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Mon. Not. R. Astron. Soc. 316, 647±656 (2000) A model of slingshot prominences in rapidly rotating stars J. M. Ferreiraw1,2,3 1 School of Physics & Astronomy, University of St Andrews, St Andrews, Fife KY16 9SS Departamento de CieÃncias AgraÂrias, Universidade dos AcËores, Terceira, AcËores, Portugal 3 Centro de AstrofõÂsica da Universidade do Porto, R. das Estrelas, 4150 Porto, Portugal 2 Accepted 2000 March 6. Received 2000 February 14; in original form 1999 October 10 A B S T R AC T We study the stability of an equilibrium model of prominences in rapidly rotating stars. A prominence is represented by an axisymmetric equatorial current sheet embedded in a massless and perfectly conducting coronal plasma. There is an equilibrium between gravity, centrifugal force and Lorentz force acting on the prominence. The energy method of Bernstein et al., in the form presented by Lepeltier & Aly, is used to derive sufficient stability conditions. Using existing observational values for the masses, dimensions and coronal locations of the prominences, we find that surface fields of a few hundred Gauss are required to hold them in a stable equilibrium. The free magnetic energy present in stable equilibrium models is sufficient to feed medium-sized flares but probably not the very energetic ones. We discuss the question of what makes stellar prominences so different from their solar counterparts and what allows them to be detected. We also examine the role of the corotation radius in their formation. Key words: stars: activity ± stars: coronae ± stars: flare ± stars: magnetic fields ± stars: rotation. 1 INTRODUCTION Most phenomena associated with magnetically driven solar activity, such as spots, flares, prominences and hot X-ray coronal loops, have been identified on other late-type stars. However, the analogy between solar and stellar activity is only partial, because the time-scales, dimensions and energies involved can be enormously different. This particular study addresses some questions related to the physics of stellar prominences. These features can generally be described as cool and dense coronal structures and, as on the Sun, are classified into two broad classes: eruptive prominences, or those related to flares, and quiescent prominences. There is evidence for both these types of prominence on magnetically active single and binary late-type stars (for recent reviews see Collier Cameron 1996 and Jeffries 1996). Interestingly, prominence-type features have apparently been detected on a T Tauri type star (Petrov 1990) and a dwarf nova (Steeghs et al. 1996). This seems to indicate that the necessary conditions for the formation of cool clouds in stellar coronae can be met in a wide range of physical systems. Although the study of stellar prominences is a fairly recent topic, it has already resulted in the discovery of a type of quiescent prominence on magnetically active rapidly rotating stars that has not been found on the Sun ± the slingshot prominence (Collier w E-mail: [email protected] q 2000 RAS Cameron & Robinson 1989). These prominences do not lie close to the surface but form instead at heights of a few stellar radii. In the particular case of single systems, they are inferred to form close to, or beyond, the corotation radius (Rk ). This is the point on the equator at which the centrifugal force balances gravity in a corotating atmosphere. The first, and best studied, star where slingshot prominences have been observed is AB Dor. Collier Cameron et al. (1990) derived a mass of 2±6 1017 g; a projected area of 1022 cm2 and a lifetime of a few days for clouds on this star. Stellar slingshot prominences are not peculiar to particular stars but rather a very common property in rapidly rotating latetype stars (Collier Cameron & Woods 1992; Jeffries 1993). Recent observations of two rapidly rotating stars, HK Aquarii (Byrne, Eibe & Rolleston 1996) and He 699 (Barnes et al. 1998) indicate the presence of low-lying absorption features, located well below Rk . At present, it is not clear whether the observed features are solar-like quiescent prominences, or a different surface phenomena. HK Aquarii also seems to have clouds at large radial distances both above and below Rk . Stellar prominences are a potentially important tool for studying the topology and strength of coronal magnetic fields on these stars. They allow one to trace closed-field structures, and the subsequent modelling may allow us to estimate the field strength required to support them in a stable equilibrium. They may also be important for powering energetic stellar flares, and for telling us something about the magnetic braking mechanism operating in these stars. 648 J. M. Ferreira Here, we construct an idealized equilibrium model for stellar slingshot prominences and study its stability properties. We start by considering simple estimates that allow us to justify our assumptions (Section 2). In Section 3 we present current sheet models of stellar prominences, while in Section 4 we analyse their stability. In Section 5 we compare the model with observations. Section 6 is devoted to the discussion of the results and their relevance to our understanding of these phenomena. Finally, we conclude in Section 7. 