Star formation and abundance gradients in the galaxy

in Figs. lll.Sa pre and b. Moreover, the functional dependences on galactocentric distance R of predicted and observed SFRs is quite different. While the observed SFR attains a very steep peak between R=4 and 6 kpc the predicted SFR is flat and nearly independent on R. How accurate are SFRs based on estimates of Lyc photon production rates? We can try to answer this question by comparing SFRs estimated by our method with SFRs estimated by other independent methods. It is therefore of interest to note that the star formation rate derived by us for the annulus between R=9-]O kpc assuming azimuthal symmetry, ~(9-]O) ~0.5 m e yr -I, is rather similar to an independent estimate by Miller and Scalo (]978), who derived for the solar vicinity ~=(3-7).]O -9 m e yr -1 pc -2, corresponding to ~(I0)=.2-.45 m e yr -I.

111.4

DIFFUSE INFRARED EMISSION AND ITS RELATION TO STAR FORMATION

From the Lyc limit at h=O.O9 ~m to the sub~m region the Galaxy emits a diffuse radiation. From 0.09 ~m to 8 ~m this diffuse emission is dominated by direct stellar radiation; at longer wavelengths, the diffuse emission comes primarily from dust grains which are heated by absorption of stellar radiation. Fig. 111.6, from Mezger (1982), shows the overall spectrum of the Galaxy from the radio range through the UV as seen by an extragalactic 'observer'. The integrated flux density of the diffuse free-free emission is always less than the flux density of the diffuse synchrotron emission, which makes it so difficult to reliably

I

I

I

0"88

I

I

C÷157 0,o63 I II

1

<

-

i",..4Yn

10-4[ lcm

CO610

~/~

/I / / \d~

I lmm

I lO01Jm

"

\

I lOtJm

I llJm k

I 0.11Jm

Fi~. III. 6: The composite continuum spectrum of our Galaxy from the radio to the optical region. The position of the four strongest atomic and ionic lines is indicated. "syn" refers to synchrotron emission, emitted by relativistic electrons gyrating in the galactic magnetic field, "f-f" refers to the free-free emission from galactic ELD HII regions. The dust continuum is composed of contributions from cold dust associated with quiescent molecular clouds (dl) which probably also emit most of the C°610#m line; medium warm dust associated with diffuse atomic hydrogen (d2), which emits the C+157pm line and possibly (i.e. the hot intercloud gas) the O°63pm line; warm dust associated with primarily the ionized gas in ELD HI.[ regions (d3), which is expected to emit the 0++88~m line, too; and hot dust associated with circumstellar shells (d~). At wavelengths <~Spm the stellar radiation dominates the spectrum (from Mezger, 1982).

Star Formation and Abundance Gradients

189

separate the two components. In the mm-range the thermal radiation from dust, heated by stellar radiation, begins to dominate. The stellar spectrum peaks at wavelengths of ~1 ~m. The near IR emission is dominated by the radiation from cool stars (Tef f ~3 0OO K). In the following we discuss the diffuse galactic FIR and NIR emission. While the far IR radiation of the Galaxy seems to be a useful tracer of the OB-star distribution and the presentday SFR, the near IR emission comes from stars now evolving as red giants (with lower-mass progenitors) and thus provides some insight into the history of star formation in the galactic disk. 111.4.1 THE THERMAL FAR INFRARED (FIR) EMISSION FROM DUST The origin of the diffuse galactic sub,m/FIR emission has been discussed in two papers by Mezger et al. (1982) and Mathis et al. (1983). An extrapolation of their results to the FIR emission from external galaxies can be found in Mezger (1982). In normal spiral galaxies we deal primarily with medium warm dust (labelled component d2 in Fig. 111.6) and with warm dust (component ds). Medium warm dust is associated with the diffuse atomic intercloud gas; the dust is heated by the general interstellar radiation field (ISRF). Warm dust is associated with ELD HII regions where most of the heating is provided by O stars. It is this latter dust component which is directly linked to the free-free emission emitted by ELD HII regions and thus allows an estimate of the SFR. Based on the IR-excess = LiR/N~ych~ ~ of our Galaxy, which expresses the IR luminosity of an HII region in terms of the energy available in form of Lyman alpha photons, Mezger (1982) derives a relation between the free-free flux density and the integrated FIR spectrum of dust component d3 ~°'IS (ff) = 1.1 10-16 fS (da) d~ FIR v with ~ in GHz, dv in Hz and S

(lll.16a)

in Jy. Integration of a typical 'ds'-spectrum relates the free-

free flux density to the maximum FIR flux density of component d3 ~°'IS (ff) = 5.5 IO-~Smax(d3)

(Ill.16b)

These relations have been derived using observational results obtained for our Galaxy. To show their usefulness, we apply them to an external galaxy, NGC 253, which is an Sc galaxy at a distance of 3.4 Mpc, with a very high activity of star formation within its central 500 pc. The maximum FIR flux density, at ~1OO ~m, is Sma x ~1OOO Jy. From Eq. (III.16b) follows ~°'ISv(ff) = 0.55 and, substituted in Eq. (III.2a)

N' =5.3 IO~3 s-I. Because of the known ' Lyc high star formation activity it is safe to assume that most of the dust is heated by O stars. We correct for the loss of Lyc photons due to absorption by dust grains or due to escape by writing in Eq. (III.2b) NLy c ~2 N'Lyc' which, substituted in Eq. (III.12) yields a SFR for the central 500 pc of NGC 253 of P(to) ~20 m@ yr -I, for a constant IMF with m ~O.I m , correlov sponding to twice the total present-day SFR of our Galaxy or about twenty tlmes the SFR in its central part. However, as shown by Rieke et al. (1980) for a star burst in M82, various observations suggest mlo w ~3.5 mo, a value which would decrease considerably the SFR in NGC253, i.ei the total mass of gas transformed into stars.

111.4.2

THE DIFFUSE NEAR INFRARED (NIR) EI~SSION

The ridge line intensity 1(£,6=O o) of the diffuse galactic emission at 2.4 ~m as observed by Oda et al. (1979) and Ito et al. (i977) is shown in Fig. III.7. While it resembles the ridge line intensities of both diffuse galactic free-free and FIR emission its interpretation in ~ A

26:a

- C

190

R. Gusten and P.G. Mezger

I 141

I

l

I

I

I

I

I

IT o,. . . . , Ito

et

al

~o e

!

e=6

t

t i

t

+r
20

30

_,__

40

50

GA[. LONGITUDES(DEG)

+ 60

70

BO

Fi~. III. 7: The ridge line intensity at 2.4 ~m as observed by Oda et al. (1979) and Ito et al. (1977) with 0~6x0~6 and 2°x2 ° square beams. Curves and shaded regions refer to model computations with different distributions of the volume emissivity E2.~um in the galactic plane and the extinction model given by Eq. (III.17) and shown in Fig. ~II. 8. The solid curve for £ ~20 ° relates to a scalin~ of the 2.4 pm volume emissivity observed in the vicinity of the sun with Innanen's mass model. The solid curve for £ <10 ° comes from a model by Serra et al. (1980) which also includes contributions from stars in the nuclear bulge. The shaded area refers to our model, where the excees ridge line intensity between £ ~5~ and 50 ° is provided by evolved stars (M giants) with masses 3 ~m/m@ %11, whose distribution follows that of 0 stars and whose total luminosity is consistent with the present-day SFR. The upper envelope of the hatched area is obtained from a slightly modified extinction model, which takes the patchiness of the CO-distribution into account. Giving less weight to the molecular gas component Eq. III. 17 is changed to 2.9 nHI.Z(R)/Z(IO) + O.07.E(CO). terms of volume emissivity as a function of

.~

1,0

I°

position in the galactic plane is complicated by extinction due to interstellar dust. Considering the contribution of dust associated with both the diffuse ISM and with molecular clouds, respectively, we use the relation

'

2.0

'

I

'

, •

T2.&pm=1.6~ ~ " - . - . . . , . . ~

.G.C.

/,"

kv(R)

-

= 1.875 nHI(R)Z(R)/Z(IO)+O. I1ER(12CO) (iii.17)

1.2\\,

"~~ ' ~

_

between visual extinction k v i n magnitudes per kpc, mean density of atomic hydrogen nHl in cm -3 and 12C0 emissivity g in K km s-1 kpc -I.

\

\--

OA

This relation assumes that the dust-to-CO -

--

-

SUN

ratio is independent on the gas metallicity, while the dust-to-HI ratio increases as md/mg =Z(R). Substitution of nHl and E(I~CO) from Table 11.2 yields an extinction at 2 . 4 ~ m

Fi~. III. 8: Curves of constant extinction optical depth, T 2 ~,_ in the galactic plane, as seen from the posztzon of the sun. Computed with Eq. (III.17) and T2.~m=Tv/13.6. •

~

in the galactic plane as shown in Fig. 111.8.

•

To convert visual extinction into extinction at 2.4 ~m we use the relation T2.~=Tv/13.6.

Star Formation and Abundance Gradients

191

The diffuse galactic NIR emission in the solar neighbourhood is mainly attributed to normal M-type giants (Tef f ~3000 K) with low-mass progenitors m ~<1-2 m®

(Hayakawa et al., 1978).

Scaling their emissivity at 2.4 pm with the galactic stellar mass distribution (e.g. Innanen (]973)), and considering the extinction in the galactic plane according to Eq. (111.17) and Fig. 111.8, yields the ridge line intensity shown as solid curve in Fig. 111.7. This predicted curve fits the observations only for galactic longitudes larger than 50 ° and less than 5 ° , but fails completely to account for the observed ridge line intensity between these longitudes. This fact was already noted by Maihara et al. (1978) and Hayakawa et al. (1977; 1978), who suggested the existence of an excess-population of M type giants and supergiants, distributed in a ring between galactic radii 4
~I0 s yr ago - was not much different from the present-day SFR.

With Tg the time which a star of mass m spends in the red giant (g) phase (Tg ~0.1 TMS) and ik(m,Tg)

the corresponding luminosity at ~=2.4 lJm (referred to by the letter k), the present

2.4 ~m luminosity per stellar mass is TMS-Tg Lk(m)dm = dm f #(m)~(to-t)Ik(m,t)dt , J

TMS

or, with Tg <
Lk(m)dm = dm ~(m)~(to-TMS(m))/~(m,t)dt

One sees immediately that the 2.4 ~m luminosity per unit stellar mass depends on (i) the galactic history, which determines the number of stars now evolving along the red giant branch, (ii) the "slope" of the mass spectrum, which may alter the contributions from different populations (e.g. giant versus supergiants) (iii) Lk(m) =Tflk(m,t)dt, the emission of a star of mass m, integrated along its post-main sequence

g evolutionary track.

