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LTE

Transfer.

Vol. 6, pp. 581-590.

Pergamon

Press Ltd.,

1%6. Printed

in Great

Britain

MODEL ATMOSPHERES WITH LINE BLANKETING REVIEW AND FUTURE PROSPECTS D. MIHALAS Princeton University Observatory. Princeton. New Jersey

IT GIVES me great pleasure to be here with you today, and I would like to thank our gracious hosts for inviting me to speak on the computation of model atmospheres including line-blanketing effects. I would like to try to summarize some of the work in this area from two points of view : computational and physical, and to criticize it from the standpoint of what is needed now to derive more realistic models. At the outset I wish to emphasize that while I will often excuse the physical inadequacy of an assumption solely on the basis of its computational expediency, I do not mean to imply approval of this approach in principle. It is no more or less than a simple question of ease of computation. Indeed in some cases, our present computers and techniques of solution are not yet fast and powerful enough nor our insight yet deep enough to allow treatment of harder problems in a reasonable amount of time. Such simplified calculations have merit only insofar as they prepare us to do more realistic calculations, and often they allow us to see quite directly the effect of removing each successive assumption. In short, while I am going to talk mostly about calculations carried out under the assumption of LTE, I would like to affirm strongly the position that the a priori assumption of LTE is physically totally unjustified, and that calculations carried out with this assumption should be regarded as only exploratory. We may hope they will give us approximate answers to sonte questions, but we should not expect them to reproduce observation in close detail. Indeed, even if they do reproduce features of the observations closely, we must beware the possibility that such agreement is not “accidental” in the sense that we have adjusted parameters until we do fit observation, but these parameters are not related to any physically relevant quantity in a meaningful way. As an example I have in mind attempting to infer boundary temperatures of atmospheres from line profiles calculated in LTE. One can always choose a run of temperature to force agreement between computed and observed profiles, but this temperature is not related to any physical property of the atmosphere. With the understanding, then, that we are only discussing prototype problems which represent high-order, and possibly unrealistic, abstractions of the problem, I would like to review a few results. Let me remind you of work that was current three or four years ago. Then we were actively interested in the solution of the energy-balance (flux-constancy) problem for model atmospheres. The problem we typically attempted to solve assumed:

1. Plane-parallel

homogeneous

layers 581

D.

582

2. Radiative 3. LTE

equilibrium-hydrostatic

MIHALAS

equilibrium

Assumption 1 is reasonable insofar as the plane-parallel geometry is concerned ; the assumption of homogeneity of the layers is at best questionable (particularly when confronted by direct observational evidence of inhomogeneities in so pedestrian a star as the sun.) The assumption of hydrostatic equilibrium is a reasonable starting point. Ultimately we might hope to include velocity fields, not only in a “kinematical” way by allowing for macroscopic velocity fields in the calculation of line profiles, but also in a “dynamical” way where we account completely for the coupling between the macroscopic velocity field and the thermodynamic state of the gas (including of course the radiation field). Such a calculation appears to be distant at present; moreover there are still many fascinating aspects of the static problem which merit exploration before we introduce the further complication of velocities. The radiative equilibrium assumption also neglects velocity fields which can dominate the energy balance condition in the outermost regions of the atmosphere. But even this .simple energy-balance condition had not yet been enforced in complete generality, and the kind of calculation we were doing was designed basically to enforce it. The LTE assumption was taken simply to avoid the much more difficult problem of solving properly the statistical equilibrium equations. Within this framework, considerable activity was shown by several groups in constructing non-grey radiative equilibrium models (for the continuum only). One great stimulus to this activity was the development of the KROOK-AVRETT procedure”’ which allowed rapid solution of the problem for a wide range of conditions. The resulting models, though still physically far from our desired goal, did represent a step forward in that they at least maintained energy equilibrium throughout-something previous models had not in general done successfully. These models were then used to study a variety of properties of the atmosphere and emergent radiation field. For example, calculations were made of the Balmer jump and of strengths of certain lines (e.g. Hy) to find how these features varied with the parameters characterizing the model. Also, on the basis of these models qualitative statements were made about the run of temperature throughout the atmosphere; a noteworthy feature was the sharp drop in the temperature of the outer layers (relative to T,,,) for effective temperatures around 25 000°K. Finally the models were used to calculate line strengths as a function of abundance and thus to carry out abundance analyses. All of these operations are perfectly permissible so long as we are talking about models. The essential leap of faith that can be severely doubted is to assert these results are relevant to stars. If we make the comparison between model and star in the spirit of determining a deJnitive result, clearly we are open to criticism. If we merely ask for rough estimates for certain features, perhaps we obtain fairly good results. These remarks particularly apply when we are discussing the properties of the continuum. In the discussion of strong lines, of course, this type of model cannot be trusted to give reliable information. Thus, if we mainly concern ourselves with estimates of colors and effective temperatures we may not do too badly. When these results had been obtained, we immediately asked if we could relax any of the assumptions. From the physical point of view it would have been most desirable to relax the assumption of LTE and to develop a physically consistent picture of the atmos-

