# Modeling of the Initial Mass Function Using the Metropolis−Hastings Algorithm

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TL;DR: In this article, a left truncated beta probability density function was introduced to demonstrate the advantage of introducing a four-parameter initial mass function for the stars and the presence of brown dwarfs.

Abstract: The initial mass function (IMF) for the stars is usually fitted by three straight lines, which means seven parameters. The presence of brown dwarfs (BD) increases to four the straight lines and to nine the parameters. Another common fitting function is the lognormal distribution, which is characterized by two parameters. This paper is devoted to demonstrating the advantage of introducing a left truncated beta probability density function, which is characterized by four parameters. The constant of normalization, the mean, the mode and the distribution function are calculated for the left truncated beta distribution. The normal-beta (NB) distribution which results from convolving independent normally distributed and beta distributed components is also derived. The chisquare test and the K-S test are performed on a first sample of stars and BDs which belongs to the massive young cluster NGC 6611 and on a second sample which represents the star’s masses of the cluster NGC 2362.

14 citations

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TL;DR: In this paper, the authors investigated the effect of the evolution of the stellar initial mass function (IMF) with increasing redshift on the formation of massive clusters in star burst galaxies.

Abstract: Theoretical and indirect observational evidences suggest that stellar initial mass function (IMF) increases with redshift. On the other hand star formation rates (SFR) may be as high as 100 $M_{\odot}$ yr$^{-1}$ in star burst galaxies. These may lead to formation of massive clusters hence massive stars to make the integrated galactic stellar initial mass function (IGIMF) top heavy (i.e. proportion of massive stars is higher than less massive stars). We investigate the joint effect of evolving IMF and several measures of SFR in dependence of galaxy wide IMF. The resulting IGIMF have slopes $\alpha_{2,IGIMF}$ in the high mass regime, which is highly dependent on the minimum mass of the embedded cluster ($M_{ecl,min}$), star formation rates and mass spectrum indices of embedded clusters (viz. $\beta$). It is found that for z $\sim$ 0 - 2, $\alpha_{2,IGIMF}$ becomes steeper (i.e. bottom heavy), for z $\sim$ 2 - 4, $\alpha_{2,IGIMF}$ becomes flatter (i.e. top heavy ) and from z $\sim$ 4 onwards $\alpha_{2,IGIMF}$ becomes again steeper. The effects are faster for higher values of $\beta$. $\alpha_{2,IGIMF}$ is flatter also for higher values of $M_{ecl,min}$. All these effects might be counted for the joint effect of increasing temperature of the ambient medium as well as varying SFR with increasing redshift.

12 citations

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TL;DR: In this article, the authors investigate the evolution of the stellar initial mass function (IMF) and varying indices of β for the integrated galactic initial mass functions, in relation to several measures of star formation rates of galaxies at various redshifts by random simulation.

Abstract: Theoretical as well as observational studies suggest that the stellar initial mass func-tion (IMF) might become top heavy with increasing redshift. Embedded cluster massfunction is a power law having index β, whose value still remains controversial. Inthe present work, we investigate the eﬀect of evolving IMF and varying indices ofβ for the integrated galactic initial mass function, in relation to several measures ofstar formation rates of galaxies at various redshifts by random simulation. The result-ing IGIMF is segmented power law at various redshifts having slopes α 1,IGIMF andα 2,IGIMF with a turnover at a characteristic mass m c ′ . These diﬀer from the stel-lar initial mass functions with slopes α 1,IMF , α 2,IMF , and characteristic masses m c for diﬀerent values of redshift z, β, minimum and maximum masses of the embeddedclusters.Key words: galaxies:star clusters: general - galaxies:evolution 1 INTRODUCTIONThe form of stellar initial mass function is of considerabledebate in the present era as it describes the nature ofstellar population, the ratio of high mass to low mass starsand inﬂuences the dynamical evolution of star clusters aswell as star formation history of the whole galaxy. Usuallyit is derived using observed luminosity function togetherwith an assumed mass-to-light ratio for the stars underconsideration. Generally, IMFs, as suggested by variousauthors, are either of Salpeter type (Salpeter 1955) orconsists of segmented power laws (Scalo 1986; Kroupa 2001;Chabrier 2003) with a turnover at some characteristic massm

10 citations

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TL;DR: In this paper, the results of numerical N-body calculations which simulate the dynamical evolution of young clusters as they emerge from molecular clouds have been presented, and the combination of these parameters which result in the production of bound stellar groups after the gas not used in star formation is completely dispersed.

Abstract: We present the results of numerical N-body calculations which simulate the dynamical evolution of young clusters as they emerge from molecular clouds. We follow the evolution of initially virialized stellar systems of 50 and, in some cases, 100 stars from the point in time immediately after the stars have formed in a cloud until a time long after all the residual star-forming gas has been dispersed from the system. By varying the star formation efficiency and the gas dispersal time for each model, we determined the combination of these parameters which result in the production of bound stellar groups after the gas not used in star formation is completely dispersed.

