Diffusion of iron-group elements in the envelopes of HgMn stars

Transcription

1 Mon. Not. R. Astron. Soc. 307, 1008±1022 (1999) Diffusion of iron-group elements in the envelopes of HgMn stars M. J. Seaton Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT Accepted 1999 April 16. Received 1999 March 19; in original form 1999 January 27 ABSTRACT The observed abundance anomalies for iron-group elements in atmospheres of HgMn stars are due to diffusive movements which are driven by radiation-pressure forces and which persist in the stellar envelopes, going down to regions with temperatures of about 10 6 K. Studies of diffusion in the envelopes are required both in order to understand the observed atmospheric abundances and in order to calculate the changes in opacities that result from changes in abundances. Let t be the Rosseland-mean optical depth. It is shown that one can define an upper boundary, t ˆ t u, such that one can obtain solutions for the diffusive movements in the region of t $ t u without any knowledge of what happens in the higher layers of t, t u. The paper is concerned with a description of the numerical methods that can be used to obtain such solutions. For Cr and Mn we are able to follow the diffusion for times of order 10 8 yr with t u ˆ 1. For Fe we are also able to obtain some estimates of abundances at t ˆ 1 allowing for diffusion processes. For Mn, Cr and Fe we attempt some comparisons of abundances computed for t u ˆ 1 with observed atmospheric abundances and obtain results that are not discouraging. For Fe and Ni, larger values of t u are required as the diffusion proceeds (after 10 8 yr we require t u. 8 for Fe and t u. 70 for Ni). For the outer regions with t, t u it will be necessary to obtain solutions allowing for outflows of iron-group elements at the stellar surfaces. In such work it should be possible to match the outer-region solutions to the envelope solutions obtained using the methods described in this paper. The diffusive movements lead to changes in Rosseland-mean opacities by factors of up to 4. For Fe there is a build-up in concentrations in the region of log T. 5:1, where the dominant ionization stages are near Ar-like. This leads to the Z-bump in opacities being shifted from its normal position at log T. 5:3 to lower values of log T and becoming more sharply peaked. There is also a large build-up in Ni concentrations in the outer parts of the envelopes, leading to opacity enhancements. In the present work we allow neither for the normal main-sequence evolution of the stars nor for the modifications in that evolution which will result from changes in opacities. Solution for both envelopes and outer regions will eventually be required. Key words: diffusion ± stars: chemically peculiar. 1 STATEMENT OF THE PROBLEM The HgMn stars show marked peculiarities in atmospheric abundances. They lie close to the main sequence, have effective temperatures within the range T eff ˆ to K and are slow rotators: the most recent review of their properties, and those of other chemically peculiar (CP) stars, is that of Smith (1996). Following the work of Michaud (1970) it is generally accepted that the chemical peculiarities are due to diffusive movements: settling due to gravitational forces expressed as gravitational accelerations g grav (acting downwards); and levitation due to radiation-pressure forces expressed as radiative accelerations g rad (acting upwards). The HgMn stars are simpler than most other CP stars in that they appear not to have strong magnetic fields, they are too hot to have superficial hydrogen convection zones, and, due to gravitational settling of He (as evidenced by depleted atmospheric He abundances), can be expected not to have He 1 convection zones. Diffusion occurs in the absence of mixing due to convection and of the stronger meridional circulations that will occur in faster rotators. q 1999 RAS

2 Diffusion in envelopes of HgMn stars 1009 Table 1. S92 abundances of iron-group elements. Element A Cr 4: Mn 2: Fe 3: Ni 1: At some point inside a star let there be a total of N atoms per unit volume and n atoms of a diffusing element. The concentration of that element is c ˆ n=n. Owing to the effects of saturation in the spectrum lines, the radiative acceleration g rad decreases as c increases. Let the diffusing atoms have diffusion velocities v: a given atom will have upward movement v. 0 if g rad k. g grav and downward movement if g rad k, g grav. Owing to the dependence of g rad on c we can, in general, find a value c stat of c such that, for c ˆ c stat, one has g rad ˆ g grav and v ˆ 0. The observed atmospheric concentrations of iron-group elements in the HgMn stars are smaller than c stat (see Smith 1996; Seaton 1996a,b). It follows that such elements will have v. 0 in their outer layers, leading to outflows at the stellar surfaces. Diffusion with surface outflow has been discussed in a number of previous papers (Michaud et al. 1983; Michaud & Charland 1986; Michaud 1991; Babel 1992; Charbonneau 1993; Babel 1995, 1996). The main new feature of the present work is to consider how the outflow is maintained by diffusive movements in deeper layers of the stellar envelopes. We use atomic data obtained in the course of the work of the Opacity Project (OP) (see The Opacity Project Team 1995): results for Rosseland-mean opacities are given by Seaton et al. (1994, to be referred to as SYMP) and those for radiative accelerations by Seaton (1997). For the first few ionization stages the OP data are supplemented by line-data from Kurucz (1990). For regions of stellar interiors with temperatures in the range of 10 5 to 10 6 K, atoms of iron-group elements are in ionization stages giving large numbers of spectrum lines and hence making large contributions to opacities (see Rogers & Iglesias 1992a; SYMP). In such regions those elements also have large values of g rad. We shall show that the abundances of irongroup elements in the atmospheres, which can be deduced from observations, are dependent on diffusive movements in those much deeper layers. A further point, already noted by Alecian, Michaud & Tully (1993), is that those movements can lead to changes in opacities and hence to changes in the structures of the stars. In the present paper we consider diffusion of the iron-group elements Cr, Mn, Fe and Ni. Let element k have abundance A(k) by number fraction. When no diffusion occurs we use the