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Accretion flows in magnetic CVs
New Astronomy Reviews 44 (2000) 75–80 www.elsevier.nl / locate / newar
Accretion flows in magnetic CVs G.A. Wynn Department of Physics and Astronomy, Leicester University, University Rd, Leicester LE1 7 RH, UK
Abstract I review recent progress in understanding the modes of accretion in the magnetic CVs, and the various spin equilibria that result. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Magnetic cataclysmic variables (mCVs) offer a unique insight into the interaction of plasma with intense magnetic fields. The large magnetic moments ( m1 | 10 32 –10 35 G cm 3 ) of the white dwarfs (WDs) in these systems result in the magnetic field having a pervasive influence on the accretion dynamics. In contrast, neutron stars in the analogous systems (pulsing X-ray binaries) generally have much lower magnetic moments (10 26 –10 30 G cm 3 ), and magnetic effects are only significant close to the stellar surface. The AM Her stars illustrate this difference most dramatically: these mCVs contain the most strongly magnetic WDs ( m1 * 10 34 G cm 3 ), and the WD spin periods are observed to be closely locked to the binary rotation (Pspin 5 Porb ). This period-locking is thought to come about because the interaction between the magnetic fields of the two stars is able to overcome the spin-up torque of the accreting matter. The synchronization condition can be expressed as (e.g. King, Frank & Whitehurst, 1991)
~ 2 m1 m2 2p Mb ]] ]] . , Porb a3
(1)
where m2 is the secondary dipole, a is the binary separation, b is the distance of the L1 point from the ~ is the accretion rate. Furthermore, WD, and M
observations have shown beyond doubt that no accretion disc is present in the AM Her stars. This is in sharp contrast to pulsing X-ray binaries, where observation clearly shows that an accretion disc is able to form. The asynchronous mCVs (DQ Her stars or intermediate polars) are believed to contain WDs with magnetic moments lower ( m1 | 10 32 –10 34 G cm 3 ) than those in the AM Hers. This has led to theories of angular momentum flows developed for the neutron star systems (which were discovered first) being applied to the asynchronous mCVs, with the implicit assumption that they are able to form accretion discs. The distinction between a disc-accreting mCV and a ‘‘discless’’ mCV is a matter of debate and deserves further qualification. I will define an accretion disc by the following criteria: (i) the disc flow is circular and Keplerian, (ii) angular momentum transport within the disc is dominated by viscous stresses, and (iii) angular momentum passing through the L1 point is returned to the binary orbit via tidal stresses at the outer disc edge. The assumption of disc formation in the asynchronous mCVs is questionable, given the large magnetic moments of the WDs. In fact, one may infer that, in at least some of these systems, the accretion dynamics may be closer to the AM Her case than that of the pulsing X-ray binaries. In this paper I discuss the possible modes of accretion in the asynchronous mCVs and
1387-6473 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 00 )00017-8
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G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
the insight that may be gained from the equilibrium values of Pspin .
ric accretion flow without significant pressure sup´ radius (e.g. port we can deduce the spherical Alfven Frank, King & Raine, 1991)
2. Magnetic accretion
R a | 5.5 3 10
The accretion flow between the two stars in a mCV is ionized by collisions as well as the strong X-ray flux from the primary, and is therefore highly conducting. The magnetic force per unit volume can be written as
where R 9 is the primary radius (10 9 cm), L33 is the system luminosity (10 33 erg s 21 ), and m30 is the primary dipole moment (10 30 G cm 3 ). Typically we find R a * a for the AM Her stars, such that magnetic effects are expected to dominate throughout the binary, as observed. In this case we have the hierarchy of timescales t mag , t dyn < t v isc at L1 . In non-magnetic CVs, the accretion stream adopts a circular, Keplerian orbit at a radius R circ from the primary (because of energy dissipation via shocks in the gas flow). An accretion disc forms by the viscous evolution of the stream from R circ in a time t v isc . If R a . R circ in a magnetic system, then we have t mag (R circ ) , t visc (R circ ), and magnetic stresses will quickly (in a time , t visc ) dissipate the stream angular momentum and no accretion disc will form. It is possible to estimate R circ from the conservation of angular momentum 2p b 2 /Porb | (GM1 R circ )1 / 2 , giving
