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The Influence of the Decay of OB Associations on the Evolution of Dwarf Galaxies
E. P. Kurbatov

arXiv:1012.1216v1 [astro-ph.GA] 6 Dec 2010

Institute of astronomy, Russian Academy of Sciences 48 Pyatnitskaya st., Moscow, Russian Federation, 119017 kurbatov@inasan.ru Abstract It is commonly b elieved that most of the stars b orn in asso ciations decaying with characteristic velo cities of stars 10 km/s. For dwarf galaxies the decay can lead to ejection of stars from the galaxy. The effect is studied for spheroidal and disk dwarf galaxies, and is shown to have substantional observational consequences for disk galaxies with escap e velo cities up to 20 km/s, or dynamical masses up to 108 M . The ejection of stars can (i) reduce the abundances of the pro ducts of Typ e Ia sup ernovae and, to a lesser degree, Typ e I I sup ernovae, in disk stars, (ii) chemically enrich the galactic halo and intergalactic medium, (iii) lead to the loss of 50% of the stellar mass in galaxies with masses 107 M and the loss of all stars in systems with masses 105 M , (iv) increase the mass-to-luminosity ratio of the galaxies. Subject headings: ISM: abundances, galaxies: abundances, galaxies: dwarf, galaxies: evolution, intergalactic medium

1.

INTRODUCTION

Mass exchange b etween a galaxy and the intergalactic medium(IGM) can influence b oth the chemical comp osition of the galactic gas and the morphology of the galaxy. Several mechanisms for gas loss by a galaxy can b e distinguished (Shustov et al. 1997): galactic wind induced by numerous sup ernova explosions, ram pressure exerted by the IGM, the tidal influence of other galaxies of the group, evap oration of gas due to interactions with the hot IGM, and blowing of the gas out of the galaxy by stellar radiation. Possible ways for the loss of stellar mass include tidal interactions, the ejection of stars due to the statistical mechanism, and the decay of stellar asso ciations. Let us consider some of these mechanisms in more detail. There have b een many studies of the effects of numerous sup ernova explosions, such as galactic fountains, sup er-bubbles, and winds, on disk galaxies (see Shustov et al. (1997); Co op er et al. (2008) and references therein). The efficiency of these pro cesses for gas ejection dep ends strongly on


­2­ the distribution of the gas. Thus, mo dels of galaxies with a stratified interstellar medium (ISM) display a much higher mass-loss efficiency than do mo dels with a continuous distribution of their ISM (Co op er et al. 2008). Computations of mo dels with a continuous gas distribution, in turn, provide different results dep ending on the distribution law (Mac Low et al. 1989). It was shown in the theoretical study of De Young and Gallagher (1990) that the fraction of exp elled gas in a 1.4 â 109 M galaxy is 0.6, but, as the authors note, the presence of dark matter was not taken into account. According to Igumenshchev et al. (1990), galaxies with masses exceeding 1012 M do not have winds, and, hence, do not lose gas via this mechanism. Based on these computations, Shustov et al. (1997) used a simple approximation for the relation b etween the mass of the galaxy and the fraction of exp elled matter in their mo dels: f
esc

= 2.4 - 0.2 lg

MG . M

(1)