2 P H Y S I C A L C O N S I D E R AT I O N S In this study we assume that the typical lifetime of a stellar prominence is sufficiently long compared with the other relevant time-scales that a static equilibrium can be considered. For the purposes of illustration we consider a solar-type star M p M( ; Rp M( with a rotation period of P 11 h so that Rk 2:5Rp : Let us first analyse the simple problem of determining the locations where material could accumulate to form a prominence. This is illustrated in Figs 1 and 2, where we calculate the points at which the combined effects of gravity and the centrifugal force (which we denote as effective gravity, ge 2GM p =r 2 er v2 r sin2 u er v2 r sin u cos u eu are in a stable balance along the direction of a prescribed field. We also assume that the field is rigid so that the Coriolis force has no component along the direction of the field and does not contribute to the perturbed force balance. The equilibrium and stability analysis is made using straightforward methods and we only present the dipole case. The field components are Br 2m1 cos u=r3 and Bu m1 sin u=r 3 ; where m 1 is the dipole moment, and the condition F k ge ´ B 0 then implies " 3 # 3 2 Rk cos u sin u 2 0: 1 2 r Figure 1. Equilibrium points for a dipole and a combined dipole and sextupole field. Shown are the equipotential lines (dashed), field lines (light full), unstable equilibrium points (dotted lines) and stable equilibrium points (thick full line). After defining the coordinate u2 sin u2 =r; which varies in the direction perpendicular to a field line, and the coordinate u1 2cos u=r 2 ; which varies along a field line, we obtain the condition for stability, namely Fk u1 ; u2 u 3 r cos u 3 sin u cos u 4 R3k u1 u1 r u1 " # u 3 2 Rk 3 2 sin u sin u 2 , 0; 2 2 u1 r with u u1 ; u2 1 3 cos2 u ; u1 r 2 sin u r u1 ; u2 1 3 cos2 u : u1 2r 3 cos u 3 From this analysis it follows that there is stability at the equatorial plane for r 3 2 r. Rk < 0:87Rk ; 4 3 while the equilibrium surface r 3 2Rk3 =3 sin2 u is unstable. This proves that it is possible to have stable equilibrium points inside the corotation surface, because of the influence of the field shape. Figure 2. Equilibrium points for a quadrupolar field. The parallel dashed lines contain the part of the stellar surface occulted by a prominence formed in the stable equilibrium region, as seen by a distant observer along that direction. If the direction of the observer changes, the projected area of the prominence on the stellar surface can either increase or decrease (cf. Section 6.2). Even in the absence of the centrifugal force, there is no need for a dip in the magnetic field in order for plasma to accumulate in a stable equilibrium when spherical geometry is adopted. It only requires the radius of curvature of the loop to be larger than the q 2000 RAS, MNRAS 316, 647±656 A model of stellar prominences radial coordinate at the loop top (see also Lepeltier & Aly 1996, hereafter LA96). Stable equilibrium points can actually be obtained graphically without requiring a stability analysis. The equilibrium curve intercepts a given field line an odd number of times, say p. This corresponds to p 2 1=2 stable points and p 1=2 unstable points. It is obvious that these stable points correspond to local maxima of the gas pressure distribution along the magnetic field. In the cases shown in Figs 1 and 2, an equilibrium point will be stable if it lies on a field line that is intercepted three times by the equilibrium curves, provided it is also the middle one. From these simple examples we infer that a stable equilibrium along the direction of the field is possible with simple potential fields. Perhaps less obvious is the fact that this can occur below Rk even in the absence of a dipped configuration (see Fig. 1a), while in other cases a stable equilibrium is only possible further way from Rk (e.g. Fig. 2). We therefore define the effective corotation radius, Rk e, as the point at which there is a stable equilibrium along the field. Assuming that prominences form on some of these equilibrium surfaces, they will, in general, have a curved shape ± with the exception of field configurations that are symmetric with respect to the stellar equator. Let us now turn our attention to the equilibrium and stability of the prominence material in the direction perpendicular to the field. At the equatorial plane, the ratio of the gas pressure gradient force to the effective gravity in the radial direction, gee, is dP/dr p RT p ; < rgee r l0 rgee l0 gee 5 where l0 is the typical radial extent of the prominence and Tp its temperature. At the corotation radius the pressure gradient is the only non-magnetic force present. However, if prominence material forms a fraction of a stellar radius away from Rk, then for typical parameters T p 104 K; l0 < 108 ±109 m the effective gravity dominates over the pressure gradient and the above ratio is much smaller than 1. We can then neglect the gas pressure gradient force from the equilibrium equation and need only look at the balance between the Lorentz force and the effective gravity. We can estimate the field strength required to hold the prominence in stable equilibrium if we assume that the local magnetic energy is greater than the generalized gravitational potential energy: 0 0 r r p B2 GM p v2 r 2 >r : 2m r0 2 r 0 Rk 6 If the prominence lies sufficiently far from Rk, the condition is similar to imposing a local AlfveÂn velocity greater than the rotational velocity, vA > vr p : Writing the unknown prominence mass density in terms of the mass density of solar prominences, r r=r( r( and taking r( 1:6 10213 g cm23 ; we obtain a minimum field strength B 3Rp 5 r=r( 1=2 G and B 4Rp 20 r=r( 1=2 G; which is similar to the field strength measured for solar prominences. Assuming a dipolar field one obtains surface fields of Bs 3Rp 135 G and Bs 4Rp 1:3 kG; respectively. 