Differential Rotation of the Active G5 V Star κ¹ Ceti: Photometry from the MOST Satellite¹

The photometric observations discussed in this paper were obtained by the MOST microsatellite (Walker et al. 2003b) between 2003 November 5 and December 5, during the commissioning phases of the mission (constraints and limitations of these phases are described in § 3.2). MOST is Canada's first orbiting space telescope, launched in 2003 June to study (1) acoustic oscillations in solar‐type and magnetic peculiar stars, (2) rapid variability in hot massive stars, and (3) reflected light from close‐in giant exoplanets. The MOST instrument is an optical telescope (aperture 15 cm) feeding a CCD photometer through a single broadband optical filter (350–700 nm). Thanks to its polar Sun‐synchronous orbit (altitude 820 km), MOST can monitor stars in a zodiacal band of sky about 54° wide for up to 8 weeks without interruption. Starlight from primary science targets such as κ¹ Ceti (i.e., brighter than V≃6) is projected onto the CCD as a fixed image of the telescope pupil, covering about 1500 pixels for high photometric stability and insensitivity to detector flatfield and radiation effects on individual pixels. As has already been demonstrated for Procyon (Matthews et al. 2004), MOST is capable of reaching a photometric precision of about ±1 part per million (ppm) in that mode to search for oscillations in very bright stars.

Despite limitations during commissioning (see § 3.2), the MOST data of κ¹ Ceti achieved a duty cycle of about 96% over a time span of 30.5 days (with only three gaps of a few hours each), with 40 s exposures obtained approximately once per minute. At the time these data were obtained, this was the longest uninterrupted observational coverage of any star other than the Sun. (MOST has since exceeded this duty cycle and time coverage on other targets, including Procyon [Matthews et al. 2004].) MOST was designed to be highly stable photometrically over periods of hours, and also self‐calibrating (see the discussion in Walker et al. 2003b); consequently, the measurements are nondifferential. For some primary science targets, fainter secondary science targets in the adjacent direct‐focus field may allow us to search for systematic drifts in the photometry, but no such stars were observed with a sufficient signal‐to‐noise ratio (S/N) for κ¹ Ceti.

The MOST photometry is performed in a single spectral band, with a filter about twice as wide as the Johnson V filter and slightly redward of it (Walker et al. 2003b), without reference to comparison stars or flux standards. Thus, there is no direct connection to any photometric system, and hence no absolute calibration of the MOST photometry. Because of this, the original data (expressed in analog‐to‐digital intensity units, or ADUs) were normalized to unity at the maximum observed flux of the star, occurring at JD 2,452,953.0. All variation amplitudes in this paper are also expressed in these relative units.

To investigate rotational modulation in κ¹ Ceti, the 40 s exposures were binned according to the 101.4 minute orbital period of the MOST satellite, increasing the S/N per data point and minimizing the influence of orbital variations in the photometric background due to stray Earth light. The electronic version of Table 1 lists the MOST orbit‐averaged photometry, and those points are plotted in the upper panel of Figure 1.

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The formal mean standard error of a single data point is 0.77 mmag (770 ppm), which is about twice the error calculated from the detected photon shot noise. The simple average of 101 individual 40 s integrations would give an error of ∼100 ppm, which is not far from what is observed as the actual scatter of about 200 ppm (which includes the intrinsic variability of the star). Some of the photometric noise is certainly intrinsic to the star (e.g., granulation variations, given the star's activity), and some is undoubtedly due to a non‐Poisson instrumental noise resulting from the significant pointing scatter and the removal of the (noisy) stray light background signal (see § 3.2).

