Initial Mass Function Of Stars Research Articles

Gravitational waves from scattering of stellar-mass black holes in galactic nuclei

Stellar-mass black holes (BHs) are expected to segregate and form a steep density cusp around supermassive black holes (SMBHs) in galactic nuclei. We follow the evolution of a multimass system of BHs and stars by numerically integrating the Fokker–Planck energy diffusion equations for a variety of BH mass distributions. We find that the BHs ‘self-segregate’, and that the rarest, most massive BHs dominate the scattering rate closest to the SMBH (10 −1 pc). BH–BH binaries form out of gravitational wave emission during BH encounters. We find that the expected rate of BH coalescence events detectable by Advanced LIGO is ∼1–10 2 yr −1 , depending on the initial mass function of stars in galactic nuclei and the mass of the most massive BHs. We find that the actual merger rate is likely ∼10 times larger than this due to the intrinsic scatter of stellar densities in many different galaxies. The BH binaries that form this way in galactic nuclei have significant eccentricities as they enter the LIGO band (90 per cent with e> 0.9), and are therefore distinguishable from other binaries, which circularize before becoming detectable. We also show that eccentric mergers can be detected to larger distances and greater BH masses than circular mergers, up to ∼700 M � . Future ground-based gravitational wave observatories will be able to constrain both the mass function of BHs and stars in galactic nuclei.

The Effect of the Initial Mass Function of Stars on the Burst Rate of GRBs

For the mechanism of production of γ-ray bursts (GRBs) it is rather generally recognized that the long-term γ-ray burst (LGRB) originates from the deaths of massive stars while the short-term γ-ray burst (SGRB) originates from the merging of close binaries. Therefore the speculation naturally follows that the number of LGRBs is directly proportional to the star formation rate (SFR). However, it is indicated from recent data analyses that this speculation does not fit the observations very well. It is considered that only massive stars with masses greater than a certain critical mass can produce the LGRB, so the initial mass function (IMF) of stars can significantly affect the production rate of LGRBs. In this paper it is considered that the IMF of stars can be used to explain the observed number distribution of the LGRBs with the redshift, and this has led to some good results.

DISK MASSES IN THE EMBEDDED AND T TAURI PHASES OF STELLAR EVOLUTION

(Abridged). Motivated by a growing concern that masses of circumstellar disks may have been systematically underestimated by conventional observational methods, we present a numerical hydrodynamics study of time-averaged disk masses (<M_d>) around low-mass Class 0, Class I, and Class II objects. Mean disk masses (\overline{M}_d}) are then calculated by weighting the time-averaged disk masses according to the corresponding stellar masses using a power-law weight function with a slope typical for the Kroupa initial mass function of stars. Two distinct types of disks are considered: self-gravitating disks, in which mass and angular momentum are redistributed exclusively by gravitational torques, and viscous disks, in which both the gravitational and viscous torques are at work. We find that self-gravitating disks have mean masses that are slowly increasing along the sequence of stellar evolution phases. More specifically, Class 0/I/II self-gravitating disks have mean masses \overline{M}_d=0.09, 0.10, and 0.12 M_sun, respectively. Viscous disks have similar mean masses (\overline{M}_d=0.10-0.11 M_sun) in the Class 0/I phases but almost a factor of 2 lower mean mass in the Class II phase (\overline{M}_d,CII=0.06 M_sun). In each evolution phase, time-averaged disk masses show a large scatter around the mean value. Our obtained mean disk masses are larger than those recently derived by Andrews & Williams and Brown et al., regardless of the physical mechanisms of mass transport in the disk.

