Galaxies Part II
Observations of Globular Clusters
King Models Ja↵e Model Stellar Dynamics and Structure of Galaxies Systems with anisotropic velocity distributions Star Clusters Modelling Cluster Evolution
Globular cluster evolution Vasily Belokurov [email protected]
Institute of Astronomy
Lent Term 2016
1/93 Galaxies Part II Outline I Observations of Globular Clusters 1 Observations of Globular Clusters
King Models The Great Debate
Ja↵e Model Galactic Structure Systems with anisotropic Properties of globular clusters velocity distributions Star Cluster Formation Modelling Cluster Evolution Star Cluster Disruption Globular cluster evolution 2 King Models 3 Ja↵e Model 4 Systems with anisotropic velocity distributions 5 Modelling Cluster Evolution 6 Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping 2/93 Galaxies Part II Outline II Observations of Globular Clusters Encounters with Binary Stars
King Models Binary Formation through inelastic encounters Ja↵e Model Finally Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
3/93 Galaxies Part II Globular Clusters Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models Globular clusters are approximately spherical, compact agglomerations of stars Ja↵e Model swarming around galaxies. They are as old as the galaxies themselves and provide Systems with anisotropic velocity distributions unique clues to how the star formation and the galaxy formation proceeded in the Modelling Cluster ancient Universe. Evolution Globular cluster evolution Being old, spherical and devoid of Dark Matter, globular clusters are perfect environments to learn about the gravity works.
4/93 Galaxies Part II
Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
M15, Hubble 5/93 Galaxies Part II
Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Top row: Messier 4 (ESO), Omega Centauri (ESO), Messier 80 (Hubble) Middle row: Messier 53 (Hubble), NGC 6752 (Hubble), Messier 13 (Hubble) Bottom row: Messier 4 (Hubble), NGC 288 (Hubble), 47 Tucanae (Hubble)
6/93 Galaxies Part II The Great Debate Observations of Globular Clusters 26 April 1920, in the Baird auditorium of the Smithsonian Museum of Natural History. The Great Debate Galactic Structure Properties of globular “The Scale of the Universe” clusters Star Cluster Formation Harlow Shapley Heber Curtis Star Cluster Disruption King Models The nebulae are part of the large Milky The nebulae are “Island Universes” located
Ja↵e Model Way system at large distances away from the Galaxy
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
7/93 Galaxies Part II The Great Debate Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models Vesto Melvin Slipher measured wavelength shifts of spiral nebulae Ja↵e Model • Systems with anisotropic Johannes C. Kapteyn advertised a small Milky Way centered on the Sun velocity distributions • Modelling Cluster Adriaan van Maanen measured apparent rotation of spiral galaxies Evolution • Knut Lundmark speculated that the dwarf novae might be so bright as to be Globular cluster evolution • detectable millions of light years from us
8/93 Galaxies Part II The Great Debate Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model Shapley believes the Galaxy is tens of kpc across. “If spiral galaxies are like Systems with anisotropic • velocity distributions islands, then the Galaxy is a continent” Modelling Cluster Evolution Curtis’s Milky Way is much smaller, only 10 kpc across Globular cluster evolution •
9/93 Galaxies Part II The Great Debate Observations of Globular Clusters Resolved F, G and K stars in globular clusters. Shapley: these are giants, Curtis: The Great Debate • Galactic Structure these are dwarfs. Properties of globular clusters Cepheids as distance indicators. Shapley: there is P-L relation. Curtis: parallaxes are Star Cluster Formation • Star Cluster Disruption too small, “more data needed”. King Models Spectroscopic parallaxes. Shapley: these are to be trusted. Curtis: to be trusted only • Ja↵e Model within 100 pc. Systems with anisotropic Interpretation of the star counts. Curtis: star counts suggest small MW, but would velocity distributions • Modelling Cluster place the dust outside the stellar disk. Shapley: did not address the issue. Evolution Distribution of spiral nebulae on the sky. Shapley: “single system” - all is possible. • Globular cluster evolution Curtis: “neither impossible nor implausible” for the MW to have a dust ring around it. Nova brightness at maximum. Shapley: the implied real brightnesses would be totally • ridiculous. Curtis: trust the calibration based on a handful of MW events. Large positive velocities of spiral nebulae. Shapley: repulsion by radiation pressure • form the MW. Curtis: “I haven’t a clue” Central location of the Sun. Shapley: an illusion caused the local star cloud. Curtis: • this is God’s own truth Rotational proper motions of spirals as measured by van Maanen. Shapley: a • strong argument against opponent. Curtis: too dodgy 10 / 93 Galaxies Part II The Great Debate Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model The outcome?
Systems with anisotropic velocity distributions Both were mainly right, but also impressively wrong in some very important points. • Modelling Cluster Right when relying on their own data, and wobbly when attempting to speculate Evolution • Globular cluster evolution based on the current theory.
