Empirical chemical stratifications in magnetic Ap stars: questions of uniqueness
aa r X i v : . [ a s t r o - ph . S R ] J a n Mon. Not. R. Astron. Soc. , 1–7 (2009) Printed 29 October 2018 (MN L A TEX style file v2.2)
Empirical chemical stratifications in magnetic Ap stars:questions of uniqueness
M.J. Stift , , G. Alecian Institut f¨ur Astronomie (IfA), Universit¨at Wien, T¨urkenschanzstrasse 17, A-1180 Wien, Austria LUTH, Observatoire de Paris, CNRS, Universit´e Paris Diderot, 5 place Jules Janssen, F-92190 Meudon, France
Accepted 2008
ABSTRACT
Over the last decades, modelling of the inhomogeneous vertical abundance distribu-tions of various chemical elements in magnetic peculiar A-type has largely relied onsimple step-function approximations. In contrast, the recently introduced regularisedvertical inverse problem (VIP) is not based on parametrised stratification profiles andhas been claimed to yield unique solutions without a priori assumptions as to theprofile shapes. It is the question of uniqueness of empirical stratifications which is atthe centre of this article. An error analysis establishes confidence intervals about theabundance profiles and it is shown that many different step-functions of sometimeswidely different amplitudes give fits to the observed spectra which equal the VIP fitsin quality. Theoretical arguments are advanced in favour of abundance profiles thatdepend on magnetic latitude, even in moderately strong magnetic fields. Includingcloud, cap and ring models in the discussion, it is shown that uniqueness of solutionscannot be achieved without phase resolved high signal-to-noise ratio (S/N) and highspectral resolution (R) spectropolarimetry in all 4 Stokes parameters.
Key words: techniques : spectroscopic – stars : abundances – stars : atmospheres –stars : chemically peculiar – stars : magnetic fields
Over the years, evidence has accumulated that a numberof (magnetic) Ap stars not only exhibit non-uniform dis-tributions of chemical elements over their surfaces – firstabundance maps within the framework of the oblique rota-tor model date back to Deutsch (1958) and to Pyper (1969)– but that vertical abundance distributions in their atmo-spheres also are non-uniform. Such stratified abundances areto be expected from diffusion theory (Michaud 1970) andare reflected in spectra that cannot be fitted by constant el-emental abundances. Depending on ionisation stage and onexcitation potential, different abundances may be needed tofit different spectral lines of a given element. Borsenbergeret al. (1981) were the first to compare (with some success)theoretical equivalent widths based on predicted stratifiedCa and Sr abundances to observed equivalent widths, andAlecian (1982) attempted to detect Mn stratification in theatmosphere of ν Her. In a recent review paper, Ryabchikova(2008) has summarised recent results on abundance stratifi-cations, illustrating her discussion with many empirical andsome theoretical profiles for various chemical elements indifferent Ap stars. The vast majority of these curves essen-tially correspond to step functions, with lower abundancesin the outer layers and a (sometimes drastic) increase to- wards the deeper layers – in a few cases just the oppositebehaviour is found. These abundance jumps can range froma few 0.1 dex to 4 dex and more; some inversion codes yieldsmooth curves, whereas in others they are assumed to bemore or less discontinuous. It has, however, always been as-sumed that the profiles do not depend on the direction or onthe strength of the local magnetic field, so that in a givenstar the same profile applies everywhere, regardless of thevariations in magnetic field direction and in field strengthas, for example, found in a dipolar geometry.On the other hand, Alecian & Stift (2006) have pre-sented snapshots of abundance increases as a function ofmagnetic field direction which reveal sometimes huge dif-ferences between vertical and horizontal fields. Alecian &Stift (2008) have also shown that, depending on the field di-rection, equilibrium stratifications can differ by several dexin the upper layers. This result is consistent with what hasalready been known about the sensitivity of the diffusion ve-locity to the horizontal component of the magnetic field (seee.g. Babel & Michaud 1991ab), and it is certainly not at vari-ance with the apparent correlations observed between abun-dance patches and magnetic geometries in magnetic obliquerotators (see e.g. Kochukhov et al. 2002).So far, no attempts have been made to reconcile the em-pirical modelling of stratifications with the sometimes com- c (cid:13) M.J. Stift, G. Alecian
Figure 1.
