4 Time-series Doppler imaging analysis

We prepared six data subsets (SS1-SS6) out of the 70-day long spectroscopic coverage. Each subset covers approximately one rotation period separated by half a rotation. The successive subsets thus overlap by approximately half of a rotation period but subsets 1, 3, and 5, as well as 2, 4, and 6 represent contiguous stellar rotations. The mid Julian dates for the individual subsets SS1-SS6 are 2450397.980, 2450408.505, 2450417.935, 2450428.410, 2450438.435 and 2450448.365, respectively. Each spectroscopic dataset is supported by simultaneous photometric observations.

4.1 The line profile inversion code

Our line-profile inversion code performs a full LTE spectrum synthesis by solving the equation of transfer through a set of Kurucz (1993) model atmospheres, at all aspect angles, and for a given set of chemical abundances. Simultaneous inversions of eight spectral lines, as well as two photometric bandpasses, were carried out using a maximum-entropy regularisation. The number of iterations was set to 15 which proved to be sufficient for a good convergence. The computations were performed on a Sun Ultra-2 workstation at Konkoly Observatory and required 20-30 min of CPU time for each Doppler map. A more detailed description of the TempMap code and additional references regarding our line-profile inversion technique can be found in Rice et al. (1989) and Piskunov & Rice (1993) and previous papers of this series (e.g., Rice & Strassmeier 1998; Strassmeier & Rice 1998).

  
4.2 Astrophysical input

 

 

Table 2: Astrophysical data for $\sigma$ Gem.

Parameter	Value
Classification	K1 III
Distance (Hipparcos)	$37.5\pm 1.1$ pc
Luminosity, L	52.5 +14.5-8.2 $L_\odot$
$\log g$	2.5 +0.23-0.42
$T_{\rm eff}$	$4630\pm 100$ K
$(B-V)_{\rm Hipparcos}$	$1.118\pm 0.006$ mag
$(V-I)_{\rm Hipparcos}$	$1.12\pm 0.05$ mag
$v\sin i$	$27.5\pm 1$ kms-1
Inclination, i	$60\hbox{$^\circ$ }\pm 15\hbox{$^\circ$ }$
Period, $P_{\rm rot}=P_{\rm orb}$	$19.60447\pm 0.00007$ days
Orbital eccentricity, e	0.0
Radius, R	12.3 $^{+1.6}_{-1.0} R_\odot$
Microturbulence for Ca, $\xi_{\rm Ca}$	0.7 kms-1
Microturbulence for Fe, $\xi_{\rm Fe}$	1.0 kms-1
Macroturbulence, $\zeta_{\rm R} = \zeta_{\rm T}$	3.0 kms-1
Chemical abundances	solar (adopted)

 

 

Table 3: Adopted atomic line parameters.

Element	Ion	$\lambda$ (Å)	$\log gf$	$\chi_{\rm low}$ (eV)
Eu	II	6437.640	-0.45	1.320
Si	I	6437.703	-2.35	5.863
Ni	I	6437.992	-0.75	5.389
V	I	6438.088	+0.25	2.684
Fe	I	6438.755	-2.48	4.435
Ca	I	6439.075	+0.10	2.526
Fe	I	6439.554	-3.05	4.473
Si	I	6440.566	-2.48	5.616
Fe	I	6429.071	-3.41	4.294
Co	I	6429.906	-2.41	2.137
V	I	6430.472	-1.00	1.955
Fe	I	6430.844	-2.00	2.176
Ca	I	6431.099	-2.61	3.910
V	I	6431.623	-1.25	1.950
Ni	I	6431.994	-1.75	3.542
Fe	II	6432.680	-3.74	2.891

Table 2 summarizes the astrophysical input parameters for $\sigma$ Gem. We adopt i=60$^\circ$ as the most likely inclination angle. This value was determined by reducing the misfit of the line profiles as a function of the inclination (for the method see, e.g., Rice & Strassmeier 2000 a.o.). Its uncertainty is estimated from the width of the $\chi^2$ minimum and is approximately $\pm$15$^\circ$.

Figure 3: Ca I 6439-Å images for the six subsets SS1-SS6. The maps are presented in a pseudo-Mercator projection from latitude -60$^\circ$ to +90$^\circ$. The line profiles and their fits are shown in the middle row, the light curves and their respective fits are plotted in the bottom two rows for the Strömgren b and y bands, respectively. The phases of the observations are marked by arrows below the maps.

