that rapidly rotating, active stars generally display polar spots, in contrast to the predominantly equatorial spots

Transcription

1 THE ASTROPHYSICAL JOURNAL, 484:855È861, 1997 August 1 No copyright is claimed for this article. Printed in U.S.A. POLAR MAGNETIC ACTIVITY AND SPIN-DOWN ON THE LOWER MAIN SEQUENCE D. L. UZASI Department of Physics, Astronomy and Geology, Valdosta State University, Valdosta, GA ; dbuzasi=valdosta.peachnet.edu Received 1996 November 19; accepted 1997 March 6 ASTRACT On the Sun, magnetic activity is essentially restricted to latitudes within 45 of the equator. However, active stars, such as RS CVn systems, and T Tauri stars generally show polar activity. I model the evolution of a toroidal magnetic Ñux tube as it rises from the base of the ection zone in various lower main-sequence stars and show that, on the main sequence, polar activity is more likely for less massive stars. For example, a 1 M star must rotate with u[5 u in order to develop signiðcant polar magnetic Ñux, while a 0.4 M _ star will generate predominantly _ polar Ñux for u[0.5 u. This leads to the expectation that essentially _ all M stars should display polar, rather than equatorial, _ magnetic activity. A similar e ect is expected for RS CVn and T Tauri systems, and for giant stars. An important consequence of this polar activity is the inhibition of angular momentum loss from these systems. On the lower main sequence, angular momentum is lost via the interaction of the stellar magnetic Ðeld with the outñowing stellar wind. However, if regions of strong magnetic Ðeld are conðned to the poles, stellar spin-down rates can be reduced dramatically, by up to a factor of 3 or more for early M stars. Examination of open clusters should show the presence of rapidly rotating M stars at ages beyond those previously expected. Subject headings: stars: activity È stars: late-type È stars: rotation 1. INTRODUCTION non (see, e.g., Weber & Davis 1967 and MacGregor & In recent years, many researchers have come to accept renner 1991) assume a homogeneous surface magnetic that rapidly rotating, active stars generally display polar spots, in contrast to the predominantly equatorial spots seen on the Sun (Strassmeier 1996, but for an opposing view see yrne 1996). The primary impetus for this model has Ðeld. However, as Giampapa (1994) has pointed out, such models predict spin-down timescales for early M stars that are far too rapid to be in accord with observations. M stars, of course, have deep ective envelopes and thus (by the come from Doppler imaging studies (see, e.g., Vogt & arguments of the previous paragraph) might be expected to Penrod 1983, Vogt & Hatzes 1991, Strassmeier et al. 1991, and Strassmeier 1996) that nearly always result in surface images displaying large spots (at least a few percent of the stellar surface area) located near or actually on the stellar pole. The size of the spots is explicable, in qualitative terms at least, by invoking the increased dynamo activity expected in these rapidly rotating systems. Recently, Schussler & Solanki (1992) demonstrated that the e ect of the Coriolis force in a rapidly rotating Sun probably would be sufficient to force all rising Ñux tubes to emerge near the poles, while in the real ÏÏ Sun, Ñux tubes move along radial lines, and thus most emerge below 30 latitude. Simple scaling arguments then suffice to demonstrate that the Coriolis e ect should be even more important display predominantly polar surface magnetic Ñux, which would impair the efficiency of angular momentum loss. The primary purpose of this work is to explore the e ects of rapid rotation on surface magnetic Ñux in lower mainsequence stars and, by extension, the e ects of rapid rotation on angular momentum loss rates in these stars. The current model for solar and stellar activity invokes a magnetic dynamo that regenerates magnetic Ðelds by using the interaction of ective Ñows and stellar rotation (Parker 1955; Leighton 1969; Durney & Robinson 1982; DeLuca & Gilman 1991). Originally, these dynamos were distributed throughout the ection zone, and while they explained certain observed phenomena well (e.g., cyclic behavior), they required radial rotational gradients at odds in stars with compact cores, such as lower main with helioseismological results (Morrow 1988). More sequence stars and giants. While this explanation is incomplete in the sense that no mechanism is supplied for the concentration of emerging Ñux tubes to form spots, it is recently, Durney (1976), followed by others (e.g., DeLuca 1986 and DeLuca & Gilman 1991), argued for placing the site of dynamo activity at the base of the ection zone, in certainly suggestive in light of the apparently frequent particular in the ective overshoot region, which occurrence of polar spots on rapidly rotating giants and subgiants (see, e.g., Table 1 in Strassmeier 1996). The site at which the surface magnetic Ñux emerges is important in contexts