Magnetically Torqued Thin Accretion Disks
We compute the properties of a geometrically thin, steady accretion disk surrounding a central rotating, magnetized star. The magnetosphere is assumed to entrain the disk over a wide range of radii. The model is simplified in that we adopt two (alternate) ad hoc, but plausible, expressions for the azimuthal component of the magnetic field as a function of radial distance. We find a solution for the angular velocity profile tending to corotation close to the central star, and smoothly matching a Keplerian curve at a radius where the viscous stress vanishes. The value of this “transition” radius is nearly the same for both of our adopted -field models. We then solve analytically for the torques on the central star and for the disk luminosity due to gravity and magnetic torques. When expressed in a dimensionless form, the resulting quantities depend on one parameter alone, the ratio of the transition radius to the corotation radius. For rapid rotators, the accretion disk may be powered mostly by spin-down of the central star. These results are independent of the viscosity prescription in the disk. We also solve for the disk structure for the special case of an optically thick alpha disk. Our results are applicable to a range of astrophysical systems including accreting neutron stars, intermediate polar cataclysmic variables, and T Tauri systems.
Subject headings:accretion disks — magnetic fields — neutron stars — pulsars: X-ray — X-rays: binaries — stars: individual ( T-Tauri, FU Orionis)
In a wide class of objects, from proto-stars to neutron stars, the central accreting object is expected to be endowed with a magnetic field of dipole strength sufficient to influence the motion of matter in the inner parts of the accretion disk. The degree of this influence depends on the (unknown) details of the interaction of the dipole field with the accreting fluid. It is not clear a priori whether the magnetosphere penetrates the disk, or whether it is capable of transmitting significant torques if it does. In the most commonly accepted model, the stellar magnetic field entrains the inner accretion disk as a result of Rayleigh-Taylor and other instabilities and exerts a torque on the disk, whose sign depends on the relative angular velocity of the disk and the star. This can result in a spin-up or spin-down torque on the star, depending on the value of the inner radius of the disk and on the accretion rate.
In this work, following the formulations of Wang (1987, 1995), Livio & Pringle (1992), and others, we adopt a simple model of distributed magnetic torques on the disk, in which their magnitude depends on the ratio of the local orbital angular velocity to the stellar rotation rate. The underlying assumption is that the external magnetic dipole penetrates the (thin) accretion disk for a wide range of radii (Livio & Pringle 1992, Wang 1996). The adopted model allows us to compute the spin-up/spin-down torques on the central star, as well as the luminosity of the disk, with no reference to the viscosity law or to the actual value of the pressure and other disk variables. If, additionally, one adopts a specific prescription for the viscosity, e.g., as in an alpha disk, it is also possible to compute the detailed radial profile of the thermodynamic variables of the disk.
The inner termination radius of an accretion disk is of considerable
astrophysical interest, as it affects the flow of energy and angular momentum
in the accretion process. In the early literature, the termination radius of the disk was thought to increase smoothly across the corotation radius as the mass accretion rate drops, with matter being ejected from the system by the super-Keplerian magnetosphere as soon as the disk is pushed out beyond the corotation radius (e.g., Davidson & Ostriker 1973, Illarionov & Sunayev 1975). More recently, it has been argued that the termination radius does not necessarily become larger than the corotation radius for the so-called “fast pulsars”
(Wang 1987), but that accretion may be stopped when “the stellar magnetic field imparts more angular momentum to the disk plasma than is removed by internal viscous forces in an unperturbed Keplerian disk” (Wang 1995). Other authors agree that the disk terminates within the corotation radius, but argue that accretion will proceed even for “fast pulsars,” because
no matter is actually ejected from the disk by the rotating magnetosphere
The plan of the paper is straightforward. After briefly reviewing some of the more relevant observations, we write down the governing angular momentum equation, we solve it, and then we discuss the implications of this solution, particularly for the torque on the central star and for the disk luminosity. In particular, in §3 we present an overview of our magnetically torqued disk solution, in §4.1 we discuss the adopted torque model, and in §4.2 we determine the inner radius of the viscous disk. In §5 we compute the torques acting on the disk as well as the concomitant luminosity. Our general results are summarized in graphical form in §6, while in §7 we give a detailed solution for a magnetically torqued alpha disk. The results are summarized and discussed in §8. An alternate -field model is discussed in Appendix B, previous work is reviewed in Appendix A, and in Appendix C a model of a magnetically dominated (zero-viscosity) Keplerian disk is presented.
