Initial Mass Function Of Stars Research Articles

Blocking Metal Accretion onto Population III Stars by Stellar Wind

Abstract Low-mass population III (PopIII) stars of could survive up until the present. The nondetection of low-mass PopIII stars in our Galaxy has already put a stringent constraint on the initial mass function (IMF) of PopIII stars, suggesting that PopIII stars have a top-heavy IMF. On the other hand, some claim that the lack of such stars stems from metal enrichment of their surfaces by the accretion of heavy elements from the interstellar medium (ISM). We investigate the effects of the stellar wind on metal accretion onto low-mass PopIII stars because accretion of the local ISM onto the Sun is prevented by the solar wind, even for neutrals. The stellar wind and radiation of low-mass PopIII stars are modeled based on knowledge of nearby low-mass stellar systems, including our Sun. We find that low-mass PopIII stars traveling across the Galaxy form a stellar magnetosphere in most of their life. Once the magnetosphere is formed, most of the neutral interstellar particles are photoionized before reaching the stellar surface and are blown away by the wind. Especially, the accretion abundance of iron will be reduced by a factor of compared with Bondi–Hoyle–Lyttleton accretion. The metal accretion can enhance iron abundance [Fe/H] only up to ∼−14. This demonstrates that low-mass PopIII stars remain pristine and will be found as metal-free stars and that further searches for them are valuable in constraining the IMF of PopIII stars.

Converging on the Initial Mass Function of Stars

Understanding the origin of stellar masses—the initial mass function (IMF)— remains one of the most challenging problems in astrophysics. The IMF is a key ingredient for simulations of galaxy formation and evolution, and is used to calibrate star formation relations in extra-galactic observations. Modeling the IMF directly in hydrodynamical simulations has been attempted in several previous studies, but the most important processes that control the IMF remain poorly understood. This is because predicting the IMF from direct hydrodynamical simulations involves complex physics such as turbulence, magnetic fields, radiation feedback and mechanical feedback, all of which are difficult to model and the methods used have limitations in terms of accuracy and computational efficiency. Moreover, a physical interpretation of the simulated IMFs requires a numerically converged solution at high resolution, which has so far not been convincingly demonstrated. Here we present a resolution study of star cluster formation aimed at producing a converged IMF. We compare a set of magnetohydrodynamical (MHD) adaptive-mesh-refinement simulations with three different implementations of the thermodynamics of the gas: 1) with an isothermal equation of state (EOS), 2) with a polytropic EOS, and 3) with a simple stellar heating feedback model. We show that in the simulations with an isothermal or polytropic EOS, the number of stars and their mass distributions depend on the numerical resolution. By contrast, the simulations that employ the simple radiative feedback module demonstrate convergence in the number of stars formed and in their IMFs.

ELEMENT ABUNDANCES IN A GAS-RICH GALAXY AT z = 5: CLUES TO THE EARLY CHEMICAL ENRICHMENT OF GALAXIES∗

ABSTRACT Element abundances in high-redshift quasar absorbers offer excellent probes of the chemical enrichment of distant galaxies, and can constrain models for population III and early population II stars. Recent observations indicate that the sub-damped Lyα (sub-DLA) absorbers are more metal-rich than DLA absorbers at redshifts 0 < z < 3. It has also been suggested that DLA metallicity drops suddenly at z > 4.7. However, only three DLAs at z > 4.5 and no sub-DLAs at z > 3.5 have “dust-free” metallicity measurements of undepleted elements. We report the first quasar sub-DLA metallicity measurement at z > 3.5, from detections of undepleted elements in high-resolution data for a sub-DLA at z = 5.0. We obtain fairly robust abundances of C, O, Si, and Fe, using lines outside the Lyα forest. This absorber is metal-poor, with [O/H] = −2.00 ± 0.12, which is ≳4σ below the level expected from extrapolation of the trend for z < 3.5 sub-DLAs. The C/O ratio is times lower than in the Sun. More strikingly, Si/O is times lower than in the Sun, whereas Si/Fe is nearly (1.2 times) solar. This absorber does not display a clear alpha/Fe enhancement. Dust depletion may have removed more Si from the gas phase than is common in the Milky Way interstellar medium, which may be expected if high-redshift supernovae form more silicate-rich dust. C/O and Si/O vary substantially between different velocity components, indicating spatial variations in dust depletion and/or early stellar nucleosynthesis (e.g., population III star initial mass function). The higher velocity gas may trace an outflow enriched by early stars.