=B 2 (B ?=)B F 5 2 ]] 1 ]]], 8p 4p
(2)
where B is the local magnetic field. The magnetic field exerts an isotropic pressure B 2 / 8p, and carries a tension B 2 / 4p along the magnetic lines of force. These magnetic stresses act as a barrier to the accretion flow, and impede the motion of plasma across field lines. The details of the motion of plasma through the magnetosphere are complex and non-linear. However, three timescales can be used to describe the accretion flow: the local dynamical timescale t dyn | (R 3 /GM1 )1 / 2 , the magnetic timescale t mag and the viscous timescale t visc . Many treatments of magnetic accretion (e.g. Arons & Lea, 1980; Aly & Kuijpers, 1990) predict that plasma penetrates the magnetosphere in the form of dense, diamagnetic filaments (t dyn , t mag ). These filaments are progressively stripped of gas as they cross field lines (either via instabilities or reconnection events). The tenuous, stripped gas is quickly magnetized (t mag , t dyn ) and forced into field-aligned flow. This magnetized flow reaches the magnetic poles of the primary star at highly ballistic speeds, and passes through a strong shock before accreting on to the WD. The hot post-shock gas is a source of intense X-ray emission, which is modulated on Pspin . Thus, mCVs allow direct observation of the spin rates of the WD primaries and offer an insight into the angular momentum flows within the binaries. An estimate of the extent of the region within which the magnetic field is expected to play an important role (i.e. where t mag , t dyn ) can be gained by comparing the ram pressure of accreting material ( r v 2 ) and the local magnetic pressure (B 2 / 8p ). These two quantities are defined to be equal at the ´ radius (R a ). Assuming a spherically symmetAlfven
S D
8
M1 ] M(
SD
R circ b ]] . (1 1 q) ] a a
1/7 22 / 7
R9
22 / 7
L 33
4/7 m 30 cm
(3)
4
(4)
which is a function of the mass ratio q 5 M2 /M1 only. In the case of the pulsing X-ray binaries one finds that R a < R circ and hence t mag (R circ ) 4 t visc (R circ ), and accretion disc formation is unaffected by the magnetic field. The inward spread of the accretion disc will eventually be halted by the magnetic field at some radius R in . The usual definition of R in is the point at which the magnetic field removes angular momentum from the disc at a greater rate than viscous stresses (e.g. Bath, Evans & Pringle, 1974) d ~ ](VK R 2 )u R 2 Bf Bz R 2 u R in 5 M in dr
(5)
where Bf ,Bz are the toroidal and poloidal field components respectively, and VK is the Keplerian angular velocity. In the case of the asynchronous mCVs we find that in many cases R a * R circ , which greatly complicates the accretion dynamics in these systems. King
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
(1993) and Wynn & King (1995) model the accretion flow in the asynchronous mCVs by assuming that material moving through the magnetosphere interacts with the local magnetic field via a velocitydependent acceleration of the general form: fmag 5 2 k[v 2 vf ] ' ,
(6)
where v and vf are the velocities of the material and field lines, and the suffix ' refers to the velocity components perpendicular to the field lines. Eq. (6) is intended to represent the dominant term of the magnetic force, with k ( | t 21 mag ) playing the role of a ‘‘magnetic a ’’. Following the argument of Wynn & King (1995) I assume that the plasma flow is inhomogeneous, diamagnetic and interacts with the field via a surface drag force, characterized by the timescale t mag | cA rb l b B
22
uv' u ]]] , uv 2 vf u '
(7)
´ speed in the medium surwhere cA is the Alfven rounding the plasma, rb is the plasma density, and l b is the typical length-scale over which fieldlines are distorted. Plasma will exchange orbital energy and angular momentum with the field on this timescale, which is dependent on Pspin since uvf u | 2p R /Pspin . Material at radii greater than 2 R co 5 (GM1 P spin / 4p 2 ) 1 / 3 ,
(8)
where the magnetosphere rotates at the local Keplerian rate, will experience a net gain of angular momentum and be ejected from the binary or captured by the secondary star. On the other hand, material inside R co will lose angular momentum and be accreted by the WD. An equilibrium will result when these angular momentum flows balance.