The 2.4 ~m luminosity contained in the excess component (i.e. the difference between full and hatched curve) in Fig. 111.7 is ~109 L 0 ~m -I , assuming radial symmetry in the stellar distribution. According to Becker's (1982) closed evolution tracks from the zero-age main sequence up to the asymptotic giant branch, the post-main sequence integrated 2.4 pm luminosity Lk(m) is roughly independent on stellar mass (i.e. within a factor of two) for 3
192

R. G~sten and P.G. Mezger

Lk(m) =

/

ik(m,T)dT ~IO 9 L@ ~m -I yr.

post-MS

The integrated luminosity of a given stellar population is then determined mainly by #(m)dm, the number of stars in the mass intervall (m,m+dm). From Eq. (III.|O) follows

[~] Lk

~60 =/Lk(m)dm =2.5.]OS.~(to). (m~ I"" -0.02)

(iii. 18)

me

for I Smc S I0 m e . With ~ ~13 me yr -I and mc=2(3) me the above equation yields L k ~10(6)

]Os L® ~m -I and a total luminosity of L(red giants) ~6(4) ]09 L@, if these red

giants emit like a 3000 K blackbody. This means that a "medium mass" red giant population (i.e. a population of red giants which have evolved from MS stars with masses m ~mc=2(3) m@), whose SFR about 109 ago was similar to the present-day SFR, can account for the luminosity of the 2.4 ~m excess ridgeline intensity shown in Fig. 111.7. Moreover, and as shown in this Figure by the shaded curve, such a population of medium mass red giants gives a reasonable fit to the observed 2.4 ~m ridgeline intensity, too, if i) the radial distribution of the medium mass red giants in the galactic plane resembles that of the present-day 0 stars; and ii) the 2.4 ~m emission from this medium mass red giants is superimposed on a contribution due to old disk population M giants (shown in Fig. 111.7 as (interrupted) solid curve).

IV.

ABUNDANCE GRADIENTS IN THE GALAXY

Abundance gradients were first detected in nearby spiral galaxies from optical spectroscopy of HII regions (see Pagel and Edmunds (1981) for recent review). Naturally, due to extinction by interstellar dust, such observations are much more difficult to make for HII regions located in the galactic plane. Therefore attempts to establish galactic abundance gradients in the galactic plane by means of optical spectroscopy are limited to "local" sources. Observations of radio recombination lines, which are of relevance for the determination of He and O abundance gradients, do not suffer from extinction and therefore allow surveys of the entire galactic plane. In the future, FIR fine structure lines will certainly play an important role in determining galactic abundance gradients. The main purpose of this paper is the development of a model for the chemical evolution of the Galaxy, which can explain the evolution of abundance gradients in the galactic disk within the constraints of gas-to-total mass ratios and present-day SFR's as derived by radioastronomical observations and discussed in the previous sections. Early work related to both optical and radio observations has been reviewed by Peimbert (1978). The most recent data on galactic abundance gradients are giyen by Shaver et al. (]983). In this section we discuss briefly the galactic abundances of tHe and leO as examples of primary elements, and the gradients of isotopic abundances of H, C and O.

Star Formation and Abundance Gradients

IV.l

193

ABUNDANCES IN THE INTERSTELLAR MATTER

Results of observationally determined number abundances and abundance gradients (excluding the inner 3 kpc region) are compiled in Table IV.I, which otherwise should be self-explanatory. In the following we discuss briefly some problems related to observations of abundance gradients.

IV.].]

ABUNDANCE GRADIENTS OF aH AND tHe

From our present knowledge of nuclear synthesis all aH and most of the tHe have been synthesized in the big bang about three minutes after the expansion of the universe has started. Since the fusion of IH into tHe is the main source of energy of stars, we expect an enrichment of the ISM in tHe as a consequence of the chemical evolution of the Galaxy. At any position in the galactic disk the total He abundance (in mass fraction) is

Y(~He) = Y

p

+ bY

ev

(IV.I)

with Y

the primordial abundance and AY the enrichment of the ISM due to chemical evolution. p ev Reasonable estimates of the primordial He production fall in the range Y ~0.22-0.25. From P the estimated abundance in the solar vicinity, Y1o ~0.28, follows AY ~0.06-0.03. ev First, due to the fact that AYev is much smaller than YP , determination of the variation in the stellar contribution to the present helium abundance requires an accuracy in the measure-

ment of the total abundance better than 0.005 (abundance by number). Secondly, in the HII regions, which are used for most He abundance determinations, He is in part neutral and in part singly ionized. Both radio and optical spectroscopy yield, however, only the ionized He abundance, while the fraction of neutral He has to be estimated indirectly. It has been found that (probably as a result of an increase in the metal abundance) the fraction of ionized helium in HII regions decreases towards the galactic center. At galactocentric distances R ~8 kpc, this latter effect begins to dominate over a possible increase in the total He abundance as discussed in Mezger (1981) and Mezger and Wink (1983). In Table IV.l and subsequent discussions, ~e use the the abundance gradient as derived by Thum et al. (1980) for 8 SR ~12 kpc. Deuterium (2H), on the other hand, is a very weakly bound nucleus and is destroyed if exposed to temperatures of ~IO e K for sufficiently long times. One expects therefore that all ~H, which has been circulated with the ISM through stars ("astrated") since the formation of the Galaxy has been destroyed. Since there is no astrophysical site known (other than the big bang, of course), where 2H could be produced in substantial quantity, one would expect that the depletion of the ISM in =H is proportiQnal to the enrichment in tHe and heavier elements. Hence, a SH gradient should have a different sign from that of the primary elements. The interstellar deuterium abundance is determined locally only by UV-observations of the diffuse gas. Due to the strong chemical fractionation of the deuterated species in cold molecular clouds, no quantitative D/H abundance ratio can be determined from molecular line studies. Recently, Vidal-Madjar et al. (1983) re-interpreted DI UV-absorption lines toward nearby luminous stars, and concluded that (due to contamination of the Dl-line with blueshifted circumstellar HI-lines) the local interstellar D/H abundance is probably ~5 I0-e, rather than ~2 IO-s as previously discussed.

194

IV.I.2

Ro G~sten and P.G. Mezger

ABUNDANCE GRADIENTS OF 160 AND OTHER ABUNDANT PRIMARY ELEMENTS

One of the basic problems in determining absolute abundances of the elements of the C,N,0 group in the ISM is posed by depletion. It is estimated that about l/4 to l/2 of the total mass fraction of these elements is used to form dust grains. While the cores of these dust grains appear to be made primarily of graphite and silicates, mantles of molecular ices tend to form on these cores within molecular clouds. As an effect of chemical enrichment during galactic evolution one would expect that the abundances of C,N,O in the ISM in the solar vicinity should be higher than the corresponding abundances in the solar system (which decoupled from the ISM about 5 lO 9 years ago). The fact that the observations compiled in Table IV.l show the reverse trend is probably due to depletion. Large-scale oxygen abundance variations have been determined from gradients in electron temperatures in HII regions, measured by radio recombination lines on the assumption that oxygen is the main coolant of the plasma (see e.g. the review by Churchwell, 1980). A strong increase in the metallicity of the ISM with steepening gradients towards the inner Galaxy is implied by these studies (Mezger et al. 1979). A recent combined optical and radio-based study of Hll-regions confirms these results, with

dlog(0/H)/dR~O.07(±O.OI5) dex kpc -1 for R >7 kpc,

by Shaver et al. (1983). The gradient in I~N/H, found in the latter investigation, appears to be considerably steeper. Due to the lack of suitable transitions in the optical range, there is little information on the large-scale distribution of the carbon abundance.

IV.I.3

ISOTOPIC ABUNDANCE GRADIENTS

Relat~ue,

isotopic abundances can be inferred from molecular line studies of the neutral gas

component. However, although probably not affected by ("selective") depletion onto dust grains, the line formation process depends strongly on the physics of the molecular cloud. Line saturation and chemical fractionatlon necessitates sometimes large corrections to the observed line ratios.

Observations of carbon and oxygen isotope ratios, as inferred from several molecular species (mainly CO and H2CO) have been reviewed recently by Penzias (]980) and Wannier (1980). Since in particular the CO line formation suffers from optical depth and fractionation effects, in the following we compare our model predictions with [12C/1sC] ratios obtained from H~CO. The formaldehyde absorption lines mainly trace higher density regimes of molecular clouds, where fractionation may be less effective. The centimeter lines of H2C0 have relatively small optical depths, which (in principle) allows a reliable derivation of the column densities. For a discussion of uncertainties (mainly due to photon trapping in the optically thick mm lines) see Henkel et al. (1980, 1983). According to these data, the mean

[12C/13C]

abundance ratio

in the local disk is 70-75, only slightly below the solar system value of 89. Excluding the galactic center sources, a small gradient in [12c/1Sc] is indicated. In any case this gradient is smaller than the large-scale variations in 160 and I~N. Due to the weakness of the rare isotope line, very few sources have been measured so far in the H2clS0 line, yielding a somewhat lower than solar system [160/lS0] abundance ratio. A positive galactic gradient is suggested by the low galactic center value. [16c/lS0] ratios based on CO observations (namely the double-isotope line ratio [12C180/13C160]) are inconclusive, as it is difficult to correct for carbon fractionation and saturation effects.

(2)

4.2(~;~).I0 -4

References:

I~N/IS N

(6) (7) (8) (9) (10) (11)

Henkel eL a i . (1979) P e n z l a s (1981) Wannier eL a l . (1981) P e n z i a s (1980) B i n e t t e eL a l . (1982) Vidal-Madjar eL a l . (1983)

saturation, fractionaL±on, use of double isotope ratios (HCN) molecular (HCN)

(14)

>(500)

(12) (13) (14) (15) (16) (17)

unobserved ionisation states, depletion shock excitation model (SNR)

optical ([NIl])

(11) (16)

?

molecular (H2CO,CO)

inferred indirectly only from T e depletion onto grains

r a d i o recomb. (HII) optical ([O11],[O111])

molecular (CO)

(12)

(1o) (11)

(9)

saturation, fractlonation

depletion onto grains, unobserved ionisatlon states

optical (ell)

(8)

m o l e c u l a r (CO,H2CO)

correction for unobserved neutral Hel contamination from PN-progenitor

r a d i o recomb. ( H e l l ) optlcal (Hell)

(8)

(7)

(6,15) (17)

strong chemical fraetionation blending with blueshifted circumstellar lines

Difficulties in Interpretation

molecular (HCN,DCN) atomic (H,D)

Observed Lines

3.5(±0.2) (13)

15o(±50)

?

20±5

?

?

?

Galactic Center

Bruston eL a l . (1981) Thum eL a l . (1980) Pelmbert (1978) Henkel e t a l . (1982) Mezger eL a l . (1979) Shaver eL a l . (]983)

(positive)

4(300)

(I)

-O.O9(±O.O15) -0.095(±0.03)

4-IO -s

(3) (2)

_<0.01

3.7

positive

- 0 08 _O[O7 (±0.015)

5 ( ± 1 ) . 1 0 -~

~3OO

<0.05

75(±5)

(i)

(I)

(3) (2)

(I)

Cameron (1980) Ross + Aller (1976) Lambert (1978) Hirayama (1971), Heasley + Milkey (1978) Kunde eL a l . (1982)

272

l.O(t.2).lO -~ . 9 ( ± . 3 ) . 1 0 -~

5.4

lSO/lV O

~NIH

490±25

8.3(+2).10 -~ 6.9(±1.3).10 -4

1sO~leO

160/H

12C/13c

(I) (2) (3) (4) (5)

(3.10-")

(3)

4.7(±1.I),10 -~

I~C/H

89±2

-0.015 -0.020

~0. I0

(4)

0.05-0.10

~He/H


positive

~2 10-s 5 10-6

(5)

3.6(~:~).I0 -s

2H/H

Disk-Gradient [dex kpc -1 ]

[OMPILATION OF SOLAR SYSTEM AND INTERSTELLAR ABUNDAN[ES

Local ISM

Isotopic Species

Solar System

TABLE I V . 1

Ln

e~

,.~ o

196

R. G~sten and P.G. Mezger

This is different for the determination of the [ISo/IVO] ratio, which according to singleisotope measurements in the optically thin 12C17(Ia) 0 lines is significantly below the solar system value and is rather constant over the galactic disk (R ~6 kpc). The intrinsic scatter in these data is small

compared with carbon-isotope measurements. This emphasizes the im-

portance of carbQn fractionation in the CO lines; for the oxygen isotopes no such fractionation process is expected (Watson,

IV.2

1979).

