THE ASTROPHYSICAL JOURNAL, 514:368372, 1999 March 20
© 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Magnetic Propeller Outflows

R. V. E. LOVELACE

Department of Astronomy, Cornell University, Ithaca, NY 14853-6801; rvl1@cornell.edu

M. M. ROMANOVA

Space Research Institute, Russian Academy of Sciences, Moscow, RussiaDepartment of Astronomy, Cornell University, Ithaca, NY 14853-6801; romanova@astrosun.tn.cornell.edu

AND

G. S. BISNOVATYI-KOGAN

Space Research Institute, Russian Academy of Sciences, Moscow, Russia; gkogan@mx.iki.rssi.ru

Received 1998 April 17; accepted 1998 October 26

ABSTRACT

     A model is developed for magnetic, "propeller"-driven outflows that cause a rapidly rotating magnetized star accreting from a disk to spin-down. Energy and angular momentum lost by the star goes into expelling most of the accreting disk matter. The theory gives an expression for the effective Alfvén radius R (where the inflowing matter is effectively stopped), which depends on the mass accretion rate, the star's mass and magnetic moment, and the star's rotation rate. The model points to a mechanism for "jumps" between spin-down and spin-up evolution and for the reverse transition, which are changes between two possible equilibrium configurations of the system. In, for example, the transition from spin-down to spin-up states, the Alfvén radius R decreases from a value larger than the corotation radius to one that is smaller. In this transition the "propeller" goes from being "on" to being "off." The ratio of the spin-down to spin-up torque (or the ratio for the reverse change) in a jump is shown to be of order unity.

Subject headings: accretion, accretion disksmagnetic fieldsplasmasstars: magnetic fieldsX-rays: stars

§1. INTRODUCTION

     Observations of some X-ray pulsars show remarkable "jumps" between states where the pulsar is spinning-down and one where it is spinning-up. Examples include the objects Cen X-3 (Bildsten et al. 1997) and GX 1+4 (Chakrabarty et al. 1997; Cui 1997). The theoretical problem of disk accretion to a rotating magnetized star has been discussed in many works over a long period (Pringle & Rees 1972; Lynden-Bell & Pringle 1974; Ghosh & Lamb 1979; Wang 1979; Lipunov 1993; Shu et al. 1994; Lovelace, Romanova, & Bisnovatyi-Kogan 1995, hereafter LRBK; Li & Wickramasinghe 1997). However, except for the work by Li & Wickramasinghe (1997), the studies do not specifically address the "propeller" regime (Illarionov & Sunyaev 1975), where the rapid rotation of the star's magnetosphere acts to expel most of the accreting matter and where the star spins-down. Recent computer simulation studies of disk accretion to a rotating star with an aligned dipole magnetic field (Hayashi, Shibata, & Matsumoto 1996; Goodson, Winglee, & Böhm 1997; Miller & Stone 1997) provide evidence of time-dependent outflows but do not give definite evidence for a propeller regime with spin-down of the star. The present work considers the propeller regime and develops a simple physical model where the energy and angular momentum lost by the rotating star goes into a magnetically driven outflow.

§2. THEORY

     We consider the problem of disk accretion onto a rotating magnetized star that has an aligned dipole magnetic field. We focus on the limit where the star is rotating rapidly and the disk-star configuration is as sketched in Figure 1 (see Fig. 3 of LRBK). We consider the flow of mass, angular momentum, and energy into and out of the annular region indicated by the box AACC in this figure, where AA is at radius r and CC is at r. Notice that at this point the values of r and r are unknown. They are determined by the physical considerations discussed here.

Fig. 1

     Consider first the outer surface CC through the disk. The influx of mass into the considered region is



where h is the half-thickness of the disk, and the subscript 2 indicates evaluation at r=r. We assume that mass accretion rate for rr is approximately constant equal to . That is, we consider that outflow from the disk is negligible for rr. The influx of angular momentum into the considered region is



Here



where T is the viscous contribution to the stress tensor, which includes both the turbulent hydrodynamic and turbulent magnetic stresses. The influx of energy into the considered region is



where v/rv/r, with v the Keplerian speed and M the mass of the star, and where w is the enthalpy. For the conditions of interest here, the disk at r is geometrically thin, so that wcv, where c is the sound speed.

