On accretion flow penetration of magnetospheres

Abstract

1 Introduction

Magnetic fields are thought to play an important role in regulation of accretion in a wide variety of astrophysical systems, ranging from protostellar systems, cataclysmic variables (CV) and neutron stars to active galactic nuclei. In many of these systems, the central accreting object is surrounded by a magnetosphere; and a key problem in understanding the accretion process in such systems is the physics of magnetosphere penetration by the accreting material, which is typically highly ionized and weakly collisional. At large distances from the central object, the plasma motion is unimpeded by the weak magnetic field: surface plasma currents screen the external magnetic field B from the plasma interior and plasma propagates ballistically (Chapman & Ferraro 1931). In this ‘diamagnetic’ mode of propagation, the plasma motion is essentially magnetohydrodynamic and can take place as long as β≡4πρV²B²>>1 (ρ and V are the density and flow velocity of the infalling matter, respectively). Penetration is commonly believed to cease in the region where β≲1 (see, e.g., Davidson & Ostriker 1973; Lamb, Pethick & Pines 1973 in the context of accretion-powered pulsars; Hameury, King & Lasota 1986; Lamb 1988 in the context of CVs). It has been argued that turbulent mixing — driven by, for example, Rayleigh—Taylor (Elsner & Lamb 1976, 1977; Arons & Lea 1976) or Kelvin—Helmholtz (Burnard, Lea & Arons 1983) instabilities — can lead to deeper penetration into the stellar magnetosphere by allowing the ionized matter to attach to magnetospheric field lines. However, the existence and the efficiency of such turbulent mixing remain uncertain, especially in cases in which attachment must occur deep within the magnetosphere (as opposed to near the magnetopause).

In many systems — in particular in binary systems with Roche lobe overflow — the accretion occurs in the form of bounded streams. For bounded plasmas it is well known (Chapman 1923; Chandrasekhar 1960; Schmidt 1960; more recently, Borovsky 1987 and references therein) that a different, ‘non-diamagnetic’ mode of propagation is possible even at low β (and in the absence of any turbulent mixing). The mechanism of this penetration is based on charge polarization that can occur in bounded plasmas moving in an ambient magnetic field; the resulting electric field causes plasma to E×B drift in the direction of the initial flow. If the plasma permittivity is sufficiently high to provide the necessary polarization charge and the depolarizing effects are negligible, charge separation can compensate for the restraining influence of the magnetic field (Rosenbluth & Longmire 1957), and plasma propagates without any significant impediment. Depolarizing effects result in drag force that decelerates the plasma flow (similarly to the effect of the Alfvén wave emission considered by Drell, Foley & Ruderman 1965); nevertheless, the penetration length can be significant. In fact, as we shall demonstrate, the penetration length can be comparable to or larger than the size of the magnetosphere in the particular case of a white dwarf magnetosphere.

This mechanism of magnetic field penetration by plasma was originally discussed in the context of geomagnetic storms by Chapman (1923), who concluded that an ionized stream impinging upon the Earth's magnetic field would be deflected by it only very slightly; a closely analogous mechanism is responsible for the instability of a plasma supported by a magnetic field against gravity (Kruskal & Schwarzschild 1954; Rosenbluth & Longmire 1957). Of particular importance for verifying the validity of this process were the theoretical investigations of Schmidt (1960) and the experiments of Baker & Hammel (1965). Schmidt showed that charge separation can explain the observations (Bostick 1956; Wetstone, Ehrlich, & Finkelstein 1960) that sufficiently dense plasma injected into a curved magnetic field propagates unaffected by it; Baker & Hammel corroborated this explanation by observing that injected plasma penetrates the magnetic field, causing only a small disturbance of the vacuum field, and that polarization charge on the plasma boundary controls plasma cross-field propagation: depolarization by means of conducting plates was seen to remove the inflowing plasma momentum. Propagation of plasma injected into the magnetic field has been studied in numerous later experiments (e.g. Meade 1965; Lindberg 1978; Leonard, Dexter & Sprott 1986; Browning et al. 1992) as well as in numerical simulations (Galvez & Borovsky 1991; Cai & Buneman 1992; Neubert et al. 1992).

