maruska zerjal- spiral structure in galaxies

8/3/2019 Maruska Zerjal- Spiral Structure in Galaxies

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Department of Physics

Seminar - 4th year

Spiral Structure in Galaxies

Author: Maruska Zerjal

Adviser: prof. dr. Tomaz Zwitter

Ljubljana, April 2010

Abstract

The most numerous among bright galaxies and the largest in the universe are

spirals and represent some of the most beautiful and spectacular phenomena dueto the presence of their remarkable spiral arms, which denote young stars and star-forming regions. We investigate the phenomenon of well-defined spiral arms throughthe basics of galactic dynamics and stellar orbits. Instead of the material arms theLin-Shu density wave theory is introduced as a well accepted theory, which dealswith a small-amplitude orbital perturbations and closed orbits in a noninertial frameof reference. We explain the corotation and ultraharmonic resonances of epicyclesand mapping of our spiral Milky Way Galaxy.


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Table of contents

1 An introduction 2

2 Galactic Dynamics 4

3 The Lin-Shu Density Wave Theory 4

3.1 Small-Amplitude Orbital Perturbations . . . . . . . . . . . . . . . . . . . . 63.2 Closed Orbits in Noninertial Frames . . . . . . . . . . . . . . . . . . . . . . 83.3 The Stability of the Spiral Structure . . . . . . . . . . . . . . . . . . . . . 103.4 Corotation and Ultraharmonic Resonances . . . . . . . . . . . . . . . . . . 10

4 Spiral Arms 11

5 Mapping the Milky Way Galaxy 12

5.1 A Large-Scale Structure of the Milky Way Galaxy . . . . . . . . . . . . . . 14

6 Conclusion 14

1 An introduction

Galaxies exhibit a rich variety of shapes from essentially spherical to flattened disk sy-stems. In 1926 Edwin Hubble proposed a morphological classification scheme, based onthe appearance of a galaxy at optical wavelengths and known as the Hubble sequenceor Hubbles tuning fork, which divides galaxies into ellipticals and spirals. A transiti-onal class between ellipticals and spirals is known as lenticulars. Unclassified galaxiesare known as irregulars. While ellipticals are the most numerous among all galaxies, themost numerous among bright galaxies [1] and the largest in the universe [2] are spirals andrepresent some of the most beautiful and spectacular phenomena due to the presence oftheir remarkable spiral arms. In consideration of presence or lack of bars and how tightlywound they are, spiral galaxies are further divided into normal spirals and barred spirals(Fig. 1). The latter represent approximately 60 % of all spiral galaxies [2].

These flattened disk galaxies comprise several distinct components (Fig. 2). Multi-component disk planes of spiral galaxies, involving thin, thick and gas disk, extends across5 100 kpc1 in diameter. Vertical scale heights of thin disk measure only a few percentsof its radii, thick disks are somewhat thicker. Thin disks are made of relatively youngstars, gas and dust. The latter is responsible for the dark lanes across the disk due to theabsorption in visible wavelenghts. In the center of the disk lies a central bulge, containing

mostly old stars. Flattened disk is surrounded by a spherical stellar halo of radii morethan 100 kpc and made up from old stars and globular clusters. It is enveloped by a darkmatter halo, which extends over 230 kpc in our Galaxy.

Masses of spiral galaxies range from 109 1012 M2 [2] and are composed of about

109 1012 stars.Most apparent feature of the disk are well-defined spiral arms, which denote young

stars and star-forming regions. They could be one of the main clues of their evolution,via the angular momentum transfer. In between the spiral arms intermediate-age starsare to be found.

1Parsec, 1 pc = 3.26 light years

2M = 2 1030 kg, one Solar mass

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Figure 1 Hubbles tuning fork divides galaxies into ellipticals and spirals [3]. A transitionalclass between ellipticals and spirals is known as lenticulars. In consideration of presence or lack ofbars and how tightly wound they are, spiral galaxies are further divided into normal spirals and barredspirals.