CORE-COLLAPSE SUPERNOVAE FROM 9 TO 120 SOLAR MASSES BASED ON NEUTRINO-POWERED EXPLOSIONS

ABSTRACT Nucleosynthesis, light curves, explosion energies, and remnant masses are calculated for a grid of supernovae (SNe) resulting from massive stars with solar metallicity and masses from 9.0 to 120 M ⊙ . The full evolution is followed using an adaptive reaction network of up to 2000 nuclei. A novel aspect of the survey is the use of a one-dimensional neutrino transport model for the explosion. This explosion model has been calibrated to give the observed energy for SN 1987A, using five standard progenitors, and for the Crab SN using a 9.6 M ⊙ progenitor. As a result of using a calibrated central engine, the final kinetic energy of the SN is variable and sensitive to the structure of each pre-SN star. Many progenitors with extended core structures do not explode, but become black holes (BHs), and the masses of exploding stars do not form a simply connected set. The resulting nucleosynthesis agrees reasonably well with the Sun provided that a reasonable contribution from SNe Ia is also allowed, but with a deficiency of light s-process isotopes. The resulting neutron star initial mass function has a mean gravitational mass near 1.4 M ⊙ . The average BH mass is about 9 M ⊙ if only the helium core implodes, and 14 M ⊙ if the entire pre-SN star collapses. Only ∼10% of SNe come from stars over 20 M ⊙ , and some of these are Type Ib or Ic. Some useful systematics of Type IIp light curves are explored.

M-dwarf binaries as tracers of star and brown dwarf formation

The separation distribution for M-dwarf binaries in the ASTRALUX survey is narrower and peaking at smaller separations than the distribution for solar-type binaries. This is often interpreted to mean that M-dwarfs constitute a continuous transition from brown dwarfs (BDs) to stars. Here a prediction for the M-dwarf separation distribution is presented, using a dynamical population synthesis (DPS) model in which "star-like" binaries with late-type primaries ($\lesssim1.5 M_{\rm sun}$) follow universal initial distribution functions and are dynamically processed in their birth embedded clusters. A separate "BD-like" population has both its own distribution functions for binaries and initial mass function (IMF), which overlaps in mass with the IMF for stars. Combining these two formation modes results in a peak on top of a wider separation distribution for late M-dwarfs consistent with the late ASTRALUX sample. The DPS separation distribution for early M-dwarfs shows no such peak and is in agreement with the M-dwarfs in Multiples (MinMS) data. We note that the latter survey is potentially in tension with the early ASTRALUX data. Concluding, the ASTRALUX and MinMS data are unable to unambiguously distinguish whether or not BDs are a continuous extension of the stellar IMF. Future observational efforts are needed to fully answer this interesting question. The DPS model predicts that binaries outside the sensitivity range of the ASTRALUX survey remain to be detected. For application to future data, we present a means to observationally measure the overlap of the putative BD-like branch and the stellar branch. We discuss the meaning of universal star formation and distribution functions.

The density variance–Mach number relation in isothermal and non-isothermal adiabatic turbulence

The density variance { Mach number relation of the turbulent interstellar medium is relevant for theoretical models of the star formation rate, eciency, and the initial mass function of stars. Here we use high-resolution hydrodynamical simulations with grid resolutions of up to 1024 3 cells to model compressible turbulence in a regime similar to the observed interstellar medium. We use Fyris Alpha, a shock-capturing code employing a high-order Godunov scheme to track large density variations induced by shocks. We investigate the robustness of the standard relation between the logarithmic density variance ( 2 ) and the sonic Mach number (M) of isothermal interstellar turbulence, in the non-isothermal regime. Specically, we test ideal gases with diatomic molecular ( = 7=5) and monatomic ( = 5=3) adiabatic indices. A periodic cube of gas is stirred with purely solenoidal forcing at low wavenumbers, leading to a fullydeveloped turbulent medium. We nd that as the gas heats in adiabatic compressions, it evolves along the relationship in the density variance { Mach number plane, but deviates signicantly from the standard expression for isothermal gases. Our main

