MAGNETIC

STARS WITH AN EXTERNAL

NON-LINEAR

FORCE-FREE

FIELD

M. A. RAADU* High Altitude Observatory, National Center for Atmospheric Research **, Boulder, Colo., U.S.A.

(Received 13 August, 1971) Abstract. The possible existence of strong magnetic fields in stars is discussed and a method of constructing highly distorted models of magnetic, rotating stars developed. For stars with both poloidal and toroidal fields at the surface a force-free outer boundary condition is necessary. Non-linear solutions of the force-free equations must be used. The force-free equations and the structure equations for a white dwarf are solved simultaneously by a finite difference method.

1. Introduction The largest magnetic field so far observed in a main sequence star is of the order of 30 k for HD215441 (Babcock, 1960). So only if a very small percentage of a star's magnetic flux emerges through the surface can the magnetic energy of main sequence stars be more than a small fraction of the gravitational energy; < 10-8 for a uniform field or ~ 10 .6 for a dipole field (Monaghan, 1968). Mestel (1965a) has argued that meridional circulation induced by rotation will tend to pull magnetic field below the surface of a star. Wright (1969) constructs models of uniformly rotating stars with a magnetic field structure such that radiative equilibrium is possible. For a rotation period of 489 days and a field of 103 G he finds a central field 1600 times the surface field as opposed to the ratio of 80 for models given by Cowling (1945). However the magnetic energy is much less than the centrifugal energy and therefore much less than the gravitational energy. In the case of white dwarfs, Mestel (1965b), has argued that if during the evolution of a star from the main sequence to a white dwarf the magnetic flux is retained but a large fraction of the mass is lost, then the ratio of magnetic to gravitational energy can be considerably increased. Recently observations have been made to detect strong fields in white dwarfs. Angel and Landstreet (1970) looked for slight circular polarization in the wings of the Balmer lines due to the Zeeman effect. They observed nine DA-type dwarfs but could detect no effect. For classification of white dwarf spectra, see Schatzman (1958). They set an upper limit of 105 G. Preston (1970) sets an upper limit of 5 x 105 G for D A stars from a comparison of the quadratic Zeeman effect with observed displacements of absorption lines. However, Kemp et al. (1970) detected a strong circular polarization of 1-3% in the visible light of a semi-DC peculiar white dwarf. They conclude that the observations indicate a mean projected * Present Address: The Astronomical Institute at Utrecht. ** The National Center for Atmospheric Research is sponsored by the National Science Foundation. Astrophysics and Space Science 14 (1971) 464-472. All Rights Reserved Copyright 9 1971 by D. Reidel Publishing Company, Dordrecht-Holland

MAGNETIC STARS W I T H AN EXTERNAL NON-LINEAR FORCE-FREE FIELD

465

B field of 107 G. For such values the magnetic energy is still only a small fraction of the gravitational energy. So far there is only one measured value of the field in a white dwarf and an upper limit has been set for a sample of DA-type dwarfs, and it would be premature to set limits on magnetic fields in all types of white dwarfs. Ostriker and Hartwick (1968) have constructed models of white dwarfs with uniform rotation and fields which vanish at the surface. Both poloidal and toroidal fields are included. However they assume a linear equation relating toroidal field to the poloidal stream function. Consequently they would have to use a linear force-free field as an outer boundary condition if the surface field was not assumed to be zero. The linear force-free field solutions must be contained within an outer spherical boundary or by a field which is finite and uniform at large distances (Liist and Schliiter, 1954). So the linear force-free fields are unacceptable as boundary conditions for magnetic stars. Ostriker and Hartwick argue that finite resistivity can lead to decay of the surface fields of a white dwarf. However their argument only applies if the surface fields are disconnected from the interior magnetic field. If the surface fields penetrate the interior, finite conductivity effects at the surface will cause the surface currents to decay, leading to an imbalance of magnetic torques. Consequently the interior field will untwist inducing currents in the outer layer. Finally all field lines which penetrate the layers of low conductivity will be untwisted and purely poloidal. When this state is reached the field in the low conductivity region will be current free and further decay will depend on conductivities in the interior. However given that there is no surface magnetic field the solutions of Ostriker and Hartwick are particularly interesting in that they can account for the large radii of Sirius B and 40 Eri B both of which are type DA in terms of an interior field which would not contradict the relatively low upper limits so far set to surface fields in white dwarfs. The choice of a zero surface field in a magnetic star with both poloidal and toroidal fields cannot be avoided if linear equations for the magnetic field are assumed. So in order to obtain physical insight into the structure of magnetic stars it is useful to relax the condition oflinearity and obtain solutions with finite surface fields. In this paper I argue that there are non-linear solutions to the force-free field equation which are simply connected and are physically acceptable as boundary conditions for a magnetic star model. As an illustration of this I simultaneously solve the equations for a highly magnetic, rotating degenerate star and a matching external non-linear force-free field. The equations are essentially non-linear everywhere and so a finite difference method is used rather than a function series expansion (cf. Ostriker and Mark, 1967). For a given mass the models obtained have radii considerably greater than that given by the classical mass-radius relation. In some cases the mass is above the Chandrasekhar limit.