The VIP based Fe stratification profile for HD 133792is plotted as logarithm of abundance ǫ (normalised to log H =12 .
0) versus log τ . The selection of perturbations to this profilewhich underly the results presented in Fig. 2 pertain to log τ = − .
701 (dot), -3.632 (dot - short dash), -1.511 (short dash), and+0.439 (dot - long dash). plex abundance structures predicted for magnetic stellar at-mospheres by theoretical studies. Kochukhov et al. (2006)(henceforth KTR06) rather have introduced a new empiri-cal approach based on a regularised solution of the verticalinversion problem (VIP). Their abundance profiles have al-legedly been derived “without making a priori assumptionsabout the shape of chemical distributions” and their “op-timum regularisation” is claimed to ensure the uniquenessof the solution. But is it really possible that this new andradically empirical approach yields the answers that theorycannot yet provide? Is the VIP method assumption-free or isit still subject to some hidden constraints? Where in the at-mosphere are the empirical stratifications well defined, andcan they be truly considered unique? How small are the de-tails that a method based on high resolution Stokes I spectracan reliably detect? Can alternative step-function like solu-tions be found for HD 133792, perhaps even solutions thatdepend on magnetic latitude in an oblique rotator model?What is the kind of information that can reliably be gleanedfrom such inversions?This paper addresses these questions (and a few more).A simple but realistic error analysis makes it possible to es-timate the interval in optical depth over which the empiricalstratification profiles are more or less well defined. Extensivenumerical modelling (involving models based on cap-, ring-and cloud-like structures) is then used in the assessment ofthe question whether uniqueness of the abundance profilescan be attained with the methods and data presently athand. Finally we advance ideas for a strategy that could re-move some non-uniqueness of the models and lead to morereliable stratification results. From the plots presented by Ryabchikova (2008) in herreview which are based on results for magnetic and non-magnetic Ap stars taken from recent literature, it emergesthat practically all empirical profiles correspond to a step-function described by 4 parameters, viz. the abundance inthe upper atmosphere, the abundance in the deep layers, the position of the jump and the width of the jump. These strat-ification profiles have always been assumed to remain con-stant over the star, regardless of the strength of the stellarmagnetic field. It is true that in stars with weak fields pro-files do not depend on magnetic latitude over large parts ofthe stellar atmosphere – in a 1 kG horizontal field Alecian &Stift (2007) have found differences compared to the zero fieldcase only for log τ < − τ < −
3) inplaces where magnetic field lines are horizontal (Alecian &Stift, 2009, in preparation) and holds true even for weakmagnetic fields. Evidence for the existence of increased Ndabundances above log τ = − . γ Equ and in HD 24712presented by Mashonkina, Ryabchikova & Ryabtsev (2005)is certainly not at variance with the theoretical predictionand so even for the 1 kG case one should expect significanthorizontal differences in the vertical stratifications of someions.We do not want to deny that many of the stratificationprofiles presented in recent times lead to improved fits tothe observed Stokes I spectra, compared to an assumed con-stant abundance with depth. But it is also a fact that noneof the predicted spectra are perfect, that residuals of 1-5%in normalised intensity – sometimes almost 10% – persistand that we stumble over lines where a stratified abundancegives a less satisfactory fit than a constant abundance. Inthe particularly well-studied star HD 133792, for example,3 out of 7 strontium lines belong to this category, and 5out of 26 chromium lines. In the same star, the stratifica-tion of calcium has essentially been derived from just 3 lines,one of which is still rather poorly fitted by the stratificationprofile. Is it possible to guarantee the uniqueness of a solu-tion when the uncertainties in the atmospheric parameters,the limited accuracy of the atomic data, and in some casesalso the unknown magnetic geometry of the star are kept inmind? Are the residuals due to the imperfect data or ratherto the imperfect (and even possibly erroneous) stratificationprofiles?In