Differential rotation

5.5 Differential rotation

Differential rotation of stars plays an important role in the generation of magnetic fields in the convection zone. In the Sun, it is involved in transformation of a weak large-scale poloidal field into a stronger toroidal component. By analogy, stellar activity is most probably also connected to differential rotation.

On the Sun the differential rotation is observed in relative motion of sunspots and can be expressed in the form

where $ψ$ denotes heliographic latitude, $Ω0$ is the rotation rate at the equator, and $Δ Ω$ is the difference in rotation rate between the pole and the equator. A strength of the differential rotation can be quantified by the rotational shear $Δ Ω$ or its reciprocal $2π ∕Δ Ω$ , which is the time the equatorial regions need to lap the pole, i.e., the lap time. It can also be characterised by the relative differential rotation rate which is expressed as the ratio of the rotational shear to the equatorial velocity

For instance, on the Sun with $Δ Ω$ = 0.055 rad d^–1, the lap time is 115 d, and $α$ = 0.2.

On stars these characteristics can be estimated from observations with various methods, for instance: Fourier analysis of light curves (Lanza et al., 1993 ), cross-correlation of successive stellar Doppler images (Donati and Collier Cameron, 1997 ), direct spot tracking (Collier Cameron et al., 2002 ), Fourier transform of rotationally broadened line profiles (Reiners and Schmitt, 2002 ), parameter fit in Zeeman–Doppler imaging (Petit et al., 2002 ), through asteroseismology (Gizon and Solanki, 2004 ) and from spectro-interferometric observations (de Souza et al., 2004 ). For slowly rotating stars of the solar type, analysis of disk-integrated Ca ii K line core emission appears to be a promising method (Donahue et al., 1996 ; Hempelmann and Donahue, 1997 ).

Long-term photometric monitoring of starspot modulation reveals changes in the seasonal rotation period which indicate the presence of differential rotation on stellar surfaces and of changes in spot latitudes (Hall, 1991a ; Henry et al., 1995 ; Messina and Guinan, 2003 ). Confronting the range of seasonal variations and the mean rotation period yields a possible correlation between them in the sense that slower rotators show larger period variations. A majority of stars show, however, a significantly smaller rotational shear than that observed on the Sun. Similar behaviour is found in periods obtained from variations of chromospheric Ca ii H & K emission-line fluxes. Over timescales of many years, the rotation period was found to show a sine-like variation which can be attributed to a solar-type activity cycle (Donahue et al., 1996 ). Note, however, that the seasonal period variations yield only lower limits of the rotational shear as they represent rotational rates spread over the range of latitudes where active regions erupt during the stellar cycle.

Using the Fourier transform method Reiners and Schmitt (2003a ,b ) derived differential rotation in terms of $α$ of a sample of rapidly rotating F0–G0 dwarfs and found that it is more common in slower rotators, in agreement with the previous findings. Moreover, the differential rotation in more active stars seems to diminish to values which cannot be measured with the Fourier transform technique.

In such cases, cross-correlation of successive stellar Doppler images is a helpful alternative. It was first performed by Donati and Collier Cameron (1997 ) for the active young dwarf AB Dor and revealed the solar type differential rotation with the equator rotating faster than the polar region (see Figure 12 ). The parameter fit using DI or ZDI developed by Donati et al. (2000 ) and Petit et al. (2002 ) is the next step in using inversion techniques. It was applied to a small sample of active G2–M2 dwarfs, all showing a solar type differential rotation (Petit et al., 2004 ; Barnes et al., 2005 ). This sample revealed a clear dependence of the rotational shear on spectral class, indicating the differential rotation to be negligible in M dwarfs and very strong in G dwarfs. The Sun, however, deviates from the correlation. In contrast to the results for early F and G dwarfs, no significant dependence of $Δ Ω$ on rotation rate was found. An interesting finding though is that the differential rotation is different when estimated from cool spots or magnetic regions, and that it undergoes temporal fluctuations on time-scales of one to a few years (Donati et al., 2003a ). Note, however, that the lack of spatial resolution and spot growth and decay as well as their systematic proper motions, can significantly affect the derived values of the differential rotation. Also, since spot latitudes strongly depend on the inclination of the rotational axis (see Section 4.2), uncertainties in the inclination may result in systematic errors in the deduced differential rotation law.

Figure 12: Two Doppler images of the young active K0 dwarf AB Dor (on the left) and a cross-correlation image (on the right) showing that near-equatorial spots rotate faster than high-latitude regions. From Donati and Collier Cameron (1997 ). See also animation by J.-F. Donati at

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