Figure 4: Fe I 6430-Å images for the six subsets SS1-SS6. Otherwise as in Fig. 3.

Our best value for the projected rotational velocity, $v_{\rm eq}\sin i$, was determined by minimizing the artificial dark or bright bands in the Doppler maps that appear if the equatorial velocity, $v_{\rm eq}$, is either too large or too small, respectively (for more details see, e.g., Strassmeier et al. 1998, 1999b). Its uncertainty depends on the S/N ratio and on the "external quality'' of the spectra (i.e. wavelength calibration, flat fielding, etc.) but also whether we have a low-gravity giant atmosphere or a high-gravity dwarf atmosphere and, last but not least, on the spot morphology on the stellar surface. In case of the K1 giant $\sigma$ Gem, we estimate an uncertainty of $\pm$1 kms^-1.

The minimum radius is computed from the assumption that the rotation is synchronized to the orbital motion. $R\sin i$ becomes then $10.65\pm 0.35$$R_\odot$ and, with $i=60\hbox{$^\circ$ }\pm15\hbox{$^\circ$ }$, $R\approx12.3_{-1.6}^{+3.2}$ $R_\odot$. This value is formally below the lower bound of radii found in the literature for K1 giants (e.g. Gray 1992). The Hipparcos parallax of $26.70\pm 0.79$ milli- $^{\prime\prime}$ (ESA 1997) and the brightest Vmagnitude observed so far, $V=4\hbox{$.\!\!^{\rm m}$ }137$ (see Sect. 3), result in an absolute visual brightness of +1 $.\!\!^{\rm m}$27 which is almost one magnitude fainter than tabulated values for K1 giants (Gray 1992). The radius from the Hipparcos distance and the effective temperature of 4630 K is 9.3 $R_\odot$ (with B.C.=-0.515; Flower 1996), and is also much smaller than for a fully developed K1 giant and even smaller than our measured minimum radius $R\sin i$. However, adopting the average surface temperature of $\approx$4200 K from our Doppler images as the effective temperature, the radii agree to within their uncertainties. O'Neil et al. (1996) obtained a photospheric temperature of 4500 K and an average spot temperature of 3850 K from measuring titanium oxide bands contemporaneous with BV photometry. The Hipparcos B-V of 1.118 indicates 4630 K (according to the calibration of Flower 1996). In any case, this temperature and the bolometric magnitude of +0 $.\!\!^{\rm m}$75 suggests that $\sigma$ Gem is beyond the base of the giant branch but not yet in the helium core-burning phase. A comparison with the evolutionary tracks from Schaller et al. (1992) for solar metallicity implies a mass of $1.5\pm 0.2$$M_\odot$ and an approximate age of 2.8 Gyr.

4.3 Atomic line-data input

In this study, we use the Ca I line at 6439 Å and the Fe I line at 6430 Å as the main mapping lines. In the (photospheric) temperature range for $\sigma$ Gem, the Ca I-6439 line is quite temperature sensitive. Therefore, cooler regions on the stellar surface will appear less pronounced because inside the spot, i.e. in the regions with lesser continuum intensity, the Ca line has a relatively large equivalent width and hence the bump produced by the spot in the Ca profiles appears less significant. As a result, the Ca I-6439 images provide inferior "resolution'' compared to the Fe I-6430 images. The iron line has the advantage that it is less sensitive to the surface temperature gradient and thus has a smaller equivalent width compared to the Ca line. Its Doppler images provide thus more surface detail. However, better fits in the sense of lower $\chi^2$ are obtained for the Ca spectra. Unfortunately, Fe I-6430 is blended by several relatively strong metal lines, most notably Fe II at 6432 Å.

In Table 3, we summarize the atomic parameters that we adopted for the Ca I-6439 and the Fe I-6430 spectral regions. By default, solar photospheric abundances were assumed which required alteration of some of the $\log gf$ values from the line list of Kurucz (1993) in order to obtain better fits to the spectra. The effect of an abundance change on the line-profile reconstruction is not distinguishable from a change of $\log gf$, as was noted in previous papers in this series and quantified in tests with artificial data by Rice & Strassmeier (2000). Also, abundances from just a single spectral line can be notoriously uncertain and we chose not to solve for the abundance but adjust the transition probability instead.

Figure 5: a) Average Ca I-6439 map, and b) average Fe I-6430 map. Both maps were obtained from the entire data set spanning 3.6 stellar rotations.

