Bimodal abundance pattern in M51: evidence for corotation resonance effects
I. A. Acharova, O.A.Galazutdinova, J. R. D. Lépine, Yu. N. Mishurov, N.A. Tikhonov
aa r X i v : . [ a s t r o - ph ] A p r Mon. Not. R. Astron. Soc. , 1–5 (2008) Printed 30 October 2018 (MN L A TEX style file v2.2)
Bimodal abundance pattern in M51: evidence for corotationresonance effects
I. A. Acharova, ⋆ O.A.Galazutdinova, J. R. D. L´epine, Yu. N. Mishurov, , and N.A. Tikhonov Space Research Department, Southern Federal University, 5 Zorge, Rostov-on-Don, 344090, Russia Special Astrophysical Observatory of Russian Academy of Sciences, Nizhnij Arkhyz, 369197, Karachaevo-Cherkessia, Russia Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, Universidade de S˜ao Paulo, Cidade Universit´aria, S˜ao Paulo, SP, Brazil
Accepted 2008 xxxx. Received 2008 xxxx; in original form 2008 xxxx
ABSTRACT
A chemical evolution model for the bimodal-like abundance distribution in the externalgalaxy M51 recently derived on the basis of HST data for more than a half million red su-pergiants is developed. It is shown that, like in our Galaxy, formation of fine structure of theradial abundance pattern – a rather steep gradient in the internal part of the disc and a plateauin the middle part – is due to the influence of the spiral arms, the bend in the slope of thedistribution being arose near the corotation resonance. Our model strongly suggests that M51is surrounded by overabundant gas infalling onto its disc.
Key words:
Galaxies: spiral - galaxies: evolution - galaxies: abundances.
Until recently, it was widely believed that the radial abundance dis-tribution (being measured in logarithmic scale) along the galacticdisc is described by a simple linear function in the Galaxy witha constant gradient. This representation is very persistent by sev-eral researches albeit some irregularities in the radial abundancedistribution along the galactic disc were pointed out by severalresearchers of H II regions and planetary nebulae. NeverthelessTwarog, Ashman & Anthony – Twarog (1997) were the firsts whotried to break this oversimplified point of view, but only after An-drievsky et al. (2002, see other papers of this series) the idea that thegradient in the galactic disc is not constant has gained substantialfoundation. Over a large number of elements the authors showedsufficiently definitely that the abundance pattern in the Galaxy isa bimodal, i.e., there is a rather steep gradient in the inner part ofthe disc and a plateau (or a shoulder) in the region including theSun, the bending (or fracture) in the slope of the distribution beinginside the solar circle.The results of Andrievsly et al. are quite reliable since theyused spectroscopic data of Cepheids which, in addition, have suf-ficiently precise distances and due to their brightness are seen in awide spatial region. In our Galaxy, the above abundance pattern isalso supported by the young planetary nebulae (Maciel & Quireza1999; Maciel, Costa & Uchida 2003), OB stars (Daflon & Cunha2004), etc. ⋆ E-mail: [email protected] (IAA); [email protected] (OAG);[email protected] (JRDL); [email protected] (YNM);[email protected] (NAT)
In a series of papers by Mishurov, L´epine & Acharova(2002), L´epine, Acharova, & Mishurov (2003), Acharova, L´epine& Mishurov (2005, hereafter ALM), etc., a theory of bimodal abun-dance structure formation was developed. It is based on the ideathat galactic density waves, responsible for spiral arms, influencethe process of enrichment of galactic disk by heavy elements, thecorotation resonance (where the rotation velocity of galactic mattercoincides with the velocity of spiral waves) playing a crucial role:the plateau-like distribution forms in the vicinity of the dispositionof the corotation circle.The above theory of ALM was applied to our Galaxy. It is wellknown, here we face the problem that owing to solar position in thegalactic disc and corresponding light extinction, the observationalsamples do not cover the most part of the galactic disc. One shouldalso bear in mind the uncertainties in distances to the objects whichcan sometimes rich tens percents. Moreover, we do not know ex-actly the number of arms in the Galaxy and the global structure ofits disc. On the contrary, in external galaxies we do not face theabove