3 AXISYMMETRIC MASSIVE CURRENT SHEET MODEL The modelling of solar prominences as infinitely thin plane sheets of mass and current was pioneered by Kippenhahn & Schluter (1957). Thereafter, several other models were developed (e.g. Malherbe & Priest 1983; Wu & Low 1987; Low 1986; LA96). q 2000 RAS, MNRAS 316, 647±656 649 Here we follow a similar approach and represent stellar prominences by an equatorial axisymmetric massive current sheet supported in a potential magnetic field. Under our assumption of a negligible gas pressure gradient force, the equilibrium is given by j B 2rgeff ; c 7 which on the equatorial current sheet reduces to s Bu lgee ; c 8 where l is the surface mass density r ld cos u=r and s is the surface current { j sd cos u=ref }: After prescribing the surface current distribution, the total magnetic field is computed by assuming that either the field is closed everywhere or it opens up at a given radius in order to simulate the effect of a wind. The surface mass density is then determined using the equilibrium equation. Writing the axisymmetric field in terms of the flux function S, 1 1 S S B ;2 ;0 ; 9 r sin u r u r and considering first the simpler case of a completely closed field, we have for a given current distribution 2p 1 S r; u s r 0 L1 r; r 0 L2 r; r 0 dr 0 ; 10 c 0 p with the kernel Li r; r 0 2 rr 0 sin u 2 2 k2i K ki 2 2E ki = 0 2 ki, i 1; 2 and k1 4rr 0 sin u= r2 r 2 2rr 0 sin u; k22 4rr 0 sin u= R2* rr 0 =Rp 2 2rr 0 sin u: K and E are the complete elliptic integrals of the first and second kind, respectively. The kernel L2 results from assuming that the field resulting from the prominence current sheet does not penetrate into the star, i.e. one must have S r Rp 0: On this field, calculated by numerically integrating equation (10), we superpose a potential field. Care must be taken in the choice of the current distribution to ensure a smooth behaviour of the surface mass density as it crosses the corotation radius, i.e. the quantity limr!Rk 2 s rBu =ge must exist. One can also construct more realistic models by including the opening of the field at some radius r d and computing the total field self-consistently (Aly 1980; Riffert 1980; Low 1986; LA96). This mathematical problem involves solving dual integral equations and the reader is referred to the above references for details. Here we consider the current profiles: s1 r s0 f 1 d 1 ; d 2 d 2 2 r r 2 d1 x r; r2 s2 r s0 f 2 d 1 ; d 2 ; Rk d 2 2 r r 2 d 1 r 2 Rk x r; 11 12 where s 0, d1, d2 are constants and f1,2 are normalizing functions given by f1 4d 1 d 2 ; d 2 2 d 1 2 f2 1 22d1 d 2 Rk h 2d 2 2 d 1 2 Rk 2 h 27 d 1 d 2 2 2Rk h; 13 14 where h d 21 d 22 Rk2 2 d 2 d 1 2 d 2 Rk 2 d 1 Rk and x r 1 650 J. M. Ferreira Figure 3. Magnetic field lines created by the current profile s 2 under the condition that no field lines cross the equatorial plane for r . d or extend to infinity. Here, the current sheet extends from d 1 2:0Rp to d 2 3:0Rp and d 4:0Rp : The massless current sheet is represented by a straight dashed line. Figure 4. Prominences as massive current sheets. The prominence is supported in a dipole field. It extends from d 1 3:0Rp to d 2 4:0Rp ; has the current profile s 1 and the field opens up at d 5:0Rp : The represented configuration is stable (cf. Section 5). for d1 , r , d2 and zero otherwise. These current profiles represent prominences extending from r d 1 to r d 2 : The current profile s 1 is the same as used by LA96, where the magnetic field is calculated in detail, while s 2 is used to represent prominences crossing Rk . The calculations of the field for the latter case are lengthy but can be obtained by application of the same procedure. The field resulting from the current profile s 2 and subject to the appropriate conditions is shown in Fig. 3. Illustrative examples of prominence models for each current profile are shown in Figs 4 and 5. The neglect of the gas pressure gradient for prominences crossing Rk can be justified on the basis that although this force is dominant over the effective gravity close to Rk , the low gas temperature makes its effect negligible on the magnetic field. Although it is desirable to build more realistic models that include the gas pressure, the simple models employed here allow us to understand some of the more basic properties of prominence equilibrium and to determine their stability conditions. 4 S TA B I L I T Y C O N D I T I O N S The energy method of Bernstein et al. (1958) has been applied successfully to the study of solar prominences embedded in a vacuum (Anzer 1969; Wu 1987). More recently, Aly, Amari & Colombi (1990) and Spruit & Taam (1990) have shown that assuming the coronal plasma to have an infinite conductivity and that the field lines are tied to the photosphere results in less stringent stability conditions. Aly, Colombi & Lepeltier (1994) and LA96 presented a rigorous derivation of the line-tying effect in Cartesian and cylindrical configurations. In a rotating system there is no energy principle, because of the effect of the Coriolis force. Nevertheless, by simply replacing gravity by ge and neglecting the Coriolis force, one can still apply the energy method to obtain sufficient, though not necessary, linear and ideal stability conditions (Rogers & Sonnerup 1986). This restriction means that if the sufficient conditions are obeyed then the equilibrium is stable. Alternatively, the equilibrium can either be stable or unstable, with the possibility of overstability. We now follow closely the analysis of LA96 in order to determine the stability conditions.Noting that in the particular models we are considering p 0; r ld cos u=r and B is current free in the corona surrounding the prominence and the massless current sheet, the second-order change in energy resulting from a perturbation j is given by d2 W d2 W p d2 W c : 15 The first term represents the change of generalized gravitational energy in the prominence and the second term is the change of energy in the corona. We then have 1 d2 W p l j ´ 7 ´ ge ´ j dS; 16 2 Figure 5. Prominences as massive current sheets. The prominence is supported in a combined dipole and sextupole field, with h ; m3 =2m1 R2p 0:55: It extends from d1 2:0Rp to d 2 3:0Rp ; has the current profile s 2 and the field opens up at d 4:0Rp : The represented configuration is stable (cf. Section 5). where dS represents an element of area of the prominence sheet. Using equation (8.64) from Roberts (1967) we obtain 1 d2 W c d2S W c d2V W c j7 j Bj2 dV 8p c 1 B2 j ´ 7j 2 j 7 ´ j ´ dS; j ´ 7B j B 2 2 8p S 17 q 2000 RAS, MNRAS 316, 647±656 A model of stellar prominences with the surface integral being performed over both sides of the surfaces of the prominence