In the present paper we investigate the plausibility of charge separation as a mechanism of penetration of astrophysical magnetospheres. While we discuss this problem generally, as a specific example we consider the case of AM Herculis cataclysmic variables. In our discussion we take advantage of the earlier work cited above, and extend it to astrophysical conditions, taking into account effects characteristic of stellar environments such as non-uniform magnetic fields, flows and gravitational fields not perpendicular to the magnetospheric fields and depolarization by leakage currents through the ambient medium.

2 General considerations

Consider accretion on a stellar-like object, of radius R_*, surrounded by a dipolar magnetic field

(1)

where R, T and φ are the spherical coordinates, the carets denote unit vectors and B_* is the field strength at the magnetic pole.

We are primarily concerned with the plasma motion at distances less than the Alfvén radius R_A(≡ the distance where β=1); for the motion at R>R_A we shall therefore adopt the conventional picture (Davidson & Ostriker 1973; Lamb et al. 1973; Hameury, King & Lasota 1986; Lamb 1988) and assume that the plasma motion is ballistic. Thus we shall consider the plasma stream as being injected at RR_A with the free-fall velocity, threaded by the magnetospheric field. For the sake of simplicity, we restrict our discussion to the case of purely poloidal injection; we shall see that this restriction means that the plasma flow is essentially two-dimensional.

In the following quantitative discussion we adopt the approach of Schmidt (1960) and employ the guiding-centre approximation to describe the motion of plasma particles. In this approximation, the zeroth-order motion u⁽⁰⁾ consists of the electric drift u_EcE×BB² and parallel flow u_∥ along the magnetic field line:

(2)

In order to simplify to essentials, we shall assume that the inflowing plasma is cold, i.e. that its sound speed is much smaller than the inflow velocity. This assumption allows us to ignore the grad-B drift, thermal expansion of the plasma stream along the field direction during infall and mirroring effects of the dipole field. The cold plasma approximation is justified in many physical situations of interest; when it is not, finite-temperature effects can be readily included, albeit at the price of increased complexity of the analysis.

With the mirroring force neglected in the cold plasma approximation, the parallel flow velocity varies along the trajectory according to

(3)

where d/dt=∂/∂t+u⁽⁰⁾·∇ is the convective derivative.

To obtain the zeroth-order transverse equation of motion requires the determination of the electric field. The latter results from the charge separation that arises as a consequence of the first-order drifts:

(4)

where Ω_sq_sBm_sc is the cyclotron frequency of species s (with charge q_s and mass m_s), g is the gravitational acceleration vector, is the unit vector in the direction of the magnetic field and is the magnetic field curvature. The first term on the right-hand side is the gravitational drift, the second is the curvature drift, and the last is the polarization drift. We have omitted here the grad-B drift, in accord with our assumptions.

The zeroth-order motion is the same for electrons and ions and will be identified with the plasma flow velocity: Vu⁽⁰⁾. First-order drifts are different for electrons and ions and they give rise to the charge polarization. The resulting electric field leads to the transverse plasma drift.

The inflowing plasma will be modelled as a narrow rectangular slab, with poloidal width 2w and toroidal width 2h≪2w≲2R (see Fig. 1). In the chosen inflow geometry, first-order drifts are primarily toroidal, resulting in the predominantly toroidal electrostatic field which gives rise to the transverse motion of the slab in the poloidal plane.

Figure 1.

Schematic geometry of the inflow.

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Introducing local cylindrical coordinates (r, T, z) such that and the z-component of equation (4) can be expressed in the form

(5)

In the same coordinate system as above, the r-component of the zeroth-order velocity is given by

(6)

while

(7)

First-order drifts give rise to the current in the z-direction

(8)

where n_s is the number density of the species s. This current leads to an accumulation of charges in regions where ∂_zJ_z≠0.