THE TURBULENT DYNAMO IN HIGHLY COMPRESSIBLE SUPERSONIC PLASMAS

The turbulent dynamo may explain the origin of cosmic magnetism. While the exponential amplification of magnetic fields has been studied for incompressible gases, little is known about dynamo action in highly-compressible, supersonic plasmas, such as the interstellar medium of galaxies and the early Universe. Here we perform the first quantitative comparison of theoretical models of the dynamo growth rate and saturation level with three-dimensional magnetohydrodynamical simulations of supersonic turbulence with grid resolutions of up to 1024^3 cells. We obtain numerical convergence and find that dynamo action occurs for both low and high magnetic Prandtl numbers Pm = nu/eta = 0.1-10 (the ratio of viscous to magnetic dissipation), which had so far only been seen for Pm >= 1 in supersonic turbulence. We measure the critical magnetic Reynolds number, Rm_crit = 129 (+43, -31), showing that the compressible dynamo is almost as efficient as in incompressible gas. Considering the physical conditions of the present and early Universe, we conclude that magnetic fields need to be taken into account during structure formation from the early to the present cosmic ages, because they suppress gas fragmentation and drive powerful jets and outflows, both greatly affecting the initial mass function of stars.

ALMA OBSERVATIONS OF A HIGH-DENSITY CORE IN TAURUS: DYNAMICAL GAS INTERACTION AT THE POSSIBLE SITE OF A MULTIPLE STAR FORMATION

Starless dense cores eventually collapse dynamically, which forms protostars inside them, and the physical properties of the cores determine the nature of the forming protostars. We report ALMA observations of dust continuum emission and molecular rotational lines toward MC27 or L1521F, which is considered to be very close to the first protostellar core phase. We found a few starless high-density cores, one of which has a very high density of ~10^7 cm^-3, within a region of several hundred AU around a very low-luminosity protostar detected by Spitzer. A very compact bipolar outflow with a dynamical timescale of a few hundred years was found toward the protostar. The molecular line observation shows several cores with an arc-like structure, possibly due to the dynamical gas interaction. These complex structures revealed in the present observations suggest that the initial condition of star formation is highly dynamical in nature, which is considered to be a key factor in understanding fundamental issues of star formation such as the formation of multiple stars and the origin of the initial mass function of stars.

Early disc accretion as the origin of abundance anomalies in globular clusters

Globular clusters (GCs), once thought to be well approximated as simple stellar populations (i.e. all stars having the same age and chemical abundance), are now known to host a variety of anomalies, such as multiple discrete (or spreads in) populations in colour–magnitude diagrams and abundance variations in light elements (e.g. Na, O, Al). Multiple models have been put forward to explain the observed anomalies, although all have serious shortcomings (e.g. requiring a non-standard initial mass function of stars and GCs to have been initially 10–100 times more massive than observed today). These models also do not agree with observations of massive stellar clusters forming today, which do not display significant age spreads nor have gas/dust within the cluster. Here we present a model for the formation of GCs, where low-mass pre-main-sequence stars accrete enriched material released from interacting massive binary and rapidly rotating stars on to their circumstellar discs, and ultimately on to the young stars. As was shown in previous studies, the accreted material matches the unusual abundances and patterns observed in GCs. The proposed model does not require multiple generations of star formation, conforms to the known properties of massive clusters forming today and solves the ‘mass budget problem’ without requiring GCs to have been significantly more massive at birth. Potential caveats to the model as well as model predictions are discussed.

METAL-MASS-TO-LIGHT RATIOS OF THE PERSEUS CLUSTER OUT TO THE VIRIAL RADIUS

We analyzed XMM-Newton data of the Perseus cluster out to $\sim$1 Mpc, or approximately half the virial radius. Using the flux ratios of Lyalpha lines of H-like Si and S to Kalpha line of He-like Fe, the abundance ratios of Si/Fe and S/Fe of the intracluster medium (ICM) were derived using the APEC plasma code v2.0.1. The temperature dependence of the line ratio limits the systematic uncertainty in the derived abundance ratio. The Si/Fe and S/Fe in the ICM of the Perseus cluster show no radial gradient. The emission-weighted averages of the Si/Fe and S/Fe ratios outside the cool core are 0.91 +- 0.08 and 0.93 +- 0.10, respectively, in solar units according to the solar abundance table of Lodders (2003). These ratios indicate that most Fe was synthesized by supernovae Ia. We collected K-band luminosities of galaxies and calculated the ratio of Fe and Si mass in the ICM to K-band luminosity, iron-mass-to-light ratio (IMLR) and silicon-mass-to-light ratio (SMLR). Within $\sim$1 Mpc, the cumulative IMLR and SMLR increase with radius. Using Suzaku data for the northwest and east directions, we also calculated the IMLR out to $\sim$ 1.8 Mpc, or about the virial radius. We constrained the SMLR out to this radius and discussed the slope of the initial mass function of stars in the cluster. Using the cumulative IMLR profile, we discuss the past supernova Ia rate.