and massless current sheets. Using the expression for ge and the equilibrium equation, one finds ! ( 2 1 Bu Br jf j2u 1 gee 2 2 d Wp j 4p p 2 gee r r r r ) 4pv2 l rj2u dS; 18 where Br Br r; u p=2 2 Br r; u p=22 2 4p=cs represents the jump in the radial field across the current sheet. In addition, the surface integral in equation (17), upon using the appropriate conditions across the interface, reduces to ! j2f 1 Br 2 Bu 2 2 2 Bu dS W c j Br ju 2 Bu Br dS; 19 8p r r r r p with no contribution from the massless current sheet as Bu 0 there. The energy principle allows us to state that there is stability if d2 W > 0; with 1 Br 1 gee 2 2 Br j2r dW 2Bu 8p gee r r p Br rBu 2 4plv j2u dS d2V W c : r r 20 According to LA96, the volume integral in the previous equation can be bounded from below by an integral over the prominence and this allows us to deduce the following sufficient stability conditions: Br 1 gee r 2 jBu j3 2Bu >0 2 Br 2 K r gee r l jBu j2 , gee > 0; 21 r jBu j 2pK Br rBu r 2 jBu jBr 2 4plv2 > 0; r r 2K 22 where K t : l 0 r 2 sin2 uB ds; 23 with the integral being performed along the field and l(t) representing the half-length of a field line connecting to the prominence. These stability criteria are similar to the ones derived for solar prominences, and in the absence of rotation they reduce to the necessary and sufficient conditions derived in LA96. Condition (21) states that the mass density per unit of magnetic flux cannot decrease in the direction of ge at a very fast rate, and this is associated with the magnetic Rayleigh±Taylor instability. Furthermore, condition (22) is related to the field curvature. The term 4plv 2 is stabilizing and allows the possibility of a stable equilibrium for r , Rk without the presence of a dip at the loop top. A simple calculation shows that in the limit of small currents and for an external dipole field this condition reduces to equation (4) above, which was deduced by a simpler method. It is convenient to compare our analysis with that of Spruit & Taam (1990). These authors considered the stability of a rotating sheet using a normal mode analysis restricted to short-wavelength q 2000 RAS, MNRAS 316, 647±656 651 perturbations. Therefore, the stability conditions they derive are necessary but not sufficient. However, if the line-tying effect is neglected, our sufficient stability conditions are equivalent to their necessary conditions [this requires the correction of an error in their equation (82) that propagates to the final condition] so that they are both necessary and sufficient. One concludes that, in this case, rotation does not add any global modes of instability. It is likely that in the presence of line-tying the conditions we have derived are also necessary. However, even if this is confirmed, it is worth noting that with more general equilibrium models (e.g. prominences located outside the equatorial plane) one only expects the use of the energy principle to provide sufficient stability conditions. 5 A P P L I C AT I O N T O O B S E R VAT I O N S 5.1 Stable configurations Here we apply the stability conditions to the equilibrium models constructed in Section 3. Some observational constraints are available for the prominences on the star AB Dor (cf. Section 1) and we take these to determine typical values. However, note that one expects the properties of the prominences to differ from star to star, and even on the same star. We consider a prominence with a mass M p 5 1017 g; located anywhere between 2 and 5Rp and with a radial extent of 0.5±1Rp. In spite of the fact that the model is axisymmetric, we assume that the prominence mass is concentrated in a region of dimension L in the azimuthal direction. The procedure that determines the minimum surface field strength necessary for a stable equilibrium is now straightforward. From the equilibrium equation (8) and our assumption about the length of the prominence, we obtain 2p L Bu s r Mp r dr: 24 c d2 gee It is convenient to define n 2ps0 R3p =m1 c: As the maximum value of n is determined from the stability conditions, equation (24) allows us to derive the surface field strength for stability r r m1 nst G gp R( B0 ; 3 B1 g( Rp n Gst R* ssrs! Mp d 2 Lst f 1;2 ; 25 17 L f st1;2 5 10 g d st2 where G G d; d 1 ; d 2 ; Rk ; m3 ; s and B1 represents the dipole surface field strength when using some standard values for the parameters of the problem. We now apply this procedure to the different current profiles. 5.1.1 Prominences above Rk Here we consider the current profile s 1 for a prominence located above Rk and supported in a dipole field, as illustrated in Fig. 4. In this case the curvature stability condition is always obeyed and the prominence is only prone to the interchange instability at its top. The dependence of the minimum field strength required for stability, B0, on the prominence radial extent, location and opening field radius is shown in Table 1. 5.1.2 Prominences crossing Rk Let us now consider the case of a prominence with the current 652 J. M. Ferreira Table 1. Minimum field strength for stability of a prominence with the current profile s 1. The prominence extends from r d1 to r d 2 and has a length L Lst 1R* in the azimuthal direction. The field is assumed open at a distance r d: The first row represents the standard case. Note that if we only impose the condition that equilibrium holds we obtain B0 39 G for the standard case. d(Rp) 1 5.0 4.0 1 5.0 4.0 1 5.0 1 6.0 d1(Rp) d2(Rp) n nst f1 f st1 G Gst B0(G) 2.5 2.5 2.5 2.5 2.5 2.5 3.0 3.0 4.0 4.0 3.5 3.5 3.5 3.0 3.0 3.0 4.0 4.0 5.0 5.0 1.0 0.67 0.34 0.82 0.64 0.49 0.53 0.27 0.19 0.08 1.0 1.0 1.0 0.5 0.5 0.5 0.9 0.9 0.8 0.8 1.0 1.0 1.1 1.8 1.9 2.0 2.3 2.5 5.1 5.7 151 B1 193 284 154 180 215 435 669 1406 2402 Table 2. Minimum field strength of the dipole component for stability of a prominence with the current profile s 2 with d 4:0Rp ; d1 2:0Rp and d2 3:0Rp : If we only impose the condition that equilibrium holds, we obtain B0 43 G for the standard case. h 0.35 0.45 0.55 0.65 n nst G Gst B0(G) 1.0 0.80 0.65 0.29 1.00 1.07 1.17 1.29 317 B1 380 462 762 profile s 2 that crosses the corotation radius. The lower part of the prominence located below Rk is in general unstable to the curvature instability if the background field only has the dipole component. We therefore follow Low (1986) and LA96, and add a sextupole field of moment m 3 that allows for the existence of dipped field lines, as illustrated in Fig. 5. Here we only consider the case d 4:0Rp ; d 1 2:0Rp and d 2 3:0Rp : We find that the equilibrium condition requires h ; m3 =2m1 R2p , 0:80; while the curvature stability is obeyed only for h . 