Initial spectrum of fluctuations -- perhaps δρ / ρ ∝ m -
(Fall & Rees 1977).
Formation in dense cloud - shocked into collapse?
E.g. thousands of new clusters have been observed to be forming in the
Antennae galaxies collision (visible as the blue dots in the image).
Credit: B. Whitmore (STScI), F. Schweizer (DTM), NASA.
106 years - newly born cluster is still mostly
hidden in cocoon of gas
This now-famous HST image shows nascent star clusters still buried in their
gaseous cocoons. But intense radiation from massive young stars can be seen to
be boiling away the cocoons at the pillar ends. Credit: J. Hester, P. Scowen (ASU), HST, NASA.
Winds from 100 M☉ stars blow
bubbles in gas.
First SNe may blow out all gas - halt further star formation.
This LMC cluster, NGC 1850, is a mere 4 million years old. The nebula on
the left is blowout from the explosion of massive stars in the cluster.
Credit for this composite HST image: M. Romaniello (ESO) et al., ESA, NASA.
107 years - no gas left. All >
10 MSun stars evolved.
Cluster not in full dynamical equilibrium yet.
Not relaxed -- may retain traces of initial state.
Unlike the Milky Way, the Magellanic Clouds are still actively forming
globular cluster-mass star clusters.
This young cluster, NGC 1818, in the Large Magellanic Cloud, formed about 40 million
years ago, but has a relaxation time longer than that -- 150-600 Myr (Elson 1987) --
so that it is not yet relaxed. Even though it is very
young, the metallicity is [Fe/H] ~ -0.8. The cluster has a mass of ~3 x 104
solar masses. Credit: D. Hunter (Lowell Obs., STScI) et al., HST, NASA.
May or may not have undergone "violent relaxation" as stars respond to
rapidly changing potential (e.g., from mass loss).
Mass loss makes cluster less bound.
Cluster expands. Depending on IMF, outer cluster may overflow eventual tidal
radius and create a halo of unbound stars.
Life cycle of a globular cluster
from Elson (1999, in Globular Cluster, eds. C. Martinez Roger, I.
Perez Fournon & F. Sanchez, Cambridge University Press).
<~ 109 years - originally fairly compact cluster core may expand
while outer, unbound cluster halo stripped away:
Outer part of surface brightness profile approaches familiar equilibrium
form with a cut-off at the tidal radius:
Cluster fades as high mass stars evolve away and as low mass stars evaporate off.
Equilibrium Shape of a Globular Cluster
rt for clusters is < 200-250 pc; typically ~
30 pc tidal radius cluster at 10 kpc covers ~300 arcmin2.
Outer parts can be studied easily from ground.
The surface brightness from the core to the tidal radius can vary by more than
five orders of magnitude.
M / L ~ 1 - 4 in globular clusters -- no dark matter.
Smallest dSph galaxies about rt ~ 500-600 pc, M/L ~
50-100!! -- presence of dark matter?
Similar absolute magnitude to very brightest globular clusters:
MV ~ -8.5 for Carina, Draco, Ursa Minor
MV ~ -9.53 for NGC 2419
MV ~ -7.06 for NGC 5466
Both globular clusters and dSph (right or wrong in the latter case)
have been fit by so-called "King models":
In 1962, King found empirically that the outer cluster looks to have a surface
density profile as follows:
where rt is the radius where f = 0
Inner cluster looks like:
where rc is the radius where f drops to
A single expression that accommodates both:
Surface density as a function of radius for a
family of analytical King profiles normalized to the core radius,
with the concentration parameter and King limiting radius shown for each curve.
King's (1962) separate empirical relations for the inner and outer parts of
the profile are shown.
Figure modified from that given in the lecture by Rebecca Elson (1999),
who presented this topic at the Tenth Canary Islands Winter School on Globular
Clusters in 1998, shortly before her untimely passing.