THE SOLAR sYSTEM ABUNDANCE AND ITS RELEVANCE FOR GALACTIC EVOLUTION MODELS

For comparison in Table IV.I, the composition of the pre-solar nebula is given. The abundances of the C,N,O-group are derived from measurements of the solar photosphere and of quiet prominences. Their isotopic ratios are from terrestrial or from meteoritic samples. The C] carbonaceous chrondrites are believed to be representative for the early solar system material (Meyer, ]979). There is considerable uncertainty in the photospheric abundances, mainly due to uncertain oscillator strengths and deviations from LTE. A reliable deuterium abundance is difficult to derive from the solar atmosphere (astration) or from terrestrial determinations (gravitational fractionation). In the Table, we give the value for D/H determined recently in Jupiter's atmosphere. Are the solar system abundances representative of the ISM composition 4.6 Gyrs ago, the time when the solar system decoupled from the interstellar gas? This question relates to the homogeneity of both the solar system material and the general ISM. Despite its well-known heterogeneities, as observed e.g. in the oxygen isotope ratios in various types of meteoretic material (Wasserburg et al. ]979), within a few percent the relevant composition of the primitive solar nebula seems rather homogeneous and well-mixed. In our chemical enrichment models of the Galaxy we shall thus assume that the bulk composition of the solar system is representative for the composition of the ISM 4.6 Gyrs ago. A matter of controversy is still the question of whether there exist chemical inhomogeneities in the interstellar medium, i.e. if element abundances vary from cloud to cloud

and from star to star. There is ~25% disper-

sion in the metallicity distribution of stars with the same age (Twarog, ]980), and obviously there are source-to-source variations in the molecular abundance between CO and H2CO and their isotope substituted species. But the latter may well be attributed to errors in correcting for line saturation and chemical fractionation. In the following

we assume that within a galactic

annulus R, R+AR the ISM is well mixed. (For a qualitative discussion of elemental mixing in the ISM see Chevalier (]979)).

V. A SELF-CONSISTENT MODEL OF STAR FORMATION AND CHEMICAL EVOLUTION OF THE GALACTIC DISK In Sect. I. we compared predictions of the closed system model with constant IMF with observations related to both element abundances and SFRs. We found that this model yields reasonable agreement with observations if applied to abundances in the solar system and the ISM in the solar vicinity. Severe disagreement between this model and observations arises, however, if the predictions are compared with observations related to: i) the radial element abundance distribution in the galactic plane (Sect. IV); ii) present-day SFRs and especially its radial dependence in the galactic plane compared to the mass distribution (Sect. 111.3.3); iii) the time evolution of abundances. Several attempts have been made to construct models of the chemical evolution of the Galaxy in order to explain the observed abundance gradients. These models are, to our knowledge, not

Star Formation and Abundance Gradients

197

consistent with the observed SFRs, net SFRs, mass distributions and abundance gradients. In this section we present a model, which we believe, avoids these difficulties.

V.l

MODIFICATIONS OF THE CLOSED SYSTEM MODEL

We introduce two modifications viz. a variable yield and a slow build-up of the galactic disk by accretion of halo gas (which leads to the well-known class of infall models).

V.l.l

A YIELD WHICH VARIES WITH GALACTOCENTRIC DISTANCE

The existence of a galactic abundance gradient in the C,N,O elements is well established (Sect. IV). However, as mentioned in Sect. 1.3 and shown in Fig. V.5, the closed system model fails to explain these gradients if the gas-to-total mass ratio, ~(R), from Table 11.2 is substituted in Eq. (1.4). As discussed in Sect. II there are relatively large uncertainties in the estimated molecular abundances, and other investigations yield higher densities . This tends to flatten ~(R) and hence the predicted metal enhancement in the inner disk would even be decreased. There is no way that closed system models with constant yield can produce the observed abundance gradients. Can infall models explain these gradients? Tinsley and Larson (1978) have shown that infall models ~an produce a modest gradient of primary element abundances during periods in the evolution of galactic disks. This case, however, requires high accretion rates which must vary strongly with galactocentric distance. In systems in highly evolved (saturated) stages with reduced infall rates, these gradients tend to flatten. If applied to the present-day state of evolution of the galactic disk, with small accretion rates and a rather constant degree of gas depletion ~(R), these models can explain neither the observed abundance gradients nor the SFRs as derived for constant IMF in Sect. III. But nevertheless any realistic attempt to follow the galactic enrichment with time has to incorporate the dynamical history of the disk, with its slow build-up by accretion of settling halo gas (see below). Therefore we include in the following discussion of the chemical evolution of the galactic disk infall of halo gas, too. We want to reiterate, however, that it is not the infall which is used to explain the observed abundance gradients across the disk. The only way to account for e.g. the strong leo gradient is to introduce a variable yield Yi(R) which increases towards the inner part of the disk. To make this modification more transparent, we discuss its consequences first in terms of the simple closed system model. The yield used in Eq. (1.4) is given by (Eq. C.l) as yi=Pi/(l-r), with m up Pi = f $(m)Am(Zi)dm

(V. la)

mlow the fraction of mass of a newly synthezised element Z. returned by a stellar generation to i the ISM, and m up r

=

f

m ~(m)(m-mf)dm

=

~P~(m)r(m)dm

(V. lb)

the dimensionless mass fraction returned by a stellar generation to the ISM, with mf the final mass of the stellar remnant, which can be a white dwarf, a neutron star or, possibly, a black hole. r(m)=(m-mf)

is the mass returned by a dying star of mass m to the ISM.

198

R. Gusten and P.G. Mezger

Results derived in the previous

sections were based on the assumption of the universality

the local IMF, which has been derived from star counts in the solar vicinity. ever, a number of observational hypothesis,

indications which have led Mezger and Smith (1977) to the

that medium and high mass stars form predominantly

mass stars form predominantly

of

There are, how-

in the interarm region.

lished which supports this hypothesis.

in spiral arms, while low-

Since then further work has been pub-

Miller and Scalo

(1978) conclude that in OB associa-

tions (which are the later stages of giant radio HII regions and, like these, are well-known spiral arm tracers)

probably stars with m 52-5 m O form only. A deficiency of low-mass

there is found by Eggen (1977) and Larson

(1982). T-Tauri stars and Herbig-Haro

are known as tracers of low-mass star formation. in T-Tauri associations,

as witnessed

by the presence of early B stars in the 0-Ophiuchus

sites of formation of low-, intermediate-

cloud.

and high-mass

of the different

In fact, there may be different stars and the local IMF would then

star formation processes.

the details we try as a first and certainly crude approximation scenario of star formation a quantitative

formulation

to in the following as hi-modal star formation:

Without knowing

to this rather complicated

of the above hypothesis, which we refer

There are two different

which describe the integrated star formation processes respectively.

objects

Only a few medium mass stars appear to form

cloud or even some 0 stars in the Orion molecular

be the result of a superposition

stars

IMFs, ~sa and ~ia,

in spiral arms and interarm regions,

We assume that both IMFs have the same slope (for which we adopt the Miller-

Scalo IMF as given in Sect. III.3.1)with a lo~mmass cut-off for the interarm region IMF at mlow=O.l m®. The deficiency of low-mass

stars in spiral arm star formation

by an increased low-mass cut-off of the IMF, mlow--mc >] m®. Some theoretical tend to support the picture of bi-modal therein).

Obviously,

spiral arms and in the interarm region. et al.

star formation

the physical conditions

(]980), in an observational

is described

investigations

(see e.g. Silk (1980) and references

in star forming clouds are quite different

in

In this context it is of interest to note that Rieke

investigation

of the physical

conditions of a star forma-

tion burst in the center of the galaxy M82, find that the IMF has to be truncated at mlow~3.5 m@ in order to match both the total IR luminosity,

the Lyc photon production rate

and the absolute magnitude at 2.2 ~m. It thus appears to be a characteristics like" star formation

of "burst-

(and as "burst-like" we can also consider the increased SFR in spiral

arms) that it produces primarily medium and high mass stars. The net effect of bi-modal low-mass

stars relative

hence results

star formation is an increase of the ratio of massive

stars to

to the local IMF, which increases the effective bulk yield Yi and

in a higher enrichment

of the ISM. Since the ratio of massive

spiral arms to massive stars formed in the interarm regions

stars formed in

increases with decreasing galac-

tocentrlc distance R, the effective bulk yield Yi(R) increases with decreasing R, too. In this way abundance gradients will form even if the gas-to-total increases

mass ratio ~(R) is constant or

slightly towards the galactic center. A quantitative

yield due to bi-model

star formation

description of the variable

is given in Sect. V.2.3. Here we want to show in rather

a qualitative way why the bulk yield Pi Yi = 1----~

increases with increasing synthesized

(V. 2)

low-mass cut-off of the spiral arm IMF. P i, the fraction of newly

element Z.I ejected into the ISM will not be affected until mlo w approaches

Stars with lower mass have MS lifetimes do not contribute

~I m®.

exceeding the age of the galactic disk and therefore

to the chemical evolution of the ISM. Fo r mlo w >I m®, as can be seen from

Fig. C.I, the production of secondary elements

like ISC and I~N will first be reduced. With

Star Formation and Abundance Gradients

199

increasing mlow, Pi of primary elements will be reduced, too. (l-r), the fraction of lockedup matter, on the other hand will steadily decrease With increasing mlo w. Evaluation of Eq. (V. lb) yields for mlow=O.l,

I and 5 m® the corresponding fractions (l-r)=0.6 , 0.23 and 0.13,

respectively. The net effect on the bulk yield (Eq. V.2) is that Yi rises sharply with increasing mlo w for primary elements from massive progenitors, but tends to flatten out for secondary elements provided predominantly by medium mass stars with m !5 m®, once mlo w >] mQ. Another desirable effect of bi-modal star formation is an increase of the ratio of Lyc photon production rate NLy c to gas consumption rate ~ for a new generation of stars. This ratio NLyc/~=~(1)P(#)

is given in Eqs.(A.3a and b) as a function of mlo w. An increase of this com-

puted quantity means that for an observationally determined Lyc photon production rate both the SFR and the net SFR (i.e. the mass of gas converted and permanently locked-up stars) is decreased. This alleviates the problem of the high SFRs derived in 111.3.3 assuming a constant IMF.

V.l.2

A REALISTIC MODEL OF THE GRADUAL BUILD-UP OF THE GALACTIC DISK BY ACCRETION OF HALO GAS

Though fragmentary, our knowledge of the history of the early Galaxy~ as obtained from the analysis of older population objects (globular clusters of the halo-, and open cluster of the diskpopulation) yield strong chronological constraints on the early dynamical and chemical evolution. Despite rather large uncertainties, nearby globular clusters(6 = o~/tdisk ~8 mOPC ~= Gyr-I, as compared to a present-day rate A(t o) ~2 m®pc -a Gyr-1.This is derived from a study of highvelocity HI clouds (Oort, 1970). But even this value is disputed since these high-velocity clouds may not relate to halo gas at all, but rather represent cooled and condensed debries of hot coronal disk matter (see Sect. II.I.I), which has been ejected into the galactic halo as suggested by the galactic "fountain model" (see York (1982) and references therein). Based on X-ray observations of the galactic corona Cox and Smith (1974) derived a present-day accretion rate which is about ten times lower than the above value. The infall pattern, or more precisely the ratio between net SFR and accretion rate

k(t) = (]-r)~(t)/A(t)

(v.3)

determines the main characteristics of infall models. Owing to the weak observational constraints on k there exists in the literature a wide range of solutions. These are summarized in Table V.I.