In this approximation, the efficiency of gas ejection b ecomes unity for galaxies with masses of 107 M , irresp ective of their morphology. According to this mo del, galaxies with such masses should not contain gas. At the same time, gas is almost completely absent only in spheroidal and elliptical dwarf galaxies, whereas it can constitute a substantial fraction of the masses of disk and irregular galaxies (Begum and Chengalur 2004; Karachentsev et al. 2004; Begum et al. 2008). On the other hand, observations have not revealed gas outflows into the IGM from galaxies with dynamical masses of 109 M (van Eymeren et al. 2009). Thus, the question of the efficiency of gas ejection due to galactic winds remains op en. Tidal interaction may b e resp onsible not only for mass exchange b etween galaxies during collisions or close fly-bys and b etween galaxies and the IGM, but also for changes in galactic morphology. According to the estimates of Tutukov (2006), every galaxy in a cluster exp eriences a collision at least once during its lifetime. In these collisions, the galaxies may merge, lose their gaseous comp onents, or b e disrupted completely. A new galaxy may also form from gas lost in galaxy collisions. The ram pressure of the IGM gas, evap oration of gas, and sweeping-out of dust are less efficient galactic mass-loss mechanisms, though they influence the chemical evolution of galaxies and enrichment of the IGM. The essence of the statistical mechanism is that, in the case of an equilibrium distribution of the stars in the gravitational p otential of the galaxy, there will always b e stars with velo cities exceeding the escap e velo city. As these stars leave the p otential well, the system relaxes to a new equilibrium state. However, the timescale for the statistical mechanism is very large close to a hundred relaxation times (Binney and Tremaine 1987), where the latter is 0.1N dyn , (2) ln N where N is the numb er of stars in the system and dyn the dynamical timescale of the system. Typical galactic dynamical timescales are 107 - 108 yr. Even for 106 M galaxies, the relaxation time exceeds the Hubble time. Other mass-loss mechanisms rewiewed by Binney and Tremaine (1987) for collisionless stellar systems are even less efficient.
relax




­3­ It is commonly b elieved that most stars are b orn in asso ciations (see however, the pap er by Elmegreen and Efremov (1996)). The lifetimes of OB asso ciations from birth to decay is short, of the order of several million years. Typical velo cities of stars acquired during the decay are of the order of 10 km/s, according to various studies (Gvaramadze and Bomans 2008) and observations (Gies 1987). Other estimates limit the velo city range to 2 - 8 km/s (Brown et al. 1997). The virial velo cities in low-mass galaxies can b e several km/s (Karachentsev et al. 2004), and the escap e velo city lower than 20 km/s (Bovill and Ricotti 2009; Dijkstra et al. 2004). In the case of disk galaxies, the ordered motions of the galactic matter may facilitate the ejection of stars. The aim of our current study is to estimate this effect and observational manifestations in dwarf galaxies. In Section 1, we compute the probability of ejection of stars from spheroidal and disk galaxies. In Section 2, we present the results of mo deling the evolution of dwarf disk galaxies taking into account the ejection of stars. Section 3 discusses our results.

2.

STELLAR EJECTION MECHANISM

To escap e its galaxy, the kinetic energy of a star must b e sufficient to bring ab out the work against the gravitational field (v + u)2 - , (3) 2 where v is the instantaneous velo city of the OB asso ciation, u the stellar velo city relative to the asso ciation, and the gravitational p otential at the lo cation of the asso ciation, with the p otential at infinity b eing zero. The dynamical timescales for asso ciations ( 108 yr) exceed their lifetimes ( 107 yr) due to the low densities of asso ciations, which are 0.1 M /p c3 (Brown et al. 1997). This means that energy equipartition for stars of different masses do es not have time to b ecome established in the asso ciation. Therefore, we will assume that the velo cities of stars of any mass 2 have isotropic Gaussian distributions with disp ersion OB ; the probability that a star moving away from the center of the OB asso ciation will overcome the p otential of its galaxy will then b e (v, -) =
(v+u)2 2

u2 d3 u exp - 2 2 2O (2 OB )3/2
-

.
B

(4)

The large-scale motion of the ISM in the galaxy increases the fraction of ejected stars due to a sort of "slingshot effect". If the ISM participates in the Keplerian motion of the galaxy with circular velo city v , the velo city of a star after decay will b e summed with the velo city of the asso ciation in the galaxy. As a result, the probability of ejection of the star b ecomes (see the App endix) 1 - - - - + - - /2 d e ( , ) = 1 + erf - erf , (5) 40 2 2


­4­ where = v
OB

=

r
2 OB

, r

=-

2 . 2 O B

(6)

Here, the velo city of the asso ciation is assumed to b e equal to the lo cal velo city of the ISM. The reason for this is that the lifetimes of asso ciations are much shorter than the galactic dynamical timescale so the velo city of the asso ciation do es not change appreciably b efore the asso ciation decays. The parameter in Eq. 5 describ es the relative velo city of the Keplerian motion in units of OB . Expression 5 is also valid for galaxies in which large-scale motions of gas are absent, such as spheroidal galaxies, but we must then adopt = 0. The value of dep ends on the mass distribution in the galaxy and the motion of the galactic gas. It is interesting to estimate the fraction of ejected stars for some typical configurations of galaxies. We will calculate this estimate as an average over the volume of the galaxy weighted by their lo cal star-formation rate : dV = V . (7) V dV Let us give estimates for a spheroidal galaxy with a Plummer density profile and a disk galaxy with an exp onential density profile. The distributions of the density and p otential in an isotropic Plummer sphere dep end on two parameters the mass M and the characteristic scale a: = 3M (1 + r 2 /a2 )- 4 a3
5/2

,

=-

GM (1 + r 2 /a2 )- a

1/2

.

