The ionization fraction in alpha-models of protoplanetary disks

We calculate the ionization fraction of protostellar alpha disks, taking into account vertical temperature structure, and the possible presence of trace metal atoms. Both thermal and X-ray ionization are considered. Previous investigations of layered disks used radial power-law models with isothermal vertical structure. But alpha models are used to model accretion, and the present work is a step towards a self-consistent treatment. The extent of the magnetically uncoupled (``dead'') zone depends sensitively on alpha, on the assumed accretion rate, and on the critical magnetic Reynolds number, below which MHD turbulence cannot be self-sustained. Its extent is extremely model-dependent. It is also shown that a tiny fraction of the cosmic abundance of metal atoms can dramatically affect the ionization balance. Gravitational instabilities are an unpromising source of transport, except in the early stages of disk formation.


INTRODUCTION
The only process known that is able to initiate and sustain turbulent transport in accretion disks is the magnetorotational instability (MRI; Hawley 1991, 1998 andreferences therein). Because of the presence of finite resistivity, however, protostellar disk applications of the MRI are not straightforward. Numerical magnetohydrodynamic (MHD) disk simulations with Ohmic dissipation (Fleming et al. 2000) show that magnetic turbulence cannot be sustained if the magnetic Reynolds number, Re M , is lower than some critical value, Re M,crit .
If there is a net magnetic flux through the disk, the simulations indicate that Re M,crit is about 100, and roughly corresponds to the Reynolds number below which the modes are linearly stable. If there is no net magnetic flux through the disk, Re M,crit is found to be much larger, on the order of 10 4 . In this case, MHD turbulence can be suppressed even if the linear modes are only slightly affected. It is important to bear in mind that the value of Re M,crit is uncertain. Recent studies of the linear stability of protostellar disks indicate that the effects of Hall electromotive forces are important, and that the actual critical Reynolds number may accordingly be smaller (Wardle 1999, Balbus & Terquem 2001).
On scales of 1 AU in a protostellar disk, Re M = 100 corresponds to ionization fraction of about 10 −12 (e.g., Balbus & Hawley 2000 and § 4 below). Although very small, this fraction may not be attained in the intermediate regions of protostellar disks. The disk zones in which Re M is lower or larger than the critical Reynolds number are usually referred to as dead and active, respectively (Gammie 1996). The active layer extends vertically from the disk surface down to some altitude, which depends upon the radial location and the disk model.
The extent of the dead zone has been modeled by several authors using different ionization agents: Gammie (1996, cosmic rays), Igea & Glassgold (1999, X-rays) and Sano et al. (2000, cosmic rays and radioactivity). All these studies were based on the minimum mass disk model of Hayashi, Nakazawa & Nakagawa (1985), in which temperature and surface mass density vary as a simple power law of the radius, and the vertical structure is isothermal. This is a rather arbitrary choice, and the large scale structure of a turbulent α disk is in fact very different (Papaloizou & Terquem 1999). There is some theoretical evidence that MHD turbulence leads to a large scale α type structure in thin Keplerian disks (Balbus & Papaloizou 1999). It is the goal of this paper to investigate the ionization fraction of an α-type disk, taking into account the vertical structure of such models. We shall consider thermal ionization and X-ray ionization, as these mechanisms are likely to be more important than cosmic rays in X-ray active young stellar objects (Glassgold, Feigelson & Montmerle 2000).
The disk ionization depends on the recombination rate of the electrons, which are removed through dissociative recombination with molecular ions and, at a much slower rate, through radiative recombination with heavy metal ions. In the previous studies of disk ionc RAS, MNRAS 000, 000-000 The ionization fraction in α-models of protoplanetary disks 3 ization by X-rays, it was assumed that all the metal atoms were locked up in dust grains most of which had themselves sedimented toward the disk midplane. However, the ionization fraction is extremely sensitive to even a very small number of metal atoms. This is because these rapidly pick up the charges of molecular ions and recombine only slowly with the electrons. In this paper, we therefore include the effect of a non zero density of metal atoms on the disk ionization.
The plan of the paper is as follows: In § 2, we describe the disk models used, and compare them with Hayashi et al. (1985). In § 3, we discuss the different ionization mechanisms.
