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.     

State-to-state rotational transition rates of the HCO+ ion by collisions with helium

The rates of rotational transitions for HCO⁺, the most abundant ion in interstellar space, induced by collision with helium are obtained for temperatures ranging from 10 to 80 K. The calculations are based on a new potential energy surface for the He–HCO⁺ interaction and on a scattering matrix whose accuracy was checked by pressure broadening and shift measurements. The rates     decrease for increasing values of j and  Δ j   , with a temperature trend depending on the energy involved in the transitions: if it is small, the rates are almost constant, while an increase with T is found for other cases. Comparison with previous and less accurate results shows an agreement within 50 per cent. Comparison between state-to-state and pressure broadening cross-sections allows us to discuss importance and influence of elastic and inelastic collisions.     

A search for absorption of Mg and Ca compounds in molecular clouds towards Galactic continuum sources

Absorption lines of MgH and CaH N  = 1 − 0 transitions were searched for in foreground molecular clouds towards the continuum sources associated with Sgr B2 (M) and W49A (N). None of these lines was detected with our sensitivity level of ∼20 mK. Millimetric absorption lines of MgO, MgOH, CaO and CaOH were also searched for towards Sgr B2 (M) without success. The fractional abundances relative to molecular hydrogen are ≲ 1.0 × 10⁻¹¹ for MgH, ≲ 7.9 × 10⁻¹³ for MgO, ≲ 1.6 × 10⁻¹⁰ for MgOH, ≲ 1.6 × 10⁻⁹ for CaH, ≲ 2.0 × 10⁻¹² for CaO, and ≲ 2.5 × 10⁻¹⁰ for CaOH, respectively. The low abundances measured in absorption indicate that a significant fraction of interstellar magnesium and calcium cannot be tied up in their monohydrides, monoxides and monohydroxides. The low abundance of MgH also implies that grain-surface chemistry involving magnesium is not efficient and that magnesium is depleted on to grains to a factor of ≳ 10^2.5 in well-shielded molecular clouds.     

Emission lines of Fe xv in spectra obtained with the Solar Extreme-Ultraviolet Research Telescope and Spectrograph

Recent R-matrix calculations of electron impact excitation rates in Mg-like Fe  xv are used to derive theoretical emission-line ratios involving transitions in the 243–418 Å  wavelength range. A comparison of these with a data set of solar active region, subflare and off-limb spectra, obtained during rocket flights by the Solar Extreme-Ultraviolet Research Telescope and Spectrograph (SERTS), reveals generally very good agreement between theory and observation, indicating that most of the Fe  xv emission lines may be employed with confidence as electron density diagnostics. In particular, the 312.55-Å  line of Fe  xv is not significantly blended with a Co  xvii transition in active region spectra, as suggested previously, although the latter does make a major contribution in the subflare observations. Most of the Fe  xv transitions which are blended have had the species responsible clearly identified, although there remain a few instances where this has not been possible. We briefly address the long-standing discrepancy between theory and experiment for the intensity ratio of the  3s^2 1S–3s3p ³P₁  intercombination line at 417.25 Å  to the  3s^2 1S–3s3p ¹P  resonance transition at 284.16 Å.     

Spectral features in gamma-rays expected from millisecond pulsars

In the advent of next generation gamma-ray missions, we present general properties of spectral features of high-energy emission above 1 MeV expected for a class of millisecond, low magnetic field (∼10⁹ G) pulsars. We extend polar-cap model calculations of Rudak & Dyks by including inverse Compton scattering events in an ambient field of thermal X-ray photons and by allowing for two models of particle acceleration. In the range between 1 MeV and a few hundred GeV, the main spectral component is the result of curvature radiation of primary particles. The synchrotron component arising from secondary pairs becomes dominant only below 1 MeV. The slope of the curvature radiation spectrum in the energy range from 100 MeV to 10 GeV strongly depends on the model of longitudinal acceleration, whereas below ∼100 MeV all slopes converge to a unique value of 4/3 (in a ν ℱ _ν convention). The thermal soft X-ray photons, which come either from the polar cap or from the surface, are Compton upscattered to a very high energy domain and form a separate spectral component peaking at ∼1 TeV. We discuss the observability of millisecond pulsars by future high‐energy instruments and present two rankings relevant for GLAST and MAGIC. We point to the pulsar J0437−4715 as a promising candidate for observations.     