(8)

Let us take the mo del of Firmani and Tutukov (1992) for the volume star-formation rate: 2 . (9)

The distribution of the dimensionless p otential over the scale r /a dep ends only on the parameter , which is defined as the ratio of the typical virial velo city in the galaxy and the velo city disp ersion in the decaying asso ciation: = 2 2 , (1 + r 2 /a2 )1/2 = GM . 2 O B a (10)

As we noted ab ove, we assume 0 for a spheroidal galaxy. Inserting these relations into Eq. 7 and integrating over the volume, we obtain the co efficient as a function of (Fig. 1, left panel, solid curve). The distribution of the surface density of an exp onential disk also dep ends on the total mass and spatial scale: M -r/a e . (11) = 2 a2


­5­

1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6

10 9 8 7 6 5 4 3 2 1

1 0.8 0.6 0.4 0.2 0 5 5.5 6 6.5 7 7.5 8 lg M25 [Msol] 8.5 9 5 5.5 6 6.5 7 7.5 8 8.5 9 lg M25 [Msol]

Fig. 1. Left panel: average probability of ejection of stars from galaxies as a function of , for disk galaxies with an exp onential density profile and circular Keplerian rotation (solid) and non-rotating spheroidal galaxies with a Plummer density profile (dashed). The central panel shows as a function of the mass of the galaxy (up to 109 M ). The mass distribution for the galaxies was taken from the catalog of nearby galaxies of Karachentsev et al. (2004). M25 is the dynamical mass within the 25m isophote. The parameter was computed as the ratio of the rotational velocity of neutral hydrogen (denoted in the catalog Vm ) and the velocity disp ersion in the decaying OB association OB . Two limiting cases were considered for the latter: OB = 2 km/s (triangles) and OB = 8 km/s (circles). The solid lines show approximations for (M25 ) for these limiting cases. The dashed lines show analogous approximations for the ejection of stars outside the dark-matter halo (see the text). Right panel: dep endence of the average probability of ejection of stars on the galaxy mass M25 . The lines corresp ond to the same cases as in central panel. Let us apply an approximate expression for the gravitational p otential, with the mass distribution taken to b e spherically symmetric: =- GM 1 - e- a r /a
r /a

.

(12)

In this approximation, as in the case of a Plummer sphere, the distributions for and dep end only on , which has the same definition: = 1 - (1 + r /a) e- r /a
r /a

,

= 2

2

1 - e- r /a

r /a

.

(13)

Finally, let us take the surface density of the starformation rate according to the SchmidtKennicutt mo del (Kennicutt 1998): 3/2 . (14) The function ( ) for the two cases is plotted in Fig. 1 (left). For disk galaxies, the values of for which the ejection of stars is p ossible are b ounded from ab ove by 4 - 5, while this limit is 2 - 2.5 for spheroidal galaxies. We used data from the Karachentsev catalog of nearby galaxies (Karachentsev et al. 2004) for galaxies with masses b elow 109 M to derive the dep endence of on the galactic mass. For each galaxy with a given mass, the value of was calculated as the ratio of


­6­ the rotational velo city of neutral hydrogen (denoted in the catalog Vm ) and the velo city disp ersion of stars in the decaying OB asso ciation OB . Since OB can b e in the range 2 - 8 km/s, the parameter is also not determined precisely. The approximate mass dep endence of this parameter is = 0.025 M25 M
0.36

O B km/s

-1

.