abundances of the S92 solar-system mix of SYMP (taken from Anders & Grevesse 1989; Grevesse & Noels 1992; Grevesse, Noels & Sauval 1993). These abundances are given in Table 1 for the iron-group elements. When abundances are modified due to diffusion processes we multiply A(k) for element k by a factor x(k) and adjust all A(`) for ` ± k so as to conserve the normalization P ma m ˆ1. Fig. 1 shows the ratios g rad /g grav, calculated using unmodified abundances, for a stellar model with T eff ˆ K, and Fig. 2 shows g rad /g grav for Mn in the K model with x ˆ 0:1, 1.0, 10 and 100. It is seen that minima in g rad /g grav occur in regions where the dominant ionization stages are near Ar-like, Ne-like and He-like. These regions will be referred to as barriers. They play an important role in the theory. It is convenient to divide a star into three regions: atmosphere, envelope and core. Let t be the Rosseland-mean optical depth. The atmosphere is the outer region, Figure 1. Ratios of radiative to gravitational accelerations, g rad /g grav, for Cr, Mn, Fe and Ni in a stellar model with T eff ˆ K and with unmodified abundances x ˆ 1. Figure 2. Ratios g rad /g grav for Mn in the K model, with Mn abundances modified by factors of x ˆ 0:1, 1.0, 10 and 100. Minima in g rad /g grav occur in regions where the dominant ionization stages of Mn are Ar-like, Ne-like and He-like. say that with t # 1; the core is the innermost region, in which energy is generated by nuclear reactions; and the envelope is the region between atmosphere and core. For the calculation of g rad we require radiative monochromatic fluxes F n. For atmospheres accurate values of F n can be obtained only by computing detailed atmospheric models for individual stars, as in the work of Alecian & Michaud (1981) 1 for Mn in HgMn stars. For envelopes the problem is simplified in that one can obtain F n on solving the equation of radiative transfer in the diffusion approximation (see 1 We note that Alecian and Michaud obtained values of c stat not very different from Mn concentrations deduced by Heacox (1979) from observations of a number of HgMn stars. However, later observational work has given concentrations smaller by about one order of magnitude. Results from UV observations are given by Smith & Dworetsky (1993) and a summary of results from optical observations is given by Adelman et al. (1996).

3 1010 M. J. Seaton Mihalas 1978). One then obtains expressions for g rad proportional to the total radiative flux F, the Rosseland-mean opacity k and a dimensionless quantity g. Both k and g depend only on the parameters of temperature, density and chemical composition, and extensive tables have been obtained and made generally available (see Seaton 1997). Use of the diffusion approximation for the solution of the equation of radiative transfer is strictly valid only for 1 but can be expected to give usable results for t $ 1: A check is provided on comparing the results of Alecian & Michaud (1981) for g rad of Mn in the region of t. 1 with those obtained using data from Seaton (1997). The agreement is reasonably close. In the region concerned, both calculations depend mainly on the use of data from Kurucz (1990). The main difference is in the calculation of F n : Alecian & Michaud (1981) use results from an atmosphere calculation whereas we use the diffusion approximation. Diffusion theory gives v to be proportional to a diffusion coefficient D which, in turn, is proportional to (1/N). As N decreases in the upper layers of a stellar atmosphere the usual equations of diffusion theory would eventually give values of v larger than thermal velocities, and these equations would then no longer be valid. Different physics, more closely related to that used in theories of stellar winds (see Babel 1995, 1996) is then required in order to obtain the diffusion velocities. 2 No difficulties arise in imposing lower boundary conditions, since at sufficiently great depths in the envelopes diffusive movements remain small over time-scales comparable with those for main-sequence evolution. The question arises: can one obtain solutions of the continuity equation for the envelopes without having to worry about obtaining solutions for the atmospheres? It will be shown that one can for the case of diffusion of iron-group elements in HgMn stars: it is possible to define values t u of t such that solutions for t $ t u depend only on conditions for t $ t u. It should eventually be possible to split the difficult overall problem into two more manageable parts: (i) first obtain solutions for the inner regions of t $ t u ; (ii) then obtain solutions for the outer regions of t # t u using the known concentrations at t ˆ t u to provide lower boundary conditions. The present paper is concerned with the development of numerical methods for obtaining the envelope solutions, and for that purpose we use some rather simple envelope models. We employ models having values of T eff and surface gravities g similar to those of observed HgMn stars, and masses obtained from approximate interpolations from results of evolutionary calculations. We take the initial conditions, at t ˆ 0, to be that no diffusive separations have occurred and then follow the diffusive movements over times comparable to main-sequence lifetimes. In this approach we allow neither for changes in stellar structures due to normal main-sequence evolution nor for further changes that result from changes in opacities. In a more complete treatment one should, of course, take the initial conditions to be at, or even before, the time when a star reaches the main sequence, and one should follow the evolution taking account of the modifications of opacities that result from diffusive movements. The diffusive movements are governed by the continuity equation (conservation of numbers of particles). Section 2 describes methods used to solve that equation. Various technical details, such as the computation and smoothing of the stellar models, expressions 2 A number of studies (Alecian & Grappin 1984; Alecian 1986, 1996) have been concerned with diffusion in stars having superficial convection zones. For such stars the problem discussed here does not arise. used for radiative accelerations and diffusion coefficients, and finite-difference formulae, are relegated to Appendix A. Results for diffusion of the iron-group elements are given in Section 3, and Section 4 gives a summary