The sky background removed from the data shown in Figure 1 varied from about 21% to 25% of the maximum flux measured from the star (on JD 2,452,953); the background variations are plotted in Figure 2. The background light is due primarily to light entering the instrument from the illuminated limb of the Earth, which is exposed to the face of the satellite with the telescope entrance aperture. During commissioning, the MOST team was still investigating the background and exploring ways to mitigate it for future targets (e.g., choosing the optimum "roll" orientation for the satellite). To correct for the sky background, which is modulated with the satellite orbital period, but varies slightly from orbit to orbit, a running averaged background that is phased with the orbital period was subtracted from the data. During the κ¹ Ceti observing run, the Moon approached to within about 14° of the MOST target field, but we observe no correlation of the sky background with the angular separation of the Moon and the target, or with the lunar phase (see the lower panels of Fig. 2). The main results of this paper are not particularly sensitive to any residual background variations that may be present. Although the sky background variability is probably the fundamental limit to the long‐term precision of the κ¹ Ceti photometry, we note that the 8.9 day period light photometric cycle was highly stable in the final light curve (§ 4.2), indicating that the background was correctly accounted for in the MOST data over the duration of the κ¹ Ceti observing run.

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A brief description of the "commissioning" observations is given below as a record and explanation of certain limitations of these observations compared to the normal scientific operations of MOST, which were initiated in 2004 January, not long after the κ¹ Ceti data were obtained.

The precise photometry that MOST was designed to obtain is based on integrating the signal in a large image of the telescope pupil (covering about 1500 pixels of the CCD) projected by a Fabry microlens and field stop enclosing the incoming stellar beam. In normal operations, a resolved subraster of the CCD with that pupil image is downloaded for every science exposure, along with Fabry images of the adjacent sky backgrounds, for processing on the ground; this is known as Science Data Stream 2, or SDS2. As a data backup, we also perform the pupil image signal integrations onboard the satellite and store that information there for several days, in case of extended loss of contact with our ground station network; these data are part of Science Data Stream 1 (SDS1).

During commissioning, there was an unrecognized problem with a small fraction of the onboard memory, which led to computer crashes occurring roughly every 1–2 days if both SDS2 and engineering telemetry were being sent to Earth. This resulted in a fraction of the data, about 21%, sent as the fully resolved SDS2 files; the rest of our κ¹ Ceti photometry is based on SDS1 data. The lack of spatially resolved Fabry image data restricts our options of sky background and stray‐light removal. However, our later experience (see Matthews et al. [2004] for an example with Procyon) shows that the quality of SDS1 photometry is comparable to SDS2, although the latter is preferred. Even with our routine of downloading only SDS1 data, the onboard memory problem led to occasional computer crashes, which interrupted the data flow for one or more satellite orbits at a time, and accounts for most of the gaps in the light curve of Figure 1. The corrupt memory locations have since been identified, and we avoid storing data there, so this problem no longer affects the MOST mission.

At the commissioning phase of the mission, we had not yet optimized the spacecraft pointing. The mean pointing errors were 45 rms in x and 62 in y, so this image wander across the Fabry microlens (with a field stop radius of 30^'') led to larger photometric errors than were later obtained during normal scientific operations. This also limited our ability to center the target within the field stop. Jitter of the star within the field stop increases noise. These are no longer serious issues for MOST. Upgrades to the attitude control system software have led to steady improvements in pointing. Observations of Procyon in 2004 January–February were made with pointing errors of 13 and 31 rms in x and y, respectively; the current pointing performance has been improved to about 10 rms in both axes.

The current explanation of the variability observed in κ¹ Ceti is that stellar spots cause light changes at the level of 0.02–0.04 mag. The changes have been noted by several ground‐based observers as having characteristic timescales of about 9 days. The most extensive studies were done by Gaidos et al. (2000) and Messina & Guinan (2003). From 20 years of literature data, Gaidos et al. (2000) found several periods ranging between 9.14 and 9.46 days, each with uncertainty at the level of 0.03–0.05 days. Messina & Guinan (2003) conducted a careful study over six seasons and determined individual periods ranging between 9.045 and 9.406 days, with typical uncertainties of 0.02–0.05 days. The simplest explanation for such period changes is that spots are forming at or migrating to different stellar latitudes that are rotating with different angular velocities.

By relating the spot modulation periods to the photometric cycle, Messina & Guinan (2003) suspected that κ¹ Ceti shows an "antisolar" pattern in which the period of the rotation modulation tends to increase steadily during the activity cycle.

