Magnetic spin–orbit coupling and mass transfer rates in the polars

The magnetic torques resulting from an asynchronous white dwarf in an AM Herculis binary are considered. The magnetic ﬁeld induced in the secondary leads to an orbital torque which affects the evolution rate of its Roche lobe. An over-synchronous primary transfers angular momentum to the orbit which signiﬁcantly reduces the rate of shrinkage of the lobe, for a range of asynchronous periods. Although the effect on the total evolution time of the binary is small, the temporarily reduced accretion rate has important consequences for the condition to approach synchronism.


I N T R O D U C T I O N
The AM Herculis binaries, also known as the polars, are an important class of magnetic cataclysmic variables. The absence of accretion discs in the AM Her systems is unique amongst close binary stars. Related to this is the orbital synchronization of the spin of their strongly magnetic white dwarf primary stars. At least four systems are known to have signi®cant degrees of asynchronism, possibly owing to the disruptive effect of recent nova explosions. However, these systems are observed to be spinning towards synchronism (e.g. Geckeler & Staubert 1997;Schwope et al. 1997).
The major luminosity source in AM Her binaries is the hot, localized accretion column which forms above the white dwarf surface. This results from magnetic ®eld channelling of the supersonic stream originating from the L 1 region of the secondary star. Incoming material passes through a standing shock at the top of the column and undergoes compressional heating. Hard X-rays are emitted from the hot post-shock material, most being reprocessed in the stellar surface to a softer component. All radiations emitted from the accretion column have their observed intensities modulated owing to the rotation of the white dwarf. This effect can be used to measure the rotation period to high accuracy.
As in all close binaries, the mechanism for maintaining mass transfer via Roche lobe over¯ow from the secondary is believed to be orbital angular momentum loss arising from gravitational waves and magnetic wind braking. The latter process generally becomes most effective at orbital periods >3 h. The continuous removal of orbital angular momentum results in a shrinkage of the Roche lobe of the secondary, enabling the L 1 region to remain in contact with the shrinking stellar surface resulting from the mass loss. It was pointed out by King (1997) that in the discless AM Her systems magnetic orbital coupling of an asynchronous primary will affect the orbital evolution and hence the mass transfer rate. The magnitude of this effect was estimated by King & Cannizzo (1998). It is shown here that, for plausible system parameters, asynchronism of the magnetic white dwarf can result in signi®cant perturbations to the mass transfer rate. The detailed analysis shows the dependence of the effect on the degree of asynchronism, with an over-synchronous primary causing a reduction in the mass transfer rate. The present paper considers such modi®cations to the gravitational radiation driven mass transfer rate.
In Section 2 the magnetic ®eld solution resulting from a asynchronous primary is presented. Section 3 applies this solution to calculate the magnetic orbital torque and the torque on the secondary star. In Section 4 the angular momentum evolution equations are formulated, and the modi®cation to the mass transfer rate is calculated for a range of degrees of asynchronism. The results are discussed in Section 5. only add small components to the torque, since their associated ®elds fall more rapidly with distance. Fig. 1 shows the orbital frame used, together with the magnetic orientation angles a; b and the spherical coordinates r; v; f centred on the secondary. The secondary and primary masses are denoted M s and M p , while Q and D are the orbital angular velocity and separation. The synodic (i.e. relative to the orbit) angular velocity of the primary is q qk, and hence b qt. The unit magnetic moment is therefore Ã m p sin a cos qt i H sin a sin qt j H cos a k; 1 where the Cartesian unit vectors are those relative to the primary origin, shown in Fig. 1. The time-varying magnetic ®eld of the primary induces a ®eld in the diffusive secondary obeying the induction equation where h is the diffusivity. The red dwarf secondary will be essentially fully convective and hence a turbulent origin for h is appropriate. It is noted that the associated magnetic force is small in the main body of the secondary so, for a tidally synchronized star, the velocity term can be ignored in the induction equation. The medium between the stars is taken to be essentially a vacuum. Allowing for a conducting magnetosphere will not change the qualitative nature of the torques, although it may enhance their magnitudes (see Campbell 1997b). The magnetic ®eld may be expressed in the generalized poloidal form where F is a poloidal scalar. This scalar can be expanded in a basis of radial functions and spherical harmonics as To make the problem mathematically tractable, the diffusion equation is solved in a sphere of radius R s equal to the mean radius of the distorted secondary. It is a reasonable approximation to expand F in terms up to l 2. The poloidal scalar of the ®eld of the primary is then F p À m 0 m p sin a 8pD 3 r 2 P 1 1 2 cos f sin qt sin f cos qt m 0 m p sin a 8pD 4 r 3 P 0 2 sin qt À P 2 2 1 2 cos 2f sin qt 1 3 sin 2f cos qt ! ; 7 where m 0 is the magnetic permeability and P jmj l v are associated Legendre functions. The magnetic ®eld exterior to the secondary resulting from its induced current source has an associated scalar F s 1 r P 1 1 cos fa 1 sin qt a 2 cos qt 1 r P 1 1 sin fa 3 sin qt a 4 cos qt 1 r 2 P 0 2 b 1 sin qt b 2 cos qt 1 r 2 P 2 2 cos 2fg 1 sin qt g 2 cos qt 1 r 2 P 2 2 sin 2fg 3 sin qt g 4 cos qt;  where a i , b i and g i are constants. The poloidal scalar in the secondary star then takes the form where subscripts s denote surface values here, and the q-dependent coef®cients A i and B i involve surface values of the functions C l r, d l r and their ®rst derivatives.

