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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

Dynamics of the interstellar matter in galaxies isolated barred spiral galaxies, cloud formation processes /

Kristen, Helmuth. January 1998 (has links)
Thesis (doctoral)--Stockholm University, 1998. / Includes bibliographical references.
2

Molecular gas in barred galaxy nuclei /

Petitpas, Glen Raymond. January 2001 (has links)
Thesis (Ph.D.) -- McMaster University, 2001. / Includes bibliographical references. Also available via World Wide Web.
3

The curious case of offset bars : markers for a baby galaxy disk or signposts of an interaction with dark matter sub halos?

Fortune, Marc Harris Yao January 2016 (has links)
>Magister Scientiae - MSc / We have used the Spitzer Survey of Stellar Structure in Galaxies (S⁴G) as a representative sample of the local universe (total of 2352 galaxies in S⁴G) to make a catalog of offset disk barred galaxies. Using the combined variation of the position angle and the ellipticity (provided by ellipse fit) and also through visual inspection, we have been able to identify all offset structures in S⁴G. While primary bars are present in 2=3 of the disk galaxies in the visible universe, offset bars have a much lower fraction. Of the ̴ 1500 (3.6µm images) disk galaxies available in S⁴G, we classified only 49 as offset barred disk galaxies. We have determined basic properties (bar to total luminosity ratio, bar length, disk scale-length and bars of offset bars shape) using GALFIT, a widely used galaxy decomposition software package. Our main conclusion is that all the offset bars are boxy, independent of their offset from the galaxy center, or the mass of the host galaxy. Additionally we find that, the early type offset bars seem to be more boxy than the late types. The comparison of our offset sample with two other samples, respectively, low mass and high mass normal barred galaxies ("normal" for bars located at the photometric center of the host galaxy), reveals them to be at an intermediate position between the two normal samples. The bar length, disk scale-length and bar to total luminosity ratio are on average larger than the low mass normal and smaller than high mass normal barred galaxies. We have found, overall, a tighter correlation between the disk and bar properties for offset bars in comparison to the two normal samples. Our explanation is that, although the offset has no visible impact on the global shape of the bars, the process responsible for these disturbances seems to affect the star formation rate such that their disk and bars are on average more active than the normal barred galaxies in the same mass range, but not enough to surpass normal barred galaxies with much higher mass.
4

Dust penetrated quantitative classifcation of nearby barred spiral galaxies

Tailor, Asha 12 September 2012 (has links)
The objective of this dissertation is twofold. Firstly an extensive yet concise literature review on the state-of-the-art of near infrared barred spiral galaxy classi cation is presented. Secondly, two quantitative approaches to galaxy classi cation at near infrared wavelengths, the relative gravitational torque method and the isophotal ellipse tting method, are applied for the rst time to a sample of selected Spitzer IRAC nearby barred galaxies. Maximum relative gravitational torques are derived for a sample of 40 nearby bright barred disk galaxies at 3.6 m and 4.5 m. These torques are compared between galaxy pairs at these passbands and we nd an excellent agreement between the 3.6 m and 4.5 m morphology. The sample used incorporates a wide range of inclination and bar strength values. The tight coupling of 3.6 m and 4.5 m morphology provides an opportunity to classify intermediate redshift galaxies that have their near-infrared rest frame emissions shifted red-ward to 4.5 m; i.e.: out to z = 0:25. We nd a greater frequency of higher maximum relative torques in our sample compared with either Block et al. (2002) or Buta et al. (2004) due to sample bias, as this dissertation is aimed at understanding quantitative methods in classifying barred galaxies. Furthermore, we compare results from applying an isophotal ellipse tting technique and a gravitational torque analysis to a common sample of 28 nearby barred S4G/Spitzer galaxies imaged at 3.6 m. These two quantitative bar strength methods are applied to images that have identical orientation and deprojection parameters for an objective comparison. We nd a strong correlation between the gravitational torque and isophotal ellipse tting methods which in principle supports a method for estimating bar potentials out to intermediate redshifts by using an isophotal ellipse tting approach. This has important implications for bar-fraction estimates and galaxy accretion/evolution scenarios.
5

Fabry-Pérot studies of the Milky Way bar kinematics, chemical composition and instrumentation /