2. Estimates and observations of torques.
Several classes of astronomical sources involve accretion onto a rotating central star supporting a strong magnetic field. These include young (proto)stellar objects, such as T Tauri stars, as well as accreting stellar remnants, i.e., white dwarfs (cataclysmic variables and polars) or neutron stars (accretion-powered X-ray pulsars).
It is now possible to directly measure the magnetic field strength in the inner parts of certain proto-stellar accretion disks, such as the one around FU Orionis (Donati et al. 2005). Accretion torques have been studied early on for white dwarfs (e.g., Lamb & Melia 1987). However, the most detailed information on accretion torques comes from studies of X-ray pulsars (Bildsten et al. 1997, and references therein). In particular, it has been found that abrupt transitions occur from spin-up to spin-down of the neutron star, with no clear change of the pulsed luminosity (which is presumed to originate in matter that is channelled to the magnetic poles), and so perhaps without a large change in the mass accretion rate. Nelson et al. (1997) take the view that in Roche-lobe overflow systems the disk may change the sense of its rotation from prograde to retrograde, but Li & Wickramasinghe (1998) deem this an unlikely possibility. While the transitions remain unexplained, we adopt the conventional view that the disk is prograde, but the magnetosphere can transmit angular momentum of either sign between the star and the disk, depending on the location of the inner edge of the disk and the mass accretion rate.
Neither the physics of angular momentum transport by the stellar magnetosphere, nor the mechanism of penetration of the magnetic field into the accretion disk is well understood. The main uncertainty concerns the radial extent of the disk region threaded by the external magnetic field, as well as the degree to which the magnetic field threading the disk is reduced in magnitude by screening currents, and yet there is a remarkable agreement as to the value of the radial distance from the star at which magnetic stresses balance hydrodynamic stresses. Various authors, using very different physical assumptions, obtain rather similar values for this radius, at least for the case when it is within the corotation radius. As we show in Appendix A, because of the rapid, , variation with distance of the squared dipole field strength, and because of a universal scaling of physical quantities in a thin rotation-supported disk, detailed estimates agree, within a factor of , with the simplest dimensionally correct formula
(e.g., Lamb, Pethick, & Pines 1973; Rappaport & Joss 1976; Ghosh & Lamb 1979; Wang 1987; Arons 1993). The modest uncertainty in the estimate of the magnetospheric radius is further reduced by a square root when one computes the “material” torque on the star
Observations of X-ray pulsars, indicate a steeper dependence of the spin-up torque on luminosity,
with (Bildsten et al. 1997), and possibly even (Parmar et al. 1989). This suggests that either the observed luminosity is not proportional to , or the torque is given by an expression different from eq. (2), or both.
In this paper, we investigate in detail the non-trivial variation with of the (magnetic and material) torques exerted on the star and of the disk luminosity that follow from the adopted model of the magnetosphere–disk interaction. We find that at high accretion rate (for the alternate model considered in Appendix B, ). At low accretion rates, the torques on the central star naturally reverse sign and yield spin-down. While the exact value of the spin-up and spin-down torques somewhat depends on the choice of the magnetic torque model, we find it to be generally true that at low accretion rates the luminosity of the disk is not proportional to the mass accretion rate. In fact, the disk luminosity is greatly enhanced by the torques on the disk whenever the star is being spun down.
3. Overview of disk solution
In the literature, one can find two approaches to the problem of an accretion disk interacting with the magnetosphere of a central rotating magnetic dipole. In the first, a standard thin disk adjusts the rotation rate of its inner edge to that of the star—and may be entrained by the external magnetic field—within a narrow boundary layer (e.g., Ichimaru 1978; Scharlemann 1978; Arons 1993). In the second, the magnetic field entrains the disk over a wider region (e.g., Ghosh & Lamb 1979; Wang 1987; Li & Wickramasinghe 1998; Rappaport, Fregau & Spruit 2004), allowing a smoother variation of the torque on the central star with the mass accretion rate. Matt & Pudritz (2005) suggest that one or the other approach may be appropriate, depending on the accretion state of the system. For the purposes of this paper, we adopt the second approach.