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phere. This, however, was not done because the introduction of the effects of line blanketing and non-radiative transfer looked interesting and, frankly, easier. Also, a calculation of the statistical equilibrium was being carried out by Lecar which showed severe difficulties of obtaining convergence, and which looked very discouraging; I will comment further on this below. Consideration of non-radiative transfer took the specific form of treating a prototype convection problem. The convection terms were put in with the mixing-length theory (which has many problems of its own), and the goal was to construct a solution with simultaneous transport of energy by non-grey radiative transport and convective transport, demanding that the sum of the two terms be constant. An approach for doing this was suggested by me, and calculations for the transition range around FO were carried out. It turned out that it was possible to construct models in this way, but that from the point of view of the emergent fluxes, convection made no important change. Recently we ha\e tried to extend these calculation? to G subdwarfs and M-stars; we found that the original approach failed miserably, while a very much simpler method allowed inclusion of convection even for the M-stars. For this group of later-type models, a \ery significant difference in the emergent flux is found between radiative and simultaneous radiativeconvective models. Introduction of the line blanketing terms have followed two basic lines of approach. The first method I will call the “direct” approach. Here we simply introduce as many frequency points as is necessary to characterize each line included. The advantage of this approach is that it allows complete specification of the details of the line profiles (computationally, at least-physically there are serious objections). The disadvantage is that we soon are considering hundreds of frequency points and the computation takes correspondingly considerable time. This method works only when the major blanketing effects are caused by relatively few strong lines. This is the situation in the case for the early B-stars and the A-stars. The kinds of information we can hope to obtain from this kind of calculation are : 1. Estimates of broad-band colors and other features of the gross properties of the energy distribution in the continuum. 2. Improved estimates of effective temperatures and bolometric corrections for early type stars. 3. The temperature distribution in the atmosphere, both at the surface and at depth. 4. Details of the line profiles. Let us ask in a qualitative way which, if any, of these features we may hope to derive information about using an LTE calculation. Very roughly speaking, the main contribution of lines to categories 1 and 2 above comes from their mere presence; that is to say. the simple blocking-out of energy transport in certain walelength bands in the continuum grossly changes broad-band cdlors, and strongly redistributes energy into other wavclength intervals. Only the total line strength is important here, not the details of the profiles. Since total equivalent widths are only moderately sensitive to the details of the model, it is at least possible that our estimates for information in these categories may not be too far wrong using LTE models. Of course the remarks made above about “accidental” agreement still apply with futt force. The influence of lines upon the temperature distribution has been divided into consideration of effects at depth and near the surface for simple physical reasons. At depth the tines are again effective mainly through the

584

D. MIHALAS

simple blocking-out of frequency bands. This causes generally higher temperatures (compared to models without line blanketing) in intermediate layers if we require that a given amount of flux be transported; this is the “backwarming” effect. Near the surface, on the other hand, the temperature is determined by the condition that the total energy emitted must equal the total energy absorbed (here we mean true absorption; the scattering terms cancel out). This condition is quite sensitive to the details of line-formation. A very clear demonstration of this has recently been given by THOMA.S,(‘)which merits repetition here. Thomas’ book, incidentally, should be read carefully by every serious student of the subject. The radiative transfer equation for continuum plus lines is:

In the case where there are no lines, the condition of radiative equilibrium reduces to JrCB,dv

= /QVdv.