Assuming that the plasma density is uniform in the z-direction except in narrow boundary layers, the charge accumulation occurs in these layers: positive charges on one side of the slab and negative charges on the other side. The electric field is uniform in the central part of the slab, but non-uniform in the boundary layer, and it vanishes outside the plasma slab (whether the outside medium is vacuum or lower-density plasma) apart from a small stray field caused by the finite slab size, which gives rise to magnetic field-aligned currents outside the slab (see later). Thus the E×B drift is confined to the slab, as is required for self-consistency of the model.

The self-consistent electric field is determined from charge conservation and the Poisson law. Assuming that the charge layer is narrow compared with the slab width h, we can approximate it by a surface charge with density σ. This charge is found by integrating the charge continuity equation, which yields

(9)

where τ_L is the time-scale of charge loss from the boundary layer (resulting from currents through the ambient medium, etc.). As, in the slab approximation, E=4πσ,equation (9) implies

(10)

Using equation (4), one obtains

(11)

where χ=4πρc²B² is the low-frequency (transverse) electric susceptibility and ρ is the mass density. If the plasma density is sufficiently high, so that χ≫1, the second term on the left-hand side of equation (11) can be neglected so that (with equations 6 and 7 and recalling that V≈u⁽⁰⁾)

(12)

where ν=1/χτ_L. Equations (3) and (12) can thus be expressed in a coordinate-independent form as

(13)

where the subscript ⊥ denotes the component perpendicular to B.

In the absence of charge losses from the boundary (ν=0), the above result implies that the plasma slab is accelerated by the gravitational field as if the magnetic field were absent. This is the reason why, in particular, the growth rate of the Kruskal—Schwarzschild instability of a plasma supported by a magnetic field against gravity is exactly the same as that of the Rayleigh—Taylor instability of a heavy neutral fluid supported against gravity by a lighter fluid (Kruskal & Schwarzschild 1954; Rosenbluth & Longmire 1957). Charge losses give rise to a drag force that decelerates the plasma, a phenomenon akin to the stabilizing effect of line-tying (Kunkel & Guillory 1966; more recently, Berk & Kotelnikov 1993 and references therein).

Observe that the E×B drift occurs on equipotential surfaces, because u_E·∇U=0. Consequently, the potential difference 2U₀ between toroidal boundaries of the slab remains constant during its motion.

We note that while the propagation mechanism discussed here requires deviation from strict charge neutrality, this deviation is exceedingly small for typical astrophysical conditions. To see this, consider the toroidal density profile near the slab edge (z∈[h,h+a]) given by nn₀[1-(z-h)/a], and the relative displacement of electron and ion slabs δ. Then n_c≡n_i-n_e=(δa)n₀<<n₀ if δa<<1. As VcEB and E=4πen_ca, it is easy to see that δr_Lχ, where r_L is the Larmor radius corresponding to the inflow velocity V. As r_L<<a in astrophysical situations of interest, n_c<<n₀ for χ>1.

3 Permittivity of the accretion stream

For the case of a purely radial infall on to an object of mass M, the second term on the left-hand side of equation (13) can be neglected if ντ_ff<<1 where . Thus a sufficient condition for deep magnetospheric penetration by an infalling plasma is χ>>1 and ντ_ff≲1 in a large fraction of the magnetosphere. It should be noted that χ>>1 at least at the entry into the magnetosphere: RR₀∼R_A. As χ can be expressed in terms of the inflow velocity V₀ into the magnetosphere, i.e. it is immediately apparent that for non-relativistic inflows χ>>1 for β∼1, i.e. R∼R_A.

Outside the magnetosphere, at R>R_A, the plasma motion is ballistic. If the accretion is caused by the Roche lobe overflow of the secondary, the gas motion can be described in the manner of Lubov & Shu (1975): the accretion occurs in the form of a stream with the cross-sectional area A and density where is the mass accretion rate. If the mass of the secondary is smaller than that of the primary, where c_s is the isothermal sound velocity, ω=2π/P and P is the orbital period (Lubov & Shu 1975). For such a steady plasma stream inflowing in the vicinity of the magnetic equator, the Alfvén radius is given by

(14)

where V_* is the escape velocity from the stellar surface. For example, for characteristic parameters of AM Herculis cataclysmic variables (M=1 M_⊙,B_*=25 MG,R_*=5×10⁸ cm,P=2 h and plasma temperature T=10⁴ K; see, e.g., Shapiro & Teukolsky 1983; Liebert & Stockman 1985 and Lamb 1988), one finds that R_A≈16R_*; it then follows that χ(R_A)≈3×10⁴.

