Monte Carlo simulations of star clusters — I. First Results

A revision of Stodoíkiewicz's Monte Carlo code is used to simulate evolution of star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. The first calculations for isolated, equal-mass N -body systems with three-body energy generation according to Spitzer's formulae show good agreement with direct N -body calculations for N  = 2000, 4096 and 10 000 particles. The density, velocity, mass distributions, energy generation, number of binaries, etc., follow the N -body results. Only the number of escapers is slightly too high compared with N -body results, and there is no level-off anisotropy for advanced post-collapse evolution of Monte Carlo models as is seen in N -body simulations for N  ≤ 2000. For simulations with N  > 10 000 gravothermal oscillations are clearly visible. The calculations of N   2000, 4096, 10 000, 32 000 and 100 000 models take about 2, 6, 20, 130 and 2500 h, respectively. The Monte Carlo code is at least 10⁵ times faster than the N -body one for N  = 32 768 with special-purpose hardware. Thus it becomes possible to run several different models to improve statistical quality of the data and run individual models with N as large as 100 000. The Monte Carlo scheme can be regarded as a method which lies in the middle between direct N -body and Fokker–Planck models and combines most advantages of both methods.     

On the presence of silicon and carbon in the pre-maximum spectrum of the Type Ia SN 1990N

The spectrum of the normal Type Ia SN 1990N observed very early on (14 days before B maximum) was analysed by Fisher et al., who showed that the large width and the unusual profile of the strong line near 6000 Å can be reproduced if the line is assumed to be due to C  ii 6578, 6583 Å and if carbon is located in a high-velocity shell. This line is one of the characterizing features of SNe Ia, and is usually thought to be due to Si  ii . A Monte Carlo spectrum synthesis code is used to investigate this suggestion further. The result is that if a standard explosion model is used, the mass enclosed in the shell at the required high velocity (25 000–35 000 km s⁻¹) is too small to give rise to a strong C  ii line. At the same time, removing silicon has a negative effect on the synthetic spectrum at other wavelengths, and removing carbon from the lower velocity regions near the photosphere makes it difficult to reproduce two weak lines which are naturally explained as C  ii , one of them being the line which Fisher et al. suggested is responsible for the strong 6000-Å feature. However, synthetic spectra confirm that although Si  ii can reproduce most of the observed 6000-Å line, the red wing of the line extends too far to be compatible with a Si  ii origin, and that the flat bottom of the line is also not easy to reproduce. The best fit is obtained for a normal SN Ia abundance mix at velocities near the photosphere (15 500–19 000 km s⁻¹) and an outer carbon–silicon shell beyond 20 000 km s⁻¹. This suggests that mixing is not complete in the outer ejecta of an SN Ia. Observations at even earlier epochs might reveal to what extent a carbon shell is unmixed.     

A comparison between terrestrial, Cytherean and lunar impact cratering records

A comparison between the terrestrial, Cytherean and lunar cratering records indicates that the large craters (diameters   D > D ₀)  on these surfaces all have cumulative numbers that are proportional to   D ^-2.59±0.05  . Atmospheres have a negligible effect on the formation of   D > D ₀  craters. It is shown that this limiting diameter is  45±3 km  in the case of Venus, and  21.0±1.5 km  in the case of Earth. In this large-diameter range, there are about  1.51±0.34  times more craters, per unit area, on Venus than on the Earth, and about  1350±310  times more craters, per unit area, on the Moon than on the Earth.     

Ages and metallicities of globular clusters in NGC 4472

We have derived ages and metallicities from co-added spectra of 131 globular clusters associated with the giant elliptical galaxy NGC 4472. Based upon a calibration with Galactic globular clusters, we find that our sample of globular clusters in NGC 4472 span a metallicity range of approximately −1.6≤[Fe/H]≤0 dex. There is evidence of a radial metallicity gradient in the globular cluster system which is steeper than that seen in the underlying starlight. Determination of the absolute ages of the globular clusters is uncertain, but formally, the metal-poor population of globular clusters has an age of 14.5±4 Gyr and the metal-rich population is 13.8±6 Gyr old. Monte Carlo simulations indicate that the globular cluster populations present in these data are older than 6 Gyr at the 95 per cent confidence level. We find that within the uncertainties, the globular clusters are old and coeval, implying that the bimodality seen in the broadband colours primarily reflects metallicity and not age differences.     