Thus the temperature parameter (entering B,) is fixed entirely by transfer in the continuum. Including now the effects of lines we find jrcCB,dv = bJ,dv-

J.,[S,-J.]dv.

Now writing rcr(v)= Crc,cp, we have

If we suppose the lines are formed in LTE, then Sr = B,. Noting further that .I, z *B, for the strong lines at the surface we see the term in the bracket is positive, and the temperature implied by the condition JK,B, dv = JrQ, dv- F Kj[B,-

JVvJv dvl

can be very substantially lower than the continuum value given above. On the other hand, S, = B, is very unrealistic for strong lines. Rather we should take s

=

I

IJvcpvdv+sB,

s

4

,

f+s

so that

Since in general s << 1, we clearly obtain a very much smaller change in temperature relative to the case without lines. Physically this is just a statement that the actual mechanism of line formation near the surface is scattering, which strongly decouples the lines from local temperatures; the only coupling is via the integrated effects of sourcesink terms over an entire thermalization length. From these arguments we see clearly that line-blanketed models in LTE give us no information whatever on the temperature distribution near the surface (even if their layers are in pure radiative equilibrium i.e. aside from questions of all other energy transport); only at depths sufficiently large to

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insure that S, + B do we begin to reach the regime where we may be able to derive information about the temperature structure. In short then, we can hope only to estimate “backwarming” effects on the temperature distribution, and even those only qualitatively. Finally, information in the last category is apparently essentiafly underivable from LTE models; the line profiles of strong lines depend sensitively upon both occupation numbers of levels and upon the transfer mechanism of line formation. Both of these features are brutally misrepresented by LTE. Even the surprisingly good agreement between observed and computed Hy profiles for Balmer-line blanketed A-star models is not necessarily an indication of the correctness of the computation; such agreement may be purely accidental in the sense described above. We therefore do not attempt here to discuss detailed line profiles. I would like now to describe some results, with which I am familiar, that have been obtained using the “direct” approach to line blanketing. The first calculation is of a B11/ model carried out by MORTONand me. t3) Morton made estimates of what lines are expected to be very strong on the range 912A I i I 16OOA from a model by Underhill with T, around 24000°K. Strong lines were found for hydrogen ithe Lyman series), argon, carbon, chlorine, iron, nitrogen, silicon and sulfur. Calculations were made of the line absorption coefficients as a function of temperature and electron pressure, and these were added to the continuum opacities to construct a model; in ail about 215 frequency points were considered. We found that the lines blocked about 40 per cent of the flux, so that the effective temperature of the final model was around 21900°K (0, = 0.23). The temperaturespressure relation in the blanketed model was virtually identical with that of an unblanketed model with 8, = 0.21; also the emergent fluxes in the visible regions were the same. Thus, for a given appearance of the spectrum in the visible, accounting for line-blanketing reduces T, by 2100°K and the bolometric correction is numerically more positive by 0.4 mag. A similar calculation has recently been carried out by T. Adams and D. Morton for a B4V star (amusingly enough a non-existent spectral type). They find that accounting for line blanketing reduces T, from 17 000°K to 16 800°K while the B.C. is more positive by 0.24 mag. These are large changes, and may be of importance in determining the relation between the observer’s color magnitude diagram and the theoretician’s HR diagram in stellar evolution studies. As to changes in the structure of the model, we found an increase of about 1400°K at intermediate depths, comparing the blanketed model with an unblanketed model of the same T,. About all else we can say from the results of this model is that certain bands in the far ultraviolet should have very strong lines; indeed Princeton rocket spectra have shown clearly some of the strong lines predicted, though no quantitative comparison has been made (and indeed we would be surprised if such a comparison agreed in.detail with calculation, even if it had been made.) Interestingly, an earlier simpler treatment of this problem by Strom and Avrett probably gives better results for the temperature distribution near the surface. Rather than include lines in the transfer problem, they merely put gaps in the spectrum at the positions of the strong lines. This accounts for the blocking and backwarming effect of the lines, but does not bias the integrals J‘KJ, dv and JK,,J, dv as including the lines in LTE does. A second group of models has been constructed for stars on the range B8-FO. Here the important blanketing agents are the Balmer lines; calculations were made including Hz-H20 in enough detail that accurate (U, B, V) colors could be computed. Typically the Balmer lines redistribute l&15 per cent of the flux. As in the case of the BlV model, there