The photometric data discussed here consist of orbit‐averaged observations (binned by intervals of 101.4 minutes), corrected for sky background, as presented in Table 1 and the upper panel of Figure 1. A double‐wave variation is clearly evident, consisting of deep minima with a depth of about 0.04 of the maximum light, interspersed with moderately shallow depressions with amplitudes of about 0.005–0.01. The shallow minima progressively move relative to the deeper ones, as if two close periodicities were involved.

The photometry only covers about three cycles of the dominant variation, so Fourier analysis is relatively ineffective at accurately identifying the periodic content of the variations. We formally recover two significant peaks at frequencies of 0.1146 and 0.2170 day⁻¹, corresponding to periods of 8.72 and 9.22 days (actually, 4.61 = 1/2 × 9.22). Because of the short duration of the observing run, these periods cannot be established to better than about ±0.1 day (Fig. 3). Instead of using the formal values of the derived periods, we use two periods, 8.9 and 9.3 days, which are derived by a simple folding of the data in a sequential decomposition of the light curve, starting from the largest and most regular variation:

• 1.  
  The most obvious is a periodic pattern reminiscent of an eclipsing light curve or a spot curve—the latter due to a spot or a group of spots with probably well‐defined edges covering a moderately large fraction of the stellar surface. This pattern has an amplitude reaching 0.04 of the maximum observed flux and a period of 8.9 ± 0.1 days. An average of three cycles gives a relatively stable light curve, as shown in the phase diagram of Figure 4.
• 2.  
  When the 8.9 day pattern is removed, another periodic variation becomes visible. This is shown in the middle panel of Figure 1. This pattern is less regular than the 8.9 day variation, with the minima becoming progressively deeper from about 0.005 to 0.01 over the three complete cycles. (There is evidence that the next cycle, which is incomplete, is shallower than the preceding one.). The three times of minima give an approximate period of 9.3 days, where the uncertainty is harder to estimate, because of the variable shapes of the dips.
• 3.  
  The middle panel of Figure 1 suggests that either the 9.3 day pattern evolved over time, or there existed residual variability from some other source, especially at the beginning of the MOST run. The variability was most probably intrinsic to the star, judging by the progression of the depths of the 9.3 day pattern and the excellent agreement between the phased minima of the 8.9 day pattern. We observe no correlation between the sky background variations (shown in Fig. 2) and the deviations from the regular 9.3 day periodic pattern, shown in the middle panel of Figure 1; this may be taken as an indication that the sky background variations were not the cause of these irregularities.

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The observed periodic variations can be explained by spots formed at different stellar latitudes and carried by the rotation of the star at different angular velocities. With single‐band photometry, little can be said about the type of spots, whether they are hotter or cooler than the surrounding photosphere. However, two of the authors did develop two simple, entirely independent geometric spot models to try to reproduce the observed light curve. The models differed in approach, in that the first used the minimum number of parameters and ended up describing only the larger spot, which apparently did not evolve during the MOST observations; only a very simplified description could be derived for the smaller spot. The second model utilized a full set of 12 parameters required to describe two spots and the geometry of their visibility on the star. The second model was underconstrained, but its basic results fully confirmed the first model in its description of the larger spot.

In the simplest terms, the larger of the two spots, with a rotational period of 8.9 days and a relative flux amplitude of 0.039, if entirely black and circular, would have a radius of 19% of the stellar radius, or an angular radius from the star center of 11°. The mean 8.9 day light curve is shown in Figure 4. Since the light variations take place over slightly more than half of the rotation, and the "flat top" is shorter than half of the rotation, the spot appears to be situated above the equator. The smaller spot, causing the 9.3 day (or slightly longer) periodicity, also if black and circular, must be about 12% of the stellar radius (angular radius 7°).