3. Spin equilibria and propeller states The spin rate of a magnetic primary accreting via a disc reaches an equilibrium when the rate at which angular momentum is accreted by the white dwarf is balanced by the braking effect of the magnetic torque on the disc close to R in . Because of the complex nature of the disc-magnetosphere interaction, most models assume that the system is axisymmetric and
77
in steady state. Such models (see e.g. Li, 1999) find R in | 0.5–1.0 R co . This implies that a small magnetosphere (low m1 ) results in fast equilibrium rotation and vice versa. Specifically, for a magnetic system with a truncated accretion disc we expect R in | R co < R circ , which translates (from Eq. (4) and Eq. (8)) into the relation Pspin /Porb < 0.1. This relation is certainly satisfied by the pulsing X-ray binaries where Pspin /Porb & 10 26 . Fig. 1 shows Pspin plotted against Porb for the asynchronous mCVs. The diagram can be roughly divided into three classes of spin equilibrium: 1. Pspin /Porb < 0.1 ( | 5 systems; circles in Fig. 1) 2. Pspin /Porb | 0.1 ( | 11 systems; stars in Fig. 1) 3. Pspin /Porb 4 0.1 (2 systems; triangles in Fig. 1). Of the systems in class 1 above, the disc equilibrium condition is clearly satisfied by at least 2: in GK Per (Pspin 5 381 s, Porb 5 48 hr) Porb is so long that R a < R circ , and in DQ Her (Pspin 5 71 s, Porb 5 4.65 hr) the very short Pspin presumably indicates a rather low magnetic field, leading to the same conclusion. It would seem then, that these two systems are the most likely mCVs to resemble the pulsing X-ray binaries, in accreting via a disrupted Keplerian accretion disc. I will label these disc-accreting, rapidrotators the DQ Her stars. A note of caution should be made here however: the even shorter spin period of AE Aqr (Pspin 5 33 s, Porb 5 9.88 hr) does not imply the presence of a Keplerian accretion disc (see below). The systems in class 2 above are very unlikely to possess accretion discs, since Pspin /Porb * 0.1 implying R in | R co * R circ . King (1993) and Wynn & King (1995) examined the regime in which t mag (R circ ) & t dyn (R circ ) utilizing the prescription (6). Their numerical calculations show that the WD attains a spin equilibrium determined approximately by the condition R co | R circ . The relation
S D
Pspin R co ]] . 11.6 ]] R circ Porb
2/3
q 0.426 , 0.05 , q , 1
(9)
then leads to equilibrium ratio Pspin /Porb | 0.1f(q), and naturally explains the systems in class 2. The exact value of the equilibrium Pspin /Porb is dependent on q, and the extent of this variation is indicated
78
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
Fig. 1. Orbital and spin periods of the asynchronous mCVs. The solid lines represent the limits of the IP spin equilibrium, the dashed line shows synchronization, and the dotted lines represent the limits of the CV period gap.
by the solid lines in Fig. 1. I will term the systems in class 2 the intermediate polars (IPs). The condition t mag (R circ ) & t dyn (R circ ) implies m30 * 10 3 for these systems, however for m30 * 10 4 condition (1) is likely to be satisfied and the system would be observable as an AM Her. The fundamental property of the accretion flow in the IPs, is not the absence of
matter surrounding the WD, but simply the nonKeplerian nature of the velocity field. The two systems in class 3, EX Hya (Pspin 5 67 min, Porb 5 98 min) and RX1238 (Pspin 5 36 min, Porb 5 90 min), cannot possibly contain accretion discs (as defined in Section 1) since their equilibrium Pspin /Porb implies R co 4 R circ . The fact that both of
Fig. 2. Equilibrium spin states for a EX Hya type system. For m30 * 10 4 the system would become an AM Her star.