11 / 93 Galaxies Part II The Great Debate Observations of Globular Clusters The State of the Art The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
12 / 93 Galaxies Part II The Great Debate Observations of Globular Clusters The State of the Art The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
13 / 93 Galaxies Part II The Great Debate Observations of Globular Clusters The State of the Art The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Trumpler’s Galaxy with the Sun in the middle of Kapteyn Universe 14 / 93 Galaxies Part II The Great Debate Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models Ja↵e Model Shapley: The Sun lies in a nondescript location in the Galaxy. Systems with anisotropic velocity distributions
Modelling Cluster This is a key part to a bigger Copernican Principle, that we live neither in the centre of Evolution the Solar System, our Galaxy, nor the Universe as a whole. Globular cluster evolution
15 / 93 Galaxies Part II The Role of Star Clusters Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models Ja↵e Model Galaxy structure Systems with anisotropic • velocity distributions Galaxy formation • Modelling Cluster Evolution Stellar evolution • Globular cluster evolution Star formation •
16 / 93 Galaxies Part II Galactic Structure Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Portegies Zwart et al, 2010
17 / 93 Galaxies Part II Galactic Structure Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
M. Irwin and PAndAS collaboration 18 / 93 Galaxies Part II Number of Globular Clusters per galaxy Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic Milky Way: 100 < N < 200 velocity distributions • Andromeda: 200 < N < 500 Modelling Cluster • Evolution M 87: N > 1000 Globular cluster evolution •
19 / 93 Galaxies Part II Number of Globular Clusters per galaxy Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
M87 globular clusters 20 / 93 Galaxies Part II Curious GC in M87 Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Caldwell et al
21 / 93 Galaxies Part II Curious GC in M87 Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Caldwell et al
22 / 93 Galaxies Part II Color-Magnitude Diagram Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
23 / 93 Galaxies Part II Stellar Luminosity Function Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
24 / 93 Galaxies Part II Color-Magnitude Diagram Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
25 / 93 Galaxies Part II Cluster Ages Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular M4 CMD clusters Star Cluster Formation Open VS globular ages. Easy! Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
10, 11, 12 and 13 Gyr isochrones
26 / 93 Galaxies Part II Cluster age resolution Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
27 / 93 Galaxies Part II Surface brightness profile Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Surface brightness of 38 Galactic globular clusters imaged with the HST 28 / 93 Galaxies Part II Metallicity distribution Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Metallicity distribution of the Galactic globular cluster system
29 / 93 Galaxies Part II Age-Metallicity relation Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Age-Metallicity for the Galactic globular cluster from Dotter et al 2011 30 / 93 Galaxies Part II Star Cluster Formation Observations of Globular Clusters The Great Debate z=0 Galactic Structure z=12 Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
Moore et al, 2006 31 / 93 Galaxies Part II Star Cluster Formation Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
O. Gnedin, A. Kravtsov 32 / 93 Galaxies Part II Star Cluster disruption Observations of Globular Clusters The Great Debate Galactic Structure Properties of globular clusters Star Cluster Formation Star Cluster Disruption King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
The tails of Palomar 5, (Odinkirchen et al)
33 / 93 Galaxies Part II King Models Observations of Globular Clusters Towards models of real star clusters. Lowered isothermal models
King Models Ja↵e Model We seek models which look like an isothermal sphere at small radii (i.e. mostly large Systems with anisotropic ), but are less dense at large radii (smaller ). velocity distributions E E Modelling Cluster Evolution 2 3 / 2 f ( )=⇢1(2⇡ ) 2 eE isothermal Globular cluster evolution E So get rid of the high v stars, since they mostly escape - and this is equivalent to choosing in the definition to give a distribution function of the form we would like. 0 E What we might try is to truncate the Gaussian we had before, so that
2 3 / 2 ⇢ (2⇡ ) 2 eE 1 > 0 f ( )= 1 E (7.1) E ( 0 ⇣ ⌘ 0 E These are the King models.
34 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
35 / 93 Galaxies Part II King Models Observations of Globular Clusters 2 3 / 2 King Models f ( )=⇢ (2⇡ ) 2 eE 1 ↵E 1 Ja↵e Model ⇣ ⌘ Systems with anisotropic As usual, find density at any radius by integrating over all velocities velocity distributions ⌦ Modelling Cluster p2 1 2 Evolution 4⇡⇢1 2 v 2 ⇢( ) = 3 exp 2 1 v dv (7.2) Globular cluster evolution (2⇡ 2) 2 Z0 " ! # p 4 2 = ⇢ exp erf 1+ (7.3) 1 2 ⇡ 2 3 2 " ✓ ◆ ! r ✓ ◆#
where x t2 erf(x)= e dt. Z0 Then Poisson: d d r 2 = 4⇡Gr 2⇢( ) (7.4) dr dr ✓ ◆ d Solve this numerically with boundary conditions = (0) and dr =0atr = 0. 36 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models d d Ja↵e Model As we integrate the ODE (7.4) outward, dr decreases because initially dr =0and 2 Systems with anisotropic d p velocity distributions dr 2 < 0. As decreases towards zero the range of values of v (0, 2 ) falls and as p Modelling Cluster 0 the number of stars 0and⇢ 0since⇢ = 2 fv 2 dv. Evolution ! ! ! 0 Globular cluster evolution Eventually at some radius rt reacheszeroandthedensityvanishes.R rt is called the tidal radius of the cluster.
From ⇢(r)=⇢( (r)) with the solution to (7.4) we can compute the mass inside the rt 2 tidal radius M(rt)=4⇡ 0 r ⇢Kdr,andhence R GM(rt) (rt)= rt
37 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution Now (0) = (r ) (0) since we set t Globular cluster evolution = 0 at the outer edge, and = +constant. The bigger the value (0) from which we start our integration, the greater will be the tidal radius, the total mass, and (0) . | |
38 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
39 / 93 Galaxies Part II King Models Observations of Globular Clusters King Models As defined before Ja↵e Model
Systems with anisotropic velocity distributions ˜r = r/r0
Modelling Cluster Evolution where Globular cluster evolution 9 2 r0 = s4⇡G⇢0
i.e. r0 stheradiusatwhichthe projected density falls to roughly half (in fact, 0.5013) of its central value. Sometimes r0 is called the core radius in analogy with the usual observational definition.