and discussion. Since the present work was started, three important papers have been published using monochromatic opacities from the OPAL project to calculate radiative accelerations. The first, by Richer et al. (1998), describes the g rad calculations; the second, by Turcotte et al. (1998a), is concerned with solar evolutionary models calculated allowing for element diffusion; and the third, by Turcotte et al. (1998b), with similar calculations for F stars (masses M of 1.1 to 1.5 M ( ). The results for g rad are discussed further in Appendix B. Turcotte et al. (1998b) find their numerical methods to be progressively more unstable for their larger values of M. Similar instability problems have been reported in a number of previous papers and have been encountered in earlier work by the present author. One of the main aims of the present work is to develop numerical methods that are stable and that can be used for more massive stars (we consider stars with masses up to 4.5 M ( ). 2 THE CONTINUITY EQUATION For diffusion in a spherically symmetric star the continuity equation is r 2 t n 1 r r2 nv ˆ0; where r is the stellar radial coordinate, n is the number density for the diffusing atoms, and v is their diffusion velocity. We assume that there are no macroscopic motions. We put n ˆ NAx; where N is the total particle density, A is the fractional abundance of diffusing atoms when no diffusion occurs and x is the factor by which their abundance is modified by the diffusion process. We change to a depth variable x: dx ˆ 2w dr with w ˆ N r=r 2 ; where R is the total radius of the star. We define F ˆ wxv; and refer to F as the particle flux (but we note that the true flux ± atoms per unit area and unit time ± is nv ˆ A R=r 2 F. For any function y of x and t we use the notations _y ˆ y= t and y 0 ˆ y= x: From equations (1) to (5) we obtain the continuity equation in the from _x ˆ F 0 : 2.1 The diffusion velocity We take the diffusion velocity, neglecting thermal diffusion (see Appendix A), to be v ˆ D m=k B T g rad 2 g grav 2 ln n=n = rš; where m is the mass of a diffusing ion, k B is the Boltzmann constant and expressions used for the diffusion coefficients D are given in Appendix A

4 Defining f ˆ wxd m=k B T g rad 2 g grav Š and b ˆ w 2 D; 8 9 Diffusion in envelopes of HgMn stars Inclusion of the CG term as a perturbation Suppose that F 0 ˆ F x; t 0 is known. We put f 0 ˆ F 0, and calculate f at t ˆ t 0 1 dt using methods described in the preceding subsection. Having obtained that solution we can then include the CG term as a perturbation: we obtain F ˆ f 1 dtbf 00 0 : 18 F ˆ f 1 bx 0 : We shall refer to f as the driving term and to bx 0 concentration gradient (CG) term. 10 as the The effect of including the CG term is to give a slow decrease in the heights of the maxima and a slow increase in the heights of the minima. 2.2 Linearization The quantity f depends on t because x depends on t. We may therefore put _f ˆ a _x; 11 where a ˆ f = x ˆ wd k B T=m x g rad 2 g grav Š= x; and hence _F ˆ a _x 1 b _x 0. Using (6) we obtain _F ˆ af 0 1 bf 00 : The quantity a depends on x and x, i.e. a ˆ a x; x. Let us suppose that x and F are known at a time t 0, i.e. x 0 ˆ x t 0 and F 0 ˆ F t 0. We use a time-interval dt, and for the time-step from t 0 to t ˆ t 0 1 dt we solve (13) with the linearization of putting a ˆ a x; x 0. This gives a new value of F, F ˆ F t. Given x 0, F 0 and F we can now solve (6) to obtain a new value of x: and then recalculate a and go to the next time-step. 2.3 Solutions neglecting the concentration-gradient term For much of the diffusion process the CG term, bf 00 in (13), is a good deal smaller than af 0. Neglecting the CG term we have _f ˆ af 0 ; 14 and we can interpret a as the velocity for movements of features in f. Fora. 0 features move outwards and for a, 0 they move inwards. The general solution of (14) is f ˆ F t 1 y where y ˆ 1=a dx 15 and where F(z) is any differentiable function of z. We can therefore put f x; t 0 1 dt ˆf x 0 ; t 0 ; where x 0 is such that dt 1 x x o 1=a dx ˆ 0: For each value of x we may therefore calculate x 0 and then calculate f(x,t) by interpolations in f(x,t 0 ). A point to note is that, when the CG term is completely neglected [b ˆ 0 in (13)], the heights of the maxima and minima in F do not change as the diffusion proceeds. 2.5 Finite-difference formulae The finite-difference formulae used are given in Appendix A. We include discussion of cases for which the CG term cannot be treated as a small perturbation, and hence for which the methods of Sections 2.3 and 2.4 cannot be used. 3 DIFFUSION OF THE IRON-GROUP ELEMENTS Unless the contrary is stated, we give all numerical values for times, t, in units of 10 6 yr. 3.1 The behaviour of f as a function of x It is important to understand the behaviour of f as a function of x. Fig. 3 gives an illustrative plot, for manganese in the K model at a depth-point with log T ˆ4:9. For x ˆ 0 we have f ˆ 0 due to the external factor of x in (8). For the iron-group elements we have g rad. g grav for small x and hence f increasing with increasing x. However, g rad decreases with increasing x due to saturation effects, and eventually a maximum value f mx of f is reached at x ˆ x mf. For sufficiently large values of x, atx ˆ x stat, we have g rad ˆ g grav and f ˆ 0 (corresponding to the concentration c stat used in Section 1). The functions x mf and f mx play an important role in the theory. We distinguish three regions. Regions A have x, x mf and hence a. 0. Features in f move outwards. Regions B have x. x mf and hence a, 0. Features in f move inwards. Regions C have x in the vicinity of x mf. In regions A and B we can calculate F using the method described in Section (2.3) provided that jaf 0 jbf 00 j. We take Regions C to be those for which that condition is not satisfied, and for which equation (13) must be solved without further approximation. 3.2 Boundary conditions If one attempts to solve an equation such as (13) in some finite range, say x u # x # x l (x u is the upper boundary, x l the lower boundary), it is in general necessary to specify values of the derivatives with respect to x at the ends of the range, x ˆ x u and x ˆ x l. We do not experience any difficulty with the lower boundary condition since we go down to depths at which all diffusive movement has practically stopped. There are more serious problems in imposing upper boundary conditions.

