1. Differences in initial conditions? Maybe Initial half mass density: M4 model NGC 6397 model 2.2x105 M 8.0x105 M compare R136 (LMC) central density > 5x104 M NGC 3603 central density > 106 M
Why are the clusters so different? 2. Fluctuations within one run? Probably not Surface brightness profile at 3 times in one model:
Why are the clusters so different? 2. Fluctuations between runs? Maybe. Surface brightness profile in four similar models: See also Hurley, 2007, MNRAS, 379, 93
Summary 1. We have constructed a Monte Carlo model of M4 which reasonably fits the surface brightness and velocity dispersion profiles, and the luminosity function at two radii. 2. According to this model, M4 is a collapsed core cluster with a measurable core radius. 3. We are attempting the same for NGC 6397. 4. Present working hypothesis: the difference between the two clusters is due to modest differences in their initial conditions and/or their detailed history.
My collaborator: Mirek Giersz
Another example of fluctuations within one run
M4
Model 2: luminosity functions
Why model globular clusters? Inferring the mass from surface brightness profile, radial velocities, 1. proper motions, mass functions. Can weigh dark components : 2. white dwarfs,... Inferring the global mass function from local mass functions: is this the same for all clusters? Inferring the primordial mass function from local, present day mass functions: correct for preferential escape of low mass stars Inferring primordial parameters of the binaries from present day abundance and period distribution: primordial abundance, period distribution, etc Measuring cluster distances by comparison of radial velocities and proper motions: correct for rotation, different observational fields, different observed components,... Determine the effect of dynamics on spectral energy distribution
Modelling globular clusters 1. Static models: Plummer's model (Plummer 1911) King's model (King 1966, Peterson & King 1975) Anisotropic models (King; Michie 1963) Multi mass models (Gunn & Griffin 1979; Meylan & Mayor et al; Pryor et al) Non parametric models (Gebhardt & Fischer 1995) Schwarzschild's method (van de Ven et al 2006) Jeans' equations (Leonard et al 1992) This list is by no means exhaustive These methods can be used to solve some problems but not all
Modelling globular clusters 2. Dynamic Evolutionary Models Gas/fluid models (Angeletti & Giannone 1980 [M3]) Fokker Planck models (Cohn and co workers 1997 [M15], 1992 [N6624]; Drukier 1993, 1995 [N6397]; Phinney 1993 [M15]) Monte Carlo model (Giersz & H 2003 [ Cen]) N body model (? 2015 [?]) Note: N body modelling of open clusters has only recently become feasible (see Hurley et al 2005 for M67; present day mass ~2000M ; initial mass 19 000 M ; 50% binaries; took 1 month.)
The Main Observational Constraints Surface brightness, star counts Radial velocities Pulsar accelerations Deep luminosity/mass functions Accurate proper motions Corrections to this list welcome von Zeipel (1908) onwards Illingworth (1976) onwards Phinney (1993), Grabhorn (1993) onwards 1995 onwards (HST) van den Bosch et al (2006)
A Monte Carlo model of M4 Collaborators Mirek Giersz (Warsaw) Jarrod Hurley (Swinburne) Simon Portegies Zwart (Amsterdam) Kristin Warnick (Edinburgh/Potsdam) Karen Meyer (Edinburgh)
The Monte Carlo code: additional dynamical processes 1) Galactic tidal field: treated as a spherically symmetric cut off 2) Binaries: Treated by reaction cross sections Binary single interactions: Spitzer 1987 Binary binary interactions: based on Mikkola 1983, 1984 Notes: There are significant differences between a tidal field and a tidal cutoff No triples, and so we cannot handle hierarchical triples Do not include physical collisions taking place during 3 and 4 body interactions Cross sections are thought to result in excessive binary activity (Fregeau)
Monte Carlo code: recent developments Addition of stellar evolution of binary and single stars Single stars: Hurley, JR, Pols, O, Tout, CA, Comprehensive analytic formulae for stellar evolution as a function of mass and metallicity, 2000, MNRAS, 315, 543 Binary stars: Hurley, JR, Tout, CA, Pols, O, Evolution of binary stars and the effect of tides on binary populations, 2002, MNRAS, 329, 897 Implemented via the McScatter interface: Heggie, DC, Portegies Zwart, S, Hurley, JR, 2006, McScatter: A simple three body scattering package with stellar evolution, NewA, 12, 20 (Also interfaces to stellar evolution package SeBa in starlab, but this is only for solar metallicity.)