Figure 2 Side-on view of spiral galaxy [4] (figure is not to scale). Multi-component disk planeextends across 5100 kpc in diameter. Vertical scale heights of the disk measure only a few percentsof its radii. The thin component of the disk is made of relatively young stars, gas and dust. Thelatter is responsible for the dark lanes across the disk due to the absorption in visible wavelenghts. Inthe center of the disk lies a central bulge, containing mostly old stars. Flattened disk is surrounded

by a spherical halo of radii more than 100 kpc and made up from old stars and globular clusters.Radius of dark-matter halo extends over 230 kpc in our Galaxy.

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Figure 3 Rotation curve for a flattened-disk spiral galaxy [1]. A dashed line is a prediction fora rotational velocity that obeys the Keplerian dynamics in a case that the visible mass of the galaxyis the total galactic mass and extends to the point A. The measured rotational curve is rather flat(solid line) and reveals unseen dark matter present well beyond the visible radius of the galaxy.

2 Galactic Dynamics

A circular Keplerian orbit at a distance r from the center of the galaxy with radius R will

have rotation velocity

v =

GM(r)

r. (1)

M(r) is the galactic mass from the center to the radius r. On Fig. 3, the visible mass of thegalaxy extends to the point A. If visible mass M(R) was total galactic mass, then v shoulddecrease as

1/r (Eq. 1) for r > R. Measurements show that for radius r > R, rotation

velocity v is approximately constant as far as 2 3R. Rotation curve reveals unseen darkmatter present well beyond the visible radius of the galaxy and which causes the flatcurve. Rotational velocities vary with the morphology of galaxy. More tightly spiral armsare wound and more prominent the bulge is, higher is galaxies rotational velocity. Mean

velocity ranges from 300 km s1 for type Sa galaxies to 175 km s1 for Sc galaxies [2].Maximum rotational velocities for irregular galaxies are significantly smaller, typicallyfrom 5070 km s1. It seems that a minimum rotation speed of roughly 50100 km s1

may be required for the development of a well-organized spiral pattern [2].

3 The Lin-Shu Density Wave Theory

Suggestion that spiral arms are material (composed of a fixed set of stars, gas and dust)and rotate with the galaxy itself, ends up with the so-called winding problem (Fig. 4).Since the disk undergoes a differential rotation, stars on more distant orbits have smallervelocities and start to lag behind those on nearer orbits. After only a few periods (forexample, period of the Sun is about 230106 yr [2]), arms would become wound too tightly

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Figure 4 The winding problem for material arms, composed of a fixed set of stars, gas anddust [2]. Since the disk undergoes a differential rotation, stars on more distant orbits start to lagbehind those on nearer orbits and spiral arms become wound too tightly.

to be observed.

In the 1960s, C. C. Lin and Frank Shu proposed that spiral arms are rather generatedby long-lived quasistatic density waves, regions in the disk with enhanced density by10 20 % that travel through the galaxy triggering star formation. Stars, gas and dusttravel on their orbits through the waves. The term quasistatic refers to the fact thatwhen viewing in a noninertial frame of reference, rotating with a globular pattern speedgp, spiral waves exhibit a stationary pattern. However, most of the stars do not rotatewith the global pattern angular speed gp.

Stars near the center have higher velocities and shorter angular frequencies than gp,but stars on the outher edge of the galaxy have angular frequencies that are larger thangp due to their smaller velocities. Inner stars would overtake the spiral arm while theouther stars would be overtaken by a spiral pattern. Critical radius Rc, where the starsand density waves would have the same angular velocity, is called the corotation radius.In noninertial frame of reference, the stars with r < Rc will appear to pass through thearms in one direction, the stars with r > Rc will appear to move in the opposite sense(see Fig. 5).

The hypothesis of quasistationary spiral structure refers only to the large-scale regularstructure that is frequently observed in galaxies [5]. Density waves leave the equatorialsymmetry unchanged, but on the other hand they are associated with density enhance-ments and rarefactions that usual break the axisymmetry of the basic state [5].

When the gas and dust clouds on r < Rc overtake the spiral arms, the enhancementof the density occurs, resulting in cloud collapse, when the Jeans criterion 3 is satisfied.