40 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Note also that r0 = rc ,whererc is the core radius6 which contains half the (projected) light, i.e. 1 where ⌃(rc )= 2 ⌃(0) (see early lecture notes).
41 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions We can define the concentration Modelling Cluster of the model Evolution Globular cluster evolution r c = log t 10 r ✓ 0 ◆ and the models are characterised by c or dimensionless (0)/ 2. For globular clusters • c =0.75 - 1.75 For elliptical galaxies c 2.2 •
42 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
43 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
44 / 93 Galaxies Part II King Models Observations of Globular Clusters NGC 2419
King Models Ja↵e Model Upper panel: V-band surface brightness Systems with anisotropic velocity distributions profile of NGC 2419 as determined by
Modelling Cluster Bellazzini (2007). The blue, short-dashed Evolution curve shows the best-fitting single-mass, Globular cluster evolution isotropic King model. This model cannot fit the “excess” light at large radii. The dotted and long-dashed red curves are fits of two S´ersic profiles that fit the outer parts of the cluster well but not the central region. The solid curve is a S´ersic profile with an added
core of size rc =14arcsec.Itprovidesagood match to the density profile and is within the reported observational uncertainties at most radii (bottom panel).
45 / 93 Galaxies Part II King Models Observations of Globular Clusters NGC 2419
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
46 / 93 Galaxies Part II King Models Observations of Globular Clusters NGC 2419
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
47 / 93 Galaxies Part II King Models Observations of Globular Clusters NGC 2419
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
48 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions The velocity dispersion in King models can also be computed by Modelling Cluster Evolution p2 1 v 2 Globular cluster evolution 2 4 0 exp 2 1 v dv v 2 = p2 ⇣ h 1 v 2 i ⌘ R exp 2 1 v 2 dv 0 2 R ⇣ h i ⌘ 2 Note that v 0atrt since the potential energy is already equal to the largest allowed energy! there.
49 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
50 / 93 Galaxies Part II King Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster In this model Evolution r = ✓ = Globular cluster evolution Observationally, however, the velocity dispersion in the outer parts is found to be 2 2 non-isotropic - there is a tendency for vr /v to increase with radius. Therefore f is not just a function of . We’ll return (briefly) to anisotropic velocity distributions later. E
51 / 93 Galaxies Part II Ja↵e Model Observations of Globular Clusters This is an application of Eddington formula (to show it can indeed be done). King Models
Ja↵e Model Consider the family of double-power law models Systems with anisotropic ⇢ velocity distributions ⇢(r)= 0 Modelling Cluster (r/a)↵(1 + r/a) ↵ Evolution
Globular cluster evolution = 4 Dehnen (Dehnen 1993) • ↵ =1, = 4 Hernquist (Hernquist 1990) • ↵ =2, =4Ja↵e (Ja↵e 1983) • ↵ =1, = 3 NFW (Navarro, Frenk & White 1993) • 1 <↵<1.5, 3fordarkhaloes • ' So, the density is given by
2 2 M r r M r 4 ⇢(r)= 1+ = J (7.5) 4⇡r 3 r r 4⇡r 3 r 2(r + r )2 J ✓ J ◆ ✓ J ◆ J J
52 / 93 Galaxies Part II Ja↵e Model Observations of Globular Clusters Then we can use Poisson’s equation to find (r): King Models GM r Ja↵e Model (r)= ln = (7.6) Systems with anisotropic rJ r + rJ velocity distributions ✓ ◆ Modelling Cluster Therefore Evolution r rJ Globular cluster evolution =exp = e r + r GM J ✓ ◆ where = rJ .Then GM r e = r 1 e J r 1 1+ = r 1 e J Hence 4 4 4 M rJ M 1 e M 1 1 2 2 ⇢( )= 3 2 2 = 3 2 = 3 e e (7.7) 4⇡rJ r (r + rJ ) 4⇡rJ e 4⇡rJ ⇣ ⌘ 53 / 93 Galaxies Part II Ja↵e Model Observations of Globular Clusters M King Models Let ⇢0 = 3 ,andthen 4⇡rJ Ja↵e Model ⇢ 1 1 4 2 2 Systems with anisotropic ⇢˜ = = e e velocity distributions ⇢0 Modelling Cluster ⇣ ⌘ Evolution
Globular cluster evolution Remember the Eddington’s formula
2 1 E d d ⇢ 1 d⇢ f ( )= + 2 2 E p8⇡ " 0 p d p d =0# Z E E ✓ ◆
d⇢ Note that d = 0, so we have =0 2 1 E d ⇢ d f ( )= +0 2 2 E p8⇡ 0 d p Z E
54 / 93 Galaxies Part II Ja↵e Model Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions 2 4 Modelling Cluster 1 d ⇢ d M 1 1 r f ( )= E +0 ⇢( )= e 2 e 2 = J Evolution ↵ p8⇡2 0 d 2 p ↵ 4⇡r 3 GM E E J ⌥ ⌅ Globular cluster evolution R ⇣ ⌘ Then⌦ we can use ⌦ ⌃ ⇧
