A Simple Law of Star Formation

We show that supersonic MHD turbulence yields a star formation rate (SFR) as low as observed in molecular clouds (MCs), for characteristic values of the free-fall time divided by the dynamical time, $t_{\rm ff}/t_{\rm dyn}$, the alfv\'{e}nic Mach number, ${\cal M}_{\rm a}$, and the sonic Mach number, ${\cal M}_{\rm s}$. Using a very large set of deep adaptive-mesh-refinement simulations, we quantify the dependence of the SFR per free-fall time, $\epsilon_{\rm ff}$, on the above parameters. Our main results are: i) $\epsilon_{\rm ff}$ decreases exponentially with increasing $t_{\rm ff}/t_{\rm dyn}$, but is insensitive to changes in ${\cal M}_{\rm s}$, for constant values of $t_{\rm ff}/t_{\rm dyn}$ and ${\cal M}_{\rm a}$. ii) Decreasing values of ${\cal M}_{\rm a}$ (stronger magnetic fields) reduce $\epsilon_{\rm ff}$, but only to a point, beyond which $\epsilon_{\rm ff}$ increases with a further decrease of ${\cal M}_{\rm a}$. iii) For values of ${\cal M}_{\rm a}$ characteristic of star-forming regions, $\epsilon_{\rm ff}$ varies with ${\cal M}_{\rm a}$ by less than a factor of two. We propose a simple star-formation law, based on the empirical fit to the minimum $\epsilon_{\rm ff}$, and depending only on $t_{\rm ff}/t_{\rm dyn}$: $\epsilon_{\rm ff} \approx \epsilon_{\rm wind} \exp(-1.6 \,t_{\rm ff}/t_{\rm dyn})$. Because it only depends on the mean gas density and rms velocity, this law is straightforward to implement in simulations and analytical models of galaxy formation and evolution.Comment: ApJ Letters - in pres

Estimation of high-resolution dust column density maps. Comparison of modified black-body fits and radiative transfer modelling

Sub-millimetre dust emission is often used to derive the column density N of dense interstellar clouds. The observations consist of data at several wavelengths but of variable resolution. We examine two procedures that been proposed for the estimation of high resolution N maps. Method A uses a low-resolution temperature map combined with higher resolution intensity data while Method B combines N estimates from different wavelength ranges. Our aim is to determine the accuracy of the methods relative to the true column densities and the estimates obtainable with radiative transfer modelling. We use magnetohydrodynamical (MHD) simulations and radiative transfer calculations to simulate sub-millimetre observations at the wavelengths of the Herschel Space Observatory. The observations are analysed with the methods and the results compared to the true values and to the results from radiative transfer modelling of observations. Both methods A and B give relatively reliable column density estimates at the resolution of 250um data while also making use of the longer wavelengths. For high signal-to-noise data, the results of Method B are better correlated with the true column density, while Method A is less sensitive to noise. When the cloud has internal heating, results of Method B are consistent with those that would be obtained with high-resolution data. Because of line-of-sight temperature variations, these underestimate the true column density and, because of a favourable cancellation of errors, Method A can sometimes give more correct values. Radiative transfer modelling, even with very simple 3D cloud models, can provide better results. However, the complexity of the models required for improvements increases rapidly with the complexity and opacity of the clouds.Comment: 14 pages, Accepted to A&

Radiative transfer on hierarchial grids

We present new methods for radiative transfer on hierarchial grids. We develop a new method for calculating the scattered flux that employs the grid structure to speed up the computation. We describe a novel subiteration algorithm that can be used to accelerate calculations with strong dust temperature self-coupling. We compute two test models, a molecular cloud and a circumstellar disc, and compare the accuracy and speed of the new algorithms against existing methods. An adaptive model of the molecular cloud with less than 8 % of the cells in the uniform grid produced results in good agreement with the full resolution model. The relative RMS error of the surface brightness <4 % at all wavelengths, and in regions of high column density the relative RMS error was only 10^{-4}. Computation with the adaptive model was faster by a factor of ~5. The new method for calculating the scattered flux is faster by a factor of ~4 in large models with a deep hierarchy structure, when images of the scattered light are computed towards several observing directions. The efficiency of the subiteration algorithm is highly dependent on the details of the model. In the circumstellar disc test the speed-up was a factor of two, but much larger gains are possible. The algorithm is expected to be most beneficial in models where a large number of small, dense regions are embedded in an environment with a lower mean density.Comment: Accepted to A&A; 13 pages, 8 figures; (v2: minor typos corrected