order to assess the question of the uniqueness of em-pirical stratification profiles it is imperative to first carry outa meaningful error analysis. Since HD 133792 is the star forwhich the most detailed, and in a certain sense, the most so-phisticated, determination of stratification profiles has everbeen attempted, the VIP based stratification profiles of thisstar are certainly well suited for this purpose. The stratification profiles for HD 133792 have been deter-mined by KTR06 under the assumptions of constancy andmaximum smoothness. Their observations extend over 10minutes and cover just 1 phase. The quality of the fit to c (cid:13) , 1–7 tratifications in magnetic Ap stars Figure 2. Lower part:
The response of 19 Fe lines (indicated by arrows and listed in Table 1) to perturbations of the abundance profileof KTR06 at a given optical depth. The effect of the 0.3 dex perturbation – see text and Fig. 1 for details – is given in the sense perturbedminus unperturbed Stokes I . Upper part:
The unperturbed spectrum with the Fe lines.ion λ | ion λ | ion λ Fe 2 5018.440 | Fe 1 5022.931 | Fe 1 5434.524Fe 2 5018.669 | Fe 1 5023.186 | Fe 2 5567.842Fe 1 5022.236 | Fe 2 5030.630 | Fe 2 5961.705Fe 1 5022.420 | Fe 2 5030.778 | Fe 2 6149.258Fe 1 5022.583 | Fe 1 5269.537 | Fe 2 6150.098Fe 1 5022.789 | Fe 2 5325.553 | Fe 1 5022.792 | Fe 1 5326.142 | Table 1.
The iron lines shown in Fig. 2, listed with increasingwavelength and indicated (from left to right) by arrows in thefigure. the observed spectrum varies between the elements. Differ-ences between observed and predicted spectra reach some2% in about 10 of the 28 iron lines analysed (we made theseestimates from the figures). The centres of weak lines areaffected to the same extent as the centres of much strongerlines; we also note that not all fits to line wings are fullysatisfactory. The other elements do not fare quite so well;whereas residuals can attain 5% for Mg and Ca and 6% forSi 6%, they exceed 9% for Sr.At this point we shall not question the atmosphericmodel which could conceivably be of different effective tem-perature and gravity (Cowley, private communication), weneither question nor even take into consideration the mag- netic field strength and geometry, we just take the pub-lished atmospheric parameters and the stratification profilesat face value. Sr is omitted in our investigation because of theexceptionally large residuals, but for the remaining elementswe first calculate the reference line spectrum predicted fromthe published stratification profiles. Then, in a controlledexperiment, we determine just how large perturbations tothese abundance profiles would have to be to lead to 1% or5% deviations from the reference line spectrum. This in turnallows us to judge the significance of any detailed structurein the stratification profiles and makes it possible to estimatethe interval over which these profiles are well defined. It isnot surprising – in view of the assumptions underlying theVIP approach – that there is little such structure and thatthe remarkable smoothness of the empirical stratifications ofMg, Si, Ca, Cr and Fe (see Fig. 5 of KTR06) is only slightlyperturbed by humps or dips. The occurrence of gradientsin the stratification profiles for log τ > We chose a straightforward approach consisting in the ap-plication of a simple perturbation to the published stratifi-cation profiles. The Atlas12 code (Kurucz 2005) was used to c (cid:13) , 1–7 M.J. Stift, G. Alecian
Figure 3. Bottom panels:
Estimated uncertainties in the stratification curves of Fe, Mg, and Si. The dashed curves correspond to a ± .
5% maximum response to the perturbation in the spectrum, the dot-dashed curves to a ±
1% maximum response. The full lines arethe respective original curves digitised from KTR06; they are plotted with dots where the 1% confidence intervals become larger than0.5 dex. Minor wiggles as in the case of Fe are due to imperfect digitisation.
Middle panels:
Possible alternative stratifications for whichthe synthetic spectra do not differ by more than 0.002 - 0.003 (rms) and by less than 0.01 (maximum) from the spectra synthesisedwith the respective original stratification curves.
Top panels:
The same alternative stratifications plotted in the sense alternative minusoriginal profile. The respective dashed and dot-dashed curves correspond to the confidence intervals displayed in the bottom panels. establish an atmospheric model for HD 133792 with T eff =9400 K, log g = 3 .
