The Many Possible Histories of Globular Cluster Cores John Fregeau (KITP), with Harvey Richer (UBC), Fred Rasio (NU), Jarrod Hurley (Swinburne)
Two Parallel Threads The strong correlation between X-ray binary number and cluster core encounter frequency for non core collapsed (CC) clusters, and the lack of correlation for CC clusters, implies that CC and non-cc clusters had distinctly different recent evolutionary histories (Fregeau 2008). Recent cluster observations are consistent with white dwarfs (WDs) receiving small (~5 km/s) kicks at birth. Such kicks could significantly alter the evolutionary history of most globular clusters (Davis, et al. 2008; Fregeau, et al. 2009).
X-Ray Binaries and Cluster Dynamics Globular clusters have been known for over 30 years to be overabundant in X-ray sources relative to the Galactic disk population, by unit mass (Clark 1975; Katz1975). Their X-ray populations include low-mass X-ray binaries (LMXBs), cataclysmic variables (CVs), millisecond pulsars (MSPs), and active binaries (ABs). It was quickly realized that the overabundance should be caused by the enhanced rate of dynamical encounters of binaries in the dense cluster cores (Verbunt & Hut 1987). Recent work has quantitatively confirmed this prediction, with the number of X-ray sources with LX >~ 4x10 30 erg/s in a cluster scaling almost linearly with its encounter frequency Γ (Pooley, et al 2003).
NX vs. Γ: Three Outliers are CC Clusters 100 6266 N X 10 Terzan 1 6397 7099 Omega Cen 6752 5904 47 Tuc 6626 6440 6093 6366 6121 1 1 10 100 Γ (normalized) Pooley, et al. 2003, Lugger, et al. 2007 (NGC 7099), Cackett, et al. 2006 (Terzan 1)
Three Phases of Cluster Evolution core contraction binary burning gravothermal oscillations
Theory Converges on the Binary Burning Heating Rate Since the binary-burning phase is so long-lived, it is widely believed that most globular clusters should currently be in this phase. Recently, two independent and very different numerical methods have been used to study cluster properties in the binary-burning phase. Remarkably they agree very well in the value of rc/rh predicted during the binary-burning phase (Heggie, et al. 2006, Fregeau & Rasio 2007). The new estimates of rc/rh in the binary burning phase are smaller than observations by a factor of >~10, implying that only the ~20% of clusters that are observationally classified as CC agree with theory.
Disagreement Between Theory and Observations in Binary Burning Phase r c /r h 1 0.1 0.01 M4 47 Tuc NGC 6397 Pal 5 Pal 13 NGC 288 M3 E3 NGC 6752 equilibrium core radius in binary burning phase 0 0.2 0.4 φ b all Galactic glob. clus. 0 5 10 15 20 N
Possible Resolutions of rc/rh Discrepancy Differing definitions of rc/rh can yield a factor of up to ~4 difference in the appropriate direction (Hurley 2007). Neglected physics in simulations important? Stellar evolutionary mass loss on long timescales (Hurley 2007), collisions of stars leading to expedited stellar evolution mass loss (Chatterjee, et al. 2007)? Additional energy sources: central IMBHs (Trenti 2006), prolonged mass segregation (Merritt, et al. 2004), evaporation of stellar-mass BH population (Mackey, et al. 2007)? Perhaps most clusters are simply not yet in the binary-burning phase (Fregeau 2008)? White dwarf birth kicks of ~5 km/s (Davis, et al. 2008; Fregeau, et al. 2009)?
The Standard Γ: NX α dnint/dt If X-ray binaries are dynamically formed, the number in a given cluster should scale with the binary interaction rate (Verbunt & Hut 1987). General interaction rate for two species: Γ dn int = n 1 n 2 σ 12 v 12 f(v 12 )d 3 v 12 d 3 r dt Approximate form: dn int ρ 2 dt crc/v 3 σ Approximate form implicitly assumes that binary fraction is constant among all clusters, all interactions are in core, compact object fraction is constant among all clusters, etc. Most importantly, this form also represents the current interaction rate.
Lifetimes of X-ray Binaries X-ray binaries are known to have finite visible lifetimes. For LMXBs, the lifetime can vary from ~10 5 to 10 7 yr for red giant donors, to ~1 Gyr for main-sequence companions, to a few Gyr for ultracompacts (Ivanova, et al. 2006). For CVs, the lifetime is roughly ~1 Gyr (Ivanova, et al. 2006). Additionally, the strong interaction that places a binary on the path to becoming an observable X-ray source typically occurs several Gyr before mass transfer starts (Ivanova, et al. 2006, 2007).
