Magnetic arcades in stellar coronae.I. Cylindrical geometryV. M. �Cade�z, R. Oliver and J. L. BallesterDepartament de F��sica, Universitat de les Illes Balears,E-07071 Palma de Mallorca, SpainAbstract. X-ray spectroscopy performed by di�erent astronomical spacecrafts hasshown that many active late-type stars possess coronae. For such reason, the mag-netic structure of stellar coronae has raised considerable interest and, by analogywith the Sun, it is generally assumed that stellar coronae are structured by magnetic�elds having the shape of arcades. Most of those coronal magnetic �eld con�gurationsassume translational symmetry and are based in planar source surfaces. However, assoon as either the length or the width of such source surfaces become not negligibleas compared to the stellar radius, the application of the cylindrical geometry seemsto be more appropriate. Then, one way of obtaining coronal magnetic con�gurationsis to deal with source domains extended over a cylindrical surface.In this paper we generate potential coronal arcades based on cylindrical sourcesurfaces with non negligible length or width compared to the stellar radius. The uxfunction, the magnetic �eld components, the shape of magnetic �eld lines and othercharacteristic magnitudes have been obtained and analyzed for both cases.1. IntroductionMany years ago, EUV and soft X-ray observations of the solar coronadone by Skylab, pointed out that it is structured by the magnetic �eld.This structuring has been strikingly con�rmed by the spectacular softX-ray pictures taken by Yohkoh spacecraft during last years. Thus,in SXT images one can observe hot and dense loops in regions withclosed magnetic �elds while coronal holes are found in regions wherethe magnetic �eld is open. For instance, Acton et al. (1992) show an X-ray image, taken on November 12, 1991, where a magnetic arcade halfa solar radius high, 400,000 km wide and 500,000 km long is present.Moreover, a large variety of similar structures, with various heights,widths and lengths, can also be seen in many Yohkoh pictures (Watariet al., 1996; Weiss et al., 1996).During recent years, X-ray telescopes onboard Exosat, Ginga andRosat spacecrafts have provided evidence about the existence of hotcoronae in magnetically-active, late-type stars. Other observations ofsuch stars (see Baliunas, 1991 for a review) have shown that the surface�elds are organised forming starspots areas, similar to sunspots. Thesestars also display other forms of magnetic activity, active cromospheres, ares and prominences, all in close analogy to the Sun. For such reason,

2 V. M. �Cade�z et al.the Sun is a good reference to study stellar activity and the magneticstructure of stellar coronae.Frequently treated con�gurations of translational symmetry basedon planar \source surfaces" (Zwingmann, 1987; Priest, 1988; �Cade�z etal., 1994) involve magnetic �elds with only two components, Bx andBz, that depend on the x- and z-coordinates but remain invariant inthe y-direction, oriented along the tunnel of the arcade. The z = 0coordinate plane is usually considered as the photospheric level withthe corona located above it, at z > 0. The magnetic �eld lines thenemerge from the photosphere into the corona, reach certain height andbend down towards the photosphere. Such assumptions simplify theproblem, although some attempts to model three-dimensional con�gu-rations have also been made (Low, 1982).This approach seems to be appropriate as long as the length andthe width of the arcade are both negligible as compared to the stellarradius. However, as soon as one of these quantities becomes comparableto the stellar radius, one can apply a proper cylindrical geometry, withmagnetic �eld con�gurations based on a cylindrical source surface. Suchapproach is useful for magnetic arcades located at or close to the stellarequator. Finally, if both length scales are of the order of the stellarradius, only the spherical geometry is appropriate.To see this point in a clearer way, Figure 1a shows the half-widths xand s of a coronal arcade in at geometry and in cylindrical geometry,respectively, taken at the stellar equator. A simple calculation showsthat the half-width of the arcade in at geometry is given byx = R� tan�0;with R� the stellar radius. In cylindrical geometry, we haves = R��0:The ratio of both half-widths is plotted in Figure 1b for di�erent valuesof �0, showing that the ratio exceeds unity with increasing �0 whichfurther means that the cartesian half-width is larger than the cylin-drical one. Such a di�erence can introduce errors in estimates of thegeometrical and magnetic magnitudes of arcades like, for example, thevalues of magnetic ux needed for an arcade to erupt, etc.To simplify the problem, we have studied the speci�c cases of poten-tial magnetic �elds in cylindrical geometry that are z-invariant and �-invariant, respectively. In addition, these �elds are two dimensional asthey lie in planes that are perpendicular to the direction of invariance.In both cases, our aim is to characterize the magnetic �eld by obtain-ing the ux function, the magnetic �eld components, the shape of thec.tex; 21/01/1997; 9:58; no v.; p.2

