Astron. Astrophys. 334, 873-894 (1998)

6. High-resolution spectra: Doppler imaging

Additional high-resolution observations collected primarily at McDonald Observatory comprise a timeseries over the rotational period of P1724. The photospheric absorption lines clearly show the types of distortions common to heavily spotted late-type stars, though the line wings tend to vary more than is usually seen. Since there is no compelling evidence in spectroscopic or photometric data sets of a companion which could cause such oscillations, we proceed under the assumption that the distortions are indeed a product of surface temperature inhomogeneities. We can also assume that P1724, as a TTS, does not rotate differentially (Johns-Krull 1996). Image reconstructions are presented herein.

6.1. Data acquisition and reduction

High resolution spectral observations were made using the Sandiford Cassegrain Echelle spectrograph at McDonald Observatory's 2.1m telescope. This instrument is a prism cross-dispersed echelle used with a Reticon Corporation pixel CCD. This combination provides a wavelength coverage of up to 1500Å at a resolving power as high as 60000 (McCarthy et al. 1993). The instrumental setup for the P1724 observations yields a wavelength coverage from 6000 to 7500Å with a resolution of 0.18Å (3 pixel slit) at 6400Å , corresponding to .

The data were reduced using the IRAF echelle package, which consisted of the standard bias subtraction, global scattered light subtraction, division by a properly calibrated flat-field frame, and order extraction. To complement this data set and provide us with more complete phase coverage, we also used the high resolution CASPEC spectra described below in Sect. 7. Table 3 summarizes the observations used in the Doppler imaging (DI).

Table 3. P1724 Doppler Imaging observing log. We list for each high-resolution observation the heliocentric Julian date, the rotational phase counted from the middle of the first of our exposures, the heliocentric radial velocity in (typically to ), the equivalent widths for the Li 6708Å , Fe 6430Å , Ca 6439 and the emission lines, as well as the (log of the) flux (in ). The last two spectra were obtained with CASPEC at the ESO 3.6m and the others with the McDonald 2.1m telescope.

Radial velocities for each individual observation are determined using a cross-correlation method similar to that outlined in Tonry & Davis (1979). Since P1724 has been classified as late G to early K, we use the NSO full disk solar spectrum (Kurucz et al. 1984) as the non-rotating template in the cross-correlation. The peaks of Gaussian fits to the cross-correlation functions define the radial velocities. The spectral regions between 6080-6150Å , 6130-6190Å , 6180-6240Å , 6380-6450Å , and 6440-6510Å contain clearly defined absorption lines and few telluric features which is what the method requires. We therefore rely on these wavelength regions in computing the cross-correlation functions.

It is expected that distortions in line profiles due to surface temperature inhomogeneities will skew the Gaussian peaks, and we measure the resulting dispersion. From our rotational timeseries, we compute a mean radial velocity of and a standard deviation of . This is consistent with the results from the low S/N high-resolution spectroscopy presented in Sect. 3.

6.2. Doppler image reconstruction

The algorithm described by Vogt et al. (1987) is used to reconstruct images of the stellar surface. The method uses a maximum entropy regularization to constrain the solution. It does not simultaneously consider broad-band continuum light curves, but we use this information in determining the final spot temperatures.

Doppler image reconstruction uses a rotational timeseries of absorption line profiles to ascertain spatial information. It requires that absorption lines are not significantly blended within the wavelength interval defined by the rotational velocity. The spectral format is scoured for such lines using the solar spectrum for reference. Some lines, though ideal, are rejected because of contamination from telluric lines. We choose the absorption lines Fe I Å , Fe I Å , Ca I Å , Ti I Å , and Ti I Å for the analysis, each defining a separate timeseries.

The method also requires knowledge of the specific intensity profiles as a function of limb angle across the stellar disk. These are obtained using the spectral synthesis package, SME (Valenti & Piskunov 1996), atomic line data from the Vienna Atomic Line Database (VALD, Piskunov et al. 1995), and the grid of Kurucz model atmospheres (Kurucz 1993). SME computes LTE models and uses a non-linear least squares algorithm to solve for any indicated free parameters. For our program star, we assume solar metallicity and a surface gravity of (as determined in Sect. 3). We let SME solve for , microturbulence, and . Since there are undoubtedly errors in all parameters, we expect to obtain slightly different results for each line as the code tries to compensate for errors in the fixed parameters, atomic data, and atmospheric models. This is acceptable since we are primarily concerned with constructing a well-matching fit to the disk integrated line profile (Stout-Batalha 1997). The solution is determined by matching the model to the average of all profiles in the timeseries. SME gives us , a mean microturbulence of , and , all consistent with the results in Sect. 3.

