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The Initial Mass Function (IMF) Continued

• IMF ψ(m, t) = number of stars formed per unit volume per unit time

• Critical measurement for interpreting observations of distant (and nearby!) galaxies

• Probably varies with environment and/or time, but: – usually assumed constant in space and time

– only meaningful when averaged over significant volume

• Nomenclature– often approximated as power law

ψ(m) dm = ψ0 m-α

or log ψ(m) d(log m) = ψ0 m-γ γ = α-1

α ~ 2.35 historical value, still considered a good value for high mass end

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• Salpeter 1955, ApJ, 121, 161 ̶ a = 2.35 M = 1-40 M

• Scalo 1986, Fund Cos Phys, 11, 1– rough power law IMF for M > 1 M

– strong flattening below 1 M (see next page)

• recent studies suggest high-mass slope closer to Salpeter

• infrared surveys confirm flattening ofIMF at low masses – no vast reservoir of free-floating

brown dwarfs, planets

– consistent with microlensing results

Kroupa, Tout, Gilmore 1990, MNRAS, 244, 76

Field Star IMF: Results

PDMF

IMF

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Scalo IMFs

The Scalo IMF has been used often in galaxy modeling but in under predicts the mass in low mass stars and may also be somewhat deficient in 10-40 solar mass stars. Be careful when choosing ingredients for galaxy modeling!

Scalo (1986)

Miller & Scalo (1979)

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IMF in Clusters

• stars are at same distance and age, simplifying analysis

• provides consistency check on field IMF

• constrains universality of IMF• limited by statistics, limited

mass range (esp. for old clusters)

• results:– most cluster IMFs consistent

with field, within small number statistics

– globular cluster IMFs also consistent

– some evidence for local variations in young clusters

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Orion Nebula Cluster and the Low-Mass IMFHillenbrand 1997, AJ, 113, 1733 (see also Luhman et al.2000 who observed a number of clusters and demonstrated that the cluster and field IMFs are very similar)

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Combining Cluster IMFs

• The combination of different limiting magnitudes and turnoff masses for the clusters may lead to distortions!

• Another problem: If upper stellar mass limit in cluster is limited by available cluster mass, combined IMF will be too steep -- cluster mass spectrum gives smaller probability for obtaining more massive stars (Reddish 1978). Discrete sampling effects need to be explicitly considered.

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IMF

some process gives scale-free hierarchical structure

“turbulence”

disk MRI

SN, SB

spiral shearing

condensations formin molecular clouds

quasi-equilibrium cores

transient, unbound

gravitationally unstable cores

instabilities in turbulent compressions

ambipolar filamentation

instabilities in expanding shells

quiescent gravitational instablity

bending mode instability

cooling instabilities

bound by external shocks

ambipolar diffusion

magnetic reconnection

gradual loss of support

lull in external energy input

turbulent dissipation

collisional coalescence and fragmentation

accretion

disruption by turbulent shearing or shocks

growth, termination by disk disruption

collapsing protostars

feedback on entire process

UV radiationwinds, jets

SNe, SBsaccretion, mass loss, dynamical interactions

Theoretical Conceptions of Processes Controlling the IMF(after a talk by John Scalo)

]

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Is the IMF Invariant?

• Need you ask after the previous chart???

• This question has been debated for a long time – Limited statistics, selection

effects may cause apparent differences

– Substantial evidence for bi-modal star formation which is a form of varying IMF

– No clear cut evidence for slope variations within given mass ranges

From Weidener and Kroupa 2005 ApJ 625 754

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Dispersal of Open Clusters

Open clusters eventually disappear as easily recognized entities

- Most clusters not massive enough to be gravitationally bound

- Differential rotation exacerbates the dispersal

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Globular Clusters• Have been a key element of many stellar population studies• Used to be a bone of contention as GC ages appeared to be at

variance with the age of the Universe • Have ~100,000s of members and usually can survive passage through

the plane of the Milky Way (but beware of mass segregation effects)

M80

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Globular Cluster Properties

• MW has ~150 globular clusters – last ones to be found were discovered at 2microns by the 2MASS survey [big ellipticals have many more GCs; M87 has over a 1000]

• Masses range from ~1000M to ~1x106M• Metallicity ranges from 0.004 of solar to about solar• Some are very strongly centrally concentrated while

others are not• Integrated light has a spectral type ranging from F3 to

G5: most of the variation is due to the variation in metallicity {redder = more metal-rich}

• Milky Way clusters have a relatively small age range (all old) but other galaxies show big spreads in ages

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Are Globular Clusters Small Galaxies?