2 2 d ⇢˜ d 2 2 = e 4e +6 4e + e d 2 d 2 2 2 =4e ⇥ e e + e ⇤ ⇥ ⇤
55 / 93 Galaxies Part II Ja↵e Model Observations of Globular Clusters 4 1 d 2⇢ d M 1 1 r King Models E 2 2 J f ( )= 2 d 2 +0 ↵⇢( )= 3 e e = GM ↵ p8⇡ 0 p 4⇡rJ ⌥ ⌅ Ja↵e Model E E ⇣ ⌘ Systems with anisotropic R GM velocity distributions With⌦ the substitution ✏ = / ,and x⌦= p2p✏ (so dx = p2 d ⌃/2p✏ and⇧ rJ Modelling Cluster 1 2 E Evolution = ✏ x ) we obtain 2 Globular cluster evolution
p2✏ 4p2 M 2 1 2 1 2 2 2✏ x ✏ 2 x ✏ 2 x 2✏ x f ( )= 3 e e e e e e + e e dx E 2 2p2⇡24⇡r 3 GM Z0 J rJ h i M ⇣ ⌘ = F (p2✏) p2F (p✏) p2F+(p✏)+F+(p2✏) (7.8) 3 3 2⇡ (GMrJ ) 2 h i where z z2 t2 F (z) e⌥ e± dt is Dawson0s integral. ± ⌘ Z0 2 F (x)= p⇡ ex erf(x) ) 2 56 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters
King Models
Ja↵e Model Systems with anisotropic The next most obvious constant of the motion for orbits in a system with spherical velocity distributions
Modelling Cluster spatial symmetry is the total angular momentum L. Evolution 2 2 2 2 2 Globular cluster evolution L = L + L + L = r v x y z | ⇥ | or 2 2 2 2 2 2 2 2 L =(rv✓) +(rv ) = r (v✓ + v )=r v ?
So consider f f ( , L2), then f depends on v through 1 (v 2 + v 2 + v 2)and ⌘ E E$2 r ✓ L2 v 2 + v 2.Bysymmetrywehavev 2 = v 2,but= v 2. $ ✓ ✓ 6 r
57 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters L2 For example, in isothermal models we can replace with 0 = 2r 2 ,wherera is King Models E E E a Ja↵e Model some fixed radius. Then Systems with anisotropic 2 velocity distributions ⇢1 L f ( , L)= 3 exp E Modelling Cluster 2 2 2 2 E (2⇡ ) 2 2ra Evolution
Globular cluster evolution 2 2 2 2 2 2 ⇢ e 2 v + v + v r (v + v ) = 1 exp r ✓ ✓ 2 3 2 2 2 (2⇡ ) 2 " 2 ra 2 #
2 2 2 2 2 ⇢1e vr r (v✓ + v ) = 3 exp 2 exp 1+ 2 2 2 2 2 r 2 (2⇡ ) ✓ ◆ " ✓ a ◆ # Using this, we can look at the transverse vs radial velocity dispersions
1 v 2f ( , L) dv dv dv 2 ✓ r ✓ v✓ = 1 E RRR 1 f ( , L) dvr dv✓ dv 1 E RRR 58 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters Then King Models 2 2 2 2 2 r v✓ r v✓ 2 1 v exp 1+ 2 2 dv✓/ 1 exp 1+ 2 2 dv✓ Ja↵e Model v ✓ r 2 r 2 ✓ 1 a 1 a = 2 2 Systems with anisotropic 2 h ⇣2 ⌘ vr i h ⇣vr ⌘ i velocity distributions vr R 1 v exp dv / R 1 exp dv r 2 2 r 2 2 r Modelling Cluster 1 1 Evolution where the velocity integralsR extend toh infinityi forR the isothermalh i model since the mass Globular cluster evolution is infinite.
v 2 1 1 1 ✓ = / = (7.9) 2 2 2 2 2 r 2 3 2(1/2 ) 1+ r vr 2 1+ 2 /2 r 2 ra a So as 4 ⇣ ⌘ 5 v 2 r 0 ✓ 1 2 ! vr ! v 2 i.e. at small radii it is isotropic,butas r increases ✓ decreases, so becomes ra 2 vr anisotropic in the sense that the orbits become more radial. 59 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters Then King Models 2 2 2 2 2 r v✓ r v✓ 2 1 v exp 1+ 2 2 dv✓/ 1 exp 1+ 2 2 dv✓ Ja↵e Model v ✓ r 2 r 2 ✓ 1 a 1 a = 2 2 Systems with anisotropic 2 h ⇣2 ⌘ vr i h ⇣vr ⌘ i velocity distributions vr R 1 v exp dv / R 1 exp dv r 2 2 r 2 2 r Modelling Cluster 1 1 Evolution where the velocity integralsR extend toh infinityi forR the isothermalh i model since the mass Globular cluster evolution is infinite.
v 2 1 1 1 ✓ = / = (7.9) 2 2 2 2 2 r 2 3 2(1/2 ) 1+ r vr 2 1+ 2 /2 r 2 ra a So as 4 ⇣ ⌘ 5 v 2 r 0 ✓ 1 2 ! vr ! v 2 i.e. at small radii it is isotropic,butas r increases ✓ decreases, so becomes ra 2 vr anisotropic in the sense that the orbits become more radial. 59 / 93 Galaxies Part II Michie Models Observations of Globular Clusters
King Models
Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
60 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters
King Models Ja↵e Model These give rise to a generalisation of the King model called Michie models, where Systems with anisotropic velocity distributions 3 2 E 2 2 L 2 Modelling Cluster ⇢1(2⇡ ) exp 2r 2 2 e 1 > 0 Evolution fM( , L)= a E E ( 0 ⇣ ⌘h i 0 Globular cluster evolution E As r this model becomes the King model, as you would expect. a !1 In the Michie model, the velocity distribution is isotropic at the centre, but nearly radial in the outer parts, with a transition region near the anisotropy radius r = ra. Is this expected? Cluster evolution - encounters tend to fling stars out, so will have radial orbits at the outside. However “encounters” mean it is not “collisionless”. ... so we do need to consider evolution of clusters in time.