From Elson, 1999
Thus - most equilibrium clusters can be described by a curve with
only two free parameters: rc and rt
Typical rc for a cluster is ~1 pc.
For a cluster at 10 kpc, this corresponds to 20 arcsec --> HST observations.
In a typical cluster rt / rc ~ 30, so second term
above is ~0.03.
So for r << rt we just recover the equation
for the inner cluster above.
For r >> rc we recover the first equation above.
The concentration of the cluster / dSph is defined by:
c = log(rt / rc)
In 1966 King developed a set of models of the surface brightness profiles of globular clusters
based on the physics of spherically symmetric, isotropic, self-gravitating systems made of stars with a single
Satisfy Boltzmann equation.
A "gas" of stars many small changes to motions due to distant encounters.
Obtain a family of models that look just like the empirical King models and characterized
by one free parameter, the central potential (Wo).
Luminosity/Mass Functions and Luminosity/Mass Segregation
Historically difficult to derive for clusters because of crowding problems
for dense clusters.
Initially astronomers concentrated on bright stars in outer parts.
Once analysis software was developed to deal with crowded,
overlapping star images, could probe closer in and fainter stars.
Note the readily identifiable parts of a cluster LF, and the
much larger density of stars on the Main Sequence (where stars
spend the most time).
In this LF, the contribution of the horizontal branch has been removed.
The M3 luminosity function including the horizontal branch (open circles).
The histogram shows a typical halo luminosity function and the filled triangles
a local subdwarf LF from high proper motion stars. From Reid & Majewski (1993,
ApJ, 409, 635).
CCD imaging means we now can
probe down the main sequence, with HST to hydrogen burning limit
for nearby clusters.
From Binney & Merrifield Figure 6.17.
As shown above and below, overall shape of globular cluster (and
halo star) LFs different than for nearby disk stars.
The latter has more high mass (younger) MS stars that "smooths over"
the bright end of the LF which is dominated by evolved stars in the
From Binney & Merrifield Figure 6.13.
There is no reason to believe that there is a universal
luminosity function for globular clusters.
Varies systematically with metallicity.
Star's luminosity depends on chemical composition as well as mass.
Thus, for a similar mass function we should expect different
luminosity functions depending on the metallicity of the cluster
This is in fact observed:
From Binney & Merrifield Figure 6.14.
Assume the stars in a cluster are formed according to a Salpeter
type mass function:
Because stars lose mass mainly after they evolve off of the main
sequence, this function should describe the LF of the main sequence.
From stellar evolution calculations, we can calculate how the
absolute magnitude of a main sequence stars depends on its mass,
We can use this to calculate a luminosity function from the
The figure below shows the calculation for a range of
and two metallicities.
From Binney & Merrifield Figure 6.15.
This procedure, with a simple power law mass function,
actually reproduces reasonably the various observed cluster
sequences shown above.
Proves that structure in the main sequence part of the LF
in globular clusters derives mainly from the luminosity-mass
relation in clusters.
Different power law indices needed for different clusters, however.
No universal power law for clusters:
is smaller for more metal rich clusters.
Implies metal rich clusters contain a larger fraction of high mass stars.
What is cause and effect:?
Extra high mass stars in some clusters may have chemically
enriched their environment of the cluster in early times,
resulting in greater overall metallicity in the other stars.
There initially was a common IMF, but evolution alters IMF.
For example, higher metallicity clusters tend to be near center of
Milky Way, crowded regions which will act preferentially on these clusters to strip
away low mass stars (Capaccioli et al. 1993).
Actually observed that the slope of the cluster mass function
is more strongly correlated with cluster's location in the Milky
Way than it is with metallicity:
Correlation of cluster mass function slope with various properties of
a globular cluster from Djorgovski, Piotto & Capaccioli
(1993, AJ, 105, 2148). Metal poor clusters are solid points, more metal rich
are open symbols. The X values are the related to
by the standard Salpeter characterization
= 1 + X.