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
these systems lie below the orbital period gap (2–3 h), leads King & Wynn (1999) to suggest that the ~ and small a in EX Hya results in the low M condition t mag (L1 ) & t dyn (L1 ) being satisfied for m30 * 10 3 . This then implies that the spin equilibrium in these systems is determined by R co | b, which is confirmed by numerical experiment and leads to an equilibrium Pspin /Porb in excellent agreement with observation (see Fig. 2). In this state EX Hya resembles an asynchronous polar, and I note that recent observational evidence in support of this is presented by Wynn, Wheatley and Maxted (in preparation). King & Wynn also show that there is a continuum of spin equilibria, for systems below the period gap, between the IP (R co | R circ ) and the EX Hya (R co | b) equilibria. Fig. 2 shows this set of spin states and the associated values of m1 . It is very likely that the system RX1238 occupies one of these states. I will label these systems EX Hya stars. The above three classes of spin equilibria do not cover the full distribution of mCV types. As mentioned above, from Fig. 1 the most likely candidate to accrete via a truncated accretion disc would be AE Aqr. However, this is not the case. Wynn, King & Horne (1997) show that the rapid spin-down of the WD in this system and its unusual Doppler tomogram could be explained by the rapidly spinning WD acting as a magnetic propeller. Most ( | 99%) of the transferred mass is centrifugally ejected from the binary on a time-scale & t dyn . The required condition for this is t mag (R circ ) & t dyn (R circ ) and R circ 4 R co , 2 implying m30 | 10 for the WD in AE Aqr. It may be that magnetic propellers are much more common amongst CVs than was hitherto realized. The short period SU UMa system WZ Sge has long posed a problem: unlike other SU UMa stars it does not show
79
outburst behaviour every few weeks with superoutbursts at intervals of several months, but only undergoes superoutbursts with a recurrence time of | 33 yr. Smak (1993) argued that the quiescent viscosity parameter (ac ) in WZ Sge must be around a 3 factor 10 smaller than other SU UMa stars, in order to accumulate enough mass to supply the outburst in a stable disc. To produce a long recurrence time with a normal value of ac , the inner, most unstable regions of the disc must be stabilized (e.g. Warner, Livio & Tout, 1996). Recently Wynn, Leach & King (2000) have shown that if the WD in WZ Sge is magnetic ( m30 & 10 2 ; see Patterson et al. (1998)) then the propeller condition will be satisfied. However, unlike AE Aqr, mass is not ejected from the system entirely but stored in a stable ring close to the tidal radius. Thus the unstable inner disc is cleared, allowing mass to accumulate on a timescale | 30 yr before outburst without resorting to low values of ac .
4. Conclusions We are now in a position to crudely classify the mCVs according to m1 and Porb (see Table 1). We see that the EX Hya systems have m1 similar to IPs above the period gap and comparable to the weakest field AM Hers below it. This indicates that mCVs above the gap will evolve to long spin periods below the gap. There is also the possibility that some of the spin states shown in Fig. 2 may be accessible to systems above gap. The link between the systems shown in Table 1 and the magnetic propellers is still not clear. Although it is likely that at least some of the mCV types shown below may well become
Table 1 Classification of mCVs Porb , 2 h 4
m30 * 10 t mag , t dyn u L 1 m30 * 10 3 t mag | t dyn u L 1 m30 * 10 t mag | t dyn u R circ m30 & 10 t mag * t v isc u R circ
Porb . 3 h AM Her 5 1; R co . a EX Hya P spin ] | 0.68; R co | b P orb IP P spin ] | 0.1; R co | R circ P orb DQ Her P spin ] < 0.1; R co < R circ P orb P spin ] P orb
4
m30 . 10 t mag , t dyn u L 1 m30 * 10 3 t mag | t dyn u R circ m30 & 10 3 t mag * t v isc u R circ
AM Her P spin ] 5 1; R co . a P orb IP P spin ] | 0.1; R co | R circ P orb DQ Her P spin ] < 0.1; R co < R circ P orb
80
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
propellers (t mag (R circ ) & t dyn (R circ ) and R circ 4 R co ) ~ cycles. by means of, for example, M
References Aly, J.J. & Kuijpers, J., 1990, A&A, 227, 473. Arons, J. & Lea, S.M., 1980, ApJ, 235, 1016. Bath, G.T., Evans, W.D. & Pringle, J.E., 1974, MNRAS, 166, 113. Frank, J., King, A.R. & Raine, D., 1991, Ap&SS, 179, 335. King, A.R., 1993, MNRAS, 261, 144.