Thermal ionization is important in the disk inner parts, whereas X-ray ionization dominates everywhere else. We also discuss the effect of the presence of heavy metal atoms on the ionization fraction. In § 4 we present results for α-type disks for a range of gas accretion rateṀ and values of the viscosity parameter α. We find that the extent of the dead zone depends very sensitively on the critical Reynolds number, the parameters of the disk model (Ṁ and α) and the density of heavy metal atoms. With no metal atoms and Re M,crit = 100, we find in most cases that the dead zone generally extends from a fraction of an AU out to 10 to 10 2 AU. With an accretion rate ofṀ = 10 −8 M ⊙ yr −1 and α = 10 −2 for instance, the dead zone extends from 0.2 to 100 AU. This is much larger than what was found by previous authors, who used a smaller value of Re M,crit . However, we also find that the dead zone disappears completely for α ≥ 10 −2 when there is even a tiny density of heavy metal atoms. For instance, a density as small as 10 −7 or 10 −6 times the cosmic abundance is enough to make a disk withṀ = 10 −8 M ⊙ yr −1 and α = 10 −2 completely turbulent. The dead zone is dramatically reduced or even disappears when the critical Reynolds number is taken to be 1, even when there are no heavy metal atoms. In § 5, we study the evolution of an α disk with a dead zone, with the aim of investigating local gravitational instability. Gammie (1996Gammie ( , 1999 and Armitage, Livio & Pringle (2001) have noted that a layered disk cannot accrete mass steadily, and that accumulation of mass in the dead zone may lead to gravitational instabilities. We find that when there is no mass falling onto the disk, the accumulation rate is too slow for gravitational instabilities to develop within the disk lifetime. Finally, in § 6 we summarize and discuss our results.

DISK MODELS
We use conventional α-disk models, as calculated by Papaloizou & Terquem (1999), to which the reader is referred for details. The disk is assumed to be in Keplerian rotation around a central mass M * = 1 M ⊙ . The opacity, taken from Bell & Lin (1994), has contributions from dust grains, molecules, atoms and ions. The values of α and mass flow rateṀ are taken to be free parameters, and determine the model uniquely. In the steady state limit,Ṁ is constant through the disk. At a given radius r, the vertical structure is obtained by solving the equations of hydrostatic equilibrium, energy conservation and radiative transport with appropriate boundary conditions. (At the temperatures of interest here, convective transport is not significant). The detailed results of these calculations for α = 10 −2 or 0.1 andṀ in the range 10 −9 -10 −6 M ⊙ yr −1 may be found in Papaloizou & Terquem (1999). In Figure 1, we plot steady state values of the midplane temperature T m and the surface mass density Σ versus r for α between 10 −3 and 10 −1 ; for illustrative purposes we have adoptedṀ = 10 −8 M ⊙ yr −1 .
All solar nebula ionization studies previous to this have used the Hayashi et al. (1985) model of the solar nebula: an useful organization of a complex problem, it certainly leaves room for other approaches to modeling the nebula. A question of some interest is how important the nebular model is to the ionization structure, which has motivated the approach presented here. For example, the Hayashi et al. (1985) model leads to a more centrally condensed disk mass than that obtained in a steady α-disk. With α = 10 −2 , a very large accretion rate ofṀ ∼ 10 −6 M ⊙ yr −1 is necessary to obtain the above Hayashi value of Σ at 1 AU. For comparison, we have plotted temperature and mass density curves of the two models in Figure 1.
Ionization by X-rays is thus much more efficient around 1 AU in an α-disk model than in the Hayashi et al. (1985) model. The theoretical justification for a minimum mass model is not strongly compelling, and it is quite incompatible with standard accretion disk theory.
In addition, as mentioned above, there is some theoretical evidence that MHD turbulence leads to a large scale α type structure in thin Keplerian disks (Balbus & Papaloizou 1999). This is the primary motivation for the work presented here.

IONIZATION AND RECOMBINATION
Protostellar disks are ionized mainly by thermal processes and by nonthermal X-rays. Cosmic rays, a classical ionization source in cool gas, have also been considered as a source of ionization (Gammie 1996, Sano et al. 2000, but the low energy particles (important for ionization) were almost certainly excluded by winds from the early solar nebula, as they are today in a far less active environment. Radioactive decay of 40 K and 26 Al has also been investigated by some authors (e.g., Consolmagno & Jokipii 1978), but their ionization effects are quite small compared to the levels of interest here (Stepinski 1992, Gammie 1996.