Note that King models of clusters that include multiple mass components (not surprisingly)
fit typical globular clusters better.
From Binney & Merrifield Figure 6.18.
These models assume that equipartition of energy should take place, and that, therefore,
lower mass stars should be less centrally concentrated than high mass stars.
We have already seen evidence for this in our discussion of binaries and
The distribution of low mass x-ray binaries (LMXRBs) in clusters also supports
If the same holds for single stars, then we should expect to see
variations in the luminosity function of globular clusters as a function
radius from the center.
Bright massive stars should be more centrally concentrated.
The luminosity function should become steeper with radius.
Observationally a difficult problem:
Need to observe a wide range of radii.
Crowding in the high density core.
Back/foreground contamination in the low density outer regions.
Nevertheless, such variations are observed and follow expectations:
From Binney & Merrifield Figure 6.19. See warnings about this
figure in the text. Crowding in the core makes the faint end
slope extremely difficult to ascertain.
HST data helps substantially in crowded cores.
The histogram is the LF in the core while the line shows the
LF for MS stars at large radii from ground-based observations.
From Binney & Merrifield Figure 6.20.
Later Evolution of a Globular Cluster
The later evolution of a cluster involves several processes that results in a changing equilibrium
Evolution through stellar evolution:
Cluster fades and loses mass as high mass stars evolve, eject gaseous material,
After > several 109 years - core may undergo
runaway collapse as dynamical energy is continually transported out of core to outer
parts of cluster (by escaping stars).
Massive stars sink to core (convert potential to kinetic energy) -- runaway
"gravothermal collapse catastrophe" if core is sufficiently compact.
In the crowded core, close encounters become more common, and this produces
fast moving stars that can escape.
These stars escape the core and remove kinetic energy from it.
Core needs to find a new equilibrium and shrinks a little.
This shrinking increases the gravitational binding energy, and the remaining
stars have to move faster to compensate for the higher potential energy.
But the heated stars are more likely to be pulled out.
Process repeats and can accelerate into a runaway situation leading
to "core collapse" (a "gravothermal catastrohe").
The cycle leading to core collapse, from http://www.khalisi.com/phdthesis/gc03.html.
The same type of collapse happens in collapsing protostars, but in the latter case the
protostar's core becomes hot and dense enough to ingnite nuclear fusion reactions, which provide a source of energy
to replace the escaping heat and halt further shrinking.
In a cluster the collapse will be halted before infinite density is reached
(or even reversed) by creating energy via the formation of hard binaries in the core by three-body interactions,
which releases energy by way of the released third star.
These hard binaries stop the collapse and help to stabilize the cluster.
"The runaway process is only halted when binary stars form at the centre through 3-body
encounters. This may happen e.g. when a single star passes two others, which may or may
not be already a physical pair. If the passage is close enough, it interacts with the
individual components separately. These interactions can be very complex in general,
resulting in the formation of a temporary triple-star system, the disruption of the former
binary, or the exchange of partners. The figure shows a typical interaction between the stars:
a single star (green) comes in from the left and meets a binary from the right (red + white).
After the dissociation of the binary, a chaotic behaviour sets in wherein different
other orbits are formed and disrupted (red + green), and in the end, a harder binary than
before (more tightly bound) is created, while one spare star escapes to infinity.
It should be stressed that the simulation provides a simplified view and real circumstances
are very much more complicated. The analysis of close encounters of a binary with one
single star does not exhaust the possible effects. During the late stages of core collapse,
binary-binary encounters are likely to predominate binary-single ones, making the
investigation more difficult. "
Figure and caption from http://www.khalisi.com/phdthesis/gc03.html.
Subsequent encounters between binary pairs and other single particles
tend to increase the binding energy of these pairs, which leads to a further
heating of the surrounding system of single particles.
A series of collapses and rebounds ("gravothermal oscillations") may follow
as the process repeats.