61 / 93 Galaxies Part II Systems with anisotropic velocity distributions Observations of Globular Clusters
King Models Ja↵e Model These give rise to a generalisation of the King model called Michie models, where Systems with anisotropic velocity distributions 3 2 E 2 2 L 2 Modelling Cluster ⇢1(2⇡ ) exp 2r 2 2 e 1 > 0 Evolution fM( , L)= a E E ( 0 ⇣ ⌘h i 0 Globular cluster evolution E As r this model becomes the King model, as you would expect. a !1 In the Michie model, the velocity distribution is isotropic at the centre, but nearly radial in the outer parts, with a transition region near the anisotropy radius r = ra. Is this expected? Cluster evolution - encounters tend to fling stars out, so will have radial orbits at the outside. However “encounters” mean it is not “collisionless”. ... so we do need to consider evolution of clusters in time.
61 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters King Models Two main approaches: Fokker-Planck equation and Direct N-body Ja↵e Model Systems with anisotropic df velocity distributions 1 Fokker-Planck equation: start from collisionless Boltzmann equation dt = 0, Modelling Cluster Evolution @f @f + v . f . =0 Globular cluster evolution @t r r @v but relax assumption ¨r = toallowforthefactthatphasespacedensityneara star can be changed due to r encounters. Find that df = (f ) =0 dt 6 where is obtained from the probability of particles being scattered out of and in to the phase space neighbourhood of a given particle. This requires knowledge of cross-sections for all types of stellar encounter.
62 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters
King Models
Ja↵e Model 2 Direct N-body methods (Aarseth 1963++++) where sum up forces on each star to
Systems with anisotropic calculate acceleration and orbit. velocity distributions Modelling Cluster Models now include stellar evolution (Tout), stellar mass loss, tidal dissipation and Evolution
Globular cluster evolution capture, collisions, binary evolution, mass transfer, gravitational radiation, disruption of binaries, tidal stripping by an external potential. To integrate orbits of stars from hard binaries to halo escapes requires very accurate N-body code. Can study small globulars correctly, and improving all the time.
N-body vital to calibrate Fokker-Planck methods. • discover any new e↵ects (since only minimal assumptions are made in advance i.e. • Newtonian gravity).
63 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters
King Models
Ja↵e Model Although N-body simulations are simple in principle, many sophisticated techniques are
Systems with anisotropic needed to make them e cient enough to be useful. velocity distributions
Modelling Cluster 1 The central core of the cluster is much denser than its outer parts, so the timestep Evolution required for stars in the core must be much shorter than the timestep for most Globular cluster evolution other stars. This problem is solved by assigning each star its own time-step. 2 Binary and triple stars can be formed with orbital periods far shorter than the crossing time. These require special treatment, by a combination of analytical solutions of the Kepler problem and other techniques. 2 3 The computation of the force by direct summation requires of order N operations, while advancing the orbits requires only of order N operations. Thus the force computation takes far more time than the orbit calculation. This problem can be addressed in either hardware or software.
64 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution The hardware solution is to develop special-purpose computers that parallelize • Globular cluster evolution and pipeline the force calculation, leaving the much easier task of advancing the orbits to a general-purpose host computer. The software solution is to separate the rapidly varying forces from the small • number of nearby particles and the slowly varying forces from the large number of distant ones, using a neighbor scheme.
65 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters Asupercomputerwiththousandsofcores
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Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution
66 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Relaxation times
King Models So far we have used the collisionless Boltzmann equation, where the assumption is that Ja↵e Model stars give rise to a smooth gravitational potential profile. This is invalidated by stellar Systems with anisotropic encounters on timescales trelax. velocity distributions
Modelling Cluster Evolution We found 0.1N Globular cluster evolution trelax tcross Relaxation times ' ln N Relaxation Escape of Stars where Core collapse 3 Mass segregation R Tidal stripping tcross Encounters with Binary ' GM Stars r Binary Formation through inelastic 5 6 encounters For a globular cluster we might take rt 50 pc, M 6 10 M , N 10 stars, and Finally 6 ⇠10 ⇠ ⇥ ⇠ so tcross 7 10 yr and trelax 5 10 yr, which is somewhat greater than the age ( 1010 yr).⇠ ⇥ ⇠ ⇥ ⇠ Those with long memories will note that this timescale is a factor of 10 or so longer than the estimate
we obtained earlier - there we used rt which was 5 smaller, and used an observed velocity! So ⇥ globular clusters as a whole may be nearly relaxed - with trelax their age. ⇠ 67 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Relaxation times
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Modelling Cluster Evolution 4 3 In the core it is less marginal. rc 1.5pc, Nc 10 , Mc 8 10 M ,andso Globular cluster evolution 7 ' ' ' ⇥ Relaxation times trelax 3 10 yr, which is much less than the age. Relaxation ⇠ ⇥ Escape of Stars 6 8 6 Core collapse For open clusters, N 100, tcross 10 yr, age 10 yr, which trelax 2 10 yr, Mass segregation ' ' ' ) ' ⇥ Tidal stripping which is considerably less than the age estimate. Encounters with Binary Stars Binary Formation So for open clusters, and at least the inner parts of globular clusters, we must take through inelastic encounters account of stellar encounters. Finally
68 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Relaxation
King Models
Ja↵e Model Systems with anisotropic What are the e↵ects of encounters on a star cluster? velocity distributions
Modelling Cluster Evolution 1. Relaxation As in a gas, we expect evolution towards a state of higher entropy. Globular cluster evolution Expect energy to be passed from “hotter” to “colder” parts of the cluster, where Relaxation times “hotter” higher velocity dispersion . A cluster can be thought of as core (with Relaxation $ Escape of Stars high v 2)andhalo(withlowv 2), so core loses energy to the halo. Core collapse Mass segregation Tidal stripping Encounters with Binary But from the virial theorem we know that the kinetic energy potential energy ,so 2 Stars 1 2 GM '| | Binary Formation 2 Mv R . If we remove energy from the core it must contract, so R ,andinso through inelastic ' # encounters doing v 2 - i.e. a self-gravitating system has negative heat capacity. There is no Finally equilibrium" configuration, and instead there is continual transfer of energy from an ever hotter, denser core to the halo (core collapse, see later).
69 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Escape of stars
King Models 2. Escape of Stars Since a globular cluster has finite mass, it has a finite escape Ja↵e Model velocity. Systems with anisotropic velocity distributions From time to time, encounters give a star enough energy that it can escape, so Modelling Cluster • Evolution have a steady evaporation of stars. Globular cluster evolution 2 Relaxation times At position r,theescapespeedve (r)= 2 (r). Relaxation Escape of Stars The mean square escape speed is Core collapse Mass segregation 2 3 Tidal stripping 2 ⇢(r)ve (r)d r Encounters with Binary < ve > = Stars ⇢(r)d 3r Binary Formation R through inelastic 3 encounters 2 ⇢(r) (r)d r Finally = R M R ⌦ = 4 M where ⌦is the potential self-energy.
70 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Escape of stars 1 2 King Models The virial theorem: ⌦=2T where T = M < v > is the kinetic energy, and hence 2 Ja↵e Model < v 2 >=4< v 2 > Systems with anisotropic e velocity distributions For a Maxwellian distribution in velocity the fraction of particles with speeds exceeding Modelling Cluster 3 Evolution twice the RMS speed is ✏ =7.4 10 (Problem set 3). Globular cluster evolution ⇥ Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
71 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Escape of stars
King Models Ja↵e Model Roughly, evaporation removes ✏N stars on a timescale trelax,so Systems with anisotropic velocity distributions dN ✏N N Modelling Cluster = = Evolution dt trelax tevap Globular cluster evolution Relaxation times 1 Relaxation tevap = ✏ trelax 136 trelax Escape of Stars ! ' Core collapse 2 Mass segregation Evaporation will set upper limit to the lifetime of a bound stellar system of 10 trelax. Tidal stripping ⇠ Encounters with Binary Stars Binary Formation Another mechanism for stars to escape is via close encounters. Rare, but can lead through inelastic • encounters to stars escaping with v >> ve rather than trickling out with E 0by Finally evaporation. ' Also supernovae result in neutron stars which can have high velocities, and so • these too escape with high velocities.
72 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Escape of stars
King Models Ja↵e Model Roughly, evaporation removes ✏N stars on a timescale trelax,so Systems with anisotropic velocity distributions dN ✏N N Modelling Cluster = = Evolution dt trelax tevap Globular cluster evolution Relaxation times 1 Relaxation tevap = ✏ trelax 136 trelax Escape of Stars ! ' Core collapse 2 Mass segregation Evaporation will set upper limit to the lifetime of a bound stellar system of 10 trelax. Tidal stripping ⇠ Encounters with Binary Stars Binary Formation Another mechanism for stars to escape is via close encounters. Rare, but can lead through inelastic • encounters to stars escaping with v >> ve rather than trickling out with E 0by Finally evaporation. ' Also supernovae result in neutron stars which can have high velocities, and so • these too escape with high velocities.
72 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse
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Ja↵e Model
Systems with anisotropic 3. Core Collapse [see Binney & Tremaine] velocity distributions A cluster evaporates on a timescale 100trelax in the absence of an external tidal field. Modelling Cluster ⇠ Evolution Total energy of cluster E = kGM2/R,wherek is a constant. Globular cluster evolution Relaxation times Most stars which evaporate have energy E 0 (i.e. they just manage to escape), so Relaxation ' Escape of Stars the cluster evolves at constant energy. Thus Core collapse Mass segregation Tidal stripping 2 Encounters with Binary R M Stars / Binary Formation through inelastic encounters and Finally M 5 ⇢ M / R3 / So as M 0, R 0and⇢ ! ! ! !1
73 / 93 Galaxies Part II Modelling Cluster Evolution Observations of Globular Clusters Core collapse
King Models
Ja↵e Model Systems with anisotropic The evolution of the mass velocity distributions distribution in an isolated cluster Modelling Cluster Evolution that began as a Plummer model. Globular cluster evolution The outer half of the cluster Relaxation times Relaxation expands, due to the gradual Escape of Stars Core collapse growth of the halo as core stars Mass segregation Tidal stripping di↵use towards the escape energy. Encounters with Binary Stars At the same time, the center Binary Formation through inelastic contracts; the central 1% of the encounters Finally mass contracts by a factor k > 30, corresponding to an increase in the central density by a factor k3 > 3 104! ⇥ B&T Figure 7.4 74 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse
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Ja↵e Model
Systems with anisotropic velocity distributions In fact what happens is not that the whole cluster collapses, but only the core does. Modelling Cluster Evolution Get core-halo structure from evaporation of stars from the core into the halo, and • Globular cluster evolution the stars ejected from the core share energy with halo stars. Relaxation times Relaxation Distribution of stars (even in the halo) at late times is governed by core Escape of Stars • Core collapse interactions. Mass segregation Tidal stripping Some core stars escape altogether but still do in work on remaining stars as Encounters with Binary • | | Stars they leave. Binary Formation through inelastic encounters Finally Transfer of heat leads to gravothermal catastrophe (negative heat capacity of the core), and core collapse follows after 10 - 20 trelax.
75 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse
King Models What halts core collapse? As the density increases, interactions become more Ja↵e Model
Systems with anisotropic important. Primordial hard binaries (if present) can prevent/delay core collapse by velocity distributions providing energy input. 3-body (rare) and (more frequent) tidal capture binaries form Modelling Cluster Evolution and halt core collapse, and also drive subsequent core expansion.
Globular cluster evolution Relaxation times Consider 3 stars with kinetic energies K1, K2 and K3,twostarsformabinarywith Relaxation Escape of Stars energy Eb < 0 and the kinetic energy of the center of mass Kb,whilethethirdstar Core collapse Mass segregation now has K30 Tidal stripping Conservation of energy requires: Encounters with Binary Stars Binary Formation through inelastic K + K + K = K + E + K 0 encounters 1 2 3 b b 3 Finally Hence Kb + K30 > K1 + K2 + K3
Therefore creation of binaries is a heat source.
76 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse King Models And so is interaction with binaries! Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
Exchange. Ejected star has speed of order of orbital velocity. 77 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse
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Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation What happens after the collapse? Escape of Stars Core collapse Binaries in the core pump kinetic energy into the cluster. The cluster expands. Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
78 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Core collapse
King Models Ja↵e Model Gravothermal oscillations Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
B&T, Figure 7.6s
79 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Mass segregation
King Models
Ja↵e Model Systems with anisotropic 4. Mass Segregation Stars in clusters do not all have the same mass - in a globular velocity distributions cluster they range from 0.2M to 0.8 M . Encounters lead to equipartition of Modelling Cluster 2 ⇠ 1 Evolution kinetic energy, so v m , and consequently massive stars (or binaries) acquire lower / Globular cluster evolution velocities and sink towards the centre, so have mass segregation. Relaxation times Relaxation Theorem: stars in cluster will end up having same average kinetic energy. Escape of Stars Core collapse Mass segregation Likewise low mass stars are preferentially sent towards the halo and eventually Tidal stripping evaporate. Since low-mass stars have high M/L, this e↵ect lowers the overall M/L of Encounters with Binary Stars Binary Formation globular clusters. through inelastic encounters Finally [In globular clusters, most stars have masses in the range 0.2 - 0.8 M - thos with masses > 0.8M have already evolved, and those with masses < 0.2M are likely to have evaporated.]
80 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Mass segregation in M 10
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
Beccari et al, 2010 81 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Mass segregation in M 10
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Ja↵e Model No primordial binaries With primordial binaries
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
Beccari et al, 2010 82 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Tidal stripping
King Models 5. Tidal Stripping Ja↵e Model
Systems with anisotropic As a cluster orbits a galaxy (e.g. a globular cluster around the Milky Way) some velocity distributions loosely bound stars may be captured by the galaxy and removed from the star cluster. Modelling Cluster Evolution The radius at which this occurs is the tidal radius.
Globular cluster evolution Relaxation times Consider a point mass galaxy, mass MG at a distance RG from the centre of a globular Relaxation Escape of Stars cluster. For a star lying along the line of centres the tidal force is Core collapse Mass segregation Tidal stripping GMG GMG Encounters with Binary Ft = 2 2 Stars R (RG + r) Binary Formation G through inelastic encounters GMG GMG r Finally 1 2 ' R2 R2 R G G ✓ G ◆ 2rGMG = 3 RG
83 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Tidal stripping
King Models Ja↵e Model At some radius rt this force balances the attraction due to the cluster i.e. Systems with anisotropic velocity distributions 2rGMG GMC Modelling Cluster 3 = 2 Evolution RG Rt Globular cluster evolution Relaxation times 1 Relaxation ) 3 Escape of Stars MC Core collapse rt = RG Mass segregation 2MG Tidal stripping ✓ ◆ Encounters with Binary Stars Binary Formation More generally, tidal stripping occurs outside a radius Rt within which the cluster through inelastic encounters density the mean density of the galaxy within the cluster orbit. Finally Evidence:⇡ tidal truncation (e.g King models fit old clusters better) • tidal tails around Pal 5 •
84 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Encounters with Binary Stars
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution 6. Encounters with Binary Stars Globular cluster evolution Relaxation times General encounters between single stars and binaries are di cult to treat. Relaxation Escape of Stars There are two broad categories: Core collapse Mass segregation Tidal stripping 1 Soft binaries (wide) Encounters with Binary Stars 2 Hard binaries (close) Binary Formation through inelastic encounters Finally
85 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Encounters with Binary Stars
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution Soft binaries (wide) Globular cluster evolution • GM 2 Relaxation times a << < v >, so star #3 is not strongly gravitationally focussed. Relaxation Escape of Stars Since star #3 travels faster than the orbital motion of #1 and #2, the encounter Core collapse Mass segregation tends to transfer energy into orbital motion of the binary (equipartition with Tidal stripping Encounters with Binary reduced particle). Hence soft binaries become softer and will ulimately dissolve Stars Binary Formation whem Ebinary > 0 (in centre of mass frame). through inelastic encounters Finally
86 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Encounters with Binary Stars
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution Hard binaries (close) • GM 2 Globular cluster evolution a >> < v >, so there is strong focussing of star #3’s orbit, and it ends up in Relaxation times the vicinity of #1 and #2 with velocity their orbital velocity. The three then Relaxation ' Escape of Stars form an unstable triple system until one star acquires enough energy to escape. Core collapse Mass segregation Since the kinetic energy of the ejected star is generally greater than that of the Tidal stripping Encounters with Binary incoming star, the energy E of the binary decreases, i.e. E increases so it Stars | | Binary Formation becomes more tightly bound. So hard binaries get harder. Note that the remaining through inelastic encounters binary can be any pairing from the three stars, not necessarily the original pair. Finally
87 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Encounters with Binary Stars
King Models Ja↵e Model Soft binaries becoming softer, and hard binaries harder, together are sometimes Systems with anisotropic referred to as Heggie’s law. velocity distributions
Modelling Cluster In globular clusters the hard/soft transition occurs at separations of about 10 au. Evolution Hard binaries act as energy sources, and thus can end core collapse. Encounters with Globular cluster evolution single stars give kinetic energy to the single star which emerges as the binary orbital Relaxation times Relaxation speed increases - also binaries are more massive and tend to sink to the core. Escape of Stars 1 2 1 5 1 2 Core collapse Cluster K.E. 2 Mv 2 10 M (10 km s ) = T Mass segregation ⇠ ⇠ 43 ⇥ Tidal stripping Binding energy T 10 J. Encounters with Binary ⇠ ' GM2 Stars The maximum energy which can be extracted from a hard binary is 2R where M is Binary Formation ⇠ through inelastic the mass of each star and R the radius for each. Putting in solar values gives us encounters 41 Finally E 10 J. So⇠ 100 binaries contain enough binding energy to disrupt a cluster! So the⇠ presence of binaries strongly a↵ects cluster evolution (but note that as the binary shrinks the e↵ective cross-section goes down).
88 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters Binary Formation through inelastic encounters
King Models 7. Binary Formation through inelastic encounters Ja↵e Model Not all binaries were formed at the the birth of the cluster. Systems with anisotropic velocity distributions Dynamical capture. Requires three independent stars in a gravitationally focussed • Gm Modelling Cluster interaction, so three stars in a region of size
King Models Right at the beginning of the course we saw that without gravitational focussing the 6 Ja↵e Model probability of a collision was about 1 per 10 stars in a cluster lifetime (this assuming a 2 Systems with anisotropic cross-section of ⇡(2R ) ). velocity distributions ⇤ Modelling Cluster If we include focussing, then conservation of angular momentum bv =2R v ,where Evolution 2 2Gm ) ⇤ ⇤ v = R ⇤ and v is the relative velocity when the two stars are far apart. Globular cluster evolution ⇤ b ⇤ v Relaxation times 2R = v⇤ i.e. the cross-section is increased by a factor Relaxation ) ⇤ Escape of Stars 2 2 Core collapse v 600 Mass segregation ⇤ = 1600 Tidal stripping v ' 15 Encounters with Binary ⇣ ⌘ ✓ ◆ Stars To form binaries by tides need R few R ,somorefrequent. Binary Formation min through inelastic ' ⇥ ⇤ encounters Binary formation manifested in low-mass X-ray binaries in globular clusters (Fabian, Finally Pringle & Rees, 1975) - close binaries involving neutron star and low-mass stellar companion. Another product of inelastic encounters are physical collisions - rarer than tidal capture. This may explain Blue Stragglers in clusters - anomalously blue stars given the turn-o↵age of the cluster. Should have evolved long ago if formed 10Gyr ago, so
could have formed more recently via main sequence star collisions. 90 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters
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Modelling Cluster Evolution 8. Other Processes Globular cluster evolution Relaxation times Stellar Evolution: Stars lose mass via stellar winds, and if the become supernovae. Relaxation 1 Ejecta travels at a few 10 km s ,andsowillescapethecluster,thusreducingthe Escape of Stars ⇥ Core collapse gravitational mass remaining in the cluster. Consequently the cluster expands in Mass segregation Tidal stripping response to this mass loss. Encounters with Binary Stars Binary Formation through inelastic encounters Finally
91 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution Globular cluster evolution Evaporation Relaxation times • Relaxation Gravo-thermal instability Escape of Stars • Core collapse Equipartition Mass segregation • Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
92 / 93 Galaxies Part II Globular cluster evolution Observations of Globular Clusters
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Ja↵e Model
Systems with anisotropic velocity distributions
Modelling Cluster Evolution
Globular cluster evolution Relaxation times Relaxation Escape of Stars Core collapse Mass segregation Tidal stripping Encounters with Binary Stars Binary Formation through inelastic encounters Finally
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