Differential rotation on rapidly rotating stars

Theories of meridional circulation and differential rotation in stellar convective zones predict trends in surface flow patterns on main-sequence stars that are amenable to direct observational testing. Here I summarise progress made in the last few years in determining surface differential rotation patterns on rapidly-rotating young main-sequence stars of spectral types F, G, K and M. Differential rotation increases strongly with increasing effective temperature along the main sequence. The shear rate appears to increase with depth in the sub-photospheric layers. Tidal locking in close binaries appears to suppress differential rotation, but better statistics are needed before this conclusion can be trusted. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stellar irradiance variations due to surface temperature inhomogeneities

Observations of rotational modulation of continuum brightness and photospheric and chromospheric spectral-line profiles of late-type stars indicate the presence of very inhomogeneous surface temperature distributions. We present three stellar examples (VY Ari, HR 7275, HU Vir) where time-series photometry is used to trace the evolution of spotted regions. Simultaneous spectroscopy and Doppler imaging for one of the three stars (HU Virgo, Fig. 1) makes it possible to compute the temperature distribution of the photosphere and the relative intensity distribution of parts of the chromosphere (from CaII K and H line profiles). The combination of time-series spot modeling and Doppler imaging enabled us to determine thesign and amount of differential surface rotation on HU Vir. We found a big, cool polar spot (see figure below) and a differential (surface) rotation law where higher-latitude regions rotate faster than lower-latitude regions (opposite to what we see on the Sun). Currently, this ensemble of techniques - time-series photometry and photospheric and chromospheric Doppler imaging - is only applicable to stars overactive by approximately a factor of 100 as compared to the active Sun, e.g. the evolved components in RS CVn-type binaries and some rapidly-rotating, single, pre-main sequence stars or giant stars. Stellar rotation is a fundamental parameter for (magnetic) activity. Starspots, or any other surface inhomogeneities, allow one to derive very precise stellar rotation rates and, if coupled with seismological observations of solar-type stars, could provide information on the internal angular momentum distribution in overactive late-type stars.To be published in Astronomy & Astrophysics.     

Differential rotation of cool active stars: the case of intermediate rotators

In this paper, we present a new method for measuring the surface differential rotation of cool stars with rotation periods of a few days, for which the sparse phase coverage achievable from single-site observations generally prevents the use of more conventional techniques. The basic idea underlying this new analysis is to obtain the surface differential rotation pattern that minimizes the information content of the reconstructed Doppler image through a simultaneous fit of all available data.
Simulations demonstrate that the performance of this new method in the case of cool stars is satisfactory for a variety of observing strategies. Differential rotation parameters can be recovered reliably as long as the total data set spans at least 4 per cent of the time for the equator to lap the pole by approximately one complete cycle. We find in particular that these results hold for potentially complex spot distributions (as long as they include a mixture of low- and high-latitude features), and for various stellar inclination angles and rotation velocities. Such measurements can be obtained from either unpolarized or polarized data sets, provided their signal-to-noise ratio is larger than approximately 500 and 5000 per 2 km s⁻¹ spectral bin, respectively.
This method should therefore be very useful for investigating differential rotation in a much larger sample of objects than what has been possible up to now, and should hence give us the opportunity of studying how differential rotation reacts to various phenomena operating in stellar convective zones, such as tidal effects or dynamo magnetic field generation.     

Rapid rotation,active nests of convection and global-scale flows in solar-likestars

In the solar convection zone, rotation couples with intensely turbulent convection to build global-scale flows of differential rotation and meridional circulation. Our sun must have rotated more rapidly in its past, as is suggested by observations of many rapidly rotating young solar-type stars. Here we explore the effects of more rapid rotation on the patterns of convection in such stars and the global-scale flows which are self-consistently established. The convection in these systems is richly time dependent and in our most rapidly rotating suns a striking pattern of spatially localized convection emerges. Convection near the equator in these systems is dominated by one or two patches of locally enhanced convection, with nearly quiescent streaming flow in between at the highest rotation rates. These active nests of convection maintain a strong differential rotation despite their small size. The structure of differential rotation is similar in all of our more rapidly rotating suns, with fast equators and slower poles. We find that the total shear in differential rotation, as measured by latitudinal angular velocity contrast, ΔΩ, increases with more rapid rotation while the relative shear, ΔΩ/Ω, decreases. In contrast, at more rapid rotation the meridional circulations decrease in both energy and peak velocities and break into multiple cells of circulation in both radius and latitude. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling the differential rotation of F stars