80 and solar metal abundances increasedby 0.5 dex. This does not constitute a perfect match to themodel established by KTR06 but is largely sufficient for ourpurposes. This model atmosphere – with 99 layers – was thenused in the COSSAM polarised spectral synthesis code (Stift1998, 2000; Wade et al. 2001) to establish theoretical spectracovering all the lines used by KTR06. The atomic line datawere taken from the VALD database (Piskunov et al. 1995;Kupka et al. 1999). The public version of COSSAM providessolely a spatial integration grid centred on the line of sightand covering the visible hemisphere of the star, but for thepresent calculations we employed a corotating grid (Stift1996) largely identical to those in general use in Dopplermapping (see Voigt, Penrod & Hatzes 1987 for details). Tak-ing the said 99 layer Atlas12 model for HD 133792, a 5-layerperturbation of { } dex was added to theempirical Fe profile of KTR06 (see Fig. 1) and a { } dex perturbation to the stratification profiles of theother elements. Applying these perturbations in turn to alldepth points, we determined the difference between originaland perturbed spectrum. Fig. 2 displays these differences for a selection of 19 Fe lines and for 10 different points in opticaldepth. For the 0.3 dex perturbation to the iron stratification– which corresponds to 1/3 of the total amplitude claimedby KTR06 and which is illustrated in Fig. 1 – the maximumeffect on the normalised spectrum is of the order of 3%.As expected, in the deeper layers the wings of strong linesprovide most of the abundance information whereas in theupper layers this is mostly done by the line cores. Outsidethe interval − . < log τ < − .
20 the spectral responsedoes not even reach 1%, dropping rapidly below 0.5% for − . < log τ and for log τ > .
0. In other words, anyattempt to reconstruct the Fe stratification profile beyondthese limits cannot possibly yield meaningful results, simplybecause for the particular atmosphere in question and theabundance profile on which our calculations are based, theselected Fe lines become insensitive to abundance changes.
The lower panels of Figs. 3a-c show “confidence intervals” –as derived with the above mentioned perturbative approach– for the stratification curves of Fe, Mg, and Si. These con- c (cid:13) , 1–7 tratifications in magnetic Ap stars fidence intervals must not be confounded with those derivedfrom rigorous statistics but result from the simple inversionof the relation perturbation vs. maximum response obtainedfrom our calculations. A 0.5% maximum response of thenormalised spectrum requires a perturbation whose size isgiven by the distance between the dashed lines and the orig-inal profile. The perturbation necessary for a 1% maximumresponse is reflected by the dot-dashed lines. It transpiresfrom these results that the stratification profiles are welldefined over relatively narrow intervals in optical depth –here they are good to 0.1 dex in the Fe case, and good to0.2 dex in the other cases – but become essentially undefinedfor log τ > +0 .
6. The narrowest such intervals with lessthan 3 decades in optical depth are found for Fe and for Si,the Mg profile is defined over about 4 decades.We very carefully checked that the vertical resolutionof the atmospheric model is good enough so that the re-sponse curves are not affected by numerical problems. Forthat purpose, the same analysis was carried out with a 199layer Atlas12 model and a 9-layer perturbation, yielding es-sentially identical results.
We have seen from Figs. 3a-c that the portions of the abun-dance profiles below log τ = +0 . τ = 2 . − tens ofdecades smaller than the contribution from the region nearlog τ = 0 . deviations from this mean abundance.Such profiles – which converge to the same abundance valuedeep in the atmosphere and in the outermost layers, andwhich simply deviate from this value in some intermediatezone – are neither in accord with the theoretical modelsof LeBlanc & Monin (2004) nor with equilibrium solutionsin the presence of magnetic fields presented by Alecian &Stift (2007, 2008). The mentioned theoretical results arealso at variance with the extreme smoothness of the strati-fication profiles. Disturbingly, Fig. 3a of KTR06 shows thatasymptotically their solution joins the mean abundance. Alarge regularisation parameter can make this asymptotic be-haviour less visible, but it still persists and is readily visiblein Fig. 5 of KTR06. Choosing the regularisation parame-ter such that one arrives at more or less the same solutionin the interval − . < log τ < . τ < −
3, see Fig. 3 of Alecian & Stift (2007). Sonobody can reasonably exclude that despite the rather mod-erate field of HD 133792, the outer parts of the stratificationprofiles of Sr, Ca, and Mg may be affected. And certainly oneshould not overlook the studies by Kurtz, Elkin & Mathys(2005) which suggest a concentration of rare earth elementsat log τ = − > β CrB, γ Equ, HD 144897, and HD 66318 to mentionjust a few of the well-studied stars. In this context it shouldalways be kept in mind that it is not the 2-fold difference infield strength between pole and equator in a dipolar obliquerotator model that is decisive, but the direction of the field.Results based on Zeeman Doppler mapping which suggestthat abundance anomalies are related to the magnetic fieldtopology (see e.g. Kochukhov et al. 2002) are certainly inqualitative accord with theoretical findings.
