5 Doppler imaging of AG Doradus

 

  
5.1 Input parameters and assumptions

The technique of Doppler imaging (see e.g. Rice 1996) offers the potential to resolve the spot distribution on the surfaces of rapidly-rotating stars. Many of the stars observed with this technique reveal spots in high latitudes, contradicting the solar paradigm. However, all of these stars have substantially larger rotation rates than the Sun because rotational broadening is required. The slowest rotator that has ever been successfully Doppler imaged is the G1.5 solar-type field star EKDra (Strassmeier & Rice 1998) with a $v\sin i$ of 17.3 kms^-1 (but a period of 2.6 days). AG Dor has an almost identical projected rotational velocity as EK Dra, and is so far the slowest-rotating K-dwarf that is being Doppler imaged.

Figure 7: Doppler images in pseudo-mercator projection for the Fe I 6546.239 Å line. a) odd spectra, b) even spectra. The arrows below the temperature maps indicate the respective phase coverage. The subpanels show the observed and computed line profiles. The lines are the fits and the bars are the observations. The length of a bar indicates the S/N per pixel. c) is the difference-map from the odd and even images. d) shows the FWHM of the cross correlation function from the odd and even map per latitude bin

Figure 8: Doppler images for Fe I 6546.239 Å line using all spectra a) in pseudo-mercator projection (the subpanel shows observed and computed line profiles) and b) in pole-on view. c) shows the average temperature distribution along constant latitudes. Arrows indicate the total phase coverage

Figure 9: Doppler image from a) the first and b) the second stellar rotation. c) shows the difference-map from a) and b)

The Doppler maps in this paper were generated using the temperature mapping code TEMPMAP (Rice et al. 1989). The updated version was presented and tested by Rice & Strassmeier (2000). We adopted a maximum-entropy regularization for all inversions. A grid of ten model atmospheres with temperatures between 3500 and 5500K in steps of 250K and $\log g = 4.5$ were taken from the ATLAS-9 CDs (Kurucz 1993). Gray (1992) lists a $\log g$ of 4.55 for a K1 dwarf. Trial inversions with other $\log g$ values did not result in better line-profile fits, thus $\log g = 4.5$ was adopted. We can make use of only one (moderately) unblended line: Fe I 6546 Å with a logarithmic transition probability ($\log gf$) of -1.35 and a lower excitation potential of 2.76 eV. For the local line-profile tabulation, we synthesize a total of five blends (including the main mapping line) as indicated in Fig. 1. The secondary star contributes about 5% to the continuum intensity which decreases the apparent line depth of the primary star. Therefore, all spectra were multiplied by a factor of 1.05 in order to correct the contribution of the secondary star.

Figure 6a shows the normalized $\chi^2$ distribution (normalized by the minimum value) from a series of test reconstructions as a function of the stellar inclination angle. We find an average minimum at around $i\approx 55\hbox{$^\circ$ }\pm10\hbox{$^\circ$ }$ from the odd, even, and full data set (see next section). Kürster (1993), Unruh & Collier Cameron (1995) and most recently Rice & Strassmeier (2000), have shown that the minimization of $\chi^2$ can be successfully used to constrain the inclination angle. Of course, whether the best attainable fit to the data is also the correct one is debatable. We also note that the surface spot distribution nevertheless remains similar, even for a change of the stellar inclination of about $\pm20\hbox{$^\circ$ }$ (Vogt et al. 1987).

Figure 6b shows the normalized $\chi^2$ distribution as a function of the projected rotational velocity. We find a clear minimum at $v \sin i = 18 \pm 1$ kms^-1. The best $v\sin i$ with the spectrum-synthesis method was $17 \pm 2$ kms^-1, while Pallavicini et al. (1992) obtained 20 kms^-1. However, the Doppler-imaging technique is very sensitive to the projected rotational velocity and provides better accuracy than any other method, as it takes into account blends in the line profile as well as line deformation due to spots. We therefore adopt $18 \pm 1$ kms^-1 as the most likely $v\sin i$ value.

The average S/N ratio of $\approx$150:1 is low but still sufficient for Doppler imaging. The resolving power of 50000 yields a velocity resolution of 6 kms^-1, offering six spectral resolution elements across the full width of the lines at the continuum. Numerical tests have shown that reliable images can be reconstructed with as low as 5 resolution elements (Piskunov & Wehlau 1990; Hatzes 1993; Strassmeier & Rice 1998). The relatively long rotational period, combined with an average integration time of 35 min, allows for a surface phase resolution of around 3.5$^\circ$ near the stellar meridian and along the direction of rotation, approximately a factor of ten smaller than the spectroscopic resolving power. The smearing due to orbital motion amounts to $\approx$3 kms^-1 for a 35 min integration, which is about half the size of the spectral resolution. We can therefore safely neglect phase smearing due to stellar rotation and orbital revolution.

