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Zeeman-Doppler Imaging

Wednesday, 6 March 2013 14:11

Stellar surface cartography with Zeeman Doppler Imaging


Improved polarized radiative transfer analysis


Our basic platform for field cartography is the ZDI code iMap (Carroll et al. 2007, 2012) which simultaneously reconstructs the magnetic field and the temperature distribution from phase-resolved Stokes spectra. This code is versatile and flexible enough to be also applicable to Zeeman-broadened Stokes I profiles to perform the desired magnetic flux density inversion for the magnetic flux (f.B). In the past, there were two major obstacles that hampered a full polarized radiative transfer driven approach to ZDI. Firstly, the expected Stokes profiles are heavily contaminated by noise, or even buried within the noise, which makes their reliable analysis and interpretation difficult. Secondly, the radiative transfer calculation presents an enormous computational burden for the inverse problem where for each iteration and each surface segment the full radiative transfer equation has to be solved.


A possible solution for the second problem was introduced by Carroll et al. (2008) where an artificial neural network is used to solve the polarized radiative transfer and calculate individual Stokes lines profiles. The approach uses Multi-Layer-Perceptrons (MLPs) to approximate the mapping between the atmospheric parameters and the resulting Stokes spectra. The data base of local line profiles is first decomposed via PCA for the purpose of dimensionality reduction. Then, a set of MLP’s is trained to calculate projection coefficients as a function of the five parameters Teff , magnetic-field strength, inclination and azimuth, and line-of-sight angle. The number of principal components of 10 is chosen in such a way that a direct reproduction of the Stokes profiles leads to an overall error of less than 10−4. Once the MLP’s have found a good approximation of the underlying functional dependence, they allow an impressively fast Stokes profile evaluation (a factor 1,000 faster compared to the conventional radiative-transfer calculation with DELO).


Inverse multi-line modeling


In Doppler imaging it is typical to use a combined long spectral line vector such that the error function, E, reads

E =1/2 ΣNp p=1 ΣNk k=1 ΣNm m=1 (Op,k,m-I(x)p,k,m)2,

where x is the model parameter vector, p the rotational phase, k the spectral line, m the wavelength or the velocity index, and Np, Nk, Nm their respective total numbers. This is the set up also for temperature DI with TempMap (e.g. Rice & Strassmeier 2000; Rice et al. 2011) but fails for ZDI due to the large noise and the increased parameter space. It demonstrates that the reduced 2 as a measure of the goodness-of-fit is sometimes of limited use.


Another improvement in the iMap code is its new inversion module. While the former versions relied on a conjugate gradient method with a local entropy regularization, the current version of iMap uses an iteratively regularized Landweber method (Engl et al. 1996). Iterative regularization for inverse problems has been the subject of various theoretical investigations over the recent years (see references in Carroll et al. 2012). The Landweber iteration rests on the idea of a simple fixed-point iteration derived from minimizing the sum of the squared errors. Our new inversion routine follows exactly this line of argumentation. It deals with four parameter spaces (temperature, radial magnetic field, azimuthal magnetic field and meridional magnetic field) each of which with a dimension equal to the number of surface segments. The inversion algorithm has to navigate through the combined parameter space to find a solution that is compatible with the data. In order to study the stability of the inversion we use a simulation that runs the inversion with the original data set but from randomly chosen starting positions. Each of the four parameter spaces is independently initialized by choosing a random value for each surface segment. This led to the spatial error maps shown in Fig. 1. What can be seen from these maps is that the error values are correlated with the field strength as well as with the temperature. For V410 Tau a temperature change of just 80K in a spotted region causes a change in the amplitude of the Stokes V signal of 4%. In the strong field regime of the polar spot this difference in amplitude is equivalent to a magnetic field of 60 G. This emphasizes once more the influence of the temperature on the magnetic field determination and leaves the frightening possibility that magnetic maps where the temperature was not included to be simply wrong.