Modes of star formation from Herschel

AbstractWe summarize some of the results obtained from Herschel surveys of nearby star forming regions and the Galactic plane. We show that in the nearby star forming regions the starless core spatial surface density distribution is very similar to that of the young stellar objects. This, taken together with the similarity between the core mass function and the initial mass function for stars and the relationship between the amount of dense gas and star formation rate, suggest that the cloud fragmentation process defines the global outcome of star formation. This “simple” view of star formation may not hold on all scales. In particular dynamical interactions are expected to become important at the conditions required to form young massive clusters. We describe the successes of a simple criterion to identify young massive cluster precursors in our Galaxy based on (sub-)millimeter wide area surveys. We further show that in the location of our Galaxy where the best candidate for a precursor of a young massive cluster is found, the “simple” scaling relationship between dense gas and star formation rate appear to break down. We suggest that in regions where the conditions approach those of the central molecular zone of our Galaxy it may be necessary to revise the scaling laws for star formation.

THE INITIAL MASS FUNCTION AND DISK FREQUENCY OF THE ρ OPHIUCHI CLOUD: AN EXTINCTION-LIMITED SAMPLE

We have completed an optical spectroscopic survey of an unbiased, extinction-limited sample of candidate young stars covering 1.3 square degrees of the Rho Ophiuchi star forming region. While infrared, X-ray, and optical surveys of the cloud have identified many young stellar objects (YSOs), these surveys are biased towards particular stages of stellar evolution and are not optimal for studies of the disk frequency and initial mass function.We have obtained over 300 optical spectra to help identify 135 association members based on the presence of H-alpha in emission, lithium absorption, X-ray emission, a mid-infrared excess, a common proper motion, reflection nebulosity, and/or extinction considerations. Spectral types along with R and I band photometry were used to derive effective temperatures and bolometric luminosities for association members to compare with theoretical tracks and isochrones for pre-main-sequence stars. An average age of 3.1 Myr is derived for this population which is intermediate between that of objects embedded in the cloud core of Rho Ophiuchi and low mass stars in the Upper Scorpius subgroup. Consistent with this age we find a circumstellar disk frequency of 27% plus or minus 5%. We also constructed an initial mass function for an extinction-limited sample of 123 YSOs (A_v less than or equal to 8 mag), which is consistent with the field star initial mass function for YSOs with masses > 0.2 M_sun. There may be a deficit of brown dwarfs but this result relies on completeness corrections and requires confirmation.

ON THE ORIGIN OF THE SALPETER SLOPE FOR THE INITIAL MASS FUNCTION

We suggest that the intrinsic, stellar initial mass function (IMF) follows a power-law slope gamma=2, inherited from hierarchical fragmentation of molecular clouds into clumps and clumps into stars. The well-known, logarithmic Salpeter slope GAMMA=1.35 in clusters is then the aggregate slope for all the star-forming clumps contributing to an individual cluster, and it is steeper than the intrinsic slope within individual clumps because the smallest star-forming clumps contributing to any given cluster are unable to form the highest-mass stars. Our Monte Carlo simulations demonstrate that the Salpeter power-law index is the limiting value obtained for the cluster IMF when the lower-mass limits for allowed stellar masses and star-forming clumps are effectively equal, m_lo = M_lo. This condition indeed is imposed for the high-mass IMF tail by the turn-over at the characteristic value m_c ~ 1 M_sun. IMF slopes of GAMMA ~ 2 are obtained if the stellar and clump upper-mass limits are also equal m_up = M_up ~ 100 M_sun, and so our model explains the observed range of IMF slopes between GAMMA ~ 1 to 2. Flatter slopes of GAMMA = 1 are expected when M_lo > m_up, which is a plausible condition in starbursts, where such slopes are suggested to occur. While this model is a simplistic parameterization of the star-formation process, it seems likely to capture the essential elements that generate the Salpeter tail of the IMF for massive stars. These principles also likely explain the IGIMF effect seen in low-density star-forming environments.

