# Magnetic Propeller Outflows

###### Abstract

A model is developed for magnetic ‘propeller’-driven outflows which 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 (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 transistion from spin-down to spin-up states the Alfvén radius decreases from a value larger than the corotation radius to one which is smaller. In this transistion 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.

MAGNETIC PROPELLER OUTFLOWS \rightheadLOVECALE, ROMANOVA, BISNOVATYI–KOGAN

Department of Astronomy,
Cornell University, Ithaca, NY 14853-6801;
\affilSpace Research Institute,
Russian Academy of Sciences, Moscow, Russia; and

Department of Astronomy,
Cornell University, Ithaca, NY 14853-6801;
\affilSpace Research Institute,
Russian Academy of Sciences, Moscow,
Russia;

Accepted to the Astrophysical Journal

accretion, accretion disks—plasmas—magnetic fields—stars: magnetic fields—X-rays: stars

## 1 Introduction

Observations of some X-ray pulsars show remarkable ‘jumps’ between states where the pulsar is spin-ning-down to one where it is spinning-up. Examples include the objects Cen X-3 (Chakrabarty et al. 1993) 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 expell 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 which 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 Figure 3 of LRBK). We consider the flow of mass, angular momentum, and energy into and out of the annular region indicated by the box in this figure, where is at radius and is at . Notice that at this point the values of and are unknown. They are determined by the physical considerations discussed here.

fig1

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

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

Here,

where 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 with the Keplerian speed and the mass of the star, and where is the enthalpy. For conditions of interest here the disk at is geometrically thin so that , where is the sound speed.

Consider next the fluxes of mass, angular momentum, and energy across the surface in Figure 1. For the physical regime considered, where ( the angular rotation rate of the star), the mass accretion across the surface is assumed to be small compared with . The reason for this is that any plasma which crosses the surface will be ‘spun-up’ to an angular velocity (by the magnetic force) which is substantially larger than the Keplerian value, and thus it will be thrown outwards. Thus the efflux of angular momentum across this surface from the considered region is The efflux of energy across this surface is 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 .

We have . Because the interaction of the star with the accretion flow is by assumption entirely across the surface , this is consistent with the spin-down of a star with constant moment of inertia ; that is, .

Next we consider the mass, angular momentum, and energy fluxes across the surfaces and in Figure 1. As mentioned, accretion to the star is small for where is the corotation radius as indicated in Figure 1. Thus, the mass accretion goes mainly into outflows, where “” stands for outflows. The angular momentum outflow across the surfaces and , , 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 , where is the speed of light. That is, The energy outflow across the and surfaces is where is the radiation energy loss rate from the disk surfaces between and , and is the rate of energy loss carried by the outflows.

Angular momentum conservation gives

We have

We can solve equation (4) for and thereby eliminate this quantity from the energy equation (5). This gives

The preceeding 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, and Contopoulos (1991, hereafter LBC). In the LBC model the outflows come predominantly from an annular inner region of the disk of radius where the disk rotation rate is . Thus we assume that the outflows come from a region of the disk which is approximately in Keplerian rotation. For the present situation, shown in Figure 1, it is clear that we must have Further, we will assume is close in value to with . For conditions where the outflow from the disk is relatively low temperature (sound speed much less than Keplerian speed), equations (16) and (18) of LBC imply the general relation This equation can be used to eliminate in favor of in equation (6). Recalling that we have

where .

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

(Shakura 1973; Shakura & Sunyaev 1973), where and . The essential change in occurs in the vicinity of so that Thus equation (7) becomes

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

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

(equation 34 of LBC), where is a dimensionless numerical constant , and is the poloidal magnetic field at the base of the outflow at . We take the simple estimate which omits corrections for example for compression of the star’s field by the inflowing plasma.