First constraints on iron abundance versus reflection fraction from the Seyfert 1 galaxy MCG–6-30-15

We report on a joint ASCA and RXTE observation spanning an ∼400 ks time interval of the bright Seyfert 1 galaxy MCG–6-30-15. The data clearly confirm the presence of a broad skewed iron line ( W _Kα ∼266 eV) and Compton reflection continuum at higher energies reported in our previous paper. We also investigate whether the gravitational and Doppler effects, which affect the iron line, may also be manifest in the reflected continuum. The uniqueness of this data set is underlined by the extremely good statistics that we obtain from the approximately four million photons that make up the 2–20 keV RXTE PCA spectrum alone. This, coupled with the high energy coverage of HEXTE and the spectral resolution of ASCA in the iron line regime, has allowed us to constrain the relationship between abundance and reflection fraction for the first time at the 99 per cent confidence level. The reflection fraction is entirely consistent with a flat disc, i.e. the cold material subtends 2π sr at the source, to an accuracy of 20 per cent. Monte Carlo simulations show that the observed strong iron line intensity is explained by an overabundance of iron by a factor of ∼2 and an underabundance of the lower- Z elements by a similar factor. By considering non-standard abundances, a clear and consistent picture can be made in which both the iron line and reflection continuum come from the same material such as, e.g., an accretion disc.     

Monte‐Carlo simulation of a possible correlation between the estimated luminosity distances and internal extinctions of type Ia supernovae

This paper describes a Monte Carlo simulation of type Ia supernova data. It was shown earlier that the data of SNe Ia might contain a possible correlation between the estimated luminosity distances and internal extinctions. This correlation was shown by different statistical investigations of the data. In order to remove observational biases (for example the effect of the detection limit of the observing instrument) and to test the reality of the effect found earlier we developed a simple routine which simulates extinction values, redshifts and absolute magnitudes for Ia supernovae. We pointed out that the correlation found earlier in the real data between the internal extinction and luminosity distance does not occur in the simulated sample. Furthermore, it became obvious that the detection limit of the observing devices used in supernova projects does not affect the far end of the redshift‐luminosity distance relationship of Ia supernovae. This result strengthens the earlier conclusions of the authors that SN Ia supernovae alone do not support the existence of dark energy. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constraints on the mass and abundance of black holes in the Galactic halo: the high-mass limit

We establish constraints on the mass and abundance of black holes in the Galactic halo by determining their impact on globular clusters, which are conventionally considered to be little evolved. Using detailed Monte Carlo simulations and simple evolutionary models, we argue that black holes with masses M _bh≳(1–3)×10⁶ M_⊙ can comprise no more than a fraction f _bh≈0.17 of the total halo density at Galactocentric radius R ≈8 kpc. This bound arises from requiring stability of the cluster mass function. A more restrictive bound may be derived if we demand that the probability of destruction of any given, low-mass M _c≈(2.5–7.5)×10⁴ M_⊙] globular cluster not exceed 50 per cent; this bound is f _bh≲0.025–0.05 at R ≈8 kpc. This constraint improves those based on disc heating and dynamical friction arguments as well as current lensing results. At smaller radius the constraint on f _bh strengthens, while at larger radius an increased fraction of black holes is allowed.     

