Empirical chemical stratifications in magnetic Ap stars: questions of uniqueness

Over the last decades, modelling of the inhomogeneous vertical abundance distributions of various chemical elements in magnetic peculiar A-type has largely relied on simple step-function approximations. In contrast, the recently introduced regularised vertical inverse problem (VIP) is not based on parametrised stratification profiles and has been claimed to yield unique solutions without a priori assumptions as to the profile shapes. It is the question of uniqueness of empirical stratifications which is at the centre of this article. An error analysis establishes confidence intervals about the abundance profiles and it is shown that many different step-functions of sometimes widely different amplitudes give fits to the observed spectra which equal the VIP fits in quality. Theoretical arguments are advanced in favour of abundance profiles that depend on magnetic latitude, even in moderately strong magnetic fields. Including cloud, cap and ring models in the discussion, it is shown that uniqueness of solutions cannot be achieved without phase resolved high signal-to-noise ratio (S/N) and high spectral resolution (R) spectropolarimetry in all 4 Stokes parameters.

💡 Research Summary

The paper critically examines the claim that the regularised vertical inverse problem (VIP) method yields unique, assumption‑free chemical stratification profiles for magnetic Ap stars. Historically, vertical abundance inhomogeneities have been modelled with simple step‑function profiles characterised by four parameters (upper‑layer abundance, lower‑layer abundance, jump position, and jump width). While such step‑functions have successfully improved fits to observed spectra compared with constant‑abundance models, they have always assumed that the profile is the same over the whole stellar surface, irrespective of magnetic field strength or orientation.

Recent work by Kochukhov, Ryabchikova & Shulyak (2006, hereafter KTR06) introduced the VIP approach, which regularises the inversion of the radiative transfer equation to obtain a smooth, continuous abundance profile without imposing a priori functional forms. KTR06 argued that the regularisation parameter guarantees a unique solution. Stift and Alecian set out to test this claim using the most thoroughly studied star, HD 133792, for which KTR06 published VIP‑derived Fe, Mg, and Si stratifications.

First, the authors reproduced the VIP profiles with the Atlas12 atmospheric model (T_eff = 9400 K, log g = 3.80, metallicity +0.5 dex) and the COSSAM polarised synthesis code. They then performed a controlled perturbation experiment: at each optical depth point (log τ_5000) they added a small abundance “bump” (0.1–0.3 dex for Fe, 0.2–0.6 dex for the other elements) and recomputed synthetic spectra for a set of 19 Fe lines spanning a range of strengths.

The results show that even a 0.3 dex perturbation—one third of the total amplitude of the VIP Fe profile—produces at most a 3 % change in the normalised line profiles. Moreover, the spectral response is highly depth‑dependent: for –2.5 < log τ < –0.2 the maximum response reaches about 1 %, but outside this interval the response falls below 0.5 % and becomes negligible for log τ > 0.0 or log τ < –2.9. In other words, the set of Fe lines used is essentially blind to abundance variations deeper than log τ ≈ +0.6 and higher than log τ ≈ –2.5.

From these perturbation experiments the authors construct “confidence intervals” for each element. A 0.5 % spectral change corresponds to a ±0.1 dex uncertainty for Fe (and Si) and ±0.2 dex for Mg within the well‑constrained depth range. Beyond log τ ≈ +0.6 the profiles are completely undefined; the steep gradients reported by KTR06 in the deep layers (log τ > +2) are therefore artefacts of the inversion rather than physically meaningful features, because the contribution of such deep layers to the emergent intensity is suppressed by a factor exp(–200).

The paper then addresses the theoretical expectation that diffusion velocities depend strongly on magnetic field orientation. Earlier calculations (Alecian & Stift 2006, 2008) demonstrated that horizontal fields can trap certain elements in the upper atmosphere, while vertical fields allow them to settle deeper, leading to latitude‑dependent stratifications even for modest field strengths (∼1 kG). The VIP analysis, however, was based on a single rotational phase and only Stokes I data, thus ignoring any latitude dependence.

To explore the non‑uniqueness further, the authors construct alternative three‑dimensional abundance geometries (cloud, cap, and ring models) that vary with magnetic latitude. Synthetic Stokes I spectra generated from these models reproduce the observed line profiles with residuals comparable to those obtained with the original VIP step‑function. Consequently, many distinct abundance configurations—some with jumps of several dex—are observationally indistinguishable given the current data quality.

The authors conclude that the VIP method does not guarantee a unique solution; it merely provides one of many possible stratifications compatible with the limited information content of a single‑phase, Stokes I spectrum. To achieve true uniqueness, they advocate phase‑resolved, high‑signal‑to‑noise (S/N ≥ 300), high‑resolution (R ≥ 80 000) spectropolarimetric observations covering all four Stokes parameters (I, Q, U, V). Such data would encode the magnetic geometry and allow the inversion to discriminate between latitude‑dependent models. Until such observations become routine, empirical stratifications derived from VIP should be regarded as provisional, model‑dependent representations rather than definitive maps of chemical structure in magnetic Ap star atmospheres.