Initial condensate composition during asteroid impacts

A model has been developed for the chemical composition of early-time condensates forming during planetary impacts at velocities sufficient for complete evaporation of the impactor (). Hydrodynamics is approximated as hemispheric expansion of a chondritic impactor described with the help of a new semi-analytic equation of state. The gas phase is a mixture of a large number of molecular and mono-atomic species, whereas the Helmholtz free energy of the condensed phase is approximated as sum of a zero-temperature contribution and an Einstein-Debye solid for the thermal part. Interpolation of the free energy between the states of dense solids and molecular gases is used over the wide range of densities and pressures. Chemical equilibrium calculations, performed in a post-processing step, begin at conditions near the coexistence curve as obtained during hydrodynamic simulations, then the condensed phase contains 31 compound species, among them the most important oxides, sulphides, silicates as well as pure elements occurring in a liquid phase at sufficiently high temperature. The composition of the resulting liquid condensate in terms of major elements is compared to that of spinel inclusions in spherules found worldwide at the K-P boundary.     

Sputtering of grains in C-type shocks

Sputtering yields are reported for the release of Mg, Fe, Si and O under impact of He, C, O, Si and Fe on grain material composed of Mg- and Fe-bearing silicates. The yields were derived using the trim code, which simulates the results of the transport of ions in matter by means of classical Monte Carlo techniques. The energetics of the sputtering process are a key factor in the sputtering calculations, and so detailed determinations have been made of the energy with which atoms are bound to the lattice, using solid-state simulation programs. The probability of ejection of an atom is computed at a given energy, for a number of angles of incidence, and integrated to obtain the mean yield at that energy. These numerical results are then fitted with a simple function of energy for convenience in subsequent applications.
A grid of C-type shock models has been computed, using our new sputtering yields, for pre-shock densities in the range 10⁴ n _H n (H)+2 n (H₂)10⁶ cm⁻³ and shock speeds 20 v _s45 km s⁻¹. Sputtered fractions can be high, exceeding 50 per cent for shock speeds in excess of approximately 40 km s⁻¹. The column densities of Si and SiO were also computed, for comparison with observations.     

On the fate of processed matter in dwarf galaxies

Two-dimensional calculations of the evolution of remnants generated by the strong mechanical energy deposited by stellar clusters in dwarf galaxies ( M  ∼ 10⁹–10¹⁰ M⊙) are presented. The evolution is followed for times longer than both the blowout time and the presumed span of energy injection generated by a coeval massive stellar cluster. The remnants are shown to end up wrapping around the central region of the host galaxy, while growing to kiloparsec-scale dimensions. Properties of the remnants such as luminosity, size, swept-up mass and expansion speed are given as functions of time for all calculated cases.   The final fate of the swept-up galactic gas and of the matter processed by the central starburst is shown to be highly dependent on the properties of the low-density galactic halo. Superbubbles powered by star clusters, with properties similar to those inferred from the observations, slow down in the presence of an extended halo to expansion speeds smaller than the host galaxy escape velocity. Values of the critical luminosity required for the superbubbles to reach the edge of the galaxies with a speed comparable to the escape speed are derived analytically and numerically. The critical luminosities are larger than those in the detected sources, and thus the superbubbles in amorphous dwarf galaxies must have already undergone blowout and are presently evolving into an extended low-density halo. This will inhibit the loss of the swept-up and processed matter from the galaxy.     

