A&A 455, 731-740 (2006)
M. Ilgner - R. P. Nelson
Astronomy Unit, Queen Mary, Mile End Road, London E1 4NS, UK
Received 29 March 2006 / Accepted 11 May 2006
Context. Recent observations of the X-ray emission from T Tauri stars in the Orion nebula have shown that they undergo frequent outbursts in their X-ray luminosity. These X-ray flares are characterised by increases in luminosity by two orders of magnitude, a typical duration of less than one day, and a significant hardening of the X-ray spectrum.
Aims. It is unknown what effect these X-ray flares will have on the ionisation fraction and dead-zone structure in protoplanetary disks. We present the results of calculations designed to address this question.
Methods. We have performed calculations of the ionisation fraction in a standard -disk model using two different chemical reaction networks. We include in our models ionisation due to X-rays from the central star, and calculate the time-dependent ionisation fraction and dead-zone structure for the inner 10 AU of a protoplanetary disk model.
Results. We find that the disk response to X-ray flares depends on whether the plasma temperature increases during flares and/or whether heavy metals (such as magnesium) are present in the gas phase. Under favourable conditions the outer disk dead-zone can disappear altogether, and the dead-zone located between 0.5 < R < 2 AU can disappear and reappear in phase with the X-ray luminosity.
Conclusions. X-ray flares can have a significant effect on the dead-zone structure in protoplanetary disks. Caution is required in interpreting this result as the duration of X-ray bursts is considerably shorter than the growth time of MHD turbulence due to the magnetorotational instability.
Key words: accretion, accretion disks - magnetohydrodynamics (MHD) - stars: planetary systems: protoplanetary disks - stars: pre-main sequence
Given that protostellar disks are cool and dense near their midplanes, there are questions about the global applicability of the MRI to these disks, as the ionisation fraction is expected to be low (Blaes & Balbus 1994; Gammie 1996). Indeed, nonlinear magnetohydrodynamic simulations of disks that include ohmic resistivity (Fleming et al. 2000) have indicated that for magnetic Reynolds numbers below a critical value , turbulence is not sustained and the disks return to a near-laminar state whose internal stresses are too small to explain the observed mass accretion rates onto T Tauri stars.
There have been a number of studies of the ionisation fraction in protostellar disks. Gammie (1996) first suggested that disks may have magnetically "active zones'' sustained by thermal or cosmic ray ionisation, adjoining regions that are "dead-zones'' where the ionisation fraction is too small to sustain MHD turbulence. Sano et al. (2000) examined this issue using a more complex chemical model that included dust grains. Glassgold et al. (1997) and Igea et al. (1999) examined the role of X-rays as a source of ionisation in protostellar disks, and highlighted doubts about whether Galactic cosmic rays could penetrate into the inner regions of protostellar disks because of the stellar wind. Fromang et al. (2002) examined the influence of gas phase heavy metals and demonstrated the potential importance of charge-transfer reactions, and Semenov et al. (2004) studied disk chemistry and the ionisation fraction using a complex reaction network drawn from the UMIST data base.
Recent observations of the X-ray emission from T Tauri stars in the Orion nebula using the Chandra observatory (COUP - Chandra Orion Ultradeep Project) have shown that in addition to providing a characteristic X-ray luminosity at a level of erg s-1, young stars emit X-ray flares whose luminosity is 100 times this value (e.g. Wolk et al. 2005; Favata et al. 2005). These flares typically last for less than a day, and are characterised by a sharp linear rise in luminosity, followed by an exponential decay. The typical recurrence time is about one week, and associated with the flares is a hardening of the X-ray spectrum indicating a rise in the plasma temperature in the stellar corona from keV to typical values of 7 keV. In this paper we address the question of what effect these X-ray flares have on the ionisation fraction and structure of dead-zones in protostellar disks.