Thermal ionization
If not condensed onto grains, the alkali ions Na + and K + will be the dominant thermal ionization source in protostellar disks (Umebayashi & Nakano 1983). At the onset of dynamically interesting ionization levels (∼ 10 −13 ), the K + ion is, with its smaller ionization potential, more important. In this regime, the Saha equation may be approximated as (Bal-

bus & Hawley 2000):
x e ≡ n e n n = 6.47 × 10 −13 a 10 −7 (1) where n e and n n are respectively the electron and neutral number densities in cm −3 , and a is the K abundance relative to hydrogen. Due to the Boltzmann cut-off factor, thermal ionization is important only in the disk inner regions, on scales less than an AU, where the midplane temperature is likely to exceed 10 3 K. (Above this temperature the alkalis will tend to be in the gas phase, making the approximation self-consistent.) If there is magnetic coupling on scales larger than this, nonthermal ionization sources are required.

X-ray ionization
Young stellar objects appear to be very active X-ray sources, with X-ray luminosities in the range of 10 29 -10 32 erg s −1 and photon energies from about 1 to 5 keV (Koyama et al. 1994, Casanova et al. 1995, Carkner et al. 1996. Glassgold, Najita & Igea (1997, hereafter GNI97, see also Igea & Glassgold 1999) pointed out that these X-rays are likely to be the dominant nonthermal ionization source in protostellar disks. These authors modeled the X-ray source as an isothermal (T = T X ) bremsstrahlung coronal ring, of radius of about 10 R ⊙ , located at a similar distance above (and below) the disk midplane. The total X-ray luminosity is L X , with each hemisphere contributing L X /2. The associated ionization rate is given by (Krolik & Kallman 1983, GNI97): Here, σ is the photoionization cross section, which is fit to a power law (Igea & Glassgold 1999): with n = 2.81 (these values apply to the case where heavy elements are depleted onto grains and get segregated from the gas). ∆ǫ = 37 eV is the average energy required by a primary photoelectron to make a secondary ionization. The dimensionless integral J is an energy integral over the X-ray spectrum involving the product of the cross section (whence the factor x −n ) and the attenuated X-ray flux. The factor τ is the optical depth at an energy of kT X ; it depends upon one's location within the disk. Generally the integral is insensitive to the lower limit threshold energy represented by x 0 , and for the large τ case of interest here, it may be asymptotically expanded. The leading order result is where A = 0.686, B = 1.778, a = 0.606, b = 0.262. In computing the optical depth, we make the approximation that the photons travel along straight lines; that is, we will neglect their diffusion both by the ambient medium and by the disk interior. Note however that scattering by the disk atmosphere would increase the ionization, both because some of the photons directed away from the disk would be scattered back, and because scattering provides pathways to the disk interior with smaller optical depths than a simple linear traversal. Note that here we do not make the approximation of GNI97 that the path of the photons inside the disk is vertical.

Recombination processes
In contrast to ionization, which is reasonably straightforward, electron recombination is greatly complicated by the presence of dust grains in the nebula. Not only are results depen-dent upon the size spectrum of the grains, imperfectly understood surface physicochemical processes will strongly influence charge capture and emission. Uncertainties attending the role of dust grains represent the greatest obstacle in estimating the solar nebula's ionization structure. Gammie (1996) and GNI97 finessed this issue by arguing that the dust will settle rapidly toward the midplane, and that the dominant recombination process will therefore be molecular dissociative recombination. While this greatly simplifies matters, it is prudent to regard the vertical distribution of the grains in magnetically coupled disk regions as an outstanding problem. Turbulence tends to mix, but the MRI is not particularly efficient at mixing vertically. Convective turbulence, should it be present, is a fine vertical mixer, and dust emission in the upper layers enhances the cooling, aiding the convection process itself. But sustaining convection without the MRI is known to be problematic.
Given the uncertainties, we take the view that it makes little sense to strive for high accuracy in a complex model. We have therefore opted for the simplicity of the previous studies, and will restrict our modeling to gas phase recombination. This may well overestimate the electron density in the regime in which adsorption by grains exceeds X-ray photo-emission, but the model is readily understood and serves as a useful benchmark. It is technically valid when the grains lie within a dead (magnetically decoupled) layer near the midplane, or if vertical mixing is inefficient. In the latter case, note that the disk is likely to develop a dead sheet at its midplane where the dust has concentrated, as in either the Gammie (1996) or GNI97 models.