(15)

In this approximation, we use the dynamical masses of the galaxies inside the 25 mag arcsec-2 isophote, denoted M25 . Using only M25 for the mass underestimates the contribution of dark matter in the outer regions of the galaxy but the approximation 15 remains valid even if we take into account the dark halo es, viz. if we increase M25 by factor equals to 1 + dm /b 6, then the co efficient on the right-hand side b ecomes 0.048. The function 15 for the limiting values of OB is shown in Fig. 1 (middle panel) for the cases when the dark-matter halo is and is not taken into account. Below, we present results of our proximation, taking into account the galactic gas using the abundances of Firmani and Tutukov (1992) and the basis for the one-zone mo del. numerical mo deling of dwarf galaxies in a single-zone apejection of stars. We traced the chemical evolution of the iron, oxygen and heavy elements. We used the mo del of studies of Shustov et al. (1997); Wieb e et al. (1998) as the

3. 3. 1.

NUMERICAL MODELING Single-zone Galactic Mo del

In a decaying asso ciation, only stars with lifetimes 107 yr end their lives in the galaxy as Typ e I I sup ernovae (SN I I). These stars have masses of ab out 13 M and higher (Tutukov and Krugel 1980). Some fraction of lower-mass stars, whose numb er is determined by , leaves the galaxy and is not able to participate in the enrichment of the ISM. However, this effect may b e small, since stars with masses b elow 8 M do not explo de as SN I I (Hashimoto et al. 1993), so that their disapp earance influences the chemical comp osition of the ISM only more weakly. On the other hand, Typ e Ia sup ernovae (SN Ia) have delay times until their explosions of 108 - 109 yr (Tutukov and Yungelson 2002), and their ejecta can strongly influence the enrichment of the ISM in metals. The mass fraction of all stars b orn as a single stellar p opulation with a given initial mass 13M function (IMF) (m) and leaving the galaxy is mmin dm ; for Salp eter IMF with index -2.35 and stellar mass range 0.1 M - 100 M this fraction is 0.88. In the single-zone approximation, the total rate of stellar mass loss due to the ejection of stars for a galaxy with a time-dep endent star formation rate (t) will b e Msej (t) = (t - s (13M ))
13M mm
in

dm ,

(16)


­7­ where s = s (m) is the lifetime of a star with mass m. The ejection of stars also affects the amount of mass participating in star formation and chemical enrichment of the ISM. The rate of gas return is f Mgb (t) =
mm mm
ax

in

dm (t - s ) 1 -

mr [1 - (13M - m)] , m

(17)

where mr = mr (m) is the mass of the stellar remnant, (x) = 0 for x 0, and (x) = 1 for x > 0. The rate of enrichment of the ISM in an element X is determined by the contributions from SN I and SN I I: 10-3 fb MX (t) = (t - M
mm
ax

SNIa

S ) PXNIa [1 - ]+

+
mm
in

dm (t - s )

1-

mr S X (t - s ) + PXNII (m, Z (t - s )) â m â [1 - (13M - m)] , (18)

where SNIa is the average delay time b etween the formation of a binary and the SN Ia explosion, S S PXNIa is the mass of element X ejected p er explosion, and PXNII (m, Z ) is the mass of element X pro duced by SN I I with pre-sup ernova mass m and initial heavyelement abundance Z . In this mo del, we assumed that SN Ia are pro duced by mergers of degenerate dwarfs in close binaries. As was shown by Tutukov and Yungelson (1994), the mean delay time b etween the formation of a binary and the SN Ia explosion is SNIa 109 yr. The rate of SN Ia explosions p er unit star-formation rate is obtained by normalizing to the mo dern SN Ia rate of 0.003 p er yr and mo dern star formation rate of 3 M p er yr (Tutukov and Yungelson 1994). We assumed that each SN SN Ia pro duces PFe Ia = 0.6 M of iron (Tsujimoto et al. 1995); we did not consider the pro duction of other elements by these sup ernovae. The yields of SN I I as functions of the mass of the star and the initial heavy-element abundance were taken from Maeder (1992). The balance of the total mass of galaxy, the mass of gas, and the masses of various elements is determined not only by star formation, but also by interaction of the galaxy with the ISM; i.e., via the galactic wind, dust ejection, and the accretion of intergalactic gas. These factors were studied previously for the same single-zone mo del (see, for instance, the study of Shustov et al. (1997)). However, accretion, dust ejection, and galactic wind were not taken into account in the present study [the last factor due to obvious shortcomings of the simple mo del for the galactic wind, Eq. 1]. As a result, the mass balances of various comp onents of a galaxy was given by the equations: Mtot = -Msej f Mg = - + Mgb fb MX = -X + MX .