King, A.R., Frank, J. & Whitehurst, R., 1991, MNRAS, 250, 152. King, A.R. & Wynn, G.A., 1999, MNRAS, 310, 203. Li, J., 1999, in: Annapolis Workshop on Magnetic Cataclysmic Variables, Coel Hellier & Koji Mukai (Eds.). Patterson, J., Richman, H., Kemp, J., & Mukai, K., 1998, PASP, 110, 403. Smak, J., 1993, AcA, 43, 101. Warner, B., Livio, M. & Tout, C.A., 1996, MNRAS, 282, 735. Wynn, G.A. & King, A.R., 1995, MNRAS, 275, 9. Wynn, G.A., King, A.R. & Horne, K., 1997, MNRAS, 286, 436. Wynn, G.A., Leach, R., & King, A.R., 2000, MNRAS, submitted.
Accretion flows in magnetic CVs G.A. Wynn Department of Physics and Astronomy, Leicester University, University Rd, Leicester LE1 7 RH, UK
Abstract I review recent progress in understanding the modes of accretion in the magnetic CVs, and the various spin equilibria that result. 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Magnetic cataclysmic variables (mCVs) offer a unique insight into the interaction of plasma with intense magnetic fields. The large magnetic moments ( m1 | 10 32 –10 35 G cm 3 ) of the white dwarfs (WDs) in these systems result in the magnetic field having a pervasive influence on the accretion dynamics. In contrast, neutron stars in the analogous systems (pulsing X-ray binaries) generally have much lower magnetic moments (10 26 –10 30 G cm 3 ), and magnetic effects are only significant close to the stellar surface. The AM Her stars illustrate this difference most dramatically: these mCVs contain the most strongly magnetic WDs ( m1 * 10 34 G cm 3 ), and the WD spin periods are observed to be closely locked to the binary rotation (Pspin 5 Porb ). This period-locking is thought to come about because the interaction between the magnetic fields of the two stars is able to overcome the spin-up torque of the accreting matter. The synchronization condition can be expressed as (e.g. King, Frank & Whitehurst, 1991)
~ 2 m1 m2 2p Mb ]] ]] . , Porb a3
(1)
where m2 is the secondary dipole, a is the binary separation, b is the distance of the L1 point from the ~ is the accretion rate. Furthermore, WD, and M
observations have shown beyond doubt that no accretion disc is present in the AM Her stars. This is in sharp contrast to pulsing X-ray binaries, where observation clearly shows that an accretion disc is able to form. The asynchronous mCVs (DQ Her stars or intermediate polars) are believed to contain WDs with magnetic moments lower ( m1 | 10 32 –10 34 G cm 3 ) than those in the AM Hers. This has led to theories of angular momentum flows developed for the neutron star systems (which were discovered first) being applied to the asynchronous mCVs, with the implicit assumption that they are able to form accretion discs. The distinction between a disc-accreting mCV and a ‘‘discless’’ mCV is a matter of debate and deserves further qualification. I will define an accretion disc by the following criteria: (i) the disc flow is circular and Keplerian, (ii) angular momentum transport within the disc is dominated by viscous stresses, and (iii) angular momentum passing through the L1 point is returned to the binary orbit via tidal stresses at the outer disc edge. The assumption of disc formation in the asynchronous mCVs is questionable, given the large magnetic moments of the WDs. In fact, one may infer that, in at least some of these systems, the accretion dynamics may be closer to the AM Her case than that of the pulsing X-ray binaries. In this paper I discuss the possible modes of accretion in the asynchronous mCVs and
1387-6473 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 00 )00017-8
76
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
the insight that may be gained from the equilibrium values of Pspin .