200

R. Gusten and P.G. Mezger

TABLE V.1

CLASSESOF INFALL MODELS Cheracteristic~ of model

~ccretiou £ 1 ~

Ys

A~

mate1 ~ r i ~ t

e

esyaptotic solution

u*O Xscc=to

Z'Yi [ I - e (1-~°*)]

Larson (1972)

constant

TwerOK (1980)

(y,A)conit,

slow, still c o n t i n u i n 8 accretion

Tosl (1982)

Tact=t°/2

slow, but l a r s e l y completed disk formation process

Zt(Yi.k÷~)" IO

$

e~p. decrease xeec~5 G~rs

~3

~2

Gdst~÷Hezger (1983)

uon~xp, decrease kl
~5

e.g. Tinsiey (1980)

closed system ("ei~le model'*)

Vader+deJong (1981~

Yi'~+Zh (Zh'O)

1.6(±5).y i

Chiosi (1980)

! &cc kl-O

rapid disk formation

~.2

Equstion (V.ll)

yi[l+ln ~]+K.zh

Z=Zo*yi ln~-~

(S. Yi)

t f o r primary non-depleted nttclel like leO. ~scc " tlmescsle of disk formation by accretion o£ halo p s , t o - "age" of the disk, ~I(A,) present-day local 8tar f o ~ t i o n

(secretion rata) i n [ ~ pc -z GYm-I], ~ - ( ] - r ) ' ¥ / A

These models range from slow and still continuing accretion (Tac c ~tdisk) with the accretion time scale comparable to the age of the disk, to a rapid (instantaneous) formation (Tacc << 0.5 tdisk )which leads to the initial conditions of the closed system model discussed in Sect. I. Intermediate are models that take into account a slow accretion process which, however, is presently completed and which have typical accretion times of ~5 Gyrs (T

acc

%0.5 tdisk ).

Another model parameter which has to be determined from observations is Zh, the metallicity of the infalling halo gas. Models of neighbourhood (Fig. V.3 ) constrain

the age-metallicity relation of stars in the solar the pre-enrichment of the halo gas to Zh S0.2 Z O. Thus

we neglect this effect in discussing the present-day abundance gradients in the disk. In realistic infall models, ~(t) is time dependent. We are doubtful about the reality of infall models which assume both constant SFRs and constant k (and hence a constant infall rate), as used by Twarog (1980) and Tosi (1982). In our model, we use a convenient analytical description of the mass flow pattern first suggested by Lynden-Bell (1975) (Moo-M~) (M~+M o) Mg(t) =

(V.4) (No+M)

It

is based on the assumption, that in a typical accretion model the

gas mass in the disk, Mg(t), rises from an initial threshold mass M ° < O . 1 M formation begins, to a maximum

, for which star

and finally declines to zero, if the mass converted into stars,

M~, approaches Moo, the asymptotic mass of the disk

{MD=CMg+M~)+Mco}.

Combining Eq. (V.4) with a relation between SFR and available gas mass in a form similar to Eq. (1.2) yields a relation,

| 1-r

dM~ dt

~(t) = T] k ( t )

(V.5)

Star Formation

!

i

and Abundance

1,0

M~

/./,/

M.

A I

I

I

~

d N

d%

.3

/ .// ' - X -

2

"\

-

\'x

/"

.1

= which connects

~

.0

X

I

I

3

a

I

6

accretion with

star formation and hence determines

R

I

I

201

Fig. V.I: Application of Lynden-Bell's '~est Accretion Model" to the gradual build-up of the galactic disk. A linear dependence of the SFR on the gas mass i8 assumed (i.e. k=1 in Eq. V. 5). In the upper diagram are shown the relative variations with time of total disk mass (MD), matter transformed into stars (M~) and mass of gas (Ma). As dotted line is shown the present-da~ local gas-to-total mass ratio u(t o) = 0.04. In the lower diagram are shown accretion rate (d/dt)MD and net star formation rate (d/dt)M~. i8 the star formation efficiency. From the intersection point of v(t o) with Ma/ M and with an adopted age of the galaDt~c disk of to=12 Gyr we determine ~= 5 10 -I°. In the right hand corner of the lower diagram are shown observational estimates of the accretion rate (HVC stands for high velocity clouds; XR stands for X-ray observations). The shaded area refers to the SFR in the solar vicinity.

i

M,,,/ ."

Gradients

5

,

the

time-scale of the build-up of the disk.

,t

In Eq.

I

12 t ISYRS]

(V.5), ~ measures

tion efficiency";

AGE OF THE OISK

the "star forma-

k, the power law ex-

ponent may vary between

I and 2 (Sect.

III). For k=l Fig. V.! shows the variation of the main parameters Bell's as function of a normalized present gas-to-total tdisk=to=]2

time Nt.

of Lynden-

"Best Accretion Model"

(Eq. (V.4))

The dotted curve in the upper diagram refers to the

mass ratio N(to) of the local galactic disk. Adopting an age of the disk

Gyr we determine

from this diagram h=5 ]0 -I° yr -I . The model shows satisfactory

agreement with observed constraints

on the local SFR as well as the X-ray limit on the accre-

tion rate. The major peak in the SFR occurred only 6 Gyr ago. This is consistent with the mean age of the stars in the solar neighbourhood.

For k=2 the phase of major disk accretion and

star formation would shift toward earlier epoches,

but still yields a reasonable description

of observations.

V.2

THE BASIC SET OF EQUATIONS

V.2.!

DESCRIBING CHEMICAL EVOLUTION

A RIGOROUS FORMULATION

In Sect. I we gave relations and instantaneous

for the chemical

evolution of a closed system with constant

return rate. Here we drop these simplifying

and derive equations, which describe

assumptions

IMF

and approximations

the chemical evolution of a galaxy for the general case

with infall, delayed mass return from evolving stars and bi-modal

star formation.

The net

SFR is 60m 8 d d--~ M~(t) = ~(t) - f ~{t-T(m)} m(t=r)

with ~(m) the mass of stellar material

*(mlr(m)dm

that is re-ejected

(V.6)

into the ISM at the star's death,

and T(m) >TMS the total lifetime of the star (Fig. A.2). m(t=T)

is the mass of those stars

202

R. Gusten and P.G. Mezger

which reach the end of their evolution at t=T. In the following we assume that stars with MS masses m ~5-8 m 0 end as white dwarfs with typical final masses ~0.6 m O (K~ster and Weidmann, 1980) while the remnants of higher mass stars are neutron stars with ~1.4 m®. The he/ g ~

consumption rate is

ddt Mg(t) = A(t) - ~t M~(t)

(V.7)

In the general case A(t)=dMD/dt allows for any type of mass exchange; in the following A(t) represents the accretion rate of halo gas on the galactic disk. The net enrichment of the ISM with an element Z. is i

d d--{ [Zi (t)Mg(t)] = -Zi(t)P(t) + Zh(t)A(t) 60m0 + f P{t-T(m)} ~(m)~r(m)-d(m)}Zi{t-T(m)}+ m(t=T) =

(v.8)

Am(Zi~

for primary element i

{ AmP~i)

with Am(Zi) AmS(zj)'Zi(t-T (m))

for secondary elements j with seed elements i.

Here the superscrlpt'p' refers to primary elements and 's' refers to secondary elements, while the subscript h refers to the halo gas, i.e. Z h is the metal abundance by mass of the infalling halo gas. The material returned from an evolved star can be enriched in newly synthesized element i (Am(Zi)~)

but it can also be depleted if the element j has been destroyed in some of the

nuclear reaction chains in the stellar interior.

In this latter case the depleted mass is

d(m)'PZ.~t-T(m)} with Z~(t-T) the abundance of the ISM at the time of the star's formation. 3 As a secondary element requires for its synthesis a seed element j its production is =Z.(t-T). 3 V.2.2

THEINSTANTANEOUS

RECYCLING APPROXIMATION

The delay T between the formation of a star and its final evolutionary stage where it returns its outer shells to the ISM

complicates Equations (V.6 through 8) in such a way,that exact

solutions can be obtained numerically only. Neglecting this time delay leads to the "instantaneous recycling approximation" and to approximate analytical solutions which are often very useful.

In this approximation Eqs. (V.6 through 8) become

dM~

dt dMg dt

=

( l-r)V

(V.9a)

=

A-(l-r)~

(V.9b)

d d-~ (ZiMg) = Pi ~-(1-r)Zi~+ZhA

(V.9c)

with (l-r) the mass fraction locked-up permanently in low-mass stars and stellar remants and the return rate r, as defined by Eq. (V. Ib). Pi is given by Eq. (V. la). Without considering accretion (A=0),and ~(t=O)=l,this set of equations has the simple solutions given in Sect. I for the closed system model. Retaining accretion, A(t) leads, after substitution of Eq. (V.9b) into Eq. (V.ga), to a "tlme-independent" formulation of Eq. (V.9c)

Star Formation and Abundance Gradients

d d d--~ (Z Mg) = yi-Zi+Zh~-~

(MD)

203

(v. 10)

With the accretion law Eq. (V.4) and Zh = const, this equation has the solution

= (M°o~M°)

IMP+ (M+Mo) in( I-

]

M. ---~--o,

(W.l la)

In late phases of the evolution of the galactic disk, when infall has stopped, Mg÷Moo-M~ and Eq. (V. 1 la) approaches asymptotically

Z(M~)-~y i

[l+ln~a-1] +Z h

(V. llb)

In Fig. V.2 solutions using the instantaI

I

I

I

PRIMARY NON-OESTROYED ELEMENT ENRICHMENT

neous recycling approximation are compared with numerical solutions of Eqs. (V.6-8)

k--2

1.0

_~- .

in Eq. (V.5). Solutions shown in Fig. V.2 are normalized to present-day abundances

'l-S/

///

for two values of the powerlaw exponent k

(i.e. ~=O.O5, to=12 Gyr and adopting Zh=O)

j

and show the relative enrichment of the

w

galactic disk with a primary non-destroyed

.S

z

iI

#.

element (such as, e.g. 160) as a function

f~

of time. We see that the instantaneous re-

INSTANT. RECYCLING

cycling approximation first tends to underestimate the enrichment of the ISM. The

I 3

I I 6 9 AGE (GYRS)

t 12

reason is that in this approximation it is assumed, that all stars with mass m 51 m® reeject the mass r(m) instantaneously after their formation. In reality, however, a

Fig. V. 2: Gomparison of numerical (exact) and analytical solutions of Eq. (V. 11) for the enrichment of the ISM as a function of time with a primary non-depleted element. For the analytical solution the instantaneous recycling approximation is used. Parameter k relates to Eq. (V. 5). Computed abundances are normalized for their Values at to=12 Gyr.

significant fraction of gas is locked up in medium-mass stars during the early evolutionary phases of the galactic disk. Hence the rigorous solution first yields a higher enrichment of the ISM. Later, when low-mass stars begin to eject their enve-

lopes of virtually unprocessed matter to the ISM, further enrichment of the ISM proceeds much more slowly than in the instantaneous recycling model.

