Dynamics of Magnetic Flux Tubes in Close Binary Stars

A&A 405, 303–311 (2003) Astronomy DOI: 10.1051/0004-6361:20030584 & c ESO 2003 Astrophysics Dynamics of magnetic flux tubes in close binary stars II. Nonlinear evolution and surface distributions V. Holzwarth1,2 and M. Schüssler1 1 Max-Planck-Institut für Aeronomie, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany e-mail: [email protected] 2 School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews KY16 9SS, UK Received 28 February 2003 / Accepted 18 April 2003 Abstract. Observations of magnetically active close binaries with orbital periods of a few days reveal the existence of starspots at preferred longitudes (with respect to the direction of the companion star). We numerically investigate the non-linear dynamics and evolution of magnetic flux tubes in the convection zone of a fast-rotating component of a close binary system and explore whether the tidal effects are able to generate non-uniformities in the surface distribution of erupting flux tubes. Assuming a synchronised system with a rotation period of two days and consisting of two solar-type components, both the tidal force and the deviation of the stellar structure from spherical shape are considered in lowest-order perturbation theory. The magnetic field is initially stored in the form of toroidal magnetic flux rings within the stably stratified overshoot region beneath the convection zone. Once the field has grown sufficiently strong, instabilities initiate the formation of rising flux loops, which rise through the convection zone and emerge at the stellar surface. We find that although the magnitude of tidal effects is rather small, they nevertheless lead to the formation of clusters of flux tube eruptions at preferred longitudes on opposite sides of the star, which result from the cumulative and resonant character of the action of tidal effects on rising flux tubes. The longitude distribution of the clusters depends on the initial parameters of flux tubes in the overshoot region like magnetic field strength and latitude, implying that there is no globally unique preferred longitude along a fixed direction. Key words. stars: binaries: close – stars: activity – stars: imaging – stars: magnetic fields – stars: starspots 1. Introduction In analogy to the “solar paradigm” for magnetic activity, it is assumed that starspots at the stellar surface are generated by Observational results indicate that in close, fast-rotating bi- erupting magnetic flux tubes which originate from the lower naries like RS CVn and BY Dra systems magnetic activity part of the outer convection zone of the star. By using non- is usually much more vigorous than in the case of the Sun linear numerical simulations we investigate the influence of (see, e.g., Strassmeier et al. 1993, and references therein). Data tidal effects on the dynamics and evolution of magnetic flux sets covering time periods much longer than the rotation pe- tubes inside the convection zone until they erupt at the stel- riod of a system reveal non-uniform longitudinal spot distri- lar surface. Particular attention is paid to the non-uniformities butions (with respect to the direction of the companion star), in the resulting surface distributions of erupting flux tubes and which indicate the existence of long-lasting spot conglomera- their dependence on initial conditions. This work complements tions at preferred longitudes where starspots persist longer or our previous studies of the equilibrium and linear stability flux eruption is more frequent than at other surface regions properties of magnetic flux tubes in the overshoot region at the (Henry et al. 1995; Jetsu 1996). The preferred longitudes are lower boundary of the convection zone (Holzwarth & Schüssler ◦ frequently about 180 apart on opposite sides of the stellar 2000, 2003, hereafter HSI). hemisphere (e.g., Berdyugina & Tuominen 1998; Berdyugina et al. 1998) and their dependence on rotation period, i.e., es- Section 2 contains a brief summary of the magnetic flux sentially on the distance between the two components, sug- tube model and the analytical approximation of the close bi- gests that they are related to the proximity of the companion nary model. In Sect. 3 we explain the accomplishment of the star (Heckert & Ordway 1995). numerical simulations as well as the implementation of initial conditions. Section 4 shows the resulting latitudinal, longitudinal and surface distributions of erupting flux tubes in the case Send offprint requests to: V. Holzwarth, of a binary systems with an orbital period of two days as well as e-mail: [email protected] a study concerning the dependence of non-uniformities on the Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20030584 304 V. Holzwarth and M. Schüssler: Flux tubes in close binaries. II. orbital period. Section 5 contains a discussion of our results expansion parameter