A&A 366, 668–675 (2001) Astronomy DOI: 10.1051/0004-6361:20010009 & c ESO 2001 Astrophysics Differential rotation of the present and the pre-main-sequence Sun M. K¨uker and M. Stix Kiepenheuer-Institut f¨ur Sonnenphysik, Sch¨oneckstrasse 6, 79104 Freiburg, Germany e-mail: [email protected]; [email protected] Received 25 September 2000 / Accepted 9 November 2000 Abstract. We present a model for the differential rotation of the present Sun as well as a solar-type star during its pre-main-sequence evolution. The model is based on the mixing-length theory of convective heat transport and a standard solar model. The resulting rotation law is in good agreement with observations and only weakly dependent on the mixing-length parameter. For the present Sun, the normalized horizontal shear decreases with increasing rotation rate, but the total shear is roughly constant. We then follow the Sun’s evolutionary track from the beginning of the contraction to the arrival on the ZAMS. While at an age of 30 Myr the total shear is very similar to that of the present Sun, it is much smaller on the Hayashi track. Key words. hydrodynamics – Sun: rotation – stars: rotation – stars: pre-main sequence 1. Introduction K¨uker et al. (1993) assumed Reynolds stress as the only source of rotational shear and found a rotation pat- The differential rotation of the solar surface is a well- tern that remarkably well agrees with observations. The known phenomenon. The rotation rate determined by model, however, predicts an increase of the latitudinal Doppler measurements is about 30 percent smaller at shear with increasing rotation rate, in clear contradiction the poles than at the equator (Snodgrass 1984), while with the observational evidence. A consistent treatment a slightly smaller difference of 20 percent has been de- of the rotation and the meridional flow yields qualitative rived from the motion of sunspots (Howard et al. 1984; agreement with observations but reproduces the solar ro- Balthasar et al. 1986). Helioseismology has revealed that tation law only if a relatively large value of the turbulence this pattern is not restricted to the surface but extends viscosity is assumed (R¨udiger et al. 1998). throughout the whole convection zone, while the radiative In the model of Kitchatinov & R¨udiger (1995, KR95) core rotates rigidly with the same period as the surface this difficulty is overcome by including the convective heat at mid-latitudes (Schou et al. 1998). Surface differential transport. Kitchatinov & R¨udiger (1993) and Kitchatinov rotation has also been found for a number of magnetically et al. (1994) have derived a consistent mixing-length active stars by long-term photometry (Hall 1991; Donahue model of Reynolds stress and convective heat transport in et al. 1996) and Doppler imaging (Collier Cameron et al. a rotating convection zone that takes into account the ef- 2000). fect of the Coriolis force on the convective motions. In the Rotational shear can generate a strong toroidal mag- KR95 model, the deviation of the heat flux from spherical netic field out of a weak poloidal field and is therefore the symmetry causes a small horizontal temperature gradient, main source of magnetic field energy in the αΩ dynamo which partly neutralizes the rotational shear as the force which is thought to be the source of magnetic activity in that drives the meridional flow. As a result, the model re- solar-type main-sequence stars. In a rigidly rotating con- produces the original K¨uker et al. (1993) result with only vection zone, a turbulent dynamo is still possible, but it one free parameter, the mixing-length parameter αMLT. must then be of the α2 type, which is known to prefer Kitchatinov & R¨udiger (1999) found the total shear, steady non-axisymmetric rather than oscillating axisym- δΩ=Ωeq − ΩPole, (1) metric fields (Moss & Brandenburg 1995; K¨uker & R¨udiger 1999). The knowledge of the internal rotation pattern is to be only weakly dependent on the rotation period. therefore essential in modeling stellar dynamos. Collier Cameron et al. (2000) have combined measure- ments of differential rotation for the pre-main-sequence stars RX J1508-4423, AB Dor, and PZ Tel with the sam- Send offprint requests to:M.Stix ple of main-sequence stars from Donahue et al. (1996) and Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20010009 M. K¨uker and M. Stix: Differential rotation of the present and the pre-main-sequence Sun 669 argue that the discrepancies between the Kitchatinov & As the mass density varies with depth but not with time, R¨udiger (1999) result and the relation between shear and mass conservation requires rotation rate derived by Donahue et al. (1996) may be due ∇· to the mix of G and K stars in the latter paper. They sug- (ρu¯)=0. (7) gest that δΩ may indeed be constant with rotation rate The meridional circulation can then be expressed by a but vary with spectral type and derive values for G and stream function A, K dwarfs that differ by a factor of three, the K dwarfs rotating more rigidly. 