Faculteç des Sciences, Deçpartement de Physique, 33 Rue Saint Leu, Amiens Cedex 80039, France. Received 1996 October 18 ; accepted 1997 May 28.
THE ASTROPHYSICAL JOURNAL, 489 : 946È950, 1997 November 10 ( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.
ON CIRCUMSTELLAR ENVELOPE FORMATION G. PASCOLI Faculte des Sciences, Departement de Physique, 33 Rue Saint Leu, Amiens Cedex 80039, France Received 1996 October 18 ; accepted 1997 May 28
ABSTRACT The origin of magnetic Ðelds observed in circumstellar envelopes and protoÈplanetary nebulae is investigated theoretically. The build-up region of these Ðelds is likely to be found deep in the progenitor (assumed to be an asymptotic giant branch red giant). With toroidal Ðelds D106 G produced by a dynamo mechanism in the degenerate core (roughly assimilated to a white dwarf of mean radius D109 cm), we Ðnd values D50 G in the convective envelope (mean radius D1013 cm) and 1 mG in the circumstellar envelope (mean radius D1016È1017 cm). A model for the ejection of massive winds is presented. We point out that the two phenomena (magnetic activity into the degenerate core and ejection of gas by the star) could be strongly interrelated. Subject headings : circumstellar matter È MHD È stars : AGB and post-AGB È stars : interiors È stars : mass loss 1.
INTRODUCTION
cyclonic component of the convection, respectively. In writing these equations, it is implicitly assumed that the magnetic Ðeld is expressed as the sum of a poloidal component H (H , H , 0) and a purely azimuthal component h H (0, 0, H p). r tThe poloidal Õ components of the Ðeld are derived from the following relationship :
The presence of magnetic Ðelds in circumstellar envelopes and in protoÈplanetary nebulae has been the subject of only a few theoretical studies (Pascoli 1992 ; Chevalier & Luo 1994), even though it seems suggested by a number of observational data (Cohen 1989 ; Nedoluha & Bowers 1992). Some important questions remain unanswered. One of these is that the region of the progenitor where these Ðelds are generated is not accurately determined in the framework of the theoretical models. (In Chevalier & Luo 1994, however, the toroidal magnetic Ðeld is produced outside the star.) In this paper, by solving the magnetohydrodynamic equations, we search for a continuous solution for the magnetic Ðeld from the center of the star up to the circumstellar region. This problem is not here sequentially examined, region by region, as in preceding works (Pascoli 1992) but rather is considered in a global manner. The only boundary conditions imposed are that the Ðeld is zero at r \ 0 and as r approaches O. Going from the degenerate core surface (mean radius D109 cm) to the circumstellar region (mean radius D1017 cm), we Ðnd that the magnetic Ðeld varies over 11 orders of magnitude (from 106 to 10~5 G) and the density over 21 orders of magnitude (from 1 to 10~21 g cm~3). The ejection of matter by an asymptotic giant branch (AGB) star is also reexamined in the framework of this model. 2.