One important addition we shall emphasize is the presence of charge exchange between molecular ions and metal atoms (Sano et al. 2000). What is of interest here is the great sensitivity of x e to the presence of even trace amounts of metal atoms, and the extreme insensitivity of the metal atoms to the spatial density of the grains. The latter is a consequence of the dual role played by the dust: they are adsorbers as well as (X-ray induced) desorbers of metal atoms.
If thermal processes prevailed in the disk, the metal atom population would be negligibly small even by these sensitive standards: the temperatures of interest are well below the condensation temperatures of most refractory elements. But just as nonthermal processes may regulate the ionization fraction of the disk, they may also regulate metal atom population.
Consider a dust grain in the presence of an X-ray radiation field. ⋆ The rate at which atoms are liberated from the grain is F σ rad y, where F is the photon number flux, σ rad is the radiation cross section, and y is the quantum yield. This is balanced by the metal adsorption rate onto the grain, n M v th σ cap s, where n M is the metal atom density, v th is the thermal velocity, σ cap is the capture cross section, and s is the sticking probability. Thus, independent of the dust density. To understand the consequences of this more clearly, we next examine the dependence of x e upon x M .
The ionization fraction x e is obtained by balancing the net rates of ionization and recombination. Electrons are captured via dissociative recombination with molecular ions (e.g., HCO + ) and radiative recombination with heavy metal ions (e.g., Mg + ). As noted, charges are also transferred from molecular ions to metal atoms, hence the level of ionization depends on the abundance of metal atoms in the gas.
Let n m , n m + , and n M + , respectively denote the density of molecules, molecular ions, and metal ions. The rate equations for n e and n m + are: where β is the dissociative recombination rate coefficient for molecular ions, β r the radiative recombination rate coefficient for metal atoms, and β t the rate coefficient of charge transfer from molecular ions to metal atoms. We have made the standard simplifying assumption that the rate coefficients are the same for all the species. For the numerical work, we take (Oppenheimer & Dalgarno 1974;Spitzer 1978;Millar, Farquahr, & Willacy 1997 for β t ): Finally, charge neutrality implies n e = n M + + n m + .
In steady-state equilibrium, equations (7), (8) and (10) lead to (Oppenheimer & Dalgarno 1974): ⋆ Cosmic rays, whose ionization effects are secondary to X-rays, may in fact be more effective at releasing metal atoms from grains. Their presence, while increasing the rate of sputtering, would not effect the qualitative point being made.
c RAS, MNRAS 000, 000-000 The ionization fraction in α-models of protoplanetary disks 9 x where x M ≡ n M /n n . The extreme sensitivity of this equation to x M is apparent upon substituting from equation (9) for β t /β and β t /β r : In the absence of metals (x M = 0), equation (11) has the simple solution used by Gammie (1996) and GNI97: a balance between the first and third terms on the left side of the equation. This case would correspond, for example, to all metals locked in sedimented grains. The opposite limit, metal domination, is equally simple: and corresponds to a balance between the second and fourth terms of the cubic. The transition from one limit to the other begins when the last two terms of the left-hand-side of the equation are comparable, that is when from which one immediately sees the extreme sensitivity to the metal atom abundance. A value of x M ∼ 10 −14 , or about 10 −7 of its cosmic abundance, might well affect the ionization balance. This, combined with insensitivity of low values of x M to the dust abundance, leads us to consider the effect of finite x M on the ionization balance in the disk.

IONIZATION FRACTION IN α-DISK MODELS
For a given disk model with fixedṀ and α, we calculate the ionization fraction x e as a function of r and z using equations (1) and (11), i.e. taking into account both thermal and X-ray ionization. In equation (11), the fraction of heavy metal atoms, x M , is varied between zero and some finite value, in order to determine the minimum value of x M for which a particular disk model is sufficiently ionized to be magnetically coupled. In § 6, we show that the values of x M we have used are at least roughly consistent with equation (6).
The Ohmic resistivity η is (e.g., Blaes & Balbus 1994): and we define the magnetic Reynolds number as: where H is the disk semithickness and c s is the sound speed. Numerical simulations including Ohmic dissipation (Fleming et al. 2000) indicate that MHD turbulence cannot be sustained and kT X = 3 keV respectively. When the entire disk is found to be active, a single curve is shown; otherwise, the difference between the two curves indicates the column density of the dead layer.