5 1012 M. J. Seaton Figure 3. The driving term f, defined by equation (8), for Mn in the K model at a depth point with log T ˆ4:9, plotted against x where x is the factor by which the Mn abundance is modified. Note that f has a maximum value of f mx at x ˆ x mf.atx ˆ x stat, f ˆ 0. At the upper boundary let the Rosseland-mean optical depth be t u. A first requirement is t u $ 1, in order that the approximations used to calculate g rad should be valid. A second requirement is that we should be in a Region A at x ˆ x u. If that condition is satisfied and if the CG term is neglected completely [bf 00 ˆ 0 in equation (13)] then there is no problem, since the solutions for x $ x u depend only on the functions for x $ x u. When the CG term is included, the solutions for x $ x u depend, in principle, on the functions for x # x u but if jaf 0 jbf 00 j at x ˆ x u ± which is the case in practice ± that dependence is very weak. We have tried two methods. In the first we estimate F 00 at x u using only information about F for x $ x u. This is satisfactory in most cases, but there are cases for which this method leads to numerical instabilities. A second, and neater, method is to multiply bf 00 by a function y(x), such as y x ˆ 2=p arctan a x 2 x u m Š with a. 0 and m $ 1, which is zero at x ˆ x u and tends to unity for large values of x, and to choose the parameters in y to be such that y is close to unity long before any Regions C are reached (it is only in Regions C and their vicinities that the CG term is of real importance). The two methods give similar results but the second is more stable. 3.3 Overview for the iron group We take the initial conditions to be 3 that x ˆ 1 at time t ˆ 0. We take the top of the envelope to be at t ˆ 1 and seek solutions for t $ 1. In some cases we have to content ourselves with solutions for t $ t u with t u. 1. At t ˆ 0 we have x ˆ 1, x 0 ˆ 0 and hence F t ˆ 0 ˆf t ˆ 0. In Fig. 4 we plot f mx and f t ˆ 0 against log(t) for the four irongroup elements in the K model. In all cases (except for that of iron at great depths) we have f, f mx, and initially all features in f move outwards. We shall consider the four elements in order of increasing complexity. Manganese. f has a maximum at log T. 5:2, that is to say in the vicinity of the Z-bump in opacities. This maximum moves 3 We shall see that there are rapid variations of F and x for small values of t. Our detailed results for this region are not of much interest, since our initial condition ± that x ˆ 1att ˆ 0 and that the radiative flux is then suddenly switched on with full power ± is not realistic. Results obtained for later times are not sensitive to details of our starting condition. Figure 4. Dashed lines: f mx against depth variable log(t) for Cr, Mn, Fe and Ni in the K model. Full lines: the driving term f at time t ˆ 0 (unmodified abundances). outwards and its height remains smaller than f mx. The entire movement is therefore in a Region A. Chromium. The maximum in f is higher than the minimum of f mx in the vicinity of the argon barrier. As this maximum moves outwards a Region C is eventually encountered. The flux can however, pass through the barrier and we can continue the integrations out to t ˆ 1. Nickel. A novel feature is that f mx becomes small in the outer regions, giving an outer barrier. In the absence of solutions for the atmospheres t, 1, at later times we can obtain solutions only for t $ t u with t u. 1. Iron. Iron differs from the other iron-group elements in that it has a much larger abundance, and the values of f mx and of f are much reduced, due to saturation effects. The maximum in f is much higher that the minimum in f mx in the region of the argon barrier. 3.4 Manganese Fig. 5 show the behaviour of F for Mn in the K model. The top part shows log(f) against log(t) for times t ˆ 0, 1, 2, 3 and 4. At t ˆ 0, F has a maximum at log T. 5:3, the region of the Z-bump. This maximum moves outwards but is slowed down in the vicinity of the Ar barrier, at log T. 4:8. Just after t ˆ 4 the maximum has passed the barrier and then moves rapidly out to the top of the envelope. The lower part of Fig. 5 shows F at later times, t ˆ 4, 5, 10 and 100. Between t ˆ 4 and 5 there is a rapid drop in F in the outer layers, while for t. 5 the rate of decrease is much slower. For all times up to t ˆ 100 (10 8 yr) there is practically no change in F in the region of the Ne barrier, at log T. 6. Fig. 6 shows the behaviour of x, for the case of Fig. 5. The upper part shows a rapid increase in x in the outer layers while the

6 Diffusion in envelopes of HgMn stars 1013 Figure 7. For Mn, the function x at the top of the envelope t ˆ 1 as a function of time t for the models with T eff ˆ 10, 11, 12, 13, 14 and 15. Figure 5. The diffusive flux F for Mn in the K model for various times t in units of 10 6 yr. The upper plot shows, as dashed lines, values of f mx and as full lines the fluxes for t ˆ 0, 1, 2, 3 and 4. The maximum in F works its way around the Ar barrier, which is at log T. 4:8. Just after t ˆ 4 that maximum has passed the barrier and reaches the top of the envelope. The lower plot shows F at later times, t ˆ 4, 5, 10 and 100. There is a sharp drop in F in the outer regions between t ˆ 4 and 5, then a much slower decline. Figure 6. The abundance multiplier x for Mn in the K model at times t as in Fig. 5. lower part shows a decrease after t. 4. The enhanced concentrations in the outer layers are at the expense of depletions in the region of log T. 5. Fig. 7 shows values of x at the top of the envelope, x t ˆ 1, in the models with T eff ˆ to K. Maxima in x(1) are reached at times just after those at which the maximum in F passes the Ar barrier. There is then a very rapid drop followed by a slow decline. Fig. 8 shows observed atmospheric abundances for Mn in the HgMn stars. The filled circles are from UV observations by Smith & Dworetsky (1993) and the open ones from optical observations by Adelman et al. (1996). The UV observations are for resonance lines formed high in the atmospheres, while the optical lines are formed at somewhat greater depths. There do not appear to be any systematic differences between the UV and optical results. 