Fallback and Black Hole Production in Massive Stars

The compact remnants of core collapse supernovae - neutron stars and black holes - have properties that reflect both the structure of their stellar progenitors and the physics of the explosion. In particular, the masses of these remnants are sensitive to the density structure of the presupernova star and to the explosion energy. To a considerable extent, the final mass is determined by the ``fallback'', during the explosion, of matter that initially moves outwards, yet ultimately fails to escape. We consider here the simulated explosion of a large number of massive stars (10 to 100 \Msun) of Population I (solar metallicity) and III (zero metallicity), and find systematic differences in the remnant mass distributions. As pointed out by Chevalier(1989), supernovae in more compact progenitor stars have stronger reverse shocks and experience more fallback. For Population III stars above about 25 \Msun and explosion energies less than $1.5 \times 10^{51}$ erg, black holes are a common outcome, with masses that increase monotonically with increasing main sequence mass up to a maximum hole mass of about 35 \Msun. If such stars produce primary nitrogen, however, their black holes are systematically smaller. For modern supernovae with nearly solar metallicity, black hole production is much less frequent and the typical masses, which depend sensitively on explosion energy, are smaller. We explore the neutron star initial mass function for both populations and, for reasonable assumptions about the initial mass cut of the explosion, find good agreement with the average of observed masses of neutron stars in binaries. We also find evidence for a bimodal distribution of neutron star masses with a spike around 1.2 \Msun (gravitational mass) and a broader distribution peaked around 1.4 \Msun.

Evidence of recent changes in the local Galactic IMF of stars and substars

This paper calculates the stellar–substellar initial mass function (IMF) based on data from low-luminosity stars and substars. We find the mass spectrum for the stars and substars less than 10 million years in age, and we estimate the low-mass part of the stellar–substellar IMF for 10 million years ago.

On the Evolution of the Dense Core Mass Function

The mass distributions of dense cores in star-forming regions are measured to have a shape similar to the initial mass function of stars. This has been generally interpreted to mean that the constituent cores will form individual stars or stellar systems at a nearly constant star formation efficiency. This article presents a series of numerical experiments evolving distributions of dense cores into stars to quantify the effects of stellar multiplicity, global core fragmentation, and a varying star formation efficiency. We find that the different evolutionary schemes have an overall small effect on the shape of the resulting distribution of stars. Our results imply that at the current level of observational accuracy, the comparison between the mass functions of dense cores and stars alone is insufficient to discern between different evolutionary models. Observations over a wide range of mass scales including the high- or low-mass tails of these distributions have the largest potential for discerning between different core evolutionary schemes.

Importance of Thermodynamics for Fragmentation and Star Formation

We discuss results from numerical simulations of star formation under various environmental conditions ranging from the turbulent interstellar medium to low-mass halos in the early universe. The thermodynamic behavior of the star-forming gas plays a crucial role in fragmentation and determines the stellar mass function as well as the dynamic properties of the nascent stellar cluster. The thermodynamic state of the gas is a result of the balance between heating and cooling processes, which in turn are determined by atomic and molecular physics and by chemical abundances. Features in the effective equation of state of the gas, such as a transition from a cooling to a heating regime, define a characteristic mass scale for fragmentation and so set the peak of the initial mass function of stars (IMF). As it is based on fundamental physical quantities and constants, this is an attractive approach to explain the apparent universality of the IMF in the solar neighborhood as well as the transition from purely primordial high-mass star formation to the low-mass mode observed today.

Formation of Stellar Clusters and the Importance of Thermodynamics for Fragmentation

AbstractWe discuss results from numerical simulations of star cluster formation in the turbulent interstellar medium (ISM). The thermodynamic behavior of the star-forming gas plays a crucial role in fragmentation and determines the stellar mass function as well as the dynamic properties of the nascent stellar cluster. This holds for star formation in molecular clouds in the solar neighborhood as well as for the formation of the very first stars in the early universe. The thermodynamic state of the ISM is a result of the balance between heating and cooling processes, which in turn are determined by atomic and molecular physics and by chemical abundances. Features in the effective equation of state of the gas, such as a transition from a cooling to a heating regime, define a characteristic mass scale for fragmentation and so set the peak of the initial mass function of stars (IMF). As it is based on fundamental physical quantities and constants, this is an attractive approach to explain the apparent universality of the IMF in the solar neighborhood as well as the transition from purely primordial high-mass star formation to the more normal low-mass mode observed today.

