arXiv subjects

Peeter Tenjes

Publications and source records attributed to Peeter Tenjes.

At least 19 recordsLinked to original sources

arXiv2025

The richest clusters in the Coma and Leo superclusters: Properties and evolution

We study the substructure, connectivity, and galaxy content of galaxy clusters A1656 and 1367 in the Coma supercluster and of A1185 in the Leo supercluster with the aim of understanding the evolution of clusters from turnaround to virialisation. We used data from the SDSS DR10 MAIN galaxy sample and from DESI cluster catalogues. The projected phase space diagram and the distribution of mass were used to identify regions of various infall stages (early and late infall, and regions of ongoing infall, i.e. regions of influence), their characteristic radii, embedded mass, and density contrasts in order to study the evolution of clusters with the spherical collapse model. We determined the substructure of clusters using normal mixture modelling and their connectivity by counting filaments in the cluster's regions of influence, analysed galaxy content of clusters, and derived scaling relations between cluster masses. All three clusters have a substructure with two to five components and up to six filaments connected to them. The radii of regions of influence are $R_\mathrm{inf} \approx 4$ Mpc, and the density contrast at their borders is $\Delta\rho_{inf} \approx 50 - 60$. The scaling relations between the masses of clusters have a very small scatter. The galaxy content of the clusters and of their regions of influence vary from cluster to cluster. In superclusters the percentage of quiescent galaxies is higher than in low-density regions between superclusters. The collapse of the regions of influence of clusters started at redshifts $z \approx 0.4 - 0.5$. Clusters will be virialised in $\approx 3.3$ Gyrs. Clusters in superclusters will not merge, and their present-day turnaround regions will be virialised in $\approx 10$ Gyrs. The large variety of properties of clusters suggests that they have followed different paths during evolution.

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arXiv2024

Superbubbles as the source of dynamical friction: gas migration, stellar and dark matter contributions

[Abridged] The gas distribution in galaxies is smooth on large scales, but is usually time-dependent and inhomogeneous on smaller scales. The time-dependence originates from processes such as cloud formation, their collisions, and supernovae (SNe) explosions, which also create inhomogeneities. The inhomogeneities in the matter distribution give rise to variations in the local galactic gravitational potential, which can contribute to dynamically coupling the gas disc to the stellar and the dark matter (DM) components of the galaxy. Specifically, multiple SNe occurring in young stellar clusters give rise to superbubbles (SBs), which modify the local acceleration field and alter the energy and momentum of stars or DM particles traversing them, in broad analogy to the dynamical friction caused by a massive object. Our aim is to quantify how the acceleration field from SBs causes dynamical friction and contributes to the secular evolution of galaxies. In order to assess this, we constructed the time-dependent density modifications to the gas distribution that mimics a SB. By evaluating the acceleration field from these density modifications, we were able to see how the momentum or angular momentum of the gas hosting the SBs changes when stars pass through the SB. Combining the effects of all the stars and SBs, we constructed an empirical approximation formula for the momentum loss in homogeneous and isotropic cases. We find that the rate at which the gas disc loses its specific angular momentum via the above process is up to 4% per Gyr, which translates to under one-half of its original value over the lifetime of the disc. For comparison, the mass transfer rate from SBs is about one order of magnitude less than from gas turbulence, and hence the SB contribution should be included to account for the gas migration rate more accurately than 10%.

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arXiv2023

Back to the present: A general treatment for the tidal field from the wake of dynamical friction

Dynamical friction can be a valuable tool for inferring dark matter properties that are difficult to constrain by other methods. Most applications of dynamical friction calculations are concerned with the long-term angular momentum loss and orbital decay of the perturber within its host. This, however, assumes knowledge of the unknown initial conditions of the system. We advance an alternative methodology to infer the host properties from the perturber's shape distortions induced by the tides of the wake of dynamical friction, which we refer to as the tidal dynamical friction. As the shape distortions rely on the tidal field that has a predominantly local origin, we present a strategy to find the local wake by integrating the stellar orbits back in time along with the perturber, then removing the perturber's potential and re-integrating them back to the present. This provides perturbed and unperturbed coordinates and hence a change in coordinates, density, and acceleration fields, which yields the back-reaction experienced by the perturber. The method successfully recovers the tidal field of the wake based on a comparison with N-body simulations. We show that similar to the tidal field itself, the noise and randomness of the dynamical friction force due to the finite number of stars is also dominated by regions close to the perturber. Stars near the perturber influence it more but are smaller in number, causing a high variance in the acceleration field. These fluctuations are intrinsic to dynamical friction. We show that a stellar density of $0.0014 {\rm M_\odot\, kpc^{-3}}$ yields an inherent variance of 10% to the dynamical friction. The current method extends the family of dynamical friction methods that allow for the inference of host properties from tidal forces of the wake. It can be applied to specific galaxies, such as Magellanic Clouds, with Gaia data.