The Turbulent ISM of Galaxies 10 Gyrs ago: Star Formation, Gas Accretion, and IMF

AbstractThe utilization of integral-field spectroscopy has led us to a new understanding of the physical conditions in galaxies within the first few billion years after the Big Bang. In this proceedings, we analyze observations of ~50 massive galaxies as seen as they were 10 Gyrs ago using SINFONI from the ESO-VLT. We show that the large line width they exhibit can be explained by the intense mechanical energy output from the young stars. We also study the influence of cold gas accretion upon these galaxies: We show that an unrealistic amount of shocked gas would be needed in order to explain the Hα emission from these galaxies through shocks from gas accretion with velocity about the Hα line widths of these galaxies. We also use DEEP2 photometric measurements for a sub-sample of 10 of these galaxies to evaluate their ratio of Hα to FUV flux as a function of their Hα and R-band luminosity surface brightnesses. Our data suggests that perhaps their initial mass function (IMF) is flatter than Salpeter at the high mass end, as has been suggested recently for some local galaxies. It may be that high turbulence is responsible for skewing the IMF towards more massive stars as suggested by some theories of star-formation. Much work is however needed to accredit this hypothesis.

Monte Carlo simulations of star clusters - VII. The globular cluster 47 Tuc

We describe Monte Carlo models for the dynamical evolution of the massive globular cluster 47 Tuc (NGC 104). The code includes treatments of two-body relaxation, most kinds of three- and four-body interactions involving primordial binaries and those formed dynamically, the Galactic tide, and the internal evolution of both single and binary stars. We arrive at a set of initial parameters for the cluster which, after 12Gyr of evolution, gives a model with a fairly satisfactory match to surface brightness and density profiles, the velocity dispersion profile, the luminosity function in two fields, and the acceleration of pulsars. Our models appear to require a relatively steep initial mass function for stars above about turnoff, with an index of about 2.8 (where the Salpeter mass function has an index of 2.35), and a relatively flat initial mass function (index about 0.4) for the lower main sequence. According to the model, the current mass is estimated at 0.9 million solar masses, of which about 34% consists of remnants. We find that primordial binaries are gradually taking over from mass loss by stellar evolution as the main dynamical driver of the core. Despite the high concentration of the cluster, core collapse will take at least another 20Gyr.

ORIGINS OF THE THICK DISK AS TRACED BY THE ALPHA ELEMENTS OF METAL-POOR GIANT STARS SELECTED FROM RAVE

Theories of thick-disk formation can be differentiated by measurements of stellar elemental abundances. We have undertaken a study of metal-poor stars selected from the RAVE spectroscopic survey of bright stars to establish whether or not there is a significant population of metal-poor thick-disk stars ([Fe/H] ≲ −1.0) and to measure their elemental abundances. In this Letter, we present abundances of four α-elements (Mg, Si, Ca, and Ti) and iron for a subsample of 212 red giant branch and 31 red clump/horizontal branch stars from this study. We find that the [α/Fe] ratios are enhanced, implying that enrichment proceeded by purely core-collapse supernovae. This requires that star formation in each star-forming region had a short duration. The relative lack of scatter in the [α/Fe] ratios implies good mixing in the interstellar medium prior to star formation. In addition, the ratios resemble that of the halo, indicating that the halo and thick disk share a similar massive star initial mass function. We conclude that the α-enhancement of the metal-poor thick disk implies that direct accretion of stars from dwarf galaxies similar to surviving dwarf galaxies today did not play a major role in the formation of the thick disk.

SUB-ALFVÉNIC NON-IDEAL MHD TURBULENCE SIMULATIONS WITH AMBIPOLAR DIFFUSION. II. COMPARISON WITH OBSERVATION, CLUMP PROPERTIES, AND SCALING TO PHYSICAL UNITS