Two Regimes of Turbulent Fragmentation and the Stellar Initial Mass Function from Primordial to Present‐Day Star Formation

The Padoan and Nordlund model of the stellar initial mass function (IMF) is derived from low order statistics of supersonic turbulence, neglecting gravity (e.g. gravitational fragmentation, accretion and merging). In this work the predictions of that model are tested using the largest numerical experiments of supersonic hydrodynamic (HD) and magneto-hydrodynamic (MHD) turbulence to date (~1000^3 computational zones) and three different codes (Enzo, Zeus and the Stagger Code). The model predicts a power law distribution for large masses, related to the turbulence energy power spectrum slope, and the shock jump conditions. This power law mass distribution is confirmed by the numerical experiments. The model also predicts a sharp difference between the HD and MHD regimes, which is recovered in the experiments as well, implying that the magnetic field, even below energy equipartition on the large scale, is a crucial component of the process of turbulent fragmentation. These results suggest that the stellar IMF of primordial stars may differ from that in later epochs of star formation, due to differences in both gas temperature and magnetic field strength. In particular, we find that the IMF of primordial stars born in turbulent clouds may be narrowly peaked around a mass of order 10 solar masses, as long as the column density of such clouds is not much in excess of 10^22 cm^-2.

Weighing the young stellar discs around Sgr A

It is believed that young massive stars orbiting Sgr A* in two stellar discs on scales of ∼ 0.1–0.2 parsec were formed either farther out in the Galaxy and then quickly migrated inwards or in situ in a massive self-gravitating disc. Comparing N-body evolution of stellar orbits with observational constraints, we set upper limits on the masses of the two stellar systems. These masses turn out to be a few times lower than the expected total stellar mass estimated from the observed young high-mass stellar population and the standard galactic initial mass function (IMF). If these stars were formed in situ, in a massive self-gravitating disc, our results suggest that the formation of low-mass stars was suppressed by a factor of at least a few, requiring a top-heavy IMF for stars formed near Sgr A*.

The Effect of Star Formation History on the Inferred Stellar Initial Mass Function

Peaks and lulls in the star formation rate (SFR) over the history of the Galaxy produce plateaus and declines in the present-day mass function (PDMF) where the main-sequence lifetime overlaps the age and duration of the SFR variation. These PDMF features can be misinterpreted as the form of the intrinsic stellar initial mass function (IMF) if the star formation rate is assumed to be constant or slowly varying with time. This effect applies to all regions that have formed stars for longer than the age of the most massive stars, including OB associations, star complexes, and especially galactic field stars. Related problems may apply to embedded clusters. Evidence is summarized for temporal SFR variations from parsec scales to entire galaxies, all of which should contribute to inferred IMF distortions. We give examples of various star formation histories to demonstrate the types of false IMF structures that might be seen. These include short-duration bursts, stochastic histories with lognormal amplitude distributions, and oscillating histories with various periods and phases. The inferred IMF should appear steeper than the intrinsic IMF over mass ranges where the stellar lifetimes correspond to times of decreasing SFRs; shallow portions of the inferred IMF correspond to times of increasing SFRs. If field regions are populated by dispersed clusters and defined by their low current SFRs, then they should have steeper inferred IMFs than the clusters. The SFRs required to give the steep field IMFs in the LMC and SMC are determined. Structure observed in several determinations of the Milky Way field star IMF can be accounted for by a stochastic and bursty star formation history.

The Variation of Integrated Star Initial Mass Functions among Galaxies

The integrated galaxial initial mass function (IGIMF) is the relevant distribution function containing the information on the distribution of stellar remnants, the number of supernovae, and the chemical enrichment history of a galaxy. Since most stars form in embedded star clusters with different masses, the IGIMF becomes an integral of the assumed (universal or invariant) stellar IMF over the embedded star cluster mass function (ECMF). For a range of reasonable assumptions about the IMF and the ECMF we find the IGIMF to be steeper (containing fewer massive stars per star) than the stellar IMF, but below a few solar masses it is invariant and identical to the stellar IMF for all galaxies. However, the steepening sensitively depends on the form of the ECMF in the low-mass regime. Furthermore, observations indicate a relation between the star formation rate of a galaxy and the most massive young stellar cluster in it. The assumption that this cluster mass marks the upper end of a young-cluster mass function leads to a connection of the star formation rate and the slope of the IGIMF above a few solar masses. The IGIMF varies with the star formation history of a galaxy. Notably, large variations of the IGIMF are evident for dE, dIrr, and LSB galaxies with a small to modest stellar mass. We find that for any galaxy the number of supernovae per star (NSNS) is suppressed relative to that expected for a Salpeter IMF. Dwarf galaxies have a smaller NSNS than massive galaxies. For dwarf galaxies the NSNS varies substantially depending on the galaxy assembly history and the assumptions made about the low-mass end of the ECMF. The findings presented here may be of some consequence for the cosmological evolution of the number of supernovae per low-mass star and the chemical enrichment of galaxies of different mass.