4 We compare the observations with abundances calculated for the bottom of the atmosphere, at t ˆ 1. The upper histogram gives calculated maximum abundances and the lower one (dashed lines) gives results at t ˆ 100. The agreement between theory and observations is encouraging. Some stars have atmospheric abundances lower than those for the upper histogram on Fig. 8 because high abundances occur only as transient phenomena (see Fig. 7). It was shown in Fig. 7 that the maxima in x(1) all occur at fairly small vales of t and it follows that the HgMn stars with high Mn abundances must be quite young, having spent no more than about 10 7 yr on the main sequence. Alecian & Michaud (1981) made detailed calculations for Mn in the atmospheres of HgMn stars, with allowance for possible departures from local thermodynamic equilibrium in calculating the populations of the Mn levels, and obtained values of Mn atmospheric abundances for static solutions, x ˆ x stat. A direct comparison of their results with those from modern observations is given in Fig. 10 of the recent paper by Jomaron et al. (1999). The trend of observed maximum abundances, against T eff,isin good agreement with the trend for x stat, but the observed maximum abundances are lower than x stat by at least 0.7 dex. 4 Since the original version of the present paper was submitted an important further paper on the determination of Mn abundances in HgMn stars has been published by Jomaron et al. (1999). For individual stars they compare abundances from earlier UV observations with those from new optical observations. They obtain agreements to within 0.2 dex except for three stars which belong to binary systems.

7 1014 M. J. Seaton Figure 8. Atmospheric abundances x for Mn in HgMn stars. Filled circles: from UV observations by Smith & Dworetsky (1993). Open circles: from optical observations by Adelman et al. (1996). Upper histogram: maximum calculated values of x at t ˆ 1 (see Fig. 7). Lower histogram: calculated values of x after diffusion for 10 8 yr. Figure 11. Values of x at the tops of the envelopes for Cr in models with T eff ˆ 10 to 15. Figure 9. Fluxes F for Cr in the vicinity of the Ar barrier, in the K model. Dashed line: the function f mx. Full lines: values of F for t ˆ 1to7 in steps of 1. At the larger depths, F decreases with increasing t. Figure 12. Atmospheric abundances for Cr in the HgMn stars. Data references as for Fig. 8. flux passes through the Ar barrier, the flux at the top of the envelope varies slowly as a function of t. Fig. 12 gives a comparison of observed atmospheric abundances for Cr with those calculated for t ˆ 1. While the agreement is not bad for the cooler HgMn stars (say T eff # K), the observed abundances are a good deal smaller than those calculated for the hotter stars. Figure 10. Abundance multipliers x for Cr in the K model. Dashed curve: values of x mf. Full curves: values of x for times as in Fig Chromium We have seen, from Fig. 4, that for Cr the maximum flux at t ˆ 0 is higher than the minimum at the bottom of the Ar barrier. As the diffusion proceeds a Region C is therefore encountered. Fig. 9 shows, for the K model, F as a function of log(t) for times t ˆ 1 to 7 in steps of 1. The barrier has not quite been reached at t ˆ 1 and has been passed by t ˆ 7. Fig. 10 gives a corresponding plot for the behaviour of x. We note that there are very rapid variations in F and x, in both depth and time, between t ˆ 6 and t ˆ 7. Fig. 11 shows log(x(1)) against log(t) for Cr. Compared with the results for Mn (Fig. 7), those for Cr show maxima that are much more flat-topped. This is because, during the time that the 3.6 Nickel The top part of Fig. 13 shows fluxes for Ni in the K model. The dashed line shows f mx and the full lines show F for t ˆ 10, 20, 30, 50 and 100. At t ˆ 0 we have F, f mx for all t $ 1 but the outer barrier is reached at t ˆ 0:09 and at all later times we can obtain solution only for t. 1. The initial maximum in F is slightly higher than the minimum of f mx in the Ar barrier, but the height of this maximum slowly decreases and F works itself around the Ar barrier without a Region C ever being encountered. The lower part of Fig. 13 shows the corresponding values of x for t ˆ 10, 20, 30, 50 and 100. In regions with log(t) less than about 5.1 there are large enhancements at all later times. These enhancements can have important consequences for the calculation of opacities. Fig. 14 shows log(x) against log(t) for the six stellar models at times t ˆ 10, 30 and 100. For Fig. 14 we use log(t) as depth variable in order to emphasize the fact that the results do not go out to t ˆ 1. Although for Ni we do not have any results to compare with observed abundances, for the sake of completeness we show, in Fig. 15, the observational results. It appears that Ni is generally depleted in the atmospheres of the HgMn stars. It