Figure 6: Differential maps. Each map is the difference between an individual map in Figs. 3 and 4 and the averge map from Fig. 5. a) for calcium, b) for iron, and c) for the average calcium and iron maps.

4.4 Doppler images for 1996/97

Figures 3 and 4 show the results for the Ca I-6439 and the Fe I-6430 line region, respectively. Six (overlapping) maps for each wavelength region are obtained (SS1-SS6). None of the maps recovered a polar cap-like spot, nor any other spots at latitudes higher than 60$^\circ$. Spots were found mainly along a belt between 30$^\circ$ and 60$^\circ$ latitude, but some weaker features were recovered in the low-latitude regions $\pm$30$^\circ$ from the equator. The main features in the medium-latitude belt have temperature contrasts of $\Delta T =T_{\rm phot}-T_{\rm spot}\approx 800$ K from the Fe map and 700 K from the Ca map, both values with an rms of around 50-100 K. The smaller structures in the equatorial regions appear consistently warmer than these larger features and have $\Delta T\approx 350\pm50$ K. Although the calcium line provides a less detailed surface temperature reconstruction than the iron line due to its intrinsically broader local line profile, there is reasonable good agreement between the respective Fe and the Ca maps. We thus conclude that even the weaker equatorial features are needed by the data and are thus likely real.

Several weak warm features with a contrast of $\Delta T\approx-100\pm50$ K are also recovered, mostly at high latitudes, but are related to dominant cool regions at similar longitudes. We attribute them to the latitudinal mirroring effect, an artifact of the Doppler-imaging technique due to the relatively high inclination value of the rotational axis of 60$^\circ$.

Although the photometric data show a changing light curve for the consecutive rotation cycles (see Fig. 1), the changing spot distribution is not obvious from the spectral line profiles. To search for signs of spot evolution during the seventy nights of observation, we first reconstruct the surface temperature distribution from the entire dataset (including all photometric and spectroscopic data in Table 1). The resulting maps in Fig. 5 - one for Ca and one for Fe - are the average surface spot distributions from 3.6 consecutive stellar rotations. A simple visual comparison immediately shows that these average images are generally smoother and have lesser surface detail than the individual images in Figs. 3 and 4. Because the phase coverage is already excellent even for the individual images, we exclude a numerical reason due to the larger number of spectra (it would have the opposite effect than is observed, i.e. with more spectra the features should become better defined). We take this as evidence for spot evolution during the full 3.6 rotation cycles.

Figure 6 shows the spot changes as a function of time by simply plotting the difference between the individual images (SSi) and the average image as consecutive differential maps (the top six panels are for the differential Ca maps, the middle six panels for the differential Fe maps, and the bottom panels for the average Ca and Fe maps). Notice that the temperature range in these maps is different to the maps in the previous figures. Further evaluation is then done by using the reconstructed average map as the input map in the forward problem and generate an artificial data set, which is then compared with the observations of the individual data sets. By determining and comparing the goodness of the line-profile fits from the individual inverse solutions with the forward computations from the average map, we find that the individual solutions fit the data on average 22% better than the average image. Table 4 summarizes our goodness of fit statistics. We calculated a (pseudo) $\chi^2$ value for all of our spectra fitted by either the maps shown in Fig. 4 or the grand-average maps shown in Fig. 5, defined as

$$\chi^2=\sum_j\frac{(M_j- D_j)^2}{\sigma_j^2}$$ (2)

with $\sigma_j=3/[(C-D_j)*{\rm SNR}]$, where C($\equiv 1$) is the continuum level, M_j and D_j are the model and the observed datapoints, respectively, while SNR is the signal-to-noise ratio of the given spectrum. In Table 4 $\bar{\chi}^2_{\rm {sub}}$, which is the mean of the $\chi^2$ values obtained for a group of spectra forming the ith subdata set SSi, is compared to $\bar{\chi}^2_{\rm {avr}}$, the mean value of the $\chi^2$ values obtained for the same subdata set fitted by the grand-average map. We find that the subdatasets are generally better fitted than the average map. Note that in Table 4 there is only one case when the average map provides a slightly (4%) better fit (for the Ca-SS6 subset). We believe this exception is the result of the poor phase coverage of the photometry for this particular data set. Generally, a photometric weight of 0.1-0.2 relative to a line profile is applied, depending upon the quality of the photometric data.