A New Γ: NX α Nint The number of visible X-ray sources should scale with the integrated number of interactions in the recent past. General interaction number for two species: Γ N int = n 1 n 2 σ 12 v 12 f(v 12 )d 3 v 12 d 3 r dt. This can be simplified as with the standard Γ, but keeping important factors and allowing for time evolution of core quantities: N int = f b f co 81 2a 4Gm t0 t l t 0 t l t x v 3 σr 1 c dt. This form is similar to standard Γ, but allows the properties of the core to vary with time, and leaves as parameters tx, the X-ray source lifetime, and tl, the lag time.
Three Phases of Cluster Evolution The three different phases of cluster evolution can be substituted into our new Γ. We exclude the gravothermal oscillation phase, since in this phase a cluster should have a core binary fraction of essentially zero, while all clusters for which it's measured show evidence for larger binary fractions. In the binary-burning phase the core properties are taken to be constant. For the core contraction phase we use simple self-similar collapse theory, in conjunction with the results of N-body simulations that show nearly universal behavior for many different binary fractions (Hurley 2007).
Binary-Burning vs. Core-Contracting The result is N int,bb N int,cc = t x 2.835 ( ) t 0.315 ( 0 t 0 t 0 +9t l ) 0.315 t 0 t 0 +9t l +9t x This expression has a minimum value of 2.0 and a maximum of 17.8 in the range tx=10-4 -3 Gyr, tl=1-10 Gyr, for t0=13 Gyr. For the canonical values of tx=1 Gyr and tl=3 Gyr with t0=13 Gyr, the value is 5.0. The three core collapsed clusters are the same three that have a significant overabundance in NX, from ~2 for M 80, to ~5 for NGC 6397, to ~20 for Terzan 1 (whose Γ value is rather uncertain, coincidentally). This suggests that the core-collapsed clusters are still in the binaryburning phase while the rest are still in core contraction.
Implications Many population studies assume constant core properties throughout cluster lifetimes. The numbers of dynamically-formed sources (e.g., blue stragglers, tidal capture binaries) can be scaled, but when feedback is important or the properties of the cluster itself are of interest (e.g., core binary fraction) a simple scaling may not work. Clusters with different initial conditions tend to approach common values of structural properties in the binary burning phase. Thus properties of observed clusters may be more strongly dependent on their initial conditions. Perhaps most anticlimactically, the many proposed explanations for cluster core energy sources are not needed, including intermediate-mass black holes, and non-traditional initial conditions.
Changing Gears... White Dwarf Birth Kicks?
White Dwarf Birth Kicks? NSs and BHs likely acquire kicks due to asymmetries during core collapse, but WDs don t undergo core collapse. WD progenitors may lose significant amounts of mass during the AGB phase, though. An asymmetry in the mass loss could increase the systemic velocity of the resulting WD. Unfortunately, the magnitude of the WD kick is likely too small for it to make an appreciable impact on WD proper motions or the WD scale height in the Galactic disk. But WD kicks would be evident in globular clusters, where vσ~few km/s.
Deep Observations of NGC 6397 (Richer, et al. 2007)
Deep, Clean Observations of NGC 6397 (Richer, et al. 2007)
MS, WD Distributions in NGC 6397 (Davis, et al. 2008)
Mass Segregation in Other Clusters (Richer, et al.) 47 Tuc M3
What Does It All Mean? Radial distributions of most stellar populations (MS, RGB, etc.) appear to agree with our understanding of mass segregation of relaxed populations. Younger WDs more radially extended in distribution than older WDs. Progenitors of WDs are ~0.8 M_sun stars, while WDs are ~0.5 M_sun. From mass segregation, expect younger WDs to be more radially concentrated than older WDs, which have had time to mass segregate. Perhaps WDs receive a systemic kick late in their evolution as stars? From kinematics of NGC 6397, inferred kick speed is ~3-5 km/s.
Where Does the Kick Come From? Asymmetric mass loss during the AGB phase? Asymmetry during the He flash? Observed WD rotation rates are consistent with non-axisymmetric mass loss at some point during evolution (Spruit 1998). Open clusters appear to be lacking WDs (e.g., Kalirai, et al. 2001), although counting of binaries containing WDs may be an issue.
Let s Test the Effect of Kicks with our Monte Carlo Code Our Monte Carlo code treats nearly all relevant cluster physics: relaxation, binary scattering, collisions, single and binary star evolution. It has been tested carefully and shows very good agreement with direct N- body, but because it is orbit-averaged is much faster. WD kicks are easy to include, since they re analogous to the NS kicks we ve been dealing with for years. We simply give a star a randomlyoriented kick of a fixed speed (2, 3,..., 9 km/s) when it becomes a WD. We expect that WDs substantially younger than the local tmass-seg will be more radially extended than the older WDs. Since the kicks are an energy source, we expect larger cluster core radii. Expect the effect to be most pronounced for clusters with vσ ~ vkick.