beyond those of starspots on rapidly rotating stars. In particular, lower main-sequence stars lose angular momentum via their winds, which are coupled to the stellar surface through the stellar magnetic Ðeld. This magnetic coupling yields angular momentum loss rates that are typically 1 or 2 orders of magnitude higher than would arise from the wind alone. Current models of this phenome- appears to solve a number of the problems that arise from the distributed dynamo. Once formed in the overshoot region, magnetic Ñux rises to the stellar surface because of magnetic buoyancy and the advection of the Ðeld by the local ective Ñow (Parker 1955). The dynamics of rising Ñux tubes in the solar interior were Ðrst explored in detail by Choudhuri & Gilman (1987) and Moreno-Insertis (1986), who considered the solar case and calculated the magnetic Ðeld strengths required at the base of the ection zone. Other models have followed 855

2 856 UZASI Vol. 484 (e.g., Chou & Fisher 1989; Fan, Fisher, & DeLuca 1993; Caligari, Moreno-Insertis, & Schussler 1995), but so far all have limited themselves to the solar case. My purpose here is not to replicate their work, although a certain repetition is unavoidable as a consistency check, but to extend it to those lower main-sequence stars in which the consequences of the Coriolis e ect on rising Ñux tubes is as yet unstudied in detail. The state of the current observations of both the Sun and the other lower main-sequence stars indicates that surface magnetic Ðelds are typically in the range of 1È5 kg (see, e.g., Saar & Linsky 1985; Saar 1994, 1996; asri & Marcy 1994; Johns-Krull & Valenti 1996). However, the strength of internal magnetic Ðelds, particularly those near the base of the ection zone, is somewhat less certain. I can estimate an approximate lower limit to this value by requiring that the Ðeld strength be sufficient to dominate local ective motions, clearly a necessary condition for the existence of a Ñux tube. Thus, º J4nov2, (1) where v can be either derived from a speciðc model inte- rior or estimated following Schatzman & Praderie (1993): A v \ F, (2) 10o1@3 where F is the ective Ñux. Using either method to estimate v gives rise to a minimum magnetic Ðeld strength of about 5000 G in the Sun, or about 1.2 ] 104 Ginan0.4 M M star. I can also estimate an upper limit to the Ðeld _ strength by assuming that the local magnetic energy density at the bottom of the ection zone (assumed to be the genesis region for the stellar dynamo) is less than the local rotational energy density. This assumption should be approximately true for a rotation-driven dynamo, since the ultimate energy source for the stellar magnetic energy is the stellar rotation. In this case, ¹ J4nor2 u2, (3) where r is the radius to the base of the ection zone and u is the stellar angular velocity. In this case, the maximum magnetic Ðeld strengths are about 105 G for the Sun and 3 ] 105 G for an 0.4 M M star with a solar rotational period. For the sake of _ this discussion, I will adopt an approximate value for the magnetic Ðeld at the base of the ection zone of 104 G for the Sun and 105 G for the M star, keeping in mind that the exact values may di er somewhat from these approximations. A Ñux tube in a rotating stellar interior encounters four main forces: magnetic buoyancy, the Coriolis force, magnetic tension, and drag. First, the magnetic buoyancy force, originally described by Parker (1955), arises when one posits equipartition of pressure within and without the Ñux tube. The buoyancy arising from this e ect can be described approximately by *o o \ 2, (4) 8nP e where P is the external gas pressure, o the external density, e and the magnetic Ðeld within the Ñux tube. This force has the dominant impact on the motion of the Ñux tube. As Parker (1955) and others (see, e.g., Choudhuri & Gilman 1987) have shown, considering magnetic buoyancy alone yields a rise velocity of A v \ 2gl *o 0.5 o as the Ñux tube rises through a length l. This yields a rise time from the base of the ection zone to the surface of about a week for the Sun, and about a month for the M star described above. Of course, these are minimum times, since I have neglected (among other items) drag, and I have assumed that the Ñux tube element rises radially. The rotating coordinate system gives rise to both a centripetal and a Coriolis force. The former can be considered as a form of reduced gravity ÏÏ but is never more than a 1% e ect, even in an extremely rapidly rotating main sequence star. Therefore, I will disregard it. The Coriolis force, however, can be important, as Schussler & Solanki have shown. The ratio of the strength of the Coriolis force to that of the magnetic buoyancy force can be written as (5) F C \ 2x  F (*o/o)g 2uvo (*og), (6) where v is the Ñux-tube rise velocity and the other symbols have their usual meanings. For the Sun, this ratio is about unity, while for the M star (with solar rotational