As the flow penetrates deeper into the magnetosphere, both the magnetic field and the plasma stream density increase. The latter dependence can be deduced from mass conservation, which for a steady-state infall implies that ρVA=constant where A=4hw is the slab cross-section and h and w are the toroidal and poloidal half-widths, respectively. For a radial infall in the poloidal plane, wR=constant. The radial dependence of the toroidal width h can be determined from the fact that the potential difference between the slab boundaries remains constant during the inflow (see earlier) so that hVB=constant. Thus ρwB=constant, which implies that χ∼R². From this it follows that for the above-quoted parameters of cataclysmic variables, χ(R_*)≈100. Thus, for these parameters, the plasma stream will penetrate to the stellar surface, as if the magnetic field were absent, provided the drag force resulting from charge losses is small.¹ As we shall see in the next section, the effect of the latter can be significant and can modify the above conclusion.

4 Depolarization effects on stream penetration

The principal question concerning the validity of the mechanism, proposed in the present paper is whether sufficient polarization can arise on the free-fall time-scale. Within the framework of the guiding-centre theory, the initial polarization is set up by the gravitational drift. By comparing the current J_geng/Ω_i due to the gravitational drift with the required charge density σVB/4πc it is easy to see that the polarization occurs on the time-scale τ_Pτ_ffχ. Thus for χ>>1, the polarization occurs on a time-scale much shorter than the free-fall time.

The next question is under what circumstances the drag force, resulting from depletion of the charge layers, is small enough to allow significant penetration. We address this question in the remainder of the present section.

Several mechanisms can cause charge loss, either by eroding the charged boundary layers or by closing the circuit between the electron and ion layers. If the charge depletion is sufficiently large, it will limit the penetration distance (see Borovsky 1987). We shall limit our discussion to processes that are relevant to the steady-state accretion problem, discussed in the previous section.

In the case of a plasma stream with a finite extent along the magnetic field, the fastest charge-loss mechanism is caused by currents flowing parallel to the magnetic field. If the total current per unit length in the flow direction (i.e. the surface current density) flowing out of each charge layer is j_L, the time-scale for charge loss from the boundary layers is given by

(15)

from which the drag coefficient can be deduced. Before, however, calculating the magnitude of the drag on the accretion stream caused by parallel currents through the ambient plasma, let us address the question of how large a magnetic field modification these currents can produce in the vicinity of the accretion stream. As the charge layer width is small compared with the plasma stream width, which in turn is small compared with both the field line length and the distance to the star, we need to compute only the magnetic field close to the two narrow current ‘sheets’ flowing parallel to the magnetic field out of charged boundary layers. The magnetic field induced in the vicinity of current sheets with surface current density j_L is δB=4πj_LcBwVτ_Lc² (cf. equation 15), where B is the unperturbed magnetic field. Expressing τ_L through the friction coefficient νB²/4πρc²τ_L, then, one finds that δBBντ_ffβw/2R, where β≡4πρV²B². As w<<R and β<1, it follows that δBB<<1, as long as ντ_ff≲1. Thus in the range of parameters for which deep magnetospheric penetration takes place, the magnetic field can be approximated by the stellar dipole field.

As the magnetic field lines are attached to a highly conducting medium (the stellar ‘surface’ or photosphere), the electrostatic potential difference between the accretion stream boundaries results in the potential drop along the magnetic field. For potential differences characteristic for astrophysical systems, discussed in the previous section, the collisional mean free path of particles accelerated by parallel electric fields would exceed the length of the field line. Thus the parallel current would not be collisionally supported; instead it would be determined by space-charge effects.