But the formation of harder binaries with greater binding energy provides
a huge reservior of energy in the cluster -- up to most of the total energy of the cluster.
When enough binaries have been formed in this way, the resulting energy production will
reverse the outward flow of energy, stops the core collapse from reaching singularity,
and after reaching a minimum radius and
maximum density, the core region will expand again.
"Core collapse, when threatened to occur by the collective effects of two-body relaxation,
can thus be narrowly averted by a handful of crucial three-body or four-body reactions
in the dense core of a nearly collapsed cluster. "
Collapse forms cusps:
Lower figure from George Djorgovski's notes: http://www.astro.caltech.edu/~george/ay20/Ay20-Lec15x.pdf.
About 20% of old Milky Way clusters are now in core collapse; 80% not. Could
be a transitory phase in many.
Comparison of clusters without (left) and with (right)
central cusps. From Binney & Merrifield.
M15 is the densest known globular cluster. This HST image of the dense
core reveals that the density of stars continues to rise toward the cluster's core,
suggesting that a sudden, runaway collapse due to the gravitational attraction of
many closely packed stars or a single central massive object, perhaps a black hole,
could account for the core's extreme density.
Credit: P. Guhathakurta (UCO/Lick, UC Santa Cruz), NASA.
Core collapse may be a transitory phenomenon, repeating in multiple "bounces".
From George Djorgovski's notes: http://www.astro.caltech.edu/~george/ay20/Ay20-Lec15x.pdf.
Some globular clusters, like M15 shown above, are suspected to contain
intermediate mass (e.g., 4,000 solar masses in the M15 case) black holes,
bridging the mass scale from supermassive black holes in the centers
of galaxies and the single star mass variety.
Could also account for cusps in GC centers.
Evidenced by dynamics suggesting extremely massive core in the
But suggested that concentrations of less massive black holes,
neutron stars or white dwarfs, congregating in the middle from the
process of mass segregation, could also account for rise in mass-to-light
ratio measured in GC centers.
x-ray and radio emission seems consistent with the IMBH hypothesis
in some cases.
Late times (> 10 Gyr) -- winnowing down of the cluster by evaporation.
Highest velocity stars continue
to escape from cluster, binding energy decreases, cluster dissipates.
Other factors (e.g., variable Galactic tidal field) may speed this process up.
CLUSTER DYNAMICAL TIMESCALES
Crossing Time (Dynamical Time)
Timescale on which orbits in a cluster mix.
Various definitions in the literature:
e.g., characteristic radius / σv
or, tcross ~
106 (rh /
vm) years, where rh is the
half-mass radius in parsecs, and vm is the mean
velocity in km/s
If virial theorem holds, i.e. a system formed by collecting together material
from a state of rest at infinity (K = W = E = 0) to an equilibrium state invests
1/2 of the gravitational energy into kinetic form (the rest in some other form):
2K + W = 0, and
where rh is in pc and M is in solar masses.
For M ~ 105M☉,
rh ~ 3 pc, tcross ~ 3.5 x
tcross (galaxy) ~
For a galaxy, the crossing time is more like 108 yr.
Timescale for Violent Relaxation
Stars respond to rapidly violently evolving potential.
Important in early life of cluster.
Dramatic mass loss from evolving high mass stars (stellar winds exceed escape velocity).
Energies of individual stars not conserved.
Violent relaxation timescale depends on how fast the background potential changes.
From time-dependent virial theorem:
where K is kinetic energy and W is potential energy.
where ρm is M☉ / pc3.
Therefore, very early in cluster life (fraction of first crossing time).
Independent of stellar mass, so will get no mass segregation. Any observed
mass segregation in young clusters would be from primordial formation processes.
Timescale for Close Stellar Encounters
Particularly relevant to formation and destruction of binary stars.
Time one must wait for a single stellar encounter that produces a 90 degree
deflection in direction of travel.