period), it is about 6. The Coriolis force therefore should be of some importance on the lower main sequence and will cause a rising Ñux tube to both twist ÏÏ (rotate in /) and turn ÏÏ (rotate in h), where the angles represent the usual spherical coordinate system, with / representing longitude and h colatitude. The remaining two forces are of much less signiðcance in understanding the motion of rising Ñux tubes. The magnetic tension force, ($ Â) F \, (7) M 4n is of some importance at the beginning of the Ñux-tube motion (at the base of the ection zone) but rapidly becomes insigniðcant (see Choudhuri & Gilman 1987). The drag force imposes a terminal velocity on the motion of the rising Ñux tube, since F drag \[1 2 C D opu2 (8) for a moving cylinder in the high Reynolds number limit, from Rouse (1978). Here p is the cross-sectional radius of the cylinder, which in general can be derived by assuming Ñux conservation: np2 \ ' ; (9) u, of course, is just the velocity of the moving cylinder, and C is the drag coefficient, taken to be 0.4 in the transverse case D and unity otherwise. I will return to the e ects of drag later on. 2. THE MODEL Following Choudhuri & Gilman (1987), let us consider an azimuthally symmetric Ñux ring embedded in the base of

3 No. 2, 1997 POLAR MAGNETIC ACTIVITY AND SPIN-DOWN 857 the ection zone. If p is small compared with the stellar radius, then the ring is essentially a line and can be described using only a pair of coordinates (r, h), giving rise to the following equation of motion for an element of the ring: o Ad dt ] 2x  \[+(P e ]P i ) ] ($Â) ]og]f, (10) 4n drag where P is the local gas pressure external to the Ñux ring and P is e the gas pressure inside the Ñux ring. The thin Ñux-tube i approximation (Schussler 1979) assumes that the radius of the Ñux tube is small compared with the local pressure scale height H. Making this approximation allows integration of the equation of motion over the Ñux ring, to obtain m i Adu dt ] 2x Âu \(m i [m e )g P P ($ Â) ] dvol ] F dvol. (11) 4n drag The last two integrals can be solved under the assumption that the magnetic Ðeld is \ e inside the ring and van- Õ ishes elsewhere. The resulting equation of motion can be written in three vector components (r, h, /), with radial component Cd2r 2m i dt2 [ radh dt 2 Ad/ [ r dt 2 sin2 h [ 2ru Ad/ dt sin2 h D \ (m [ m )g [ '2 i e 2np2 sin h ] D r. (12) Here ' is the transverse magnetic Ñux through the ring, and D is the radial component of the drag. Note the contribu- r tion of the Coriolis force (the term involving u), the buoyancy force (the Ðrst term on the right), and the magnetic tension force (the second term on the right). The h equation is C d2h 2m r i dt2 ] 2 dr dh dt dt [ rad/ sin h cos h dt2 sin h cos h D \[ '2 [2ru Ad/ dt while the / equation reduces to m i C r d2/ dt2 2np2 cos h ] D h, (13) dr d/ dh d/ sin h ] 2 sin h ] 2r dt dt dt dt cos h A ]2u r dh dr hd cos h ] dt dt sin \ D. (14) Õ Once again, Coriolis terms can be identiðed by the presence of u, and magnetic tension terms by the presence of '. These equations are amenable to solution only if accurate expressions for p and *o/o are known. Let us assume that r,h the ection zone is adiabatically stratiðed and neglect the superadiabatic gradient (we will justify this neglect later). Using the perfect gas law P \ Ro T gives e e e T \ T [ Ac [ 1 gr AR [ R _ 0, (15) e 0 c R R where use of the subscript 0 indicates the value at the base of the ection zone (the starting point ÏÏ). Using the usual polytropic expressions for pressure and density gives P e \ P 0 AT e T 0 c@(c~1), (16) o e \ o 0 AT e T 0 1@(c~1). (17) If the Ñux ring starts in temperature equilibrium with its surroundings, then p2 p2 \ o 0 R 0 sin h 0 or sin h 0 \ C 1 [ Ac [ 1 c gr R [ R D~1@(c~1)AR sin h _ R T R R sin h 0 (18) During the above discussion of magnetic buoyancy, I assumed that *o/o \ 2/8nP. However, this is only a Ðrst- e order approximation. More accurately, *o o \ 2 C1 AT (2~c)@(c~1)A R sin h 2 8nP c T R sin h R +*T R dr. (19) T _ R0 Here the Ðrst term represents adiabatic movement of the Ñux tube, while the second term reñects the superadiabatic gradient contribution, and +*T has its usual meaning. In general, then, expressions for *o/o and p can be calculated if a model of the ection zone is available. I have used model interiors calculated by D. A. Vandenerg (1996, private communication), which use current OPAL opacities, model the ective overshoot region, and reñect main sequence stars with solar abundances aged to approximately 5 ] 108 yr. I solved the equations of motion using a Ðnite di erence algorithm (see, e.g., Press et al. 1988), subject to the boundary condition that the emergent magnetic Ðeld strength be approximately 2000 G; this boundary condition was met by shooting. While large changes in this condition signiðcantly alter the detailed results discussed below, more reasonable variations (1È3 kg) have only a minimal e ect. I have run models for a range of magnetic Ñuxes (1017È1021 Mx), and the results are stable over this range. In general, the model results discussed below assume ' \ 1018 Mx, in accord with observed magnetic Ñuxes in the smallest observable solar Ñux tubes (Stix 1989). Model calculations were carried outward radially until the thin Ñux-tube approximation no longer held, which was generally quite near the surface. Originally, I ran a suite of models assuming that u \ ] 10~6 rad s~1 D u(r, h). However, more accu- rate models for the Sun should reñect the available helio- ] A 1 [ 1 cd ] P