(19)

The details of this numerical mo del are presented in Firmani and Tutukov (1992); Shustov et al. (1997); Wieb e et al. (1998).


­8­ 3. 2. Results of the Computations

We can see from the dep endence (M25 ) (Fig. 1) that the decay of OB asso ciations essentially do es not lead to mass loss from spheroidal galaxies, even those with the lowest masses. For this reason, we restricted our numerical analysis to disk galaxies.We computed four series of mo dels for the evolution of galaxies with masses from 106.5 M to 108.5 M and various values of OB . For comparison, we also computed a series of closed mo dels. The radii of the galaxies corresp onded to the relation M R2 . To avoid a large influence of the initial conditions on the burst of star formation, the semi-thickness of the protogalactic disk was set to 10 kp c in all computations. The remaining parameters for the computed series of mo dels are given in the table. In series A, OB = 2 km/s, while in series B and C OB = 8 km/s. In series C, we allowed for the influence of the dark-matter halo [see the comments concerning Eq. 15]. Thus, the series B computations enabled us to estimate the effects of ejecting stars from the disk into the galactic halo, while series C illustrated the effects of ejecting stars from the halo into the IGM. lg M /M 6.5 7 7.5 8 8.5 R, p c 79.5 141.4 251.5 447 795
OB

= 2 km/s (series A) 0.21 0.09 0.012 0 0



OB

= 8 km/s (series B) 0.92 0.743 0.481 0.21 0.075

2.73 4.14 6.26 9.48 14.25

0.70 1.07 1.62 2.45 3.72

= 8 km/s +dark halo (series C) 1.38 0.6 2.08 0.32 3.16 0.13 4.78 0.035 7.23 0
OB

Figure 2 shows the integrated characteristics of the mo del galaxies at the end of the computations. As exp ected, including the ejection of stars in the series A mo dels did not result in significant deviations of the integrated characteristics from those for the closed mo del. In the series B and C mo dels, for the lowest mass galaxies, the ratio of the ejected mass and the dynamical mass can range from 1.5 - 2 (for matter ejected from the halo into the IGM) to 6 (for matter ejected from the disk into the halo). The amount of ejected mass obtained in the series B mo dels provides some idea of the extent to which the morphology of a galactic disk can vary. For a galaxy with dynamical mass 107 M , the stellar mass in the halo exceeds the mass of the disk by a factor of 1.5 - 2; i.e., such a galaxy cannot b e considered a disk galaxy. Galaxies with final dynamical masses of 106 M can have a mass of 4.5 â 106 M in the halo and 1.5 â 106 M in the IGM. The mass­luminosity relation shows that, compared to the closed mo del, the total luminosity of the lowest-mass galaxies can b e half an order of magnitude lower, and the disk luminosity an order of magnitude lower. The mass-to-luminosity ratio itself increases by more than a factor of 2.5 for the low-mass galaxies of series B. For series B galaxies with masses b elow 107 M , the total luminosity of stars ejected into the halo is a factor of two to three higher than the disk luminosity (Fig. 2, middle panel in the


­9­ upp er row). This also provides evidence for the transformation of the galactic morphology from disk to spheroidal.
7 6 5 4 3 2 1 0 5 5.5 6 6.5 7 7.5 8 8.5 9 Ms /M
ej

20 18 16 14 12 10 8 6 4 2 0 5 5.5 6 6.5 7 7.5 8 8.5 9 Ls /L
ej

18 16 14 12 10 8 6 5 5.5 6 6.5 7 7.5 8 8.5 9 M/L

0.5

[Z]g

0.4

[O/H]g

0.5

[Fe/H]g

0.4

0.3

0.4

0.3

0.2

0.3

5

5.5

6

6.5

7

7.5

8

8.5

9

5

5.5

6

6.5

7

7.5

8

8.5

9 -0.4 -0.5 -0.6

5

5.5

6

6.5

7

7.5

8

8.5

9

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 5 5.5 6 6.5 7 7.5 8 8.5 9 [Z]s

-0.1 -0.2 -0.3

-0.7 -0.4 -0.8 -0.5 -0.6 5 5.5 6 6.5 7 7.5 8 8.5 9 [O/H]s -0.9 -1 5 5.5 6 6.5 7 7.5 8 8.5 9 [Fe/H]s

Fig. 2.