problem since we see them entirely. Hence, we can hope toconstruct the radial abundance pattern in the whole galactic disc inspite of all uncertainties in distances to them. So, it would be veryinteresting to apply ALMs theory to external spiral galaxies.Unfortunately the corresponding observational data were veryfragmentary. Perhaps the most complete sample of abundances de-rived over HII regions in external galaxies was presented by Zarit-sky, Kennicutt & Hucra 1994. In their sample, there are severalspiral galaxies that demonstrate bimodal radial abundance distri-bution. But as it was shown by Dutil & Roy (2001) that to drawdefinite conclusions about fine features in chemical abundance gra- c (cid:13) I. A. Acharova et. al. dient in a galaxy one needs sufficiently large number of objects(more than 16 HII regions).A completely different observational material was analyzedby Tikhonov, Tikhonov & Galazutdinova (2007, hereafter TTG).For the galaxy M51 they carried out the stellar photometry of HSTimages and determined magnitude I and color index V − I for morethan 0.5 million stars. It enables to derive the distribution of thecolor index of red supergiants along the galactic radius. Accordingto TTG the distribution of the above index along the galactic radiusdemonstrate a bimodal-like structure. Since the color index of redsupergiants depends on their metallicity the radial distribution ofthe last value will demonstrate the same radial distribution.The goal of the present paper is to apply the theory of ALM toexplain the formation of bimodal-like radial abundance pattern inthe external spiral galaxy M51. To take into account the effects of spiral arms on the galactic discenrichment by heavy elements we use equations from ALM: ∂µ s ∂t = (1 − R ) ψ, (1) ∂µ g ∂t = − (1 − R ) ψ + f, (2) µ g ∂Z∂t = P Z ψ + f ( Z f − Z ) + 1 r ∂∂r ( rµ g D ∂Z∂r ) , (3)where Z is the mass content (or abundance) of heavy elements, µ s and µ g are the surface densities for the stellar and gaseous discscorrespondingly, ψ is the star formation rate (SFR; we use the in-stantaneous recycling approximation), R ≈ . is the stellar massfraction returned into interstellar medium (ISM), f is the infall rateof matter onto the galactic disc, Z f is its abundance, P Z is the ratefor enrichment of galactic disc by heavy elements (see below), r isthe galactocentric distance, t is time. The last term in equation (3)describes the heavy elements diffusion due to turbulent motions inISM. Following Mishurov et al. (2002), for the diffusion coefficient D we use the gaskinetic estimate, modeling the turbulent ISM bya system of clouds and supposing the values for them close to theones in our Galaxy (see details in ALM).For the SFR Schmidt-like approximation was used: ψ = βµ kg , (4)where β is a normalizing factor, the exponent k was adopted k = 1 . (Schuster et al., 2007, Kennicutt 1998). The coefficient β is a constant both in r and t . It is fitted so as to derive a plausi-ble final (i.e., present) density distributions for stellar and gaseouscomponents.In the classical paper by Tinsley (1980), the coefficient P Z was considered as a true constant that means the stellar mass frac-tion ejected into ISM (per unit of time and unit of surface in galacticplane) as newly synthesized heavy elements. It is obvious, such in-terpretation was based on a tacit assumption that sources of heavyelements are uniformly distributed in galactic azimuth. However,following Oort (1974) we believe that the sources are concentratedin spiral arms. Hence the rate for enrichment of the ISM by heavyelements, P Z , is to be proportional to the frequency at which anyelementary volume of gas enters spiral arms and occurs close to thesources of heavy elements, i.e.: P Z = η | Ω( r ) − Ω P | Θ , (5)where Ω( r ) is the angular rotation velocity of matter in galacticdisk, Ω P is the angular rotation velocity for spiral density wavepattern, η is the constant for the rate of enrichment the galactic discby heavy elements, Θ is some cut-off factor (see below).It is well known that the spiral wave pattern rotates as a solid,i.e. Ω P is a constant (Lin, Yuan & Shu 1969). The location of thecorotation resonance ( r c ) in a galaxy is determined by: Ω( r c ) = Ω P . (6)From the above equations it is seen: in the vicinity of the