V.2.3

A VARIABLE YIELD AS A CONSEQUENCE OF BI-MODAL STAR FORmaTION

Bi-modal star formation and its effect on the yield have been discussed qualitatively in Sec. V.I.I. For an incorporation in our model of the chemical evolution of the galactic disk we derive a quantitative formulation of the yield Yi(R) as a function of galactocentric distance R by defining a new IMF,~, whose shape is that of the Miller-Scalo IMF ~ms, but whose lowmass cut-off is mlow=O.! m 8 in the interarm region and mlow=mc 51 m@ in the spiral arm region. The spiral arm IMF will be weighted with the factor av(R), which gives an adequate description

204

R. Gusten and P.G. Mezger

of the observed ratio of Lye photon production rates (and hence of the SFR of massive stars) in spiral arms and interarm regions, respectively.

(See Sect. 111.2.3).

~(R,mc,m) ffiN-I {~ia(m ~O.I me) + ~v(R)~Sa(m ~mc) }

(V.12)

with the normalization factor m up N(R,mc) = f @(R,m)mdm mlow

Note that ~ in (V. 12) is the normalized Miller-Scalo IMF as given in Eq. (lll.lOa,b). As previously we use mupffi60 m 8 in all numerical expressions. Substitution of Eq. (V.12) into Eq. (V. la) yields

P(R,m c) = N-I

60 60 1 f @(m)Am(Zi)dm + ev(R) f @(m)Am(Zi)dm 0.1 me

(v

+

Herein, the yields Pi are those derived for the normalized continuous or truncated Miller-Scalo

I I ~ , r e s p e c t i v e l y and given, f o r continuous I I ~ , in Table C . I . For primary elements i

with

massive progenitors m ~mc, p~a becomes independent on m e and hence pSa=piaffip. . . (Table C. I) i i i

(V.13b)

P(R'mc) = N-I Pi {I+~v(R)}

Substitution of Eq. (V.12) in Eq. (V. lb) yields the fraction of matter locked-up permanently in stars (I ~mc/m e !6)

l-Vt(R,mc) ffiN-I

f ~(m)mdm + 0.6 IO.l

~(m)dm + ~v(R) f ~(m)d mc

6o

+ ].4(l+otV(R)) f ~(m)dm 6

1

= N-I F(R,mc)

(v.14) r(R,mc) should not be confused with r(m) = m-mf used in Sect. V.2.1. Here 0.6 m e relates to the remnants of medium mass stars and 1.4 me relates to the remnants of high mass stars, respectively (see Sect. V.2.1). Substitution of Eqs. (Vl3a and 14) in Eq. (V.2) gives the effective yield as a function of R and m

Yi(R,mc) =

P(R'mc) ]_~(R,mc )

=

c

ia sa Pi +~v(R)Pi F(R,mc )

(V.15a)

which, with Eq. (V.]3b) and an analytical approximation of the numerically evaluated function F(R,mc) , becomes ]+~v(R) =

Yi(R'mc )

(V.]5b)

Pi

{0.6 +~v(R) [O.14m-I"~+O.O08]} e

Here, numerical values, of Pi can be taken from Table C. ]. For cxffiO(i.e. no star formation in spiral arms, ~=~bla) yi=Pi/O.6 , the relation used in Sect. I for a constant Miller-Scalo IMF.

Star Formation and Abundance Gradients

205

Table V.2: The variable yield Yi(R,mc) for the case of bimodal star formation (with e=1). Pi is the production efficiency, as calculated for a continuous standard IMF (Table C.]). Yi/Pi R/kpc

=~(R)

3

5

1

2.7

3.1

3.2

3.14

3.9

5.8

6.2

! .46

! .84

I .92

10

4.5

mc/m e = I

Yi(4.5; m c) =

Yi(10; m c) Values of Eq. (V.15b) are given in Table V.I, and have to be compared with yi(a=O)/Pi=(l-r) -I = 1.7, the value that applies for a constant (continuous) IMF. The modified yield increases with both ~ and m c , and relative abundance enhancements by factors of J2 between the inner disk and the solar vicinity are easily obtained, for m ~3 and a ~]. Further increasing these c parameters, however affects the yield only moderately, which indeed is rather insensitive on m

~3 and ~ ~1. C

V.3

PREDICTIONS OF THE MODEL OF BI-MODAL STAR FORMATION COMPARED ~ T H

OBSERVATIONS

As discussed in Sect. 1.3 the predictions of the closed system model with constant IMF (or constant yield, respectively) are in severe disagreement with observations related to 1) the chemical evolution of the ISM as a function of the age of the galactic disk; 2) abundance variations with galactocentric distance (abundance gradients); 3) SFRs and the stellar mass distribution.

In this section we compare predictions of the model of bi-modal star formation

and infall of halo gas with observations related to these three points and obtain good agreement in all cases.

V.3.1

THE CHEF~CAL EVOLUTION OF THE ISM IN THE SOLAR VICINITY

Time variation of the metal abundance ("metallicity") can be derived from observations of stars (G and M dwarfs) with MS lifetimes longer than the age of the galactic disk. Metal abundances are derived from line-blanketing indices such as the ultraviolet excess 6(U-B) or the StrSmgren parameter Am I. Naturally, these observations are limited to the solar vicinity. In the cumulative metallicity distribution of stars, the fraction S(Z)/S I of a sample of S I stars with metal abundance SZ is determined. This fraction is related to the SFR ~(t) by t S(Z)/S I ~ f ~ ( t ' ) d t '

o with t=t'(Z)

the time at which t h e ISM had t h e m e t a l l i c i t y

B e l l (1975) a l r e a d y ,

Z. I t has been shown by Lynden-

t h a t h i s "Best A c c r e t i o n Model" d i s c u s s e d in S e c t . V.1.2 and i n c o r p o -

rated in our model, can well reproduce the observed cumulative metallicity distribution. Twarog (1980) has refined this type of analysis by determining in addition to the metallicity of F dwarfs their ages from four-color and H8 photometry. His observed age-me~llici~y

re-

lation (AMR) in Fig. V.3 shows a sharp rise during the early phase of galactic evolution, starting ~12 Gyrs ago, but tends to flatten out during the last ~5 Gyrs, about the time the solar system was formed. As discussed by Twarog this time dependence is typical for (unJPv^ 26:3 - v

R,

206

I

I

I

I

Gusten and P.G. Mezger

I

0.1

Fig. V. 3: Age-metallicity relation (AMR) as obtained by Twarog(1980) for F dwarfs in the solar vicinity (filled squares and error boxes). Curves are A ~ s as predicted by our model for the chemical evolution of the ISM in the solar vicinity. The shaded curve has as its limits solutions for Zh=O and 0.2 Z@, respectively. saturated) infall models with constant ratio

-0.1

of net SFR to accretion rate, ~(t)=const. (Eq. V.3). In this case Eq. (V.9c) has the -o.~

simple solution

Z

-05

=

(V.16)

(yik+Zh) [I-(~/~o )(k-')-1]

with ~ the present-day gas-to-total mass ratio and ~o the initial ratio at the time 0

,~

8 12 STELLAR A6E IN GYRS

16

accretion started. The fact, however) that this relation fits the observed AMR can not be used as an argument for ~(t)fficonst.

In

Fig. V.3 are shown two AMRs, calculated from our model and two different values of the metal abundance of the infalling halo gas, Zh. For these numerical solutions of the set of equations (V.4,6,7 and 8) an exponent k=].5 (not to be confused with k in Eq.

V.16) has been

adopted, intermediate between the (extreme) cases k=] and 2 discussed in Sect. III. In spite of the strong time variations of both accretion rate and SFR (see Fig. V.], lower diagram), our predicted AMR for Zh=0.2 Z8 gives an excellent fit to the observed relation. But regarding the uncertainties in the age-scaling of the stars, satisfactory agreement may be obtained also for the case of accretion of unenriched halo gas. It should be stressed that this good agreement between observed and predicted time variation of the metal abundance of stars in the solar vicinity is a characteristic of infall models, and does not require bi-modal star formation.

V.3.2

THE CHEMICAL EVOLUTION OF THE ISM IN THE GALACTIC DISK

As shown in Sect. V.2.3 a yield which varies with galactocentric distance R is a consequence of bi-modal star formation. In this section we compare the abundance variations predicted by the model of bi-modal star formation for a number of primary and secondary elements with observations. Model parameters are k=l.5, me=3 m 8 and a=l. Production efficiencies P(R,mc) , analogue to F4~V. 13a) but without the instantaneous recycling approximation, have been computed with the IMF weighted production of elements Z. as given in Fig. C.I. Predicted and obi served abundances are shown in Fig. V.4. It should be noted that we have adjusted all predicted abundances to agree with abundances observed for the ISM in the solar vicinity. Absolute abundances of 160 (and other primary elements) predicted by our model are usually by a factor of two higher than the observed abundances. As shown in Table 1.1 this disagreement exists already for the closed system model, but is aggravated for the bi-modal star formation model as shown in Table V.I. We do not attach importance to this point, as several uncertain parameters enter the calculation of absolute abundances such as stellar yields, the state of gas depletion ~(R), the galactic history etc.

Star Formation and Abundance Gradients

[i0-~,] '

1

I

[,60/HI

I

'

I

1

IwO/HII

207

I

I

I

(b)

i~'He/HI

(a)

x

-÷-

(+) ~ b l e

10

I

'

x

yield

0.12

xl x M x

x

0.08

constant,ie~7- - - "~T--'"- ~ + 0.0/* I

,

[

I

90 60

,

I

[12Z/13[1

I

i

J I

I

(c)

[]

~

I

,

I

I

,

I

[180/1701

(d)

e

rl

m

consi'anf Y~mld~. -. -

I

1 k

30 I

i

2

I

i

6

I

10

,

I

I

,

1/, 2 GALACTIC RADIUS R IN kpc

I

,

6

I

10

I

1~

Fig. V. 4: Observed and predicted present-day element abundances as a function of galactocentric distance R. Solid lines refer to predictions of the hi-modal star formation model with parameter ~=1, mc=3m @. Shaded curves have as their limits solutions for ~z (from Table II. 2) and ~=const.=O.04, respectively. Dashed curves are the predictions of our model for ~ 0 (i.e. no star formation in spiral arms and continuous I~). Symbols o and ® refer to predicted and observed solar system abundances. Predicted abundances have been adjusted to agree with abundances observed for the ISM in the solar vicinity. a) Symbols + (Mezger et al., 1979) and o (Shaver etal., 1983) refer to observed 160 abundances. b) Shaded curve shaws the variation of the ionized abundance y+ with R of a group of giant radio HII regions. Open dots refer to high-resolution radio recombination line observations yielding lower limits to the number abundance y(ISM) ~y+ (Thum et al., 1980). Crosses refer to He-abundances in planetary nebulae yielding upper limits y(ISM) ~y(PN)=y++y ++. Solid curve is the He abundance Y=Y~+~Yev, with yD=O.075 the primordial abundance by number and AYev, the contribution from stellar nucleos~nthesi8, predicted by our model. c) Crosses refer to observations by Henkel et al. (1980, 1983) of H2CO isotopic radio lines. d) Crosses refer to observations by Penzias (1981) of CO isotopic radio lines. Triangles show changes in the predicted variation for standard depletion (&, see Table C.I) and increased depletion ( A and dotted square) of ISo.

Figure V.4, together with its figure caption, should be self-explanatory. limit our discussion to a few comments. abundance is excellent.