0:33: Naturally, these values are a function of the current sheet strength, but for the values of n allowed by condition (22) this dependence is negligible and we omit presenting it. Table 2 gives the minimum value of B0 as a function of h for stability to be attained. Here, the prominence is prone to this instability both at its top and its bottom. These lie below and above Rk , respectively. For the case considered, the condition is broken first at the prominence upper end. The examples presented show that it is possible to construct stable field configurations, capable of supporting a prominence, for reasonable field strengths. The fact that the model with s 2 requires higher field strengths can easily be understood if we remember that in this configuration there is a neutral point below the prominence. As a consequence, Bu is comparatively smaller at the location of the prominence, which implies a smaller stabilizing effect. 5.2 Can a slingshot prominence eruption drive very energetic flares? The energy of the flares observed on these stars can be as high as 5 1034 erg (e.g. Vilhu et al. 1993), although flares with much lower energies are also commonly observed. We now consider the possibility that the eruption and probable ejection of these titanic prominences could be the source of the flare energy. Let us consider the simpler case of d ! 1 so that coronal electric currents are restricted to flowing only in the prominence. The magnetic energy is given by 1 1 E B2 dV B2 B2np 2Bnp ´ Bp dV 8p V 8p V p 1 B2 B2np dV; 26 8p V p where Bp and Bnp represent the potential and non-potential components of the field, respectively, and we have assumed that the field resulting from the prominence current, Bnp, does not penetrate the stellar surface. Let us assume that after a prominence eruption the field relaxes to the potential configuration Bp. Then, the energy released, or free magnetic energy, is simply 1 DE B2 dV: 27 8p V np Using the virial relation, we can obtain a simpler expression for evaluation of this free energy: 1 1 r ´ { 7 B B} dV B2np dV 8p V 4p 1 B2 2Bnp ´ Bp r ´ dS: 28 8p S np The first integral on the right-hand side can also be simplified to an integral along the prominence, s 22p rBu r 2 dr: 29 c Using the equilibrium relation, one can also express this last integral as 2W 2 2K; where W is the gravitational energy and K the kinetic rotational energy. The u -component of Bnp on the stellar surface can easily be obtained after using equations (9) and (10). We find that the free magnetic energy in stable equilibrium models of prominences is of the order of a few times 1030 ±1031 erg depending on the prominence location and radial extent and increasing linearly with the prominence mass. If we only impose equilibrium but not stability then the free energy increases by roughly one order of magnitude. By comparison, the kinetic energy of rotation is typically a few times 1032 erg. This indicates that in these prominences there is enough energy to explain the low- and medium-energy flares but probably not the very energetic ones. Nevertheless, one cannot rule out the possibility that the eruption of a slingshot prominence can trigger energy release in the very large volume around and underneath the prominence. However, this requires the presence of electric currents outside the prominence, which have not been considered in the present work, and further modelling is required to test this hypothesis. Therefore, the strong flares result either from the interaction of magnetic fields near the surface or from the ejection of prominences located far from the surface. Note that although q 2000 RAS, MNRAS 316, 647±656 A model of stellar prominences the field is weaker far from the surface, the eruption can comprise a very large volume. 6 6.1 DISCUSSION Comparison with observations In spite of the many simplifying assumptions, our model supports the view that stable equilibrium of prominences is indeed possible. The analysis presented in Sections 4 and 5 shows that the equilibrium models of stellar prominences are only prone to local instabilities of the type similar to their solar counterparts. Average surface field strengths of a few to several hundred G are required for stability against the interchange type instability. Owing to the high magnetic activity observed on these types of stars, the required values are not unrealistic. In the particular case of AB Dor, Zeeman Doppler imaging indicates that about 20 per cent of its surface is covered with high magnetic flux with a typical value of 500 G and peaks of up to 1.5 kG (Donati & Collier Cameron 1997). It must be stressed that if a prominence is inferred to lie at large radii but below Rk , it cannot be concluded that rotation is unimportant for its formation and equilibrium. Besides any observational errors in determining the prominence location (e.g. uncertainties in the stellar radius or inclination; the effect of the finite extent of the prominences in the different directions), it may be the case that it has formed in a large-scale dipole-type field where mass can accumulate below Rk . In the particular case of HK Aquarii, material can accumulate in a dipole field for r . 