Download A model of slingshot prominences in rapidly rotating stars

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 ˆ M( ;
Rp ˆ M( † with a rotation period of P ˆ 11 h so that Rk ˆ 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 ˆ 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 ˆ 2m1 cos u=r3 and Bu ˆ m1 sin u=r 3 ; where
m 1 is the dipole moment, and the condition F k ˆ ge ´ B ˆ 0 then
implies
"
3 #
3 2
Rk
cos u sin u 2
ˆ 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 ˆ sin u2 =r; which varies in the
direction perpendicular to a field line, and the coordinate u1 ˆ
2cos u=r 2 ; which varies along a field line, we obtain the condition
for stability, namely
­Fk …u1 ; u2 †
­u
3
­r
ˆ 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
ˆ
;
­u1
r 2 sin u
­r…u1 ; u2 † 1 ‡ 3 cos2 u
ˆ
:
­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 ˆ 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
;
<
ˆ
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 ˆ 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 ˆr p
B2
GM p v2 r 2
‡
>r
:
2m
r0
2 r 0 ˆRk
…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 ˆ …r=r( † r( and taking r( ˆ 1:6 10213 g cm23 ; we obtain a
minimum field strength B…3Rp † ˆ 5…r=r( †1=2 G and B…4Rp † ˆ
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 † ˆ 135 G and Bs …4Rp † ˆ 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 ˆ 2rgeff ;
c
…7†
which on the equatorial current sheet reduces to
s
Bu ˆ lgee ;
c
…8†
where l is the surface mass density ‰r ˆ ld…cos u†=rŠ and s is the
surface current { j ˆ ‰sd…cos u†=rŠef }:
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
Bˆ
;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† ˆ
s…r 0 †‰L1 …r; r 0 † ‡ L2 …r; r 0 †Š dr 0 ;
…10†
c 0
p
with the kernel Li …r; r 0 † ˆ 2 …rr 0 sin u†‰…2 2 k2i †K…ki † 2 2E…ki †Š=
0
2
ki, i ˆ 1; 2 and k1 ˆ 4rr 0 sin u=…r2 ‡ r 2 ‡ 2rr 0 sin u†; k22 ˆ
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 ˆ Rp † ˆ 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…r†Bu =ge must
exist.
One can also construct more realistic models by including the
opening of the field at some radius r ˆ 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† ˆ s0 f 1 …d 1 ; d 2 †
…d 2 2 r†…r 2 d1 †
x…r†;
r2
s2 …r† ˆ 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 ˆ
4d 1 d 2
;
…d 2 2 d 1 †2
f2 ˆ
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 ˆ …d 21 ‡ d 22 ‡ Rk2 2 d 2 d 1 2 d 2 Rk 2 d 1 Rk † and x…r† ˆ 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 ˆ 2:0Rp to d 2 ˆ 3:0Rp
and d ˆ 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 ˆ 3:0Rp to d 2 ˆ 4:0Rp ; has
the current profile s 1 and the field opens up at d ˆ 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 ˆ d 1 to r ˆ 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 ˆ 0; r ˆ 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 ˆ 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 ˆ
l‰…j ´ 7† ´ ge Š ´ 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 ˆ 0:55: It extends from d1 ˆ 2:0Rp to d 2 ˆ 3:0Rp ; has the
current profile s 2 and the field opens up at d ˆ 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 ˆ d2S W c ‡ d2V W c ˆ
j7 …j B†j2 dV
8p c
…
1
B2
‡
‰j ´ 7j 2 j…7 ´ j†Š ´ 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 Š jf j2u
1 ­gee 2
2
‡ ‡
d Wp ˆ
j
4p p
2
gee ­r r
r
r
)
‡ 4pv2 l…r†j2u
dS;
…18†
where ‰Br Š ˆ Br ‰r; u ˆ …p=2†‡ Š 2 Br ‰r; u ˆ …p=2†2 Š ˆ 2…4p=c†s
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 Š 2 ­Bu
2
2
2 Bu
dS W c ˆ
j ‡
‰Br Šju 2 Bu ‰Br Š
dS; …19†
8p
­r r
­r
r
p
with no contribution from the massless current sheet as Bu ˆ 0
there. The energy principle allows us to state that there is stability
if d2 W > 0; with
…
1
­‰Br Š
1 ­gee
2
2
‰Br Š j2r
dWˆ
2Bu
8p
gee ­r
­r
p
‰Br Š ­…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 Š
1 ­gee
r 2 jBu j3
2Bu
>0
2
‰Br Š ‡ 2
K
­r
gee ­r
­
l
jBu j2
,
gee ‡
> 0;
…21†
­r jBu j
2pK
‰Br Š ­…rBu †
r 2 jBu j‰Br Š2
‡ 4plv2 ‡
> 0;
r
­r
2K
…22†
where
K…t† :ˆ
…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 ˆ 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 ˆ
r dr:
…24†
c d2
gee
It is convenient to define n ˆ 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 ˆ B1
g( Rp
n Gst
R*
ssrs!