coro-tation resonance the enrichment of ISM by heavy elements is de-pressed since the galactic matter here moves in phase with spiralwaves, so, in the corotation vicinity, the most part of interstellargas (that at the initial moment of time was beyond spiral arms) willnever enter the arms. Hence, the corresponding gas will be far fromsources of heavy elements and will not be enriched by them. That iswhy we expect some irregularity in radial distribution of abundance(say, bending or fracture) close to the corotation.Modeling the abundance evolution in the disk of our Galaxy,ALM referred mainly to oxygen. The fact is that oxygen is pro-duced by short-lived SNe II whereas they synthesize only about30 % of iron, 70 % of that being produced by SNe Ia. SNe II arestrongly concentrated in spiral arms. On the contrary, for a longtime it was believed that SN Ia are long-lived objects and they donot keep the memory of the fact that they were born in spiral arms.So we can expect that oxygen is the best indicator of spiral arminfluence on abundance distribution in galactic discs. This is whyALM concentrated on explanation of oxygen distribution.However, new data show that there are 2 populations of pro-genitors for SNe Ia - short-lived and long-lived (Mannicci, DellaValle & Panagia 2006; Matteucci et al. 2006). The first type is tobe concentrated in spiral arms, the second subgroup is not stronglyconfined to arms (by the way, perhaps these new data explain theold result of Bartunov, Tsvetkov & Filimonova 1994, who revealedthat SNe Ia also demonstrate correlation with spiral arms, not sosharp as SNe II and SN Ib do, but the correlation is clear). Eachtype of SNe Ia produces about equal part of iron (Matteucci et al.2006). In total, SN II and short-lived group of SNe Ia synthesize ∼
65% of iron. So, the total (O + Fe + ...) content of heavy element,i.e., the value Z in our notation, may be considered as a sufficientlygood indicator of spiral arms influence on fine radial abundancedistribution in galactic disc although not so sharp as oxygen. Nev-ertheless the research on nucleosynthesis in galactic disc should becontinued since a part of heavy elements sources does not concen-trate in spiral arms (see Acharova, Mishurov & L´epine 2008) .Several words about the cut-off factor Θ . In the density wavetheory of Lin et al. (1969) the wave zone is restricted by inner andouter Lindblad resonances, the locations for the resonances beingdetermined by the condition ν = ± , where ν ( r ) = m (Ω − Ω P ) /κ is the dimensionless wave frequency, m is the number of arms (forM51 m = 2 ), κ is the epicyclic frequency (see details in Lin et al.,1969). So we adopt: Θ = 1 if | ν m | and Θ = 0 otherwise.Following Lacey & Fall (1985) and Portinari & Chiosi (1999)the infall rate of matter onto the galactic disk is: f ( r, t ) = f exp( − r/ ∆ − t/τ ) , where f is the central rate, ∆ and τ arethe space and time scales (in order to reduce the number of freeparameters we suppose τ to be independent of r ).The above system of equations splits into 2 groups – the firstone that describes the evolution of stellar and gaseous densities(equations 1, 2 and 4) and the second one that describes the evo- c (cid:13) , 1–5 imodal abundance pattern in M51: evidence for corotation resonance effects lution of abundance (equations 3, 5). The first group of equationsrepresents a system of ordinary differential equations that can besolved independently. Equation (3) contains SFR function ψ ( r, t ) and gaseous density µ g ( r, t ) that must be computed before solv-ing it. Unlike the previous group of equations, the last one is apartial differential equation. Besides the initial conditions we haveto superimpose the boundary conditions. For that we use the typi-cal conditions for the diffusion processes: the absence of diffusionflows at the galactic center and at the outer end (the result doesnot distinctly depends on value of galactic radius – 15 kpc or 25kpc). They simply guarantee the abundance finiteness here. As aninitial conditions for densities we adopt µ g = µ S = 0 , for con-tent Z = 0 . Z ⊙ (according to Asplund, Grevesse & Sauval 2005 Z ⊙ = 0 . ).In Fig. 1 are shown the present radial profiles of gaseous,stellar and full densities in M51 computed for the followingset of parameters: β = 0 .