2:8Rp in spite of the fact that Rk 3:2Rp : This effect can be enhanced because of the fact that the weight of the prominence material accumulating below Rk will cause sagging of the field lines and compression of the field underneath it. This, in turn, could allow new material to accumulate in a stable equilibrium at even smaller radial distances. 6.2 Why can we observe stellar prominences? If we assume an analogy between solar and stellar prominences, then we expect stellar prominences to be thin structures. If so, how can they be observed? The key may be the fact that if prominences 653 have a significant radial extent, their projected area on the stellar surface along the direction of the observer is a substantial fraction of the total stellar surface area. Hence a significant fraction of the chromospheric flux is blocked (see Fig 2). Furthermore, note that in the region where stellar prominences form the pressure scaleheight is about an order of magnitude larger than the solar value. Thus one expects stellar prominences to have a greater width than solar ones (Fig. 6). It is also possible that the assumptions made in the present work are incorrect and stellar prominences are very different from solar quiescent prominences in that they may be very dynamical phenomena, which are never in a state of (quasi-)equilibrium. One possibility is that these clouds could be, or result from, huge Ha post-flare loops (Jeffries 1993). Although the plausibility of this scenario still needs to be considered in detail,1 we briefly point out the implications based on two different, but not mutually exclusive, possibilities. (i) Dynamical phenomena. (a) Cool post-flare loops are thermally destroyed by heating and conduction on a short time-scale. As the reconnection region moves outwards, new cool loops can be formed above older ones so that at different rotation rates one may be seeing different Ha loops. (b) For loops below Rk e one expects small (undetectable?) downflows if only the top part of the loop is observed, or flows as high as 100 km s21 if the lower parts are visible and the velocity is close to free-fall. (c) There is no preferential location for the Ha clouds to be observed. (ii) Formation mechanism for quiescent prominences. (a) Cool post-flare loops can evolve into a prominence in quasiequilibrium to which our model can be applied. (b) This mechanism provides an attractive method for feeding mass from the chromosphere to the summits of large loops on realistic time-scales (shorter than the radiative time-scale of the order of 1 d). (c) If the magnetic field evolves further into an unstable configuration, then this could lead to a new flare. (d) Prominences are preferentially located near or above Rk e. Recently, Eibe et al. (1999) found evidence for a large-scale mass downflow on the late-type fast rotator BD 228 4409. These flows were interpreted as being caused by cool material falling in post-flare loops. They argue that these observations support the view that slingshot prominences cannot exist outside the equatorial plane. In contrast, we point out that if a prominence were to form in a quadrupolar field on this star, and assuming its inclination to be i 508; it would not be observable. In Appendix A we present a simple model of a prominence as a circular line current located outside the equatorial plane and in the presence of a quadrupolar field. We show that stable equilibrium is possible if the surface field is not too weak, even though the model does not address the problem of local instabilities. Whether these intermediate-latitude prominences occur on this or other stars depends on the geometry of the field at the time of observation. Figure 6. Schematic representation of a prominence extending from 2.2Rp to 3.2Rp and embedded in a dipole field of our standard star. Its width at each field line is twice the pressure scaleheight at T p 10 000 K: q 2000 RAS, MNRAS 316, 647±656 1 Recent work has questioned the existence of such large L > Rp loops, claimed to be present in some previous analyses of flares on several stars (Favata, Micela & Reale 2000). 654 J. M. Ferreira During most of the solar cycle there is a broad distribution of coronal mass ejections with latitude (Hundhausen 1993), indicating that the large-scale field is not dipolar. If we use the Sun as a guide, we expect these large-scale stellar prominences to form in a broad latitude interval and not just at the equator.2 7 CONCLUSIONS In this work we have studied the equilibrium and stability of an equatorial current sheet model of a large-scale stellar prominence. We have derived sufficient stability conditions that have the same physical nature as the necessary and sufficient conditions found for solar prominences. That is, the mass density per unit flux cannot decrease in the direction of the effective gravity at a very fast rate, a condition associated with a magnetic Rayleigh±Taylor type instability, and the field curvature has to allow for stability along the direction of the field. Using observational constraints on the properties of the prominences, we find that field strengths of a few to several hundred Gauss are required to hold them in a stable equilibrium. This condition appears to be easily obeyed for these fast rotators according to the available observations. We also find that the free magnetic energy of stable equilibrium prominence models is over two orders of magnitude lower than the more energetic stellar flares detected on these stars. Nevertheless, it is possible that their eruption acts as a trigger for the energy release associated with volume currents in the surrounding coronal plasma. The majority of the observed stellar prominences differ from their solar counterparts in that they are located several stellar radii above the surface. If rotation plays a crucial part in their force balance then one does not expect to observe these absorbing clouds on stars with low inclination angles. However, we argue that there is no reason why slingshot prominences should not be observable at