The primary torque
The foregoing ®eld solution was used by Campbell (1983) to ®nd the torque on the primary star and to relate this to the dissipation of currents in the secondary. The torque is given by where B s r p is the induced ®eld of the secondary at the position of the primary. The ®eld solution yields the torque, averaged over a synodic period 2p=q, as with F 1 and F 2 functions of the surface values C ls , C H ls and d H ls . Fig. 2 shows f q=h, illustrating that it has a single maximum. The dimensionless function f q=h depends purely on the similarity variable It follows that a s , t d =P syn 1=2 , where t d is the magnetic diffusion time through the secondary and P syn 2p=q. Synodically averaged torques are appropriate for t sn q P syn , where t sn is the synchronization time-scale given by with I the moment of inertia of the primary.

The orbital torque
The present paper calculates the orbital magnetic torque and the torque on the secondary, and relates these to T mp . The consequences for mass transfer can then be investigated. The spatial dependence of the secondary's ®eld B s results in a net force on the primary star given by It follows from (1) that Ã m p´Bs sin a cos qt i H´B s sin a sin qt j H´B s cos a k´B s ; and the synodic time average of the last term vanishes, since it contains only linear terms in sin qt and cos qt. Substituting the ®rst two terms in (15), using (8) and (16) and time-averaging, leads to the force components at r; v; f D; p=2; 0 as with F v 0. An equal and opposite force is exerted on the centre of mass of the secondary.
The force F has a non-central component F f which leads to an orbital torque. Adding the primary and secondary orbital torques gives a total magnetic torque about the binary centre of mass of T mo ÀDj H^F : 19 Noting that at the centre of the primary Ã r Àj H and Ã f i H , it follows that The continuity of B at the surface of the secondary requires continuity of F and ¶F= ¶r at r R s . Applying these conditions, using (7)±(9), leads to the relations Using these in (18) and (20) gives The forms of A 2 and B 2 then lead to where the q-dependent functions in the brackets are the same as those occurring in (12) for T mp , apart from their coef®cients.

The secondary torque
The magnetic torque acting about the centre of mass of the secondary is given by where V s is the stellar volume. It follows from (2)±(4) that Substituting this in (23), and expanding the vector products, yields Equations (3) and (4) give Using (26)±(28) in (25), and noting that only the k-component integral will be non-vanishing, leads to The torque integral in (29) is evaluated by noting that each harmonic of F in (4) has the property L 2 F lm ll 1F lm , and that multiplication of F by i gives a phase shift of p=2 in the exponential. Then, using (9) for F, evaluating the angular integrals noting the orthogonality of Y m l v; f, and time-averaging over 2p=q, yields where A i and B i are the q-dependent coef®cients occurring in (9). The functions C l r have the property (see Campbell 1983) where the subscript s denotes the stellar surface. Using this and the forms of A i and B i in (30) gives the secondary magnetic torque where the functions F 1 and F 2 are the same as those occurring in (12) and (22). Adding the primary, secondary and orbital magnetic torques given by (11), (22) and (32) yields This relation gives the magnetic coupling, owing to an asynchronous primary, between the stellar spins and the orbital motion. It states that no angular momentum is lost from the binary as a result of the action of these torques. The spin±orbit coupling enables the primary star to exchange angular momentum and energy with the orbit as it spins towards synchronism. A similar torque balance to (33) was found by King, Frank & Whitehurst (1990) for the synchronous dipole interaction between intrinsic ®elds of the primary and secondary. As in the dissipative asynchronous case considered here, the non-central force on the primary arises from the spatial dependence of the ®eld of the secondary in which it lies.