Due to a finite time of star formation (105 yr for a 15 M star [2]), the starburst willappear downstream the arm in the edge of the wave. Since the massive O and B stars4

have relatively short lifetime in comparison with their existence within the density wave,they die out before leaving the arm. Only longeval red and less luminous stars remainand are passed by the wave. They are found in between the spiral arms. However, thereexist local maxima of old red dwarf stars due to the minima of a gravitational potentialdwell of the spiral arm with enhanced density.

3The Jeans criterion tells the minimum cloud mass Mc, required for its collapse: Mc >5kBTGmH

3/2 3

40

1/2, where kB is Boltzmann constant, T gas temperature, G gravitational constant,

a mean particle mass, mH mass of a hydrogenium atom and 0

is a cloud density [2].4O and B stars are young, hot, very massive and luminous. Their lifetime is relatively short, 107 yr.

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Figure 5 (a) Trailing spiral arms in an inertial frame of reference (S) [2]. A velocity of aquasistatic density wave is equal to a global angular pattern speed gp. Star A rotates with a speedA > gp, star B with B < gp and star C corotates with a density wave (C = gp). (b) Innoninertial frame of reference (S), star A seems to take over the density wave, star B lags behindthe wave and star C corotates with the wave itself.

3.1 Small-Amplitude Orbital Perturbations

In order to reveal the resulting spiral shape from the orbits of the stars, let us determinethe axial symmetric gravitational potential and orbital motion of the stars. We takeinto account that for the statistical majority of galaxies, the underlying potential thatis associated with the grand-design spiral structure is stationary in a suitable rotatingframe. We consider only a stellar component of the galaxy (N = 1011 stars) - the numberis large enough for the smoothness of the local potential instead of its natural graininess.Another presumption says that the stellar component can be considered collisionless. Weneglect also the potential of the spiral waves.

In the cylindrical coordinates (r,,z) we introduce an effective gravitational potentialas

eff

(r, z) = (r, z) +J2z

2r2. (2)

Jz = r2 is a constant for the stars motion and denotes the z component of the orbital

angular momentum per unit mass of the star. (r, z) is defined as (r, z) = U/m, where

md2r

dt2= U(r, z), (3)

m is a mass of the star and U(r, z) is the gravitational potential of the disk. The minimumof the eff occurs, when z = 0 and the orbit of the star is perfectly circular. Theperturbation analysis of the first order gives us the following effective potential:

eff(r, z) 0eff +

12

22 + 12

2z2. (4)

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0eff denotes the minimum of unperturbed potential at 0eff(Rm, 0), = Rm r, Rm is

the radius at 0eff. We defined the constants

2 2eff

r2|m,

2 2eff

z2|m. (5)

and get the equations of harmonic motion:

2, z 2z (6)

with the solution

(t) = AR sin t, z(t) = Az sin(t + ). (7)

We proclaim to be the epicycle frequency and for the oscillation frequency. is ageneral phase shift between (t) and z(t). AR and Az are amplitudes of the oscillation.The star oscilates around the equilibrium position (Rm, 0), which is stable point androtates on a circular orbit with the angular frequency .

Now we search for the difference (t) between the azimuthal position of the star andthe equilibrium point. We know that

=v

r(t)=

Jzr(t)2

(8)

and r(t) = Rm + (t) = Rm(1 + (t)/Rm). For (t) Rm, we use the binomial expansionand it follows that

Jz

R2m

1 2

(t)

Rm

. (9)

After integration we get

(t) = 0 + Jz

R2mt + 2Jz

R3mAR cos t = 0 + t + 2

RmAR cos t, (10)

where Jz/R2m was introduced. The last term in Eq. 10 acts as the oscillation of

the star about the equilibrium point in the direction. By defining the difference inazimuthal position between the star and the equilibrium point

(t) ((t) (0 + t)) Rm, (11)

by the perturbation theory of the first order it follows that

(t) =2

AR cos t. (12)

Equations (7) and (12) represent the oscillation of the star around its equilibrium point,which moves in circle around the center of the galaxy with angular speed [2].

In an inertial frame of reference, stars orbit is not closed, but reminds of the rosettepattern (Fig. 6), which is to be explained with help of epicycles. The center of the epicyclecorresponds to the equilibrium position. An axial ratio of an oval shaped epicycle isgiven by the ratios of the amplitudes of the oscillations in (t) and (t),

=maxmax

=2

. (13)

The center of the (, ) epicycle rotates around the galactic center with the angularvelocity of the equilibrium point, (Fig. 6).