Rangwala, Naseem, January 2009 (has links)
Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references.
6

Evidence For Increased Star Formation in Barred Galaxy Centres

Laing, Jennifer M January 2023 (has links)
Galactic bars play an important role in the dynamical evolution of their host galaxy, but their own evolution and impact on the local gas reservoir and star formation rate are still open questions. Recent work by the Physics at High Angular resolution in Nearby GalaxieS (PHANGS) collaboration found higher molecular gas surface densities and velocity dispersions in barred galaxies compared to unbarred galaxies. The higher turbulence found in bars is expected to create the observed increases. In this work, I explore bar turbulence in molecular gas using published high resolution measurements of CO(2-1) from the PHANGS-ALMA survey. I compare properties of the molecular gas, such as surface density, velocity dispersion and star formation rate, in the centres of barred and unbarred galaxies. I consider the effect of galaxy environment on these properties from a local perspective (at cloud scales, ~100 pc) for galaxies with and without an AGN. On global scales, I consider these properties in the context of the environment in which a galaxy lives, whether in a cluster or in the field. All three quantities (gas surface density, velocity dispersion, and star formation rate) are found to be enhanced in barred galaxy centres, even without an AGN and regardless of global environment. / Thesis / Master of Science (MSc)
7

Gradientes de Cor e o Cenário de Evolução Secular em Galáxias Espirais de Tipo Tardio / Color Gradients and the Secular Evolutionary Scenario in Late-Type Spiral Galaxies

Gadotti, Dimitri Alexei 19 October 1999 (has links)
Nós realizamos um estudo estatístico do comportamento de perfis de cor em bandas largas (UBV) para 257 galáxias espirais do tipo Sbc, ordinárias e barradas, utilizando dados obtidos através de fotometria fotoelétrica de abertura, disponíveis na literatura (Longo & de Vaucouleurs 1983,1985). Nós determinamos os gradientes de cor (B-V) e (U-B) para as galáxias da amostra total, bem como os índices de cor (B-V) e (U-B) de bojos e discos separadamente, utilizando métodos estatísticos robustos. Utilizamos uma técnica de decomposição bi-dimensional para modelar os perfis de brilho de bojos e discos em imagens dos arquivos do ``Digitised Sky Survey' (DSS), obtendo parâmetros estruturais característicos para 39 galáxias. A aquisição de imagens de 14 galáxias no Laboratório Nacional de Astrofísica permitiu-nos realizar um estudo fotométrico comparativo, e atestar a validade dos resultados obtidos neste estudo. Entre os principais resultados obtidos, destacam-se: (i) - 65% das galáxias possuem gradientes de cor negativos (mais vermelhos no centro), 25% possuem gradientes nulos, e 10% apresentam gradientes positivos; (ii) - galáxias que apresentam gradientes de cor nulos tendem a ser barradas; (iii) - os índices de cor ao longo das galáxias com gradientes nulos são similares aos índices de cor dos discos das galáxias com gradientes negativos; (iv) - confirmamos a correlação entre os índices de cor de bojos e discos, já obtida por outros autores; (v) - a ausência de correlação entre os gradientes de cor e de metalicidade sugere que o excesso de galáxias barradas com gradientes de cor nulos ou positivos reflete uma diferença no comportamento da idade média da população estelar ao longo de galáxias barradas e ordinárias; (vi) - galáxias com gradientes de cor nulos ou positivos têm uma leve tendência a apresentar bojos maiores e com maior concentração central de luz; e (vii) - confirmamos a correlação entre as escalas de comprimento de bojos e discos, já obtida por outros autores. Estes resultados são compatíveis e favoráveis ao cenário de evolução secular, no qual barras produzem fluxos radiais de massa para as regiões centrais de galáxias, não somente homogeneizando as populações estelares ao longo de galáxias, produzindo discos e bojos com índices de cor semelhantes, mas também contribuindo para a formação e/ou construção de bojos. / We have done a statistical study of the behaviour of the broadband color profiles (UBV) for 257 Sbc galaxies, barred and unbarred, collecting data obtained through photoeletric aperture photometry, available in the literature (Longo & de Vaucouleurs 1983,1985). We have determined (B-V) and (U-B) color gradients for the total sample of galaxies, as well as (B-V) and (U-B) color indices of bulges and disks separately, using robust statistical methods. Applying a bi-dimensional decomposition technique to model the brightness profiles of bulges and disks in images from the Digitised Sky Survey (DSS), we obtained characteristic structural parameters for 39 galaxies. The acquisition of images for 14 galaxies in the Laboratório Nacional de Astrofísica (Astrophysics National Laboratory) allowed us to do a comparative photometric study, and verify the validity of the results obtained in this work. Among the main results obtained, we point out: (i) - 65% of the galaxies have negative color gradients (reddish inward), 25% have zero gradients, and 10% show positive gradients; (ii) - galaxies that show zero color gradients tend to be barred; (iii) - the color indices along the galaxies with zero color gradients are similar to the color indices of the disks of the galaxies with negative color gradients; (iv) - we confirm the correlation between the color indices of bulges and disks, already found by other authors; (v) - the absence of correlation between color and metallicity gradients suggests that the excess of barred galaxies with zero or positive color gradients reflects a difference in the behaviour of the mean age of the stellar population along barred and unbarred galaxies; (vi) - galaxies with zero or positive color gradients show a slight tendency of having larger bulges, with a greater central concentration of light; and (vii) - we confirm the correlation between the scale lenghts of bulges and disks, already found by other authors. These results are compatible and favourable to the secular evolutionary scenario, in which stellar bars induce radial mass fluxes to the central regions of galaxies, not only turning homogeneous the stellar populations along the galaxies, producing disks and bulges with similar color indices, but also contributing to the formation and/or building of galactic bulges.
8