Therefore, we

The agreement between predicted and observed 160

The stellar contribution to the He abundance, AYev , is that pre-

dicted for a non-depleted primary element and added to a primordial abundance of yp=O.075 (by number) or Y =0.23 (by mass). IZC follows closely the 160 enrichment but is somewhat less P effectively enriched (~-]0%) due to depletion into 14N during the CNO cycle. The isotope ratio 12C/13C is the ratio of a primary to a secondary element, where 13C is burned on 12C. Eqs.

(1.4 and 5) for the closed system evolution predict zP/z s = | / y S

and we

therefore expect that the gradient of 12C/13C should decrease toward the inner disk. Predicted and observed isotope ratios are in good agreement.

That the decrease of I~c/Iac is

less steep than the increase of 160 is due to the fact, that the yield of a secondary element from lower mass stars (~m c) does not fully benefit from the increase due to the bimodal star formation,

as discussed in Sect. V.].].

208

R. GSsten and P.G. Mezger

In the case of the isotope ratio lSo/IVO, both nuclei are probably secondary elements, so that no gradient at all would be exprected across the galactic disk. Observations seem in fact to confirm this, but the well established difference in the isotopic ratio between the solar system (5.4) and the ISM (3.7) is in conflict with this simple analysis. The observed strong variation of the ISO/IvO abundance ratio

may however in part be explained by strong

depletion of leO due to astration in lower mass (longer-living) stars. While the enrichment of a secondary (non-depleted) element proceeds =~y~y~ (in~) 2 (in terms of the simple model, Eq. 1.5), depletion leads to a less steep increase with ~ (and hence with time) during advanced evolution stages. Let the (negative) yield y~ describe the depletion of the secondary element j during subsequent astration, then Eq. (V. IO) becomes (for closed system evolution) d Zs d in~ = -ySzp + y-Zs

which is solved by zS = - ySyp (y_) 2

ll+y-lm/ - ~Y- 1

(V. 17)

Obviously in late phases and for effective depletion Z s =In~, and thus even the ratio between (here) two secondary elements (e.g. Is0/170) can show strong variations (=in~) if depletion for one of the isotopes is important.

The solid line in Fig. V.4 is calculated with y-(leO)

from Dearborn et al. (Table C.I). In case of leO this flattening is further enhanced by dilution of the ISM with the (leO-depleted) ejecta of low-mass long-livlng stars, which at present provide major contributions to the amount of gas in the ISM. Here we do not want to discuss details of the ISo production, for which we refer to Glisten and Mezger (1983). As a last point we want to discuss present-day and solar system deuterium abundances and their relation to the primordial abundance of this isotope. Deuterium is not produced but only destroyed during astration and therefore the primordial 2H abundance X

should decrease with P increasing metal enrichment of the ISM. With our model we derive for the present-day local

ISM a depletion of XIsM/X p ~0.05 (for ~=O.04) and for the local ISM at the time when the solar system decoupled a depletion of X@/Xp ~O.]7. With [2H/IH]@= 3.6 ]0-s for the solar -6 system and ~5 10 for the ISM in the solar vicinity (Table IV.I) we estimate with these depletion factors primordial deuterium abundances of [2H/IH]

~(I-2) 10-4. These values are P considerably higher than earlier estimates based on the closed system model with constant IMF, which yields according to Eq. (1.6) a depletion of XIsM/Xp=O.2 only. The predicted de-

crease of the solar system deuterium abundance to the present-day ISM composition,

[Xe/XIs M]

~3.5, is consistent with recent observations by Vidal-Madjar et al. (]983). The enhanced astration in the inner Galaxy due to the increased number of short-living stars per stellar generation in our bi-modal star formation model leads to a decrease in XIS M by a factor of 1.5-2.O between R=9.5 kpc and 4.5 kpc.

V.3.3

STAR FORMATION RATES DERIVED FOR BI-MODAL STAR FORMATION

As discussed earlier and shown in Fig. 111.4, SFRs derived from observed Lye photon production rates adopting a constant IMF are about ten times higher then SFRs predicted from time averaged SFRs. As mentioned in Sect. V.I.1 one effect of bi-modal star formation is to increase the Lye photon production rate per unit mass, NLyc/~ , since the bi-modal star formation process produces more massive stars relative to low mass stars. Hence we expect that SFRs derived from Lye photon production rates adopting bi-modal star formation will be lower than the corresponding SFRs derived for a constant IMF.

Star Formation and Abundance Gradients

I

'

I

'

I

,t._.

I I/

o

I~obsLl-r]konst,tlF)

.,..o

/ ! i

I

.,

~

~)obs II-~lbimd.

IHF)

•

~!~!~!~!~!~..~.~!~!~ii~!~!~:i~t~irell-r] I

4.

i

I

Fig. V. 5: Observed and predicted present-day net SFRs, i.e. the amount of matter permanently locked-up in stars. ~obsEl-r] (const. IMF) refers to net SFRs obtained for a constant IMF from the observed Lye photon production rates (see also Fig. III. 5 and Sect. III. 3.3). ~re[1-r] are the net SFRs which are c~nsistent with the present stellar mass distribution (see text). ~obs[1-~] (bi-mod. IMF) refers to the net SFR obtained for the case of bimodal star formation from the observed Lyc photon production rates. This is demonstrated in Fig. V.5. Shown

.

/

209

I

l

8 12 GALACTIC RADIUS R IN kpc

here are the net SFRs (i.e. that part of the ISM which is converted permanently into stars, normalized to the total present-day mass of stars to<~(l-a)> (with to=tdisk the age of the galactic disk and <~(l-r)> the

average net SFR). In the case of continual star formation observed and predicted net SFRs should coincide. The gross discrepancy between predicted and observed SFRs, if the observed SFRs are derived for a constant IMF from the observed Lyc photon production rates, has already been mentioned in Sect. 111.3.3 and Fig. 111.5 and has been noted before by Cass& et al. (1979) and Talbot (]980). This discrepancy between observed and predicted net SFRs disappears, however, if net SFRs are derived for hi-modal star formation from the observed Lyc photon production rates. The reason is that in spiral arm star formation the fraction of mass that is permanently locked-up in stars is much smaller (due to the increased mlow=mc >I m@) than through star formation in the interarm region with ~low ~0.I m@. As a quantitative demonstration we compare SFRs and net SFRs for the galactic disk as derived from the observed total Lyc photon production rate given in Eq. (111.4). We obtain for the present-day SFR in the galactic disk ~(const. IMF) ~]3 m@ yr -I and

~(bi-mod. IMF) ~6 m e yr.

For the net SFR (i.e. the mass of ISM which per year is permanently locked-up in stars) we derive ~[l-r]

(const. IMF) ~9 m@ yr -I as compared to ~[l-r] (bi-mod. IMF) ~1.5 m e yr -I. The

specific parameters of the model of hi-modal star formation, ~=l and m c ~3 m@, give reasonable fits to the observations discussed in Sect. V.3.2 and this section. At this point we would llke to come back to the galactic diffuse near infrared (NIR) emission discussed in Sect. 111.4.2. There we encountered a similar problem as with SFRs derived for a constant IMF: Both NIR emission and SFRs do not scale with the preseNt-day mass distribution of stars in the disk. To obtain a fit for the ridge line intensity of the observed NIR emission we had to add a hypothetical population of M-giants with masses m Z(2-3)m®, whose distribution follows that of O stars. We showed that the luminosity of these M-giants, integrated over the galactic disk, is consistent with a SFR for stars m ~(2-3)m8 which ~IO s yr ago (the time, when the progenitors of the present-day M-giants formed) was not much different from the present-day SFR for the O stars. This result is easily explained by the model of bi-modal star formation, where the spatial distribution of stars with m Z(2-3)m@ (and hence of O stars and the progenitors of M-giants) is similar. The IMF for bi-modal star formation, Eq. (V.12), yields the same spatial distribution for O stars and the progenitors of M giants (m ~2-3 m e ) but a different spatial distribution for lower-mass stars, which determine the mass distribution of stars in the galactic disk.

210

R. GSsten and P.G. Mezger

ACKNOWLEDGEMENTS It is our pleasure to thank C.M. Walmsley and T.L. Wilson for reading the manuscript. Our special thanks go to Mrs. Breitfeld and Mrs. M~rz who typed this manuscript on a very tight time schedule.

Star Formation and Abundance Gradients

APPENDIX A:

211

PRODUCTION RATE OF LYMAN CONTINUUM PHOTONS FROM EARLY-TYPE STARS

In Sect. III we use observationally determined Lyc photon production rates to estimate the SFR in the galactic disk. The Lyc photon production rate of a star of mass m and main sequence (MS) age T is

NLyc(m,%) = 4~-R~(T)-nLyc(Te(T),g(l~,m)) , with R~ the stellar radius, g the surface gravity and T photon flux

Fig.

nLyc(Cm-as-1)

while R,, T

spheres,

at

the stellar

as function of m,T are derived from models of stellar evolution. In

e temperatures

A.I effective

surface

the effective temperature. The Lyc e is obtained from models of stellar atmo-

and stellar

radii

are compiled for the zero-age

('ZAMS')

and terminal-age ('TAMS') MS from recently computed stellar evolution tracks. The MS lifetime TMS ( F i g .

A.2)

is defined

here as the phase of core-hydrogen

burning;

it

ends after

the

core exhaustion of hydrogen, when the star begins to contract (first T -minimum) and shell e b u r n i n g d e v e l o p e s . F o r m o s t t r a c k s Y ~ 0 . 3 a n d Z=O.03 h a s b e e n a s s u m e d a n d no m a s s l o s s h a s been taken into account. Stars with less enriched initial composition

Z ~0.02 Z e are

I

I

•

I zAMs .08 06

% ±TAMS

-BO -BI

=

-B2 -BS

2

I 25

~z4 Z3

TAMS.~

T

p

Fi~. A.I: Effective temperatures and stellar radii for zero-age ('ZAMS') and terminal-age ('TAMS') main-sequence stars according to stellar evolution tracks from Alcock and Paczynski (1978, e), Becker (1981,x), Brunish and Truran (1982,o, Z=O. 02), de Loore et al. (1977,A), Ezer and Cameron (1967,A, Z=O.02), Maeder (1981,+), and Stothers (1972,o). If not indicated otherwise, an initial composition of Z=O.03 and Y ~0.3 has been assumed. Spectral type (Te) and stellar mass (m) are connected following Cont i (1975), Hutchings (1980) and Remie and

illlili] I

6O

20 40 STELLARMASSm IN me

I

109

"\

I

I~~MS=

I

1

Fig. A.2: Main-sequence lifetime versus stellar mass. For references see Fig. A.1. Simple analytical fits for m ~ 8 m@ and Z=O.03 are given. Tracks computed for a less enriched initial composition (i.e. Z =0.02) yield shorter MS lifetimes.