ric accretion flow without significant pressure sup´ radius (e.g. port we can deduce the spherical Alfven Frank, King & Raine, 1991)
2. Magnetic accretion
R a | 5.5 3 10
The accretion flow between the two stars in a mCV is ionized by collisions as well as the strong X-ray flux from the primary, and is therefore highly conducting. The magnetic force per unit volume can be written as
where R 9 is the primary radius (10 9 cm), L33 is the system luminosity (10 33 erg s 21 ), and m30 is the primary dipole moment (10 30 G cm 3 ). Typically we find R a * a for the AM Her stars, such that magnetic effects are expected to dominate throughout the binary, as observed. In this case we have the hierarchy of timescales t mag , t dyn < t v isc at L1 . In non-magnetic CVs, the accretion stream adopts a circular, Keplerian orbit at a radius R circ from the primary (because of energy dissipation via shocks in the gas flow). An accretion disc forms by the viscous evolution of the stream from R circ in a time t v isc . If R a . R circ in a magnetic system, then we have t mag (R circ ) , t visc (R circ ), and magnetic stresses will quickly (in a time , t visc ) dissipate the stream angular momentum and no accretion disc will form. It is possible to estimate R circ from the conservation of angular momentum 2p b 2 /Porb | (GM1 R circ )1 / 2 , giving
=B 2 (B ?=)B F 5 2 ]] 1 ]]], 8p 4p
(2)
where B is the local magnetic field. The magnetic field exerts an isotropic pressure B 2 / 8p, and carries a tension B 2 / 4p along the magnetic lines of force. These magnetic stresses act as a barrier to the accretion flow, and impede the motion of plasma across field lines. The details of the motion of plasma through the magnetosphere are complex and non-linear. However, three timescales can be used to describe the accretion flow: the local dynamical timescale t dyn | (R 3 /GM1 )1 / 2 , the magnetic timescale t mag and the viscous timescale t visc . Many treatments of magnetic accretion (e.g. Arons & Lea, 1980; Aly & Kuijpers, 1990) predict that plasma penetrates the magnetosphere in the form of dense, diamagnetic filaments (t dyn , t mag ). These filaments are progressively stripped of gas as they cross field lines (either via instabilities or reconnection events). The tenuous, stripped gas is quickly magnetized (t mag , t dyn ) and forced into field-aligned flow. This magnetized flow reaches the magnetic poles of the primary star at highly ballistic speeds, and passes through a strong shock before accreting on to the WD. The hot post-shock gas is a source of intense X-ray emission, which is modulated on Pspin . Thus, mCVs allow direct observation of the spin rates of the WD primaries and offer an insight into the angular momentum flows within the binaries. An estimate of the extent of the region within which the magnetic field is expected to play an important role (i.e. where t mag , t dyn ) can be gained by comparing the ram pressure of accreting material ( r v 2 ) and the local magnetic pressure (B 2 / 8p ). These two quantities are defined to be equal at the ´ radius (R a ). Assuming a spherically symmetAlfven
S D
8
M1 ] M(
SD
R circ b ]] . (1 1 q) ] a a
1/7 22 / 7
R9
22 / 7
L 33
4/7 m 30 cm
(3)
4
(4)
which is a function of the mass ratio q 5 M2 /M1 only. In the case of the pulsing X-ray binaries one finds that R a < R circ and hence t mag (R circ ) 4 t visc (R circ ), and accretion disc formation is unaffected by the magnetic field. The inward spread of the accretion disc will eventually be halted by the magnetic field at some radius R in . The usual definition of R in is the point at which the magnetic field removes angular momentum from the disc at a greater rate than viscous stresses (e.g. Bath, Evans & Pringle, 1974) d ~ ](VK R 2 )u R 2 Bf Bz R 2 u R in 5 M in dr
(5)
where Bf ,Bz are the toroidal and poloidal field components respectively, and VK is the Keplerian angular velocity. In the case of the asynchronous mCVs we find that in many cases R a * R circ , which greatly complicates the accretion dynamics in these systems. King
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
(1993) and Wynn & King (1995) model the accretion flow in the asynchronous mCVs by assuming that material moving through the magnetosphere interacts with the local magnetic field via a velocitydependent acceleration of the general form: fmag 5 2 k[v 2 vf ] ' ,
(6)
where v and vf are the velocities of the material and field lines, and the suffix ' refers to the velocity components perpendicular to the field lines. Eq. (6) is intended to represent the dominant term of the magnetic force, with k ( | t 21 mag ) playing the role of a ‘‘magnetic a ’’. Following the argument of Wynn & King (1995) I assume that the plasma flow is inhomogeneous, diamagnetic and interacts with the field via a surface drag force, characterized by the timescale t mag | cA rb l b B
22
uv' u ]]] , uv 2 vf u '
(7)
´ speed in the medium surwhere cA is the Alfven rounding the plasma, rb is the plasma density, and l b is the typical length-scale over which fieldlines are distorted. Plasma will exchange orbital energy and angular momentum with the field on this timescale, which is dependent on Pspin since uvf u | 2p R /Pspin . Material at radii greater than 2 R co 5 (GM1 P spin / 4p 2 ) 1 / 3 ,
(8)
where the magnetosphere rotates at the local Keplerian rate, will experience a net gain of angular momentum and be ejected from the binary or captured by the secondary star. On the other hand, material inside R co will lose angular momentum and be accreted by the WD. An equilibrium will result when these angular momentum flows balance.