Dep endence of integrated parameters on the disk masses (on a logarithmic scale in units of M ) in four series of models. Solid diamonds corresp ond to the closed galaxy model, op en diamonds to the model with OB = 2 km/s (series A), solid triangles to the model with OB = 8 km/s plus a dark-matter halo (series C), and op en triangles to the model with OB = 8 km/s without a dark-matter halo (series B). The upp er row of plots shows the ratio of the ejectedmass and diskmass, the ratio of the luminosities of the ejected stars and the disk, and the mass-to-luminosity ratio for the disk (in solar units). The panels in the middle and lower rows show the abundances of various elements in the gas and stars (averaged over the stellar p opulation).

Since we did not consider galactic wind and accretion in our mo dels, the star-formation history in the galaxies consisted of a single burst which depleted most of the gas: in all the galaxies, the


­ 10 ­ mass fraction of the gas at the end of the computations was 5 - 10%. The relative abundances of elements in the gas decrease weakly with the mass of the galaxy and with increasing OB (Fig. 2, middle row of plots). One exception is the mo dels at the low-mass end of series B, for which the relative abundances of element grow with the galaxy mass by 0.1 dex, compared to the closed mo dels. As the plots show, in all series of mo dels, the mass dep endence of the abundances converges to a minimum for some mass in all series of mo dels. The reason for this may b e a comp etition of two pro cesses determining the abundances in the gas: the enrichment of gas by sup ernovae and the return of gas p o or in elements by explosions of new low-mass stars. The influence of the latter pro cess may b e diminished if a galaxy loses a considerable fraction of its low-mass stars. The elemental abundances in the stars systematically decrease with the mass of the galaxy and with increasing OB (Fig. 2, lower row of plots). The ejection of stars influences the iron abundance most. In a galaxy with dynamical mass 106 M , the iron abundance can decrease by 0.5 dex in the disk stars (series B) and by more than 0.3 dex in the halo stars (series C). Figure 3 shows the distribution of the relative numb ers of stars over the abundances of iron, oxygen, and heavy elements for all the computed series of mo dels. Series B (second column of plots) differs most from the closed mo del. Along with the shift of the distribution toward lower abundances, stars heavily enriched in iron app ear, which formed in the first few billion years after the SN Ia explosions. The characteristic app earance of the stellar distributions is due to the fact that the galaxies in this mo del exp erience only one burst of star formation, which o ccurred in a low-metallicity gas.

4.

DISCUSSION AND CONCLUSIONS

We have studied the influence of the loss of stellar mass on the evolution of dwarf spheroidal and disk galaxies. The decay of OB asso ciations was considered as a p ossible mass-loss mechanism, with the decay enabling some stars to obtain velo cities sufficient to escap e their galaxy. The decay of asso ciations is essentially of no imp ortance for the evolution of spheroidal galaxies. The effect is also small for disk galaxies with OB = 2 km/s. Since a value of 10 km/s is thought to b e typical, we fo cused our analysis on mo dels with OB = 8 km/s. The results of our analysis are as follows. 1. During the lifetime of an able to enrich the ISM in galaxy do not contribute elements for the halo or I OB asso ciation ( 107 yr), the most massive SN I I ( 13 M ) are the pro ducts of their explosions. Lower-mass stars that leave their to the enrichment of the disk ISM, but instead serve as a source of GM. The same is true of SNIa.