5.109. m-Z'7*l.Z.107

~I0 B t-o

~',

g

/ I~Ms=1.Z.109.m-t8%3.106

systematically hotter, more luminous and evolve faster. The simple analytical ap-

:E

10~

proximations of TMS(m) given in this figure refer to stellar evolution tracks with Z=O.03. For reasonable, viz. obserI

I

I

I

I

I

3

ID STELLARMASSmINme

30

60

vatlonally well-determined mass loss rates of lO-(v tO6)m® yr-1 for 30-40 m@

212

R. GUsten and P.G. Mezger

MS stars (Abbott et al., 1981, Garmany et al., 1981) T and R~ differ only moderately from e values obtained for constant-mass tracks (Brunish and Truran, 19821Maeder, 1981). Hence, as shown below, the IMF weighted Lyc photon production rate of a given stellar population is underestimated by only ~10-20% and thus is comparable to the uncertainty introduced by assumptions of the initial composition. Therefore, we neglect mass loss effects in the following discussion. At any rate, we feel that the biggest uncertainties in our computations of the Lyc photon production rate come from the adopted relation between stellar atmospheres and stellar masses, which is for massive stars m ~30 mQ entirely based on theoretical investigations. Only for less massive objects have observations of wide binary systems shown satisfactory agreement (Hutchings,

1980). Other uncertainties in the computed Lyc photon

production rates come from the models of stellar atmospheres. Since the time of earlier compilations of Lyc photon production rates (Panagia, 1973; Mezger et al., 1974) results of more realistic llne-blanketed LTE model atmospheres have been published (Kurucz, 1979) which predict moderately reduced Lyc photon production rates and less He ionizing Lyc photons. For surface gravities g, appropriate for massive stars, values of nLyc(Te,g) are shown in Fig. A.3. Kurucz's models are computed for solar system abundances only and with increasing metal abundance (and hence line opacity) a lower Lyc photon production rate is to be expected for a star of given mass. While this effect is not considered here, the (probably much stronger) effect of a decrease of the upper mass limit (mup) of the IMF with increasing Z is taken into account.

In Fig. A.4 we give the Lyc photon production rate as a function of stellar mass for both the ZAMS and TAMS

points on the evolution tracks. The total number of Lyc photons produced

by a star of mass m during its lifetime is obtained by integration

NLyc(m) = j" NLyc(m,T)dT TMS

SPECTRAL TYPE B5 I

I

B2

B1

I

I

BO I I

(A. 1)

and given in Fig. A.5. For two

08 i

06 I

mass ranges analytical approxi-

I

mations are also given in this Figure. We see that only stars A

.t-~/

with masses >10 m@ contribute significantly to the integrated Lyc photon production rate and after weighting with the IMF -

1022

the "typical" ionizing 0 star is

u

found to be in the mass range 30

102o x

C=, 10~8

, ~

~4.5° x I 2

I I 3 4 TEMPERATURETeff (lOZ'K)

I 5

Fig. A.3: Lye photon flux densities, given as function of surface temperature (Te) and gravity (g), according to the LTE line-blanketed models of Kurucz (1979), and solar system abundances (Z ~0.02) of the stellar atmospheres. For comparison Panagia's flux densities are given, derived from earlier stellar model atmospheres and for log g=4.

Star Formation and Abundance Gradients

I

u

U

213

I

u

n

48

It.x-~ 46 u

~ /,4 ,,,,It

42 o

I I0

,

I 30

ZAMS '( TAMS o I 50

,

,

STELLAR MASS m IN me

Fi~. A.4: Lyc photon production rate NL (m,T) in s -I for zero-age (ZAMS) and terminal-age (TAMS) YCain-sequence stars as function of stellar mass. I

I

I

I

I

01

NLYC('l:,m)d1:

/ /

MS

/ / / I I

C~

/

3.0

i I

8

/o I

2.75 106z(m-261°'6~,,,, / .ii / £'/

Z.O

/ / iI

~

iI le ii

1 . 0

/ I

lO

,- .,t

,J ~

3.5 10ss ,m 5 I

I

I

30

50

70

STELLAR MASS m IN me

Fi~. A.5: The total number of Lye photons emitted by a star of mass m durin E its main-sequence evolution. Simple analytical approximations are given for m ~ 30 m@.

214

R. GHsten and P.G. Mezger

to 40 me, corresponding to a spectral type 06. Relation (A.]) weighted with the IMF, yields the quantity

e(~)

= f ~(m) .NLyc(m)dm

(A.2a)

which in this form is independent on the normalization of the IMF (especially on mlow) but depends on both the shape of the IMF, ~(m), and on m up . We have evaluated eq. (A.2a) for both the Salpeter (sal) and Miller-Scalo (ms) IMF. For mup 235 m 8 P(@) may be approximated by =

pt~ms)-.~ .-

p(~sal)

3.5.106o.[1-l.l.lOa.m

=

-1-97

+

up

1.2.10O.m-2.av+4.0.10".m

up

-s-97]

up

2.9.1061.[l_13.5.m-O.V + 94.m-1.7 + 261.m-2.v] up up up

(A.2b)

Substitution of eqs. (A.2b) in eq. (111.12) yields a relation between Lyc photon production rate NLy c and SFR NLyc/~ = ~(1).P(~)

(A.3a)

which for Salpeter and Miller-Scalo IMF, respectively, has the form

~sal ms

• A54(

NLye/~ = 3.2 lu

--0,35_

mlo w

--0.3Sx--1

mup

)

(0.I056-1.43 m-~7+9"g4up m-l"7)up

NLyc/~ = f-12.5 I0S6(4.71410-4-0.508 m-l"gV+5"69up m-2"SV)up

with 0.1
4.44 - 2.5 mlo w°'4

i2.75 f(mlow,mup) =

m~-O.808

- 16.8 m -1"62 up

1.O
5.1 mZl"2~-71ow.210-3-16.8 m -1"62up I0 --1.62

16.8 (mlow

--1,62

-mup

)

(A.3b) 30

As usual all stellar masses are expressed in solar masses. APPENDIX B:

DECONVOLUTION OF THE DIFFUSE GALACTIC FREE-FREE EMISSION ASSUMING AZIMUTHAL SYMMETRY

After separation of the thermal and non-thermal components, one obtains the diffuse galactic free-free emission as a function of galactic longitude I and latitude b. The ridge line intensity S(~,b=O °) is shown in Fig. 111.2 in units of Jy/beam area. Eq. (111.3) relates the flux densities at a distance D to the volume emissivity e. The telescope beam with which the diffuse emission e was observed had HPBW's of AlxAb, corresponding to a beam solid angle = ].]33AZ.Ab. As outlined in Sect. III the free-free volume emissivity is directly rem

lated to the Lyc photon production rate and hence to the distribution of 0 stars in the galactic disk. Substitution in Eq. 111.3 of the volume emissivity eff(R,@,z), given as a function of the cylinder coordinates (R,@,z) (see Fig. B.]) yields for the ridge line flux density

Star Formation and Abundance Gradients

S(Z,b=O) ~m

1

@

215

Z

m m Rout ~-- f dO f dR R ~ dz ~(R,C),z)/4~T(D2+z 2) mo R.inn -z m

(B.I)

Note that z is the coordinate perpendicular to the galactic plane and 6 = arc tan (z/D). The integration limits O ~z ~z

= D tan(Ab/2) max

:0

o

Rin~ R(O,/-AZ/2) ~R ~Rou t = R(O,/+AZ/2) 0 ~@ ~@max :

G.C.

@(/'Rmax)--

depend on both Rmax, the cut-off radius of the freefree emission, and on the telescope HPBW's, AZ and ~b. If Ab is small as compared to the latitude distribution of the free-free emission,

(i.e. if

Dmax'tan(Ab/2) < Zo, with z° the scale height of the

\

free-free emission) and if we assume azimuthal symmetry for the distribution of 0 stars (i.e, Eff(R,Q,z)÷Eff(R,z))

SUN

then eq. (B.I) reduces to D max

~

:

Definition of coordinate

S (/'b=o°) m

I

~

I

J

I

x'" "x

12 •

i rn

x-.

~-9

,

,.-,,

i

Y

i=~

=--6

l xl \

i +-+,

/f

I-=-

I=~ I----

l ,,_~I~

i

I

/

'."

'÷

E (R)dD.

'

(B.2)

I

x WESTERHOUT ~,0+ MAIHEWSONet oL.~"O

X~

A

f o

'~

',

\ ,,

1,, \ \,..-~.

x%%

÷

%,

i..,..i z

,-3 ZXZ

r,,,."

co

I

I0

i

I

~

I

30 50 LONGITUDE,~ IN DEG

,

+

70

Fig. B. 2: Ridge line brightness temperature of the diffuse thermal background according to Westerhout (1958) and Mathewson et al. (1962). Observing frequencies and angular resolution were (1.39 GHz, 34') and (1.44 GHz, bO'), respectively.

216

R. G{isten and P.G. Mezger

We neglect free-free emission outside the solar circle. Hence R

max

= I0 kpc and

D = I0 cos(l) ±[(R 2 - lOesina(/)]-°'s d D = ±(R 2 _ 10 2 sin2(/))_o.s R dR

with ± corresponding

to @ ~ 90 ° . Substitution

in the above relation yields

R

S(l'b=o°)

2"~axe(R)

~m and, with new variables

R (R 2 - 10 2 sin2(/)) -°'s dR

10sin(l)

u = R2 - R 2 and Q = R 2 - 102. sin2(/) max max Q S(Q) = Se(u ) (Q _ u)_O.S du m o

(B. 3a)

This integral equation can be solved by a Laplace transformation,

yielding the solution

u

s

=÷

(B.3b)

(u

m o

Obviously,

the accuracy of the deconvolution

depends critically on the accuracy with which

the derivative of the ridge line intensity S(/), i.e. the quantity dS(Q)/d~

in eq.

(B.3b),

can be determined.

Fig. B.2 shows the ridge line intensities diffuse galactic free-free emission,

(expressed

in brightness

as derived from observations

Mathewson et al. (1962). The uncertainties

in these low-resolution

temperatures) by Westerhout observations,

of the (]958) and which hardly

fulfill condition B.2 are large. To follow how errors in the ridge line intensities into the deconvolved

radial volume emissivities,

cedure. Herein e(R,z=O)

is approximated

and for 'm' independent

line-of-sights

we developed a numerical

by 'n' radial

propagate

deconvolution

intervals of constant

pro-

emissivity el'

along the galactic plane eq. (B.l) becomes a system

of m times n linear equations. n S(/j) = E e "F i=] i ij '

! ~j Sm

which is inverted numerically.

The radial emissivity and Mathewson

e(R) as derived from the average of the measurements

of Westerhout

et al. is shown in Fig. B.3a. The error bars measure the uncertainty

radial distribution

as estimated from the differences

in the

between the two sets of brightness

temperatures.

The corresponding Lyc photon production rates, derived from g(R) as described 111.2.2 are shown in Fig. B.3b.

in Sect.

Star Formation

and Abundance

I

I

I

Gradients

I

217

I

b 50 • •

.:.:..'.:.:.

......

::::::::::~:~

.':.;...,,.,.;.~,..'

Z

~...-,~

~- 30

: .'!:~:::...~

r-~

~%'~ ~

r~

:.,. ,~.:::. "•":i

I

I0 :~. ..:.

I

I

I

Z

4

I

I

_6

I0

8-

N -rI

4

>

- -

m

LZD

m

I

I

2

4

I

I

I

6 8 RADIUS(kpc}

I0

Fi~. B. 3a : shows the radial distribution of the free-free emissivity ~(R) in the galactic plane, as deconvolvedfrom the ridge line brightness temperatures shown in Fig. B.2. For a gaussian z-distribution, the Lye photon production rates, integrated over 1 kpc wide annuli, are derived in Fi E. B. 3b.