In table 1, we summarize the dead zone properties for the different models considered.
The quantity x M,min is the value of x M above which the dead zone disappears, and r max is the radius at which the vertical column density of the dead zone is largest. The final column is the percentage of the vertical column density occupied by the dead zone at r = r max .
Figures 3, 5, and 7 show that as x M increases, the dead zone shrinks both radially (with its outer edge moving inwards), and vertically. Its inner edge does not move outwards, because it is located at the radius at which the temperature drops below the level needed for thermal ionization to be effective, which is independent of x M . For a given α, this radius increases withṀ, as the disk becomes hotter. Similarly, this radius increases when α decreases forṀ fixed.
The existence of a local maximum in the column density of the active layer at r = r max is easily understood. The α-disk models have an almost uniform surface mass density over a broad radial range that depends onṀ and α (see fig. 1). As we move away from the star remaining in this region, the X-ray flux decreases because of simple geometrical dilution.
Since the column density along the path of the photons stay about the same, the active layer becomes thinner. It starts to thicken only when the surface mass density in the disk begins to drop. Table 1 gives the value of x M above which the disk is completely active. This should be compared with the cosmic abundance of the metal atoms present in a disk, which is 2 × 10 −6 for Na and 3 × 10 −5 for Mg (Anders & Grevesse 1989, Boss 1996. When Re M,crit = 100 and for a disk model with α = 10 −2 andṀ = 10 −8 M ⊙ yr −1 , we see that an abundance of only 10 −6 or even 10 −7 of cosmic (depending on the species) is needed for the disk to be active.
When α ≤ 10 −3 , the disk is never fully active, even for large values of x M . This is because the disk is now very massive (about 0.3 solar mass within 100 AU forṀ = 10 −8 M ⊙ yr −1 ), and there is a zone in the disk where X-rays simply cannot penetrate. Increasing x M clearly cannot prevent this!

EVOLUTION OF A DISK WITH A DEAD ZONE
The fate of the magnetically coupled upper disk layers is far from clear. This relatively low density region may emerge in the form of a disk wind; it is also possible that the layers will accrete. This is the assumption of Gammie (1996), but it awaits verification by MHD simulations of stratified, partially ionized disks.
Should it occur, layered accretion in a disk will not be steady (Gammie 1996). The regions of the disk where there is an inactive layer act like a bottleneck through which the accretion is slowed and where matter therefore accumulates. Here we investigate whether enough mass can accumulate to make the disk locally gravitationally unstable. Of course, our calculation is in some sense not self-consistent, since it is based upon the assumption that the disk structure is a constant α model which we then show cannot be maintained! However, our interest is limited to estimating the extent to which upper layer accretion can build up the column density of the inner disk to the point of gravitational collapse. For this purpose, the precise vertical structure is likely to be a correction of detail, not a fundamental change. Armitage et al. (2001) have considered a disk embedded in a collapsing envelope, so that mass is continuously added to the disk at a rate larger than the accretion through the disk itself. This strongly favors the development of gravitational instabilities. Here, we are concerned with the later stages of disk evolution, when there is no longer any infall. The inner regions are then supplied only by material coming from further out in the disk plane.
In a fully turbulent disk, the surface mass density evolves with time according to the following equation (Balbus & Papaloizou 1999): where W is the horizontal stress tensor per unit mass averaged over the azimuthal angle, W = ρW dz/Σ is the density-averaged of W over the disk thickness and the prime denotes derivative with respect to radius. For an α disk model, W is given by (Shakura & Sunyaev 1973): where c s is the sound speed † . There is no source term in equation (18) as there is no infall onto the disk.
In a layered disk, the right-hand-side of equation (18) also gives the variation per unit time of the surface mass density of the active zone, Σ a , provided we replace Σ by Σ a and W by W a . (The notation X a = a ρXdz/Σ a ; the integration is over the active zone).
But on the left-hand-side, Σ must be replaced by Σ d , the surface density of the dead zone. This is because, at a given location in the disk, Σ a is fixed, and determined by the X-ray source. Therefore, the mass which is added or removed at some radius is actually added to or removed from the dead zone. The evolution of the disk, in the parts where there is a dead zone, is then formally described by the two equations: Note that if Σ d given by equation (21) becomes negative at some location, then it means the dead zone disappears, and equation (18) should be used instead to evolve Σ, which is indistinguishable from Σ a . Since W a does not vary with time in our models, Σ d given by equation (21) varies linearly with time.