Any model must face the limitation that the parameter without any constraint is the unperturbed level of the light curve, l₀; we arbitrarily assumed it to be at the highest observed level (see Fig. 1), but some small spots may have been present on the surface at all times. While the spot longitude λ can be removed from the parameter list by an appropriate adjustment of the phases, even the simplest model involves several adjustable parameters: the rotation axis inclination i, the limb‐darkening coefficient u, the latitude β, the size or the angular radius as seen from the center of the star r, in addition to the darkness t of the spot. The darkness (0≤t≤1) of the spot cannot be estimated from the single‐color data, so that the minimum size of the spot can be estimated assuming a black spot (t = 0). Even such a highly simplified model, with several parameters fixed, contains at least three fully adjustable geometrical parameters: i, β, and r. The best model of the larger spot fits the primary variation to better than 0.001 (Fig. 4). The relevant parameters for the star and the larger spot (where the symbol ≡ means a fixed parameter) are P = 8.9 ± 0.1 day, i = 70° ± 4°, β = +40° ± 7°, r = 111 ± 06, λ≡0°, t≡0, and u≡0.8. The 8.9 day spot is relatively large and may be larger if not entirely black, so the assumption of it being circular—especially in view of the clear evidence of the shearing differential rotation—is highly disputable. Therefore, this solution should be taken as indicative, rather than of much physical significance.

The second model assumed two independent spots with different rotational periods. It was specified by the rotational inclination i of the star and the limb‐darkening parameter u, with all parameters describing the two spots: the radii and flux contrastsread and the latitude, initial longitude, and the rotation period for each spot. Such a model required extensive tests of different combinations of parameters. While the results are inconclusive in details because of the large number (12) of weakly constrained parameters, they are fully consistent with the minimum‐parameter model described above in the description of the first (8.9 day) spot. The second spot in the extended model appears to have a longer period, perhaps as long as 9.7 days, although uncertainty of the period is large and results from the obvious evolution of the second spot over time and a complex coupling with values of the remaining parameters.

The models have been unable to define accurate latitudes of both spots, although the second, more complex model suggested a higher latitude for the smaller, 9.3 day spot—a situation that would correspond to the solar surface differential rotation pattern. However, there is no question from both independent analyses that the two spots have unambiguously different periods, supporting the interpretation of SDR with ΔΩ/Ω≃4%. This is clear evidence of surface differential rotation on κ¹ Ceti, and only the second after EK Dra (Messina & Guinan 2003) in which two spots were also simultaneously visible. All other inferences were based on apparent period changes (Hall & Busby 1990; Henry et al. 1995; Gaidos et al. 2000; Messina & Guinan 2003), the interpretation of subtle broadening effects in spectral lines, so far detectable only in the most rapidly rotating stars (Donati & Collier Cameron 1997; Donati et al. 1999), or Doppler imaging from epoch to epoch (Collier Cameron 2002). (For general reviews of these techniques and their limitations, see Rice [2002], Gizon & Solanki [2004], and Strassmeier [2004]).

The MOST orbit‐averaged data were also searched for short‐lived variations that might be associated with transits of a close‐in giant planet or with white‐light flares. The latter seemed more likely, given the history of activity of κ¹ Ceti. No candidates for transits or flares were found in the data. We can set useful limits on such events over the month that we observed the star. There were no flare events or transits with peak‐to‐valley variations of more than 2 mmag with durations longer than 200 minutes, and none with variations greater than 3 mmag with durations between 2 and 200 minutes.

For planetary transits, this limits the size of any possible transiting bodies with orbital periods of about 30 days or less to approximately 0.045–0.055 of the stellar radius, about half the size of Jupiter. However, our spot modeling and measurements of Vsin i indicate that the rotational inclination of the star is not close to 90°, so it is not expected that the orbital plane of any planets in the κ¹ Ceti system would be coincident with the line of sight.

Several of us (D. B., S. R., E. S., and G. W.) monitored the Ca ii H and K reversals of a number of 51 Peg systems (Shkolnik et al. 2003), in addition to several active, single stars using the Gecko spectrograph on the Canada‐France‐Hawaii 3.6 m Telescope (CFHT). The intention was to detect chromospheric activity synchronized with the planetary orbits. As a by‐product, we determined radial velocities for these stars, with σ_RV≤20 m s⁻¹ (Walker et al. 2003a). Here we present the various results for κ¹ Ceti.