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Figure 6 A nonclosed rosette pattern is formed (solid line) by a stars orbital motion whenviewing in an inertial frame of reference [2]. The motion reveals the epicyclic behaviour when onlythe first-order approximation is contributed. The center of the epicycle (dashed line) rotates aroundthe center of the galaxy with .

3.2 Closed Orbits in Noninertial Frames

The number of oscillations per orbit is equal to N = 1/2 or

N =

. (14)

If N is an integer, the orbit is closed. Most stellar orbits in an inertial frame are notclosed and the rosette pattern results, while in a noninertial frame of reference, rotatingwith the local angular pattern speed lp = , relative to the inertial frame, the orbit isclosed and appears at the Rm (Fig. 6).

In a noninertial frame of reference, a closed orbit would complete n orbits and mepicycle oscillations (m and n are integers). Its valid to choose m( lp) = n or

lp(r) = (r) n

m(r). (15)

At radius r, only a small number of values m and n would give a substantial enhancementsin mass density [2], which means that only selected modes ((n, m) = (1, 2)) are mostcommon to be observed.

When looking in a noninertial frame, that is rotating with the global angular patternvelocity gp and when lp = lp(r), we can set gp = lp. From the Earth, such closedorbital trajectories (for (n, m) = (1, 2), for example) could be nested with their majoraxis aligned (Fig. 7a). In case we orient such oval-shaped orbit in a way its major axis isrotated slightly relative to the one immediately interior to it, we get a trailing two-armedgrand-design spiral wave pattern (Fig. 7b). Rotation in the opposite sense would give

leading arms (Fig. 7c). For example, M51 is a grand-design trailing-armed spiral with(n, m) = (1, 2) (Fig. 8) and M101 is a four-armed galaxy with trailing arms and with(n, m) = (1, 4).

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Figure 7 Nested orbits (a) [6] in a galaxy with gp = /2 (i.e. ((n,m) = (1, 3))) ina frame of reference rotating with gp. When rotating the axes of ellipses, there comes to trailingarms (b) or leading arms (c), if rotating in the opposite side.

Figure 8 The grand-design Whirlpool Galaxy M51 [7] in interaction with an irregular galaxyNGC 5195. This is a trailing spiral pattern with (n,m) = (1, 2) modes and corresponds to regionsof active star formation [2]. Dust and gas clouds are spread on the inner edges of the arms as wellas in the surroundings of the galaxies. Due to the relatively large size of the galaxies, compared tothe average distance between them, the encounters with other galaxies are very likely to occur atleast a few times over their immense lifetimes [8] and play an important role in the evolution of the

galaxies.

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Figure 9 The Bahcall-Soneira model of our Galaxy [2] for lp = nm for various n,m.

Curve for /2 is nearly flat over a large range of Galactic disk and consequently systems withm = 2 are most frequently found.

3.3 The Stability of the Spiral Structure

The stability of the (n, m) = (1, 2) structure depends mostly on whether

lp(r) = (r)

(r)

2 (16)

is actually independent of r or not, that is, whether there is an appropriate gp or not.The most frequent systems are two-armed with m = 2, probably due to the fact that

their rotation velocity is flat over a wide range of radii, just like the lp(r) = (r) (r)2

(Fig. 9).Observations show that in the statistical majority of galaxies, the dynamics of the disk

is dominated by one mode ((n, m) = (1, 2) is most frequent) or by a very small numberof modes [5]. We also know that the presence of gas is essential for spiral structure, bymeans of self-regulation, due to the fact that collapsing gas clouds result in a new-bornstars which illuminates the spiral arms.

3.4 Corotation and Ultraharmonic Resonances

Now we neglect our previous assumption of insignificance of the potential of the arms.It follows that when the star encounters a density wave with its maximum value of thedifference between the equilibrium and azimuthal position max, a resonance developsand epicycle oscillation amplitudes AR and Az are considerably increased. If a star is atits maximum max each time it enters the density wave, the perturbation of the densityenhancement and gravitational potential will always be in the same sense. Perturbationswill therefore cumulate.