4.1 Penetration in ‘vacuum’

Let us first consider the case when effects of the ambient plasma are not important (we shall presently make this statement more precise). We shall estimate the magnitude of the steady-state space-charge-limited current in the one-dimensional (1D) approximation, from the Child—Langmuir law (Child 1911; Langmuir 1913). As in astrophysical situations of interest α≡eU₀m_pc²>1 (m_p is the proton mass) for even relatively narrow plasma streams (in the example in the last section, this occurs for h>2 km at RR_A), the space-charge limited flow of both electrons and ions is ultrarelativistic. In such a case the 1D space-charge limited current density is given by (Litwin & Rosner 1998)

(16)

where d is the distance between the electrodes; we shall identify it with the distance between the accretion stream and the stellar surface along the magnetic field line: d≈1.4R, for the geometry specified in Fig. 1. The above current is much larger than the space-charge limited current when only one charge-particle species is involved [which is formally given by equation (16) with α=1/2, cf. Jory & Trivelpiece (1969)]; this is caused by the space-charge neutralizing effect of the second species.

From equations (15) and (16), with j_LaJ_sc(a<<h is the charge layer width) and recalling that σVB/4πc, the friction coefficient ν (=1/χτ_L) can be readily found, yielding

(17)

Let us again consider a purely radial inflow in the vicinity of the magnetic equator. Recalling that hVB=constant and ρwB=constant, it follows that νλxR_Aψ²V_A, where μa_Aw_A,Ω_AeB_Am_pc,xRR_A,ψVV_A and the subscript ‘A’ denotes the value at RR_A. Here and in the following we assume that ah=constant, i.e. that the shape of the density profile remains constant. The equation of motion (13) then becomes

(18)

We solve the above equation in the region 0<x1 for various values of parameter λ, with the initial condition ψ(1)=1, corresponding to the free-fall flow velocity at RR_A. We find that for λλ_crit≈1.66 the flow is initially decelerated and subsequently reaccelerated by the gravitational field; for λ>λ_crit the flow is continuously decelerated until it stops at a distance R>R_crit≈0.51R_A without reaching the stellar surface. In Fig. 2 we show examples of the two classes of solutions and compare them with the free-fall solution.

Figure 2.

Solution of equation (18), normalized to the free-fall solution for λ=1 (long-dashed line), 1.66 (solid line), 1.67 (short-dashed line) and λ=2 (dotted line).

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A deep penetration thus requires λ≲1.6. This is equivalent to

(19)

Thus a deep penetration occurs if the accretion stream is sufficiently narrow; the maximum stream width h_crit depends, in particular, on the charge layer aspect ratio μ(≡a_Aw_A<<1) at RR_A. As a_A<<h_A<<w_A, we assume μ to be in the range ∼10⁻²—10⁻¹. For the case of accretion in a cataclysmic variable, discussed in the previous section, equation (19) implies that the charge-loss-induced drag force can be neglected if the stream cross-field width at the magnetospheric boundary 2h_A<4×10⁴ km (for μ=0.05); the latter value is in particular larger than the estimated stream width ∼c_s/2ω≈5×10³ km.

4.2 Penetration in ambient plasma

The situation can be different if ambient plasma of sufficient high density n_a is present, specifically if n_a is so high that the Bohm current J_Ben_ac_s (Bohm 1949) exceeds J_sc(U₀, d). In this case, an electrostatic sheath is established along the magnetic field, with the width l<d such that J_scl(U₀,l)≈0.6J_B (Lieberman & Lichtenberg 1994).

The drag coefficient is obtained from equation (15), with j_L=0.6J_Ba; one then finds, following reasoning similar to before, the equation of motion

(20)

where ξ=2.4πμR_AJ_BcB_A. Assuming that J_B=constant (see however the later discussion), numerical solutions of equation (20) are found to be qualitatively similar to those shown in Fig. 2; one finds that deep magnetospheric penetration occurs if ξ<2.4, i.e. if

(21)

For the case of accretion on a cataclysmic variable that we discussed earlier, the above result implies that the plasma flow is unimpeded, e.g. (again assuming μ=0.05) if the ambient plasma temperature T_a=10⁴ K (i.e. comparable to the temperature of the accreting stream material) and n_a<n_crit≈10⁸ cm⁻³.