1 ∂A −1 ∂A u¯m = , , 0 . (8) While this picture is quite appealing, the three stars ρr2 sin θ ∂θ ρr sin θ ∂r mentioned above are very rapidly rotating pre-main- sequence stars and therefore differ in structure and spec- The azimuthal component of the curl of (3) describes the tral type from main-sequence stars of the same mass. One meridional motion: must therefore ask whether the same relation between ∂ω 1 ∂Ω2 = − ∇× ∇(ρQ) + r sin θ differential rotation and spectral type holds for main- ∂t ρ ∂z sequence and PMS stars. φ In this paper, we study the differential rotation of 1 ∇ ×∇ + 2 ( ρ p)φ, (9) the present Sun and a solar-type pre-main-sequence star ρ (or the PMS Sun) at different ages. We first address the where ω =(∇×u¯)φ is the curl of the meridional flow present Sun to compare the results from our model with velocity and ∂/∂z =cosθ · ∂/∂r − sin θ/r · ∂/∂θ is the those from previous work and observations. The model is gradient along the axis of rotation. then applied to a series of PMS models for the Sun at With the expression for the specific entropy of a perfect different ages from an evolutionary track as computed by gas, Ahrens et al. (1992). p s = C ln (10) 2. The model v ργ 2.1. The transport equations and the relation for hydrostatic equilibrium, 2.1.1. Equation of motion ∇p = −gρ, (11) We use the mean-field formulation of hydrodynamics, i.e. the last term on the RHS of (9) may be rewritten in terms apply an appropriate averaging procedure to split the of the specific entropy, velocity field into a mean and a fluctuating part, 1 1 g ∂s 0 ∇ ×∇ − ∇ ×∇ ≈− u = u¯ + u . (2) 2 ( ρ p)φ = ( s p) , (12) ρ Cpρ rCp ∂θ The mean velocity field, u¯, is then governed by the where the approximation is justified though the fact that Reynolds equation, in the unperturbed (non-rotating) state the entropy is con- ∂u¯ stant while the pressure is a rapidly varying function of ρ +(u¯ ·∇)u¯ = −∇ · (ρQ) −∇p + ρg + ∇·π, (3) ∂t the radius. h 0 0 i The stream function and the vorticity ω are related via where Qij = uiuj is the correlation tensor of the velocity 0 the equation fluctuations, u . In stellar convection zones, the molecular stress tensor π is usually about 10 orders of magnitude 1 ∂ρ ∂A 1 ∂ρ DA − − A = −ρω, (13) smaller than the Reynolds stress and can be neglected. ρ ∂r ∂r ρr ∂r As the density distribution is spherically symmetric, and the gas motion is dominated by the global rotation, where D =∆− 1/(r2 sin2 θ). we assume axisymmetry for the mean velocity and tem- The correlation tensor Q consists of a viscous and a perature. The velocity field can then be described as a non-viscous part: superposition of a rotation and a meridional flow: ν Λ Qij = Qij + Qij , (14) u¯ = r sin θ Ω φˆ + u¯m, (4) (R¨udiger 1989). The first term on the RHS of (14), the ˆ where φ is the unit vector in the azimuthal direction. The turbulence viscosity, azimuthal component of the Reynolds equation expresses the conservation of angular momentum: ∂u¯k Qν = −N , (15) ij ijkl ∂x ∂ρr2 sin2 θ Ω l + ∇·t =0, (5) ∂t has been calculated by Kitchatinov et al. (1994), while a where detailed theory of the second term, the Λ effect, m h 0 0i Λ t = r sin θ ρr sin θΩu¯ + ρ uφu . (6) Qij =ΛijkΩk, (16) 670 M. K¨uker and M. Stix: Differential rotation of the present and the pre-main-sequence Sun (where Ω =Ωzˆ) has been derived by Kitchatinov & where R¨udiger (1993). The Λ effect is proportional to the an- l = α H (26) gular velocity itself rather than its gradient and thus re- MLT p distributes angular momentum even in case of initially is the mixing length and rigid rotation. All components of the correlation tensor − p depend on the angular velocity as well as on the convec- Hp = ∂p (27) tive turnover time,τcorr, via the Coriolis number, ∂r the pressure scale height. The convection velocity and the Ω∗ =2τ Ω. (17) corr mixing length together determine the convective turnover In the limiting case of very slow rotation, Ω∗ 1, the time through the relation viscous stress becomes isotropic and reduces to the well- l τ = .