The actual profile of angular velocity in the entrainment region is not known, but it is clear that the angular velocity has a maximum, at a radius . Most authors assume that the disk is nearly Keplerian up to , and that radius is taken to be close to the magnetospheric radius, . The viscous torque is usually taken to vanish only at . It is understood that this procedure is not consistent, i.e., the angular velocity cannot be both Keplerian ( for ), and have a maximum () at . The inferred magnetic stress at varies by a factor of four, depending on whether the former or the latter is taken to hold (e.g., Li & Wang 1996), because the magnetic stress is taken to be proportional to (see eq. 16).
In this paper we adopt a model for the magnetic torques on the disk and then proceed to solve the azimuthal component of the equation of motion. We assume that the viscous torque vanishes at a certain radius , and that the disk is Keplerian at least for all . This implies that . The value of follows from the adopted model of the magnetic torque. The viscous torque can vanish at because the magnetospheric stresses are sufficiently high to remove angular momentum (from a thin annulus at this radius) at the rate required to sustain the prescribed, constant and uniform mass accretion rate. The magnetospheric stresses on the disk can only increase inwards from . Accordingly we take the viscous torque to vanish for all as well. No other assumptions are needed to solve for , once the magnetic torques are specified. It is easy to find an angular velocity profile that smoothly matches the stellar rotation rate at small radii and the Keplerian curve at (§4.2).
An additional point should be mentioned. If the stellar magnetic field entrains the disk, it exerts a torque on it, i.e., it either deposits angular momentum in the disk, or removes some angular momentum from it. This affects the amount of angular momentum transported in the disk by the viscous torques, and hence the rate of energy dissipation and the luminosity of the disk (§5.2).
4. Magnetically Torqued Thin Disks
4.1. Angular momentum transport
We start by defining a few fiducial radii within the accretion disk which we will utilize in this work. The corotation radius is taken to be the radial distance at which the Keplerian angular frequency, , is equal to the rotation frequency of the central star, :
where is the mass of the central star. We also define a convenient dimensional magnetospheric radius:
where is the steady-state mass accretion rate, and is the magnetic dipole moment of the central star. Here and throughout this work we take eq. (5) to be a formal definition of , however, our value of is identical to the inner disk radius in the model of Arons (1993). Because the ratio of these two fiducial radii appears quite often, we define a dimensionless parameter, , which relates all four system parameters of the problem ():
For a fixed system, this parameter is a measure of the mass accretion rate: , for so we may define a fiducial accretion rate through
Finally, we define as the radial inner boundary of the region where the accretion disk is Keplerian and the viscous torque is non-vanishing. The “fastness” parameter corresponding to this transition radius is defined as
The equation governing conservation of angular momentum for a thin accretion disk subjected to distributed magnetic torques from the central star is:
where is the radial coordinate, is the local orbital angular frequency of the disk material, is the full disk thickness (which is a function of ), is the viscous torque per unit volume, and is the magnetic torque on the disk per unit volume. Here, the advection of angular momentum by matter flowing inwards is driven by the viscous and magnetic torques, where a positive sign for indicates that angular momentum is deposited in the disk. The vertically averaged viscous torque per unit volume can be written as:
where is the vertically averaged– (and is the height integrated–) - component of the viscous stress-energy tensor; the two are related through .
We take the magnetic torque on the disk to be distributed over a range of radial distances with the magnetic torque per unit volume given by:
so that the vertically averaged magnetic torque is
where, in the last term, the magnetic field is evaluated at the top “surface” of the disk, i.e., at .