The claim that a model derived with the “optimum regular-isation” is unique has to be understood in the sense that itis unique within the framework of VIP (Kochukhov, privatecommunication). These seems to be a reasonable claim, butwe really want to have a look at the uniqueness of empir-ical stratification profiles in a more general sense. Lookingat HD 133792, are there other abundance profiles that re-produce the observed spectrum as well as the VIP solution?Faced with an infinity of possible profile shapes and profiledistributions, we started with the simplest case, viz. a stepfunction whose shape stays constant over the star, irrespec-tive of magnetic latitude.A 4 parameter step function is defined by the “outer”and the “inner” abundances, and by the position and widthof the transition region which connects the “inner” and the“outer” parts. We made a reasonably extended, almost ex-haustive search for such step functions in the case of Mg, Si,Ca, and Fe, by calculating a dense grid covering all possibleparameter values. Step function solutions were consideredacceptable whenever the rms deviation of the alternativespectrum from the normalised spectrum calculated with theVIP solution did not exceed 0 . − .
003 and when the max-imum deviation was less than 1%. Given the 2 - 6% maxi-mum size of the residuals of the VIP fits for the elements inquestion, this is indeed an extremely strong constraint. The c (cid:13) , 1–7 M.J. Stift, G. Alecian
Figure 4. (a)
The smooth original digitised VIP stratification curve for Fe together with a selection of global polygonal curve step-function alternatives. The curves are plotted with dots where the confidence interval exceeds 0.5 dex (as explained in Fig. 3). (b)
Thesame for global cloud models and (c) for clouds confined to a polar cap. middle panels of Figs. 3a-c display a selection of acceptableglobal step function solutions. For clarity they are plottedagain in the top panels in the sense alternative minus VIPsolution, and in addition, confidence intervals overlay thecurves. At this point we want to remind the reader againthat we are in no way looking for alternative fits to the realobserved spectrum , but only for excellent fits to the
VIPbased synthetic spectrum . So one would not expect the al-ternative stratifications to differ completely from the VIPprofiles.In the case of Fe, the differences between step functionsolutions and the VIP curve tend to stay within the 0.5% re-sponse curve. Still, for the large number of acceptable stepfunction models we find a noticeable spread in amplitudeand also in the width of the transition region, due to thevirtual lack of response to perturbations for log τ > . Many more simple abundance profiles can be explored whichconstitute a zero-order approximation to equilibrium strati-fication profiles found for magnetic stellar atmospheres. We decided to have a look at one particular family of modelswhich are based on the assumption that a cloud of increasedelemental abundance hovers somewhere in the atmosphere.This cloud is either distributed uniformly all over the star, orrestricted to a cap around one magnetic pole, or located ina ring around the magnetic equator. There are 2 transitionregions, one towards the upper part of the atmosphere, thesecond towards the bottom, outside of which the abundanceis constant and not necessarily solar. As it turns out, allthese geometries can lead to excellent fits to the VIP basedsynthetic spectrum. Figs. 4a-c compare the VIP iron profileto global step-function (jump) solutions as discussed above,to global cloud models, and to clouds confined to a polarcap. In all 3 classes of models shown, the abundances in thedeeper layers are not particularly well defined, in contrastto the abundances in the outer layers. In the global step-function case (Fig. 4a) the 0.3 dex spread in abundances atthe bottom of the atmosphere corresponds to a remarkable50% of the minimum possible value of the jump; in the casesof a global cloud (Fig. 4b) and of a cloud in a polar cap(Fig. 4c) the spread at the bottom reduces to 0.2 dex – theminimum amplitude remaining at 0.6 dex. For the ring casewe only dispose of a few hundred models and therefore it isnot possible to give definitive values for the total possiblespread in amplitude.We did not delve into an exhaustive search for cap mod-els at all possible phases but concentrated on cap modelsseen pole-on and on ring models seen equator-on. In bothcases we found a rather narrow