Initial and Final Models Initial model King W0=7.5 rvir = 5 pc N=3x10 5 1% binaries. Final model vσ, rc/rh (sort of), fb, fb,c consistent with NGC 6397 observations rh ~ 4x too large.
Long Term Evolution of Clusters with WD Kicks 0.25 0.2 0.15 6 km/s kick r c /r h 4 km/s kick 0.1 no kick 0.05 0 0 5000 10000 15000 t [Myr]
Radial Distributions of WDs, Generally Speaking no kick 6 km/s kick
Projected Radial Distribution of WDs in NGC 6397 Field (4km/s Model)
Salient Points Radial distributions of old vs. young WDs in models with kicks agree well with observations. (And don t agree for models with no kicks.) WD kicks comparable to the cluster velocity dispersion yield cluster core sizes ~10x larger than models without kicks at late times. In typical non- core-collapsed clusters WD kicks thus represent a possible resolution of the factor of ~10 discrepancy between observations and theory in rc/rh. However, quantitative statements about statistical significance require more detailed modeling of NGC 6397 and Monte Carlo sampling.
Predictions, Objections, Complications If WD kick speed is not correlated with cluster properties, expect: vkick<vσ (e.g., 47 Tuc): little to no effect observed vkick~vσ: older WDs more centrally concentrated than younger WDs, core larger than no-kick case vkick>vσ (e.g., NGC 288): large cluster core, missing WDs AGB wind speeds can be ~10 km/s (Marshall, et al. 2004). If the wind is retained by the cluster the effect of kicks may be weaker. AGB timescale can be >10 7 yr, but typical orbit timescale is ~10 5 yr. If kick occurs during AGB phase, the dynamical result would not be that of a kick, but a weaker, adiabatic modification of the orbit. Upcoming WD proper motion observations should help clarify the issue.
Many Possible Histories WD kicks Prolonged stellar evolution (Hurley 2007) cr soft binary destruction Standard evolution Initial expansion (Giersz & Heggie 2009) t
Summary Recent observations of XRBs and WDs in globulars suggest that a cluster can have had one of several possible recent evolutionary histories. The time integrated interaction rate (which should scale linearly with the number of visible XRBs, collisional blue stragglers, etc.) over the last ~Gyr can vary by a factor of >10 among possible histories. X-ray observations can be used to probe the evolutionary history, but are expensive (optical data also required). By their natural abundance, collisional blue stragglers offer insight into the evolutionary histories of clusters, if only the details of their formation and evolution were better understood.
Recent Wrinkles Current cluster core binary fractions are difficult to determine, but can be measured via, e.g., main-sequence fitting. The binary fraction ranges from a few % to ~30%. On the other hand, the observed binary fraction in low density environments, like the field or open clusters, is rather large (>~50%). MS method in NGC 6397 (Davis, et al. 2008) Recent simulations show that the core hard binary fraction generically increases dramatically with time (Fregeau, Ivanova, & Rasio 2009).
The Importance of Soft Binaries Hard binaries: binaries with binding energy greater than the local kinetic energy; tend to become more bound (harden) in scattering interactions. Soft binaries: binaries with binding energy less than the local kinetic energy; tend to become less bound (soften) in scattering interactions. Observed field binaries have orbital periods up to ~10 7 days. Binaries with orbits longer than ~10 3 days are soft in globular cluster cores. An initial fb=0.5 with a field distribution of orbital properties quickly becomes fb=0.15, so computational theorists tend to ignore soft binaries, since they re dynamically irrelevant. However, soft binary disruption is an energy sink, and will cause a cluster core to contract significantly. A simple analytical argument shows that for typical initial conditions, a cluster with core radius of ~2 pc can shrink to ~0.2 pc in just a few Myr.
The Importance of Soft Binaries r c [pc] 0.5 0.4 0.3 0.2 0.1 0 f b,c r c 0 0 10 20 30 40 50 60 70 80 90 100 t [Myr] 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 f b,c
Agreement Between N-Body and Monte Carlo 0.1 MC (N=100k) NB (N=16k) r c /r h 0.01 0.001 0 50 100 150 200 t [t rh ]
MC Code: Now With Stellar Evolution! r c /r h 0.5 0.4 0.3 0.2 NB, f b =0% NB, f b =5% NB, f b =10% MC MC (no SE) MC (no tide) 0.1 0 0 5000 10000 15000 t [Myr]
Neutron Star Birth Kicks Radio pulsar proper motions imply a large kick speed. Hobbs, et al. (2005)
Black Hole Birth Kicks Can evolve BH XRB backward in time to constrain BH kick speed. Kick clearly required for XTE J1118+480. Fragos, et al. (2008)