Next we consider the torque on the star . Because most of the matter inflowing in the accretion disk at is driven off in outflows at distances , the stress is necessarily due to the magnetic field. The magnetic field in the vicinity of 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 time scale of these processes is . We make the estimate of the torque where 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 quantites at . The magnetic field components, , with necessarily, must be of magnitude less than or of the order of the dipole field at . The fact that has the opposite sign to that of is due to the fact that the field arises differential rotation between the region , which rotates at rate , and the region , which rotates at rate . Also, it is reasonable to assume . Therefore,

where (the time average of ) is a dimensionless constant analogous to the parameter of Shakura (1973) and Shakura & Sunyaev (1973). (Note that because , and are of the same order of magnitude.)

Substituting equations (10) and (11) into equation (9) gives

Here, we have introduced two characteristic radii – the corotation radius,

with the pulsar period and . [For a young stellar object, , where is the period in units of days.] The second is the nominal Alfvén radius

where the accretion rate with and with the magnetic field at the star’s equatorial surface . [For a young stellar object , where the normalization corresponds to a stellar radius of cm, a surface magnetic field of G, and an accretion rate of .] The corotation radius is the distance from the star where the centrifugual force on a particle corotating with the star () balances the gravitational attraction (). The Alfvén radius is the distance from a non-rotating 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 (12) require .

Notice that (or ) is the effective Alfvén radius for a rotating star. It depends on both and in contrast with the common notion that the Alfvén radius is given by even for a rotating star. ¿From equation (11), the spin-down rate of the star is , where is the moment of inertia of the star (assumed constant). Thus the spin-down rate depends on both and .

Figure 2 shows the dependence of on and 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. Due to 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, (for ), so that the torque on the star is . Thus the braking index is , where is defined by the relation . Numerically,

where . For , the mass accretion rate to the star is small compared with , but some accretion may occur due to ‘leakage’ of relatively low angular momentum plasma across field lines near (Arons & Lea 1976).

As decreases, the spin-down torque on the star increases. Over a long interval, will decrease to a value larger than 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 , , and , where is given by equation (11). In this limit the accretion luminosity is , where 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 right-most 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’ of the disk rotation curve calculated by LRBK, the star spins-up at the rate , and most of the disk accretion falls onto the star. The dashed horizontal line in Figure 2 indicates which is necessarily less than .

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

With the system on the line, it evolves to the left. Because , there must be an upward jump from point to point . For this case the torque ratio is given by equation (14) with . From point the system evolves to the right. For the example shown in Figure 2, the torque ratio is for whereas it is for . The vertical line is at the left-most position allowed for the considered conditions, but this transistion could also occur if the line is shift to the right. The line can be displaced slightly to the right or it can be displaced to the left to be coincident with the 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 . The smaller the horizontal separation of the and the lines, the shorter is the time interval between jumps.

Summarizing, we can say that the horizontal locations of the transistions, , and , are indeterminate within a definite range. The locations of the jumps in the plane may in fact be a stochastic or chaotic in nature and give rise to chaotic hysteresis in the of 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 ). 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 transistions between spin-down and spin-up and the reverse transistions may be described by an analogous model.

The allowed values in Figure 2 have , where . 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 (15) can result for pulsars which accrete from a stellar wind (Bisnovatyi-Kogan 1991). [For a young stellar object, equation (15) gives .]

## 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 (equation 12) and the spin-down torque on the star (equation 11). Our work is in qualitative accord with that of Li and Wickramasinghe (1997) who also consider the propeller effect of Illarionov and 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 depends not only on , , and , but also on the star’s rotation rate . Because decreases as decreases, there is a minimum value of or a maximum value of the pulsar period . 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-down to spin-up transition, for example, the effective Alfvén radius decreases by an appreciable factor going from to . The propeller goes from being “on” to being “off” in this transistion, which is a change between two possible equilibrium configurations. The transistions 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-down to spin-up torques (or the ratio for the reverse transition) is found to be of order unity (equation 14). This agrees with observations of for example Cen X-3 (Chakrabarty et al. 1993) and GX 1+4 (Chakrabarty et al. 1997; Cui 1997).

We thank Dr. 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 No. RP1-173 of the U.S. Civilian R&D Foundation for the Independent States of the Former Soviet Union. The work of RVEL was also supported in part by NASA grant NAG5 6311.

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