range of extensions, capscovering the star up to 50 − ◦ from one magnetic pole,and rings being confined to 25 − ◦ from the magneticequator. Cap solutions consistently give narrower transitionregions than both global step-function and global cloud so-lutions; for a given optical depth, the abundance is substan-tially higher in the interval − . < log τ < c (cid:13) , 1–7 tratifications in magnetic Ap stars the remarkable spread in abundance at the bottom of theatmosphere – reveals just how uncomfortably large the un-certainties in the empirical profiles really are. On a positive note, our modelling confirms that it is possibleto unequivocally establish the presence of chemical stratifi-cations and that the sense of the abundance change is al-ways clear. There can be thus no doubt that the decreasewith depth of the Mg abundance in HD 133792, for example,is indeed a decrease, and that the abundance of Fe increaseswith depth in this star. The respective orders of magnitude of the abundance jumps are quite well defined.Our findings however reveal that none of the presentlyused approaches is capable of providing unique stratificationprofiles. Taking the slowly rotating Ap star HD 133792, wehave shown that for several chemical elements, many differ-ent global step-function-like solutions can be found whichperfectly reproduce the VIP based synthetic spectrum; therespective amplitudes of the jumps however can differ sub-stantially among each other. For Fe the step-function am-plitudes are invariably smaller than the VIP value. We havefurther shown that the quality of the VIP fit is also well-matched by either global cloud-like solutions, by clouds in acap around a magnetic pole or by clouds in a ring about themagnetic equator. Both for cap and ring models, we finda certain spread in amplitudes and in addition a narrow-ing of the transition region. Even when, as in the case ofHD 133792, profiles of 28 Fe lines are used in the inversion,there appears to be no way to distinguish between the var-ious possible stratification profiles (at least not with Stokes I only).Cap and ring geometries can be seen as rough approx-imations to the results of equilibrium stratification calcu-lations by Alecian & Stift (2008) who have demonstratedthat in magnetic fields of 5 kG and more, stratification pro-files become strongly dependent on the field angle. Whilewe consider that in strongly magnetic stars like HD 66318and HD 144897 these models could possibly come slightlycloser to reality than models which assume globally con-stant profile shapes, nobody can guarantee uniqueness oreven correctness. Only with excellent phase coverage insteadof observations at just 1 phase, and with high quality ob-servations in all 4 Stokes parameters will it perhaps becomepossible to distinguish between the rival models. Ideally onewould have to reconstruct the run of abundance with depthand the magnetic field vector at each point of the stellarsurface. Whether such an extremely ill-defined problem canever be solved is hard to predict.Thus, at present, surveys of (magnetic) Ap stars inview of empirically establishing the extent of the stratifi-cation phenomenon are invaluable for our understanding ofradiatively driven diffusion and its dependence on stellarparameters including magnetic fields. Empirical inversionsreveal the sense and the order of magnitude of an abun-dance change with depth, but not the exact amplitude, northe precise location of the transition region, and certainlynot any fine structure in the stratification profile. Usuallyempirical stratifications are only defined over a rather re-stricted interval in optical depth, apparently never beyond log τ > +0 .
6, and for the reasons discussed above, theyare expected to be particularly unreliable in strongly mag-netic Ap stars. They cannot therefore in the foreseeable fu-ture provide the desired strong constraints to theoreticaldiffusion modelling.
ACKNOWLEDGEMENTS
MJS acknowledges support by the
Austrian Science Fund(FWF) , project P16003-N05 “Radiation driven diffusion inmagnetic stellar atmospheres”. Dr. Shulyak generously pro-vided his Linux version of the Atlas12 code and kindlyhelped with the installation. Thanks also go to Dr.Kochukhov for most interesting discussions and a numberof clarifications. Helpful comments by the referee have im-proved the manuscript.This paper has been typeset from a TEX/ L A TEX file preparedby the author.
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