2. The Equation of Magnetostatic Balance The equation for magnetostatic balance of a uniformly rotating star is, - v p + e v (~ + ~ )

+ j ^ B = o,

(1)

466

M.A.I ~ D U

where 7r is the axial distance, r sin0, q5 the gravitational potential, f2 the angular velocity, and j is the electric current. The magnetic field is given by toroidal and poloidal components B = Bp + B t ,

(2)

defined by B, -

Vr A t - - ,

(3)

B,

,4)

where t is a unit toroidal vector (Chandrasekhar, 1961), and 0 the stream function. The toroidal component of Equation (1) gives the condition that in axisymmetry the magnetic field exerts no torques, (r

~B t = f

(5)

Now the pressure is given in terms of enthalpy e by ode = dp.

(6)

Then the poloidal component of Equation (1) is, + ~d {89

V -0 rc2V ~

} =_

47r7~2Q d~ __

(7)

where c = e - q5 - 89

.

(8)

Outside the star it is assumed that there is a very low density conducting medium, such as the solar corona. Near the surface of the star the dynamic effects should be small and the field will be close to a force-free field configuration given from Equation (7) by (V~t)

d rl 2-2~

7~2V ~ -

+~tzTr

zh ) = 0 .

(9)

The condition 5 hotds both in the force-free region and within the star. Solutions to Equations (5) and (9) have been found in the cases t h a t f (r is a linear function or when it is chosen in such a way that separation of variables can be used (Liist and Schlfiter, 1954 and 1955). However, as described in the introduction these solutions are not suitable descriptions of the field outside a magnetic star. I choose the function f (r in such a way that at large distances the field is close to a dipole field. The conditions are, for small values of the stream function r ~

sin 2 0 /.

,

(10)

and B~<< Bp.

(11)

MAGNETICSTARSWITHAN EXTERNALNON-LINEARFORCE-FREEFIELD

467

These conditions can be reduced to the form f(0)<<02

as

(12)

0~0.

Hence, for the toroidal term in Equation (9), d

F(0) = ~

{89~z2B2} <<~/3

as

~ --* 0.

(13)

In the present calculations I choose, F (~b) = c@4 ,

(14)

where • is a parameter describing the twist of the field. Consequently, the Equation (7) is now everywhere non-linear. 3. The Stellar Structure Equations The present analysis holds whenever pressure can be expressed as a unique function of density. Then through Equation (6) enthalpy can be written as a function of density. This relation replaces the energy equation. In the case of polytropic models of main sequence stars the structure is physically incorrect for properties relating to energy transfer although the mass distribution may be well represented. In the case of a white dwarf the matter is degenerate and most of the pressure is due to electron degeneracy which is a function of density only. So if the small thermal pressure is neglected the zero temperature models are still a close approximation to the true structure, at least in the interior. In the case of degenerate matter the enthalpy is given by

meC2

D2/~2~2hl[2 - 1}

e = -{(1 + /~m~

JtFl,,,e~, )

(15)

where P r is the momentum at the Fermi surface. This is related to the electron density by He : T

"

(16)

The mass of an electron is in e and of a proton m~. The mean atomic weight per unit of charge is denoted by #. (See, for example, Mestel, 1965b.) For the gravitational field Poisson's equation holds:

V2~b "~ 4zcGQ = 0.

(17)

Equations (5), (7), (17) together with the enthalpy density relation defined through Equations (15) and (16) are a complete set for the problem. 4. Boundary Conditions and the Numerical Method The stream function is set equal to zero at infinity and on the axis of symmetry. The gravitational potential is set equal to a constant at infinity and the normal deviatives

468

M.A.RAADU

on the axis are set equal to zero. The constant at infinity is chosen in such a way that the central value of the potential is a fixed parameter. A system of cylindrical coordinates u, v is employed. These are defined in terms of the usual cylindrical polar coordinates by r =

(1 -

u)/u,

(18)

z = v/(1 - v2).