Implications of bias evolution on measurements of the integrated Sachs–Wolfe effect: errors and biases in parameter estimation

The subject of this paper is a quantification of the impact of uncertainties in bias and bias evolution on the interpretation of measurements of the integrated Sachs–Wolfe (ISW) effect, in particular on the estimation of cosmological parameters. We carry out a Fisher matrix analysis for quantifying the degeneracies between the parameters of a dark energy cosmology and bias evolution, for the combination of the PLANCK microwave sky survey with the EUCLID main galaxy sample, where bias evolution   b ( a ) = b ₀+ (1 − a ) b_a   is modelled with two parameters b ₀ and   b_a   . Using a realistic bias model introduces a characteristic suppression of the ISW spectrum on large angular scales, due to the altered distance-weighting functions. The errors in estimating cosmological parameters if the data with evolving bias is interpreted in the framework of cosmologies with constant bias are quantified in an extended Fisher formalism. We find that the best-fitting values of all parameters are shifted by an amount comparable to the statistical accuracy: the estimation bias in units of the statistical accuracy amounts to 1.19 for Ω_m, 0.27 for σ₈ and 0.72 for w for bias evolution with   b_a = 1  . Leaving   b_a   open as a free parameter deteriorates the statistical accuracy, in particular on Ω_m and w .     

Calculated rotational and vibrational excitation rates for electron–HeH+ collisions

Molecular R-matrix calculations are performed at a range of energies to give rotational and vibrational excitation and de-excitation cross-sections and, hence, rates for electron collisions with HeH⁺ up to electron temperatures of 20 000 K. Critical electron densities are also given. The rotational calculations include the Coulomb–Born completion of the cross-sections for high l values. Rates for the transition j  = 0 → 2, which have previously been assumed to be negligible, are found to be up to half those for j  = 0 → 1, raising the prospect of observing the HeH⁺ j  = 2 → 1 emission line at 74.8 μm.     

On the robustness of the ammonia thermometer

Ammonia inversion lines are often used as probes of the physical conditions in the dense interstellar medium. The excitation temperature between the first two para-metastable (rotational) levels is an excellent probe of the gas kinetic temperature. However, the calibration of this ammonia thermometer depends on the accuracy of the collisional rates with H₂. Here, we present new collisional rates for ortho- and para-NH₃ colliding with  para-H₂( J = 0)  , and investigate the effects of these new rates on the excitation of ammonia. Scattering calculations employ a new, high-accuracy, potential energy surface computed at the coupled-cluster CCSD(T) level with a basis set extrapolation procedure. Rates are obtained for all transitions involving ammonia levels with   J ≤ 3  and for kinetic temperatures in the range 5–100 K. We find that the calibration curve of the ammonia thermometer – which relates the observed excitation temperature between the first two para-metastable levels to the gas kinetic temperature – does not change significantly when these new rates are used. Thus, the calibration of ammonia thermometer appears to be robust. Effects of the new rates on the excitation temperature of inversion and rotation–inversion transitions are also found to be small.     

The survival and disruption of cold dark matter microhaloes: implications for direct and indirect detection experiments

If the dark matter particle is a neutralino, then the first structures to form are cuspy cold dark matter (CDM) haloes collapsing after redshifts   z ≈ 100  in the mass range  10⁻⁶–10⁻³ M_⊙  . We carry out a detailed study of the survival of these microhaloes in the Galaxy as they experience tidal encounters with stars, molecular clouds, and other dark matter substructures. We test the validity of analytic impulsive heating calculations using high-resolution N -body simulations. A major limitation of analytic estimates is that mean energy inputs are compared to mean binding energies, instead of the actual mass lost from the system. This energy criterion leads to an overestimate of the stripped mass and an underestimate of the disruption time-scale, since CDM haloes are strongly bound in their inner parts. We show that a significant fraction of material from CDM microhaloes can be unbound by encounters with Galactic substructure and stars; however, the cuspy central regions remain relatively intact. Furthermore, the microhaloes near the solar radius are those which collapse significantly earlier than average and will suffer very little mass-loss. Thus, we expect a fraction of surviving bound microhaloes, a smooth component with narrow features in phase space, which may be uncovered by direct detection experiments, as well as numerous surviving cuspy cores with proper motions of arcminutes per year, which can be detected indirectly via their annihilation into gamma-rays.     