     Consider next the fluxes of mass, angular momentum, and energy across the surface AA in Figure 1. For the physical regime considered, where v(r)/r (=the angular rotation rate of the star), the mass accretion across the AA surface is assumed to be small compared with . The reason for this is that any plasma that crosses the AA surface will be "spun-up" to an angular velocity (by the magnetic force) that is substantially larger than the Keplerian value, and thus it will be thrown outward. Thus the efflux of angular momentum across this surface from the considered region is =-T. The efflux of energy across this surface is =-T, where is the angular rotation rate of the star and the inner magnetosphere as shown in Figure 1. For the conditions considered, the star slows down and loses rotational energy, so that T>0.

     We have /=. Because the interaction of the star with the accretion flow is, by assumption, entirely across the surface AA, this is consistent with the spin-down of a star with constant moment of inertia I; that is, E/L=I/(I)=.

     Next we consider the mass, angular momentum, and energy fluxes across the surfaces AC and AC in Figure 1. As mentioned, accretion to the star is small for rr where r is the corotation radius as indicated in Figure 1. Thus, the mass accretion goes mainly into outflows, , where "out" stands for outflows. The angular momentum outflow across the surfaces AC and AC, , must be the difference between the angular momentum lost by the star and the incoming angular momentum of the accretion flow. The angular momentum carried by radiation from the disk is negligible, because (v/c)1, where c is the speed of light. That is, =-. The energy outflow across the AC and AC surfaces is +=-, where is the radiation energy loss rate from the disk surfaces between r=r and r and is the rate of energy loss carried by the outflows.

     Angular momentum conservation gives



We have



We can solve equation (5) for T and thereby eliminate this quantity from the energy equation (6). This gives

     The preceding equations are independent of the nature of the outflows from the disk. At this point we consider the case of magnetically driven outflows as treated by Lovelace, Berk, & Contopoulos (1991, hereafter LBC). In the LBC model the outflows come predominantly from an annular inner region of the disk of radius R, where the disk rotation rate is =. Thus we assume that the outflows come from a region of the disk that is approximately in Keplerian rotation. For the present situation, shown in Figure 1, it is clear that we must have r<R<r. Further, we will assume R is close in value to r with (R-r)/R1. For conditions where the outflow from the disk is of relatively low temperature (sound speed much less than Keplerian speed), equations (16) and (18) of LBC imply the general relation =-3GM/(2R). This equation can be used to eliminate in favor of in equation (7). Recalling that = we have



where R/r<1.

     The energy dissipation in the region of the disk from r=r to r heats the disk, and this heat energy is transformed into outgoing radiation . Thus we have



(Shakura 1973; Shakura & Sunyaev 1973), where (r)= and (r)0. The essential change in (r) occurs in the vicinity of R, so that (3GM/2)(R-r). Thus equation (8) becomes



Thus the power from the spin-down of the star T must be larger than a certain value in order to drive the outflow.

     For magnetically driven outflows, the value of can be written as



(eq. [34] of LBC), where is a dimensionless numerical constant 0.234 and B is the poloidal magnetic field at the base of the outflow at r=R. We take the simple estimate B=/R, which omits corrections for example for compression of the star's field by the inflowing plasma.