Magnetic arcades in stellar coronae. I. Cylindrical geometry 3

Figure 1. (a) Comparison between the arcade half-width in cartesian geometry (x)and in cylindrical geometry (s). (b) The ratio x=s as a function of �0.magnetic �eld lines, etc., and to compare such results to those obtainedwhen cartesian geometry is assumed.2. The z-invariant caseThis case considers magnetic arcades with baselines curved in the �-direction of the cylindrical coordinate system, that extend vertically inthe radial r-direction and with the tunnel of the arcade running alongthe cylinder, in the z-direction. This con�guration is then applicable toc.tex; 21/01/1997; 9:58; no v.; p.3

4 V. M. �Cade�z et al.

Figure 2. Magnetic con�guration of a z-invariant (i.e., short and wide) coronalarcade. The dashed line represents the stellar equator.wide but short arcades located at or close to the stellar equator (Figure2).The related magnetic �eld is z-invariant and its curvature is feltalong the width of the arcade only. Such a �eld can be considered asan improved model of a cartesian arcade with its baseline much largerthan the length of the tunnel.2.1. General equations and solutionsThe potential magnetic �eld of an arcade oriented perpendicular to theaxis of the cylinder has only two z-invariant components, Br(r; �) andB�(r; �), and satis�es the condition r� ~B = 0, which givesr� ~B = 1r �@(rB�)@r � @Br@� � ez = 0: (1)Introducing the magnetic vector potential ~A = A(r; �)ez in ~B =r� ~A, we get ~B = r� ~A = rA� ez; (2)which further means that A(r; �) is the ux function, which remainsconstant on each magnetic surface. Thus, the shape of the magnetic�eld lines is obtained from A(r; �) = const:, while the �eld componentsfollow from (2),c.tex; 21/01/1997; 9:58; no v.; p.4

Magnetic arcades in stellar coronae. I. Cylindrical geometry 5Br = 1r @A@� ; B� = �@A@r : (3)The equation for A is now obtained from (1) and (3) as@2A@r2 + 1r @A@r + 1r2 @2A@�2 = 0: (4)Taking into account the natural periodicity of A, namely A(r; �) =A(r; �+2�), and the requirement for a �nite A at r� R�, the solutionof (4) is derived by separation of variables and takes the following formA(r; �) = 1Xn=0 1rn hD(1)n cos(n�) +D(2)n sin(n�)i ; (5)where D(1)n and D(2)n are the integration constants, that have to bedetermined from the appropriate boundary condition speci�ed at thesurface of the star, r = R�, and imposed either on the ux functionor on one of the magnetic �eld components. Each of these possibilitieswill be investigated separately.The singularity in (5) occurring at r = 0, is not important as thesolution (5) refers to the domain of the stellar corona only, i.e. at r �R�.Finally, by knowing the ux function one immediately obtains from(3) the vertical and horizontal magnetic �eld components, Br and B�,Br(r; �) = 1Xn=0 nrn+1 hD(2)n cos(n�)�D(1)n sin(n�)i ; (6a)B�(r; �) = 1Xn=0 nrn+1 hD(1)n cos(n�) +D(2)n sin(n�)i : (6b)The solutions (6) are expressed as a superposition of Fourier com-ponents B(n)r and B(n)� ,B(n)r = nrn+1 hD(2)n cos(n�)�D(1)n sin(n�)i ;B(n)� = nrn+1 hD(1)n cos(n�) +D(2)n sin(n�)i ;which are mutually related byB(n)� = � 1n @@�B(n)r : (7)c.tex; 21/01/1997; 9:58; no v.; p.5