2. Disk galaxies that had at the onset of their star formation masses of 3 â 107 M contain half of their mass in disk stars and the other half in the halo. The halo luminosities in such galaxies exceed the disk luminosities by a factor 1.52. We can thus infer that galaxies with masses 107 M that were initially disk galaxies change their morphology to spheroidal. According to the classificaiton of de Vaucouleurs et al. (1991), spiral galaxies are assigned morphological indices T = 4 (see also Corwin et al. (1994)). In the catalog of nearby galaxies


­ 11 ­
6.5

M = 10 0.3 [Fe/H] 0.2

M

sol

OB = 2 km/s 0.3

M = 10

6.5

M

sol

OB = 8 km/s 0.3

M = 10

7.5

M

sol

OB = 8 km/s 0.3

M = 10

8.5

M

sol

OB = 8 km/s

[Fe/H] 0.2 0.2

[Fe/H] 0.2

[Fe/H]

0.1

0.1

0.1

0.1

0 -2.5 0.3 [O/H] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [O/H] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [O/H] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [O/H] 0.2 -2 -1.5 -1 -0.5 0 0.5

0.1

0.1

0.1

0.1

0 -2.5 0.3 [Z] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [Z] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [Z] 0.2 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 0.3 [Z] 0.2 -2 -1.5 -1 -0.5 0 0.5

0.1

0.1

0.1

0.1

0 -2.5 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 -2 -1.5 -1 -0.5 0 0.5

0 -2.5 -2 -1.5 -1 -0.5 0 0.5

Fig. 3. Distribution of the relative number of stars over the current abundances of iron, oxygen, and heavy elements. In the left column, the solid curves corresp ond to galaxies with masses of 106.5 M in the series A models, and the thin dotted curve to the closed model. In the other columns, the solid curves corresp ond to galaxies with masses of 106.5 M , 107.5 M , and 108.5 M in the series B models, the dashed curves to galaxies with the same masses in the series C models, and the thin dotted curves to the closed model. (Karachentsev et al. 2004), disk galaxies have absolute magnitudes not exceeding -13m (for morphological indices from 0 to 7; i.e., including lenticular and irregular galaxies that are closest to disk galaxies); this magnitude corresp onds to a luminosity 107 L and a galactic mass 108 M . Lower-mass galaxies are classified as spheroidal and irregular. This is confirmed by the computations for the mo dels we have adopted here. 3. In systems with masses 105 M , a large fraction of the stellar mass leaves not only the disk of the galaxy, but also the halo (Fig. 1, right panel). Thus, if extremely low-mass galaxies can form at all, they can lose almost all their stellar p opulation to the IGM after their first burst of star formation. As a result, a dark-matter halo enriched in gas should b e left. This scenario may b e imp ortant for the problem of missing satellites of the Galaxy. 4. The ejection of stars increases the mass-toluminosity ratio. For galaxies with total masses (disk + halo) of 107 M , this ratio increases by a factor of 2 - 2.5 (Fig. 2, upp er right


­ 12 ­ panel). 5. The ejection of stars may result in strong variations of elemental abundances 2,middle row of plots): along with the systematic decrease of the abundances the elemental abundances in the lowest-mass galaxies can increase by 0.1 - abundances in stars systematically fall with the galaxy mass decrease, by 0.2 in the gas (Fig. by 0.05 dex, 0.15 dex. The dex.

This study was supp orted by the Federal Agency on Science and Innovation (state contract no. 02.740.11.0247), the Federal Education Agency (contract RNP-2.1.1-1937), the Program of State Supp ort for Leading Scientific Scho ols of the Russian Federation (grant no. NSh-4354.2008.2), and the Russian Foundation for Basic Research (pro ject nos. 08-02-91321-IND and 07-02-00454).

APPENDIX We will obtain an expression for the probability of ejection of a star from an OB asso ciation moving along a Keplerian orbit in a galaxy with a given gravitational p otential (r ). A general expression for this probability is (v, -) =
(v+u)2 2

u2 d3 u exp - 2 2O (2 OB )3/2
-

.
B

(20)

The velo city vector for a circular Keplerian orbit is v = e v , v= r , r (21)

where e is a unit vector in the azimuthal direction. The integration domain is in this case describ ed by the equation (22) u2 + u2 + 2v u + v 2 + 2 0 , where u is the mo dulus of the velo city orthogonal to e . Solving this equation leads to the condition for the velo city comp onent u u (-, (u
-

)] [(u

+

), +) ,

u

±

= -v ±

-2 - u2 .

(23)

Since the comp onents