On the escape of particles from cosmic ray modified shocks

Stationary solutions to the problem of particle acceleration at shock waves in the non-linear regime, when the dynamical reaction of the accelerated particles on the shock cannot be neglected, are known to show a prominent energy flux escaping from the shock towards upstream infinity. On physical grounds, the escape of particles from the upstream region of a shock has to be expected in all those situations in which the maximum momentum of accelerated particles,   p _max  , decreases with time, as is the case for the Sedov–Taylor phase of expansion of a shell supernova remnant, when both the shock velocity and the cosmic ray induced magnetization decrease. In this situation, at each time t , particles with momenta larger than   p _max( t )  leave the system from upstream, carrying away a large fraction of the energy if the shock is strongly modified by the presence of cosmic rays. This phenomenon is of crucial importance for explaining the cosmic ray spectrum detected at the Earth. In this paper, we discuss how this escape flux appears in the different approaches to non-linear diffusive shock acceleration, and especially in the quasi-stationary semi-analytical kinetic ones. We apply our calculations to the Sedov–Taylor phase of a typical supernova remnant, including in a self-consistent way particle acceleration, magnetic field amplification and the dynamical reaction on the shock structure of both particles and fields. Within this framework, we calculate the temporal evolution of the maximum energy reached by the accelerated particles and of the escape flux towards upstream infinity. The latter quantity is directly related to the cosmic ray spectrum detected at the Earth.     

The epoch of helium reionization

We study the reionization of He  ii by quasars using a numerical approach that combines 3D radiative transfer calculations with cosmological hydrodynamical simulations. Sources producing the ionizing radiation are selected according to an empirical quasar luminosity function, and are assigned luminosities according to their intrinsic masses. The free parameters associated with this procedure are (1) a universal source lifetime, (2) a minimum mass cut-off, (3) a minimum luminosity cut-off, (4) a solid angle specifying the extent to which radiation is beamed, and (5) a tail-end spectral index for the radiative energy distribution of the sources. We present models in which these parameters are varied, and examine characteristics of the resultant reionization process that distinguish the various cases. In addition, we extract artificial spectra from the simulations and quantify statistical properties of the spectral features in each model.
We find that the most important factor affecting the evolution of He  ii reionization is the cumulative number of ionizing photons that are produced by the sources. Comparisons between He  ii opacities measured observationally and those obtained by our analysis reveal that the available ranges in plausible values for the parameters provide enough leeway to provide a satisfactory match. However, one property common to all our calculations is that the epoch of He  ii reionization must have occurred at a redshift in the range  3≲ z ≲4  . If so, future observational programmes will be able to directly trace the details of the ionization history of helium and to probe the low-density phase of the intergalactic medium during this phase of the evolution of the Universe.     

High-resolution optical spectroscopy of the sharp-lined B-type star HD 83206

Very-high-resolution ( R ∼160 000) spectroscopic observations are presented for the early B-type star, HD 83206. Because it has very sharp metal lines, this star affords an opportunity to test theories of model atmospheres and line formation. Non-LTE model atmosphere calculations have been used to estimate the atmospheric parameters and absolute metal abundances (C, N, O, Mg and Si); an LTE analysis was also undertaken to investigate the validity of this simpler approach and to estimate an iron abundance. For the non-LTE calculations, there is excellent agreement with observations of the Balmer lines H α and H δ and the lines of Si  ii and Si  iii for atmospheric parameters of T _eff≃21 700±600 K and log  g ≃4.00±0.15 dex. The agreement is less convincing for the LTE calculations, and a higher gravity is deduced. Careful comparison of the metal line profiles with non-LTE calculations implies that the projected rotational and microturbulent velocities have maximum values of ≃5 and ≃2 km s⁻¹, respectively. The latter value is smaller than has often been adopted in LTE model atmosphere analyses of main-sequence stars. Non-LTE absolute metal abundances are estimated, and a comparison with those for normal B-type stars (deduced using similar non-LTE techniques) shows no significant differences. A comparison of the abundances deduced using non-LTE and LTE calculations implies systematic differences of 0.1–0.2 dex, showing the importance of using a non-LTE approach when accurate absolute abundances are required. Its location in the Hertzsprung–Russell diagram and normal metal abundance lead us to conclude that HD 83206 is probably a main-sequence B-type star. As such, it is among the sharpest-lined young B-type star discovered to date.     