Without a full, self-consistent magnetohydrodynamic solution of the disk equations, there is no simple way of determining . In principle, the magnetic torque on the accretion disk should be computed self consistently, but for purposes of gaining insight into the global effects of magnetic torques we prefer to adopt a reasonable analytic model rather than working with a more complex problem that is not necessarily any more valid. To obtain results that would be applicable to any thin disk model, we simply adopt the following somewhat ad hoc but physically plausible prescription for at (Livio & Pringle 1992; Wang 1995; Rappaport, Fregeau, & Spruit 2004):
which we take to hold for (see also, Wang 1987; 1995; 1996 for more sophisticated and physically motivated versions of this relation). Here and in the following we assume, without loss of generality, that . The expression for given in eq. (13) has the property that the magnetic torque vanishes at the corotation radius, while is comparable to at large distances. The latter is compatible with the fact that cannot be larger than over significant distances, for reasons of equilibrium and stability of the field above the disk plane (see discussion in Rappaport et al. 2004). As the field is wound up by differential rotation between the star and the disk, the azimuthal component first increases, but the increasing energy in the azimuthal field component pushes the field configuration outward into an open configuration when the azimuthal component becomes comparable to the poloidal component. This was proven by Aly (1984, 1985) in a rather general context, and worked out in some detail for the case of a disk around a magnetic star by Lynden-Bell & Boily (1994). For radial distances inside the corotation radius () we adopt one of two plausible expressions for :
where the first version, , is just an analytic continuation of the expression given in eq. (13), while the second version, , has the aesthetic advantage that as , as opposed to becoming progressively (for Keplerian ). However, has the disadvantage that the functional form for the magnetic torque must be switched at . We utilize both of these functions in this work, but give results in the text for only since these are algebraically less messy. The corresponding results for are given in Appendix B. We note that a form similar to was discussed extensively by Erkut & Alpar (2004), who also derive an equation very similar to our eq. (B5).
These two prescriptions (eq. ) for the azimuthal field can be discussed in terms of the magnetic diffusion coefficient . As argued by Livio & Pringle (1992) the azimuthal field is created by vertical shearing. The poloidal field is stretched by a factor of in a time , i.e., the azimuthal field is created at the rate . In steady state this has to be balanced by the diffusion rate , resulting in . Campbell (1992) derived the last relation from the induction equation, while Wang (1987) obtained it from considerations of buoyancy. Our two prescriptions for then correspond to , or , respectively. Close to the corotation radius both prescriptions coincide. In the second prescription, the diffusivity is larger by a factor than the largest possible value of the coefficient of kinematic viscosity in subsonic turbulence, , where is the speed of sound, i.e., we do not necessarily assume that the magnetic diffusivity and the disk viscosity are of the same origin (cf. Campbell 2000).
We note that Agapitou & Papaloizou (2000) use a diffusivity prescription resulting in a torque that differs from our prescription by a constant factor (their eq. ). Their discussion suggests that a large value of is more physical than the value . However, the effect of smaller assumed diffusivity seems to be offset by the effects of an inflated field (see below). We further note that in the following, a rescaling of the diffusivity by a constant factor corresponds to rescaling of by , and of by . Wang (1995) showed that the torque on the central star is independent of , when expressed in units of (see eq. , where we have neglected the contribution in eq. [5.3]).
We consider a central dipole aligned with the stellar spin axis, and an accretion disk that is axisymmetric and perpendicular to the spin axis. In the following we will assume that the poloidal field is given by the dipole formula . We expect that the results obtained in this paper will not differ qualitatively if this assumption is relaxed. For example, the expression of Arons (1993) for the inner radius of the disk, , which we reproduce in eq. (A12) of Appendix A, is based on an exact solution of Aly (1980) for the field structure in the presence of a diamagnetic disk (in which, close to the inner edge of the disk the field is enhanced by a factor of ), and yet, it agrees with our result for : for ; see eqs. (17), (B7), and Fig. 2. Likewise, considering numerical solutions of realistic disks, Bardou & Heyvaerts (1996) and Agapitou & Papaloizou (2000) show that the field lines may be inflated outside the corotation radius, leading to a reduction in the value of the poloidal field relative to the dipole value that is assumed here. However, this does not seem to strongly affect the results discussed in this paper. Eventually, Agapitou & Papaloizou (2000) find values of torque which are the same as ours, within a factor of a few [see our §5.3].