8 Diffusion in envelopes of HgMn stars 1015 Figure 13. Diffusion of Ni in the K model. The upper plot shows f mx as a dashed line and F as full lines for times t ˆ 0, 10, 20, 30, 50 and 100. In the outer layers F increases with increasing t. The calculations are stopped when the calculated values of f approach the values of f mx of the outer barrier. The lower plot shows x mf as a dashed line and values of x as full lines for t ˆ 10, 20, 30, 50 and 100. In the outer layers x increases with increasing t. should be of interest to make calculations for the atmospheres and outer parts of the envelopes, using methods similar to those discussed here to obtain the fluxes coming from the deeper layers. With unmodified abundances, Ni makes contributions to Rosseland-mean opacities which are significant although not large. We see from Figs 13 and 14 that diffusion can lead to increases in Ni abundances by factors of up to 100 in the outer layers log T # 5:2; log t # 5Š. Fig. 16 shows, for the K model, opacities with normal abundances x ˆ 1 and after diffusion of Ni for times of t ˆ 30 and t ˆ 100. It is seen that some substantial enhancements occur. Figure 14. Abundance multipliers x for Ni in models with T eff ˆ to K and for times t ˆ 10, 30 and 100. The depth variable used here is log(t) where t is the Rosseland-mean optical depth. The outer barrier for Ni prevents us from continuing the calculations out to t ˆ 1. Full lines, t ˆ 10: short-dashed lines, t ˆ 30; long-dashed lines, t ˆ Iron There is an initial rapid variation of F for iron. At t ˆ 0, with x ˆ 1, we have F, f mx for all t $ 1 (except for very great depths). Fig. 17 shows f mx, and F at t ˆ 0 out to t ˆ 1. It is seen that F has a maximum at log T. 4:7, in the region between the outer barrier and the Ar barrier. This maximum quickly moves out and reaches the outer barrier, forcing us to adopt larger values of t u. Fig. 17 shows the fluxes for t ˆ 0:0, 0.2, 0.4 and 1.0. At t ˆ 1:0 we have t u ˆ 15:2 and we cannot, at subsequent times, go back to a smaller value. Fig. 18 shows the behaviour of F in the vicinity of the Ar barrier, for t ˆ 10, 30 and 100. The barrier is reached just after t ˆ 10. Fig. 19 shows the corresponding behaviour of x. In the Region C of the Ar barrier, x has a peak with a maximum value of Figure 15. Obseved atmospheric abundances for Ni. Data references as for Fig. 8. Owing to the presence of an outer barrier for Ni we do not have any calculated results to compare with the observations. about 200 at t ˆ 100. Fig. 20 shows log(x) against log(t) for t ˆ 10, 30 and 100 in the six stellar models. The full line in Fig. 21 shows opacities with normal abundances, for the K model, against log(t). The maximum in the vicinity of log t. 5 log T. 5:3 is the Z-bump and is produced by iron and, to a lesser extent, the other irongroup elements. The effect of diffusion is to reduce the iron abundance in the vicinity of the Z-bump and to give enhancements

9 1016 M. J. Seaton Figure 16. Roseland-mean opacities in the K model with modified Ni abundances. The opacities k are in the usual units of cm 2 g 21. The full lines shows opacities with normal abundances; the short-dashed line after Ni diffusion for a time t ˆ 30; and the long-dashed line for t ˆ 100: Figure 19. Fe in the K model. Dashed lines: values of x mf. Full lines: values of x for t ˆ 10, 30 and 100. In the vicinty of the peak in x, at log T. 5:15, x increases with increasing t. Figure 17. Initial movements of F for Fe in the K model. The dashed line shows f mx and the full lines show F for t ˆ 0:0, 0.2, 0.4 and 1.0. The curve for t ˆ 0 goes out to the top of the envelope, at t ˆ 1, and has a maximum between the outer barrier and the Ar barrier. This maximum moves outwards and the presence of the outer barrier forces us to curtail the calculations at values of t greater than 1. Figure 18. Fluxes for Fe in the K model in the vicinity of the Ar barrier. Dashed line: values of f mx. Full lines: fluxes F for t ˆ 10, 30 and 100. For the larger values of log(t), F decreases with increasing t. by factors of up to 100 in the vicinity of log t ˆ5:1. We do not obtain increases in total Rosseland-mean opacities by factors of order 100 because the enhanced iron is in a region, near to the Ar barrier, where the iron opacity cross-section is much reduced. Figure 20. Values of x for Fe in models with T eff ˆ to K, for times t ˆ 10, 30 and 100. Full lines, t ˆ 10: short-dashed lines, t ˆ 30; long-dashed lines, t ˆ 100. Fig. 21 shows results for t ˆ 30 and t ˆ 100. The Z-bump is replaced by a Z-spike, with opacity enhancements by factors of up to 4. Fig. 22 shows the behaviour of f mx in the upper parts of the envelopes of the various models. With the exception of the K model, we see that f mx has minima at values t m of t near the

10 Diffusion in envelopes of HgMn stars 1017 Figure 21. Rosselan-mean opacities in the K model with modified abundances for Fe. Units and full line as for Fig. 16. Short-dashed lines, after diffusion of Fe for a time t ˆ 30; long-dashed lines for t ˆ 100. Figure 23. Atmospheric abundances x for Fe in HgMn stars. Observed values as in Fig. 8. The histogram shows abundances calculated using methods described in the text. Figure 22. Values of f mx for Fe in the outer parts of the envelopes for the various stellar models. Curves labelled with values of T eff. tops of the envelopes log t m. 1 for the K model). At the tops of the envelopes, t ˆ 1, we would therefore expect to obtain Regions A. The initial rapid variations in F (see Fig. 17) prevent us from reaching these regions at later times, but we can obtain some estimates of values of x at t ˆ 1. From Figs 9 and 18 we see that, when F is passing through a barrier, at depths somewhat less than that of the minimum in the barrier, F has a nearly constant value, nearly equal to the value of f mx at the bottom of the barrier. We therefore assume that, in the outer regions of t, t m, F will be approximately equal to f mx at t ˆ t m, and using these values of F we calculate x at t ˆ 1. For T eff ˆ K we make the ± possibly dubious ± assumption that F t ˆ 1 ˆf mx t ˆ 1. Fig. 23 compares the results obtained with observed atmospheric Fe abundances. The measure of agreement is not too bad. The coolest stars considered have abundances larger than those calculated. We note that stars with T eff, K may have outer convective zones. 