Mp
d 2 Lst f 1;2
;
…25†
17
L f st1;2
5 10 g d st2
where G ˆ 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 ˆ d1 to r ˆ d 2 and has a length L ˆ Lst ˆ 1R* in the
azimuthal direction. The field is assumed open at a distance
r ˆ d: The first row represents the standard case. Note that if
we only impose the condition that equilibrium holds we
obtain B0 ˆ 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 ˆ 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 ˆ
4:0Rp ; d1 ˆ 2:0Rp and d2 ˆ
3:0Rp : If we only impose the
condition that equilibrium holds,
we obtain B0 ˆ 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 ˆ 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 ˆ 4:0Rp ; d 1 ˆ 2:0Rp and d 2 ˆ 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
Eˆ
B2 dV ˆ
…B2 ‡ B2np ‡ 2Bnp ´ Bp † dV
8p V
8p V p
…
1
ˆ
…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 ˆ
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 ˆ
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 ˆ 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 ˆ 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 ˆ 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. D., 1989, MNRAS, 236, 57
Collier Cameron A., Woods J. A., 1992, MNRAS, 258, 360
Collier Cameron A., Duncan D. K., Ehrenfreund P., Foing B. H., Kuntz
K. D., Penston M. V., Robinson R. D., Soderblom D. R., 1990,
MNRAS, 247, 415
Donati J.-F., Collier Cameron A., 1997, MNRAS, 291, 1
Eibe M. T., Byrne P. B., Jeffries R. D., Gunn A. G., 1999, A&A, 341, 527
Favata F., Micela G., Reale F., 2000, A&A, 354, 102
Ferreira J. M., Jardine M., 1995, A&A, 298, 172 (Paper I)
Hundhausen A. J., 1993, J. Geophys. Res., 98, 13177
Jeffries R. D., 1993, MNRAS, 262, 369
Jeffries R. D., 1996, in Strassmeier K., Linsky J., eds, IAU Symp. 176,
Stellar Surface Structure. Kluwer, Dordrecht, p. 461
Kippenhahn R., SchluÈter A., 1957, Z. Astrophys., 43, 36
Kuperus M., Raadu M., 1974, A&A, 31, 184
Lepeltier T., Aly J. J., 1996, A&A, 306, 645 (LA96)
Low B. C., 1986, ApJ, 310, 953
Malherbe J. M., Priest E. R., 1983, A&A, 123, 80
Petrov P. P., 1990, Ap&SS, 169, 61
Riffert H., 1980, Ap&SS, 71, 195
Roberts P. 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 ­
Bˆ 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†
ˆ2
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 ˆ Rp † ˆ 0: Here, t ˆ …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
Aˆ
2 p ;
…A3†
c 2 rb sin uk1 2 rb sin uk2
where k21 ˆ 4rb sin u=‰r2 ‡ h2 ‡ b2 2 2r…h cos u 2 b sin u†Š; k22 ˆ
4rb sin u=‰r 2 …b2 ‡ h2 †=R2p ‡ R2p 2 2r…h cos u 2 b sin u†Š; T…k† ˆ
‰…2 2 k2 †K…k2 † 2 2E…k2 †Š 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 ˆ
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 ˆ …I=c†2 R2 …r†;
with
T…k2 †
…R2p 2 r2 † E…k22 †
2
2
K…k
†
:
…A5†
‡ 2Rp 2
R2 …r† ˆ
2
Rp
…R p ‡ r 2 †2 1 2 k22
It has the properties
r ˆ 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 ˆ b; is given by
Fˆ
M
I m1
g 2
‡ …F r1 ‡ F r2 † ˆ 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 ˆ 100 G; M ˆ 5 1017 g and e ˆ 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 ˆ 14:1
r ˆ 10:8
r ˆ 5:8
r ˆ 12:9
r ˆ 9:8
r ˆ 5:3
r ˆ 15:0
r ˆ 11:4
r ˆ 6:2
r ˆ 13:6
r ˆ 10:4
r ˆ 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 ˆ B ´ dS ˆ 22pr…A ‡ Ad †jrˆb2
rˆRp a constant;
…A9†
…A10†
with Ad ˆ 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 ˆ B0 Rp
;0 ;
…A11†
;
r4
r4
and a loop carrying current I of radius b at position (r, u ) so that
b ˆ 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 ˆ m
‡ F ur21 ˆ 0;
Figure A2. Current loop in a quadrupolar field for B0 ˆ 100 G; M ˆ
5 1017 g and e ˆ 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 ˆ 0 (thick full lines).
…A12†
and
F u2 ˆ 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
ˆ 0;
…A13†
where we have defined m ˆ M=2pb; and u1 ˆ …3 cos2 u 2 1†=r3 ;
u2 ˆ 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 ˆ B ´ dS;
…A16†
and if the normal to this surface is parallel to eu this can be
expressed as
F ˆ 22pr sin u…A ‡ Aq †jxx21 ;
…A17†
where x1 ˆ …Rp ; u† is the line resulting from the intersection of
this surface with the stellar surface and x2 ˆ …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 ˆ 6:1
r ˆ 5:1
r ˆ 3:4
r ˆ 5:7
r ˆ 4:8
r ˆ 3:2
considering, after the filament is perturbed in the direction
perpendicular to the field, moving to x3 ˆ …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 ˆ f …x3 ; x1 † 2 f …x2 ; x1 † ˆ f …x3 ; x4 † 2 f …x2 ; x1 †
‡ f …x4 ; x1 † ˆ d‰22pr sin u…A ‡ Aq †jxx21 Š ‡ dFs ;
…A18†
where f (xi, xj) represents the magnetic flux through points xi and
xj, x4 ˆ …Rp ; u ‡ du† denotes the auxiliary point on the stellar
surface and dFs ˆ 22pB0 R2p …3 cos2 u 2 1†sin 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