4.2. Matching solutions in the sub-Keplerian regime
Unlike most other workers, to describe the steady disk we use the same disk equation for all radii (eq. [4.1]). Since the model torques depend on the radial profile of the angular velocity in the disk, and the angular velocity profile depends on the torques, the azimuthal equation of motion of the accreting fluid must be solved in a self-consistent manner. Assuming no mass loss from the disk, we find a solution which asymptotically matches the rotation rate of the star at small radii, and smoothly matches a viscous Keplerian disk at a certain radius, . The value of this radius also follows from the same eq. (4.1).
We begin by arguing that the disk may be taken to be Keplerian where the viscous torque is non-vanishing, with the corollary that for . In a steady, thin accretion disk, the radial pressure gradients are a factor of smaller than gravity (Shakura and Sunyaev 1973; is the dimensionless disk thickness), and hence, in the absence of strong external disturbance the disk is rotation supported, i.e., nearly Keplerian everywhere. On the other hand, if the disk extends within the corotation radius, its angular velocity must reach a maximum (at a smaller value ) before it can match the lower angular velocity of the star. At the maximum, the viscous torque must vanish. We expect the viscous torques to be zero also for all ; this is consistent with the fact that magneto-rotational instability (Balbus & Hawley 1991), which is thought to be responsible for the presence of an effective viscosity in the disk, does not operate in the region of radially increasing angular velocity . Hence, the viscous torque vanishes for all , for a certain .
Accretion can proceed in the region only because the magnetic torques are sufficiently high to remove angular momentum at the requisite rate. In fact, by the argument in the previous paragraph, in the sub-Keplerian region the magnetic torque must be dominant, the viscous term in eq. (4.1) only becoming important when , where is the Keplerian orbital frequency. To keep the algebra clean we assume that , i.e., the viscous torque vanishes already in the Keplerian region. Since eq. (4.1) does not admit Keplerian solutions for , we must then have . This is not a very restrictive assumption. If, instead of assuming , we were to take the viscous torque to remove one-half of the angular momentum at the matching boundary, , the estimate of in eqs. (17) - (19) would change by a factor of for , i.e., by less than 15%, and not at all in the limit . In short, we assume if and only if .
The equation for angular momentum transport (eq. [4.1]) in the region , where viscous stresses are zero, is our starting point for exploring formal flow solutions which make the transition from Keplerian rotation at to corotation with the central star as :
This being a first order ordinary differential equation, a single initial condition (at a fixed radius) specifies the solution. However, using the freedom of choosing the matching radius, it is possible to match both the slope and the value of the solution with a Keplerian one. Indeed, we first find the radius at which the solution matches Keplerian rotation by substituting for the derivative on the left hand side, and then use as the single initial condition to solve for for all .
With the described substitution, eq. (16) reduces to an algebraic equation for :
This is essentially the same as the expression for the inner edge of the disk found by Wang (1995), and also used by Kenyon et al. (1996). For any choice of value for the dimensionless parameter this equation can be solved numerically for the inner radius, in units of . The asymptotic limits for large and small values of are:
Note that , always. Plots of and are shown in Fig. 2. For a given system, the high accretion rate limit of eq. (17) is , by eqs. (7), (18). Of course, the disk cannot penetrate the surface of the star or extend far within the marginally stable orbit (ISCO) predicted by general relativity (e.g., Kluźniak & Wagoner 1995), so eqs. (17-19) and the solutions discussed below are valid only for .
For any non-vanishing mass accretion rate, , eq. (17) allows us to eliminate the unobservable quantity from all the equations, so that the ratio is the only remaining parameter in the expressions for various physical quantities. Specifically, the first equality in eq. (17) can be rewritten as
The limit of no magnetic field corresponds to , , , . The limit of corresponds to , and more specifically .
The counterpart of eq. (17) for our (alternate) prescription (eq. [B7] in Appendix B) has also been derived by Matthews et al. (2005). However, their interpretation of is different from ours. For Matthews et al., (their ) is the inner edge of the magnetically dominated disk, where the surface density vanishes (i.e., for ), and where the radial inflow velocity is large. In our approach, is the outer edge of the magnetically dominated inner disk; the disk is Keplerian at , and therefore the radial velocity must still be much smaller than the value of at this radius. Furthermore, there is no physical reason for the density to vanish at .