The angular momentum equations
The angular momentum evolution equations for the secondary, primary and orbit are Ç L s T ms T tid ; 34 These equations are appropriate when the primary is asynchronous and are time averages over a synodic period 2p=q. The accretion torque is then where A is the distance of the centre of the primary from the L 1 point, being related to the orbital separation D by the ®tted formula A 0:50 À 0:23 logM s =M p D 38 (Plavec & Kratochvil 1964). The tidal torque T tid only requires a small degree of asynchronism of the secondary, and couples its spin to the orbital motion. The gravitational radiation torque is given by Lifshitz 1951), where c is the speed of light. This torque drains angular momentum from the orbit, causing a decrease in D and hence an increase in Q.
In binaries containing accretion discs, the rate of transfer of orbital angular momentum owing to mass transfer from the secondary, via L 1 , is balanced by a back¯ow of angular momentum from the disc at the tidal radius. No such orbital feedback occurs in a discless AM Her binary, so there is a net transfer of orbital angular momentum to the primary, and a consequently large accretion torque. This transfer is represented by the accretion torque terms appearing in (35) and (36), with a negative contribution in the latter orbital equation.
The magnetic wind braking torque, believed to be effective at orbital periods >3 h, is not considered in the present paper. Its form is less certain than that arising from gravitational radiation, partly due to lack of knowledge of the dependence of the secondary's surface magnetic ®eld on its rotation rate (Campbell 1997a). Magnetic torques resulting from the interaction of the primary with a permanent (i.e. non-induced) magnetic ®eld emanating from the secondary do not appear in (34)±(36). Such ®elds may be important in the maintenance of synchronism of the primary. However, the asynchronous situation is considered here and the torques resulting from permanent ®elds have vertical components with vanishing synodic averages (Campbell 1997b).
The magnetic torque T ms appearing in (34) will cause some spin evolution of the secondary away from synchronism. However, the tidal torque will become operable and cancel T ms , keeping the star rotating close to the orbital angular velocity Q, with Ç L s . 0. Hence (34) becomes T ms T tid 0; 40 with the vector notation now dropped since all torques are in the z-direction. Eliminating T ms between (33) and (40) yields Use of this in (36) then gives the orbital equation Ç L orb T gr À T a À T mp : 42