The inclination angle is computed assuming a rotational period of 5.679 days, a radius of (cf. Sect. 5), and , which is our best estimate of the rotational velocity (cf. Sect. 5). This gives an inclination angle of .

6.3. Doppler imaging results

Our line fits for the Doppler imaging are presented in Fig. 7a,b. The final Doppler image of P1724 is shown in Fig. 8 as a polar projection down to a latitude of . The assumed inclination of the star's rotational axis, , does not allow us to see below this latitude. The image is actually the average of the five individual images reconstructed from each of the five timeseries (one for each spectral line selected). While we do not display all images here, we note their high degree of correlation as quantified by the linear Pearson correlation coefficients listed in Table 4. Individual image reconstructions and a more thorough comparative analysis can be found in Stout-Batalha (1997).

Fig. 7a and b. Doppler imaging line fits. The observed absorption lines (open circles) in the timeseries are plotted with the model fits (solid lines) for each of the five lines used in this analysis. Also plotted are the residual vectors (in the sense: ) to illustrate that no systematic features appear above the noise level that the code may have failed to fit. The average of all residuals is plotted along the bottom. This figure is available by ftp from CDS

Fig. 8. Doppler image. The Doppler image reconstruction of P1724 as a polar projection down to a latitude of  S latitude. The image shown is the average of the images from each timeseries. Tick marks around the periphery note the rotational phases at which the star was observed. The thick line marks the stellar equator

Table 4. P1724 Doppler imaging linear correlation coefficients.

As expected, we find evidence of low to intermediate latitude surface features as demanded by the large velocity excursions of the absorption line wings. More specifically, we find a predominant spot (or spot group ⁶) near phase 0.75 and centered at a latitude of approximately . This feature is easily seen as a distortion near the line center of the absorption lines taken at phase 0.71. There is also evidence of lower level features between 0 and which persist upon image averaging. However, most of these features are located at sub-observer longitudes and are therefore likely artifacts of the algorithm, an artifact referred to as phase-ghosting which is discussed in more detail in Stout-Batalha (1997). The size of the spotted area is of the visible disk.

The average temperature of the predominant spot group is obtained by comparing the V -band photometric observations with the artificial light curve which our image reconstruction produces. We find that a local temperature cooler than the surrounding photosphere (taken to be ) reproduces the observed light curve amplitude best. This is only a decrease relative to the minimum temperature found in the average image reconstruction. The total spot area assigned this temperature is represented in Fig. 8 by the black-shaded region.

From the temperature of the spot and the physical parameters of the surrounding photosphere, we can estimate the strength of the magnetic field assuming equipartition and the ideal gas equation. With and as obtained above for P1724, along with a value of µ = 0.6 for the mean molecular weight, we obtain a gas pressure of which then corresponds to a magnetic field of for P1724. This is quite similar to those measured in other TTS (Basri et al. 1992, Guenther & Emerson 1996) though perhaps somewhat larger.

Fig. 9a shows the modeled versus the observed light curve. The photometric observations that are closer in time to the Doppler imaging observations (the RW data set) have been phased relative to the HJD at mid-exposure of our first high-resolution observation. In generating the model RV curve, the re-constructed spot distribution was first converted into a two temperature distribution. All image pixels more than cooler than the photospheric temperature were assigned a temperature cooler than the photospheric value; all image pixels less than cooler than the photospheric temperature were replaced by the photospheric value. This was done primarily because the re-construction process produces a smooth temperature variation from spot to photosphere, even if the real distribution is a discrete distribution. Furthermore, the maximum entropy method tends to make spots warmer than they actually are. Finally, a two temperature distribution is more consistent with the solar analog. The agreement between the predicted and observed light curve is excellent.