The specific angular momentum of the matter flow, , is governed by the equation valid for all (essentially a rewritten version of eq. ):
which can be cast in completely dimensionless form by using rescaled variables , and
This equation admits to analytic solutions which involve incomplete gamma functions, but instead of reproducing the formulae we exhibit the solution graphically, after simply integrating this equation numerically. The results are shown in Fig. 2, together with the Keplerian curve. Note that the solutions smoothly match a Keplerian at , reach a maximum at a lower value of , and finally decrease smoothly until they level off asymptotically toward .
Neither the value of the maximum angular frequency, , nor the value of the radius at which this maximum is attained, plays any role in the following considerations. This is because the viscous torque is non-zero only for radii . There is no viscous dissipation, and no angular momentum can be transmitted upstream through the fluid, for any . Once the accreting fluid passes through the (imaginary) cylinder at , all its angular momentum is transmitted to the star, at the rate .
5. Torques and Luminosity
Now that we have found a solution in which the viscous torque vanishes at and the disk is Keplerian for all we can use eq. (4.1) to compute the viscous torque in the disk, and the luminosity of the disk. We are also in a position to compute the total torque on the star. These results follow from our model assumptions alone, and are independent of the detailed structure of the thin accretion disk (equation of state of the fluid, the viscosity prescription, disk height, etc.).
5.1. Viscous Torque
If we integrate both sides of eq. (4.1) with respect to , starting at , the largest radius where (inner edge of the viscous accretion disk), we find:
where for the component of the magnetic field we have taken to represent an unscreened dipole field whose axis is aligned approximately along the spin axis of the central star and perpendicular to the accretion disk. Finally, we can carry out the remaining integral and solve for the vertically integrated component of the stress tensor at an arbitrary radial distance:
Here we have made use of our result from Section 4.2 that the disk matter is in Keplerian orbital motion everywhere for .
5.2. Viscous Heating of the Disk
We now utilize our result for the vertically integrated stress to compute the local viscous heating in the disk. The viscous heating per unit volume is given by , and the corresponding viscous heating per radial interval is . From this we find:
The first term on the right hand side of the equation is the usual luminosity that results from the redistribution by viscous torques of the released gravitational potential energy (Shakura & Sunyaev 1973). The second term represents the viscous dissipation of mechanical energy that is input to the disk via the magnetic torques. By utilizing the definition of eq. (5), , we can cast eq. (25) into a somewhat more aesthetically pleasing form:
Finally, eq. (26) can be analytically integrated to yield the total power released by viscous processes in the disk:
The first term represents the gravitational power that would be released in the disk in the absence of magnetic torques. Clearly, in this model the disk may be powered in part by the central star—this occurs whenever , or, in terms of the fastness parameter, when . In general, after the constraint of eq. (20) is used, eq. (28) may be rewritten as
This quantity is positive for all .
The total mechanical power injected into the viscous disk and the part of the magnetosphere entraining it at , is given by
where the second term on the right hand side is the rotational frequency of the central star times the total magnetic torque on the disk (see eq.  below). For , this expression reduces to:
Eq. (31) can be rewritten as
a positive quantity for all .
It is not clear what fraction of the difference is released in the disk, and what fraction is released in the magnetosphere (perhaps as non-thermal radiation). In the limit of , , and , where .
5.3. Magnetic torques on the disk and the neutron star
The total magnetic torque on the accretion disk, , is given by the magnetic contribution to evaluated as (this is equivalent to the integral on the right hand side of eq. [5.1] integrated to ). This quantity can be found from the second term on the right side of eq. (24), multiplied by , and in the limit :
It may be interesting to note that the net torque on the disk vanishes, i.e., , when . In the limit of , .