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arXiv2023

Galaxy groups and clusters and their brightest galaxies within the cosmic web

Our aim is to combine data on single galaxies, galaxy groups, their BGGs, and their location in the cosmic web, to determine classes of groups, and to obtain a better understanding of their properties and evolution. Data on groups and their BGGs are based on the Sloan Digital Sky Survey DR10 MAIN spectroscopic galaxy sample. We characterize the group environments by the luminosity-density field and their filament membership. We divide BGGs according to their star formation properties as quenched, and red and blue star-forming galaxies. We apply multidimensional Gaussian mixture modelling to divide groups based on their properties and environments. We analyse the offset of BGGs with respect to the group centre, and the relation between the stellar velocity dispersion of BGGs and the group velocity dispersions. We show that the groups in our sample can be divided into two main classes: high-luminosity rich groups and clusters, and low-luminosity poor groups with threshold luminosity $L = 15 \times 10^{10} h^{-2} L_{sun}$ and mass $M = 23 \times 10^{12} h^{-1} M_{sun}$. In rich clusters approximately 90% of the BGGs are red and quenched galaxies, while in poor groups only 40- 60$% of BGGs are red and quenched, and the rest of the BGGs are star-forming, either blue (20 - 40% of BGGs) or red (17% of BCGs). Rich groups and clusters are located in global high-density regions in filaments or filament outskirts, while poor groups reside everywhere in the cosmic web. Our results suggest that group and cluster properties are modulated by their location in the cosmic web, but the properties of their BGGs are mostly determined by processes within group or cluster dark matter halo. We emphasize the role of superclusters as a special environment for group growth.

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arXiv2022

Death at watersheds: galaxy quenching in low-density environments

Our aim is to understand the effect of environment to galaxy quenching in various local and global environments. We focus on galaxies with very old stellar populations (VO galaxies), typically found in the centers of clusters and groups, and search for such galaxies in the lowest global density environments, watersheds between superclusters. We use the Sloan Digital Sky Survey MAIN galaxy sample to calculate the luminosity-density field and get global density field, to determine groups and filaments, and to obtain data on galaxy properties. We divide groups into low- and high-luminosity groups based on the highest luminosity of groups in the watershed region, $L_{gr} = 15 \times10^{10} h^{-2} L_{sun}$. Our study shows that the global density is most strongly related to the richness of galaxy groups. Its influence on the overall star formation quenching in galaxies is less strong. Correlations between the morphological properties of galaxies and the global density field are the weakest. The watershed regions are populated mostly by single galaxies (70% of all galaxies there), and by low-luminosity groups. Still, approximately one-third of all galaxies in the watershed regions are VO galaxies. They have lower stellar masses, smaller stellar velocity dispersions, and stellar populations that are up to 2Gyr younger than those of VO galaxies in other global environments. In higher density global environments, the morphological properties of galaxies are very similar. Differences in galaxy properties are the largest between satellites and brightest group galaxies. Our results suggest that galaxy evolution is determined by the birthplace of galaxies in the cosmic web, and mainly by internal processes which lead to the present-day properties of galaxies. This may explain the similarity of (VO) galaxies in extremely different environments.