12 ( M ⊙ pc − ) − k Gyr − , f =600 M ⊙ pc − Gyr − , ∆ = 3 . kpc , τ = 3 Gyr . In all our mod-elling we suppose that the distance to M51 is 9.6 Mpc and the ageof the galactic disc is 10 Gyrs.The final stellar mass in M51 happens to be of the order of . · M ⊙ , gaseous . · M ⊙ , present SFR is M ⊙ yr − .The global parameters are close to the observed ones for M51(Schuster et al. 2007). Radial abundance pattern in the disc of M51
The above theory refers to the radial abundance distribution. Thatmeans that the corresponding equations are derived from morecommon equations by means of their averaging over the azimuth.However, usually the abundances in external galaxies are derivedover HII regions which are strongly concentrated in spiral arms.Hence, HII regions do not represent the abundance distribution overthe whole galactic disc.As we wrote in the Introduction, for M51 TTG used com-pletely different objects: red supergiants (the corresponding imageswere obtained by HST). These stars are found both in spiral armsand in the interarm regions. So, we can average the abundance overthe azimuthal angle in the galactic plane. To do this, the galacticplane was divided into circular rings (ellipses for the visible disc) ofwidth about − . kpc and the mean color index V − I was derivedfor each ring over stars that have fallen into the corresponding ring.Further, using isochrones of Bertelli et al. (1994) we transformedthe color index into abundance and derived the radial distribution of Z in M51. The result is shown in Fig. 2. Here we use a customaryrepresentation for abundance in logarithmic scale normalizing it tothe solar value Z ⊙ : [ Z ] = log ( Z ) − log ( Z ⊙ ) . (7)First of all we would like to stress that the abundance in M51is higher than the one in our own Galaxy by about . dex . Data ofother authors confirm this conclusion. Indeed, comparing the radialdistribution of oxygen in M51 (Boissier at al., 2004) and in the discof our Galaxy (Portinari & Chiosi 1999) one can see that the contentin M51 is systematically higher than the oxygen abundance in ourGalaxy by about the same value. What is the reason for such dif-ferent abundance in M51 and in the Galaxy? Zaritsky et al. (1994)connect it with some global properties of a given galaxy. We showbelow that it demands that the gas infalling onto the disc of M51 isoverabundant relatively to the gas infalling onto our Galaxy. Another interesting feature is the bending of the slope of dis-tribution at r ≈ kpc clearly seen in this figure. Indeed, inside r ≈ kpc there is a gradient of the order of − . dex kpc − . Be-tween r ≈ kpc and kpc there is a plateau (or a shoulder) like inthe Galaxy (see the cited papers of Andrievsky et al., and ALM).The explanation of this structure is the target of the present paper.It is worth-while to notice, according to our equations, the con-tent Z means the abundance of interstellar gas. Hence the theoret-ical abundance at present time must be compared with the one ofyoung objects. In this connection, it is important that stars in TTGsample have ages within 40 Myrs. Rotation curve
For computation of abundance evolution we need the data on the ro-tation curve in M51. Several rotation curves were derived, e.g., So-fue (1994), Garc´ıa-Burillo, Gu´elin & Cernicharo (1993), etc. Thesecurves differ in the middle and outer parts of M51 (see discussionin Schuster et al., 2007). In the middle part, the rotation velocity de-rived by Sofue is several tens km/sec larger than the one of Garcia-Burillo et al. But in the outer part, the behavior of these curves dif-fers fundamentally: the curve of Garc´ıa-Burillo et al. is flat whereasthe one of Sofue declines too fast – faster than for the Keplerianlaw. Such behavior of the rotation curve is interpreted by Sofue asa consequence of the wrap of the galactic disc due to interactionwith the M51 companion.In our modeling we treated 2 types of rotation curves. In theinner part of M51 ( r < kpc ) the curves are close to the one ofSofue (1994) for both the cases. In the outer part they differ: in thefirst case, the curve is flat ( V rot is at level 250 km/s), in the secondcase the curve corresponds to opposite limiting case - it follows theKeplerian law. The rotation curves used in the present paper areshown in Fig. 3. Independent determination of location of corotationresonance
Since we connect the peculiarities in the radial abundance patternwith the corotation resonance it would be useful to have some in-dependent indications of a possible location of it. In an externalgalaxy it can be determined by means of several methods: i ) kine-matic method of Tremaine & Weinberg (1984); ii ) analyzing theradial distribution of young bright objects; iii ) studying the radialvariation of relative star formation efficiency (Cepa & Beckman,1989), etc. In our paper, we will refer to the first two methods.Using a modified Weinberg – Tremaine method, Zimmer,Rand & McGraw (2004) processed the field of velocities of HIin M51 obtained over 21 cm emission and derived Ω P = 38 ± km s − kpc − . For the adopted rotation curves this leads to lo-cation of the corotation from r c ≈ . kpc to about kpc .On the other hand, we would like to pay attention to a dip inthe radial distribution of blue supergiants (i.e., very young objects)in M51 at r ≈ kpc seen in Fig. 8 of TTG. This is a direct indi-cation of the location of corotation. Indeed, according to Roberts(1969) galactic spiral shocks (arising when interstellar gas flowsthrough spiral arms) are the triggering mechanism of star forma-tion, at least of massive bright stars which are strongly concen-trated in spiral arms. However, near the corotation, the intensity ofthe shocks becomes small since the relative velocity of interstellargas and spiral pattern tends to zero in its vicinity ( | Ω − Ω P | → ).Hence, near the corotation the stimulating effect of spiral arms on c (cid:13) , 1–5 I. A. Acharova et. al. star formation will be