We model stellar differential rotation based on the mean-field theory of fluid dynamics. DR is mainly driven by Reynolds stress, which is anisotropic and has a non-diffusive component because the Coriolis force affects the convection pattern. Likewise, the convective heat transport is not strictly radial but slightly tilted towards the rotation axis, causing the polar caps to be slightly warmer than the equator. This drives a flow opposite to that caused by differential rotation and so allows the system to avoid the Taylor-Proudman state. Our model reproduces the rotation pattern in the solar convection zone and allows predictions for other stars with outer convection zones. The surface shear turns out to depend mainly on the spectral type and only weakly on the rotation rate. We present results for stars of spectral type F which show signs of very strong differential rotation in some cases. Stars just below the mass limit for outer convection zones have shallow convection zones with short convective turnover times. We find solar-type rotation and meridional flow patterns at much shorter rotation periods and horizontal shear much larger than on the solar surface, in agreement with recent observations. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spot patterns and differential rotation in the eclipsing pre-cataclysmic variable binary, V471 Tau

We present surface spot maps of the K2V primary star in the pre-cataclysmic variable binary system, V471 Tau. The spot maps show the presence of large high-latitude spots located at the sub-white dwarf longitude region. By tracking the relative movement of spot groups over the course of four nights (eight rotation cycles), we measure the surface differential rotation rate of the system. Our results reveal that the star is rotating rigidly with a surface shear rate,  dΩ= 1.6 ± 6 mrad d⁻¹  . The single active star AB Dor has a similar spectral type, rotation period and activity level as the K star in V471 Tau, but displays much stronger surface shear  (46 < dΩ < 58 mrad d⁻¹)  . Our results suggest that tidal locking may inhibit differential rotation; this reduced shear, however, does not affect the overall magnetic activity levels in active K dwarfs.     

Modelling surface magnetic field evolution on AB Doradûs due to diffusion and surface differential rotation

From Zeeman–Doppler images of the young, rapidly-rotating K0 dwarf AB Doradûs, we have created a potential approximation to the observed radial magnetic field and have evolved it over 30 d subject to the observed surface differential rotation , meridional flow and various diffusion rates. Assuming that the dark polar cap seen in Doppler images of this star is caused by the presence of a unipolar field, we have shown that the observed differential rotation will shear this field to produce the observed high-latitude band of unidirectional azimuthal field. By cross-correlating the evolved fields with the initial field each day we have followed the decay with time of the cross-correlation function. Over 30 d it decays by only 10 per cent. This contrasts with the results of Barnes et al. , who show that on this time-scale the spot distribution of He699 is uncorrelated. We propose that this is due to the effects of flux emergence changing the spot distributions.     

Further images of α Persei G dwarfs

We present two images of intermediate and low axial inclination G dwarfs (AP 149 and AP 193) in the young open cluster α Persei, and compare these with previous images of intermediate and high axial inclination objects in this cluster. All stars show starspots at high latitudes, with one star exhibiting a strong polar spot. Although low-latitude features are found on all stars to some degree, the detection of spots on AP 193 is marginal. The apparent difference in starspot morphology from one object to the next is probably the result of a stellar magnetic cycle, although the exact effect on the starspot distribution throughout a cycle is unknown.
Polar spots are found in many Doppler images of rapidly rotating cool stars. In the past, their existence has been called into question, and it has been suggested that they could be the manifestations of NLTE (e.g. chromospheric filling in of line profiles) effects rather than real photospheric features. We assume the polar spots to be real photospheric features, and conclude that the flat-bottomed nature of the profile shape can be attributed to photospheric polar spots. The degree of truncation of the profile depends not only on spot size and strength, but also on the effective foreshortening of the polar region, a function of axial inclination.
H α is in emission on AP 149 which shows a double peak at most phases. The time-series of the profile shows an s-wave pattern as the position of these peaks changes throughout the rotation cycle. We attribute this to coronal clouds located above the stellar surface in synchronous orbit. A maximum-entropy tomogram is derived revealing four distinct emission regions located near and above the corotation radius.     