3. Spin equilibria and propeller states The spin rate of a magnetic primary accreting via a disc reaches an equilibrium when the rate at which angular momentum is accreted by the white dwarf is balanced by the braking effect of the magnetic torque on the disc close to R in . Because of the complex nature of the disc-magnetosphere interaction, most models assume that the system is axisymmetric and
77
in steady state. Such models (see e.g. Li, 1999) find R in | 0.5–1.0 R co . This implies that a small magnetosphere (low m1 ) results in fast equilibrium rotation and vice versa. Specifically, for a magnetic system with a truncated accretion disc we expect R in | R co < R circ , which translates (from Eq. (4) and Eq. (8)) into the relation Pspin /Porb < 0.1. This relation is certainly satisfied by the pulsing X-ray binaries where Pspin /Porb & 10 26 . Fig. 1 shows Pspin plotted against Porb for the asynchronous mCVs. The diagram can be roughly divided into three classes of spin equilibrium: 1. Pspin /Porb < 0.1 ( | 5 systems; circles in Fig. 1) 2. Pspin /Porb | 0.1 ( | 11 systems; stars in Fig. 1) 3. Pspin /Porb 4 0.1 (2 systems; triangles in Fig. 1). Of the systems in class 1 above, the disc equilibrium condition is clearly satisfied by at least 2: in GK Per (Pspin 5 381 s, Porb 5 48 hr) Porb is so long that R a < R circ , and in DQ Her (Pspin 5 71 s, Porb 5 4.65 hr) the very short Pspin presumably indicates a rather low magnetic field, leading to the same conclusion. It would seem then, that these two systems are the most likely mCVs to resemble the pulsing X-ray binaries, in accreting via a disrupted Keplerian accretion disc. I will label these disc-accreting, rapidrotators the DQ Her stars. A note of caution should be made here however: the even shorter spin period of AE Aqr (Pspin 5 33 s, Porb 5 9.88 hr) does not imply the presence of a Keplerian accretion disc (see below). The systems in class 2 above are very unlikely to possess accretion discs, since Pspin /Porb * 0.1 implying R in | R co * R circ . King (1993) and Wynn & King (1995) examined the regime in which t mag (R circ ) & t dyn (R circ ) utilizing the prescription (6). Their numerical calculations show that the WD attains a spin equilibrium determined approximately by the condition R co | R circ . The relation
S D
Pspin R co ]] . 11.6 ]] R circ Porb
2/3
q 0.426 , 0.05 , q , 1
(9)
then leads to equilibrium ratio Pspin /Porb | 0.1f(q), and naturally explains the systems in class 2. The exact value of the equilibrium Pspin /Porb is dependent on q, and the extent of this variation is indicated
78
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
Fig. 1. Orbital and spin periods of the asynchronous mCVs. The solid lines represent the limits of the IP spin equilibrium, the dashed line shows synchronization, and the dotted lines represent the limits of the CV period gap.
by the solid lines in Fig. 1. I will term the systems in class 2 the intermediate polars (IPs). The condition t mag (R circ ) & t dyn (R circ ) implies m30 * 10 3 for these systems, however for m30 * 10 4 condition (1) is likely to be satisfied and the system would be observable as an AM Her. The fundamental property of the accretion flow in the IPs, is not the absence of
matter surrounding the WD, but simply the nonKeplerian nature of the velocity field. The two systems in class 3, EX Hya (Pspin 5 67 min, Porb 5 98 min) and RX1238 (Pspin 5 36 min, Porb 5 90 min), cannot possibly contain accretion discs (as defined in Section 1) since their equilibrium Pspin /Porb implies R co 4 R circ . The fact that both of
Fig. 2. Equilibrium spin states for a EX Hya type system. For m30 * 10 4 the system would become an AM Her star.