depressed. So, here we have to observe thereduced number of bright stars. Therefore, the dip in the radial dis-tribution of blue supergiants indeed indicates the location of coro-tation resonance.Taking this fact into account, in our modelling, we will con-sider the location of the corotation resonance between r = 5 . kpc and . kpc . Results of our modeling of radial abundance pattern evolution inM51 for the above 2 types of rotation curves and 2 locations ofcorotation resonance are shown in Figs. 4, 5 (in our experiments,we found η ≈ . · − Gyr − ). From these figures it is seen:the radial abundance distribution in the disc of M51 indeed reflectsthe influence of spiral arms. In all cases, a bimodal – like radialdistribution of abundance forms with a gradient in the inner partof the galactic disc close to the observed one and the plateau inthe region − − kpc . The theoretical fine structure of the radialabundance pattern is close to the observed one, the bend of theslope of abundance distribution being situated in the vicinity of thecorotation resonance. Hence, like in our own Galaxy, the bend canbe considered as an indicator of position of corotation in externalgalaxies.In Sec 3.1 we noticed that M51 is overabundant relative toour own Galaxy. As a consequence, to get good agreement withobservations we had to adopt the abundance of infalling onto thegalactic disc gas to be Z f ≈ . dex , that is about 7 times higherthan usually adopted in modelling of our Galaxy (see ALM).The question arises: can we explain the observed radial abun-dance pattern by means of any other combination of free param-eters, supposing that Z f is several times smaller than the abovevalue, but the rate–constant η is larger (correspondingly the rateof heavy elements production is higher) and the initial abundanceis much higher than the solar abundance? The answer is negative:none of our numerous experiments with other various combinationsof the free parameters gave the abundance pattern which would co-incides with the observed one. In order to feel the above statementin Fig. 6 we show the results of our computations for Z f = Z ⊙ / , Z ( t = 0) = 10 Z ⊙ and η = 4 · − Gyr − and · − Gyr − (in the both cases the flat rotation curve was used and the corotationresonance was supposed to be situated at r = 5 . kpc ). From thisfigure it is seen: ) high initial value of abundance does not automatically mean thatthe final abundance (i.e., the abundance of young objects at presenttime) will also be high since heavy elements are mainly consumedby low mass stars in preceding times and will never be returned tointerstellar gas; ) since the rate of heavy elements enrichment depends on the dif-ference | Ω − Ω P | the increasing of the rate – constant η leads to avery steep gradient much steeper than observed.In other words, we have to recognize that M51 is surroundedby an overabundant gas.In our simplest theory we do not derive an abrupt decline ofthe distribution in the outer part of the galaxy, beyond 8 kpc, al-though within errors our model coincides with observations. One Notice that galactic wind takes away all components of ISM and does notchange the relative abundance of heavy elements, i.e., Z . of the way to explain this fact is to adopt that the abundance of in-falling gas reduces in the outer part of the galactic disk. To demon-strate this effect we considered the following model: for r kpcZ f = 0 . , but beyond kpc Z f is 10% smaller. The result isshown in Fig. 7. From here the reader can see: it is possible to ex-plain rapid falling down of galactic abundance in the outskirt of thedisc supposing that the infalling gas is only slightly depleted there. The theory by Acharova, L´epine & Mishurov (2005) which ex-plains the formation of bimodal radial abundance distribution in agalactic disc under the influence of spiral arms, especially due to ef-fects of corotation resonance, was applied to external galaxy M51.As the observational data we used the modified results of Tikhonov,Tikhonov & Galazutdinova (2007) on color index V − I of morethan 0.5 million red supergigants in M51 derived by means of treat-ment of HST images. To adjust them for our theory we convertedthe color index into abundance.The observed radial abundance distribution in M51 demon-strates the following fine structure: i) the bimodal radial distribu-tion with a gradient of the order of − . dex kpc − in the innerpart of the galaxy ( r < kpc ) and the plateau between about and kpc (hence, there is a bend in the slope of the distribution at r ≈ kpc ); ii) M51 is an overabundant relative to our Galaxy onabout . dex and more.Our theory strongly suggests that like in the Galaxy, the for-mation of the bimodal radial abundance pattern in M51 is con-nected with the influence of spiral arms on the process of enrich-ment the galactic disc by heavy elements, the bend in the slope ofthe distribution being formed in the vicinity of the corotation reso-nance.Our modelling have led to an unambiguous conclusion: it isimpossible to simultaneously explain the overabundance in M51(relative to our Galaxy) and the fine structure of the radial abun-dance distribution by internal properties of the galaxy, say, byhigher rate for production of heavy elements in the galactic discand predominantly enriched protogalactic gas. We have to supposethat the gas, surrounding M51 and infalling onto its disc, is over-abundant. How can it be?After ALM, we would like to attract attention to the problemof derivation of observational information on chemical distributionin different objects with different ages, say, HII regions, supergi-gants, planetary nebulae, open clusters in discs of close galaxies.This will enable us to derive more refined information on galacticevolution. ACKNOWLEDGMENTS
The work was supported in part by the grant of Federal Agencyfor Education of the Ministry of Education and Science of RussianFederation and the grant of Southern Federal University (Russia).