4 858 UZASI Vol. 484 seismological data, so in some cases (noted below) I have adopted the Libbrecht & Morrow (1991) internal solar rotation model l (R, h) \ A ] cos2 h ] C cos4 h, (20) with A \ 461 nhz, \[60.5 nhz, and C \[75.4 nhz. The Libbrecht & Morrow model certainly is not the most current such model (see, e.g., Scherrer et al. 1996), but it reñects the main features of more recent and more accurate measurements of the internal solar rotation, and is thus useful for demonstrating the main e ects of di erential rotation on rising Ñux tubes. More importantly, there is no a priori reason that a solar-like rotation law should hold for stars other than the Sun, and therefore no reason to adopt a complex description of internal di erential rotation that might give rise only to a spurious accuracy. The overall problem of stellar internal di erential rotation is discussed in some detail below. 3. RESULTS 3.1. T he Sun Figure 1 summarizes the results for a Ñux tube starting at 10 latitude and rising through the solar interior, which generally accord with results of other investigators (Choudhuri & Gilman 1987; Caligari et al. 1995). The model age of 500 Myr was chosen for consistency with later models of less massive stars, which are intended to represent moderately aged, open cluster stars. For rigid internal solar rotation at the observed surface rate, Ñux tubes rise essentially along radial vectors, with tubes starting at 10 latitude diverging from the radial by only about 1. In order to obtain 2 kg surface Ðeld strengths, the required magnetic Ðeld at the base of the ection zone is 8.8 ] 104 G, in the range spanned by the lower and upper limits calculated above. Note that the Ñux-tube paths do not quite reach the solar surface, in accord with the results of Caligari et al. (1995). Rather than reñecting the true physical reality, this situation reñects the failure of the thin Ñux-tube approximation at a depth of some 15,000 km below the solar surface. A quick check of the validity of the model can be made by examining the geometry of the emergent Ñux tube relative to the solar surface. Solar model results predict that the emergent Ñux tube should be inclined so as to trail the solar rotation by about 4. Howard (1991) has found that actual Ðeld regions on the Sun tilt so as to trail the rotation by 3 È9, which lends some support to the model results. Adopting a more detailed (and realistic) rotation law does little to change the situation, as can be seen in Figure 1. Even an extreme (and highly unrealistic) rotation law that supposes the observed equatorial rotation rate decreases to zero at the poles does not have a signiðcant e ect. However, signiðcant changes in the path of the rising Ñux tube can be observed if the solar rotation rate is increased. At 10 times the observed solar rotation rate (neglecting di erential rotation), divergence from radial rise becomes signiðcant (Fig. 2), reaching 14 for a Ñux tube starting at latitude 10 for a solar rotation rate of 5 times the actual solar rate, and increasing rapidly thereafter. In fact, one can deðne an u, beyond which the paths of rising Ñux tubes crit undergo a qualitative change from rising radially to rising parallel to the rotation axis. For the Sun, u 3.6 crit ] 10~5, which corresponds to a rotational period of about 2 days. For the current solar rotation rate, tubes that start at 10 reach the surface in some 5.94 ] 105 s, or about a week. The maximum radial velocity achieved by the rising tube occurs just below the surface and is 230ms~1. Such a tube emerges some 7 in longitude (in the antirotational direction) from its starting point. Flux tubes in more rapidly rotating models rise somewhat more slowly (partly because of the longer path they travel), so that in the u \ 10 u _ FIG. 1.ÈTrajectories of rising Ñux tubes in the ective envelope of a star of mass 1.0 M and age 500 Myr. The solid lines indicate paths for models assuming a _ Libbrecht & Morrow (1991) di erential rotation law, while the dot-dashed lines indicate paths for a model with extreme ÏÏ di erential rotation, as discussed in the text. The trajectories are labeled by rotational velocity, and all begin at h \ 10. FIG. 2.ÈTrajectories of rising Ñux tubes in the ective envelope of a star of mass 1.0 M and age 500 Myr. The paths are labeled by rotational velocity, and all begin _ at h \ 10.