It is very tempting to think of globular clusters as an extension of the mass sequence of elliptical galaxies….ButSome GCs clearly form as part of the initial sequence in the formation of larger galaxies so they may be remnants of cloud collapse.It is also not at all certain how many GCs have black holes in their centers.GCs interact with their parent galaxies so their properties may be strongly modified relative to an isolated dwarf elliptical.GCs are virtually all quite spherical with little flattening unlike dwarf ellipticals (most oblate are the equivalent of E3; flattening is due to rotation).

Frogel, Persson, & Cohen 1980

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More HR Diagrams

Metallicity

Lower Higher

Open Cluster

Sharpness of the turn-off from the MS onto the subgiant branch indicates that the stars in a cluster have a very small age spread (~2%).

Horizontal branch is bluer for lower metallicities but HBs show a range of properties at a given metallicity (related to “2nd parameter” issue)

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WD Cooling Curve

• HST photometry has revealed the white dwarf population in GCs (about 10% of the total mass of a GC).

• The WD sequence is not a mass sequence but rather a cooling sequence.

• Might be possible to use the WD sequence as a check on cluster ages.

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Distribution in the Milky Way

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MetallicitiesOne definition of metallicity:Another scheme:(X,Y,Z) = fractional abundances by

mass of H, He, everything elseSolar = (0.70,0.28.0.02)

Fe Fe

H H

N NFe log logH N N

⎛ ⎞ ⎛ ⎞⎡ ⎤ = −⎜ ⎟ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎝ ⎠

Detailed studies reveal that α-process elements from core collapse SN are more enriched than Fe => GCs enriched preferentially by massive stars

Horizontal Branch14 Gyr isochrones

Line width indicates Z: .0001,.001,.006

Solid have Y=0.2, dotted have Y=0.3

Spread on HB implies a spread in mass loss.

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Ages• As recently as 1996 entire

Annual Reviews articles were dedicated to discussions of GC ages which ranged from 10 to 20 Gyr.

• Ages are now important as markers of halo collapse

• Accurate ages require good metallicity data, distances, and careful fitting of isochrones.

• Age spread greatest among more metal-rich clusters (~4Gyr?) which are ~2 Gyr younger than metal-poor clusters

• MS turn-off best age indicator – note similarity of shapes (and hence colors) of the giant branch.

MV(TO) = 2.70 log(t/Gyr) + 0.30[Fe/H] + 1.41

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GC Luminosity Functions• Source confusion has plagued attempts to

produce reliable LFs• Converting LFs to mass functions reveals slope

variations from cluster to cluster with shallower slopes for more metal-rich clusters

• Slopes most strongly correlated with cluster location in the galaxy => low mass stars may be stripped away by interactions with the disk so a universal IMF may hold.

GCField

Deficit in GCs as these luminosities are on the giant branch

Metal-poor

Metal-rich

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GC Luminosity Profiles• Multi-mass King model fits the data but

has a number of parameters • Can test the multi-mass King model by

looking for other evidence of energy equipartition

If equipartition prevails, more massive stars will be moving more slowly and found preferentially near the centers of clusters. The LFs in the figure to the right were measured at three different radii and the larger radii clearly have more low mass stars. Note that mass segregation enhances the loss of low mass stars due to tidal stripping.

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Central Cusps

Globular clusters have sufficiently high core stellar densities and are old enough that it is surprising that only about 20% of GCs show central cusps. Naively one would expect that all would have experienced core collapse.

What “re-inflates” the cores?

Could black holes cause the central cusps?