The dates of the observations are listed in Table 2. They were acquired on four observing runs in 2002 and 2003 with the Gecko coudé spectrograph, which was fiber‐fed from the Cassegrain focus of the CFHT. Single spectra were recorded at a resolution of 110,000 spanning 60 Å and centered at 3947 Å (see Fig. 5). The S/N was ≈500 per pixel in the continuum and 150 in the H and K cores. Full details of the data reduction are given in Shkolnik et al. (2003).

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κ¹ Ceti shows moderate broadening of lines due to rotation at the rate of about 9 days. The broadening is detectable in our high‐resolution spectroscopy. Combined with the rotation period and the value of the radius, Vsin i can be used to set a constraint on the rotational axis inclination i, which is one of the free parameters in the spot models. The previous determinations of Vsin i were 3.9 km s⁻¹ (Fekel 1997), 4.3 km s⁻¹ (Benz & Mayor 1984), 5.6 km s⁻¹ (Soderblom et al. 1989), and 3.8 km s⁻¹ (Messina et al. 2001).

We determined the value of Vsin i by analyzing the broadening of the photospheric lines in the region of Ca ii λλ3933/3968 in our CFHT spectra. We used the broadening function formalism (Rucinski 1999, 2002) of the deconvolution of photospheric spectra of rotating stars with spectra of sharp‐line, slowly rotating stars. The approach leads to the removal of all common agents of line broadening, not only of the instrumental component, but also of the tangential turbulence and of all other components that are the same for the program and the template stars. For κ¹ Ceti, we used τ Ceti, which is a slightly cooler dwarf (G8 V) that rotates very slowly (Vsin i = 0.6 km s⁻¹; Fekel 1997).

For the determination of the broadening function, the spectra must be rectified and normalized. As shown in Figure 5, the rich photospheric spectrum in both κ¹ Ceti and τ Ceti is superimposed on strongly variable far wings of the Ca ii absorption. We removed sections of the spectrum where the wings fall below 1/3 of the maximum flux, and applied a simple, upper‐envelope rectification between high points of the quasi continuum. Figure 6 shows the broadening function for one of the CFHT spectra, obtained on 2003 August 26 (see Table 2). Instead of least‐squares fits of the limb‐darkened rotational profile (Gray 1992) to the broadening functions (with the limb‐darkening coefficient assumed to be u = 1.0, appropriate for the UV in late‐type stars), we matched the half‐widths at half‐maximum; such Vsin i determinations are less sensitive to the residual uncertainties at the base of the broadening function. Our average Vsin i = 4.64 ± 0.11 km s⁻¹ is compatible with previous determinations. We note that Gaidos et al. (2000), using an almost identical value of Vsin i = 4.5 km s⁻¹, interpreted it as implying 40°<i<90°. The main spot in both model fits is located at a moderate stellar latitude above the equator. If the period of 8.9 ± 0.1 days were ascribed to the stellar equator, then the best‐fitting inclination of i = 60° ± 5° and the measured Vsin i combine to give a stellar radius of R≃0.95 ± 0.10 R_⊙, which is certainly reasonable for a G5 V star.

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The 7 Å region centered on the K line that was used in the analysis is shown for a single spectrum in Figure 7, where spectral intensity is normalized to unity at the wavelength limits. In Figure 8 the residuals from the average normalized spectra are shown for each night (where pairs of spectra on the same night have been averaged). Activity is confined entirely to the reversals, within about 0.94 Å around the center of the emission core, or within ±35.8 km s⁻¹. If this broadening is caused by the velocity field, it is some 7.7 times larger than Vsin i. To visualize the changes in the reversals, the integrated flux residuals shown in Figure 9 are expressed in Å relative to the restricted continuum height shown in Figure 7. This height corresponds to about one‐third of the local photospheric continuum around the Ca ii lines, so that the integrated residuals are proportional, but not identical, to the emission equivalent width. They are listed in column (4) of Table 2.