Impact parameter = p0 =
G(m1 + m2) /
vrel2, where vrel is the relative
Cross-section for encounter:
where n is the density of stars (# / pc3) and tce
is the timescale for a close encounter.
Set the relative velocity in terms of mean velocity:
In globular, assume Saltpeter MF with
0.2 M☉ low mass
cutoff: Get m1 ~
m2 ~ 0.3 M☉ as
vm ~ 5 km/s, n ~ 103
tce ~ 1.7 x 1010
- very long.
Timescales much longer than present age. However, may be important in densest cluster cores
(where n is larger).
Binary formation and destruction accelerated in cluster cores.
See blue stragglers, binaries, LMXRBs in core...
Accelerated in dense clusters, e.g. those undergoing runaway core collapse.
Two Body Relaxation Timescale
Most important timescale for globular cluster evolution.
Time for star to experience net deflection of 90 degrees due to
Equivalently, timescale for equipartition of energy and mass segregation.
Also timescale for a star to change velocity by an amount equal to its
where N = # of stars in cluster.
More massive stars relax more quickly (inversely proportional to square of
Local quantity -- inversely proportional to local density -- inner regions relax sooner.
Therefore convenient to discuss the two-body timescale for the mean density
inside half-mass radius.
t2body ~ 108 years at core.
t2body ~ 109 years at
Typical dSph, very low density t2body
~ 1013 years.
E galaxy, t2body ~ 1018 years.
Thus, for dSph and E galaxies, something else creating their relaxed appearances -- e.g., violent
Disk Shocking Timescale
Heats cluster by compressing it in the direction perpendicular to disk.
Effects outer parts of cluster most, causing halo of cluster to puff up.
Accumulation of 2-body encounters in cluster - equipartition
of energy 1/2 M* v2.
Low mass stars reach escape velocity - evaporate - decreased cluster
mass lowers escape velocity.
Note: Change in mass / luminosity distribution
Core Collapse (Spitzer 1969)
"Hotter" stars removed from core - mass segregation.
Core collapse, dynamical energy transported out by 2-body
Increases evaporation rate several times (Lee & Goodman 1995).
Rotation of cluster also helps (Longaretti & Lagoute 1997).
Mass Loss by Stellar Evolution
Supernovae explosions (also, disruptive?)
Ejection of planetary nebulae.
High mass stars - occurs early.
Decreases cluster binding energy.
Each mechanism is enhanced in presence of Galactic potential -- "tidal
truncation" -- radius where Galactic force = cluster force.
rtide = R (Mcluster / 3 MGalaxy (R) )
External Mechanisms (outside forces that act on clusters)
Disk Shocking (Spitzer 1987, Chernoff et al. 1986)
Compression of cluster by disk gravity.
Kinematic "heating" by "compressive shock".
From Galactic bar - effect likely small (Long et al. 1992).
Selectively acts on clusters with highly eccentric orbits
(Aguilar et al. 1992).
Tidal Gravitational Shocking (Spitzer 1987)
When cluster passes another spherical mass (generally giant
molecular cloud complexes, dark matter blobs??;
but effect of other clusters small).
Open clusters can be destroyed in one encounter; globulars
hardier, takes repeated "hits".
Acts preferentially on orbits near plane.
Tidal force of Milky Way overcoming binding force.
Escapees form leading and trailing streams of debris.
Grillmair (1998) et al. have now found 20 Milky Way and M31
clusters with some evidence for "tails".
Dynamical Friction (Chandrasekhar 1943a, b)
From Frank Shu, The Dynamical Universe: An Introduction to Astronomy, C. 1982.
Cluster continuously loses orbital energy to field stars
and dark matter.
Spirals in, tidal stripping and evaporation increase.
Acts preferentially to deplete Galactic center of most
massive clusters (Tremaine et al. 1975).