     Next we consider the torque on the star T. Because most of the matter inflowing in the accretion disk at r=r is driven off in outflows at distances r>r, the stress is necessarily due to the magnetic field. The magnetic field in the vicinity of r has an essential time-dependence owing to the continual processes of stellar flux leaking outward into the disk, the resulting field loops being inflated by the differential rotation (LRBK) and the reconnection between the open disk field and the closed stellar field loops. The timescale of these processes is t2r/v(r). We make the estimate of the torque T=-2r(2z)BB/(4), where z is the vertical half-thickness of the region where the magnetic stress is significant, and where the angular brackets denote a time average of the field quantities at rr. The magnetic field components, B, B, with B-B necessarily, must be of magnitude less than or of the order of the dipole field B=/r at r=r. The fact that B has the opposite sign to that of B is due to the fact that the B field arises from differential rotation between the region r<r, which rotates at rate , and the region r>r, which rotates at rate (r)<. Also, it is reasonable to assume zr. Therefore



where 1 [the time average of (t)] is a dimensionless constant analogous to the -parameter of Shakura(1973) and Shakura & Sunyaev (1973). [Note that becauserR, , and =(r/R) are of the same order of magnitude.]

     Substituting equations (11) and (12) into equation (10) gives



Here we have introduced two characteristic radii: The first is the corotation radius,



with P(2/)/1 s the pulsar period and MM/M. (For a young stellar object, r1.36×10MP, where P is the period in units of 10 days.) The second is the nominal Alfvén radius,



where the accretion rate /(10 g s), with 10¹⁷ g s1.6×10 M yr^-1, and /10 G cm, with the magnetic field at the star's equatorial surface /r=10 G (10 cm r). [For a young stellar object r1.81×10 cm /(M), where the normalization corresponds to a stellar radius of 10¹¹ cm, a surface magnetic field of 3×10 G, and an accretion rate of 1.6×10 M yr.] The corotation radius is the distance from the star where the centrifugal force on a particle corotating with the star (r) balances the gravitational attraction (GM/r). The Alfvén radius r is the distance from a nonrotating star where the free-fall of a quasi-spherical accretion flow is stopped, which occurs (approximately) where the kinetic energy density of the flow equals the energy density of the star's dipole field. Note that the assumptions leading to equation (13) require R>r.

     Notice that R (or rR) is the effective Alfvén radius for a rotating star. It depends on both r and r, in contrast with the common notion that the Alfvén radius is given by r even for a rotating star. From equation (12), the spin-down rate of the star is I(d/dt)=- /R, where I is the moment of inertia of the star (assumed constant). Thus the spin-down rate depends on both r and r.

     Figure 2 shows the dependence of R on r and r for a sample case. For conditions of a newly formed disk around a young pulsar, the initial system point would be on the upper left-hand part of the curve. Because of the pulsar slowing down (assuming and constant), the system point would move downward and to the right, as indicated by the arrow. In this region of the diagram, Rr/r (for 1), so that the torque on the star is T(GM)/. Thus the braking index is n=-, where n is defined by the relation =-const. . Numerically,



where II/(10 g cm). For Rr, the mass accretion rate to the star is small compared with , but some accretion may occur because of "leakage" of relatively low angular momentum plasma across field lines near r (Arons & Lea 1976).

Fig. 2

     As R decreases, the spin-down torque on the star increases. Over a long interval, R will decrease to a value larger than r, but not much larger. In this limit, mass accretion to the star may become significant. Our treatment can be extended to this limit by noting that =-, =r-T, and =(-GM/2r)-T, where T is given by equation (12). In this limit the accretion luminosity is GM/r, where r is the star's radius. Figure 2 is not changed appreciably for <.

     Further spin-down of the star will cause the system point in Figure 2 to approach the rightmost part of the curve. Further spin-down of the star is impossible. At this point of the evolution, the only possibility is a transition to the spin-up regime. In this regime the effective Alfvén radius is the "turnover radius" r of the disk rotation curve calculated by LRBK, the star spins-up at the rate Id/dt=(GMr) , and most of the disk accretion falls onto the star. The dashed horizontal line in Figure 2 indicates r, which is necessarily less than r.