6 V. M. �Cade�z et al.Expression (7) has a practical application as it allows the calculationof one magnetic �eld component when the other one is known at anarbitrary r.2.2. Boundary conditionsAs already mentioned, to determine the integration constants D(1)n andD(2)n a boundary condition has to be imposed either on the ux functionor on one of the magnetic �eld components. In what follows, we shallconsider all three possibilities.2.2.1. Boundary condition applied to the ux functionLet us �rst introduce the boundary condition by prescribing the distri-bution of the ux function A at the stellar surface as a known function�(�) that is periodic in �, with period 2�, in order to avoid multivaluedsolutions. According to (5), this yields� = A(R�; �) � 1Xn=0 1Rn� hD(1)n cos(n�) +D(2)n sin(n�)i : (8)On the other hand, the periodic distribution (8) can also be expandedin a Fourier series,� = 1Xm=0 [am cos(m�) + bm sin(m�)] ; (9)where the coe�cients am and bm are given asam = 1� Z 2�0 �(�) cos(m�)d�; (10a)bm = 1� Z 2�0 �(�) sin(m�)d�: (10b)By comparing the expressions (8) and (9), one immediately obtainsthe coe�cients D(1)n and D(2)n :D(1)n = Rn�� Z 2�0 �(�) cos(n�)d�; (11a)D(2)n = Rn�� Z 2�0 �(�) sin(n�)d�; (11b)where the relations (10) have been taken into account.c.tex; 21/01/1997; 9:58; no v.; p.6

Magnetic arcades in stellar coronae. I. Cylindrical geometry 7The coe�cients (11) uniquely determine the solutions (5) and (6)for the ux function and the magnetic �eld components respectively,that satisfy the boundary condition (8) imposed on A at r = R�.2.2.2. Boundary condition applied to BrFor practical purposes, it is often more convenient to prescribe one ofthe magnetic �eld components, rather than the ux function, at r = R�,the reason being that the magnetic �eld components are, in principle,easier to be obtained by various observational techniques.In this paragraph, we shall consider the case when the vertical com-ponent of the magnetic �eld is prescribed as some periodic function�r(�) at r = R�. To obtain the corresponding solutions (5) and (6) oneonly has to calculate the coe�cients D(1)n and D(2)n .According to the �rst equation in (6) we have�r = Br(R�; �) � 1Xn=0 nRn+1� hD(2)n cos(n�)�D(1)n sin(n�)i : (12)On the other hand, the periodic function �r(�) can also be expressedas a Fourier series,�r = 1Xm=0 ha(r)m cos(m�) + b(r)m sin(m�)i (13)with the coe�cients a(r)m and b(r)m given asa(r)m = 1� Z 2�0 �r(�) cos(m�)d�; (14a)b(r)m = 1� Z 2�0 �r(�) sin(m�)d�: (14b)By comparing expressions (12) and (13), one obtains the coe�cientsD(1)n and D(2)n in terms of �r,D(1)n = �Rn+1��n Z 2�0 �r(�) sin(n�)d�; (15a)D(2)n = Rn+1��n Z 2�0 �r(�) cos(n�)d�; (15b)where the relations (14) have been taken into account.c.tex; 21/01/1997; 9:58; no v.; p.7