Evolution of barred galaxies and associated structures

Kruk, Sandor J. January 2018 (has links)
Bars are common in disc galaxies along with many associated structures such as disc-like bulges, boxy/peanut bulges, rings, etc. They are a sign of maturity of disc galaxies and can play an important role in their evolution. In this thesis, I investigate the specific role bars play in quenching the star formation in, and shaping of their host galaxies. In order to test how bars affect their host galaxies, I study the discs, bars and bulges of what is currently the largest sample of barred galaxies (~3,500), selected with visual morphologies from the Galaxy Zoo project. I perform multi-wavelength and multi-component photometric decomposition, with the novel GALFITM software. With the detailed structural analysis I obtain physical quantities such as the bar- and bulge-to-total luminosity ratios, effective radii, Sérsic indices and colours of the individual components. I find a clear difference in the colours of the components, the discs being bluer than the bars and bulges. An overwhelming fraction of bulge components have Sérsic indices consistent with being disc-like bulges. I compare the barred galaxies with a mass- and environment-matched volume-limited sample of unbarred galaxies, finding that the discs of unbarred galaxies are bluer compared to the discs of barred galaxies, while there is only a small difference in the colours of the bulges. I suggest that this is evidence for secular evolution via bars that leads to the build-up of disc-like bulges and to the quenching of star formation in the galaxy discs. I identify a subsample of unbarred galaxies that are better fitted with an additional component, identified as an inner lens/oval. I find that their structural properties are similar to barred galaxies, and speculate that lenses might be former bars. Using the decompositions, I identify a sample of 271 late-type galaxies with curious bars that are off-centre from the disc. I measure offsets up to 2.5 kpc between the photometric centres of the stellar disc and stellar bar, which are in good agreement with predictions from simulations of dwarf-dwarf tidal interactions. The median mass of these galaxies is 10<sup>9.6</sup> M<sub>⊙</sub>, and they are similar to the Large Magellanic Cloud, which also has an offset bar. Very few high mass galaxies with significant bulges show offsets, thus I suggest that the self-gravity of a significant bulge prevents the disc and bar from getting displaced with respect to each other. I conduct a search for companions to test the hypothesis of tidal interactions, but find that a similar fraction of galaxies with offset bars have companions within 100 kpc as galaxies with centred bars. Since many of these galaxies appear isolated, interactions might not be the only way to produce an offset bar. One suggested alternative is that the dark matter haloes surrounding the galaxies are lopsided, which distorts the potential, and imprints the lopsidedness and offsets onto the galaxy discs. I investigate the asymmetries in the kinematics of a subsample of such galaxies using data from the MaNGA survey, and find that the perturbations in the haloes are ~ 6% for both galaxies with off-centre and centred bars. I also measure the amplitude of non-circular motions in the outer discs due to an oval potential and find only minor departures from circularity, suggesting that the dark matter haloes are consistent with being spherical (axis ratio q ≳ 0.96). Therefore, the lopsidedness of the dark matter haloes cannot be the origin of the offsets. Either small companions are missed due to the incompleteness of the Sloan Digital Sky Survey spectroscopic survey, or interactions with dark matter satellites might explain the offsets. Modeling the kinematics of these galaxies, I find that the Hα gas rotation is centred closer to the centre of the bar than the centre of stellar rotation, suggesting that, in general, the bars are located closer to the dynamical centres of these galaxies than the discs. This implies that the discs are offset in these galaxies, not the bars. If offsets are characteristic of low mass galaxies only, high mass galaxies show vertically extended bars, known as boxy/peanut bulges. I investigate, for the first time, the formation and evolution of these structures associated to bars, from z≈0 to z=1. I compare two samples of moderately inclined galaxies with masses M<sub>*</sub> > 10<sup>10</sup> M<sub>⊙</sub>, imaged by the Sloan Digital Sky Survey and the Hubble Space Telescope. Using a novel technique to classify bar isophotes, and based on the visual inspection of three expert astronomers, I find an evolving fraction of galaxies having boxy/peanut bulges from 30% at z≈0 to ~ 0% at z=1, and a strong correlation with stellar mass. I find 26 galaxies (15 at higher redshifts) in the phase of bar buckling, the mechanism proposed to form boxy/peanut bulges. The peak redshift of buckling is z≈0.75, where the bar buckling fraction is 4 times higher than in the local Universe. My observations suggest that many, if not all, of the boxy/peanut bulges are formed via buckling, ~ 2 Gyr after bar formation, with the buckling phase lasting for approximately 0.8 Gyr. I discuss my findings in the context of the evolution of barred galaxies and propose ideas for future work - applying similar decomposition techniques to higher redshift, and better resolution datasets, using integral field spectroscopic data to study the stellar populations of barred galaxies in greater detail, as well as a novel project to identify large nuclear discs in galaxies.
9