The degeneracy between dust colour temperature and spectral index. Comparison of methods for estimating the beta(T) relation

Sub-millimetre dust emission provides information on the physics of interstellar clouds and dust. Noise can produce anticorrelation between the colour temperature T_C and the spectral index beta. This must be separated from the intrinsic beta(T) relation of dust. We compare methods for the analysis of the beta(T) relation. We examine sub-millimetre observations simulated as simple modified black body emission or using 3D radiative transfer modelling. In addition to chi^2 fitting, we examine the results of the SIMEX method, basic Bayesian model, hierarchical models, and one method that explicitly assumes a functional form for beta(T). All methods exhibit some bias. Bayesian method shows significantly lower bias than direct chi^2 fits. The same is true for hierarchical models that also result in a smaller scatter in the temperature and spectral index values. However, significant bias was observed in cases with high noise levels. Beta and T estimates of the hierarchical model are biased towards the relation determined by the data with the highest S/N ratio. This can alter the recovered beta(T) function. With the method where we explicitly assume a functional form for the beta(T) relation, the bias is similar to the Bayesian method. In the case of an actual Herschel field, all methods agree, showing some degree of anticorrelation between T and beta. The Bayesian method and the hierarchical models can both reduce the noise-induced parameter correlations. However, all methods can exhibit non-negligible bias. This is particularly true for hierarchical models and observations of varying signal-to-noise ratios and must be taken into account when interpreting the results.Comment: Submitted to A&A, 18 page

Constrained simulations of the Antennae Galaxies: Comparison with Herschel-PACS observations

We present a set of hydro-dynamical numerical simulations of the Antennae galaxies in order to understand the origin of the central overlap starburst. Our dynamical model provides a good match to the observed nuclear and overlap star formation, especially when using a range of rather inefficient stellar feedback efficiencies (0.01 < q_EoS < 0.1). In this case a simple conversion of local star formation to molecular hydrogen surface density motivated by observations accounts well for the observed distribution of CO. Using radiative transfer post-processing we model synthetic far-infrared spectral energy distributions (SEDs) and two-dimensional emission maps for direct comparison with Herschel-PACS observations. For a gas-to-dust ratio of 62:1 and the best matching range of stellar feedback efficiencies the synthetic far-infrared SEDs of the central star forming region peak at values of ~65 - 81 Jy at 99 - 116 um, similar to a three-component modified black body fit to infrared observations. Also the spatial distribution of the far-infrared emission at 70 um, 100 um, and 160 um compares well with the observations: >50% (> 35%) of the emission in each band is concentrated in the overlap region while only < 30% (< 15%) is distributed to the combined emission from the two galactic nuclei in the simulations (observations). As a proof of principle we show that parameter variations in the feedback model result in unambiguous changes both in the global and in the spatially resolved observable far-infrared properties of Antennae galaxy models. Our results strengthen the importance of direct, spatially resolved comparative studies of matched galaxy merger simulations as a valuable tool to constrain the fundamental star formation and feedback physics.Comment: 17 pages, 8 figures, 4 tables, submitted to MNRAS, including revisions after first referee report, comments welcom