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arXiv2022

The evolution of high-density cores of the BOSS Great Wall superclusters

High-density cores (HDCs) of galaxy superclusters that embed rich clusters and groups of galaxies are the earliest large objects to form in the cosmic web, and the largest objects that may collapse in the present or future. We study the dynamical state and possible evolution of the HDCs in the BOSS Great Wall (BGW) superclusters at redshift $z \approx 0.5$ in order to understand the growth and evolution of structures in the Universe. We derived the density contrast values for the spherical collapse model in a wide range of redshifts and used these values to study the dynamical state and possible evolution of the HDCs of the BGW superclusters. The masses of the HDCs were calculated using stellar masses of galaxies in them. We found the masses and radii of the turnaround and future collapse regions in the HDCs and compared them with those of local superclusters. We determined eight HDCs in the BGW superclusters. The masses of their turnaround regions are in the range of $M_{\mathrm{T}} \approx 0.4 - 3.3\times~10^{15}h^{-1}M_\odot,$ and radii are in the range of $R_{\mathrm{T}} \approx 3.5 - 7 h^{-1}$Mpc. The radii of their future collapse regions are in the range of $R_{\mathrm{FC}} \approx 4 - 8h^{-1}$Mpc. Distances between individual cores in superclusters are much larger: of the order of $25 - 35h^{-1}$Mpc. The richness and sizes of the HDCs are comparable with those of the HDCs of the richest superclusters in the local Universe. The BGW superclusters will probably evolve to several poorer superclusters with masses similar to those of the local superclusters. This may weaken the tension with the $\Lambda$CDM model, which does not predict a large number of very rich and large superclusters in our local cosmic neighbourhood, and explains why there are no superclusters as elongated as those in the BGW in the local Universe.

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arXiv2021

Non-equilibrium in the Solar Neighbourhood using dynamical modelling with Gaia DR2

Matter distribution models of the Milky Way galaxy are usually stationary, although there are known to be wave-like perturbations in the disc at $\sim10\%$ level of the total density. Modelling of the overall acceleration field by allowing non-equilibrium is a complicated task. We must learn to distinguish whether density enhancements are persistent or not by their nature. In the present paper, we elaborate our orbital arc method to include the effects of massless perturbations and non-stationarities in the modelling. The method is tested by modelling of simulation data and shown to be valid. We apply the method to the Gaia DR 2 data within a region of $\sim 0.5$ kpc from the Sun and confirm that acceleration field in the Solar Neighbourhood has a perturbed nature -- the phase space density along the orbits of stars grow in the order of $h\lesssim 5\%$ per Myr due to non-stationarity. This result is a temporally local value and can be used only within the timeframe of a few Myrs. An attempt to pinpoint the origin of the perturbation shows that the stars having larger absolute angular momentum are the main carriers of the local perturbation. As they are faster than the average thin disc star, they are either originating further away and are close in their pericentre or they are perturbed locally by a fast co-moving perturber, such as gas disc inhomogenities.

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arXiv2021

The Corona Borealis supercluster: connectivity, collapse, and evolution

We present a study of the Corona Borealis (CB) supercluster. We determined the high-density cores of the CB and the richest galaxy clusters in them, and studied their dynamical state and galaxy content. We determined filaments in the supercluster to analyse the connectivity of clusters. We compared the mass distribution in the CB with predictions from the spherical collapse model and analysed the acceleration field in the CB. We found that at a radius $R_{\mathrm{30}}$ around clusters in the CB (A2065, A2061, A2089, and Gr2064) (corresponding to the density contrast $\Delta\rho \approx 30$), the galaxy distribution shows a minimum. The $R_{30}$ values for individual clusters lie in the range of $3 - 6$ $h^{-1}$ Mpc. The radii of the clusters (splashback radii) lie in the range of $R_{\mathrm{cl}} \approx 2 - 3$ $R_{\mathrm{vir}}$. The projected phase space diagrams and the comparison with the spherical collapse model suggest that $R_{\mathrm{30}}$ regions have passed turnaround and are collapsing. Galaxy content in clusters varies strongly. The cluster A2061 has the highest fraction of galaxies with old stellar populations, and A2065 has the highest fraction of galaxies with young stellar populations. The number of long filaments near clusters vary from one at A2089 to five at A2061. During the future evolution, the clusters in the main part of the CB may merge and form one of the largest bound systems in the nearby Universe. Another part of the CB, with the cluster Gr2064, will form a separate system. The structures with a current density contrast $\Delta\rho \approx 30$ have passed turnaround and started to collapse at redshifts $z \approx 0.3 - 0.4$. The comparison of the number and properties of the most massive collapsing supercluster cores from observations and simulations may serve as a test for cosmological models.