We consider as initial conditions a particular disk model characterized by the parameter α and a uniformṀ . For illustrative purposes, we choseṀ = 10 −8 M ⊙ yr −1 and α = 3 ×10 −2 and 10 −2 . The total surface mass density of such a disk is shown in Figure 8, where we also display the Toomre parameter, Q(r) = Ωc s /[πGΣ(r)], where G is the gravitational constant.
To get a lower limit for Q, we evaluate c s at the disk surface. Because mass is going to accumulate at some intermediate radii in the disk, Q is going to decrease there. We now investigate whether it can decrease enough for gravitational instabilities to develop, i.e. for Q to drop to values ∼ 1. Note that forṀ = 10 −7 M ⊙ yr −1 and α = 10 −2 , the initial disk model is already gravitationally unstable beyond about 30 AU (the total mass enclosed in this radius being about 0.03 M ⊙ ), which is why we consider here a smaller value ofṀ .
We compute the extent of the disk dead zone for Re M,crit = 100 and x M = 0 (see § 4).
To discuss the evolution of the disk, we need to consider the accretion rate through the disk, which is obtained from (Lynden- Bell & Pringle 1974): (The a subscripts should be dropped in the absence of a dead zone.) Here,Ṁ does not change with time. In Figure 8 we plotṀ versus r at the radii where there is a dead zone.
We see thatṀ has a minimum at some location r 0 . For r < r 0 ,Ṁ decreases with radius in the active region of the disk. The mass initially within r 0 in the dead zone will thus gradually be accreted onto the star, and whatever mass arrives at r 0 will subsequently be accreted as well. On the other hand,Ṁ increases with radius beyond r 0 in the active layer. Thus, in this region mass accumulates in the dead zone according to: where Σ i is the initial value of Σ at r, and equations (21) and (22) have been used. The evolution of the Q parameter is then obtained simply from where the constant is simply the initial value of the product. For the models whose initial conditions are shown in Figure 8, Q reaches values around unity first at a radius of a few AU after a time of 2 × 10 7 years. This is longer by about an order of magnitude than the disk lifetime inferred from observations (Haisch, Lada & Lada 2001). Note that this timescale is set by the largest value of Σ reached at the end of the evolution and is thus insensitive to the starting conditions. A simple combination of layered accretion and gravitational instability does not appear to be sufficient to truncate our fiducial disk's lifetime rapidly enough.

DISCUSSION AND CONCLUSION
We have calculated a model for the ionization fraction of an α-type disk, taking into account the full vertical structure of the disk. In common with other investigations (Gammie 1996, Glassgold et al. 1997), we have not included grain physics in our ionization model, which is an important limitation. Both the extent and the thickness of the dead zone depend sensitively on the parameters of the disk model, the gas accretion rateṀ and α. For a critical Reynolds number of 100 and parameters believed to be typical of protostellar disks, the dead zone is found to extend from a fraction of an AU to 10-100 AU. ForṀ = 10 −8 M ⊙ yr −1 and α = 10 −2 for instance, the dead zone extends from 0.2 to 100 AU. For comparison, Igea & Glassgold (1999), who used a smaller value of Re M,crit , calculated that the dead zone extended to about 5 AU. (Also, since in this model the mass is more centrally condensed than in ours, the dead zone at 1 AU occupies a larger fraction of the disk vertical column density.) But for α = 0.1 andṀ < 10 −7 M ⊙ yr −1 , the disk can be magnetically coupled over its full extent. This is because the surface mass density is low in this case. Models with α = 10 −3 have a very large and thick dead zone. When Re M,crit = 1, we have found that the dead zone is dramatically reduced or even disappears. Clearly, uncertainty in the input parameters profoundly influences the extent of magnetic coupling in protostellar disks, and full coupling to the field in some stages of disk evolution currently remains a possibility.
We have found that the characteristics of the dead zone are ultra-sensitive to the presence of heavy metal atoms in the disk. This is because the metal atoms are rapidly charged by molecular ions and recombine comparatively slowly. Molecular dissociative recombination is otherwise very rapid. For α = 10 −2 andṀ = 10 −8 M ⊙ yr −1 for instance, an abundance of only 10 −6 -10 −7 of cosmic is all that is needed in our model for disk to be completely active, even when Re M,crit = 100.