Tähtienvälisten pilvien säteilynkuljetusmallinnus

Almost all information on astrophysical objects is obtained through observation of electromagnetic radiation. The observed radiation has been altered in interactions with matter, and understanding the transport of radiation is a key prerequisite for understanding the physical conditions in the observed objects. The transport of radiation is described by the radiative transfer equation. Owing to its complex nature, solving the radiative transfer equation is difficult, and it is usually necessary to resort to numerical calculations. In this thesis, the focus is on the modelling of radiation transport in interstellar clouds. The dense gas and dust in interstellar clouds scatter, absorb, and emit radiation, and understanding the radiative transfer effects is crucial in the interpretation of observations. Four of the five articles that are contained in this thesis concern various applications of radiative transfer modelling. Two articles focus on the modelling of spectral line radiation. We study the use of OH Zeeman splitting observations in the determination of magnetic field strengths in molecular clouds. The role of magnetic fields in the process of star formation is still largely an open question with two competing models: the turbulence dominated scenario where magnetic fields are weak, and the ambipolar diffusion driven model with stronger magnetic fields. By combining magneto-hydrodynamical calculations with radiative transfer simulations, we show that the turbulence dominated scenario is consistent with the observed magnetic field strengths. Two articles concern the dust radiative transfer. We study the dust density distribution and grain properties in the dust envelope surrounding the carbon star IRC +10216. By modelling the surface brightness distribution of the scattered light in the dust envelope, we can infer the mass-loss history of the star and improve models of newly formed dust grains. In another article we use magneto-hydrodynamical calculations and radiative transfer simulations to study the reliability of cloud core mass estimates. Observations of dust thermal emission at the far-infrared and sub-millimetre wavelengths are commonly used to determine the masses of molecular cloud cores. By constructing synthetic observations of a model cloud and comparing the estimated masses to the true masses that are obtained directly from the cloud model, we can determine the robustness of mass estimates. Instead of focusing on the applications of radiative transfer modelling, one article describes new numerical methods for efficient radiative transfer simulations. We describe new algorithms for radiative transfer on hierarchical grids. The new algorithms, in particular the use of sub-iterations, are faster by a factor of several compared to the old methods.Lähes kaikki tieto tähtitieteellisistä kohteista saadaan havaitsemalla sähkömagneettista säteilyä. Havaittava säteily on muuttunut vuorovaikutuksissa aineen kanssa, joten havaintojen tulkinta edellyttää säteilyn siirtymisen ymmärtämistä. Säteilyn siirtymistä kuvataan säteilynkuljetysyhtälöllä. Yhtälön monimutkaisuuden vuoksi ratkaisussa joudutaan yleensä käyttämään numeerisia menetelmiä. Tässä väitöskirjassa keskitytään mallintamaan säteilynkuljetusta tähtienvälisissä pilvissä. Niiden tiheä kaasu ja pöly sirottavat, absorboivat ja emittoivat säteilyä, joten säteilynkuljetusmallinnus on avainasemassa pilvien tutkimisessa. Neljä väitöskirjan viidestä artikkelista käsittelee säteilynkuljetusmallinnuksen sovelluksia. Kaksi artikkelia käsittelee spektriviivasäteilyä. Niin sanotun Zeemanin ilmiön ansiosta hydroksyyliradikaalin (OH) spektriviivahavainnoista voidaan johtaa magneettikentän voimakkuus tähtienvälisissä pilvissä. Magneettikenttien rooli tähtien synnyssä on yhä suurelta osin avoin kysymys, joten magneettikenttien mittaus on hyvin tärkeää. Yhdistämällä magnetohydrodynaamiset simulaatiot ja säteilynkuljetusmallinnuksen osoitamme, että malli jonka mukaan pilvien magneettikentät ovat heikkoja ja melko merkityksettömiä tähtien syntyprosessissa sopii kirjallisuudessa esitettyihin spektriviivahavaintoihin. Kaksi artikkelia käsittelee tähtienvälistä pölyä. Tutkimme valonsirontaa hiilitähteä IRC +10216 ympäröivässä pölykuoressa. Mallintamalla kuoren kirkkausjakaumaa voimme selvittää pölykuoren rakennetta sekä pölyhiukkasten ominaisuuksia. Toisessa artikkelissa magnetohydrodynaamisia malleja säteilynkuljetuslaskuja käytetään pilviytimien massa-arvioiden luotettavuuden tutkimiseen. Viides artikkeli keskittyy sovellusten sijasta tehokkaiden säteilynkuljetusalgoritmien kehitykseen. Esittelemme uusia menetelmiä, joilla säteilynkuljetusongelma voidaan ratkaista murto-osassa vanhojen menetelmien vaatimasta ajasta