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arXiv2020

Tidal forces from the wake of dynamical friction: warps, lopsidedness and kinematic misalignment

A galaxy moving through a background of dark matter particles, induces an overdensity of these particles or a wake behind it. The back reaction of this wake on the galaxy is a force field that can be decomposed into an effective deceleration (called dynamical friction) and a tidal field. In this paper we determine the tidal forces, thus generated on the galaxy, and the resulting observables, which are shown to be warps, lopsidedness and/or kinematic-photometric position angle misalignments. We estimate the magnitude of the tidal-like effects needed to reproduce the observed warp and lopsidedness on the isolated galaxy IC 2487. Within a realistic range of dark matter distribution properties the observed warped and lopsided kinematical properties of IC 2487 is possible to reproduce (the background medium of dark matter particles has a velocity dispersion of $\lesssim 80\,{\rm km\,s^{-1}}$ and the density $10^4-10^5~{\rm M_\odot\,kpc^{-3}}$, more likely at the lower end). We conclude that the proposed mechanism can generate warps, lopsidedness and misalignments observed in isolated galaxies or galaxies in loose groups. The method can be used also to constrain dark matter spatial and velocity distribution properties.

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arXiv2020

Multiscale cosmic web detachments, connectivity, and preprocessing in the supercluster SClA2142 cocoon

We study the properties, connectivity, and galaxy content of groups and filaments in the low-density region (cocoon) around A2142 supercluster (SClA2142). We traced the SClA2142 cocoon boundaries by the lowest luminosity-density regions that separate SClA2142 from other superclusters. We determined galaxy filaments and groups in the cocoon and analysed the connectivity of groups, the high density core (HDC) of the supercluster, and the whole of the supercluster. We compared the distribution and properties of galaxies with different star-formation properties in the supercluster and in the cocoon. SClA2142 and the long filament that is connected to it forms the longest straight structure in the Universe detected so far, with a length of $75$ $h^{-1}$ Mpc. The connectivity of the supercluster is C = 6 - 7; poor groups have C = 1 - 2. Long filaments around the supercluster's main body are detached from it at the turnaround region. Galaxies with very old stellar populations lie in systems across a wide range of richness from the richest cluster to poorest groups and single galaxies. They lie even at local densities as low as $D1 < 1$ in the cocoon and up to $D1 > 800$ in the supercluster. Recently quenched galaxies lie in the cocoon mainly in one region and their properties are different in the cocoon and in the supercluster. The star-formation properties of single galaxies are similar across all environments. The collapsing main body of SClA2142 with the detached long filaments near it are evidence of an important epoch in the supercluster evolution. Further studies are needed to understand the reasons of similarity of galaxies with very old stellar populations in extremely different environments. The presence of long, straight structures in the cosmic web may serve as a test for cosmological models.

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arXiv2020

Quantifying torque from the Milky Way bar using Gaia DR2

We determine the mass of the Milky Way bar and the torque it causes, using Gaia DR2, by applying the orbital arc method. Based on this, we have found that the gravitational acceleration is not directed towards the centre of our Galaxy but a few degrees away from it. We propose that the tangential acceleration component is caused by the bar of the Galaxy. Calculations based on our model suggest that the torque experienced by the region around the Sun is $\approx 2400\, km^2 s^{-2}$ per solar mass. The mass estimate for the bar is $\sim 1.6\pm0.3\times10^{10} M_\odot$. Using greatly improved data from Gaia DR2, we have computed the acceleration field to great accuracy by adapting the oPDF method (Han et al. 2016) locally and used the phase space coordinates of $\sim 4\times10^5$ stars within a distance of 0.5 kpc from the Sun. In the orbital arc method, the first step is to guess an acceleration field and then reconstruct the stellar orbits using this acceleration for all the stars within a specified region. Next, the stars are redistributed along orbits to check if the overall phase space distribution has changed. We repeat this process until we find an acceleration field that results in a new phase space distribution that is the same as the one that we started with; we have then recovered the true underlying acceleration.

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arXiv2019

The influence of dark matter halo on the stellar stream asymmetry via dynamical friction

We study the effect of dynamical friction on globular clusters and on the stars evaporated from the globular clusters (stellar streams) moving in a galactic halo. Due to dynamical friction, the position of a globular cluster (GC) as a stream progenitor starts to shift with respect to its original position in the reference frame of initial GC orbit. Therefore the stars that have evaporated at different times have different mean position with respect to the GC position. This shifting results in a certain asymmetry in stellar density distribution between the leading and trailing arms of the stream. The degree of the asymmetry depends on the characteristics of the environment in which the GC and the stream stars move. As GCs are located mainly in outer parts of a galaxy, this makes dynamical friction a unique probe to constrain the underlying dark matter spatial density and velocity distributions. For a GC NGC 3201 we compared our theoretical shift estimates with available observations. Due to large uncertainties in current observation data, we can only conclude that the derived estimates have the same order of magnitude.