Ambipolar diffusion is important in redistributing magnetic flux and in damping Alfven waves in molecular clouds. The importance of ambipolar diffusion on a length scale $\ell$ is governed by the ambipolar diffusion Reynolds number, $\rad=\ell/\lad$, where $\lad$ is the characteristic length scale for ambipolar diffusion. The logarithmic mean of the AD Reynolds number in a sample of 15 molecular clumps with measured magnetic fields (Crutcher 1999) is 17, comparable to the theoretically expected value. We identify several regimes of ambipolar diffusion in a turbulent medium, depending on the ratio of the flow time to collision times between ions and neutrals; the clumps observed by Crutcher (1999) are all in the standard regime of ambipolar diffusion, in which the neutrals and ions are coupled over a flow time. We have carried out two-fluid simulations of ambipolar diffusion in isothermal, turbulent boxes for a range of values of $\rad$. The mean Mach numbers were fixed at $\calm=3$ and $\ma=0.67$; self-gravity was not included. We study the properties of overdensities--i.e., clumps--in the simulation and show that the slope of the higher-mass portion of the clump mass spectrum increases as $\rad$ decreases, which is qualitatively consistent with Padoan et al. (2007)'s finding that the mass spectrum in hydrodynamic turbulence is significantly steeper than in ideal MHD turbulence. For a value of $\rad$ similar to the observed value, we find a slope that is consistent with that of the high-mass end of the Initial Mass Function for stars. However, the value we find for the spectral index in our ideal MHD simulation differs from theirs, presumably because our simulations have different initial conditions. This suggests that the mass spectrum of the clumps in the Padoan et al. (2007) turbulent fragmentation model for the IMF depends on the environment, which would conflict with evidence ...

EXTREMELY METAL-POOR STARS IN THE MILKY WAY: A SECOND GENERATION FORMED AFTER REIONIZATION

Cosmological simulations of Population III star formation suggest an initial mass function (IMF) biased toward very massive stars (M ≳ 100 M☉) formed in minihalos at redshift z ≳ 20, when the cooling is driven by molecular hydrogen. However, this result conflicts with observations of extremely metal-poor (EMP) stars in the Milky Way (MW) halo, whose r-process elemental abundances appear to be incompatible with those expected from very massive Population III progenitors. We propose a new solution to the problem in which the IMF of second-generation stars formed at z ≳ 10, before reionization, is deficient in sub-solar mass stars, owing to the high cosmic microwave background temperature floor. The observed EMP stars are formed preferentially at z ≲ 10 in pockets of gas enriched to metallicity Z ≳ 10−3.5 Z☉ by winds from Population II stars. Our cosmological simulations of dark matter halos like the MW show that current samples of EMP stars can only constrain the IMF of late-time Population III stars, formed at z ≲ 13 in halos with virial temperature Tvir ∼ 104 K. This suggests that pair instability supernovae were not produced primarily by this population. To begin probing the IMF of Population III stars formed at higher redshift will require a large survey, with at least 500 and probably several thousand EMP stars of metallicities Z ≈ 10−3.5 Z☉.

Constraining the initial mass function of stars in the Galactic Centre

(abridged) Here we discuss the question whether the extreme circumstances in the centre of the Milky Way may be the reason for a significant variation of the IMF. By means of stellar evolution models using different codes we show that the observed luminosity in the central parsec is too high to be explained by a long-standing top-heavy IMF, considering the limited amount of mass inferred from stellar kinematics in this region. In contrast, continuous star formation over the Galaxy's lifetime following a canonical IMF results in a mass-to-light ratio and a total mass of stellar black holes (SBHs) consistent with the observations. Furthermore, these SBHs migrate towards the centre due to dynamical friction, turning the cusp of visible stars into a core as observed in the Galactic Centre. For the first time here we explain the luminosity and dynamical mass of the central cluster and both the presence and extent of the observed core, since the number of SBHs expected from a canonical IMF is just enough to make up for the missing luminous mass. We conclude that the Galactic Centre is consistent with the canonical IMF and do not suggest a systematic variation as a result of the region's properties such as high density, metallicity, strong tidal field etc.

Probability distribution of terrestrial planets in habitable zones around host stars

With more and more exoplanets being detected, it is paid closer attention to whether there are lives outside solar system. We try to obtain habitable zones and the probability distribution of terrestrial planets in habitable zones around host stars. Using Eggleton's code, we calculate the evolution of stars with masses less than 4.00 \mo. We also use the fitting formulae of stellar luminosity and radius, the boundary flux of habitable zones, the distribution of semimajor axis and mass of planets and the initial mass function of stars. We obtain the luminosity and radius of stars with masses from 0.08 to 4.00 \mo, and calculate the habitable zones of host stars, affected by stellar effective temperature. We achieve the probability distribution of terrestrial planets in habitable zones around host stars. We also calculate that the number of terrestrial planets in habitable zones of host stars is 45.5 billion, and the number of terrestrial planets in habitable zones around K type stars is the most, in the Milky Way.