3.8 Modified opacities Figs 16 and 21 illustrated the changes in opacity that occur due to diffusion of Ni and Fe. There are also some transient changes which occur for smaller values of t as enhancements of Cr and Mn abundances reach the upper parts of the envelope. In Fig. 24 we show changes in opacity due to diffusion of all four iron-group elements. Let k be the unmodified opacity and dk(k) the change Figure 24. Changes in opacity, for the K model, due to diffusion of all four iron-group elements, Cr, Mn, Fe and Ni. The unmodified opacity is k and that after diffusive movements is k Dk. The figure shows Dk/k at times t ˆ 10 (short-dashed line), t ˆ 30 (long-dashed line) and t ˆ 100 (full line). in opacity due to diffusion of element k. We assume the changes to be additive, which is not strictly correct but is adequate for our present purpose. The combined modified opacity is then k 1 Dk where Dk ˆ Pkdk k. Fig. 24 shows Dk/k for the K model at times t ˆ 10, 30 and 100 [the plot is made only for log t $ 2 because, at the later times, we are unable to obtain Ni abundances for smaller values of log(t)]. The changes in opacities can lead to changes in the structures of the stellar models. At t ˆ 10 no large changes in opacity have yet occurred and hence there should not be much modification in structure up to the times at which the maximum abundances of Cr and Mn reach the tops of the envelopes (Figs 7 and 11). Larger changes occur at later times. The spike at log t. 4:25 log T. 5:15Š is due to a pile-up of Fe in the vicinity of the Ar barrier, and opacity enhancements at smaller values of log(t) are mainly due to a pile-up of Ni before its outer barrier. In future work it will be of interest to recalculate the models allowing for the changes in opacities. It will be necessary to consider whether the opacity spike leads to the introduction of a new convection zone. It will also be of interest to consider possible implications for pulsation properties. 3.9 Escape from the atmospheres For atoms with number density n, the column density in the

11 1018 M. J. Seaton atmosphere is 1 q ˆ n dr; 19 r 1 where r 1 is the value of r at t ˆ 1. The number of atoms entering the atmosphere from the envelope, after a time t, is p ˆ t 0 nv tˆ1 dt: 20 For Mn in the K model we obtain an initial value of q ˆ 1: cm 22 (with x ˆ 1) and p ˆ 4: cm 22 after t ˆ 10 7 yr. If there were no escape at the top of the atmosphere we would obtain a mean atmospheric concentration enhancemnent, after time t, of x ˆ p=q ˆ 2: Since the observed Mn atmospheric concentration is never much larger than x ˆ 100 we conclude that there must be escape of Mn at the top of the atmosphere. Similar results are obtained for other atoms. For iron the flux reaching the atmosphere is approximately equal to the flux f mx at the bottom of the outer barrier. Without escape of iron we would obtain a mean atmospheric concentration of x ˆ 6: after 10 7 yr compared with the observed concentration of x, SUMMARY AND COMMENTS 4.1 Numerical methods We use a depth variable x and a time variable t and take the concentration x(x,t) of a diffusing element to be such that x ˆ 1 when no diffusion occurs. For any function y(x,t) we put _y ˆ y= t and y 0 ˆ y= x. The continuity equation is _x ˆ F 0 where F is the diffusive flux. We have F x; x ˆf x; x 1 b x x 0 where f is the driving term, proportional to x g rad x; x 2 g grav x Š, and bx 0 is the concentration gradient (CG) term. We obtain _F ˆ a x; x F 0 1 b x F 00 as the equation to be solved for F, where a ˆ f = x. For the time-step t 0 to t 0 1 dt we put a ˆ a x; x t 0, obtain a new value of F at t 0 1 dt, and then use _x ˆ F 0 to obtain a new value of x, and hence a new value of a. We define x mf to be such that at x ˆ x mf we have a ˆ 0: for x, x mf we have a. 0; and for x. x mf we have a, 0. Regions A have x, x mf and jaf 0 jbf 00 j. A first approximation is obtained on solving _f ˆ af 0, i.e. neglecting the CG term. We can then interpret a as the velocity for movement of features in f. Features move outwards in regions A. The CG term can be allowed for in Regions A using perturbation methods. Regions C are those for which af 0 is not large compared to bf 00. Solutions of _F ˆ af 0 1 bf 00 can be obtained using stable numerical methods and boundary conditions in x determined by solutions in neighbouring Regions A. 4.2 The upper boundary The top of the envelope is defined to be at t ˆ 1 where t is the Rosseland-mean optical depth. We seek solutions for the envelopes which are independent of those for the atmospheres (regions with t, 1). For envelopes let x u be the smallest value of x that puts us in a Region A. If b ˆ 0atxˆx u the solutions for x $ x u are entirely independent of the behaviour of the solutions for x, x u ; and they are practically independent if b is sufficiently small at x u. We can obtain solutions for envelopes, independent of those for atmospheres, for values of x $ x u. In some cases x u corresponds to an optical depth t u. 1. All of the present calculations give bf 00 atx ˆ x u to be very small. We cannot, of course, exclude the possibility that inclusion of atmospheric