A photometric and spectroscopic study of the brightest northern Cepheids — I. Observations

We present simultaneous UBV uvby photometry for the 18 brightest northern Cepheids carried out between 1995 and 1997. Additionally, two fainter stars have been observed in the Johnson system only. The whole photometric data base contains about 3500 individual data points for 20 stars. The accuracy has been carefully tested with different methods. A serious systematic difference has been found between the present data set and the Stro¨mgren photometry available in the literature, which has probably been caused by the peculiar filter set used in the earlier study.   As an extension to the photometry, we took high-resolution optical spectra at David Dunlap Observatory in the red spectral region (λ/Δλ≈ 40 000, in the interval of 6200 and 6600 Å including Hα). The spectroscopic programme contained 12 stars from the photometric programme, the newly discovered bright classical Cepheid CK Cam and two double-mode Cepheids (TU Cas and CO Aur). New radial velocities obtained with the cross-correlation technique are presented. We found significant velocity differences between two cross-correlated spectral regions (6188–6220 and 6405–6435 Å) as large as 0.8–1.2 km s⁻¹, which show very characteristic phase dependence in certain Cepheids.   Finally, recent period variations are briefly discussed in terms of phase jumps and duplicity.     

Pulsar spin–velocity alignment from single and binary neutron star progenitors

The role of binary progenitors of neutron stars (NSs) in the apparent distribution of space velocities and spin–velocity alignment observed in young pulsars is studied. We performed a Monte Carlo synthesis of pulsar populations originated from single and binary stars with different assumptions about the NS natal kick (kick–spin alignment, kick amplitude and kick reduction in electron-capture supernovae in binary progenitors with initial main-sequence masses from the range  8–11 M_⊙  which experienced mass exchange due to Roche lobe overflow). The calculated spin–velocity alignment in pulsars is compared with data inferred from radio polarization measurements. The observed space velocity of pulsars is found to be mostly affected by the natal kick velocity form and its amplitude; the fraction of binaries is not important here for reasonably large kicks. The natal kick–spin alignment is found to strongly affect the spin–velocity correlation of pulsars. Comparison with the observed pulsar spin–velocity angles favours a sizeable fraction of binary progenitors and kick–spin angles  ∼5°–20°  .     

The influence of binary interactions on infrared passbands of populations

In our evolutionary population synthesis models, samples of binaries are reproduced by a ' patched ' Monte Carlo simulation and the stellar masses, integrated   J ,  H ,  K ,  L ,  L 2  and M magnitudes, mass-to-light ratios and broad colours involving infrared bands are presented for an extensive set of instantaneous-burst binary stellar populations. In addition, the fluctuations in the integrated colours, which have been given by Zhang et al., are reduced.
By comparing the results for binary stellar populations with (Model A) and without (Model B) binary interactions, we show that the inclusion of binary interactions makes the stellar mass of a binary stellar population smaller (  ∼3.6–4.5  per cent during the past 15 Gyr), magnitudes greater (except   U , ∼ 0.18 mag  at the most), colours bluer (∼0.15 mag for   V − K   at the most) and mass-to-light ratios greater (∼0.06 for K band) except those in the U and B passbands at higher metallicities. Binary interactions make the V magnitude less sensitive to age, and R and I magnitudes more sensitive to metallicity.
Given an age, the absolute values of the differences in the stellar mass, magnitudes and mass-to-light ratios (except those in the U and B bands) between Models A and B reach a maximum at   Z = 0.0001  , i.e. the effects of binary interactions on these parameters reach a maximum, while the differences in some colours reach a maximum at   Z ∼ 0.01–0.0004  . In contrast, the absolute value of the difference in the stellar mass is minimal at   Z = 0.03  ; those in the   U ,  B ,  V   magnitudes and the mass-to-light ratios in the U and B bands reach a minimum at   Z ∼ 0.01–0.004  .     

How rapidly do neutron stars spin at birth? Constraints from archival X-ray observations of extragalactic supernovae

Traditionally, studies aimed at inferring the distribution of birth periods of neutron stars are based on radio surveys. Here we propose an independent method to constrain the pulsar spin periods at birth based on their X-ray luminosities. In particular, the observed luminosity distribution of supernovae (SNe) poses a constraint on the initial rotational energy of the embedded pulsars, via the     correlation found for radio pulsars, and under the assumption that this relation continues to hold beyond the observed range. We have extracted X-ray luminosities (or limits) for a large sample of historical SNe observed with Chandra , XMM and Swift , which have been firmly classified as core-collapse SNe. We have then compared these observational limits with the results of Monte Carlo simulations of the pulsar X-ray luminosity distribution for a range of values of the birth parameters. We find that a pulsar population dominated by millisecond periods at birth is ruled out by the data.     