The globular cluster M4 Distance from sun Distance from GC Mass 1.72 2.2kpc 5.9kpc 63 000M Core radius Half light radius Tidal radius Relaxation time (Rh) 0.53 pc 2.3 pc 21 pc 660 Myr Binary fraction [Fe/H] Age AV 1 15% 1.2 12Gyr 1.33 Selected as a MODEST target (Hamilton 2004)
A galactic orbit for M4 = M4 From Dinescu, DI, Girard, TM, van Altena, WF. 1999, Space Velocities of Globular Clusters. III. Cluster Orbits and Halo Substructure, AJ, 117, 1792
Model 1 t = 0: M = 2.49 105, fb = 0.1, = 1.3, rt = 47.3pc, rh = 1.80pc t = 12Gyr: M = 1.09 105, fb = 0.0844, = 1.06, rt = 35.9pc, rh = 4.89pc
Model 1: luminosity function Question: should we trust any model that fits only the surface brightness and velocity dispersion, i.e. any model before 1995?
Model 2 temp: Colour magnitude diagram
Model 2: photometric binaries
Model 2: binary semi major axes
Model 2: spatial distributions
Model 2: distributions of bright objects
Model 2: Summary and comparison Quantity Mass rh Model 2 temp 5.07 104 3.23pc Richer et al 6.25 104 3.06pc L MWD 2.63 104 2.00 104 6.25 104 MNS 3.55 103 } 3.25 104 Core collapse occurs at ~8Gyr; present core radius sustained by binary burning.
Finding initial conditions Each model takes a few days. You need A good starting guess A fast way of improving the model (e.g. scaled small models) The starting guess: use Quick Cluster Evolution. Given the initial conditions, you can Compute M(t) from formulae of Lamers et al, generalised to other Z and other IMF Compute r (t) from simple dynamical theory h Compute rc (t) from fit to Baumgardt & Makino and simple theory Compute (t) from M(t) via Baumgardt & Makino Now invert this process to iterate on initial conditions given data at 12Gyr
Conclusions 1. Monte Carlo models can provide similar results to N body models, with similar physics (binaries, stellar evolution, etc.), except for Use of a tidal cutoff Use of static tide (curable) Rotation Use of cross sections for triple/quad interactions (curable) Neglect of triples (?curable) 2. Monte Carlo models are feasible in reasonable time for globular clusters, which are too large for N body models. Yield predictions for mass segregation, distributions of binary parameters,... The only comparable method is the hybrid code of Giersz & Spurzem ( A stochastic Monte Carlo approach to modelling real star cluster evolution III, 2003, MNRAS, 343, 781, and references therein)
Future work 1. General repairs 2. Improved treatment of tide 3. A collapsed core cluster: NGC 6624 4. Replace cross sections by directly integrated binary single and binary binary encounters 5. Astrophysical implications: CVs, MS pulsars, etc We welcome any observational and theoretical comparisons for our models
The globular cluster M4 Distance from sun Mass 2.2kpc 63 000M Core radius Half mass radius Tidal radius Relaxation time (Rh) 0.53 pc 2.3 pc 21 pc 660 Myr
Recent work with the Warsaw Monte Carlo code Background (papers by M. Giersz) 1998, MNRAS, 298, 1239 Monte Carlo simulations of star clusters I. First Results 2001, MNRAS, 324, 218 Monte Carlo simulations of star clusters II. Tidally limited, multimass systems with stellar evolution 2006, MNRAS, 371, 484: Monte Carlo simulations of star clusters III. A million body star cluster The code incorporates many refinements invented by J.S. Stodołkiewicz (1982 6), which was in turn based on the method of Hénon (1971 5)
Monte Carlo code: calibration 2 2 G m n ln N Mass Time unit depends on relaxation 0.065 v 3 time tr = not well known for unequal masses. Half mass radius Calibration against N body model suggests 0.005 0.01 cf. Hénon 1975 Time (Myr)
Monte Carlo: comparison with N body model of M67 Source: Hurley, JR, Pols, OR, Aarseth, SJ, Tout, CA, 2005, A complete N body model of the old open cluster M67, MNRAS, 363, 293 Initial model: Number of single stars Number of binaries IMF Initial tidal radius Initial total mass 12000 12000 KTG 32.2 pc 19 340 M
Monte Carlo: comparison with M67 Data at 4Gyr M/M N body 2037 Binary frac 60% Half mass rad 3.8 pc Run time 1 month Monte Carlo 1974 48% 4.3 pc 68 min Remark: Low binary fraction because of excessive binary activity through use of cross sections?