5 No. 2, 1997 POLAR MAGNETIC ACTIVITY AND SPIN-DOWN 859 FIG. 3.ÈTrajectories of rising Ñux tubes in the ective envelope of a star of mass 0.4 M and age 500 Myr. The paths are labeled by rotational velocity, and all begin _ at h \ 10. case, the rising tube takes 1.43 ] 107 s, or nearly 6 months, to reach the surface, and it achieves a maximum radial velocity of only 130 m s~1. This tube moves some 22 from its starting point in longitude L ower Main Sequence The calculation of paths for stars less massive than the Sun requires the use of more assumptions. Accurate values of ' for stellar Ñux tubes are unknown, as are rotation laws. I will begin by assuming that ' ', or about 1018 Mx for a typical Ñux tube, although, * as for _ the solar case, the particular value of ' chosen has no dramatic e ect on the results. The choice of is an issue, however, and I have chosen so that emerging 0 Ñux tubes reach R/R º 0.95, as in the solar 0 case. I will discuss the e ect of relaxing * this constraint. Figure 3 summarizes the results of calculations for a 0.4 M star for a range of u. Such a model requires \ 9.6 ] _ 104 G in order to produce magnetic Ðeld strengths 0 of 2 kg at its highest point, some 1.2 ] 104 km below the stellar surface, where the thin Ñux-tube approximation fails. Rise times are signiðcantly di erent from the solar case, ranging from 3.09 ] 107 s (about a year) for the u \ 0.05 u case to 5.6 ] 107 s (nearly 2 yr) for the u \ u _ case. Maximum rise velocities are much slower than the _ solar case, about 2È3 ms~1, while longitudinal movement is generally more signiðcant than it is for the Sun, ranging from about 8 (for u \ 0.05 u ) to over 42 (for u \ u ). The most noticeable di erence _ between this case _ and that of the Sun, however, is in the emergence latitude. For Ñux tubes starting at 5 latitude, the rise paths deviate signiðcantly (about 6 ) from radial vectors, even for u \ 0.05 u. _ In fact, for a 0.4 M star, u ¹ 2 u, so Ñux tubes rise _ crit _ parallel to the rotation axis for rotational periods shorter than 2È3 weeks. Thus, while rapidly rotating G V stars might be expected to show predominantly polar activity, essentially all early M V starsèexcept for the slowest rotatorsèshould show signs of polar activity. Note that this conclusion is independent of the choice of dynamo model, since for u[u all Ñux tubes are forced to emerge crit near the poles, no matter where they start from. Figure 4 FIG. 4.ÈA summary of model results for a range of stellar masses. In each case, the emergence latitude for a Ñux tube that began at h \ 10 is shown. The open crosses indicate 1.0 M stars, the open squares 0.9 M, the open triangles 0.8 M, the open circles 0.7 M, the Ðlled squares 0.6 M, the Ðlled triangles 0.5 M, and the Ðlled circles _ 0.4 M.