intermediate latitudes, as appears to be the case in AB Dor if its inclination is indeed close to 608 as proposed by Collier Cameron & Foing (1997). Further observations of prominence complexes in a broader sample of stars are required to answer some pertinent questions. What allows these absorbing clouds to be formed so far away from the surface, a phenomenon never observed on the Sun? Are we seeing the same type of event on all stars? What is the role played by rotation in the formation and mechanical equilibria of these prominences? REFERENCES Aly J. J., 1980, A&A, 86, 192 Aly J. J., Amari T., Colombi S., Dubois M. A., Bely-Dubau F., 1990, Plasma Phenomena in the Solar Atmosphere, Proc. of the CargeÁse Workshop, Editions de la Physique, Les Ulis Aly J. J., Colombi S., Lepeltier T., 1994, ApJ, 432, 793 Anzer U., 1969, Solar Phys., 8, 37 Barnes J. R., Collier Cameron A., Unruh Y. C., Donati J. F., Hussain G. A. J., 1998, MNRAS, 299, 904 Bernstein I. B., Frieman E. A., Kruskal M. D., Kulsrud R., 1958, Proc. R. Soc. London, Ser. A, 244, 17 Bratygin V. V., Toptygin I. N., 1964, Problems in Electrodynamics. Academic Press, London Byrne P. B., Eibe M. T., Rolleston W. R. J., 1996, A&A, 311, 651 Collier Cameron A., 1996, in Strassmeier K., Linsky J., eds, IAU Symp. 176, Stellar Surface Structure. Kluwer, Dordrecht, p. 449 Collier Cameron A., Foing B. H., 1997, Obs., 117, 218 Collier Cameron A., Robinson R. 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H., 1967, An Introduction to Magnetohydrodynamics. Longmans, London Rogers B., Sonnerup B. N., 1986, J. Geophys. Res., 91, 8837 Spruit H. C., Taam R. E., 1990, A&A, 229, 475 Steeghs D., Horne K., Marsh T. R., Donati J. F., 1996, MNRAS, 281, 626 Vilhu O., Tsuru T., Cameron A. C., Budding E., Banks T., Slee B., Ehrenfreund P., Foing B. H., 1993, MNRAS, 258, 360 Wu F., 1987, ApJ, 320, 418 Wu F., Low B. C., 1987, ApJ, 312, 431 AC K N O W L E D G M E N T S APPENDIX A: LINE CURRENT MODELS OF STELLAR PROMINENCES Most of this work was done while I held a PPARC postdoctoral position at the University of St Andrews, for which I am very grateful. I would like to thank Dr Moira Jardine for constructive comments on the manuscript. I am grateful for the warm hospitality of the Centro de AstrofõÂsica in Oporto, where the writing up was completed. I acknowledge the useful remarks of the referee, which have helped to improve the contents of the paper. Finally, I am very grateful to Dr David Brooks for his thorough corrections to the paper. A prominence is represented by a massive circular line current of radius b, with its centre at the rotation axis. Its magnetic field can be expressed in terms of the f -component of the magnetic potential vector 1 1 B 2 r sin uA; 0 ; r sin uA; 2 A1 r sin u u r sin u r which obeys the equation 2 Some of these fast rotators show evidence for polar spots. If these spots are unipolar and contain a significant fraction of the total surface magnetic flux, one expects the summits of the large loops to lie near the equator. In this case stellar slingshot prominences should be confined close to the equatorial plane. 72 A 2 r2 A 4p d r 2 t 2 d cos u 2 cos u0 ; I sin u c r A2 subject to the boundary condition that no flux penetrates into the star, A r Rp 0: Here, t h2 b2 1=2 and h is the distance q 2000 RAS, MNRAS 316, 647±656 A model of stellar prominences 655 from the line current to the equatorial plane. This has the solution 4Ib T k1 T k2 p A 2 p ; A3 c 2 rb sin uk1 2 rb sin uk2 where k21 4rb sin u=r2 h2 b2 2 2r h cos u 2 b sin u; k22 4rb sin u=r 2 b2 h2 =R2p R2p 2 2r h cos u 2 b sin u; T k 2 2 k2 K k2 2 2E k2 and K, E are the complete elliptic integrals of the first and second kind, respectively. The first term represents the potential resulting from the line current and the second term the potential resulting from the induced image currents on the stellar surface (cf. Ferreira & Jardine 1995, hereafter Paper I). We now consider this line current in the presence of a background potential field and study its equilibrium and stability properties. A1 Equilibrium and stability in a dipole field This particular case was studied in Paper I and it was concluded that a stable equilibrium is possible for typical surface fields and prominence locations. However, we incorrectly neglected the magnetic repulsion of the circular loop on itself. This force per unit length can be obtained using the concept of self-inductance (e.g. Bratygin & Topygin 1964) and is given by 2 I 1 8b F r1 21 ; A4 log c b e where e is the small but finite radius of the current loop e ! Rp ; and, because of our assumption of small radius e , we have neglected the contribution from the internal magnetic field, which depends strongly on the current profile inside the loop. One may argue that this force appears only because of the assumed azimuthal symmetry, and that real stellar prominences are unlikely to have this property. However, once we have assumed the model geometry we have to include it for consistency. There is another repulsion force resulting from the induced surface currents. At the equator, this is given by F r2 I=c2 R2 r; with T k2 R2p 2 r2 E k22 2 2 K k : A5 2Rp 2 R2 r 2 Rp R p r 2 2 1 2 k22 It has the properties r Rp h; r @ Rp ; h ! Rp ; F r2 < F r2 < 2 I 1 ; c h 2 3 I pR p ; c r4 A6 A7 so that in the limit where the current loop is close to the surface one recovers the plane geometry case (Kuperus & Raadu 1974), while at large distances this repulsion force decreases very rapidly and is much smaller than Fr1. The equilibrium equation for a circular loop on the equatorial plane, with r b; is given by F M I m1 g 2 F r1 F r2 0; 2pb ee c b3 A8 where M is the total mass of the prominence (assumed constant) and m 1 is the dipole moment. The first term represents the generalized gravity, the second term is the force resulting from the q 2000 RAS, MNRAS 316, 647±656 Figure A1. The double-valued equilibrium current as a function of r (in Rp) for B0 100 G; M 5 1017 g and e 1024 Rp (full line). The dashed line represents the equilibrium current when the repulsion forces are neglected. Table A1. The maximum equilibrium values of r for which the stability conditions are obeyed, using the equations both with and without the repulsion forces. e (Rp) B0(G) 1022 exact 1024 exact 1022 approx. 