1 1 H \ L (A cos h) , H \ [ L (rA ) . r r cos h h ' h r r ' The quantities occurring in equations (1) and (2) and in the regenerative terms c and ! are the magnetic di†usivity g, the turbulent velocity v , and the nonuniform velocity v ; L , L , Õ L/Lh, t r and L denote the tpartial derivatives L/Lt, L/Lr, and h respectively. We conjecture that the mass-loss process by an evolved red giant is closely connected to the magnetic activity in the dense core. A large body of observational data (molecular emission lines) shows that the formation of a circumstellar envelope surrounding an AGB star, with mass D1 M , _ results from the ejection of material by this star in the form of a massive and slow wind. These winds are characterized by two quantities about which we have good information : the mass-loss rates and the expansion velocities. The massloss rates M0 are highÈ10~5 M yr~1Èand the expansion _ the typical duration of velocities v are low [10 km s~1, w the mass-loss process being of the order of 104 yr. In addition, it is found that the density in the extensive circumstellar envelopes is well described by spherically symmetric outÑow at both constant mass loss and expansion velocity (see Habing 1996 for a review). Observations of the giant halo corroborate this scheme. The mean density of the wind can be modeled by a simple o D r~2 law, characteristic of a steady ejection process (Bassgen & Grewing 1989). The fact that the winds are produced with a smooth gradient of density is also supported by theoretical models (Kwok, Purton, & Fitzgerald 1978 ; Frank 1994 ; Icke 1994). If the ejection is continuous and stationary over a period D104 yr, then, in the same way, we can reasonably assume that the stellar characteristics remain globally unchanged during such a relatively long period. One can seriously oppose the argument that protoÈ planetary nebulae and planetary nebulae very often appear
METHOD OF CALCULATION
The basic equations describing the generation of axisymmetric magnetic Ðelds in the stellar core are (Levy & Rose 1974 ; Parker 1979, ° 19.4)
A
B
1 A \ !H , L A [g *[ ' t ' r2 cos2 h '
A
(1)
B
1 1 H [ L gL (rH ) L H [g *[ t ' r2 cos2 h Õ r r r ' [
1 L gL (H cos h) \ cH , r r2 cos h h h '
(2)
where the r- and h-dependent functions c \ L v [ v /r and r ' and ' the ! P cr(v /v )2 represent the nonuniform rotation t ' 946
CIRCUMSTELLAR ENVELOPE FORMATION with various density enhancements (blobs, Ðlaments, etc.), unexplained by a smooth gradient of density. In fact, it is important to notice that when we observe a planetary nebula, we are usually seeing a small fraction of the mass lost by the progenitor. In the framework of the interacting wind model, this matter has been ““ snowplowed ÏÏ (i.e., shocked, and swept up, or bulldozed) and diversely processed with strong departure from the homogeneous state initially prevailing in the slow wind (Bryce, Balick, & Meaburn 1994). In planetary nebulae (PNs), the appearance, at a large scale, of Ðlaments and other peripheral structures could simply be due to projection e†ects reinforced by ionization processes in prolate envelopes, the latter initially possessing a very smooth equatorial-to-polar density contrast : D1.5È2 (Pascoli 1990, 1993). On the contrary, in cool (unprocessed) circumstellar envelopes, density and velocity gradients both appear smooth to a large extent (Cohen 1989 ; Bowers 1990), even if an increasing degree of asymmetry is clearly apparent toward the innermost regions of these envelopes (Bowers & Johnston 1994). As a matter of fact, at the very end of the slow wind ejection, complicated phenomena exist like emissions of bipolar jets or blobs of gas by highly evolved stars. Internal processes also occur within the wind (e.g., formation of dust at a certain distance from the star accompanied by hydrodynamical instabilities (Morris 1992). But all these phenomena are of little concern here. Eventually, we can realistically assume that the major facts of the massive winds expelled from AGB stars can be understood in the framework of a stationary model of gas ejection presumably correlated with a steady magnetic activity in the dense core, over a D104 yr period. In view of the above considerations, the magnetic Ðeld being in a steady state during the mass-loss event, we will investigate stationary solutions of equations (1) and (2). We have tried solutions of the type A (r, h) \ A a(x)j(h) \ H r a(x)j(h) , ' c pc c H (r, h) \ H b(x)k(h) . ' tc The index c labels the quantities at the surface of the degenerate core and x 4 r/r . c considered a quadrupole-like Ðeld, We have exclusively that is, H (r, h) \ H (r, [h) ' ' thus, we have set