6 860 UZASI Vol. 484 summarizes results for stars in the range 0.4 \ M/M \ 1.0. _ Relaxing the requirement that Ñux tubes reach R º 0.95R does not change this situation. In fact, lowering to 2.9 * ] 104 G, which causes the model to terminate at 0 R \ 0.91R, only reduces u by a factor of 2 from the earlier case. * crit Further exploration of equations (4)È(6) yields clues about why the behavior of the rising tubes is so di erent in the two cases. I can combine these equations to obtain F C \ AlP 1@2 8un1@2 e. (21) F g If I choose l proportional to R, and take values for the other parameters at the base of * the ection zone, then F /F is about 6 times greater for a 0.4 M star than for the Sun C (if and u are equal). The di erence _ arises primarily from the e ect of the increased depth of the ection zone on the values of (T, P, o), all of which are greater in the overshoot region of the e M star. As in the solar case, changing the internal di erential rotation from solid body to a more realistic (for the Sun!) rotation law has little e ect on the results. However, as Choudhuri & Gilman (1987) have pointed out, internal rotation laws of the form u(r, h) P 1 r2 sin2 h (22) will have the dramatic e ect of causing all rising Ñux tubes to move radially, since such a rotation law enforces an angular momentum per unit volume that is radius independent, so that rising Ñux tubes must move radially to conserve angular momentum. In light of the model results reviewed above, the implications of this are highly signiðcant: observed equatorial activity in rapidly rotating, lower main-sequence stars implies radial di erential rotation qualitatively unlike that seen in the Sun. 4. DISCUSSION Over the past several decades, the observational picture of the rotational evolution of single low-mass stars on the main sequence has become relatively clear. Observations of open clusters such as a Per (Stau er et al. 1985; Stau er, Hartmann, & Jones 1989; Prosser 1992), the Pleiades (Stau er 1994; Stau er & Hartmann 1987; Soderblom et al. 1993), and the Hyades (Radick et al. 1987) have been largely responsible for this pleasant state of a airs. The resulting picture of cool star rotational evolution can be summarized as follows: 1. The very youngest clusters, such as a Per, contain rapid rotators (v sin i [ 100 km s~1) at all spectral types from M through F. 2. Somewhat older clusters, such as the Pleiades, contain rapidly rotating K and M dwarfs, but essentially no rapidly rotating G stars. 3. In the oldest clusters, all G and K dwarfs are slow, solar-like rotators, and rapid rotators are to be found only among the M stars. 4. At all ages, there is a signiðcant representation (up to 50%) of slow, solar-like rotators. The goal of any theory of stellar angular momentum evolution is to explain simultaneously the longer spin-down timescales for some later type stars and the presence of slow rotators in clusters of all ages. The simple Skumanich (1972) relation does not satisfy this requirement (see, e.g., Simon 1992), and thus more complex models have been developed. Essentially all current models of angular momentum evolution on the lower main sequence (see, e.g., Endal & SoÐa 1981, MacGregor & renner 1991, and Li & Collier Cameron 1993) use some variant of the Weber-Davis wind (Weber & Davis 1967) to remove angular momentum from the star. The wind is spherically symmetric and yields angular momentum loss rates of the form dj dt \ 2 3 ur 2 M0, (23) A where r is the Alfve n radius and M0 is the stellar mass-loss rate. The A factor of 2 arises from the fact that those parts of the stellar wind arising 3 at higher latitudes are less efficient angular momentum loss mechanisms, by a factor (for a spherically symmetric wind) related to sin h. Such a wind (MacGregor & renner 1991) predicts spin-down timescales of q \ 3 J J 2 M0 r2. (24) A MacGregor & renner have concentrated on modeling the Sun, and Giampapa (1994) has pointed out a signiðcant difficulty with these models for low-mass stars. In particular, if we apply the deðnition of the Alfve n radius (essentially the radius where the kinetic energy density of the