Gradientes de Cor e o Cenário de Evolução Secular em Galáxias Espirais de Tipo Tardio / Color Gradients and the Secular Evolutionary Scenario in Late-Type Spiral Galaxies

Dimitri Alexei Gadotti 19 October 1999 (has links)
Nós realizamos um estudo estatístico do comportamento de perfis de cor em bandas largas (UBV) para 257 galáxias espirais do tipo Sbc, ordinárias e barradas, utilizando dados obtidos através de fotometria fotoelétrica de abertura, disponíveis na literatura (Longo & de Vaucouleurs 1983,1985). Nós determinamos os gradientes de cor (B-V) e (U-B) para as galáxias da amostra total, bem como os índices de cor (B-V) e (U-B) de bojos e discos separadamente, utilizando métodos estatísticos robustos. Utilizamos uma técnica de decomposição bi-dimensional para modelar os perfis de brilho de bojos e discos em imagens dos arquivos do ``Digitised Sky Survey' (DSS), obtendo parâmetros estruturais característicos para 39 galáxias. A aquisição de imagens de 14 galáxias no Laboratório Nacional de Astrofísica permitiu-nos realizar um estudo fotométrico comparativo, e atestar a validade dos resultados obtidos neste estudo. Entre os principais resultados obtidos, destacam-se: (i) - 65% das galáxias possuem gradientes de cor negativos (mais vermelhos no centro), 25% possuem gradientes nulos, e 10% apresentam gradientes positivos; (ii) - galáxias que apresentam gradientes de cor nulos tendem a ser barradas; (iii) - os índices de cor ao longo das galáxias com gradientes nulos são similares aos índices de cor dos discos das galáxias com gradientes negativos; (iv) - confirmamos a correlação entre os índices de cor de bojos e discos, já obtida por outros autores; (v) - a ausência de correlação entre os gradientes de cor e de metalicidade sugere que o excesso de galáxias barradas com gradientes de cor nulos ou positivos reflete uma diferença no comportamento da idade média da população estelar ao longo de galáxias barradas e ordinárias; (vi) - galáxias com gradientes de cor nulos ou positivos têm uma leve tendência a apresentar bojos maiores e com maior concentração central de luz; e (vii) - confirmamos a correlação entre as escalas de comprimento de bojos e discos, já obtida por outros autores. Estes resultados são compatíveis e favoráveis ao cenário de evolução secular, no qual barras produzem fluxos radiais de massa para as regiões centrais de galáxias, não somente homogeneizando as populações estelares ao longo de galáxias, produzindo discos e bojos com índices de cor semelhantes, mas também contribuindo para a formação e/ou construção de bojos. / We have done a statistical study of the behaviour of the broadband color profiles (UBV) for 257 Sbc galaxies, barred and unbarred, collecting data obtained through photoeletric aperture photometry, available in the literature (Longo & de Vaucouleurs 1983,1985). We have determined (B-V) and (U-B) color gradients for the total sample of galaxies, as well as (B-V) and (U-B) color indices of bulges and disks separately, using robust statistical methods. Applying a bi-dimensional decomposition technique to model the brightness profiles of bulges and disks in images from the Digitised Sky Survey (DSS), we obtained characteristic structural parameters for 39 galaxies. The acquisition of images for 14 galaxies in the Laboratório Nacional de Astrofísica (Astrophysics National Laboratory) allowed us to do a comparative photometric study, and verify the validity of the results obtained in this work. Among the main results obtained, we point out: (i) - 65% of the galaxies have negative color gradients (reddish inward), 25% have zero gradients, and 10% show positive gradients; (ii) - galaxies that show zero color gradients tend to be barred; (iii) - the color indices along the galaxies with zero color gradients are similar to the color indices of the disks of the galaxies with negative color gradients; (iv) - we confirm the correlation between the color indices of bulges and disks, already found by other authors; (v) - the absence of correlation between color and metallicity gradients suggests that the excess of barred galaxies with zero or positive color gradients reflects a difference in the behaviour of the mean age of the stellar population along barred and unbarred galaxies; (vi) - galaxies with zero or positive color gradients show a slight tendency of having larger bulges, with a greater central concentration of light; and (vii) - we confirm the correlation between the scale lenghts of bulges and disks, already found by other authors. These results are compatible and favourable to the secular evolutionary scenario, in which stellar bars induce radial mass fluxes to the central regions of galaxies, not only turning homogeneous the stellar populations along the galaxies, producing disks and bulges with similar color indices, but also contributing to the formation and/or building of galactic bulges.
10

Ανίχνευση και μελέτη φαινομένων μεσοαστρική ύλης / Detection and study of the interstellar media