of the approximation and a the distance and Sect. 6 gives our conclusions. between the two components. For a weakly deformed star it can be assumed that the value of any stellar quantity, f , (pressure, density, ...) is still only a function of the effective potential, 2. Model assumptions Ψeff =Ψ +Ψtide, where Ψ is the gravitational potential of the Ψ 2.1. The flux tube model star and tide the potential of the tidal perturbation (see Eq. (2) in HSI). Following Eq. (5) in HSI, the (Eulerian) perturbation We study the spot distribution on active binary stars by ap- of the stellar structure, ∆ f , can be written as plying the “solar paradigm” for magnetic activity, assuming ∆ f Ψ r that starspots are due to erupting magnetic flux tubes (e.g., ∝ tide ∼ 3q cos 2φ, (2) Schüssler et al. 1996). In this model, the magnetic field is f Ψ H f stored in the form of toroidal flux tubes inside the subadia- where the undisturbed value, f , and its local scale height, H , batic overshoot-region below the convection zone at the inter- f are determined from an unperturbed stellar model. The leading face to the radiative core. Once a critical magnetic field strength orders of the tidal effects given in Eqs. (1) and (2), respectively, is exceeded, perturbations initiate the onset of an undulatory both show a π-periodicity in the azimuthal direction φ. (Parker-type) instability and the growth of flux loops, which Using Kepler’s third law, the order of magnitude of tidal rise through the convection zone and lead to the formation of ff bipolar active regions and starspots upon emergence at the stel- e ects becomes lar surface. 3 −1 −2 3 −2 q r M T The numerical code we have used for the non-linear sim- q ∼ 10 , (3) 1 + q R M d ulations is explicated in Moreno-Insertis (1986) and Caligari et al. (1995). All calculations are carried out in the framework while owing to the ratio r/H f in Eq. (2), the perturbation of of the “thin flux tube approximation” (Spruit 1981) describing the stellar structure also depends on the scale height of the con- the evolution of an isolated magnetic flux tube embedded in sidered quantity. It is enhanced if f changes in a relative small a field-free environment. The surrounding convection zone is stellar layer, like the superadiabaticity, δ = ∇−∇ad, inside the treated as an ideal plasma with vanishing viscosity and infinite thin overshoot region below the convection zone. For the sys- conductivity, implying the concept of frozen-in magnetic fields. tem considered here (T = 2d, M = M, q = 1 and r ≤ R), 3 −3 The dynamical interaction between the flux tube and its envi- Eq. (3) yields q ∼ 10 (with a ∼ 8 R). ronment is taken into account by a hydrodynamic drag force acting upon a tube with circular cross-section which moves relative to its environment. The radius of the cross-section is 3. Simulations determined according to the amount of magnetic flux and the The linear stability analysis in HSI describes the response of lateral balance between the external gas pressure and the inter- a toroidal equilibrium flux tube to small perturbations inside nal total (gas and magnetic) pressure. Since the radiative time the overshoot region and its evolution prior to its penetration scale in the convection zone typically exceeds the rising time to the convection zone above. In the evolution of rapidly ris- of flux tubes, we assume the adiabatic evolution of flux tubes. ing flux loops in the superadiabatical region non-linear effects of the tube dynamics become important, which have to be fol- 2.2. The binary model lowed by numerical simulations. According to the results in HSI, we consider flux tubes with initial magnetic field strengths 4 5 We consider a detached binary system with two solar-type between 7 × 10 G ≤ Bs ≤ 2 × 10 G. The initial flux rings are components, which move on circular orbits with an orbital pe- located in the middle of the overshoot region at latitudes be- ◦ ◦ riod of two days. For the active star we assume synchronised, tween 0 ≤ λs ≤ 80 parallel to the equatorial plane and liable solid-body rotation, with the spin axis orientated perpendicu- to an undulatory (Parker-type) instability (e.g., Ferriz-Mas & lar to the orbital plane. Its internal structure is described by a Schüssler 1995). The simulations start with a perturbation of a tidally deformed solar model, whereas the companion star is flux ring and cover up to 32 yrs of its evolution in the convec- considered as a point mass. The model includes both the tidal tion zone. In order to sample the influence of tidal effects on force and the deviation of the stellar structure from spherical rising flux tubes, we carry out a large number of simulations in shape, which are sufficiently small to be treated in lowest-order which each tube is subject to a perturbation localised around a perturbation theory (see HSI, Sect.

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