Profiles of interstellar cloud filaments. Observational effects in synthetic sub-millimetre observations

Sub-millimetre observations suggest that the filaments of interstellar clouds have rather uniform widths and can be described with the so-called Plummer profiles. The shapes of the filament profiles are linked to their physical state. Before drawing conclusions on the observed column density profiles, we must evaluate the observational uncertainties. We want to estimate the bias that could result from radiative transfer effects or from variations of submm dust emissivity. We use cloud models obtained with magnetohydrodynamic simulations and carry out radiative transfer calculations to produce maps of sub-millimetre emission. Column densities are estimated based on the synthetic observations. For selected filaments, the estimated profiles are compared to those derived from the original column density. Possible effects from spatial variations of dust properties are examined. With instrumental noise typical of the Herschel observations, the parameters derived for nearby clouds are correct to within a few percent. The radiative transfer effects have only a minor effect on the results. If the signal-to-noise ratio is degraded by a factor of four, the errors become significant and for half of the examined filaments the values cannot be constrained. The errors increase in proportion to the cloud distance. Assuming the resolution of Herschel instruments, the model filaments are barely resolved at a distance of ~400 pc and the errors in the parameters of the Plummer function are several tens of per cent. The Plummer parameters, in particular the power-law exponent p, are sensitive to noise but can be determined with good accuracy using Herschel data. One must be cautious about possible line-of-sight confusion. In our models, a large fraction of the filaments seen in the column density maps are not continuous structures in three dimensions.Comment: 12 pages, 14 figures, accepted to A&

Composite biasing in Monte Carlo radiative transfer

Biasing or importance sampling is a powerful technique in Monte Carlo radiative transfer, and can be applied in different forms to increase the accuracy and efficiency of simulations. One of the drawbacks of the use of biasing is the potential introduction of large weight factors. We discuss a general strategy, composite biasing, to suppress the appearance of large weight factors. We use this composite biasing approach for two different problems faced by current state-of-the-art Monte Carlo radiative transfer codes: the generation of photon packages from multiple components, and the penetration of radiation through high optical depth barriers. In both cases, the implementation of the relevant algorithms is trivial and does not interfere with any other optimisation techniques. Through simple test models, we demonstrate the general applicability, accuracy and efficiency of the composite biasing approach. In particular, for the penetration of high optical depths, the gain in efficiency is spectacular for the specific problems that we consider: in simulations with composite path length stretching, high accuracy results are obtained even for simulations with modest numbers of photon packages, while simulations without biasing cannot reach convergence, even with a huge number of photon packages.Comment: 12 pages, accepted for publication in A&

Self-Consistent Analysis of OH-Zeeman Observations: Too Much Noise about Noise

We had recently re-analyzed in a self-consistent way OH-Zeeman observations in four molecular-cloud envelopes and we had shown that, contrary to claims by Crutcher et al., there is no evidence that the mass-to-flux ratio decreases from the envelopes to the cores of these clouds. The key difference between our data analysis and the earlier one by Crutcher et al. is the relaxation of the overly restrictive assumption made by Crutcher et al, that the magnetic field strength is independent of position in each of the four envelopes. In a more recent paper, Crutcher et al. (1) claim that our analysis is not self-consistent, in that it misses a cosine factor, and (2) present new arguments to support their contention that the magnetic-field strength is indeed independent of position in each of the four envelopes. We show that the claim of the missing cosine factor is false, that the new arguments contain even more serious problems than the Crutcher et al. original data analysis, and we present new observational evidence, independent of the OH-Zeeman data, that suggests significant variations in the magnetic-field strength in the four cloud envelopes.Comment: 8 pages, 3 figures, MNRAS in pres