outñowing wind and the magnetic energy density are equal), 2 A 8n \ 1 2 o A u A 2, (25) where o is wind density and u is wind velocity at the a a Alfve n radius, to the formalism above and assume both a steady mass-loss rate (M0 \[4nor2u) and magnetic Ñux conservation, we Ðnd du dt \[u S 2R * 4, (26) Iu A where I is the moment of inertia and is the surface magnetic Ðeld strength. Substituting values S appropriate for early M and G dwarfs results in the conclusion that the M dwarf should spin down several times more rapidly than the solar-type star, a conclusion severely at odds with observations. However, if angular momentum loss is via magnetically coupled corotation of the stellar wind, and the surface stellar magnetic Ðeld is not homogeneously distributed, then the assumption that angular momentum loss rates vary over the stellar surface simply as cos h is clearly incorrect. In particular, in light of the models discussed above, angular momentum loss should become less efficient for rapidly rotating stars, since the wind will corotate only over those portions of the stellar surface with a signiðcant magnetic Ðeld. I can deðne a latitude l, where Ñux tubes that start min near the equator in rapidly rotating stars (u º u ) end up. crit

7 No. 2, 1997 POLAR MAGNETIC ACTIVITY AND SPIN-DOWN 861 Such Ñux tubes move parallel to the rotation axis, so l \ cos~1 AR core. (27) min R * For a rapidly rotating G2 V star, this latitude is about 45, so all magnetic Ñux must emerge poleward of this point. However, the cone mapped out inside this angle is close enough to the stellar equator that angular momentum loss is only minimally impaired. Furthermore, since such stars have u º u, corresponding to a two-day rotational period for the crit Sun, they are quite rare at this spectral type. Of course, the distribution of surface magnetic Ñux within the cone h º l is dependent on the speciðcs of the stellar dynamo. The solar min dynamo mechanism currently generates magnetic Ñux within the range 0 \ l \ 45, with the exact latitude band depending on the particular time in the solar cycle. If the dynamo mechanism were to retain the same form with the Sun spun up to u, then the observed mag- netic Ñux would occupy a band crit with range 45 \ l \ 60, and spin-down efficiency would be inhibited by about 50%. In the case of M stars, however, l [ 60, because of min their small cores, and the consequent trapping of surface magnetic Ñux near the poles can have a signiðcant e ect on angular momentum loss rates. In fact, if the emergent wind is spherically symmetric, the decrease in angular momentum loss efficiency is about a factor of 2.5È3 for an M4 V star with u º u. Such a rotation rate corresponds to a crit period of 2 weeks or more, so large numbers of M stars should show this e ect. A variety of authors have noted the presence of polar spots on rapidly rotating active stars. Donati et al. (1990, 1992) have used Zeeman-Doppler imaging (Semel 1989) to examine the surface magnetic Ðeld structure on the primary star in the RS CVn system HR 1099 and report that it appears to emerge toroidally rather than radially, as one might naively expect were buoyancy the dominant force acting on rising magnetic Ñux tubes. We can explore this situation by combining equations (3) and (21) to obtain A F C \ 4 lp e. (28) F or2 g1@2 Neglecting factors of order unity and substituting approximate expressions for P (P \ GM2/R4) and g give e e F C 1 AlM1@2. (29) F R2 o Note that u has dropped out of this equation, since we have assumed P u in equation (3). Thus, the essential require- ments for F [ F are a relatively small radiative core and a relatively low C density at the base of the ection zone. oth of these are present in subgiants and giants, and so the emergence of polar spots in these stars is to be expected. Recently, and concurrently with this modeling work, asri & Marcy (1995) and Jones, Fischer, & Stau er (1996) have used the Keck telescope to extend the range of known rotational velocities in the Pleiades to V \ 21, corresponding to approximately 0.1 M. The result of these observ- _ ations is that all M dwarfs in