Άκρας, Σταύρος 07 July 2010 (has links)
Ο σκοπός της παρούσας διδακτορικής διατριβής ήταν η μελέτη δύο σημαντικών φαινομένων μεσοαστρική ύλης όπως είναι τα Πλανητικά Νεφελώμάτα (ΠΝ) και η ράβδος των σπειροειδών γαλαξιών. Πιο συγκεκριμένα, μελετήθηκαν 44 ΠΝ στην περιοχή του Γαλαξιακού σφαιροειδούς (Boumis et al. 2003; 2006) και προσδιορίστηκαν οι φυσικοί παράμετροι τους, όπως είναι η ενεργός θερμοκρασία και η λαμπρότητα του κεντρικού αστεριού, η πυκνότητα και η θερμοκρασία των ηλεκτρονίων, η αφθονία των Ηe, N, O, S κτλ., χρησιμοποιώντας το μοντέλο φωτο-ιονισμού Cloudy (Akras et al. 2010a). Επίσης, μελετήθηκε η ράβδος των γαλαξιών, χρησιμοποιώντας το μοντέλο διάδοσης ακτινοβολίας CRETE, με σκοπό να ερευνηθεί πώς η ύπαρξη της ράβδου επηρεάζει την μορφολογία των γαλαξιών και την σκόνη τους. Ταυτόχρονα, προσδιορίστηκαν οι παράμετροι της ράβδου, όπως το μήκος, το ύψος, η γωνία κλίσης και η γωνία θέσης της για 4 ραβδωτούς σπειροειδείς γαλαξίες (NGC 4013, UGC 2048, IC 2531 και το Γαλαξία μας). Στο πρώτο μέρος της μελέτης, διαχωρίστηκαν τα ΠΝ ανάλογα με την μορφολογία τους σε σφαιρικά, ελλειπτικά και διπολικά και βρέθηκε ότι η αφθονία του Ηe και του Ν είναι μεγαλύτερη στην τελευταία κατηγορία σε σχέση με τις υπόλοιπες. Επιπλέον, η χρήση του στατιστικού εργαλείου PCA, έδειξε ότι τα κυκλικά και τα ελλειπτικά ΠΝ διαχωρίζονται από τα διπολικά, βάσει της τιμής του λόγου log(Ν/Ο), ο οποίος παίρνει αρνητικές και θετικές τιμές για τα διπολικά, ενώ μόνο αρνητικές τιμές στις υπόλοιπες κατηγορίες. Η κρίσιμη τιμή βρέθηκε ίση με -0.18 dex και αντιστοιχεί σε μάζα προγενέστερου αστεριού ίση με 2.6 (Akras & Boumis 2007). Στο δεύτερο μέρος, περιγράφηκε η ράβδος των γαλαξιών χρησιμοποιώντας την συναρτησιακή ελλειψοειδής υπερβολής. Μεταβάλλοντας την γωνία κλίσης του γαλαξία, η δομή σχήματος «Χ», η οποία παρατηρείται στις ράβδους, μπορεί να παρατηρηθεί μόνο για γωνίες μεγαλύτερες από 60ο. Επιπρόσθετα, στην περίπτωση του Γαλαξία μας, βρέθηκε ότι η γωνία θέσης της ράβδου είναι γύρω στις 25ο και το μήκος της 3.75 kpc (Akras et al. 2010b). Λαμβάνοντας υπόψη ότι η μέση τιμή του μήκους της ράβδου στους γαλαξίες είναι μεταξύ 3.0 και 4.0 kpc, προκύπτει ότι για τον NGC 4013 η γωνία θέσης του είναι μεταξύ 5 και 10 μοίρες, για τον UGC 2048 μεταξύ 40 και 50 μοίρες, για τον IC 2531 35 και 45 μοίρες ενώ τέλος για το Γαλαξία μας υπολογίζεται μεταξύ 20 και 30 μοίρες (Akras et al. 2010b). / The aim of this PhD thesis was the study of two very important interstellar medium phenomena like the Planetary Nebulae (PNe) and the stellar bar in spiral galaxies. In particular, we studied 44 PNe in the Galactic bulge region and we determined their physical parameters, like the effective temperature and luminosity of the central star, the electron temperature and density, the abundance of He, N, O, S etc., using the photo-ionization model “Cloudy” (Akras et al. 2010a). It was also pursued to study the stellar bar component using the 3D radiative transfer model CRETE, in order to investigate the effects of a stellar bar component to the morphology of the galaxy and its dust content. In addition, the parameters of the bar component such as the length, the height, the inclination angle and the position angle were determined for four spiral galaxies (NGC 4013, UGC 2048, IC 2531 and our Galaxy). In the first part of the thesis, the PNe were separated according to their morphology (spherical, elliptical and bipolar shape) and we found that the He and N abundances are greater in bipolar PNe. Moreover, by using the statistical tool PCA, it was found that the circular and the elliptical PNe are different from the bipolar, since the log(O/N) takes negative values in the first two and positive or negative values in the bipolar PNe. The critical value was found at -0.18 dex, which corresponds to a stellar mass of 2.6 (Akras & Boumis 2007). In the second part, we managed to accurately describe the morphology of the bar component by using the function of hyperbolic ellipse. For different inclination angles of the observed galaxies, it was found that the “X-shape" feature can be seen only in the case where the inclination angle is greater than 60 degrees. In the case of Milky Way, it was found that the position angle of the bar is approximately 25 degrees and the length equals to 3.75 kpc (Akras et al. 2010b). Considering that the mean length of the bar component is equal to 3.0-4.0 kpc, it was determined that the position angle of a) NGC 4013 takes values between 5 and 10 degrees, b) UGC 2048 takes values between 40 and 50 degrees, c) IC 2531 takes values between 35 and 45 degrees, and our Galaxy takes values between 20 and 30 degrees (Akras et al. 2010b).

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