The stellar mass spectrum from non-isothermal gravoturbulent fragmentation

Identifying the processes that determine the initial mass function of stars (IMF) is a fundamental problem in star formation theory. One of the major uncertainties is the exact chemical state of the star forming gas and its influence on the dynamical evolution. Most simulations of star forming clusters use an isothermal equation of state (EOS). However, theoretical predictions and observations suggest that the effective polytropic exponent gamma in the EOS varies with density. We address these issues and study the effect of a piecewise polytropic EOS on the formation of stellar clusters in turbulent, self-gravitating molecular clouds using three-dimensional, smoothed particle hydrodynamics simulations. To approximate the results of published predictions of the thermal behavior of collapsing clouds, we increase the polytropic exponent gamma from 0.7 to 1.1 at some chosen density n_c, which we vary. The change of thermodynamic state at n_c selects a characteristic mass scale for fragmentation M_ch, which we relate to the peak of the observed IMF. Our investigation generally supports the idea that the distribution of stellar masses depends mainly on the thermodynamic state of the star-forming gas. The thermodynamic state of interstellar gas is a result of the balance between heating and cooling processes, which in turn are determined by fundamental atomic and molecular physics and by chemical abundances. Given the abundances, the derivation of a characteristic stellar mass can thus be based on universal quantities and constants.

The Pleiades mass function: Models versus observations

Two stellar-dynamical models of binary-rich embedded proto-Orion-Nebula-type clusters that evolve to Pleiades-like clusters are studied with an emphasis on comparing the stellar mass function with observational constraints. By the age of the Pleiades (about 100 Myr) both models show a similar degree of mass segregation which also agrees with observational constraints. This thus indicates that the Pleiades is well relaxed and that it is suffering from severe amnesia. It is found that the initial mass function (IMF) must have been indistinguishable from the standard or Galactic-field IMF for stars with mass m? 2 M ○. , provided the Pleiades precursor had a central density of about 10 4.8 stars/pc 3 . A denser model with 10 5.8 stars/pc 3 also leads to reasonable agreement with observational constraints, but owing to the shorter relaxation time of the embedded cluster it evolves through energy equipartition to a mass-segregated condition just prior to residual-gas expulsion. This model consequently preferentially loses low-mass stars and brown dwarfs (BDs), but the effect is not very pronounced. The empirical data indicate that the Pleiades IMF may have been steeper than the Salpeter for stars with m? 2 M ○. .

On the determination of age and mass functions of stars in young open star clusters from the analysis of their luminosity functions

Based on the CCD observations of remote young open clusters NGC 2383, NGC 2384, NGC 4103, NGC 4755, NGC 7510 and Hogg 15, we constructed their observed luminosity functions (LFs). The observed LFs are corrected for field star contamination determined with the help of galactic star count model. In the case of Hogg 15 and NGC 2383 we also considered the additional contamination from neighbouring clusters NGC 4609 and NGC 2384 respectively. These corrections provided the realistic pattern of cluster LF in the vicinity of the MS turn on point and at fainter magnitudes, revealed the so called H-feature arising due to transition of the Pre-MS phase to MS, which is dependent on the cluster age. The theoretical LFs were constructed representing a cluster population model with continuous star formation for a short time scale and a power law Initial Mass Function (IMF) and these were fitted to the observed LF. As a result we are able to determine for each cluster a set of parameters, describing cluster population (the age, duration of star formation, IMF slope and percentage of field star contamination). It was found that in spite of the non-monotonic behaviour of observed LFs, cluster IMFs can be described as the power law functions with slopes similar to Salpeter's value. The present MS turn on cluster ages are several times lower than those derived from the fitting of theoretical isochrones to the turn off region of the upper Main Sequences.