Here, we can compare our results with the numerical work of Agapitou & Papaloizou (2000). If the disk extended only between and , as Agapitou & Papaloizou (2000) assume in their calculations, we would have obtained
instead of eq. (34). Agapitou & Papaloizou take (see our eq. ) with found numerically to be reduced relative to the dipole value that we assume. The torque they find in their numerical solution is a few times larger than , with for —see eq.  and Fig. 14 of Agapitou & Papaloizou (2000). Thus, our simple model yields results for the torque which are not too different from those found for more realistic magnetic field configurations. Since in reality , we note that rescaling of the product of the toroidal and poloidal magnetic fields, , by a constant factor, , affects our results only to the extent that varies, so that the values of are rescaled (eqs. , ). At the same time the constraint of eq. (17) guarantees that there is no explicit dependence on in eq. (35), and in many of our other equations. Formally, and are invariant under the rescaling and , see eq. (5).
The magnetic torque on the central star is just the negative of eq. (34). To find the total torque on the central star we add the material torque:
where is the radius of the central star. (Note that is taken to be negligible in the production of our graphical results.) For high accretion rates, the material torque is the leading term in eq. (5.3), .
If we combine eqs. (5.3), (5), and (20), we can find an analytic expression for the condition that there is zero torque on the central star (under the approximation that we neglect the term in eq. [5.3], cf., eq. ). We find , and , yielding and . The corresponding values for our alternate model (Appendix B) are very similar: and .
6. Results in Graphical Form
The inner disk transition radii, , as given by eqs. (17), (B7), are shown in Fig. 2. The four plotted curves are in units of , and in units of , for both of our prescriptions for . These radii are plotted as a function of the parameter . Some fiducial dimensionless values of and can be found in Table 1.
|eq. (8)||eq. (17)||eq. (7)|
In the slow rotator limit (i.e., weak or high ; large ), , as expected, while asymptotically approaches for the model, and is larger for the model, slowly increasing with increasing . In the opposite limit of a rapid rotator (i.e., strong B or low ; small ), we see that as , as discussed extensively by Rappaport et al. (2004). In that paper, the authors suggest that for reasonably fast central rotators, the matter is not ejected via a propeller mechanism, but rather the accretion disk adjusts its structure to allow matter to reach (see also §4.2 in this article). Note that for this and other reasons, varies quite strongly with , in contrast with the Ghosh & Lamb (1979) model, where the same ratio, , is a constant equal to about 0.4. In particular, we find that as .
In Figures 3 through 5 we plot the viscous luminosity in the disk per radial interval, , eq. (26), with three different normalizations. In the first of these (Fig. 3), is normalized to the differential gravitational luminosity at , viz, . Fig. 4 is multiplied by a factor of to yield the viscous luminosity per natural logarithmic radial interval. In Fig. 5, is normalized to the same quantity for a Shakura & Sunyaev disk. In all cases, the pattern of the curves indicates how the inner radius of the disk moves outward as the parameter increases (i.e., as the central object becomes a faster rotator), and ultimately asymptotically approaches . For very rapid rotators, the plots clearly indicate that the disk energetics are dominated by the mechanical energy being pumped in by the magnetic field, and dissipated (non-locally) by viscous stresses. For the rapid rotators, the extra energy being pumped into the disk is powered by a spin-down of the central star. Finally, in Fig. 6 we show the integral viscous disk luminosity profiles, i.e., released for radii .
We note that in Figs. 3–6, due to the way the curves have been normalized, the large disk luminosities for rapid rotators are only with respect to in the absence of magnetic torques. The absolute values would steadily decrease as grows, since is decreasing in the normalization factor . The absolute value of the disk luminosity (i.e., not normalized to ) is shown in Fig. 7 as a function of . In order to make these plots as generic as possible we have normalized the luminosities to , where is fixed for a given system, and is normalized to , i.e., a value that is also fixed for a given system. The three luminosities shown in the figure are (i) , as given by eq. (28), and which includes the mechanical energies put into the disk via the gravitational field and the magnetic torques; (ii) which results from the deposited gravitational energy alone; and (iii) the total energy released in the system, including that due to an equivalent “frictional” dissipation via the magnetic field (see eq. ). The effects of magnetic heating of the disk are very apparent for low values of , i.e., the fast rotator case. The effects of magnetic “cooling”, for slow rotators, are much less apparent.