and
A (r, h) \ [A (r, [h) ; ' '
k(0) \ 1,
L k(0) \ 0, j(0) \ 0, L j(0) \ 1 . h h For the other physical quantities g, c, and !, one may write g(r, h) \ g g(x)u(h) , c(r, h) \ c c(x)v(h) , c c !(r, h) \ ! !(x)q(h) , c imposing the normalization conditions u(0) \ 1 ,
L u(0) \ 0 , v(0) \ 1 , h L v(0) \ 0 , q(0) \ 0 , L q(0) \ 1 . h h We shall adopt now the following scaling law for the turbulent velocity : v D xp . (3) t The exponent p was chosen to simultaneously Ðt the turbulent velocity within the core (v D 1 km s~1) and in the t
947
low-density envelope. The turbulent velocity is quasiconstant in the major part of this envelope and of the order of 5È10 km s~1 (see the models of Soker 1992). On the other hand, in the case of nonuniform rotation with angular velocity declining outward, we write r v \ ) r2 v(h) . (4) ' c c (r ] r )2 c The actual form of this law is, in fact, unknown. There is no rigorous theory to predict it in an evolved red giant. (In any case it is a very complex problem.) Fortunately, the phenomena envisioned in this paper are independent of the details of the distribution of nonuniform rotation, the single ingredient of our models being that this nonuniform rotation is essentially concentrated in the innermost regions of the star. The simple and convenient law we have chosen in equation (4) assumes that the angular velocity ) is nearly constant in the dense core (0 ¹ r ¹ r ) and decreases as r~2 within the low-density envelope (r c\ r ¹ r*), but another law with c ) D r~n (n º 1) is also suitable. In this way one obtains for the dimensionless functions c(x) and !(x) : c(x) \
x and !(x) \ x2p(1 ] x)e~(x~1)2 , (1 ] x)3
with ! \ [(2l2 )/(r ) ) and c \ [2) . c tc c c to the c vicinity c of the equatorial Eventually, restricting plane, equations (1) and (2) can be expressed in dimensionless variables, where the prime denotes the derivative d/dx, aA \ bA ]
A
2 1 !(x) a@ ] [j~1L2 j(0) [ 1]a ] a b\0 , h x x2 g(x)
B G
(5)
H
2 g@ [L2 k(0) [ 1] g@ h b ] b@ ] ] x g x2 xg ]b
where
A B
1 a\0 , (1 ] x)3g
(6)
v 2 [2) r2 H [2) r2 H tc c c pc . c c tc and b \ v g H g H c tc c pc 'c We label the set of equations (5) and (6) system 1 in the following discussion. At Ðrst appearance, many parameters have to be determined in order to solve these equations. In fact, these parameters cannot be taken independent of each other. On the contrary, they are strongly interrelated, and their e†ective determination concerns only a few of them. Table 1 collects the values of the physical quantities taken at the core surface and occurring in the coefficients a and b. The mean magnetic Ñux lost by the star '0 is supplied by * ; Nedoluha & the observational data (Shepherd et al. 1990 Bowers 1992, and references therein). Equating this stellar magnetic Ñux loss with the core Ñux loss '0 gives c '0 D '0 D 1020 Mx s~1 . c * We have implicitly assumed here that, Ðrst, the dynamo mechanism essentially operates into the dense core and not in the low-density envelope of the star. Second, throughout this very large envelope, the ohmic dissipation is negligible, and the turbulent di†usivity ensures simple transportation a\
948
PASCOLI
Vol. 489
TABLE 1 CHARACTERISTIC VALUES AT SURFACE OF DEGENERATE CORE r D r M c (M ) _ 1.0
M env (M ) _ 0.1
c
r c (cm)
) c (s~1)
v tc (cm s~1)
H pc (G)
H tc (G)
g (cm2 s~1)
p
109
10~2
105
104
106
1.5 ] 1012
0.2
NOTE.ÈMagnetic Ðeld values D106 G are of the order of these detected in the magnetic white dwarfs (which have radii D10~2 R ). The timescale q D _ [(M ) )/(r H H )] for the spin-down of the rotating degenerate core is D105 yr, c c c pc tc greater than or equal to the typical time corresponding to the nebular ejection.