APPENDIX C:

SITES OF NUCLEOSYNTHESIS

In this appendix we give a short summary of the stellar nucleosynthesis elements

and isotopes, which are incorporated

sented before by e.g. Audouze et al. mechanisms

Such discussions

for the

have been pre-

(]975); ours is an update which includes new production

and more accurate estimates of yields. While there is general agreement

nuclear processes controversy

in our model.

processes

that synthesize

the elements and isotopes discussed here, there is some

about which astrophysical

for general references). Perhaps more surprising the predictions

about the

objects are the main sources

For example the origin of rare isotopes

(see e.g. Truran

is the fact that there has been considerable

of nuclear astrophysics

and models of chemical

(1977)

such as leo is unclear. disagreement

between

evolution of the Galaxy,

in

218

R. GUsten and P.G. Mezger

particular concerning the nucleosynthetic sites of the conm~on species of the CNO-group (and Helium).In addition to the physical conditions in the synthesis regions (temperature, density, reaction rates, etc.), a major uncertainty is how to dredge-up the freshly made elements from their production sites deep in the star's interior to the envelope. From the envelope they may be ejected into the ISM - via (steady) mass loss, planetary nebula-like ejection and super nova explosion.

In the following we briefly summarize the astrophysical environment in which the nucleosynthesis of various elements and isotopes and their liberation into the ISH may occur. To obtain the timescale of the enrichment, special attention must be paid to the mass (and hence age) of the stellar progenitor and to the question whether the synthesis process is primary or not. Species like 160, that in a given star can be synthesized from primordial hydrogen and helium (perhaps in a network of processes) are termed primary elements. Those like ISc, which are built up from a pre-existing abundance of elements such as 12C, which are produced in a past generation of stars, are of s e c o n d l y

origin. Consequently the en-

richment in secondary relative to that in primary nuclei is delayed (see Sect. V). The production rates, compiled in Figure C.I and Table C.I from recent yield calculations, are based on single-object tracks only, and may have to be modified if interaction in a binary system is taken into account. Although 2H is a basic step in normal hydrogen burning, its destruction proceeds much faster than its formation. Hence in stellar nucleosynthesis, 2H is not produced. Furthermore, the reaction 2H(p,y)3Hestarts at temperatures ~(0.5-].0)]0 s K. This temperature is generally exceeded in stellar interiors, even during pre main-sequence evolution, so that the pre-stellar (primordiaD Deuterium cannot survive. As mentioned before, the amounts of freshly made ~He (the end product of simple H-burning), 12C and 16C (products of He-burning) as well as 13C, I~N, and IVO (from the CNO-cycle) that are liberated into the ISM are rather uncertain. Hence these production rates depend on details of the late evolution of the stellar progenitor (i.e. mass loss, mixing efficiencies along the giant branch etc.). Earlier investigations of the pre-supernova composition of massive He-cores ~3 m e (corresponding to total mass ~I0 m~), without mass loss have been made by Arnett (]978). More recent closed evolution tracks (including the hydrogen envelope) with mass loss yield somewhat lower metal-productlon

(Maeder, ]98]). It is shown that various rates of (mainly post main-

sequence) mass loss affect the net yields of He and "metals" only slightly. Although with increasing mass loss rates, stellar winds from evolved cores (WN, WC-stars) become an important ejection mechanism, their contribution to the enrichment of the interstellar gas with 12C, I~N and IVO is only a fraction of that expected from medium mass ~ stars (see Fig.

C.1). 160

is ejected into the ISM from stars m ~I0 m@ only. Otherwise it is kept locked in a stel-

lar carbon-oxygen core that never exceeds the Chandrasekhar-limlt.

For most of the other

elements however, their major contribution is from medium mass stars. As pointed out by

Stars m >8 m@ ignite carbon burning in a non-degenerate core, and develop the well-known shell-structure during their subsequent burning phases. Thus their evolution is quite different from that of stars of lower mass, and we follow the c o l o n classification into massive (>8 m 8) and medium mass (!
hot CN

ISO(a,v)laC

incomplete CNO

15N

le0

170

(CHO)

References:

(I) (2) (3) (4) (5)

A r n e t t (1978) R e n z i n i + Voll (1981) Woosley + Weaver (1980) Dearborn e t a l . (1978) Thlelemann + Arnold (1978)

I~N

Is O

laC

12C,I~N

CNO

a-capture CNO

~ac(~eO)

~aC

~He

1~N(a,y)InF

a-capture

CNO

CNO

l~N(~,T)leF(8+)leO(a,y)a2Ne

CNO

I~N

Is 0

CNO

13C

incomplete

triple'a'

laC

a-capture

a-capture

pp,CNO

"He

Seed-nuclei

Nucleosynthesis Process

Destruction

aD(p,¥)SHe

Production

(6) (7) (8) (9) (IO)

IHS

MS

IMS

(Ms)

MS

IMS/MS?

IbiS

IMS

IMS

7.9 ( - 4 )

-3(-7)?

?

?

~2(-5)

2(-6)

6.4-9.2(-3)

?

6=1.5 3.1 ( - 3 ) 2. I ( - 3 ) p r i m .

6ffiO

6=1.5 1.6(-4) 0.9 (-4)prim.

in m a s s i v e s t a r s

d u r i n g q u i e t c o r e H e - b u r n i n g i n m
e x p l o s i v e H e - b u r n i n g (SN?)

incomplete CNO-burning with deep mixing along AGB, no hot-bottom burning included hot CNO-burning in novae/supernovae

enriched winds from WN-stars

helium-burning

hot CN-cycle (T>2-108) in novae (or supernovae)

major sink in CNO-cycle; s e v e r a l d r e d g i n g up p h a s e s a l o n g AGB, h o t - b o t t o m b u r n i n g in c o n v e c t i v e e n v e l o p e

low-temperature 'hot-bottom t burning

6=0

4.4 (-5)

incomplete CNO-burnlng in Red Giant envelopes

in Red G i a n t s

4,5

4 6,7 6

9

1,3.8

7,9

2,4

2

1,3,10

WR-stars

Helium-burning, partly dredge-up

1.5 ( - 3 )

2

1,8.10

References

dredge-up in Red Giants 6-mixing length ratio

winds (WR-stars)

of all masses

hydrogen-burning, 50Z in s t e l l a r

completely destroyed in s t a r s

Notes

3.9 ( - 3 ) 6ffil.5 2 . 3 ( - 3 ) 8ffiO

IMS MS

1.3(-2) 6=o 1.8 (-2) 6=1.5 1 . 9 ( - 2 )

-0.6

Net Bulk Y i e l d s

MS

all

Stellar ~ss Ran~

Howard e t a l . (1971) Audouze e t a l . (1978) blaeder (1981) Woosley + Weaver (1980) Maeder ( 1 9 8 2 ) , Abbott (1982)

(ter.)

sec.

sec. (prim.)

prim.

sec.

(prim.)

sec.

(prim.)

sec.

prim.

prim.

Type

YIELDSOF NUCLEOSYNTHESIS PROCESSES

2H

Species

TABLE C.1

e~

> o"

~rj 0

220

R. GUsten and P.G. Mezger

Iben and Truran (1978), during their post main-sequence evolution (first ascent along the red giant branch) these stars develop deep convective envelopes, that even penetrate to the outer layers of the hydrogen burning shell, carrying fresh products of CN0-burning (ItN, the and some 13C, Iv0) to the surface ('first dredge-up' phase). In stars more massive than~3 me further stirring occurs after ignition of the He-burning shell ('second dredge-up' phase) and during their unstable asymptotic giant branch ('AGB') evolution (third dredge-up). In this last phase these stars undergo He-shell flashes, that stimulate periods of convective mixing between the base of the He-burning shell and the stellar envelope, dredging up fresh the (from the H-burning shell) and lmc (from the He-burning shell; see Iben and Truran (1978), Wood (1981) and references therein)). Depending on the details of the mixing process, the matter released during the AGB-evolution (via stellar winds or planetary nebula ejection) is significantly enriched in ImC and tHe. Further modification of the envelope composition could be caused by hot-bottom CN-burning during the quiescent interflash phase. This occurs when the temperature is sufficiently high at the base of the convective envelope to start the (cold) CN-cycle, and convert part of the fresh ImC, that has been dredged-up from the He-shell, into 13C and ItN (see Renzini and Voli, 1981). The efficiency of this burning process depends sensitively on the temperature (and hence the treatment of convection) in these envelope layers. For illustration in Fig. C.I, we show results for two extreme cases, i.e. for 8=0 and 1.5. The parameter 6 is the ratio of mixing length to pressure scale height. Although hard to quantify, hot-bottom burning is nevertheless of special importance for our chemical evolution models, since part of the 13C and ItN (and probably Iv0, which has not been included by Renzini and Voli) is of

primary origin in this process (Fig. C.1). Clearly, observational constraints on the 'primary' tiN-yields are needed, either from abundance determinations in evolved AGB-stars (Wood etal., 1981; Kaler, 1982) or from chemical enrichment models as described in Sect. V. The nucleosynthetic sources of 15N and leo are even more uncertain, and during normal (cold) equilibrium CNO-burning both are destroyed. A possible site of 15N (and IVO) production seems to be hot CNO-burning in novae (and supernovae), because at high temperatures the final abundances in the cycle are changed in favor of the rare 15N (Iv0) isotopes.

For example, for

T >2 10s K the 15N/tiN ratio may increase to unity, compared with 10-t in the cold cycle. There are several potential sites for the nucleosynthesis of Is0. However the problem is to protect 180 from destruction in further CNO- or He-processing [Is0(a,y)22Ne]. One promising production scheme is explosive He-burning on 1iN in the He-rich shells during supernova-explosions (e.g. Howard etal.,

1971). However, using recent results on the Is0(a,y)2mNe cross-

section, Thielemann and Arnould (1978) suggested that even during "quiet core" and/or shell He-burning in medium

mass stars (2 ~m N5 me), significant amounts of leo could be pro-

duced. What fraction of this survives further He-shell flash-burning (most likely in the lower mass objects with lower flash temperatures) and can be mixed to the surface is uncertain. In these scenarios 15N and leo are likely to be of secondary origin. They are synthesized from ItN, which itself is produced in situ (from ImC) in this stellar generation. If, however, burned on 1iN, produced in an earlier stellar generation, part of the 180 may be even a "tertiary" product. In Table C.I the net bulk yield Yi of a stellar generation is calculated, which is defined as the fraction Pi of mass of newly synthesized element 'Z.',I per fraction of matter locked up in long-lived objects Yi

Pi f~(m)Am(Z)dm (l-r) = (l-r)

(C.1)

Star Formation

Am(Z)

and Abundance

is the mass of the newly synthesized

Net bulk yields

Gradients

221

element Z that is returned

in Table C.] are given for the Miller-Sealo

into the ISM (Fig. C.I).

IMF with

(]-r) = 0.6

I

1

[1¢]

I

' 10s211

16 0

2

,

13 C

I

I I-.

E
E

.I

,

e

10

Z

I

I

I

30

--

i10-3]

T

4He

"~i i

5 z,

i l

1.0

l

(MS t MS_. O I

5

-r-

t

10

10

"'

2O

3O

I

I

i

,,

•

14N

3 2 1

ti i I

I

20

10

![10-4]

I

f

2.0 C)

I

70

50

-- PRIMARY

--

-

"',.

_

i

PRIMARY

)

I

10

20

12 C

[10_1, l >

I

'

• ! o • T

I

;

I

10

i

20

I

ARNETT (1978) MAEDER [1981) DEARBORN et al (1978) WOOSLEY+WEAVER (1980) RENZINI+VOLI (1981) : a = o 6

15

I

30 STELLARMASS(too)

Fi9. C.I: I ~ weighted production of some primary and secondary elements. Am(Z) is the mass of a newly synthesized element Z that is ejected into the ISM by a star of mass m. Symbols refer to model computations by different authors.

222

R. G~sten and P.G. Mezger

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