In a recent paper (Ilgner & Nelson 2006a) we compared the predictions made by a number of chemical reaction networks about the structure of dead-zones in standard -disk models. This study included an examination of the reaction scheme proposed by Oppenheimer & Dalgarno (1974), and more complex schemes drawn from the UMIST data base (Le Teuff et al. 1996). In a follow-up paper (Ilgner & Nelson 2006b) we examined the role of turbulent mixing in determining the structure of dead-zones in -disk models using chemical reaction networks drawn from (Ilgner & Nelson 2006a). In this paper we continue with our work on the ionisation structure within protoplanetary disks and examine the effect that X-ray flares have on dead-zones in -disk models using reaction networks drawn from Ilgner & Nelson (2006a).
In general we find that X-ray flares can have a fairly dramatic effect on the ionisation structure in disks, especially if the X-ray spectrum hardens during flares and/or trace quantities of heavy metals (magnesium) are present in the gas phase. Our disk models can be divided into three distinct regions: an inner region where the disk is always active due to thermal ionisation; a central region in which the dead-zone formally decreases in depth substantially or disappears altogether during X-ray flares, but which returns to being a deep dead-zone in between flares; an outer region beyond R=2 AU in which the dead-zone depth does not change in time, and which can become very thin or disappear altogether in the presence of heavy metals and an increasing plasma temperature during outbursts.
This paper is organised as follows. In Sect. 2 we describe our modelling procedure, including the disk model, the chemical models, and our method for simulating X-ray flares. In Sect. 3 we present the results of our models, and discuss the effects of changing the model parameters. In Sect. 4 we discuss our results in the context of turbulent protostellar disks, and in Sect. 5 we summarise our findings.
Apart from the ionisation rate which is discussed in a separate section below, all the other parameters are taken from Ilgner & Nelson (2006a).
|Figure 1: Time sequence of the X-ray flares . The perturbation is periodic in time with a periodicity while the width of a single perturbation is .|
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We now adopt a model in which the X-ray luminosity maintains a base value, , on top of which are superposed X-ray flares with peak luminosity . In order to maintain the compatibility with our previous work (Ilgner & Nelson 2006a,b) we use .
We approximate the flare temporal morphology by a sequence of outbursts that arise periodically in time. The shape of a single flare is modelled by using a Gaussian profile. We note that the symmetric profile thus obtained is simpler than the observed flare morphology which is more accurately characterised as a linear rise and exponential decay, but we believe that our model captures the essentials of how X-ray flares affect the ionisation structure in the disk independently of such details. We assume that one flare occurs per week, and each outburst lasts for 24 h. These values are similar to those observed during the Chandra Orion Ultradeep Project (COUP) as reported by Wolk et al. (2005).
For a given total X-ray luminosity
one can calculate the ionisation rate
due to X-rays at each position in the disk model.
This requires an integration along each line of sight through
the disk model to the X-ray source, and our method for this is
described in Ilgner & Nelson (2006a).
In this paper we consider both models in which the plasma temperature
remains constant, and models in which the plasma temperature varies
along with the X-ray luminosity.
When the plasma temperature remains constant the attenuation of
the X-rays is also constant, so the temporal morphology of the
X-ray luminosity and the local ionisation rate are
characterised by the same mathematical function.
in the local ionisation rate due to an X-ray flare is
then given by the following time-dependent ionisation rate
Observations indicate that the plasma temperature
as the X-ray luminosity does during a flare. This means that
we need to take account of the fact that the penetration depth
also varies with time, due to the hardening of the X-ray spectrum.
As we will see in Sect. 3.3, this can
have quite dramatic effects on the local ionisation rate.
To model this we consider a minimum plasma temperature
which applies when the X-ray luminosity is at its base value
and calculate the ionisation rate
at each position in the disk. When the X-ray luminosity has reached its
we assume that the plasma temperature has reached
its maximum value
and calculate the ionisation
at each position in the disk.