(19)

The system of equations was solved iterativdy in finite difference form. At each iteration non-linear terms were held constant and Equations (7) and (17) solved by the alternating direction implicit method. By constant rescaling of the stream function and the source term in Equation (7) the total magnetic flux was made equal to a fixed parameter, c was assumed to be a linear function of the stream function although this was not essential to the success of the method. With these constraints and boundary conditions good convergence was gained after five to ten iterations. The equations were solved on a 66 x 66 grid. Most of the errors are expected to arise through the representation of the boundary conditions. However these errors mostly affect the parameters of the model rather than the structure. The accuracy was estimated directly by changing the length scale for the calculation and noting any change in the calculated modds. Errors estimated in this way were less than 1 ~ . The method breaks down for large values of the twist parameter =. However in the models obtained the toroidal and poloidal fields are comparable at the surface of the star. TABLE 1 The parameters for which model stars are calculated. Within each group either the twist parameter or the rotation rate is varied. The models in group 5 have both rotation and magnetic flux. Group

1/7o ~ Flux (cm2 (3)

1 2 3 4 5 6

0.1 1.63 • 1029 0.05 1.63X 10z9 0.1 1.15 x 10g9 0.05 1.15• 1029 0.1 1.63 • 10z9 0.05 -

ct

Rotation period (seconds)

0, 2.5, 5.0

oo

-

12.0 m, 12.0, 8.5

The parameters used to define the models are given in Table I. The models are grouped into sets of three within which either the twisting parameter ~ (given in dimensionless units) or the rotation period was varied. The degeneracy parameter is that given by Chandrasekhar (1939) and defined to be

1 ( l.tmHeo'~_ -- 1 + - - s ) 7z

,

where eo is the central value of the enthalpy.

(20)

MAGNETICSTARSWITH AN EXTERNALNON-LINEARFORCE-FREEFIELD

469

The mean atomic weight per unit charge # was taken to be 2.0. It is unlikely that white dwarfs contain any appreciable amount of hydrogen (Mestel, 1965b). However, pressure induced nuclear reactions could increase the value o f kt in the interior (Schatzman, I958). 5. Results and Discussion

Figure I shows the result of a calculation for a rotating magnetic white dwarf from group 5 with a twisting parameter e = 5.0. Poloidal field lines and surfaces of constant enthalpy are plotted. The high distortion of the models considered is evident.

(I) X

<

EQUATORIAL P L A N E Fig. 1. The poloidal field lines and constant enthalpy surfaces for a magnetic star containing 1.63 • 1029 crn 2 G of magnetic flux with twisting parameter cu The central degeneracy is 0.1 and the rotation period 12 s. Only the northern hemisphere is shown and the plots are axisymmetric about the vertical.

The structure of the white dwarf models is described in Figures 2 and 3. Figure 2 gives the mass as a function of the equatorial radius. The arrows indicate the direction of increasing e. The presence of a strong magnetic field leads to an increase in the possible mass for a star of a given radius and some of the models are above the Chandrasekhar limiting mass. As the magnetic field is twisted the mass of a model must be reduced to maintain the same central degeneracy. This indicates that twisting of the field causes a net compression which in the extreme could lead to prolate models such as those considered by Ostriker and Hartwick (1968). In contrast, rotation increases the mass for a given central degeneracy showing that rotation helps to support material against gravity.

470

M.A.R A A D U

I.I

I

i

I

? 1.0

to to

OCb 4

I

0.9COO 3

0'8 t 0.71

~ "

O... b

I

I

I 4.0

-

~

5.0

EQUATORIAL RADI US Fig. 2. The mass is given as a function of equatorial radius for the computed model stars. The numbers give the groups listed in Table I. The bold arrow indicates the direction of increasing twist parameter ct. For group 6 the small arrows indicate increasing rotation rate. Points a and b give the parameters for undistorted models with central degeneracy 0.05 and 0.1 respectively. The mass is given in terms of the classical limiting mass. The equatorial radius is in megameters.

I

J

I

I

I

/ /

5.0

/

be. / /

-

/

/ / /

to t~3
tr

/

/

3G

/ /

4.0

/ /

J O 13_

oOoO

/

I~

/

5

/

_

/

,,/ /

3.0

dY1

46)~

/ ,

/ /

/

2

I 3.0

I

I

t

4.0

I 5.0

EQUATORIAL RADIUS Fig. 3. The polar and equatorial radii (in megameters) are given for the computed model stars. The numbers refer to the groups listed in Table I. The bold arrow indicates the direction of increasing cc The small arrows for group 6 indicate increasing rotation rate. Points a and b are for undistorted models with degeneracy 0.05 and 0.1.