Because is actually not exactly the same over the entire disk, i.e. = (r), thereexist only certain radii where lp = gp. From the Earth, such orbits are closed and aresonant amplitude amplifications are possible. When gp = lp = /2 (n/m = 1/2),

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Figure 10 Positions of the resonance radii. Their existence depends on the shape of thegalactic rotation curve and on the global angular pattern velocity gp [2]. Normally there may existone or two inner Lindblad radii (ILR), but none can be found if gp is sufficiently large. In case thatgp = , a corotation resonance (CR) exists and stars velocity is equal to the velocity of the spiral

wave. If gp = + /2, it may come to an outer Lindblad resonance (OLR).

an inner Lindblad5 resonance occurs and is possible at several radii (zero, one or twoinner Lindblad radii - Fig. 10b) in dependence of the rotation curve of the galaxy. Incase that stellar velocity is equal to the velocity of the spiral wave ( gp = ), a corotationresonance exists. If gp = + /2, it may come to an outer Lindblad resonance (Fig.10). It is possible for ultraharmonic resonance to develop when gp = /4.

In resonance, it is more likely for gas clouds to collide and for the dissipation ofthe energy, which results in damping in spiral waves, unless there is another mechanismsustaining the waves.

4 Spiral Arms

Among others, the most prominent variation of spiral structure is in number and shapeof its arms. The most majestic spiral galaxies are so-called grand-design spirals, showingonly two very symmetric arms. About 10 % of spirals are considered grand-design spirals,60 % are multiple-armed galaxies and the remaining 30 % present flocculent galaxies,which do not possess well-defined spiral arms that are traceable over a significant angulardistance [2].

At visible wavelengths, these galaxies are dominated by their spiral pattern due to

the presence of very luminous O and B main-sequence stars and HII regions are locatedwithin the arms. In comparison with the rotational period of the galaxy (23 107 yr),lifetime of those stars is very short (107 yr), ending with conclusion that spiral patterncorresponds to regions of active star formation [2]. Dust and gas clouds are spread on theinner edges of the arms.

An intuitive guess suggests that spiral arms are trailing (Fig. 5), which means thatthe tips of arms point to the opposite side of rotation of the galaxy. In one case, (galaxyNGC 4622), the Dopper effect seen in the ground-based spectrum [9] showed that at leastone of the spiral arms must be leading. An important role in galactic evolution also playsthe galactic bar.

5Bertil Lindblad, Swedish astronomer, 1895-1965

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5 Mapping the Milky Way Galaxy

Surprisingly, among the relatively well-grounded findings about the other galaxies, thelarge-scale structure of our Milky Way Galaxy still remains somehow misterious dueto the fact that we are positioned inside the galaxy itself. Uncertainty in most of thelight at optical wavelengths arises from the obscurity of the extensive dust clouds alongthe Galactic disk [10]. However, there have been successful surveys carried out, oneof them are recent GLIMPSE surveys, that have archived over 100 million stars whichplay an important role in tracing a large-scale Galactic structure [11]. About 90 % ofcatalogued stars are mostly red-clump giants, which have relatively small range in intrinsicluminosities and therefore act as a standard candles, used in a distance determination.Mapping of the Milky Way basically includes measuring stellar distances, their radialvelocities, revealed by the Doppler effect in their spectra and investigation of the density,which is performed by combining the star count per surface area and their distances.

Figure 11 Indication of the Milky Waysbar [11] - there is a hump at 12 mag in thenorth that is absent in the south and is a signa-ture of the northern arm of the central galacticbar.

Figure 12 Number of stellar sources perdeg2 in dependence of a Galactic longitude [11].The large excess at the Galactic center is due tothe central bulge. A Scutum-Centaurus arm canbe seen, while the Sagittarius arm is not visible.

Figure 11 shows the power-law exponent of stars per magnitude per square degreeversus magnitude at 4.5 m and as a function of a Galactic longitude l6 (plotted for threelines of sight). We note the appearance of a bump at 12 mag and l = 15.5 and itsabsence at l = 344.5. The hump is interpreted as a strong evidence for a northern arm

of a central bar as a major feature of our Galaxy, with radius 4.4 0.5 kpc and rotatedfor about 44 10 to the Sun-Galactic bulge line [11].