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Figure 1.
Final profiles of surface densities – gaseous ( µ g ), stellar ( µ s ) and full ( µ g + µ s ) in the disc of M51 at present time. The age of the disc was adoptedto be 10 Gyrs. c (cid:13)000
Final profiles of surface densities – gaseous ( µ g ), stellar ( µ s ) and full ( µ g + µ s ) in the disc of M51 at present time. The age of the disc was adoptedto be 10 Gyrs. c (cid:13)000 , 1–5 imodal abundance pattern in M51: evidence for corotation resonance effects Figure 2.
Radial distribution of abundance (in logarithmic scale) in M51 normalized on the solar abundance (see text). The bimodal structure (in terms ofAndrievsky et. al. 2002) with a gradient of the order − . dex kpc − in the internal part of the disc (for r < kpc ) and a plateau between 5 and 8 kpc isclearly seen.c (cid:13) , 1–5 I. A. Acharova et. al.
Figure 3.
Adopted rotation curves: solid line corresponds to the flat curve in the outer part of the disc, crosses – to Keplerian law. The dashed line correspondsto Ω P = 39 . km sec − kpc − . For this value of the angular rotation velocity of spiral wave pattern the corotation resonance happens to be situated at r = 6 . kpc . The locations of inner and outer lindblad resonances can be derived from intersections of the dashed line with the lines corresponding to Ω ∓ κ/ . c (cid:13)000
Adopted rotation curves: solid line corresponds to the flat curve in the outer part of the disc, crosses – to Keplerian law. The dashed line correspondsto Ω P = 39 . km sec − kpc − . For this value of the angular rotation velocity of spiral wave pattern the corotation resonance happens to be situated at r = 6 . kpc . The locations of inner and outer lindblad resonances can be derived from intersections of the dashed line with the lines corresponding to Ω ∓ κ/ . c (cid:13)000 , 1–5 imodal abundance pattern in M51: evidence for corotation resonance effects Figure 4.
Comparison of theoretical radial abundance distribution formed under the influence of spiral density waves ( solid and dashed lines) for 2 locationsof the corotation resonance with the observed pattern in the case of flat rotation curve. The observed data are showed by filled squares with error bars.c (cid:13) , 1–5 I. A. Acharova et. al.
Figure 5.
The same as in Fig. 4 but for the Keplerian law. c (cid:13)000
The same as in Fig. 4 but for the Keplerian law. c (cid:13)000 , 1–5 imodal abundance pattern in M51: evidence for corotation resonance effects Figure 6.
The same as in Fig. 4 but for the case when the infalling gas abundance is Z f = Z ⊙ / , the initial abundance is Z ( t = 0) = 10 Z ⊙ , the values ofthe rate–constant η are shown in the figure and the corotation resonance is situated at r = 5 . kpc . The observed abundances are given by filled squares witherror bars.c (cid:13) , 1–5 I. A. Acharova et. al.
Figure 7.
The same as in Figs. 4 and 5 but for the case when the infalling gas abundance in the outer part of galactic disc ( r > kpc ) is 10% less than for r < kpc . For the both cases the corotation resonance is situated at r = 5 . kpc . c (cid:13)000