     The location of the turnover line r in Figure 2 suggests the possible evolution shown by the sequence of points abcda. The system can jump down from point a where the star spins-down to point b where it spins-up. The spin-down torque at a is -/R (where R is the effective Alfvén radius at point a), whereas the spin-up torque at b is (GMr) . The magnitude of the ratio of these torques is



With the system on the r line, it evolves to the left. Because rr, there must be an upward jump from point c to point d. For this case the torque ratio is given by equation (15) with RR. From point d the system evolves to the right. For the example shown in Figure 2, the torque ratio is 3.45 for ab, whereas it is 0.87 for cd. The vertical line cd is at the leftmost position allowed for the considered conditions, but this transition could also occur if the line is shifted to the right. The line ab can be displaced slightly to the right, or it can be displaced to the left to be coincident with the cd line. In the latter case the torque ratio for the spin-down to spin-up jump is approximately equal to the torque ratio for the spin-up to spin-down jump and is 0.87. The smaller the horizontal separation of the ab and the cd lines, the shorter is the time interval between jumps.

     Summarizing, we can say that the horizontal locations of the transitions, ab and cd, are indeterminate within a definite range. The locations of the jumps in the (R, r) plane may in fact be a stochastic or chaotic in nature and give rise to chaotic hysteresis in the spin-down/spin-up behavior of the pulsar. The jumps could be triggered by small variations in the accretion flow (, for example) and magnetic field configuration (the time-dependence of in the torque T). Analysis of the accreting neutron star system Her X-1 (Voges, Atmanspacher, & Scheingraber 1987; Morfill et al. 1989) suggests that the intensity variations are described by a low-dimensional deterministic chaotic model. The transitions between spin-down and spin-up and the reverse transitions may be described by an analogous model.

     The allowed values in Figure 2 have r/rconst.k, where k. This corresponds to pulsar periods



For some long-period pulsars such as GX 1+4, this inequality points to magnetic moment values appreciably larger than unity. Periods much longer than allowed by equation (16) can result for pulsars that accrete from a stellar wind (Bisnovatyi-Kogan 1991). [For a young stellar object, eq. (16) gives P8 days (/M)(/).]

§3. DISCUSSION

     This work presents a new investigation of the propeller regime of disk accretion to a rapidly rotating magnetized star. The work considers the field configuration proposed by LRBK; the theory of LBC on magnetically driven jets; and the conservation of mass, angular momentum, and energy to derive an expression for the effective Alfvén radius R (eq. [13]) and the spin-down torque on the star T (eq. [12]). Our work is in qualitative accord with that of Li & Wickramasinghe (1997), who also consider the propeller effect of Illarionov & Sunyaev (1975). Our Figure 1 is similar to Figure 4 of Li and Wickramasinghe for the spin-down regime, and our earlier work on the spin-up regime (LRBK) agrees with their Figure 3. We find that R depends not only on , , and M, but also on the star's rotation rate . Because R decreases as decreases, there is a minimum value of or a maximum value of the pulsar period P=2/. The model points to a mechanism for jumps between spin-down and spin-up evolution (and the reverse transition). In our picture, in a spin-downspin-up transition, for example, the effective Alfvén radius decreases by an appreciable factor, going from >r to <r. The propeller goes from being "on" to being "off" in this transition, which is a change between two possible equilibrium configurations. The transitions may be stochastic or chaotic in nature, with triggering due to small variations in the accretion flow or in the magnetic field configuration. The ratio of the spin-downspin-up torques (or the ratio for the reverse transition) is found to be of order unity(eq. [15]). This agrees with observations of Cen X-3 (Bildsten et al. 1997), for example, and GX 1+4 (Chakrabarty et al. 1997; Cui 1997).

ACKNOWLEDGMENTS

     We thank Wei Cui for valuable comments on this work. This work was supported in part by NSF grant AST-9320068. Also, this work was made possible in part by Grant RP1-173 of the US Civilian R&D Foundation for the Independent States of the Former Soviet Union. The work of R. V. E. L. was also supported in part by NASA grant NAGW 2293.

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FIGURES

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