of magnetic Ðeld from the dense core to the stellar surface. (Turbulence by itself does not disipate magnetic Ðelds ; Hoyng 1987.) As it is well known, the toroidal magnetic Ðeld is regenerated by stretching of the poloidal Ðeld lines (Parker 1979, ° 19.4). In the stationary case, this regeneration process within the dense core has to be equilibrated by the Ñux loss at the core surface, so that
Just above the stellar surface, the mean velocity of gas is not zero. Assuming the Ñow is steady, we have the equation :
r2 H D c pc D '0 , c q reg where the regenerative time q D (2n/*) ). Adopting the reg by equation c (4), we have di†erential rotation law given \ 0.25) . In other respects, the regen*) D r (L)/Lr)o c time of magnetic Ñux tubes c time c is equal r/rc to the uplift erative through the core region, that is, of the order of r /v , where c A(Parker the Alfven speed v \ [H /(4no )1@2]. We have then A tc c 1979, ° 8.7)
where v is the velocity of gas, o is the density, P is the gas pressure, H is the mean magnetic Ðeld, and % isgthe turbulent stress tensor. M (\M ] M ) is the mass of the star. c usedenv As the equation of state, we the ideal gas law with an adiabatic index \ 5/3, that is P D o5@3. The turbulent stress g tensor % is given by
'0
reg
r (4no )1@2 c D c . reg H tc Solving these equations with r D 109 cm, o D 1 g cm~3, c H D 104 cG and H D ) D 10~2 s~1 immediately gives c pc tc 106 G. Much higher Ðelds can, however, be generated when equating the Coriolis force and the magnetic stresses. In this case, we have q
[(¿ Æ $)¿ [ GM
r [ $P g r3 ]
1 ($ Â H) Â H ] $ Æ % \ 0 , 4n
(7)
%\% ]% . t mag Assuming turbulence to be isotropic, it follows that 1 (% ) \ [ ol2 d and t ij 3 t ij
(%
h2 ) \[ d . mag ij 8n ij
In other respects, equipartition between turbulent magnetic energy and turbulent kinetic energy seems relevant here. This yields 1 h2 ov2 D . 3 t 8n
4nr o ) v D (H H ) . c c c tc pc tc ' It is very often believed that when these forces become comparable, the Ñuid motion is inhibited (Bullard & Gellman 1955). Using this relationship gives
Furthermore, in the outermost region of the convective envelope of the red giant, we have adopted the natural choice
(H H ) D 1012È1013 G2 . pc tc ' This critical value is 2 or 3 orders of magnitudes higher than the one we have previously obtained (D1010 G2). Such high Ðeld intensities seem unlikely to be caused by the existence of the much more constraining process described above ; that is, the rise of Ñux tubes by magnetic buoyancy. Such a very efficient mechanism limits the magnetic Ðeld intensity well before the critical value is reached. In this context, a possible feedback process by Ñuid motion inhibition is unlikely. (In other words, magnetic Ðelds are of little importance in the evolution of the dense core which is much more a simple hydrodynamics problem than a magnetohydrodynamics problem. The magnetic Ðeld is likewise passive in the innermost regions of the low-density envelope. The inÑuence of this Ðeld is essentially felt in the outermost regions of this envelope and especially at the stellar surface.)
Finally, equation of continuity and induction equation are, respectively,
HDh .
$ Æ (o¿) \ 0 ,
(8)
$ Â [g($ Â H)] \ 0 .
(9)
Equations (7), (8) and (9) (labeled system 2) are solved using the values given in Table 2. It is important to specify that the value for H appearing in this table is not taken in an ad hoc manner *but is directly issued from calculation, especially by solving equations (5) and (6) choosing a continuity solution for the functions o, v, and H and their Ðrst derivatives H between the outermost region of the star and the Ñow. Starting from r \ r , the solution of the system 1 was obtained by integratingc this point both inward and outward, with the boundary conditions that a, b ] 0 when
No. 2, 1997
CIRCUMSTELLAR ENVELOPE FORMATION
949
TABLE 2 CHARACTERISTIC VALUES AT STELLAR SURFACE r D r M (M ) _ 1.1
*
o * (G cm~3)
r (cm)
v a t* (cm s~1)
H * (G)
h * (G)
3.5 ] 10~10
3 ] 1013
7 ] 105
40
40
NOTE.ÈMagnetic Ðeld value is provided assuming the continuity of the magnetic Ðeld and of its Ðrst derivative at the stellar surface, identifying these quantities with the corresponding ones obtained in solving the system 1. a This value of v is of the order of the convective velocity estit mated in the subphotosphere of a red giant (Schwarzschild 1975).
r ] 0 and O. This objective is achieved by performing a series of iterations with di†erent variations of g, kA, and jA until these boundary conditions are satisÐed (the functions kA and jA give the latitudinal dependence of the magnetic Ðeld in the vicinity of the equatorial plane ; Fig. 1). An important condition is that, when r º r , the velocity of gas * is other than zero. On that account, eventually systems (1) and (2) have to be self-consistently solved in an iterative manner with a smooth Ðtting in the vicinity of the stellar surface. 3.