the Pleiades are rapid rotators. In addition, Jones et al. Ðnd a gradual decrease in spindown with decreasing mass. These observational results are consistent with the model results reported above. I acknowledge support for this research from NASA grant NAGW Don Vandenerg helpfully provided the stellar interior models. In addition, I gratefully acknowledge discussions with, and helpful advice from, Heather L. Preston, Graham A. Murphy, and Harold L. Nations. Je rey Linsky, the referee, helped to improve signiðcantly the presentation of this paper. REFERENCES asri, G., & Marcy, G. W. 1994, ApJ, 431, 844 Prosser, C. 1992, AJ, 103, 488 ÈÈÈ. 1995, AJ, 109, 762 Radick, R. R., Thompson, D. T., Lockwood, G. W., Duncan, D. K., & yrne, P , in Stellar Surface Structure, ed. K. G. Strassmeier & J. L. aggett, W. E. 1987, ApJ, 321, 459 Linsky (Dordrecht: Kluwer), 299 Rouse, H. 1978, Elementary Mechanics of Fluids (New York: Dover) Caligari, P., Moreno-Insertis, F., & Schussler, M. 1995, ApJ, 441, 886 Saar, S. 1994, in IAU Symp. 154, Infrared Solar Physics, ed. D. M. Rabin, Chou, D., & Fisher, G. H. 1989, ApJ, 341, 533 J. T. Je eries, & C. Lindsey (Dordrecht: Kluwer), 437 Choudhuri, A. R., & Gilman, P. A. 1987, ApJ, 316, 788 ÈÈÈ. 1996, in Stellar Surface Structure, ed. K. G. Strassmeier & J. L. DeLuca, E. E. 1986, Ph.D. thesis, NCAR Linsky (Dordrecht: Kluwer), 237 DeLuca, E. E., & Gilman, P. A. 1991, in Solar Interiors and Atmospheres, Saar, S. H., & Linsky, J. L. 1985, ApJ, 299, L47 ed. A. N. Cox, W. C. Livingston, & S. Matthews (Tucson: Univ. Arizona Schatzman, E., & Praderie, F. 1993, The Stars (erlin: Springer) Press), 275 Scherrer, P., et al. 1996, AAS, 28, 4 Donati, J.-F., rown, S. F., Semel, M., Rees, D. E., Dempsey, R. C., Mat- Schussler, M. 1979, A&A, 71, 79 thews, J. M., Henry, G. W., & Hall, D. S. 1992, A&A, 265, 682 Schussler, M., & Solanki, S. K. 1992, A&A, 264, L13 Donati, J.-F., Semel, M., Rees, D. E., Taylor, K., & Robinson, R. D. 1990, Semel, M. 1989, A&A, 223, 456 A&A, 232, 1 Simon, T. 1992, in ASP Conf. Ser. 26, Cool Stars, Stellar Systems, and the Durney,. R. 1976, ApJ, 204, 589 Sun, Seventh Cambridge Workshop, ed. M. S. Giampapa & J. A. ookbinder (San Francisco: ASP), 3 Durney,. R., & Robinson, R. D. 1982, ApJ, 253, 290 Endal, A. S., & SoÐa, S. 1981, ApJ, 243, 625 Skumanich, A. 1972, ApJ, 171, 565 Fan, Y., Fisher, G. H., & DeLuca, E. E. 1993, ApJ, 405, 390 Soderblom, D., Stau er, J., MacGregor, K., & Jones,. 1993, ApJ, 409, 624 Giampapa, M. S. 1994, in ASP Conf. Ser. 64, Cool Stars, Stellar Systems, Stau er, J. 1994, in ASP Conf. Ser. 64, Cool Stars, Stellar Systems, and the and the Sun, Eighth Cambridge Workshop, ed. J.-P. Caillault (San Francisco: ASP), 509 ASP), 163 Sun, Eighth Cambridge Workshop, ed. J.-P. Caillault (San Francisco: Howard, R. F. 1991, Sol. Phys., 134, 233 Stau er, J., & Hartmann, L. 1987, ApJ, 318, 337 Johns-Krull, C. M., & Valenti, J. A. 1996, ApJ, 459, L95 Stau er, J. R., Hartmann, L. W., urnham, J. N., & Jones,. F. 1985, ApJ, Jones,. F., Fischer, D. A., & Stau er, J. R. 1996, AJ, 112, , 247 Leighton, R , ApJ, 156, 1 Stau er, J. R., Hartmann, L. W., & Jones,. F. 1989, ApJ, 346, 160 Li, J., & Collier Cameron, A. 1993, MNRAS, 261, 766 Stix, M. 1989, The Sun: An Introduction (erlin: Springer) Libbrecht, K. G., & Morrow, C. A. 1991, in Solar Interiors and Atmo- Strassmeier, K. G. 1996, in Stellar Surface Structure, ed. K. G. Strassmeier spheres, ed. A. N. Cox, W. C. Livingston, & S. Matthews (Tucson: Univ. & J. L. Linsky (Dordrecht: Kluwer), 299 Arizona Press), 479 Strassmeier, K. G., et al. 1991, A&A, 247, 130 MacGregor, K., & renner, M. 1991, ApJ, 376, 204 Vogt, S. S., & Hatzes, A. P. 1991, in The Sun and Cool Stars: Activity, Moreno-Insertis, F. 1986, A&A, 166, 291 Magnetism, Dynamos I, ed. D. Tuominen, D. Moss, & G. Rudiger Morrow, C. A. 1988, Ph.D. thesis, Univ. Colorado (erlin: Springer), 297 Parker, E. N. 1955, ApJ, 121, 491 Vogt, S. S., & Penrod, G. D. 1983, PASP, 95, 565 Press, W. H., Flannery,. P., Teukolsky, S. A., & Vetterling, N. T. 1988, Weber, E. J., & Davis, L. 1967, ApJ, 148, 217 Numerical Recipes in C (Cambridge: Cambridge Univ. Press)