8 V. M. �Cade�z et al.2.2.3. Boundary condition applied to B�A procedure analogous to that in the previous paragraph, this timeapplied to the horizontal magnetic �eld component B�, yields the fol-lowing expressions for D(1)n and D(2)n ,D(1)n = Rn+1��n Z 2�0 ��(�) cos(n�)d�; (16a)D(2)n = Rn+1��n Z 2�0 ��(�) sin(n�)d�: (16b)Here�� = B�(R�; �) � 1Xn=0 nRn+1� hD(1)n cos(n�) +D(2)n sin(n�)i (17)is the initially speci�ed periodic function describing the angular depen-dence of the horizontal magnetic �eld component at r = R�.2.3. A particular exampleAs an example, we consider the case when the boundary condition onthe vertical magnetic �eld component is prescribed at r = R�.Let then the distribution �r � Br(R�; �) in (12) be given by�r = �B� sin(n0�); (18)where n0 is an integer as Br(R�; �) should be a single valued functionof �. The coe�cients (15) then becomeD(1)n = B�Rn+1�n �n;n0 and D(2)n = 0and the solutions (5) and (6) reduce toA(r; �) = B�R�n0 �R�r �n0 cos(n0�) (19)and Br(r; �) = �B� �R�r �n0+1 sin(n0�); (20a)B�(r; �) = B� �R�r �n0+1 cos(n0�); (20b)c.tex; 21/01/1997; 9:58; no v.; p.8

Magnetic arcades in stellar coronae. I. Cylindrical geometry 9respectively.Expression (19) can now be used for plotting the magnetic �eld linesde�ned by A(r; �) = const:The resulting magnetic �eld components (20) indicate that the mag-netic �eld strength depends on the radius r only,B � �B2r +B2��1=2 = B� �R�r �n0+1 : (21)2.3.1. The embedded arcadeThe obtained �eld (20) can now be applied to an isolated arcade thatis in static equilibrium with the surrounding non-magnetized plasma.This obviously means that the magnetic �eld ceases to be potential atthe boundary of such an arcade, where the external gas pressure pe isbalanced by the magnetic pressure pm � B2=2�0.We shall now consider an arcade embedded in a �eld-free stellarcorona with a boundary surface described by A(r; �) = A0. The �eldinside the arcade remains potential, given by expressions (20), whilethe boundary surface contains a current-sheet whose Lorentz force bal-ances the external gas pressure. To simplify the problem, an isothermalcorona, Te = const:, is assumed. Furthermore, we shall include the vari-ation of gravitational acceleration g with the radius r, which is givenby g(r) = g�R�r (22)in cylindrical geometry.The external pressure pe follows from the hydrostatic balance equa-tion, @pe@r + �e(r)g(r) = 0;which, together with (22) and the perfect gas low pe = R�eTe, yieldspe = pe� �R�r �R�=H� ; (23)where H� � RT�=g� is the pressure scale-height at r = R� and T� = Teis the temperature of the isothermal corona.On the other hand, the magnetic pressure at the boundary of thearcade follows from (21) and is given bypm � B22�0 = pm� �R�r �2(n0+1) ; (24)c.tex; 21/01/1997; 9:58; no v.; p.9

10 V. M. �Cade�z et al.with pm� � B2�=2�0.From (23) and (24) we see that the boundary condition pe = pm willbe satis�ed on the surface A = A0 provided thatn0 = R�2H� � 1; (25)where n0 is an integer. The relation (25) represents the condition foran embedded arcade to occur as a stable con�guration. Then, to obtaina particular value of n0, we must impose the adequate coronal temper-ature which, together with the stellar radius, determines the pressurescale-height H�.Next, inserting (19) into equation A(r; �) = A0 for the externalboundary we obtainB�R�n0 �R�r �n0 cos(n0�) = A0: (26)Introducing the angular width of the arcade �0 de�ned by A(R�; �0) =A0 or cos(n0�0) = A0n0B�R� ;equation (26) can be written as�R�r �n0 cos(n0�) = cos(n0�0): (27)This expression now shows that the external boundary of the arcadereaches the stellar surface r = R� at � = ��0 and has its maximumheight rmax at � = 0, rmax = R�[cos(n0�0)]�1=n0 (28)In order to compare with the at geometry case, we consider a solarpotential arcade with a half-width of 150; 000 km. In cartesian geom-etry, the maximum height of such arcade is around 276; 000 km (seePriest, 1988, who considered a similar arcade model). In cylindricalgeometry, to have a realistic coronal temperature we use n0 = 6 (whichresults in Te ' 985; 000 K), and using the same half-width we obtainfrom (28) a maximum height close to 698; 000 km. Hence, the cylindri-cal arcade is much higher than the at geometry arcade.There are also some additional interesting characteristics of suchan embedded arcade. For example, we can consider the magnetic �eld ux F that emerges through the base of the arcade at r = R�, withinthe interval z0 � z � 0 along the tunnel. Only the emerging uxc.tex; 21/01/1997; 9:58; no v.; p.10