The density of metal atoms depends on the rate at which they are captured by grains and the rate at which they are liberated when an X-ray hits a grain. (Cosmic rays, which have been ignored here, may be important for this process.) The resulting density is very insensitive to the amount of dust present. If we assume that σ rad ∼ σ cap in equation (6), then n M = ǫL X /(4πr 2 v th kT X ) at the distance r from the star, where ǫ is an efficiency factor (equal to the ratio of the quantum yield to the sticking probablity which were introduced in eq. [6]). With L X and kT X in the range 10 29 -10 31 erg s −1 and 1-5 keV, respectively, v th ∼ 1 km s −1 , ǫ ∼ 10 −6 (Bringa 2001, private communication), we get a maximum value for n M which is about 200 cm −3 at r = 1 AU. For our fiducial valuesṀ = 10 −8 M ⊙ yr −1 and α = 10 −2 , the number density near the disk midplane at 1 AU is about n = 6 × 10 13 cm −3 .
The maximum value of the relative density of metal atoms is therefore ∼ 10 −12 , some 10 −7 -10 −6 of the cosmic abundance. This crude estimate shows that the density of metal atoms needed for magnetic coupling is within range reached, independently of the number of grains present. The presence of grains would certainly affect the ionization fraction. This is difficult to evaluate in a protostellar environment dominated by X-rays, and we have not attempted the calculation in this work. But it is important to note that the value of n M does not depend on the density of grains. It remains unchanged even in a disk where almost all the grains have sedimented toward the midplane. This is likely to be the situation in the later stages of disk evolution, as the sedimentation timescale for micron-size (or larger) grains is short (Nakagawa, Nakazawa & Hayashi 1981). Grain sedimentation and growth cause characteristic changes to the radiative spectral energy distribution. Such changes are in fact observed, though other interpretations cannot at present be ruled out (e.g., Beckwith, Henning & Nakagawa 2000). Sano et al. (2000) investigated a finite density of metal atoms in a calculation of disk ionization (in their model produced by cosmic rays and radioactivity), together with the electrons-grains and ions-grains reactions. However, they assumed that the fraction of metal atoms in the gas phase was rather high, equal to 2 × 10 −2 . In their model, the relative abundance of metal atoms is therefore around 10 −6 (they used a cosmic abundance of about 8 × 10 −5 for the refractory heavy elements), several orders of magnitude higher than what we consider here. Their conclusion that the disk is almost completely turbulent when grains are sufficiently depleted depends very much upon this assumption.
Finally, we have estimated the evolutionary timescale against gravitational collapse of an α disk with a dead zone. In our fiducial model, we find that ∼ 10 −3 M ⊙ accumulates between 2 and 20 AU, in regions of low optical depth at millimeter wavelengths. This would be easily detectable. The disk persists beyond 10 7 years, an order of magnitude in excess of observational constraints. These results are rather robust. The difficulty suggests that a more elaborate model of the dead zone is required, or that disks are more ionized than our model indicates, or more broadly that simple layered accretion models may not be capturing the essential dynamics of disk accretion. A deeper understanding of disk ionization physics clearly is essential for further progress in this domain of star formation theory.  Table 1. Characteristics of the dead zone. Column 1 contains Re M,crit . In columns 2 and 3 are listed the parameters which characterized the disk, i.e. α andṀ in M ⊙ yr −1 . Column 4 contains x M,min , which is the value of x M above which there is no dead zone. The following columns contain the parameters which describe the dead zone for x M = 0: its inner radius (which is actually insensitive to x M ), its outer radius, the radius rmax at which its vertical column density is maximum, and the percentage of the vertical column density it occupies at this radius.  For comparison, the model used by Igea & Glassgold (1997) is also displayed (dotted-dashed lines).    c RAS, MNRAS 000, 000-000

ACKNOWLEDGMENT
The ionization fraction in α-models of protoplanetary disks 23   BecauseṀ increases with r beyond some radius, mass is going to accumulate in the dead zone at some intermediate radii during the disk evolution, and Q is going decrease there. However, it does not reach the limit for gravitational instabilities within the disk lifetime.