MAGNETIC STARS W I T H AN EXTERNAL NON-LINEAR FORCE-FREE FIELD

471

Figure 3 gives the equatorial and polar radii of the models. The magnetic field has greatest effect on the polar radius in contrast to rotation which mainly affects the equatorial radius, for a given central degeneracy. The magnetic field greatly distorts the inner regions of the models. However the greatest effect of rotation is away from the axis near the equator. So for rotation the increase of equatorial radius is understandable. Table II gives the values of the maximum poloidal field and the surface values of the poloidal and toroidal fields. The interior field is of the same order of magnitude as the fields in Ostriker and Hartwick's models. However the surface values are TABLE II The computed values of the surface components of the magnetic field at the equator and the maximum poloidal field strength are tabulated. Comparison of the surface toroidal and poloidal fields shows the significance of increasing the parameter c~. 1011 G Group

1

2

3

4 5

a

Bp max

Bpsurf

Btsurf

0 2.5 5.0 0 2.5 5.0 0 2.5 5.0 0 2.5 5.0 0 2.5 5.0

90.6 87.5 83.6 180. 173. 165. 69.5 67.2 64.1 137. 132. 125. 88.9 85.6 81.6

5.60 5.33 4.90 9.26 8.79 8.00 3.76 3.56 3.29 6.17 5.87 5.30 5.35 5.06 4.63

0 1.94 2.99 0 3.33 5.26 0 1.43 2.22 0 2.45 3.95 0 1.88 2.87

several orders of magnitude greater than any of the observational estimates so far made. The largest twisting parameter used gives poloidal and toroidal fields of comparable strength at the surface. This sets the limit to e for which models can be obtained. The percentage of flux emerging from the surface of the models considered decreases systematically with increasing c~. However the interpretation of this effect is not obvious. In all the models I have chosen the function c (~) defined by Equation (8) to be linear. In general a sequence of models with different values of e will not have the same distribution of material over the field lines. Instead the matter distribution will be such as to make the function c ( 0 ) linear. So it is not clear that twisting alone can lead to a decrease of emerging flux.

472

M.A.RAADU

6. Conclusion In this paper I have shown that self-consistent models of magnetic stars can be found with non-zero poloidal and toroidal fields at the surface which match onto an external non-linear force-free field. All the external field lines originally emerge from the surface of the star and at large distances the field is close to a dipole structure. Thus the bulk of the currents giving rise to the field are within a finite volume and the field structure is physically acceptable. The method of solving the system of equations by iteration and finite differencing works well even though the equations are non-linear. It should be possible to extend the method to deal with similar problems of axisymmetric equilibrium. The particular calculations for magnetic white dwarfs clearly show up the physical significance of a toroidal field component, although observations may rule out the extremely distorted models considered here. It is important to investigate the stability of the constructed models. I f an instability is found it would be interesting to see whether it could destroy the surface fields leading to a magnetic star with a purely internal field or whether the total field could be expelled. Thus a stability analysis could lead to an explanation of the relatively low observed field strengths.

Acknowledgements I would like to thank Professor L. Mestel for originally drawing my attention to the relevance of force-free fields to the boundary value problem for magnetic stars. During the progress of this work I had many useful discussions with Dr E. TandbergHanssen. I am indebted to the High Altitude Observatory for sponsoring my visit.

References Angel, J. R. P. and Landstreet, J. D.: 1970, Astrophys. J. Letters 160, L147. Babcock, H. W. : 1960, Astrophys. J. 132, 521. Chandrasekhar, S.: 1961,Hydrodynamic andHydromagnetic Stability, Clarendon Press, Oxford, p. 622. Cowling, T. G.: 1945, Monthly Notices Roy. Astron. Soe. 105, 166. Kemp, J. C., Swedlund, J. B., Landstreet, J. D., and Angel, J. R. P. : 1970, Astrophys. J. Letters 161, L77. Liist, R. and Schlfiter, A. : 1954, Z. Astrophys. 34, 263. Liist, R. and SchliJter, A.: 1955, Z. Astrophys. 38, 190. Mestel, L.: 1965a, in R. Liist (ed.) Stellar and Solar Magnetic Fields, North Holland Pub. Co., Amsterdam, p. 87. Mestel, L. : 1965b, 'Theory of White Dwarfs', in Stars and Stellar Systems (Ed. by L. H. Aller and D. B. McLaughlin), University of Chicago Press, Chicago, p. 297. Monaghan, J. J. : 1968, Z. Astrophys. 69, 146. Ostriker, J. P. and Mark, J. W.-K. : 1968, Astrophys. J. 151, 1075. Ostriker, J. P. and Hartwick, F. D. A.: 1968, Astrophys. J. 153, 797. Preston, G. W. : 1970, Astrophys. 9". Letters 160, L143. Schatzman, E. : 1958, White Dwarfs, North Holland Pub. Co., Amsterdam, p. 104. Wright, G. A. E.: 1969, Monthly Notices Roy. Astron. Soc. 146, 197.