The torques on the central star are shown in Fig. 8. The four curves are combinations of and vs. or . Here and are defined by eqs. (28) and (31), respectively. The torques, and , are defined in eqs. (34) and (5.3), respectively, where . For high values of the disk luminosity we see that the magnetic torques are typically an order of magnitude smaller than the matter accretion torques. At lower luminosities, the magnetic torque switches sign at a luminosity of about (1.2 - 1.3) , while the total torque (including that due to accretion) reverses sign only at luminosities about an order of magnitude lower (cf. Table 1). Note that the negative torques tend to a constant value, as the disk luminosity decreases (i.e., as the mass accretion rate drops).
In terms of the spin-up/spin-down dependence on the mass accretion rate, our results are qualitatively similar to those of Wang 1987, 1996, and quite different from those of Ghosh & Lamb (1979). With the help of eq. (17), and neglecting the term , the torque of eq. (5.3) can be expressed as a dimensionless function of the fastness parameter:
or, with a different normalization,
As in Wang’s (1996) dimensionless torque, is a fairly flat function up to the zero at , in general agreement with the observations of Finger et al. (1996) and in contrast to Ghosh & Lamb’s (1979) steeply decreasing torque which vanishes at . Remarkably, the corresponding function in our alternate model, although also fairly flat with a value higher by a factor , has a first zero at a very similar value of (eq. [B15]).
As discussed in § 5.3 above, the condition for zero torque on the central star is () for both of our models. This corresponds to , a value which coincidentally happens to be in approximate agreement with the Ghosh & Lamb (1979) model prediction of for any torque (Fig. 2). However, the value of fastness at this point, is quite different from the Ghosh & Lamb (1979) value of .
The zero torque constraint on leads to a relation between the equilibrium spin of a neutron star at fixed and the surface field: , where is the neutron star magnetic moment in units of G cm, and is the mass accretion rate in units of g s, or approximately the Eddington limit for a neutron star. The equilibrium spin period turns out to be
where is the surface value of the magnetic field in units of G. Unfortunately, at present it is not clear how close to equilibrium the various X-ray pulsars are.
Observations of X-ray pulsars indicate a bimodal behavior of spin-up and spin-down, with a rapid change between two states of similar magnitude, but of opposite sign of the rate of change of the pulsar spin, (Bildsten et al. 1997). The change between the two states can be quite abrupt, and occurs without a large change in (pulsed) luminosity. In our model, a change in luminosity over a fairly narrow range of luminosity could, indeed, lead to a change from spin-up torque to a spin-down torque of comparable magnitude (or vice versa), see Fig. 8. Further, the spin-down torque quickly tends to a constant value, as the luminosity drops. This is in rough qualitative agreement with the observations, however the model seems to admit a wider range of torques, particularly of spin-up torques, than is oberved. In particular, recent observations suggest very little change, if any, of the magnitude of the torque during the observed transition from spin-up to spin-down in 4U 1626-67 (Krauss et al. 2007, see their Fig. 5).
7. Complete Thin Alpha-Disk Solutions
Our expression for the vertically integrated viscous stress (eq. ) can be used to derive the full set of thin accretion disk variables if we specify a prescription for the stress. The most often utilized of these is the “alpha-prescription” of the Shakura-Sunyaev model (1973) in which is taken to be , where is the disk pressure in the midplane, and is a dimensionless parameter which specifies the strength of the viscous forces. Equation (24) then directly yields:
Here, the first term in parentheses in the expression for is the usual factor in the Shakura-Sunyaev solution. If it were not for the constraint of eq. (17), putting one would recover the Shakura-Sunyaev solution, but this would be inconsistent as is no longer an arbitrary constant, and instead must tend to zero with . After eliminating from eq. (42) with the help of eq. (20), we obtain
in the domain , . Here, , and is defined in eq. (8). The limit of vanishing field, is given by
because in this limit (eq. ). For a fixed value of (or one bounded from below), , , in the same limit.
Three of the four remaining SS-disk equations remain unchanged. For the equation of state we take: , where is the mean molecular weight per particle. The vertical force balance equation is given by: