douglas heggie university of edinburgh, ukross/nbody-2015/slides/heggie.pdf · douglas heggie...

16 September 2015 Lund 1

12 things they don't tell you about the dynamics of star clusters

Douglas HeggieUniversity of Edinburgh, UK

With apologies to Ha-Joon Chang


The things (minus the small print)1. Black holes don't escape

2. Low-mass stars don't escape preferentially

3. Primordial binaries don't matter

4. Neutron stars matter

5. There is no equipartition between stellar masses

6.. High-concentration clusters are nowhere near core collapse

7. Star clusters don't fill their tidal radius

8. Escapers don't escape

9. There is no such thing as tidal heating

10. Stars don't escape on the relaxation time scale

11. Lagrange points don't exist

12.. The size of a cluster isn't set at perigalacticon


Thing 1.Stellar-mass Black holes don't all

escape in all clusters

What they say

1.Black holes segregate to

the centre

2.They then behave like an

isolated cluster

3.They have a small

relaxation time

4. 2+3 ⇒ the black holes escape very fast


What they don't say

1. Isolated clusters also expand on the relaxation time scale

2. Expansion counteracts tendency to segregate further

3. Black holes sit inside a much deeper potential well than an isolated cluster.

4. Would-be escapers cannot escape so easily

5. The would-be escapers donate their escape energy to the other stars

6. The black holes drive the expansion of the whole star cluster

7. They escape on the long relaxation time scale of the whole cluster

See Breen & H (2013)


Numerical illustrations

Mackey, Wilkinson, Davies, Gilmore (2008)

0 10Gyr

Giersz &H (2014)

Numbers of BH and BH binaries

M4, NGC 6397, 47 Tuc

M22Additional remarks

1. Disappearance of BH coincides with “second core collapse”

2. Stellar-mass BH expected in uncollapsedclusters, and not in post-collapse clusters


Thing 2. Low-mass stars don't always

escape preferentiallyWhat they say

1. Stars escape in two-body encounters

2. Heavy stars tend to lose energy, low-mass stars tend to gain energy in encounters (tendency to equipartition of energy)

3. It is easier for low-mass stars to gain energy above the escape energy


What they don't say

1. Heavy stars tend to sink to the centre

2. At high central densities, binary stars form from the heavy stars by three-body encounters

3. One of the stars in a three-body encounter gains high energy

4. Heavy stars escape preferentially in such circumstances

See Kruijssen (2009)

When the upper stellar mass is large enough,the low-mass stars are least rapidly removed


Thing 3. Primordial binaries don't matter much, at least for uncollapsed clusters

What they say

1. In binaries with P ⪅ 10 year, binary components have more energy than single stars

2. Time scale for changing energy of stars in binary-single interactions ~ relaxation time/binary fraction

3. “Heating” of single stars causes expansion on time scale of a few relaxation times

4. “Heating” by binaries sets the core radius in post-collapse evolution


What they don't say

1. It's a second-order effect

2. Expansion of the cluster is powered mostly by central stellar-mass black holes

3. Binaries gradually take over as black holesescape, approaching (second) core collapse

4. Questionable if primordial binaries set the post-collapse core radius

5. They affect the time to core collapse a lot 47 Tuc (fb = 0.018)

Evolution of core and half-mass radiiGiersz & H (2011)

Note: interactions affect the binaries a lot


Thing 4. Neutron stars matter for some cluster models

What they say

1. Neutron stars are about 2% of cluster mass

2. Few-percent effect on relaxation time, escape time scale, etc.


What they don't tell you

1. Presence or near-absence of NS depends on natal kicks (typically ≫ escape speed from cluster)

2. Presence or absence can change lifetime by factor ~4 (Contenta, Varri, H 2015)

3. Clusters dissolve by

two processes

a) Two- and three-body

encounters (long lifetime)

b) Mass-loss of stellar

evolution (short lifetime)

4. The models (from Baumgardt

& Makino 2003) sit on a separatrix

where these processes are finely

balanced


Thing 5. There is no equipartition between stellar masses among visible stars in

globular clusters

What they say

1. Two-body encounters lead to equipartition in a few relaxation times


What they don't say

1. In equipartition low-mass stars would evaporate too quickly

2. Multi-mass King models do not give equipartition (Miocchi 2006)

3. N-body models do not give equipartition (Trenti & van der Marel 2013)

4. Fokker-Planck models do not give equipartition (Inagaki & Saslaw 1985), except at highest masses


Shogo Inagaki ?1948-2015

2008


Thing 6. High-concentration clusters are nowhere near core collapse

What they say

1. Clusters are evolving from low concentration (large cores) to high concentration (small cores, c ≈ 2.5)

2. 47 Tuc has c = 2.07, a small

dense core, and is undergoing

rapid evolution towards core

collapse

Harris catalogue


What they don't say

1. Models imply that core collapse will take at

least another 20 Gyr


Thing 7. Star clusters don't fill their tidal radius

What they say

1. The Galactic tide strips off stars beyond a “tidal radius”

2. King models have a finite “tidal radius” to incorporate this effect

3. Globular clusters fit King models quite well

4. Therefore the radius of globular clusters is determined by the Galactic tide

5. We can use radii of globular clusters to estimate strength of tidal field


What they don't say

1. Edge radius may be set by initial conditions

2. If a cluster starts smaller than its tidal radius

i. It first expands so that its relaxation time is of order its age, until its radius

equals the tidal radius

ii.After that it contracts,

and its relaxation time

is of order its remaining

lifetime (Henon 1961;

Gieles, H & Zhao 2011)

3. GHZ say 2/3 are (i), 1/3

are (ii)

4. Gives a reinterpretation

of the “survival triangle”

Gnedin & Ostriker 1997

Thing 8. Some Escapers don't escape

What they say

1. For a cluster on a circular

Galactic orbit, switch to rotating

frame centred at the cluster

2.Combined potential

(cluster, tidal field,

centrifugal acceleration)

has last closed

equipotential Vcrit

3. Stars with higher energy escape

What they don’t say

1. The condition E > Vcrit is necessary, not sufficient

2. There’s also a Coriolis acceleration, which can

keep high-energy stars inside the cluster

3. There is a population of

“potential escapers”

4. Can reach up to 10% of

cluster members (Baumgardt

2001)

5. Ignored in all snapshot

modelling of globular clusters

Henon 1970

Thing 9. There is no such thing as tidal heating on a circular Galactic orbit

What they say

1. Clusters exhibit non-Keplerian

velocity dispersion profile

2. Tidal field is time-dependent

(“bulge shocking”, “disk

shocking”) and strongest

at large radii

3. Velocity dispersion elevated by

“tidal heating”, or even something “non-Newtonian”

Drukier et al 1998 (M15)

What they don't say

1. For a cluster on a circular

Galactic orbit, in the

rotating frame the

potential is static.

2. Energy is conserved (Jacobi

integral), therefore there is

no tidal heating.

3. Velocity dispersion is

elevated by potential

escapers (see Thing 8)

Kuepper et al 2010

Thing 10. Stars don't escape on the relaxation time scale except for very large N

What they say

1. Stars escape by two-body encounters, which may elevate the energy of one star above the escape energy.

2. Two-body encounters change stellar energies on the relaxation time scale

3. Stars escape on the relaxation time scale

What they don't say1. Stars with E > Vcrit remain

as potential escapers for a

time

2. During this time E changes

on relaxation time scale, and

escape becomes easier as E

increases

3. The balance of these

processes gives an escape

time scale (i.e. time scale for

loss of mass) ∝ tr/N1/4

(Baumgardt 2001)

4. Eventually this must turn over

to ∝ tr

Thing 11. Lagrange points don't exist for clusters on elliptical orbits

What they say

1. There is a point where

the attraction of the

cluster and Galaxy are

in balance.

2. It is an equilibrium point

3. It is a critical point of the “potential”

4. Stars beyond this point are escaping

What they don't say

1. For a cluster on an elliptical Galactic orbit,

there is a point where the forces balance, but

it's a moving point,

not an equilibrium

2. There is no potential,

and no critical point

3. The nearest analogue

to the Lagrange point of

the circular problem is

a periodic orbit

“Lagrange” periodic orbit for variouspower-law Galactic potentials

Thing 12. The size of a cluster isn't set at perigalacticon

What they say

1. The tidal radius is smallest at perigalacticon

2. Stars beyond the tidal radius escape

3. Cluster members are non-escapers, and must lie inside the tidal radius at perigalacticon

What they don't say

1. It's not (Kuepper+ 2010), at least up to e = 0.5.Fit a King profile to an N-body simulation, and compare edge radius (red) with tidal radius (dots) for e = 0 (left), 0.5 (right).

Edge radius agrees with mean tidal radius to about 10%.

Summary

What they say

“...underlying it all is a basic dynamical structure that is very simple”

What they don't say

“Nothing is rich but the inexhaustible wealth of nature. She shows us only surfaces, but she is a million fathoms deep.” - Emerson

With thanks to my collaborators on these topics

Phil Breen

Mirek Giersz

Filippo Contenta

Anna Lisa Varri

Shogo Inagaki

Mark Gieles

Holger Baumgardt

Toshi Fukushige

Ben Bar-Or

Kate Daniel

Andreas Küpper

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