At this point, we note that the main spotted regions in the average Doppler maps were also recovered by a fit to the photometric data alone (Sect. 3). The best fit to the light and color curves was achieved with a spot temperature contrast of 600$\pm$100 K, in agreement with the temperatures from the Doppler images. The photometric solution also agrees with the Doppler maps in the sense that no spots were recovered above +60$^\circ$ latitude. However, the spot at a longitude of $\approx$240$^\circ$ (SPOT 3 in Fig. 2) was reconstructed at a latitude higher by $\approx$25$^\circ$ than in the Doppler maps. We attribute this difference to the combined effect of sparse sampling of the consecutive light curves and the mathematical ambiguity that arises from the inversion of a disk-integrated one-dimensional data set into a surface map. Being aware of this intrinsic latitude ambiguity, we nevertheless conclude that the time-series analysis of light curves is a very useful tool for studying starspot and is supportive of the Doppler-imaging results.

 

 

Table 4: A comparison of the $\bar{\chi}^2$ values for the fits to the subdata sets by the individual images ( $\bar{\chi}^2_{\rm {sub}}$) and by the average map ( $\bar{\chi}^2_{\rm {avr}}$).

Ca I-6439	SS1	SS2	SS3	SS4	SS5	SS6
$\bar{\chi}^2_{\rm {sub}}$	4.492	5.262	5.483	3.976	2.772	3.483
$\bar{\chi}^2_{\rm {avr}}$	5.747	6.206	7.103	4.735	3.312	3.331

Fe I-6430	SS1	SS2	SS3	SS4	SS5	SS6
$\bar{\chi}^2_{\rm {sub}}$	3.575	2.981	3.822	3.699	2.886	4.804
$\bar{\chi}^2_{\rm {avr}}$	4.608	4.473	5.252	3.968	3.300	5.363

4.5 Differential surface rotation and spot evolution

To isolate a possible surface migration pattern and to obtain a quantitative description of it, we first cross correlate the consecutive Doppler images with each other but for both lines separately (for a detailed description of the method see Collier Cameron (2001) and previous papers in this series). We only cross-correlate the phase independent contiguous images, i.e., Corr{SS1/SS3}, Corr{SS2/SS4}, Corr{SS3/SS5}, and Corr{SS4/SS6} so that any cross-talk from a signature in the same line profile is excluded. This restricts the time resolution to one stellar rotation. The resulting cross-correlation-function maps (ccf-maps) for the Ca and Fe lines are displayed in Figs. 7a and b, respectively.

In most cases a coherent latitude dependency of the correlation signal is seen in maps from both spectral lines. Nevertheless, the results in Fig. 7 are inconclusive. Clear equatorial deceleration within an approximately $\pm30$$^\circ$ range around the equator is seen in Corr{SS2/SS4} and Corr{SS3/SS5}, no convincing pattern seems to be evident in Corr{SS4/SS6}, and Corr{SS1/SS3} recovers a pattern with a longitudinal shift direction that appears mostly reversed with respect to the other three correlation maps in Fig. 7. We also cross-correlated the two images with the largest time span in between, i.e. Corr{SS1/SS6}, but the ambiguity remained.

Figure 7: The cross-correlation functions (ccf) from the independent contiguous Doppler images. a) displays the ccf-maps from the Ca line and b) those from the Fe line. The x-axis plots the longitudinal shifts in degrees. The correlation coefficient (grey scale) is weighted by the difference of the average temperature per latitude strip to the effective temperature.

We then computed an average cross-correlation function from all individual ccfs shown in Fig. 7 and searched for a correlation peak in each latitude strip (for a description of the procedure see e.g. Paper V by Weber & Strassmeier 1998). These peaks are then fitted with a bi-quadratic differential rotation law between the $[-30\hbox{$^\circ$ },+70\hbox{$^\circ$ }]$latitude range (i.e. the most reliable part of our Doppler maps). Despite the very weak statistical significance, it suggests a solar-like acceleration in two nearly symmetric bands around the equator for $\sigma$ Gem (the $\chi^2$ for the bi-quadratic fit was 9.31 and 10.22 was achieved for a quadratic fit with just one term, while we got 17.55 for a linear fit, i.e. the same longitudinal shift for all latitudes). Nevertheless, we consider this as too weak a detection. Obviously, a simple interpretation with a latitude-dependent surface differential rotation pattern is not straightforward from our data. We believe our analysis is hampered by two facts. Firstly, the spot pattern may not have changed enough during the time of our observations in order to resolve a coherent differential-rotation signature and, secondly, the cross correlations may have been masked by individual spot evolution.