When looking along the Galactic midplane toward the Scutum-Centaurus arm tan-gency (see Fig. 13) at 306 < l < 313, there is an enhancement of stars due to theincreased line if sight path length through the arm and the increased number of stars inthe arm [11]. However, at the opposite side at 54 < l


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Figure 13 Observations in radio, infrared, and visible wavelengths show that our Milky Way

Galaxy is a grand-design two-armed barred spiral with the Scutum-Centaurus and Perseus arms andseveral secondary arms (Sagittarius, Norma, the outer arm, and the 3 kpc expanding arm) [11].

5.1 A Large-Scale Structure of the Milky Way Galaxy

To begin with, let us thread on some of the most fundamental physical parameters of ourGalaxy. The Milky Way Galaxy is composed of 1011 stars. Stellar mass of the thindisk in our Galaxy is about 1010 1011 M [2]. Multi-component disk plane of our spiralMilky Way Galaxy, involving thin, thick and gas disk, extends across 50 kpc in diameter.Our distance to the Galactic center is believed to be 8 8.5 kpc [11], but there are somesuggestions for the value of 7.62 0.32 kpc [11], which has yet to be confirmed by anindependent analysis.

Vertical scale height of thin disk is 350 pc (1.4 % of its radii), thick disk is moreextensive (1 kpc or 4 % of disks radii). Flattened disk is surrounded by a spherical haloof radii more than 100 kpc and made up from old stars and globular clusters. Radius ofdark-matter halo extends over 230 kpc.

Among other various possibilities, spiral structure is best defined by detection of the HII regions and ionizing radiation, which indicates young, hot and luminous OB-type starsthat are situated in high mass star forming regions (spiral arms) in the Galactic plane.The red light spiral characteristics of older, low-mass stars are less pronounced. Newresults using VLBA (Very Long Baseline Array) show that estimates of the fundamental

parameters of the Milky Way, R0 and 0 indicate a rotation speed of 0 = 254 km s1,that is 15 % faster than previous results [10].

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Our Galaxy probably belongs to type SBbc galaxies [2]. Among its morphologicalfeatures are the Central Galactic bar and the Long bar. The first one is much morevertically extended than the thin stellar disk [11], the second one is thinner and non-axisymmetric. There are Near and Far 3-kpc Arms, associated with gas flow roughlyparallel to the bar. The Milky Way Galaxy has four principal spiral arms, which arenamed after the constellation in which they are observed: Norma, Sagittarius, Perseusand Scutum-Centaurus spiral arms.

6 Conclusion

We showed that the theory of a material arms, made of stars, gas and dust, ends upwith the winding problem and can not explain the occurence of the examined pattern.The Lin-Shu density wave theory was introduced and helps to explain the structure of awell-defined spiral patterns in a flat-disk galaxies very well. We made a small-amplitudeorbital perturbations. In the first order, orbital motions of the stars are described bythe epicycles. There exist several radii, where the resonance occurs in dependence of the

mode and the spiral waves could be damped unless there is another mechanism sustainingthe waves. The theory explains the shape of about 70% of all galaxies.

Although the existence of difficulties with revealing the spiral pattern of our Galaxy,its shape is relatively well-known. Its is classified as the SBbc galaxy - spiral, barred andwith a relatively weak wounded spiral arms. Hovewer, the misteries still remain in ourgalactic shape and dynamics as well as in the other spiral galaxies.

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versity Press

[2] B. W. Caroll and D. A. Ostlie, 2007, An Introduction to Modern Astrophysics,Second edition, Addison Wesley

[3] Galaxy classification - Ellipticals, Lenticulars, Spirals, Irregulars, Pro-blems with visual classification, Morphologydensity relation, Fig. 1., b/a,http://www.jrank.org/space/pages/2353/galaxy-classification.html, [16.3.2010]

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of the Day (2002 January 25), [31.3.2010]

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[10] M. J. Reid et al., 2009, Structure and Dynamics of the Milky Way: an Astro2010Science White Paper, arXiv:0902.3928v2 [astro-ph.GA]

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