1024 approx. 1000 500 100 r 14:1 r 10:8 r 5:8 r 12:9 r 9:8 r 5:3 r 15:0 r 11:4 r 6:2 r 13:6 r 10:4 r 5:5 dipole field and the last term combines the repulsion forces on the circular loop resulting from itself and the image surface currents. The equilibrium equation has two solutions, as the particular case shown in Fig. A1 illustrates. To study the stability of these equilibria, one ignores the effect of the Coriolis force and determines sufficient stability conditions for perturbations along and perpendicular to the background field, taking into account the variation in the current as the prominence is perturbed. This variation is determined by the condition that magnetic flux is conserved between the stellar surface and the surface of the filament. Therefore, one requires dF , 0; db e F B ´ dS 22pr A Ad jrb2 rRp a constant; A9 A10 with Ad m1 sin u=r 2 : We find that thepequilibrium is stable in the direction along the field for r . Rk 3 2=3 (cf. Section 2). The sufficient condition for stability perpendicular to the field is obeyed for the lower equilibrium branch solution only until a maximum height (as summarized in Table A1). Beyond this height the condition is not obeyed, and this is true also for the upper equilibrium branch. Although we have neglected the stabilizing effect of the Coriolis force, we expect most of the equilibria for which the condition is not obeyed to be unstable. This is because the magnetic forces start to dominate over the inertial forces. One concludes that inclusion of the repulsion force Fr1 only changes the results obtained in Paper I significantly for the cases 656 J. M. Ferreira where the equilibrium height is close to its maximum value, and for the upper equilibrium branch. A2 Equilibrium and stability in a quadrupole field It is desirable to show that a prominence, represented by a current loop, can exist in stable equilibrium at intermediate stellar latitudes. For this purpose we consider a quadrupolar field 2 4 3 cos u 2 1 2 cos u sin u B B0 Rp ;0 ; A11 ; r4 r4 and a loop carrying current I of radius b at position (r, u ) so that b r sin u: The equilibrium equations along and perpendicular to the background field are GM p 3 cos2 u 2 1 2 mV2 r sin 2 u 5 cos2 u 2 1 r2 2 I 1 8b 2 2 1 5 cos2 u 2 1 sin u log c b e F u1 m F ur21 0; Figure A2. Current loop in a quadrupolar field for B0 100 G; M 5 1017 g and e 1024 : The equilibrium path as the current increases is shown (dashed line). The field lines are also shown (thin full lines), as are the points for which ge ´ B 0 (thick full lines). A12 and F u2 mV2 r sin u 5 cos2 u 2 3 cos u m GM p sin 2u r2 B0 Rp4 I 5 cos4 u 2 2 cos2 u 1 r4 2 I 1 8b 2 1 5 cos2 u 2 3 cos u log c b e F ur22 0; A13 where we have defined m M=2pb; and u1 3 cos2 u 2 1=r3 ; u2 cos u sin u=r 2 as coordinates along and perpendicular to this u field. Also, F r2 1;2 represents the repulsion force resulting from the induced surface currents along the direction u1,2. The solution of these coupled equations shows that the current is a double-value function of r (similar to the case of a dipole) but that there is only one equilibrium solution at a given point (r, u ), as shown in Fig. A2. To address the question of stability, we consider perturbations along and perpendicular to the background field and impose the constraint that the magnetic flux between the stellar surface and the surface of the filament must remain constant: Fu1 , 0; u1 F constant; A14 Fu2 , 0; u2 F constant: A15 The magnetic flux through a given open surface is given by F B ´ dS; A16 and if the normal to this surface is parallel to eu this can be expressed as F 22pr sin u A Aq jxx21 ; A17 where x1 Rp ; u is the line resulting from the intersection of this surface with the stellar surface and x2 r 2 e; u is at the boundary of the filament. In the particular case we are Table A2. The maximum equilibrium values of r for different values of B0 and e in a quadrupole field, for which the sufficient stability conditions are obeyed. e (Rp) B0(G) 1022 1024 1000 500 100 r 6:1 r 5:1 r 3:4 r 5:7 r 4:8 r 3:2 considering, after the filament is perturbed in the direction perpendicular to the field, moving to x3 r dr; u du; the normal to the surface is no longer eu because x1 must remain fixed. The change of flux can still be calculated using dF f x3 ; x1 2 f x2 ; x1 f x3 ; x4 2 f x2 ; x1 f x4 ; x1 d22pr sin u A Aq jxx21 dFs ; A18 where f (xi, xj) represents the magnetic flux through points xi and xj, x4 Rp ; u du denotes the auxiliary point on the stellar surface and dFs 22pB0 R2p 3 cos2 u 2 1sin udu represents the flux through the stellar surface between x4 and x1. The first condition is obeyed at every equilibrium point except for those located close to the equator, as the quadrupolar field is nearly radial there. In this region the repulsion forces are negligible and there is stability for r > 1:14Rk : The other condition is never obeyed for the equilibrium region, where r(u ) is a decreasing function. In addition, as in the dipole case, the upper part of the equilibrium region, where r is an increasing function of u , does not obey the stability condition. Beyond these maximum values one expects these equilibria to be unstable. These results are summarized in Table A2 for some representative parameters. We conclude that equilibrium models of stellar prominences located either at the equatorial plane or outside it, and modelled as circular current loops, are stable if the surface field is not too weak or the prominence does not lie too far from the surface. This paper has been typeset from a TEX/LATEX file prepared by the author. q 2000 RAS, MNRAS 316, 647±656