G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
these systems lie below the orbital period gap (2–3 h), leads King & Wynn (1999) to suggest that the ~ and small a in EX Hya results in the low M condition t mag (L1 ) & t dyn (L1 ) being satisfied for m30 * 10 3 . This then implies that the spin equilibrium in these systems is determined by R co | b, which is confirmed by numerical experiment and leads to an equilibrium Pspin /Porb in excellent agreement with observation (see Fig. 2). In this state EX Hya resembles an asynchronous polar, and I note that recent observational evidence in support of this is presented by Wynn, Wheatley and Maxted (in preparation). King & Wynn also show that there is a continuum of spin equilibria, for systems below the period gap, between the IP (R co | R circ ) and the EX Hya (R co | b) equilibria. Fig. 2 shows this set of spin states and the associated values of m1 . It is very likely that the system RX1238 occupies one of these states. I will label these systems EX Hya stars. The above three classes of spin equilibria do not cover the full distribution of mCV types. As mentioned above, from Fig. 1 the most likely candidate to accrete via a truncated accretion disc would be AE Aqr. However, this is not the case. Wynn, King & Horne (1997) show that the rapid spin-down of the WD in this system and its unusual Doppler tomogram could be explained by the rapidly spinning WD acting as a magnetic propeller. Most ( | 99%) of the transferred mass is centrifugally ejected from the binary on a time-scale & t dyn . The required condition for this is t mag (R circ ) & t dyn (R circ ) and R circ 4 R co , 2 implying m30 | 10 for the WD in AE Aqr. It may be that magnetic propellers are much more common amongst CVs than was hitherto realized. The short period SU UMa system WZ Sge has long posed a problem: unlike other SU UMa stars it does not show
79
outburst behaviour every few weeks with superoutbursts at intervals of several months, but only undergoes superoutbursts with a recurrence time of | 33 yr. Smak (1993) argued that the quiescent viscosity parameter (ac ) in WZ Sge must be around a 3 factor 10 smaller than other SU UMa stars, in order to accumulate enough mass to supply the outburst in a stable disc. To produce a long recurrence time with a normal value of ac , the inner, most unstable regions of the disc must be stabilized (e.g. Warner, Livio & Tout, 1996). Recently Wynn, Leach & King (2000) have shown that if the WD in WZ Sge is magnetic ( m30 & 10 2 ; see Patterson et al. (1998)) then the propeller condition will be satisfied. However, unlike AE Aqr, mass is not ejected from the system entirely but stored in a stable ring close to the tidal radius. Thus the unstable inner disc is cleared, allowing mass to accumulate on a timescale | 30 yr before outburst without resorting to low values of ac .
4. Conclusions We are now in a position to crudely classify the mCVs according to m1 and Porb (see Table 1). We see that the EX Hya systems have m1 similar to IPs above the period gap and comparable to the weakest field AM Hers below it. This indicates that mCVs above the gap will evolve to long spin periods below the gap. There is also the possibility that some of the spin states shown in Fig. 2 may be accessible to systems above gap. The link between the systems shown in Table 1 and the magnetic propellers is still not clear. Although it is likely that at least some of the mCV types shown below may well become
Table 1 Classification of mCVs Porb , 2 h 4
m30 * 10 t mag , t dyn u L 1 m30 * 10 3 t mag | t dyn u L 1 m30 * 10 t mag | t dyn u R circ m30 & 10 t mag * t v isc u R circ
Porb . 3 h AM Her 5 1; R co . a EX Hya P spin ] | 0.68; R co | b P orb IP P spin ] | 0.1; R co | R circ P orb DQ Her P spin ] < 0.1; R co < R circ P orb P spin ] P orb
4
m30 . 10 t mag , t dyn u L 1 m30 * 10 3 t mag | t dyn u R circ m30 & 10 3 t mag * t v isc u R circ
AM Her P spin ] 5 1; R co . a P orb IP P spin ] | 0.1; R co | R circ P orb DQ Her P spin ] < 0.1; R co < R circ P orb
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G. A. Wynn / New Astronomy Reviews 44 (2000) 75 – 80
propellers (t mag (R circ ) & t dyn (R circ ) and R circ 4 R co ) ~ cycles. by means of, for example, M
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