Mass loss and orbital period decrease in detached chromospherically active binaries

The secular evolution of the orbital angular momentum (OAM), the systemic mass  ( M = M ₁+ M ₂)  and the orbital period of 114 chromospherically active binaries (CABs) were investigated after determining the kinematical ages of the subsamples which were set according to OAM bins. OAMs, systemic masses and orbital periods were shown to be decreasing by the kinematical ages. The first-order decreasing rates of OAM, systemic mass and orbital period have been determined as     per systemic OAM,     per systemic mass and     per orbital period, respectively, from the kinematical ages. The ratio of d log  J /d log  M = 2.68, which were derived from the kinematics of the present sample, implies that there must be a mechanism which amplifies the angular momentum loss (AML)     times in comparison to isotropic AML of hypothetical isotropic wind from the components. It has been shown that simple isotropic mass loss from the surface of a component or both components would increase the orbital period.     

Contribution to diffuse gamma-rays in the Galactic Centre region from unresolved millisecond pulsars

Diffuse gamma-rays in the Galactic Centre region have been studied. We propose that there exists a population of millisecond pulsars in the Galactic Centre, which emit GeV gamma-rays through synchrotron-curvature radiation as predicted by outer gap models. These GeV gamma-rays from unresolved millisecond pulsars probably contribute to the diffuse gamma-ray spectrum detected by EGRET which displays a break at a few GeV. We have used a Monte Carlo method to obtain simulated samples of millisecond pulsars in the Galactic Centre region covered by EGRET  (∼ 15)  according to the different period and magnetic field distributions from observed millisecond pulsars in the Galactic field and globular clusters, and superposed their synchrotron-curvature spectra to derive the total GeV flux. Our simulated results suggest that there probably exist about 6000 unresolved millisecond pulsars in the region of angular resolution of EGRET, the emissions of which could contribute significantly to the observed diffuse gamma-rays in the Galactic Centre.     

Confidence intervals for the correlation between the gamma-ray burst peak energy and the associated supernova peak brightness

A strong correlation between the gamma-ray burster peak energy and the peak luminosity of the associated supernova was discovered by Li for four GRBs. Despite the fact that the formal significance level of the correlation is 0.3 per cent, the smallness of the data set requires careful further evaluation of the result. Subject to the assumption that the data are bivariate Gaussian, a 95 per cent confidence interval of  (−0.9972, 0.02)  for the correlation is derived. Using data from the literature, it is shown that the distribution of known peak GRB energies is not Gaussian if X-ray flashes are included in the sample. This leads to a proposed alternative to the bivariate Gaussian model, which entails describing the dependence between the two variables by a Gaussian copula. The copula is still characterized by a correlation coefficient. The Bayesian posterior distribution of the correlation coefficient is evaluated using a Markov chain Monte Carlo method. The mean values of the posterior distributions range from −0.33 to about zero, depending on the specifics of the supernova (SN) peak brightness distribution. The implication is that the existing data favour a modest correlation between the GRB peak energy and the SN peak brightness; confidence intervals are very wide and include zero.     

Monte Carlo simulations of star clusters – II. Tidally limited, multimass systems with stellar evolution

A revision of Stodółkiewicz's Monte Carlo code is used to simulate evolution of large star clusters. The new method treats each superstar as a single star and follows the evolution and motion of all individual stellar objects. A survey of the evolution of N -body systems influenced by the tidal field of a parent galaxy and by stellar evolution is presented. The process of energy generation is realized by means of appropriately modified versions of Spitzer's and Mikkola's formulae for the interaction cross-section between binaries and field stars and binaries themselves. The results presented are in good agreement with theoretical expectations and the results of other methods (Fokker–Planck, Monte Carlo and N -body). The initial rapid mass loss, resulting from stellar evolution of the most massive stars, causes expansion of the whole cluster and eventually leads to the disruption of less bound systems ( W ₀=3). Models with larger W ₀ survive this phase of evolution and then undergo core collapse and subsequent post-collapse expansion, like isolated models. The expansion phase is eventually reversed when tidal limitation becomes important. The results presented are the first major step in the direction of simulating evolution of real globular clusters by means of the Monte Carlo method.     