Differential rotation on both components of the pre-main-sequence binary system HD 155555

We present the first measurements of surface differential rotation on a pre-main-sequence binary system. Using intensity (Stokes I) and circularly polarized (Stokes V) time-series spectra, taken over 11 nights at the Anglo-Australian Telescope (AAT), we incorporate a solar-like differential rotation law into the surface imaging process. We find that both components of the young, 18 Myr, HD 155555 (V824 Ara, G5IV + K0IV) binary system show significant differential rotation. The equator–pole lap times as determined from the intensity spectra are 80 d for the primary star and 163 d for the secondary. Similarly, for the magnetic spectra we obtain equator–pole lap times of 44 and 71 d, respectively, showing that the shearing time-scale of magnetic regions is approximately half of that found for stellar spots. Both components are therefore found to have rates of differential rotation similar to those of the same spectral-type main-sequence single stars. The results for HD 155555 are therefore in contrast to those found in other, more evolved, binary systems where negligible or weak differential rotation has been discovered. We discuss two possible explanations for this: first that at the age of HD 155555 binary tidal forces have not yet had time to suppress differential rotation and secondly that the weak differential rotation previously observed on evolved binaries is a consequence of their large convection zone depths. We suggest that the latter is the more likely solution and show that both temperature and convection zone depth (from evolutionary models) are good predictors of differential rotation strength. Finally, we also examine the possible consequences of the measured differential rotation on the interaction of binary star coronae.     

Can variable meridional flows lead to false exoplanet detections?

The search for habitable exoplanets centers on planets with Earth-like conditions around late type stars. Radial velocity searches for these planets require precisions of 1 m/s and better. That is now being achieved. At these precisions stellar surface motions might lead to false detections. Of particular interest are variable meridional flows on stellar surfaces. I review the available observations of solar surface meridional flows using both Doppler shift and local helioseismology techniques. Interpretation in terms of Doppler shifts in integrated starlight leads to estimates of the likelihood of false detections. It is unlikely that these false detections occur in the habitability zones of exoplanets. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sounding stellar cycles with Kepler – I. Strategy for selecting targets

The long-term monitoring and high photometric precision of the Kepler satellite will provide a unique opportunity to sound the stellar cycles of many solar-type stars using asteroseismology. This can be achieved by studying periodic changes in the amplitudes and frequencies of the oscillation modes observed in these stars. By comparing these measurements with conventional ground-based chromospheric activity indices, we can improve our understanding of the relationship between chromospheric changes and those taking place deep in the interior throughout the stellar activity cycle. In addition, asteroseismic measurements of the convection zone depth and differential rotation may help us determine whether stellar cycles are driven at the top or at the base of the convection zone. In this paper, we analyse the precision that will be possible using Kepler to measure stellar cycles, convection zone depths and differential rotation. Based on this analysis, we describe a strategy for selecting specific targets to be observed by the Kepler Asteroseismic Investigation for the full length of the mission, to optimize their suitability for probing stellar cycles in a wide variety of solar-type stars.     

Starspots and relativity: Applied Doppler imaging for the Gravity Probe B mission

We present Doppler images and surface differential rotation measurements for the primary of the RS CVn binary IM Pegasi, the guide star for the Gravity Probe B experiment. The data used is a subset of that taken during optical support of the mission and was obtained almost nightly over a near three year period from the Automatic Spectroscopic Telescope operated by Tennessee State University. Using the technique of least-squares deconvolution to increase the signal-to-noise ratio of the data, we have reconstructed 31 maximum entropy Doppler images of the star. The images show that the spot features are relatively stable for over a year (and possibly longer) with both a polar spot and lower latitude features. The most intense features are located on the side facing the secondary. In addition, we have incorporated a solar-like differential rotation law into the imaging process to determine the level of surface differential rotation for IM Peg for 22 epochs. A weighted least-squares average of the measurements gives a surface shear of 0.0142 ± 0.0007 rad/d, meaning that the equator takes ∼440 ± 20 days to lap the poles. Although the level of surface differential rotation was shown to vary over the period of the observations, this may indicate an underestimate in the errors of the method rather than any temporal evolution in the differential rotation. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Doppler images and chromospheric variability of TWA 17

We present Doppler imaging and a Balmer line analysis of the weak-line T Tauri star TWA 17. Spectra were taken in 2006 with the University College London Echelle Spectrograph on the Anglo-Australian Telescope. Using least-squares deconvolution to improve the effective signal-to-noise ratio, we produced a Doppler map of the surface spot distribution. This shows similar features to maps of other rapidly rotating T Tauri stars, i.e. a polar spot with more spots extending out of it down to the equator.
In addition to the photospheric variability, the chromospheric variability was studied using the Balmer emission. The mean Hα profile has a narrow component consistent with rotational broadening and a broad component extending out to 220 km s⁻¹. The variability in Hα suggests that the chromosphere has at least one slingshot prominence  3 R _*  above the surface.     