Figure 1. Error maps from a multi-line inversion of CFHT/Espadons data of V410 Tau (V 11.5 mag) with iMap. Left: errors of the temperature map; right: errors of the respective magnetic-component maps. Note the correspondence of the magnetic-field errors with the temperature.



A recent application to V410 Tauri


V410 Tauri is a weak-lined T Tauri stars that is dubbed a ”young Sun”. The ZDI magnetic field structures obtained recently by Carroll et al. (2012) show a good spatial correlation with the surface temperature and are dominated by a strong field within the cool polar spot (Fig. 2). Earlier, Rice et al. (2011) presented temperature maps from atomic and molecular lines from the same data. The ZDI maps exhibit a large-scale organization of both polarities around the polar cap in the form of a twisted bipolar structure. The magnetic field reaches a value of 1.9 kG within the polar region but smaller fields are also present down to lower latitudes. Within the polar spot the two polarities appear separated by a sharp, pole-crossing, neutral sheet. In total the field forms an S-shaped structure centered at the rotation pole. The topology is predominantly radial but 1 kG meridional components coexist very close to the pole. An azimuthal component, on the other hand, is restricted to certain regions and also reaches 1 kG at one point while the rest of it mostly remains near 500 G. Expressed in terms of the poloidal and toroidal components, we found that 73% of the surface field is poloidal and just 27% is toroidal. The pronounced non-axisymmetric field structure and the non-detection of a differential rotation for V410 Tau (dΩ = 0.007±0.009) supports the idea of an underlying 2Ω-type dynamo, which is predicted for weak-lined T Tauri stars. Following Küker & Rüdiger (1999) for an 2Ω-dynamo the field topology at the surface can be expected to be nonaxisymmetric with a typical S1- or A1-type geometry and appears to be in excellent agreement with the ZDI map.



In Fig. 3, we compare the magnetic map from iMap with the independent ZDI map of Skelly et al. (2010), based on the code by Brown et al. (1991). The eye immediately catches some significant differencesy. Skelly et al. (2010) found that 50% of the total field has a toroidal component and the other 50% are poloidal and that the majority of the field shows strongly inclined large-scale azimuthal fields which are distributed over one entire hemisphere, whereas our reconstructed topology has only 27% toroidal fields and 73% poloidal fields with a strong radial and bipolar component around the polar region. In contrast to our reconstructed map where the strong magnetic fields are well correlated with the temperature, the fields of Skelly et al. (2010) seem to show almost no correlation with their brightness maps, i.e. a proxy of the temperature distribution. We note that our data were taken with CFHT/Espadons during December 5, 2008 to January 14, 2009. The fact that the Skelly et al. data were observed at almost the same time as ours at the Telescope Bernard Lyot with the Narval spectropolarimeter (January 2 to January 17, 2009), is certainly a reason for a closer inspection and comparison of the two results. We can only speculate about the reasons for the disparity of the two reconstructions but we want to emphasize that in contrast to Skelly et al. (2010) our approach makes no assumptions on the surface topology in terms of a spherical harmonic decomposition or potential field structure. It is furthermore fully based on polarized radiative transfer and line profile modeling instead of assuming a Gaussian local line profile and it pursues a strategy which simultaneously invert the temperature and magnetic field. This gives us enough confidence to believe in the validity of our new iMap results.



Figure 3. A comparison of two independent ZDI maps of the WTTS V410 Tauri. Top row: Skelly et al. (2010) map based on observations with TBL/Narval in the period of January 2009 and the Brown et al. (1991) code. Lower panel: Carroll et al. (2012) map based on observations with CFHT/Espadons in the period of December-January 2009 and the iMap code. The iMapZDI is from Stokes V&I and SSRM, the Skelly et al. ZDI is from Stokes V alone and LSD. Notethat the Skelly et al. map is shown for the entire surface while the Carroll et al. map is shown just down to the stellar equator.




Last Updated on Wednesday, 6 March 2013 14:30