Field Star Diffusion
Diffusion through clusters, energy given to cluster (dynamical
friction), heating increases, evaporation increases (Peng & Weisheit
Acts preferentially on disk clusters where density of
field stars is high.
But - tangible number of field star captures - anomalous
star source? Not "pristine" SSPs.
Given all of the sources of mass loss in globulars that are expected to be
at play, many efforts have gone into searching for the expected tails of
Example 1: ω Cen
One of the first "best" claims for a globular cluster tidal tail was that
by Leon et al. (2000, A&A, 359, 907) ω Cen tidal tail:
ω Cen may be expected to have tidal tails because
of its unusual properties suggesting that it may be the residual
nucleus of an accreted dwarf spheroidal (Lee et al. 1999, Majewski et al. 2000,
ω Cen resembles M54
(second most massive M.W. "G.C.") now
recognized as nucleus (or fallen into nucleus) of tidally disrupted Sgr dSph.
It is one globular cluster that has a considerable internal
age range (at least several Gyr).
It has a VERY large internal metallicity spread.
It is s-process enhanced (to enrich the cluster this way
the ejecta from low mass stars that evolve into AGB stars on timescales
of a Gyr had to be retained by the cluster and recycled into
the next generations of stars -- pointing to a prolonged star
Its orbital period is so small, and its orbit so extreme
(retrograde through the disk) that it is hard to understand how
the cluster could have retained its gas for multiple populations
unless it was originally massive.
The presently destructive orbit may have been a recent
phenomenon: The object may have originally evolved far from the Milky Way
which allowed a complex SFH, but then the orbit decayed to its
present one. However, for dynamical friction to work requires
a large initial mass -- that of a dSph.
Freeman (1993) -- nucleated dE galaxies might winnow down to
GC-like objects after disruption.
Walker et al. (1996) - disruption models yield G.C.-like remnants
Unfortunately, a reanalysis of the Leon et al. tails by UVa undergrad
David Law, SRM and MFS (Law et al. 2003, AJ, 126, 1871) using 2MASS
shows that variable dust extinction is likely to be the culprit in
producing the apparent tails by Leon et al.
BUT: Majewski et al. (2012) new results!!
Example 2: Palomar 5
At last, with the first results from the Sloan Digital Sky Survey, which
just happened to go through the wimpy globular cluster
Pal 5 (on the celestial equator), a great
example of a cluster tidal tail was found!
The "archetype" of globular cluster tidal tails -- those found by
the digital sky survey on the globular cluster Palomar 5. Upper panels
from Odenkirchen et al. (2001, AJ, 548, L165) showing initial discovery in the SDSS
equatorial strip data. Lower panel is an extended view of 10 degree
tails from Sloan in Odenkirchen et al. (2003, AJ, 126, 2385).
Cluster Vital Diagram
Below from Gnedin & Ostriker (1997), but other versions in the literature.
For present clusters:
Aguilar et al. (1988) - 4 doomed per Hubble time
Lee & Goodman (1995) - 30 post-core-collapse will be gone in next Hubble
time, perhaps more.
Gnedin & Ostriker (1997) - new code, extra relaxation effects - 50 to
> 90% of present clusters will be gone in next Hubble time.
Original cluster population may have been quite larger than seen today.
Destroyed cluster stars are now in field star populations.
The destruction mechanisms act selectively by orbit, mass; generally
small or highly eccentric orbits.
Do present clusters trace field stars? Original cluster family?.
Expect -- and SEE -- cluster properties to vary by Galactic orbit/position.
From George Djorgovski's notes: http://www.astro.caltech.edu/~george/ay20/Ay20-Lec15x.pdf.
Proper Motion Analysis of Palomar 13 (Siegel, Majewski, Cudworth & Takamiya 2001)
- large B.S.S. population and double SGB supports idea of severe stripping.
NOTE: Near perigalacticon now - found also in Pal 12, E3 - missing outer halo Pal