586 is a

D. MIHALAS

backwarming effect, but here it typically ranges from 300% to 400°K. Comparisons with measured energy distributions for Vega showed good agreement, but again this may be accidental. One interesting and possibly fairly reliable result was the calculation of the (U, B, I/)colors; it was found that the colors computed for the models agree very well in the (U-B) vs. (B-l’) diagram with the colors of actual stars, after allowance is made empirically for the effects of metal-line blanketing according to the techniques of MELBOURNE and WILDEY, BURBIDGE, SANDAGE, and BURBIDGE.(~) If one accepts the empirical method of allowing for metal lines, it is possible to estimate an effective temperature scale for stars on this range which compares well with other estimates by a variety of methods. The second approach to line blanketing is the statistical approach. Here we deal with say, thousands of lines, and try to characterize them by as few parameters as is reasonable to give an accurate description of the energy transfer through the atmosphere. In essence this approach is a generalization of Chandrasekhar’s picket-fence method, and as such it does not give details of the line profiles but only energy fluxes in relatively broad. bands. In the light of what we have argued above, however, this is the most meaningful information LTE models can offer anyway, so in practice this is an excellent technique. The statistical approach is the only feasible one when the blanketing is due to numerous faint to medium-strong lines as in the case of solar-type stars or due to molecular bands as in the case of M-stars. Considerable work remains to be done on such models, again in the spirit of exploration to find what physical processes may be important. Our colleagues at Harvard are working to apply this method to the Sun and solar-type stars. Dr. Auman at Princeton is working on blanketing by H,O in M-stars. Unlike the cases which can be treated by the direct approach, it is not clear how we will be able to remove the LTE assumption, at least in the near future, when literally thousands of lines are involved. Perhaps future work will show simplified approaches that will permit this. From what I have said this far, it should be clear that in my opinion, the central problem in the theory today is the relaxation of the LTE assumption. With a few exceptions, we have already enough LTE calculations to cover the entire range of parameters (T, log g) of astrophysical interest. These models are mathematically accurate; any discrepancy they show with observation is of physical significance-it will be removed only by more accurate physics, not by refined computing techniques. Indeed, the LTE models will have served their purpose nicely if they lead to clear contradiction of the observations, for if they don’t contradict observation, we must still carry out the more accurate analysis on a sound physical basis to remove the possibilities of what I have called accidental agreement. Already there is some evidence that the LTE models do disagree with observation; in particular the Balmer jump predicted (even after allowing for line-blanketing) appears too large if a fit is to be made to the slope of the energy distribution in the. visible. The central question remaining is “How do we best proceed to remove the assumption of LTE?” It has long been clear that this question is equivalent to “How do we handle the bound-bound transitions?’ The bound-bound transitions are extremely difficult to treat because the line transfer problems are coupled the line source-functions have a large scattering term and because consideration of the lines immediately requires a very large number of frequency points in the calculation which correspondingly becomes quite timeconsuming. An attempt was made a few years ago by Lecar to calculate an A0 model

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atmosphere, solving the radiative equilibrium and statistical equilibrium equations’ simultaneously. After an enormous effort to find the solution, Lecar had to give up because the solution did not converge. His approach had been a relatively straightforward one, indeed it is probably just the approach many of us would have taken, and it was not clear for a while why it had failed. Only in retrospect is it easy to see that it unfortunately was doomed to failure from the outset. Lecar’s calculation can be described (or at least paraphrased) as follows : 1. From an LTE model in radiative equilibrium, calculate the radiation field. 2. Use this radiation field to calculate occupation numbers (or departure coefficients bj) from the statistical equilibrium equations. 3. Re-compute the radiation field and proceed to step (2) until convergence is obtained for the occupation numbers. This is clearly a perfectly straightforward and reasonable approach. The essential reason it fails is due to the treatment of the bound-bound above scheme essentially writes the line source-functions as