A question thus arises: what is the ambient plasma density compared with n_crit? It is generally assumed in other theoretical works (e.g. see Joss, Katz & Rappaport 1979; Campbell 1983) that the region outside the accretion stream is essentially a vacuum, which a magnetosphere filled with plasma of density ∼n_crit is definitely not. Unfortunately, this question cannot be resolved, even qualitatively, on the basis of existing observations. One therefore has to resort to theoretical considerations. A fully reliable calculation — involving at minimum two-fluid magnetohydrodynamics in three dimensions — is, however, beyond present capabilities, and we therefore have to rely on less certain model estimates.

First we note that if the ambient plasma were in hydrostatic equilibrium with the primary photosphere, its density in the vicinity of the Alfvén radius would be much smaller than n_crit because of its small scaleheight. If the companion star possessed a corona similar to the solar one, its scaleheight would be much larger; nevertheless, because of its low pressure and high ionization degree, this unbounded coronal plasma is not expected to penetrate the strong magnetic field (> 1 kG) of the primary near the Alfvén radius.

Another possibility is that the ambient plasma is not in hydrostatic equilibrium but is instead constantly fed by the mass loss from the accretion stream. We discuss this now.

4.3 Plasma filling of the magnetosphere

In the context of the model discussed in the present paper, it is possible to estimate the ambient plasma density on the basis of mass loss from the accretion stream. This mass loss is caused by the shearing-off of the boundary layers: because the electric field is non-uniform in the charge layer, the E×B flow velocity is non-uniform as well. As discussed at the end of Section 3, for the density profile n(z)=n₀[1-(z-h)T(z-h)/a] (where zh+a,a<<h and T is Heaviside's function), E=4πeδn(z) (recall that δV₀/Ω₀χ₀) and therefore

(22)

here and in the following the subscript 0 denotes quantities in the plasma bulk. The portion of the stream within the sonic points V(z)≳c_s] will penetrate the magnetosphere; however, plasma with a cross-field drift velocity much slower than the sound speed [say, with V(z)<c_s/2] will primarily spread along the magnetic field. This occurs for z>z_s, where z_s is determined from the condition V(z_s)=c_s/2 or

(23)

This implies that

(24)

Thus the mean density of plasma escaping the accretion stream is given by

(25)

and its flux density by The escaping plasma spreads along the magnetic field, reducing the gap between the accretion stream and the stellar surface; the current from the plasma to the star increases until it becomes so large that it erodes the plasma front: J_sc≈J_B where J_B is the Bohm current. As before, we find the density of the surface current flowing out of the charge layer, the corresponding charge-loss time-scale is given by equation (15). The equation of motion then becomes

(26)

where

(27)

As before, we solve the above equation numerically for various values of ζ. We find that a deep penetration occurs for ζ<0.72 or, alternatively, for

(28)

For significantly higher temperatures the flow is strongly decelerated in the vicinity of the Alfvén radius. This behaviour is illustrated in Fig. 3.

Figure 3.

Solution of equation (26), normalized to the free-fall solution for ζ=0.6 (long-dashed line), 0.718 (solid line), 0.719 (short-dashed line) and 0.9 (dotted line).

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Thus, for typical parameters of AM Her cataclysmic variables, a deep penetration can occur if the plasma temperature is less than T_crit∼10⁴ K. This temperature is comparable to the temperature near the Lagrange (L1) point (cf. Schmidt, Liebert & Stockman 1995) and thus can be expected to be the accreting gas temperature in the outer magnetosphere. In the closer proximity of the primary surface, the accreting plasma temperature may be higher, which would prevent further cross-field penetration. Thus, the ‘attachment’ of the accretion stream to the magnetic field occurs at a radius determined by the temperature profile rather than at the Alfvén radius; in the range of parameters of AM Her cataclysmic variables, both the attachment near the Alfvén radius and the penetration to the close vicinity of the stellar surface appear possible.