 

 

Table 7: Adopted Doppler-imaging parameters for AG Dor

Parameter	Value
Tphot	4900 K
$\log g$	4.5
$v\sin i$	18 kms-1
i	55$^\circ$
$P_{rot} \equiv P_{orb}$	2.562 days
Micro turbulence $\xi$	2.0 kms-1
Macro turbulence $\zeta_{R} = \zeta_{T}$	4.0 kms-1
[Fe/Fe$_\odot ]$	-0.45

5.2 A Doppler image for November 1992

We obtained 38 spectra from 6 consecutive nights and split them into two blocks of 19 spectra each, taking odd numbers for the first block and even numbers for the second (in chronological order). Therefore, we obtain two Doppler images from two independent data sets - but from the same epoch - and compare them as a consistency check (see also e.g. Barnes et al. 1998). However, our final maps are computed from the combined data.

Figure 7 compares the maps from the odd and even spectra. It demonstrates that both data sets reconstruct a very similar spot configuration with much detail in common. A bridge between the low-latitude spot at $\ell\approx300$$^\circ$ and the polar appendage at $\approx$350$^\circ$ is visible in both cases (it is a bit more emphasized in the even map). Also an equatorial spot at $\ell\approx180$$^\circ$ and an asymmetry of the polar spot at $\ell\approx100$$^\circ$ is visible. A third low-latitude feature at $\ell\approx50$$^\circ$ appeared split into two cool features in the even-spectra image, while single and of lower contrast in the odd-spectra image. The combined image still recovers a split feature with both parts of lower contrast compared to the even-spectra image. The difference between the odd and the even map in Fig. 7c shows an average divergence of $\pm$200K. This is mostly due to shifts of the spot location between the even and the odd map (above all from the bright patches, see below). Figure 7d gives the FWHMof the cross-correlation of the odd and even map as a function of stellar latitude. As expected, the equatorial latitudes show more details in the surface structure, resulting in a smaller FWHM of the cross-correlation peak.

The observed and computed line profiles for the combined spectra, along with our final Doppler image, are plotted in Figs. 8a and  8b, respectively. We find a polar spot with two cool appendages having a temperature difference of 1000-1500K with respect to the adopted photospheric temperature of 4900K. A total of three low-latitude spots or spot groups are recovered with a temperature difference of $\approx$800K. Two or possibly three high-latitude bright patches ($T\approx$ 4860-4930K) seem to occur at the same Doppler shifts as the three cool equatorial spots. The difference map from the odd and even spectra in Fig. 7c shows the largest differences at the location of two of these bright patches, indicating that these features are very likely artificial due to the combined effects of small $v\sin i$, limited S/N, and a mirroring with the cool spots.

Figure 8c shows the latitudinal temperature distribution. It was derived by binning all surface pixels along constant latitudes. The average temperature in each bin is generally below the expected photospheric temperature of 4900K. A relatively sharp drop in temperature occurs at a latitude of approximately 50-60$^\circ$, close to the projected angle of inclination. However, this is merely coincidental and due to the two polar appendages. The polar feature itself extends on average to a latitude of +70$^\circ$ (better seen in the pole-on projection in Fig. 8b), but it is clear that the high latitudes block most of the otherwise "missing flux''.

The overall goodness of fit for the entire data set was $\chi^2_{\rm tot}$ = 0.2235 (for the odd-spectra fit 0.1101; even-spectra fit 0.1077). These can be considered excellent fits for the S/N ratio given. Some line profiles could not be fitted to the level we would have liked, notably phases 0 $.\!\!^{\scriptscriptstyle\rm p}$116, 0 $.\!\!^{\scriptscriptstyle\rm p}$210, 0 $.\!\!^{\scriptscriptstyle\rm p}$846 and 0 $.\!\!^{\scriptscriptstyle\rm p}$875 in Fig. 8a. It is possible that the spot configuration slightly evolved within the six nights of our observations, i.e. two and a half stellar rotations. Therefore, we generated another set of two Doppler images, using the first 19 spectra (nights 1-3) and the second 19 spectra (nights 4-6). The maps and the phase coverages are shown in Figs. 9a and b. Unfortunately, the phase gaps become very large (up to 120$^\circ$), and it is not straightforward to compare the images in the presence of such large gaps (see Rice & Strassmeier 2000). We can not draw any firm conclusions whether the differences are solely due to the poor reconstruction within the phase gaps or due to partial spot migrations on the stellar surface. Cross-correlating the two surface maps also did not lead to a useful correlation because of the masking with the phase gaps.