Orbital evolution and mass transfer
In this section the orbital evolution equation is used to calculate the mass loss rate from the secondary. The orbital angular momentum can be expressed as For a mass ratio range of 0:1 < M s =M p < 0:8, the mean radius of the Roche lobe of the secondary obeys the equation to good accuracy (Paczynski 1967). Differentiating (43) and (44) The mass transfer time-scale is and is comparable to the orbital evolution time-scale of L orb =j Ç L orb j. The simple mass±radius relation R s R ( n M s M ( 47 will be adopted here, with n . 1:1 (Kippenhahn & Weigert 1990).
Equations (43), (44) and (47) enable the orbital angular momentum to be expressed as A lobe-®lling secondary has R s R L , and (45) and (47) give The terms in the orbital angular momentum equation (42) are now expressed in terms of masses in order to derive an expression for Ç M s . The Keplerian equation together with (44) and (47), gives  where the asynchronous dependence is contained in a s . A relation between the mass of a lobe-®lling secondary and the orbital period follows from (44), (47) and (50) as : 57 Using this and (47) to eliminate R s in (13) gives where jqj=Q is the degree of asynchronism of the primary. Substituting (53)±(56) in (42), and simplifying, gives the mass loss rate of the secondary as Ç M s À 4:68´10 À10 m 2=3 1 À m 2 À 5:73n 1=2 Qf a s n 4 m 4=3 44 À 7m À 3f1 À 0:45 logm=1 À mg 2 M ( yr À1 ; 59 where m M s =M and Q B 0 p 40 MG 2 R p 9:7´10 6 m 6 M 0:79 M ( À4 : 60 Examples of the variation of Ç M s with asynchronism are shown in Figs 3 and 4, normalized by the standard gravitational radiation driven accretion rate Ç M gr À8:0´10 À11 1 À m 2 4 À 7mm 2=3 M ( yr À1 : 61 For the parameters considered, which are typical values, the mass transfer rate is signi®cantly reduced for a range of q=Q around the value at which f q=h has its maximum (cf. Fig. 2). For higher and lower values of q=Q the ratio j Ç M s j=j Ç M gr j exceeds unity. This is caused by the effect of the accretion torque term in (42) which extracts angular momentum from the orbit, owing to the absence of a disc, and hence enhances j Ç M s j. It is noted that the stellar surface in the L 1 region adjusts on the local dynamical time-scale, t dyn , determined by the sound speed. Hence the mass transfer rate can adjust essentially instantaneously to changes in the Roche lobe size on time-scales signi®cantly longer than t dyn (e.g. Lubow & Shu 1975;Edwards & Pringle 1987).
The typical spin evolution time-scale for the primary to move along the asynchronous curves shown in Figs 3 and 4 is t sn , 10 7 yr, where t sn is the synchronization time given by (14). Since the mass transfer time-scale is t M , 10 9 yr, the effect of the lowered accretion rate on the binary evolution is small. However, its effect on the synchronization process can be signi®cant. This is because a critical value of j Ç M s j exists, denoted Ç M c , above which the accretion torque exceeds the synchronizing torque jT mp j for all values of q=Q. At j Ç M s j Ç M c the curves of T mp and T a touch at the maximum of f q=h. Because j Ç M s j is signi®cantly reduced in this region by the effect considered here, it is less likely to exceed Ç M c . It is noted that the condition j Ç M s j < Ç M c is necessary for spin-down towards synchronism, but does not ensure the attainment of synchronism. The right-hand intersection of the torque curves shown in Figs 5 and 6 is an unstable spin equilibrium. If the angular velocity of the primary lies beneath this point, its spin will evolve towards the left-hand intersection which represents a stable spin equilibrium and a slightly asynchronous state. As explained by Campbell (1997b), certain conditions are needed to attain synchronism, including a stable, nondissipative locking mechanism and a suitable magnetic diffusivity in the secondary. It should be emphasized that such conditions are independent of the mass transfer rate lowering effect considered in the present paper.
For an under-synchronous primary the magnetic spin±orbit coupling acts to increase the inertial space angular momentum and energy of the star, and spin it towards synchronism. The transfer of orbital angular momentum will enhance the accretion rate and associated torque which, in this case, aids the synchronization process.
It is noted that the inclusion of magnetospheric currents¯owing between the stars is likely to enhance the strength of the magnetic torques considered here. However, they should not affect the qualitative nature of the torque dependences on the degree of asynchronism. In particular, the induced torque magnitudes should decrease at high and low degrees of asynchronism.
As the primary approaches synchronism, provided that the attainment conditions outlined above are met, the induced magnetic torque reaches zero and hence Ç M s and the accretion torque become independent of q. Corotation must be maintained by a non-dissipative torque which cancels T a in a stable manner. The synodically time-averaged angular momentum equations (34)±(36) do not then apply. Torque balance ensures that Ç L s . 0 and Ç L p . 0, while Ç L orb evolves under the action of T gr and possibly a magnetic wind braking torque for longer period systems.

C O N C L U S I O N S
The work presented here shows that, for asynchronous white dwarfs in discless binary systems, the mass transfer rate can have a signi®cant dependence on the degree of asynchronism. For an over-synchronous primary, this facilitates the approach to synchronism by lowering j Ç M s j for a range of q=Q about the value at which the induced magnetic torque has its maximum. In the case of an under-synchronous primary j Ç M s j is enhanced, but this also aids the approach of the star to corotation. The torque becomes independent of q as corotation is attained. Hence different forms are appropriate for the accretion torque in the cases of synchronous and asynchronous primaries.

AC K N OW L E D G M E N T
The author thanks the Astrophysics Institute, Potsdam, for ®nancial support and hospitality during the period over which this work was done.