Particle acceleration by ultrarelativistic shocks: theory and simulations

We consider the acceleration of charged particles near ultrarelativistic shocks, with Lorentz factor     . We present simulations of the acceleration process and compare these with results from semi-analytical calculations. We show that the spectrum that results from acceleration near ultrarelativistic shocks is a power law,     , with a nearly universal value     for the slope of this power law.
We confirm that the ultrarelativistic equivalent of the Fermi acceleration at a shock differs from its non-relativistic counterpart by the occurrence of large anisotropies in the distribution of the accelerated particles near the shock. In the rest frame of the upstream fluid, particles can only outrun the shock when their direction of motion lies within a small loss cone of opening angle     around the shock normal.
We also show that all physically plausible deflection or scattering mechanisms can change the upstream flight direction of relativistic particles originating from downstream by only a small amount:     . This limits the energy change per shock crossing cycle to     , except for the first cycle where particles originate upstream. In that case the upstream energy is boosted by a factor     for those particles that are scattered back across the shock into the upstream region.     

Inefficient star formation: the combined effects of magnetic fields and radiative feedback

We investigate the effects of magnetic fields and radiative protostellar feedback on the star formation process using self-gravitating radiation magnetohydrodynamical calculations. We present results from a series of calculations of the collapse of  50 M_⊙  molecular clouds with various magnetic field strengths and with and without radiative transfer. We find that both magnetic fields and radiation have a dramatic impact on star formation, though the two effects are in many ways complementary. Magnetic fields primarily provide support on large scales to low-density gas, whereas radiation is found to strongly suppress small-scale fragmentation by increasing the temperature in the high-density material near the protostars. With strong magnetic fields and radiative feedback, the net result is an inefficient star formation process with a star formation rate of  ≲10  per cent per free-fall time that approaches the observed rate, although we have only been able to follow the calculations for 1/3 of a free-fall time beyond the onset of star formation.     

Si x emission lines in spectra obtained with the Solar EUV Rocket Telescope and Spectrograph (SERTS)

New R -matrix calculations of electron impact excitation rates for transitions among the 2s²2p, 2s2p² and 2p³ levels of Si  x are presented. These data are subsequently used, in conjunction with recent estimates for proton excitation rates, to derive theoretical electron density sensitive emission-line ratios involving transitions in the ∼253–356 Å wavelength range. A comparision of these with observations of a solar active region and subflare, obtained during the 1989 flight of the Solar EUV Rocket Telescope and Spectrograph ( SERTS ), reveals that the electron densities determined from most of the Si  x line ratios are consistent with one another for both solar features. In addition, the derived densities are also in good agreement with the values of N _e estimated from diagnostic lines in other species formed at similar electron temperatures to Si  x , such as Fe  xii and Fe  xiii . These results provide observational support for the general accuracy of the adopted atomic data, and hence line ratio calculations, employed in the present analysis. However, we find that the Si  x 256.32-Å line is blended with the He  ii transition at the same wavelength, while the feature at 292.25 Å is not due to Si  x , but currently remains unidentified. The intensity of the 253.81-Å line in the SERTS active region spectrum is about a factor of 3 larger than expected from theory, but the reason for this is unclear, and requires additional observations to explain the discrepancy.     

Low- and intermediate-mass close binary evolution and the initial–final mass relation

Using Eggleton's stellar evolution code, we carry out 150 runs of Population I binary evolution calculations with the initial primary mass between 1 and 8 M_⊙, the initial mass ratio     between 1.1 and 4, and the onset of Roche lobe overflow (RLOF) at an early, middle or late Hertzsprung-gap stage. We assume that RLOF is conservative in the calculations, and find that the remnant mass of the primary may change by more than 40 per cent over the range of initial mass ratio or orbital period, for a given primary mass. This is contrary to the often-held belief that the remnant mass depends only on the progenitor mass if mass transfer begins in the Hertzsprung gap. We fit a formula, with an error less than 3.6 per cent, for the remnant (white dwarf) mass as a function of the initial mass M _1i of the primary, the initial mass ratio q _i and the radius of the primary at the onset of RLOF. We also find that a carbon–oxygen white dwarf with mass as low as 0.33 M_⊙ may be formed if the initial mass of the primary is around 2.5 M_⊙.