Magnetic arcades in stellar coronae. I. Cylindrical geometry 11is taken into acount, so the ux should be calculated in the domainwhere Br(R�; �) > 0 or, according to expression (20), for the intervalof angles 0 � � � ��0. Thus,F = R� Z 0��0 d� Z z00 Brdz = B�R�z0n0 [1� cos(n0�0)]or, introducing F1 � F=z0 as the magnetic ux per unit length z0, oneobtains cos(n0�0) = 1� F1n0B�R� ; (29)which determines the angular half-width �0 of the arcade in terms of theparameters F1, n0 and B�. The magnetic �eld B� can now be expressedin terms of the external gas pressure pe� through the boundary condi-tion pe�= pm�� B2�=2�0.Next, one can investigate the variation of the vertical extent (28) ofthe arcade when the parameters in (29) change. In particular, when(I) The ux F1 through the base is allowed to vary while n0 and pe�remain �xed.(II) The external gas pressure pe� is allowed to vary while n0 andF1 are �xed.(III) The parameter n0 is allowed to vary while pe� and F1 are �xed.In case (I), the half-width �0 of the arcade increases with F1, asseen from (29), and so does the height of the arcade rmax in (28). Thearcade erupts, i.e. rmax becomes in�nite, for F1= B�R�=n0 when thecondition n0�0 = �=2 is achieved in (28).As to case (II), the external gas pressure pe� is �rst expressed interms of B� and then, taking into account (29), we can writepe� = B2�2�0 = F 21 n202�0R2�[1� cos(n0�0]2 : (30)The critical values for the external gas pressure and the magnetic �eldthat cause the eruption of the arcade follow from (30) by taking n0�0 =�=2. They are pe�= F 21 n20=(2�0R2�) and B�= F1n0=R�, respectively.Finally, case (III) considers the e�ects arising from the variationof the parameter n0 that describes the rate at which the magnetic�eld (20) decreases with height. According to (28) and (29), the arcadeerupts if the condition n0 = B�R�=F1 is satis�ed, keeping in mind thatn0 must be an integer. The angular half-width of the arcade is thensimply given by �0= �=(2n0)= �F1=(2R�B�).Now we can compare the magnetic uxes causing the eruptions ofpotential arcades in cylindrical and at geometry models. The magneticc.tex; 21/01/1997; 9:58; no v.; p.11

12 V. M. �Cade�z et al. ux through the base of a potential arcade in cartesian coordinates is(Priest, 1988) F (car)1 = B�l [1� cos(lx0)] ;with l the magnetic scale-height. Applying the condition for the erup-tion of the arcade and taking the same value for B� in both models,one obtains F (cyl)1 = F (Car)1 R�ln0 ;where F (cyl)1 is the magnetic ux per unit length in cylindrical coordi-nates. From (28), R�l � 1 = n0 andF (cyl)1 = F (car)1 n0 + 1n0 :hence, the magnetic ux required to produce the eruption of an arcadein cylindrical geometry is always greater than that required for atgeometry. 3. The �-invariant caseIn this section we consider cylindrical geometry with the arcade baselinetaken in the z-direction, extending vertically in the r-direction and withthe closed arcade tunnel running in the azimuthal �-direction. Themagnetic �eld is assumed �-invariant. Such a con�guration introducescurvature e�ects along the tunnel and is related to a very long butnarrow source surface (Figure 3).3.1. Equations and boundary conditionsThe potential magnetic �eld in this case has two components, Br(r; z)and Bz(r; z), and satis�es the condition r� ~B = 0, which in cylindricalcoordinates gives r� ~B = �@Br@z � @Bz@r � e� = 0: (31)Introducing the magnetic vector potential ~A = A(r; z)e� into ~B =r� ~A we get ~B = r� ~A = 1rr(rA)� e�; (32)c.tex; 21/01/1997; 9:58; no v.; p.12

