5 Differential surface rotation and meridional flows

Figure 9: Differential rotation on the surface of KU Peg. Plotted are the longitudinal shifts per latitude bin from a cross correlation of maps from Rotation 1 and 2. a) Symbols are the shifts measured and the lines are $\sin^2 b$-fits (b latitude) to the shifts. The full line is for the entire latitude range, the dashed line just for the equatorial $\pm$25$^\circ$ range. b) The average of the shifts derived from individual lines on top of a greyscale image of the cross correlation of the two average maps. The phase lag of the higher latitude regions possibly indicates that higher-latitude regions rotate slower than the equator.

By cross correlating longitudinal strips at successive latitudes from the two maps in Figs. 5a, b, we can derive the differential surface rotation on KUPegasi. Since the two maps are from consecutive rotations it is save to assume that the surface features in the individual maps are still the same (but see previous section). In such a case the spots can be used as tracers for surface velocity fields, although the criterion $\tau_{\rm spot\, lifetime}>\tau_{\rm observation}$is not necessarily a stringent requirement if there are many spots with the same general trend of migration. Of course, the interpretation is still hampered by the possibility of a coincidental spot alignment that mimics a latitude-dependent migration pattern. At this point, we simply caution the reader that our data are just a snapshot and will be masked by spot evolution. We applied the fxcor routine of the IRAF package (for details see the IRAF manual at iraf.noao.edu) to fit Gaussians to the peak of the cross-correlation functions for the Ca I 6439 image, the Fe I 6430 image, and the average of those two images. The result in Fig. 9 shows a complex surface differential-rotation function: the shifts within $\pm25\hbox{$^\circ$ }$ of the equator are tidily fitted with a solar-like differential-rotation law proportional to $\sin^2 b$, but between +25$^\circ$ and +45$^\circ$ (and possibly also between -25$^\circ$ and -45$^\circ$) the function changes its sign and thus these regions appear to accelerate again. Above $\approx$+50$^\circ$ the width of the cross-correlation function increases rapidly due to the decreased surface resolution and the longitudinal shifts cannot be measured very reliably there. Its error bars from the Gaussian fits are $\approx$3 times larger than near the equator. Despite this limitation there is some evidence though that the rotation decelerates again above $\approx$ +50$^\circ$. This is certainly inconclusive from our two stellar rotations but should not go unnoticed.

We tried to fit the cross correlations with a simple $\sin^2 b$ law for the entire latitude range (full line in Fig. 9). Such a fit is obviously not a good representation of the data (rms of 0.29) but is intended to obtain a save lower limit for the magnitude of the differential rotation. We also did a $\sin^2 b$ fit restricted to shifts within $\pm$25$^\circ$ latitude (dashed line in Fig. 9) and thereby also obtain an estimate of the external error per latitude range. Its rms is accordingly better, 0.11. The first fit leads to the following differential rotation law for -40$^\circ$ to 65$^\circ$

 

$$\Omega(b) \Omega_0 - \Omega_1 \sin^2 b$$ =

$$15.41 - 1.38 \sin^2 b \ \ \ \ [\hbox{$^\circ$ }\,{\rm day}^{-1}].$$ = (2)

The second fit for $\pm$25$^\circ$ gives

 

$$\Omega(b) \Omega_0 - \Omega_1 \sin^2 b$$ =

$$15.63 - 5.32 \sin^2 b \ \ \ \ [\hbox{$^\circ$ }\,{\rm day}^{-1}].$$ = (3)

The differential rotation coefficient $\alpha=(\Omega_0-\Omega_{\rm pole})/\Omega_0 = \Omega_1/\Omega_0$ is from Eq. (2) $\approx$0.09 and $1/\Delta\Omega$, i.e. the time the equatorial region laps the pole, $\approx$260 days. This is comparable to the solar value of 120 days. If we only take the latitudes below $\pm$25$^\circ$ into account (Eq. (3)), we get values of $\alpha=0.34$ and $1/\Delta\Omega\approx70$ days.

Figure 10: a) Meridional changes on the surface of KU Peg. Plotted are the latitudinal shifts per longitudinal bin from a cross correlation of the hemisphere above the equator from Rotation 1 and 2. The two bumps at 40$^\circ$ and 330$^\circ$ possibly indicate a poleward flow of the magnitude of $\approx$0.4$^\circ$days-1. b) The redistribution of spots from Rotation 1 (filled symbols) to Rotation 2 (open symbols). The individual spots are listed in Table 3.

Figure 11: The average Doppler image from the first rotation in orthographic projection at phase zero (left) and phase 180$^\circ$ (right). The size and the direction of the arrows indicate the surface flows resulting from our cross correlations. The longitudes and latitudes at which the arrows are drawn are arbitrary because the net shifts were computed from all longitudes and latitudes ( $\ell =0\hbox {$^\circ $ }$ and $\pm$60$^\circ$, and $b=50\hbox {$^\circ $ }$ are shown here). Note that the poleward flow is strongest on the opposite side of the polar-spot appendage.

To quantify latitudinal changes on the stellar surface, we now cross correlate the maps along meridional circles. We just adopt the "northern'' hemisphere, i.e. the hemisphere that is fully in view (all pixels with positive latitude), and cross correlate its longitudinal strips of the Ca I 6439 images, the Fe I 6430 images, and the average of those two images, respectively. The result in Fig. 10a clearly shows that there were two latitudinal shifts on the surface of KU Peg within one stellar rotation that consistently appeared in both spectral lines. One event at phase $\approx$0.1 (40$^\circ$) and another at $\approx$0.9 (330$^\circ$), with an average magnitude of $+0.3\pm0.1$ (rms)$^\circ$day^-1 and $+0.4\pm0.1$ (rms)$^\circ$day^-1, respectively. The intermittent longitudes on the opposite side of the visible pole may show a reversed shift of magnitude -0.2$^\circ$day^-1, but this may be overinterpretation given the large error bars and the inconsistency at phases from approximately 135$^\circ$ to 180$^\circ$. The latter is most likely caused by the well-known "north-south'' mirroring of the polar appendage at $\ell\approx 160\hbox{$^\circ$ }$.

As for the previous cross correlations the error bars per bin are estimated from a Gaussian fit of the FWHM of the cross correlation function. Both shifts have a positive sign and thus indicate a polar-directed change. We interpret these shifts as a meridional change of magnetic flux and, since it seems to be a local event on the stellar surface, tentatively suggest that magnetic reconnection and its associated plasma motions may be the underlying cause rather than a global meridional flow.