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arXiv2018

A method to calculate gravitational accelerations within discrete localised regions in the Milky Way

We present a method to calculate gravitational potential gradients within regions containing few tens of thousands stars with known phase space coordinates. The central idea of the method is to calculate orbital arcs for each star within a given region for a certain parametrised potential (gravitational acceleration) and to assume that position of each star on its orbital arc is a random variable with a uniform probability density in time. Thereafter, by combining individual probability densities of stars it is possible to calculate the overall probability density distribution and likelihood for a given region as a function of gravitational acceleration parameters. The likelihood has a maximum if the calculated probability distribution and the observed distribution of stars in phase space are consistent. This allows us to constrain gravitational accelerations and potential gradient values. The method assumes that phases of stars are mixed within the regions where stellar orbits are calculated. We tested the method for 12 small rectangular regions within simulated disc galaxy from Gaia Wiki. Tests show that even with a rather simple acceleration form the calculated accelerations in galactic plane coincide with their true values from simulation about 5 per cent, misalignment between the calculated and true acceleration vector directions is less than 1 degree (median values). The model can be used with the Milky Way Gaia complete solution data.

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arXiv2017

A method to calculate the local density distribution of the Galaxy from the Tycho-Gaia Astrometric Solution data

New and more reliable distances and proper motions of a large number of stars in the Tycho-Gaia Astrometric Solution (TGAS) catalogue allow to calculate the local matter density distribution more precisely than earlier. We devised a method to calculate the stationary gravitational potential distribution perpendicular to the Galactic plane by comparing the vertical probability density distribution of a sample of observed stars with the theoretical probability density distribution computed from their vertical coordinates and velocities. We applied the model to idealised test stars and to the real observational samples. Tests with two mock datasets proved that the method is viable and provides reasonable results. Applying the method to TGAS data we derived that the total matter density in the Solar neighbourhood is $0.09\pm 0.02 \text{M}_\odot\text{pc}^{-3}$ being consistent with the results from literature. The matter surface density within $|z|\le 0.75 \text{kpc}$ is $42\pm 4 \text{M}_\odot\text{pc}^{-2}$. This is slightly less than the results derived by other authors but within errors is consistent with previous estimates. Our results show no firm evidence for significant amount of dark matter in the Solar neighbourhood. However, we caution that our calculations at $|z| \leq 0.75$ kpc rely on an extrapolation from the velocity distribution function calculated at $|z| \leq 25$ pc. This extrapolation can be very sensitive to our assumption that the stellar motions are perfectly decoupled in R and z, and to our assumption of equilibrium. Indeed, we find that $ρ(z)$ within $|z|\le 0.75$ kpc is asymmetric with respect to the Galactic plane at distances $|z| = 0.1-0.4$ kpc indicating that the density distribution may be influenced by density perturbations.

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arXiv2017

Spiral arms and disc stability in the Andromeda galaxy

Aims: Density waves are often considered as the triggering mechanism of star formation in spiral galaxies. Our aim is to study relations between different star formation tracers (stellar UV and near-IR radiation and emission from HI, CO and cold dust) in the spiral arms of M31, to calculate stability conditions in the galaxy disc and to draw conclusions about possible star formation triggering mechanisms. Methods: We select fourteen spiral arm segments from the de-projected data maps and compare emission distributions along the cross sections of the segments in different datasets to each other, in order to detect spatial offsets between young stellar populations and the star forming medium. By using the disc stability condition as a function of perturbation wavelength and distance from the galaxy centre we calculate the effective disc stability parameters and the least stable wavelengths at different distances. For this we utilise a mass distribution model of M31 with four disc components (old and young stellar discs, cold and warm gaseous discs) embedded within the external potential of the bulge, the stellar halo and the dark matter halo. Each component is considered to have a realistic finite thickness. Results: No systematic offsets between the observed UV and CO/far-IR emission across the spiral segments are detected. The calculated effective stability parameter has a minimal value Q_{eff} ~ 1.8 at galactocentric distances 12 - 13 kpc. The least stable wavelengths are rather long, with the minimal values starting from ~ 3 kpc at distances R > 11 kpc. Conclusions: The classical density wave theory is not a realistic explanation for the spiral structure of M31. Instead, external causes should be considered, e.g. interactions with massive gas clouds or dwarf companions of M31.