Fig. 9. Observed V band and radial velocity variations compared to the DI model predictions. The differential photometry by RW is shown in the top panel, and in the lower panel we include the CfA velocities (open circles) and those from the spectra used for the Doppler imaging reconstruction (filled circles) listed in Table 3. The full line is for the synthetic Fe I 6430Å line only, and the broken line for Ca I 6439Å

Shown in Fig. 9b are the measured radial velocities plotted as a function of rotational phase, together with those predicted by the same `thresholded' image used in generating the predicted light curve. This image was used to generate a set of synthetic Fe I 6430Å and Ca I 6439Å profiles with good phase coverage that were then used for computing the radial velocity. Again, there is good qualitative agreement in the shape of the variation, although the predicted amplitude is less than the observed one.

The reader is cautioned, however, that the predicted RV curve should be regarded more as a qualitative one for the following reason: The DI algorithm cannot predict with great accuracy the amplitude of the distortions in the absorption lines. This amplitude depends on exactly what the line profile and continuum look like in the spotted regions, which we do not know with certainty. The DI algorithm makes the simplifying assumption that the equivalent width of the line is the same in the spot and photosphere. It determines the continuum value in the spot simply by scaling the photospheric continuum flux (at the appropriate limb angle) by a black-body relation to reflect the spot temperature. These two assumptions work to under-estimate the amplitude of the distortion in the line profile. The smaller distortions manifest themselves in the predicted RV curve by decreasing the amplitude of the variations. We see exactly that in Fig. 9b. However, the shape of the predicted curve is in very good agreement with the observations.

For the sake of argument we mention here that it is still possible that there is a close, possibly even sub-stellar, spectroscopic companion which produces part of the radial velocity variations. We have tested this possibility by subtracting the DI model prediction on the RV variability (specifically for the Fe line as plotted above in Fig. 9b) from our actual RV data to obtain a residual variability. We then searched for periodic signals in these residual velocities, and found a 6.5 day period. Assuming for the moment that this excess variation is caused by a companion, an orbital fit through these residuals gives a semi-amplitude of and a high eccentricity of 0.64. The corresponding minimum mass of such a companion would be . ⁷ However, the rms scatter around this orbit is fairly large, about . Since this is about the same as the scatter from a Keplerian orbit fit to the raw CfA velocities ( as given above), we do not find the companion hypothesis very compelling. The uncertainties in the DI model prediction of the RV curve are probably large enough to explain the difference with the observational data without the need for additional components. Our analyses suggest that spottedness is most likely the sole cause of the variability. If there is a close companion to P1724, we do not find conclusive evidence for it in our data.

The low latitude () of the predominant spot group on this object stands in contrast to those found in other image reconstructions of TTS. HD 283572 (Joncour et al. 1994b), V410 Tau (Joncour et al. 1994a, Strassmeier et al. 1994, Hatzes 1995, Rice & Strassmeier 1996), and Sz 68 (Johns-Krull & Hatzes 1997) all show evidence of high latitude features (above ). One parameter separating these objects from P1724 is the rotational period. All three previously analyzed TTS have rotational periods days while P1724 rotates with a period of 5.7 days. The theoretical models of Schüssler et al. (1996) predict that magnetic flux tubes will emerge at higher latitudes on the stellar surface for higher rotation rates. They also predict that the emergence latitude depends strongly on the depth of the convection zone in that a deeper convection zone leads to higher latitude emergence. Among the TTS imaged, P1724 is the most massive with (see Sect. 5). Though placement in the HR diagram (see Sect. 5) puts it close to the stellar birth line, stars of this mass quickly develop a radiative core thereby reducing the depth of the convection zone. These factors may contribute in producing the low latitude surface features.

Even if the phasing appears to be stable for years, it is still possible that some small differential rotation and spot migration are present, but counteracting; we believe, though, that this is rather unlikely. The dominant spot we see on P1724 may also be indicative of a global (oblique) dipole field. As mentioned earlier, there have been a number of reports regarding spots on late-type stars lasting for a decade or even longer (e.g., Hall & Busby 1990, Oláh et al. 1997), so that P1724 does not appear to be such an extreme case.

We conclude that the modulations in the high resolution absorption line spectra of P1724 are well explained by surface temperature inhomogeneities, the distribution of which is weighted at phase 0.75 and latitude. These results have expanded the parameter space for theoretical work on rapidly rotating, spotted stars.

© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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