Magnetic arcades in stellar coronae. I. Cylindrical geometry 13

Figure 3. Magnetic con�guration of a �-invariant (i.e., long and narrow) coronalarcade. The dashed line represents the stellar equator.which further means that the quantity rA(r; z) remains constant on amagnetic surface. The �eld components follow from (32),Br = �@A@z ; Bz = 1r @@z (rA): (33)Substituting these expressions into (31), one obtains the equation forA, 1r @2@r2 (rA)� 1r2 @@r (rA) + @2A@z2 = 0: (34)The above relation can be Fourier transformed in the variable z,which yields the following modi�ed Bessel equation,r2@2Ak@r2 + r@Ak@r � (1 + k2r2)Ak = 0; (35)where Ak(r) = 1p2� Z +1�1 A(r; z)e�ikzdzis the Fourier transform of A.The solution of (35) that does not diverge at r ! +1 and whichsatis�es a prescribed boundary condition at the stellar surface r = R�is thenc.tex; 21/01/1997; 9:58; no v.; p.13

14 V. M. �Cade�z et al.Ak(r) = Ak(R�) K1(kr)K1(kR�) ; (36)where K1 is the modi�ed Hankel function and Ak(R�) is the Fouriertransform of the boundary condition A(R�; z) imposed on A(r; z) atr = R�, Ak(R�) = 1p2� Z +1�1 A(R�; z)e�ikzdz: (37)Performing the inverse Fourier transform of Ak from (36) we obtainthe required solution A in the following form,A(r; z) = 1p2� Z +1�1 Ak(R�) K1(kr)K1(kR�)eikzdk; (38)where Ak(R�) is given by (37).The magnetic �eld components then follow from (33) and (38) asBr(r; z) = � ip2� Z +1�1 Ak(R�) K1(kr)K1(kR�)eikzkdk; (39a)Bz(r; z) = 1p2� Z +1�1 Ak(R�)� 1K1(kR�) dK1(kr)dr + K1(kr)rK1(kR�)� eikzdk: (39b)We have so far assumed that the ux fuction A is prescribed atthe surface r = R�. However, it is also possible to specify one of themagnetic �eld components, Br(R�; z) or Bz(R�; z), instead. In this caseAk(R�) that enters solutions (38) and (39) has to be expressed in termsof either Br(R�; z) or Bz(R�; z). This can be easily done by consider-ing the solutions (39) as the inverse Fourier transforms of the relatedmagnetic �eld components Brk(r) and Bzk(r)Br(r; z) = 1p2� Z +1�1 Brk(r)eikzdk; (40a)Bz(r; z) = 1p2� Z +1�1 Bzk(r)eikzdk; (40b)where Brk(r) and Bzk(r) are de�ned asBrk(r) = 1p2� Z +1�1 Br(r; z)e�ikzdz; (41a)c.tex; 21/01/1997; 9:58; no v.; p.14

Magnetic arcades in stellar coronae. I. Cylindrical geometry 15Bzk(r) = 1p2� Z +1�1 Bz(r; z)e�ikzdz: (41b)Comparing (39) and (40) and taking r = R�, one obtainsBrk(R�) = �ikAk(R�); (42a)Bzk(R�) = Ak(R�) �K 01(kR�)K1(kR�) + 1R� � ; (42b)where K 01(kR�) � dK1(kr)=dr at r = R�.Finally, we can write two relations that follow from (42) and (40) atr = R�, Ak(R�) = ip2�k Z +1�1 Br(R�; z)e�ikzdz; (43a)Ak(R�) = 1p2� R�K1(kR�)K1(kR�) +R�K 01(kR�) Z +1�1 Bz(R�; z)e�ikzdz: (43b)Therefore, if one of the magnetic �eld components, Br(R�; z) orBz(R�; z), is given as the boundary condition, then the correspondingexpression (43) rather than (37) has to be used for Ak(R�) in solutions(38) and (39).3.2. A particular exampleAs an example we shall consider the case when the boundary conditionis imposed on the radial component of the magnetic �eld by prescribingBr(R�; z) = �B� sin(k0z): (44)Then Ak(R�) follows from (43),Ak(R�) = �p�B�p2k [�(k � k0)� �(k + k0);while the solutions (38) and (39) becomeA(r; z) = � B�K1(k0r)k0K1(k0R�) cos(k0z); (45a)Bz = B�k0K1(k0R�) cos(k0z) �K1(k0r)r + dK1(k0r)dr � : (45b)c.tex; 21/01/1997; 9:58; no v.; p.15