7 citations

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TL;DR: In this paper, a stochastic model has been developed to study the hierarchical fragmentation process of young massive clusters in external galaxies considering close binary components along with individual ones, and the resulting mass spectra computed at different projected distances, show signature of mass segregation.

Abstract: A stochastic model has been developed to study the hierarchical fragmentation process of young massive clusters in external galaxies considering close binary components along with individual ones. Stellar masses for individual ones have been generated from truncated Pareto distribution and stellar masses for close binary components have been generated from a truncated Bi-variate Gumbel Exponential distribution. The above distribution is identified by fitting the observed bi-variate distribution of masses of eclipsing binary stars computed from the light curves catalogued in the package Binary Maker 3.0. The resulting mass spectra computed at different projected distances, show signature of mass segregation. Degree of mass segregation becomes reduced due to the inclusion of binary fraction. This might be due to the reduction of massive stars and inclusion of less massive stars rather than inclusion of single massive stars and the effect of line of sight length projected to an observer.

5 citations

##### References

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TL;DR: In this article, the spectral evolution of stellar populations at ages between 100,000 yr and 20 Gyr at a resolution of 3 A across the whole wavelength range from 3200 to 9500 A for a wide range of metallicities.

Abstract: We present a new model for computing the spectral evolution of stellar populations at ages between 100,000 yr and 20 Gyr at a resolution of 3 A across the whole wavelength range from 3200 to 9500 A for a wide range of metallicities. These predictions are based on a newly available library of observed stellar spectra. We also compute the spectral evolution across a larger wavelength range, from 91 A to 160 micron, at lower resolution. The model incorporates recent progress in stellar evolution theory and an observationally motivated prescription for thermally-pulsing stars on the asymptotic giant branch. The latter is supported by observations of surface brightness fluctuations in nearby stellar populations. We show that this model reproduces well the observed optical and near-infrared colour-magnitude diagrams of Galactic star clusters of various ages and metallicities. Stochastic fluctuations in the numbers of stars in different evolutionary phases can account for the full range of observed integrated colours of star clusters in the Magellanic Clouds. The model reproduces in detail typical galaxy spectra from the Early Data Release (EDR) of the Sloan Digital Sky Survey (SDSS). We exemplify how this type of spectral fit can constrain physical parameters such as the star formation history, metallicity and dust content of galaxies. Our model is the first to enable accurate studies of absorption-line strengths in galaxies containing stars over the full range of ages. Using the highest-quality spectra of the SDSS EDR, we show that this model can reproduce simultaneously the observed strengths of those Lick indices that do not depend strongly on element abundance ratios [abridged].

9,707 citations

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TL;DR: In this paper, the evolutionary significance of the observed luminosity function for main-sequence stars in the solar neighborhood is discussed and it is shown that stars move off the main sequence after burning about 10 per cent of their hydrogen mass and that stars have been created at a uniform rate in a solar neighborhood for the last five billion years.

Abstract: The evolutionary significance of the observed luminosity function for main-sequence stars in the solar neighborhood is discussed. The hypothesis is made that stars move off the main sequence after burning about 10 per cent of their hydrogen mass and that stars have been created at a uniform rate in the solar neighborhood for the last five billion years. Using this hypothesis and the observed luminosity function, the rate of star creation as a function of stellar mass is calculated. The total number and mass of stars which have moved off the main sequence is found to be comparable with the total number of white dwarfs and with the total mass of all fainter main-sequence stars, respectively.

8,092 citations

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TL;DR: This new edition contains five completely new chapters covering new developments and has sold 4300 copies worldwide of the first edition (1999).

Abstract: We have sold 4300 copies worldwide of the first edition (1999). This new edition contains five completely new chapters covering new developments.

6,854 citations

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01 Jan 2005

3,063 citations

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TL;DR: In this paper, the authors quantify the complex interdependence of stellar binarity, the stellar mass-luminosity relation, the mass function, the colour-magnitude relation and the Galactic disc structure, all of which must be understood when analysing star-count data and stellar luminosity functions.

Abstract: We quantify the complex interdependence of stellar binarity, the stellar mass-luminosity relation, the mass function, the colour-magnitude relation and Galactic disc structure, all of which must be understood when analysing star-count data and stellar luminosity functions. We derive a mass-M V relation and a model for the change of stellar luminosity with changes in chemical abundance and age. Combination of this with detailed modelling of all astrophysical and observational contributions to the Malmquist scatter allows us to model star-count data without approximating Malmquist corrections. We show for the first time that a single mass function and normalization explain the stellar distribution towards both Galactic poles, as well as the distribution of stars within a distance of 5.2 pc of the Sun

1,646 citations