RESULTS
Figure 2 shows the variations of the toroidal and poloidal magnetic Ðeld components. Starting with values D106 and D104 G for these components, respectively, at the surface of the degenerate core (r \ 109 cm), we see that the poloidal c a very steep decrease outside the component (H ) presents r degenerate core (the dynamo region). In the convective envelope, the Ðeld is essentially toroidal with values D50 G, and its variation is rather smooth. Away from the star, the Ðeld (component H ) is approximately decreasing as r~1 Õ cm, H D10~5 G at r D 1018 cm). (H D 0.1 G at r D 1014 The observational estimates of Nedoluha & Bowers (1992), 15 mG \ H \ 1 mG at r D 8000 AU, and of Cohen (1989), 6 cm, seem consistent with these results. 1 mG at r D 1016 The combination of equations (7), (8), and (9) gives the velocity distribution outside the star. The solution is unique and physical when this one passes through a critical point
FIG. 1.ÈTurbulent magnetic di†usivity g plotted against the distance (r \ 109 cm). The functions kA \ L2k(0) and jA \ j~1L2 j(0) express the c h latitudinal dependence of the Ðeld in hthe vicinity of the equatorial plane.
FIG. 2.ÈPoloidal H and toroidal H magnetic Ðeld components and density o as functions ofrdistance (x \ r/rÕ, r \ 109 cm). c c
that we have located, by an iterative method, at r D 6r . * The calculations give for the thickness of the interface between the stellar surface (mean velocities of gas D0) and the wind (mean velocities of gas D1 km s~1) * D r . * * Figure 3 shows the gas velocity as a function of distance. Four solutions corresponding to di†erent variations of v are given. All these curves show a rapid increase over at distance D10r and then stabilize at approximately con* stant velocity 10È20 km s~1. The Ðnal velocities are found equal to about half the escape velocity from the star (M D M and v D30 km s~1). This result also appears in a _ esc agreement with the observational data quantitative (Habing 1996). Many physical mechanisms have been proposed to explain the large mass loss by AGB stars : radiation pressure acting on dust, radial pulsations, Alfven or sound waves (Hearn 1990). However, in the present model, the problem is di†erently understood because the mass loss is intrinsically linked to the production of magnetic Ðeld into the degenerate core. If the magnetic Ñux produced by the dynamo into the core is not evacuated through the stellar surface, the magnetic Ðeld increases without limit, and no steady solution can be obtained. In that sense the magnetic Ðeld production deep in the star would induce the mass loss
FIG. 3.ÈGas velocity v as a function of distance (x \ r/r , r \ 3 * of * the ] 1013 cm). Each curve is labeled according to the dependence turbulent velocity with x or, equivalently, according to the degree of turbulent energy dissipation : v D 1 (v constant), v D 1/x, etc. t t t
950
PASCOLI
process at the stellar surface in a natural way, the latter phenomenon appears simply as a direct consequence of the former one. 4.
CONCLUSION
A quantitative model for the ejection of gas by a red giant on the AGB branch has been constructed. It is found that turbulent and magnetic pressures can drive the gas to the
observed expansion velocity D10 km s~1. Resulting magnetic Ðelds in the circumstellar envelope are estimated to be 10~3È10~5 G with distances varying from 1016 to 1018 cm. Following this view, the primary cause of the ejection of massive winds by an AGB red giant would possibly be the magnetic activity present in its degenerate core. Eventually, the magnetic Ðeld in the low-density envelope is found essentially toroidal and its intensity is of the order of 50 G.
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