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Night‐time science with large solar telescopes: The magnetic Sun through time

Today the Sun has a regular magnetic cycle driven by a dynamo action. But how did this regular cycle develop? How do basic parameters such as rotation rate, age, and differential rotation affect the generation of magnetic fields? Zeeman Doppler imaging (ZDI) is a technique that uses high‐resolution observations in circularly polarised light to map the surface magnetic topology on stars. Utilising the spectropolarimetric capabilities of future large solar telescopes it will be possible to study the evolution and morphology of the magnetic fields on a range of Sun‐like stars from solar twins through to rapidly‐rotating active young Suns and thus study the solar magnetic dynamo through time. In this article I discuss recent results from ZDI of Sun‐like stars and how we can use night‐time observations from future solar telescopes to solve unanswered questions about the origin and evolution of the Sun's magnetic dynamo (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On axisymmetric hydromagnetic equilibria

The physical characteristics of possible axisymmetric equilibria are examined on the basis of the integrals of hydromagnetic equations. It is shown for nearly spherical configurations that a surface differential rotation is possible only in the absence of a meridional circulation with either purely toroidal or purely poloidal magnetic field. In the presence of a meridional circulation, it is shown that no surface rotation or constant rotation is possible if the magnetic field is purely toroidal, and that no rotation is possible if the magnetic field is purely poloidal. A brief discussion is given on the possible solutions including the case of stellar winds with force-free magnetic fields.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

On Turbulent Heat Transport in Rotating Convective Zones

This paper analyses the tensorial aspect of heat conduction which certainly becomes important for gravitationally stratified and rotating turbulent media. The structure of the eddy conductivity tensor is indeed a non-trivial one for turbulences with correlation times not too short in comparison with the rotation period. Since this situation may be realized in the solar convection zone, a more complex formulation of solar heat-flux problems is required rather than that with a scalar heat conductivity. We still neglect here all other energy transporters, i.e. radiation and meridional circulation, and treat a solar heat-flux problem with a heuristic mean-field ansatz of the eddy conductivity tensor. Its rotationally created latitude dependence always leads to a pole-equator surface temperature difference, which proves to be negligibly small, however, if realistic assumptions for the outward decrease of density and correlation time are involved. As a final rough estimation shows, this screening effect of the outer radiating surface should work for all late-type stars of the main sequence hence a “differential temperature” (in analogy to “differential rotation”) should not be expected for those stars.     

An Evolving Synoptic Magnetic Flux map and Implications for the Distribution of Photospheric Magnetic Flux

We describe a procedure intended to produce accurate daily estimates of the magnetic flux distribution on the entire solar surface. Models of differential rotation, meridional flow, supergranulation, and the random emergence of background flux elements are used to regularly update unobserved or poorly observed portions of an initial traditional magnetic synoptic map that acts as a seed. Fresh observations replace model estimates when available. Application of these surface magnetic transport models gives us new insight into the distribution and evolution of magnetic flux on the Sun, especially at the poles where canopy effects, limited spatial resolution, and foreshortening result in poor measurements. We find that meridional circulation has a considerable effect on the distribution of polar magnetic fields. We present a modeled polar field distribution as well as time series of the difference between the northern and southern polar magnetic flux; this flux imbalance is related to the heliospheric current sheet tilt. We also estimate that the amount of new background magnetic flux needed to sustain the `quiet-Sun' magnetic field is about 1.1×10²³ Mx d^–1 (equivalent to several large active regions) at the spatial resolution and epoch of our maps. We comment on the diffusive properties of supergranules, ephemeral regions, and intranetwork flux. The maps are available on the NSO World Wide Web page.     

Doppler Images of ζ Andromedae

Doppler images are presented for the RS CVn-type binary ζ And. Our upgraded Doppler imaging code TempMap_ε takes into account the distorted geometry of the primary giant component. On the maps several low latitude spots are restored with a temperature contrast of about 1000 K. Some weak polar features are also found. Cross-correlation of the consecutive Doppler-maps suggests solar-like differential surface rotation.     

Doppler imaging of BD+22°4409 (LO Peg) using least-squares deconvolution

We present maximum-entropy reconstructions of the rapidly rotating dwarf single star BD+22°4409 (LO Peg) from observations at the William Herschel Telescope in 1993 August. Since this star is too faint to use the conventional single- or three-line Doppler imaging methods, we make use of the novel method of least-squares deconvolution, which utilizes the large number of photospheric lines in an echelle spectrum to produce a single high signal-to-noise ratio profile.
The star-spot distributions from the image reconstructions show cool features at both high and low latitudes, in contradiction to recent theoretical predictions of the dynamo behaviour in rapidly rotating stars. Cross-correlation of the images from consecutive nights shows a good correlation from the small-scale structures, but no evidence of surface differential rotation. From the cross-correlation of the high-latitude spot we are able to reject the period of 9.22 h of Jeffries et al. in favour of their preferred period of 10.17 h, confirming the result of Robb & Cardinal.