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The best‐fitting periodic variation for the K residuals in Table 2 over the four observing runs in 2002 and 2003 has a period of 9.332 ± 0.035 days. The residuals are plotted as a function of phase in Figure 9, with the arbitrary initial epoch of the minimum residual flux at T₀(HJD) = 2,452,485.117 (the minimum flux epoch can be estimated to ±0.1 days). The data from 2002 are shown as open symbols, and those from 2003 September are solid. Although the data from 2002 cover only three rotations, with the data from 2003, the span in the rotations is extended to 44.

Since the period is largely defined by the 2002 observations, Figure 9 implies that the rotation associated with the enhanced Ca ii activity has remained quite stable. The 9.332 day period is entirely compatible with that found by Baliunas et al. (1983; 9.4 ± 0.1 day), suggesting that the enhanced activity has in fact been stable at the same latitude for at least 36 yr (over 1400 rotations). This period is also compatible with the 9.3 day period of the less prominent spot that we see in the MOST observations (see § 4.2). Given this apparent stability, it is reasonable to extrapolate variations in the K‐line residuals to the epoch of the MOST photometry that corresponds to only seven additional rotations. The bottom panel of Figure 1 shows the K‐line flux changes that can be related to the photometric variations due to the 9.3 day spot seen in the MOST photometry. The maxima of the two curves coincide. The interpretation of this coincidence is not obvious, as the enhanced Ca ii could be expected to accompany the dark spot, unless what we see as the 9.3 day photometric signal is actually a bright spot, and the minima are due to the disappearance of the bright plage region.

Radial velocities were estimated from ≈20 Å of the photospheric spectrum (3942–3963 Å; see Fig. 5) using the fxcor routine in IRAF,⁵ with Th/Ar comparison lines providing dispersion correction (more detail is given in Walker et al. 2003a). The first spectrum of the series in 2002 acted as the template from which differential radial velocities (ΔRV) were measured for all the spectra. The ΔRV values are listed in Table 2, plotted against the 9.332 day rotational phase determined from the K‐line residuals in Figure 9, where the open symbols correspond to data from 2002, and the solid points to 2003 September. There is a remarkable consistency between the ΔRVs from both years, suggesting that the underlying ΔRV technique is highly stable even without an imposed fiducial (such as iodine vapor).

From the 2002 data, we derive σ_RV = 21.8 m s⁻¹, and 23.6 m s⁻¹ when combined with 2003. These values are very similar to the σ_RV = 24.4 m s⁻¹ found over 11 yr by Cumming et al. (1999). For comparison, a planet that induces a reflex radial velocity variation below 50 m s⁻¹ and is tidally synchronized to the star would have Msin i≃0.74M_J and a = 0.084 AU. Consequently, κ¹ Ceti could still harbor a giant planet in a tidally locked orbit, which we would not have detected.

In 2003 the ΔRVs appear significantly different between the two nights, which seems to be reinforced by the consistency within the pairs of ΔRVs. While a planetary perturbation cannot be ruled out by the 2002 data and other PRV studies, the difference in 2003 might be associated with the velocity field of the star itself. The increase of velocity with increasing K‐line strength is consistent with the extreme event in 1988.5 seen by Walker et al. (1995). However, the data are insufficient to further discuss the point here.

We thank the anonymous referee for excellent suggestions, which helped us improve the quality and consistency of the paper.

The MOST project is funded and supervised by the Canadian Space Agency. We are grateful to the engineers at Dynacon, Inc., the Space Flight Laboratory of the University of Toronto Institute for Aerospace Studies, the University of British Columbia Department of Physics and Astronomy, and the University of Vienna and Austrian Technical University for their contributions to the success of the MOST mission. D. B. G., J. M. M., A. F. J. M., S. M. R., E. S., and G. A. H. W. were supported by the Natural Sciences and Engineering Research Council of Canada. A. F. J. M. is also supported by FCAR (Québec). W. W. W. received support from the Austrian Space Agency and the Austrian Science Fund (P14984).

This research has made use of the SIMBAD database, operated at the CDS in Strasbourg, France, and accessible through the Canadian Astronomy Data Centre, which is operated by the Herzberg Institute of Astrophysics, National Research Council of Canada.