Why did it fail? transitions. The

2hv3biehv,kT_-l 1

Sij = ~

c2 [ bj which is rigorous transfer equation

(assuming

cp,, z j,,). The b’s from the nth iteration

p$-

= I-S

(*I

1 are inserted,

and the

is solved for I (or J)

which is used to calculate new b’s. This procedure is correct in principle and will ultimately converge (in principle). The essential flaw is in writing the source function in the above form. The reason is that scattering becomes the dominant transfer mechanism in the outer layers as is readily seen from

(**) Lecar’s method starts with a thermal (LTE) solution for S; the scattering is allowed for only implicitly, so that a single cycle allows the scattering information to propagate only over a distance of Ar z 1 (since the Lecar procedure essentially performs a A iteration). But scattering in the line is the essential physical mechanism over an entire thermalization length T + l/s (for doppler broadening) so that, very roughly speaking, l/s iterations are required to propagate the effects of scattering from the point where the thermal solution is valid to the surface. As we know, E is a very small number and therefore a very large number of iterations is required. Lecar’s solution has the advantage of taking account of changes in the source-sink terms very fastidiously from iteration to iteration. This advantage, however, is insignificant in light of the difficulties mentioned above. A far superior approach is to use the source-function (**) (with suitable extra terms to allow for coupling to all levels), and thus account for scattering quite explicitly. One then solves the transfer problem (using perhaps the integral equation approach or Rybicki’s method) allowing fully for scattering. This strongly revises the population ratios, and propagates the information from depth to surface in a single iteration. Of course the new population

D. MIHALAS

588

ratios will in general imply changes in the source-sink terms over the thermalization length and iteration is still required, but the fundamental change due to explicit rather than implicit treatment of scattering in the lines means the difference between success and failure. Full calculations have not yet (to my knowledge) been undertaken and solved; but extremely interesting problems have been studied, e.g. by Johnson, Avrett and Kalkofen, and Cuny as reported in the Second Harvard-Smithsonian Conference on Stellar Atmospheres. Offhand an approach like this seems very promising. Yet we may fairly ask if there is not a simpler approach, or more to the point, whether the final solution might not be gained in steps. I think the answer here is a strong “yes”. Recently Kalkofen and Strom have explored the construction of model atmospheres solving the combined radiative transfer and statistical equilibrium equations with an intriguing simplification: if Rij represents a net radiative rate from level i to j and Cij a net collisional rate, and if ICdenotes continuum then the statistical equilibrium equations are :

C

(Rij+Cij)+Rix+Ci,

=

0,

(i = I, 2,. . .)

j+i

Kalkofen and Strom proceed simply by setting the term CR,, to zero [i.e. NRB The assumption of detailed balance in the lines essentially requires that

z 01.

is so small that it can be neglected in comparison with other terms in the equation. Physically this is a very interesting approach because we in this way are allowing completely for departures from LTE due to free diffusion to the boundary of photons in the continuum. The continuum is the most transparent region of the spectrum and has by far the largest unsaturated bandwidth, so that the largest deviations from LTE at the greatest depths are due to just such diffusion. The problem posed by Strom and Kalkofen is vastly simpler to solve than the problem including bound-bound radiative transitions. In my opinion it represents a very logical first step in relaxing the LTE assumption. Once such a solution is obtained, it could be used as a starting solution for the case where lines are included. The essential question is where do the Kalkofen-Strom assumptions break down. The condition that Rij is essentially zero is satisfied at depths T greater than a few thermalization depths. This might require T 2 j/c. Now from the point of view of the energy-balance condition in A-stars, the most important lines are the Balmer lines, so we mainly wish to find out when the above condition is violated for them. For the Balmer lines E is about O-1 (here strictly should refer to photoionizations not collisions, but we use the notation loosely), so that T should be 230. But the opacity in the Balmer lines 104-lo5 greater than in the continuum, so we are talking about continuum optical depths of 0903. Now actually the above expression is a bit optimistic since the Baimer lines have strong wings. We might instead adopt T’/’ 2 3ali2/c where a - 0.5. Then Tz 500, so that the continuum optical depth might be as high as 0.05. These estimates seem to indicate that we will obtain essentially the correct solution for the statistical equilibrium problem for T k 0405 to O-05. Thus if we calculate a model based on these assumptions, and include the Balmer lines using the b-values we derive (it is essential to include the