The Effect of Method of Constructing the Mass Distribution on Single Stellar Populations

We use two methods of constructing the initial mass distribution, the traditional way and Monte Carlo simulation, to obtain integrated U - B, B - V, V-R and V-I colours and absorption-line indices denned by the Lick Observatory image dissector scanner (referred to as Lick/IDS), for instantaneous burst solarmetallicity single stellar populations with ages in the range 1-15 Gyr. We find that the evolutionary curves of all colours obtained by the traditional method are smoother than those by Monte Carlo simulation, that the U - B and B - V colours obtained by the two methods agree with one another, while the V - R and V - I colours by the traditional method are bluer than those by Monte Carlo simulation. A comparison of the Lick/IDS absorption-line indices shows that the variations in all the indices by the traditional method are smoother than that for the Monte Carlo simulation, and that all the indices except for TiO1 and TiO2 are consistent with those for the Monte Carlo simulation.     

Searching for cluster substructure using APM and ROSAT data

We present a detailed study of the morphological features of 22 rich galaxy clusters. Our sample is constructed from a cross-correlation of optical     data with X-ray (0.1–2.4 keV) ROSAT pointed observations. We systematically compare cluster images and morphological parameters in an attempt to reliably identify possible substructure in both optical and the X-ray images. To this end, we compute various moments of the optical and X-ray surface-brightness distribution such as the ellipticities, centre-of-mass shifts and ellipsoidal orientations. We assess the significance of our results using Monte Carlo simulations. We find significant correlations between the optical and X-ray morphological parameters, indicating that in both parts of the spectrum it is possible to identify correctly the dynamical state of a cluster. Most of our clusters (17/22) have a good one-to-one correspondence between the optical and the X-ray images, and about 10 appear to have strong indications of substructure. This corresponds to a minimum percentage of order ∼45 per cent, which is in very good accordance with other similar analyses. Finally, five out of 22 systems (∼22 per cent) seem to have distinct subclumps in the optical which are not verified in the X-ray images, and thus are suspect of being due to optical projection effects. These results will serve as a useful guide in interpreting subsequent analyses of large optical cluster catalogues.     

Keck spectroscopy of the faint dwarf elliptical galaxy population in the Perseus Cluster core: mixed stellar populations and a flat luminosity function

We present the result of a photometric and Keck low-resolution imaging spectrometer (LRIS) spectroscopic study of dwarf galaxies in the core of the Perseus Cluster, down to a magnitude of   M _B =−12.5  . Spectra were obtained for 23 dwarf-galaxy candidates, from which we measure radial velocities and stellar population characteristics from absorption line indices. From radial velocities obtained using these spectra, we confirm 12 systems as cluster members, with the remaining 11 as non-members. Using these newly confirmed cluster members, we are able to extend the confirmed colour–magnitude relation for the Perseus Cluster down to   M _B =−12.5  . We confirm an increase in the scatter about the colour–magnitude relationship below   M _B =−15.5  , but reject the hypothesis that very red dwarfs are cluster members. We measure the faint-end slope of the luminosity function between   M _B =−18  and −12.5, finding  α=−1.26 ± 0.06  , which is similar to that of the field. This implies that an overabundance of dwarf galaxies does not exist in the core of the Perseus Cluster. By comparing metal and Balmer absorption line indices with α-enhanced single stellar population models, we derive ages and metallicities for these newly confirmed cluster members. We find two distinct dwarf elliptical populations: an old, metal-poor population with ages ∼8 Gyr and metallicities  [Fe/H] < −0.33  , and a young, metal-rich population with ages <5 Gyr and metallicities  [Fe/H] > −0.33  . Dwarf galaxies in the Perseus Cluster are therefore not a simple homogeneous population, but rather exhibit a range in age and metallicity.