The Initial Mass Function and Stellar Content of NGC 3603

We present UBVRI and Hα photometry of the extremely compact, starburst cluster NGC 3603. Ground-based images, Hubble Space Telescope (HST) archival data, as well as Chandra X-ray data have been used for this study. We present, for the first time, optical color-magnitude diagrams (CMDs) for the stars in the cluster core down to V = 22 mag. A well-defined main sequence (MS) as well as low-mass pre–main-sequence (PMS) stars can clearly be seen in the CMDs. This result confirms the finding by Eisenhauer et al. that low-mass stars are forming in the starburst cluster. We also derive an age (1 ± 1 Myr) and distance of the cluster (V0 - MV = 14.2 ± 0.2 mag, i.e., d = 6.9 ± 0.6 kpc). The interstellar reddening shows a minimum value [E(B-V) = 1.25 mag] at the core (r ≤ 02) and increases rapidly up to E(B-V) ≈ 1.8 mag or even higher with increasing distance from the center, which suggests the presence of a wind-driven cavity near the cluster center as found by Frogel et al. and Clayton. We also determine the radius of the cluster (r ≈ 2') from the surface-density profile of bright stars and X-ray sources. To investigate the initial mass function (IMF) of the cluster, we performed a completeness test for the core region. Using the theoretical mass-luminosity relation of MS as well as PMS stars, we derived the IMF of stars imaged with the HST/PC1. The surface density of stars in the core is enormously high and decreases gradually with increasing distance from the center. The slope of the IMF also shows radial variation: fairly flat in the core (Γ = -0.5 ± 0.1 at r ≤ 01) and gradually steepening (Γ = -0.8 ± 0.2 at r = 01 ~ 02 and Γ = -1.2 ± 0.2 at r > 02 of HST/PC1). The stars in the halo region of NGC 3603 (outside of HST/PC1) seem to be slightly older (age ≈ 5 Myr) and presumably formed before the stars in the core.

The Variation of the Initial Mass Function in Clusters

I review recent measurements of the initial mass function of stars and brown dwarfs in star-forming regions and open clusters and summarize the implications of these data for theories of star formation.

The Mass Spectrum of Metal‐free Stars Resulting from Photodissociation Feedback: A Scenario for the Formation of Low‐Mass Population III Stars

The initial mass function (IMF) of metal-free stars that form in the initial starburst of massive (virial temperatures >10^4K) metal-free protogalaxies is studied. In particular, we focus on the effect of H2 photodissociation by pre-existing stars on the fragmentation mass scale, presumedly determined by the Jeans mass at the end of the initial free-fall phase, i.e., at the so-called ``loitering phase,'' characterized by the local temperature minimum. Photodissociation diminishes the Jeans mass at the loitering phase, thereby reducing the fragmentation mass scale of primordial clouds. Thus, in a given cloud, far ultraviolet (FUV) radiation from the first star, which is supposedly very massive (about 10^3Msun), reduces the mass scale for subsequent fragmentation. Through a series of similar processes the IMF for metal-free stars is established. If FUV radiation exceeds a threshold level, the star-forming clumps collapse solely through atomic cooling. Correspondingly, the fragmentation scale drops discontinuously from a few time 10Msun to sub-solar scales. In compact clouds (>1.6kpc for clouds of gas mass 10^8Msun), this level of radiation field is attained, and sub-solar mass stars are formed even in a metal-free environment. Consequently, the IMF becomes bi-modal, with peaks at a few tenths of Msun and a few times 10Msun. The high-mass portion of the IMF is found to be a very steep function of the stellar mass, xi_high(m) being proportinal to m^{-5}. Therefore, the typical mass scale of metal-free stars is significantly smaller than that of the very first stars. Also we study the thermal instability in collapsing primordial prestellar clumps, and discuss why the thermal instability occuring during the three-body H2 formation does not appear to manifest itself in causing further fragmentation of such clumps.