16 V. M. �Cade�z et al.The solutions (45) indicate that the magnetic pressure pm � B2r+B2zdepends also on z, in addition to the radial variable r, in the caseof a �-invariant arcade. As a consequence, if such a magnetic arcadeis embedded in a non-magnetized plasma, it will not be in a staticequilibrium with the surroundings, because the external gas pressureis only r-dependent and cannot balance pm(r; z) everywhere on theboundary of the arcade.4. ConclusionsAs an intermediate case between coronal magnetic arcades generatedby planar and spherical source surfaces, we have generated potentialcoronal arcades based on cylindrical source surfaces with either thelength or the width being comparable to the solar radius. In both cases,the ux function, the magnetic �eld components, the shape of magnetic�eld lines and other characteristic magnitudes have been obtained.The magnetic �eld in the z-invariant case decreases in the radialdirection at a rate �xed by n0, which can be �xed by imposing a realisticcoronal temperature and using the known values for the stellar radius,the gravitational acceleration and the mean molecular weight. In the�-invariant case, the decrease of the magnetic �eld components in theradial direction is linked to the modi�ed Hankel function K1(k0r) andits �rst derivative.Embedded arcades can exist in static equilibrium only if they arez-invariant, that is, for arcades whose height is much larger than theirhalf-width. A comparison with arcades generated in at geometry indi-cates that the height of a z-invariant cylindrical arcade is much biggerthan that of the cartesian one having the same half-width. In addition,the estimated values of the magnetic ux needed for an embedded z-invariant arcade to erupt are much larger than those needed for theeruption of a cartesian arcade.To summarize, the introduction of the cylindrical geometry to modelpotential coronal arcades in stellar atmospheres points out the existenceof some di�erences with respect to those generated in at geometry.Such di�erences could be important for understanding the behaviourof coronal arcades generated by source surfaces whose magnetic �eldsare modelled using cylindrical geometry.

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Magnetic arcades in stellar coronae. I. Cylindrical geometry 17AcknowledgementsV.M. �Cade�z wishes to acknowledge the �nancial support received fromDGICYT SAB95 - 0334. R. O. and J. L. B. acknowledge �nancial sup-port from DGICYT under grant PB93 - 0420ReferencesActon, L., Tsuneta, S., Ogawara, R., Bentley, R., Bruner, M., Can�eld, R., Culhane,L., Doschek, G., Hiei, E., Hirayama, T., Hudson, H., Kosugi, T., Lang, J., Lemen,J., Nishimura, J., Makishima, K., Uchida, Y. and Watanabe T.: 1992, Nature258, 618Baliunas, S. L.: 1991, in The Sun in Time, C. P. Sonnett, M. S. Giampapa and M.S. Matthews (Eds.), The University of Arizona Press, 809�Cade�z, V. M., Oliver, R. and Ballester, J. L.: 1994, Astron. Astrophys. 282, 934Low, B. C.: 1982, Astrophys. J. 263, 952Priest, E. R.: 1988, Astrophys. J. 328, 848Watari, S., Detman, T. and Joselyn, J. A.: 1996, Solar Phys. 169, 167Weiss, L. A., Gosling, J. T., McAllister, A. H., Hundhausen, A. J., Burkepile, J. T.,Phillips, J. L., Strong, K. T. and Forsyth, R. J.: 1996, Astron. Astrophys. 316,384Zwingmann, W.: 1987, Solar Phys. 111, 309

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