5 Discussion

In this paper we have argued that plasma accreting on to an object surrounded by a magnetosphere can, under well-defined circumstances, penetrate deeply into the magnetosphere, well past the Alfvén radius (and even to the close proximity of the stellar surface), if its permittivity is sufficiently high. The particular mechanism responsible for cross-field propagation is E×B drift, caused by the electric field arising from polarization currents in the accretion stream. An essential aspect of such a penetrating accretion flow is its finite extent in the direction perpendicular to the stream velocity, i.e. the accretion flow must have the geometric characteristic of a slab or beam, with cross-field dimension smaller than the magnetosphere characteristic spatial scales. For this reason, our solution is not relevant to, for example, uniform spherical accretion, or to the problem of plasma penetration associated with magnetospheres embedded in an ambient wind (such as planetary magnetospheres); in both of these cases, the charge separation necessary to create the electric fields driving the cross-field motions cannot occur without the intercession of further instabilities.² In contrast, examples of systems where such accretion via cross-field penetration might be of considerable interest include accretion on to the primary in AM Her cataclysmic binary systems, and other systems in which accretion occurs in the form of a finite cross-section stream.

A crucial requirement for the above mechanism to function is that the depolarizing effect of the surrounding medium (in particular that of the central object itself) should be sufficiently weak. We showed that these effects result in a drag on the plasma flow, similarly to the effect of Alfvén wave emission on the motion of a satellite in the ionosphere (Drell et al. 1965). Estimating the drag force resulting from magnetic field-aligned currents, we determined the criteria for deep magnetosphere penetration; in particular, we found that deep penetration results for sufficiently narrow streams (such as suggested by the analysis of Lubov & Shu 1975) and for sufficiently low ambient plasma densities and temperatures.

We have employed our model to determine the ambient plasma density on the basis of mass loss from the accretion stream. We then found that deep penetration will result if the accreting plasma temperature is lower that a certain critical temperature T_crit. For the specific example of radial accretion on to a magnetized white dwarf, with a dipolar magnetic field and parameters characteristic of AM Herculis cataclysmic variables, T_crit∼10⁴ K, which is similar to the temperature near the Lagrange (L1) point (Schmidt et al. 1995); thus it can be expected to be the accreting gas temperature, at least in the outer magnetosphere. In the closer proximity of the primary surface, the accreting plasma temperature may be higher, which would prevent further cross-field penetration. In such a situation, the accretion stream would become ‘attached’ to the magnetic field at a radius smaller than the Alfvén radius; the cross-field penetration can be either deep or shallow, depending on the temperature profile.

It is quite striking that, despite an admittedly crude model, the plasma temperature at which deep penetration is found to occur is, in the case of AM Her CVs, comparable to the temperature expected to be the accreting gas temperature in the outer magnetosphere. We therefore conclude that both deep penetration and ‘attachment’ near the Alfvén radius are plausible in the range of parameters characteristic for AM Her CVs. Indeed, observations indicate that while in some systems a deep penetration (even to the close vicinity of the stellar surface) takes place, in other systems the ‘attachment’ occurs far from the surface of the primary. The latter appears to be the case in HU Aquarii (Schwope, Mantel & Horne 1997); the former might be the origin of the accretion ring in V1500 Cygni discussed by Schmidt & Stockman (1991).

We have focused our discussion on steady-state radial accretion on to a magnetized white dwarf with a dipolar magnetic field. In reality, radial accretion will not, in general, occur: the impact parameter of the inflow is finite, owing to the non-vanishing specific angular momentum of the accreting matter. Nevertheless, this example demonstrates that, for realistic parameters, the plasma stream can penetrate very deeply into the magnetosphere. The accreting flow can reach field lines that are never near the magnetopause boundary.

While the general formalism discussed here is valid for an arbitrary time-dependent flow, our specific examples were limited to the steady state. In a non-steady situation, other charge-loss mechanisms than those discussed here have to be taken into account. On the other hand, the inductive effects, which are known to impede discharging of charged boundary layers (e.g. Baker & Hammel 1965), are likely to be important. The analysis of these effects is relegated to future publications.

Acknowledgments

The authors thank Russell Kulsrud and Stephen Libby for helpful discussions. This research has been supported by the DOE ASCI/Alliances Program at the University of Chicago.

References