Galactic‐Field Initial Mass Functions of Massive Stars

Over the past years observations of young and populous star clusters have shown that the stellar initial mass function (IMF) appears to be an invariant featureless Salpeter power law with an exponent ? = 2.35 for stars more massive than a few M?. A consensus has also emerged that most, if not all, stars form in stellar groups and star clusters and that the mass function of young star clusters in the solar neighborhood and in interacting galaxies can be described, over the mass range of a few 10 to 107 M?, as a power law with an exponent ? ? 2. These two results imply that galactic-field IMFs for early-type stars cannot, under any circumstances, be a Salpeter power law, but that they must have a steeper exponent, ?field 2.8. This has important consequences for the distribution of stellar remnants and for the chemodynamical and photometric evolution of galaxies.

Galactic Stellar and Substellar Initial Mass Function11The page charges for this Review were partially covered by a generous gift from a PASP supporter.

ABSTRACTWe review recent determinations of the present‐day mass function (PDMF) and initial mass function (IMF) in various components of the Galaxy—disk, spheroid, young, and globular clusters—and in conditions characteristic of early star formation. As a general feature, the IMF is found to depend weakly on the environment and to be well described by a power‐law form for m≳1 M⊙ and a lognormal form below, except possibly for early star formation conditions. The disk IMF for single objects has a characteristic mass around mc ∼ 0.08 M⊙ and a variance in logarithmic mass σ ∼ 0.7, whereas the IMF for multiple systems has mc ∼ 0.2 M⊙ and σ ∼ 0.6. The extension of the single MF into the brown dwarf regime is in good agreement with present estimates of L‐ and T‐dwarf densities and yields a disk brown dwarf number density comparable to the stellar one, nBD ∼ n* ∼ 0.1 pc−3. The IMF of young clusters is found to be consistent with the disk field IMF, providing the same correction for unresolved binaries, confirming the fact that young star clusters and disk field stars represent the same stellar population. Dynamical effects, yielding depletion of the lowest mass objects, are found to become consequential for ages ≳130 Myr. The spheroid IMF relies on much less robust grounds. The large metallicity spread in the local subdwarf photometric sample, in particular, remains puzzling. Recent observations suggest that there is a continuous kinematic shear between the thick‐disk population, present in local samples, and the genuine spheroid one. This enables us to derive only an upper limit for the spheroid mass density and IMF. Within all the uncertainties, the latter is found to be similar to the one derived for globular clusters and is well represented also by a lognormal form with a characteristic mass slightly larger than for the disk, mc ∼ 0.2–0.3 M⊙, excluding a significant population of brown dwarfs in globular clusters and in the spheroid. The IMF characteristic of early star formation at large redshift remains undetermined, but different observational constraints suggest that it does not extend below ∼1 M⊙. These results suggest a characteristic mass for star formation that decreases with time, from conditions prevailing at large redshift to conditions characteristic of the spheroid (or thick disk) to present‐day conditions. These conclusions, however, remain speculative, given the large uncertainties in the spheroid and early star IMF determinations.These IMFs allow a reasonably robust determination of the Galactic present‐day and initial stellar and brown dwarf contents. They also have important galactic implications beyond the Milky Way in yielding more accurate mass‐to‐light ratio determinations. The mass‐to‐light ratios obtained with the disk and the spheroid IMF yield values 1.8–1.4 times smaller than for a Salpeter IMF, respectively, in agreement with various recent dynamical determinations. This general IMF determination is examined in the context of star formation theory. None of the theories based on a Jeans‐type mechanism, where fragmentation is due only to gravity, can fulfill all the observational constraints on star formation and predict a large number of substellar objects. On the other hand, recent numerical simulations of compressible turbulence, in particular in super‐Alfvénic conditions, seem to reproduce both qualitatively and quantitatively the stellar and substellar IMF and thus provide an appealing theoretical foundation. In this picture, star formation is induced by the dissipation of large‐scale turbulence to smaller scales through radiative MHD shocks, producing filamentary structures. These shocks produce local nonequilibrium structures with large density contrasts, which collapse eventually in gravitationally bound objects under the combined influence of turbulence and gravity. The concept of a single Jeans mass is replaced by a distribution of local Jeans masses, representative of the lognormal probability density function of the turbulent gas. Objects below the mean thermal Jeans mass still have a possibility to collapse, although with a decreasing probability.