LTE model atmospheres

with line blanketing

review and future prospects

589

lines in the transfer problem they block energy and thus affect the overall thermal structure of the atmosphere), we appear to obtain a self-consistent picture (of the continuum only) up to depths which are optically thin in the continuum. This has two important consequences : 1. We probably obtain the correct emergent fluxes in the continuum. 2. The energy balance is correct for ~~ _> 0.005 to 0.05 and changes in the solution for smaller T'S will probably have no further effect at greater depths hence source-sink terms there are probably already fairly well determined. It thus appears that the energy balance problem can be solved in two steps also: first at depth and then near the surface. In general, it appears to me that it is a very rational approach to allow for deviations due first to the most transparent bands where the photons diffuse most freely, and then add successive series of lines in order decreasing thermalization lengths measured in units of the continuum. Supposing that one proceeds in this way it is not obvious whether problems will arise due to coupling from one line of a series to another or because of coupling from series to series. One way of looking at this problem is to consider it as a situation where there are several relevant diffusion lengths of the same order of size. As is known from solutions of other diffusion type problems, compensation and cancellation can occur in such situations, and it becomes necessary to treat all of the relevant processes simultaneously (as opposed to iteratively). In principle one should be able to cope with these problems, but in practice it may be difficult. The Kalkofen-Strom method has already been applied by them to calculate a few late-B and early-.4 models. The most worrisome limitation in their calculation is the restricted number of levels they consider; this however can easily be relaxed. At Princeton we hope to try this approach soon for hydrogen-helium models, hopefully up through the O-star range. In conclusion, I apologize for having turned a talk about LTE models into a critique of their weaknesses. I have done so only because it is clear that they have served their purpose and with the exception of a few exploratory types of calculation, probably should no longer be pursued. The conceptual ground has been broken for us by Thomas, Jefferies, Pecker, and Hummer, to mention only a few, and it is time for those of us who actually run the computers to catch up and begin to try to apply their concepts, which hitherto have mainly been used to study well-posed demonstration-type problems and problems of primarily solar interest, to problems of interest to stellar spectroscopists in general.

Acknowledgemmt-This tract AF 49(638)-1555.

work was supported

in part by the

Air Force OtIic~ of Scientific Research

under con-

REFERENCES I. E. H. AVRE-rT and M. KROOK. Asrrophys. J. 137, 874 (1963). 2. R. N. THOMAS, Non-Equilibrium Thermodynamics in the Prcwnce qf (I Radiation Field, University of Colorado Press (1965). 3. D. M. MIHALAS and D. C. MORTON, Asrrophys. .I. 142, 253 (1965). 4. R. L. WILDEY, E. M. BURBIDGE, A. R. SANDAGE and G. R. BL.RBIDGE, Asrrophys. .I. 135, 94 (1962).

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DISCUSSION ANNE B. UNDERHILL:The placing of real stars in the HR diagram derived from stellar structure studies such as those by STOTHERSand IBEN (Astrophys. .I. 141-143) can be performed rather consistently with the help of model atmospheres. The chief problem is to identify a given mass with a given spectral type. The identifications can be made using masses known from spectroscopic binaries. The bolometric corrections given in column 3 are from model atmospheres with line blanketing, where types and Tefl are the same as those in columns 4 and 2. Then for point 2 on Iben’s tracks (= average main-sequence star) one finds the visual absolute magnitudes given in the 6th column. These agree fairly well with observed values such as shown in the HR diagram by PETRIE(P&l. Am. Sm. Pacif, 77). Stellar structure

Star masses 30 15 9 5 3

Binaries

Model atm

Equivalent Telf 4oooo” 29 600 23 400 17400 12600

M 801

type

- 8.4 -6.4 -47 -2.5 0.4

06 09, BO B2 B5,6 B8,9

B.C. (-3Q) -2.4 -2.0 -1.0 -0.4

M” -

5.4 4.0 2.1 1.5 0.0

Observed

Observed

M”

type

-4.0 - 2.0 -0.9 0.0

SO,09 B2, B3 B5, B6 B8, B9

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