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arXiv2016

Stellar kinematics using a third integral of motion: method and application on the Andromeda galaxy

We probe the feasibility of describing the structure of a multi-component axisymmetric galaxy with a dynamical model based on the Jeans equations while taking into account a third integral of motion. We demonstrate that using the third integral in the form derived by G. Kuzmin, it is possible to calculate the stellar kinematics of a galaxy from the Jeans equations by integrating the equations along certain characteristic curves. In cases where the third integral of motion does not describe the system exactly, the derived kinematics would describe the galaxy only approximately. We apply our method to the Andromeda galaxy, for which the mass distribution is relatively firmly known. We are able to reproduce the observed stellar kinematics of the galaxy rather well. The calculated model suggests that the velocity dispersion ratios $σ_z^2/σ_R^2$ of M31 decrease with increasing R. Moving away from the galactic plane, $σ_z^2/σ_R^2$ remains the same. The velocity dispersions $σ_θ^2$ and $σ_R^2$ are roughly equal in the galactic plane.

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arXiv2010

SDSS surface photometry of M 31 with absorption corrections

The objective of this work is to obtain an extinction-corrected distribution of optical surface brightness and colour indices of the large nearby galaxy M 31 using homogeneous observational data and a model for intrinsic extinction. We process the Sloan Digital Sky Survey (SDSS) images in ugriz passbands and construct corresponding mosaic images, taking special care of subtracting the varying sky background. We apply the galactic model developed in Tempel et al. (2010) and far-infrared imaging to correct the photometry for intrinsic dust effects. We obtain observed and dust-corrected distributions of the surface brightness of M 31 and a map of line-of-sight extinctions inside the galaxy. Our extinction model suggests that either M 31 is intrinsically non-symmetric along the minor axis or the dust properties differ from those of the Milky Way. Assuming the latter case, we present the surface brightness distributions and integral photometry for the Sloan filters as well as the standard UBVRI system. We find the following intrinsic integral colour indices for M 31: (U-B)_0=0.35; (B-V)_0=0.86; (V-R)_0=0.63; (R-I)_0=0.53; the total intrinsic absorption-corrected luminosities of M 31 in the B and the V filters are 4.10 and 3.24 mag, respectively.

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arXiv2009

Dust-corrected surface photometry of M 31 from the Spitzer far infrared observations

We create a model for recovering the intrinsic, absorption-corrected surface brightness distribution of a galaxy and apply the model to the M31. We construct a galactic model as a superposition of axially symmetric stellar components and a dust disc to analyse the intrinsic absorption efects. Dust column density is assumed to be proportional to the far-infrared flux of the galaxy. Along each line of sight, the observed far-infrared spectral energy distribution is approximated with modified black body functions considering dust components with different temperatures, allowing to determine the temperatures and relative column densities of the dust components. We apply the model to the nearby galaxy M31 using the Spitzer Space Telescope far-infrared observations for mapping dust distribution and temperature. A warm and a cold dust component are distinguished. The temperature of the warm dust in M31 varies between 56 and 60 K and is highest in the spiral arms; the temperature of the cold component is mostly 15-19 K and rises up to about 25 K at the centre of the galaxy. The intensity-weighted mean temperature of the dust decreases from T ~32 K at the centre to T ~20 K at R ~7 kpc and outwards. We also calculate the intrinsic UBVRIL surface brightness distributions and the spatial luminosity distribution. The intrinsic dust extinction in the V-colour rises from 0.25 mag at the centre to 0.4-0.5 mag at R = 6-13 kpc and decreases smoothly thereafter. The calculated total extinction-corrected luminosity of M31 is L_B = (3.64 pm 0.15) 10^10L_sun, corresponding to an absolute luminosity M_B = (-20.89 pm 0.04) mag. Of the total B-luminosity, 20% (0.24 mag) is obscured from us by the dust inside M31. The intrinsic shape of the bulge is slightly prolate in our best-fit model.

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