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Internal Kinematics

• Because of the spherical shapes and modest number of stars, globular clusters were a target for some of the first N-body simulations

• Star densities in the core are sufficiently high a number of interesting dynamical effects may be seen:– Energy transfer between binaries and the cluster system– Core collapse– Mass segregation– Merging of stars:

• Blue stragglers have masses of ~1.3M so they should not be present given the current ages of clusters.

• Blue stragglers are concentrated towards the centers of clusters• Models of stellar collisions resulting in remnants match the properties

of blue stragglers

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Correlation of Properties w/ Location

• Initial supposition was the GCs formed during the initial collapse of the proto-Milky Way (ELS model)

• More metal-rich clusters are concentrated towards the plane and have a significant component of rotation in their motions.

• More metal-rich clusters also appear to be slightly younger than more metal-poor clusters

• Cluster formation no doubt traced some of the early collapse of the galaxy, but the picture is more complicated and evidence from GC systems around other galaxies suggests that clusters can form in major star formation events (eg., starbursts, mergers)

• Also need to consider possibility that a cluster might be the remnant of accretion of a satellite galaxy

Zinn 1985 ApJ 293 424

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Do GCs Harbor Black Holes?

• GCs could anchor the low mass end of the M-σ relation

• In principle, 3-D velocity distributions could be observed for GCs

• So far best data come from HST proper motion measurements

(McLaughlin et al. 2006)• Upper limit on MBH for

47 Tuc is ~1000M

25M15 result has been retracted.

X

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Globular Clusters in other Galaxies• GCs form when galaxies have starbursts, not in more passive star

formation as seen in disks so they should reflect major episodes of star formation in the past and the formation of the most massiveparts of galaxies, the spheroids.

• Many extragalactic globular cluster systems have been observed (see Brodie and Strader ARAA 2006)

M104’s clustersystem from Spitler et al. 2006. Blue crosses denote metal-poor clusters and red circles denote metal-rich clusters.

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Bimodal Distributions Common• Just as the MW globular population shows blue and red (metal-

poor and metal-rich) groups, so do many populations around other galaxies.

GCs around M87Examples from M31’s GC population

Intermediate age & metallicity

Old & metal-rich

Old & metal-poor

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Bigger Galaxies = More Globular Clusters

• T is the number of clusters per 109 M so the relative numbers of clusters increase with galaxy mass, not just the absolute number.• Why there are more GCs per unit mass in more massive galaxies is unclear:Does this rule out formation of giant ellipticals from mergers of disk galaxies? Is this the result of more early cluster formation in the deeper DM haloes that produce giant ellipticals? Does environment play a role?

Squares = cluster galaxies, circles = field

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Relationship to Galaxy Mass/Luminosity

• Metallicity of globular clusters is correlated with the mass of the parent galaxy.

• This correlation provides a significant constraint on galaxy formation models.

Peak GC metallicity vs galaxy MB for red and blue clusters.

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Radial Distribution of GCs

• Just as in the MW, more metal rich clusters tend to be more centrally concentrated around the parent galaxy

• Metal-rich clusters follow the light distribution closely suggesting that these clusters formed at about the same time as the bulk of the stars.

Filled = metal-rich globulars

Open = metal-poor

NGC 1399

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Formation of GCs

• Low metallicity clusters must have formed very early, around z~10-15 and their formation must have been truncated by reionization. The minimum masses that could cool and collapse, ~108 M => ~106 M in stars, are strongly suggestive of present day GCs.

• DM simulations can reproduce many of the properties of low metallicity cluster populations

• High metallicity clusters are much more complicated –formation of clusters in mergers and starbursts suggests that these clusters have a varied formation history

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Antennae Clusters

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DM Simulation

• Green in this simulation at z=12 represents regions with sufficiently high density that they can cool and form stars

• Boxes show such peaks and where they end up at z=0

Z=12Z=0

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DM Simulation of MW Clusters

Radial distribution of metal-poor clusters in the MW can be reproduced if GCs form in >2.5σ peaksHave a mass > 2x108 MFormed stars by z~10

Interesting note: JWST may have sufficient sensitivity to detect newly formed GCs

From Moore et al. 2006