solutions will lead to larger values of bf 00 at x ˆ x u. 4.3 Diffusive movements We start with x ˆ 1 at time t ˆ 0 and attempt to follow the diffusive movements of the iron-group elements for times of 10 8 yr. Initially F has maxima in regions of log T. 5:3, where the dominant ionization stages are intermediate between Ar-like and Ne-like. Those maxima move outwards but are slowed down in the vicinity of the Ar barrier (dominant stages near Ar-like) Mn For Mn the entire movement of F, out to the top of the envelope, is in a Region A. The calculated abundances at t ˆ 1 (Fig. 7) can be compared with observed atmospheric abundances (Fig. 8). We obtain enhancements, by factors of up to 100, when the maximum in F reaches the top of the envelope. This occurs at times never significantly larger than 10 7 yr, indicating that the HgMn stars are fairly young Cr For Cr, Regions C are encountered in the vicinities of the Ar barriers but the solutions can still be continued out to t ˆ Ni For Ni there is an outer barrier (small values of x mf in the outer parts of the envelope) which prevents us from obtaining solutions out to the top of the envelope. Large enhancements for Ni are obtained before the barrier is reached Fe For Fe the maximum in F has not passed a very deep Ar barrier after diffusion for 10 8 yr. An outer barrier is encountered for Fe but we are able to obtain some estimates of values of x for Fe at t ˆ 1, and hence make comparisons with atmospheric abundances (Fig. 23). 4.4 Opacities The diffusive movements for the iron-group elements lead to changes in opacities which are particularly important at later times, say after 10 7 yr. The normal Z-bump at log T. 5:3 is replaced by a Z-spike at log T. 5:1 (Fig. 21), which results from a pile-up of Fe in the region of the Ar barrier. There are also opacity enhancements in the region of log T, 5 due to a pile-up of Ni against its outer barrier. Fig. 24 shows the enhancements in opacities that occur allowing for diffusion of all four iron-group elements. There is a need for further work allowing for evolution of the models, both normal main-sequence evolution and changes in structure which result from changes in opacities due to both diffusive movements of the iron-group elements and to gravitational settling of He.

12 Diffusion in envelopes of HgMn stars The atmospheres It is shown that, if there were no escape from the tops of the atmospheres, the accumulation of iron-group elements would give atmospheric abundances much larger than those observed. Selective escape of iron-group elements must occur at the tops of the atmospheres. There is a need for further study of diffusion processes in the atmospheres, using as lower boundary conditions the fluxes from the envelopes, calculated using methods similar to those described here, and with upper boundary conditions allowing for escape at the tops of the atmospheres. Seaton M. J., 1996a, Ap&SS, 237, 107 Seaton M. J., 1996b, Phys. Scr., T65, 129 Seaton M. J., 1996c, MNRAS, 279, 85 Seaton M. J., 1997, MNRAS, 289, 700 Seaton M.J., Yu Yan, Mihalas D., Pradhan Anil K., 1994, MNRAS, 266, 805 (SYMP) Smith K. C., 1996, Ap&SS, 237, 77 Smith K. C., Dworetsky M. M., 1993, A&A, 274, 335 The Opacity Project Team, 1995, The Opacity Project, Vol. 1. 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We integrate inwards with specified values for the stellar radius R, effective temperature T eff, and stellar mass M. The integrations are not extended to regions where nuclear energy generation occurs. We use 5 ln(t) as independent depth variable. A fine depth-mesh is required for the present work and we therefore use a small interval of 0.01 in log(t) for the computation of the models. We use opacities 6 from Seaton (1996c) with the mixture S92 of SYMP. We do not include any convection zones, it being assumed that sufficient gravitational settling of He has occurred for the elimination of a He 1 ±He 12 convection zone (consistent with observed surface depletions of He in the HgMn stars). We do not, however, allow for any He depletions in calculating opacities. A1.1 Parameters of the models We compute models for T eff ˆ 10, 11, 12, 13, 14 and 15. In all cases we take the surface gravity to be log g sfce ˆ3:87, which is the mean spectroscopic value for HgMn stars obtained by Smith & Dworetsky (1993) [they do not find any systematic dependence of log(g sfce )ont eff ]. We use values of stellar masses M from approximate interpolations in the evolutionary results of Bertelli et al. (1986). Adopted values of M/M ( are given in Table A1. We use a perfect-gas equation of state: effects of electron degeneracy are not important in regions where there are significant diffusive movements for the iron-group elements. A1.2 Smoothing of the models In order to have a complete resolution of all features in the monochromatic opacities it would be necessary to use some 10 7 or even 10 8 frequency points, which would be quite prohibitive. We therefore employ data calculated using techniques of opacity sampling, with 10 4 frequency points for opacities (see SYMP) and 10 5 points for radiative accelerations (Seaton 1997). Use of those techniques can lead to errors of about 1 or 2 per cent. For the present work we require all data for for the models to be smooth to a much higher accuracy. Some further smoothing of opacities and radiative accelerations is effected using methods described by Seaton (1993) but they are not adequate to give the degree of 5 We consistently use the notations of ln for log e and log for log Tables of opacities, and codes for interpolation and smoothing, are available from CDS. See Seaton (1996c).