The time dependent perturbation to the local ionisation rate is then
|Figure 2: Model1 - column densities of the whole disk (solid line) and of the active zones (dashed and dotted lines) - corresponding to magnetic Reynolds numbers greater than 100 - for two different heavy metal abundances . The left panel is for and the right panel is for . The upper and lower dashed lines refer to simulations with constant ionisation rate and while the long-dashed line refers to simulation with . The dotted line refers to the column density of the active zone at obtained by assuming a time-dependent ionisation rate . The time of the plot is at the mid-point between two X-ray flares.|
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We have evolved the disk chemistry using the kinetic models
model1 and model3 by taking the perturbation
of the ionisation rate due to X-ray flares into account. As in our previous
studies we wish to determine which parts of the disk are sufficiently
ionised for the gas to be well coupled to the magnetic field, and thus
able to maintain MHD turbulence, and which regions are too neutral
for such turbulence to be maintained. Again, we refer to those regions
as being "active'' and "dead'' zones respectively, with the region
bordering the two being the "transition'' zone. The important
discriminant that determines whether the disk is active or dead is
the magnetic Reynolds number,
Our approach to modelling the chemistry in disks with X-ray flares is as follows. First, we evolved models keeping the ionisation rate constant in time. We considered three cases with , , and The kinetic equations are solved for a time interval of . Hence, the ionisation fraction is a function of time t, and in principle so is the location of the transition zone. However, for these models which assume a time-independent ionisation rate , the change in the vertical location of the transition zone at all cylindrical radii in the computational domain was below the grid resolution for .
Using the abundances obtained from the model at as initial abundances, we also calculated the ionisation fraction by considering a time-dependent ionisation rate . The kinetic equations here were solved for a time interval of . We restricted the time integration of the kinetic equations by assuming a finite maximum absolute step size in order to resolve the flares in time. Because of the adopted value of the perturbation , models with time-independent ionisation rates and represent the limiting cases for models with .
Oppenheimer & Dalgarno model
The results obtained for model1 are presented in Fig. 2, which shows the column density of the whole disk plotted as a function of radius using the solid line, and the column density of the active zone using either dashed lines (for which , , and ) or dotted lines referring to . The dotted line corresponds to a time which is halfway between two X-ray flares. The left panel shows cases for which the heavy metal abundance and the right panel shows cases with .
As expected, Fig. 2 shows that a time-independent increase in the X-ray luminosity leads to a corresponding decrease in the depth of the dead-zone (note that the disk inner regions are fully active because of thermal ionisation of potassium, Ilgner & Nelson 2006a). Interestingly, even an increase in X-ray luminosity by a factor of 100 is insufficient to fully ionise the disk, and a dead-zone remains beyond R> 0.5 AU. When we consider the X-ray flaring model, we find that the behaviour of the ionisation fraction and dead-zone depends on radial position within the disk. The region between 0.5 < R < 1.2 AU has the same dead-zone structure as the model with constant X-ray luminosity set at the base level . As will be discussed in more detail later on, this is because the recombination time in this region is shorter than the period of the X-ray outburst cycle, so the disk ionisation fraction remains in phase with the X-ray luminosity. Further out in the disk beyond R > 2 AU, we find that the dead-zone structure is very close to that obtained when the X-ray luminosity takes a constant value that is equal to the time-averged value of the X-ray flaring model (i.e. ). This is because the recombination time is now longer than the period of the X-ray flaring, so the response of the ionisation fraction lags the instantaneous ionisation rate. Over long evolution times the disk responds to the average ionisation rate.
|Figure 3: Model3 - column densities of the whole disk (solid line) and of the active zones (dashed and dotted lines) - refering to magnetic Reynolds numbers greater than 100 - for two different elemental abundances . The left panel is for and the right panel is for . The upper and the lower dashed lines refer to simulations assuming a constant ionisation rate and , while the long-dashed dotted line refers to simulation with . The dotted line shows the column density of the active zone at assuming a time-dependent ionisation rate . The amplitude of the perturbation is caused by increasing the X-ray luminosity at the flare peak.|
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The right-hand panel in Fig. 2 shows the effect of introducing
a small abundance (
of heavy metals. As described in
Fromang et al. (2002) and Ilgner & Nelson (2006a), the introduction of
heavy metals to the gas phase
is expected to increase the free-electron fraction
because of charge-transfer reactions with molecular ions.
Figure 2 shows that an increase in constant X-ray luminosity
by a factor of 100 causes the dead-zone to disappear beyond 2 AU.
A constant ionisation rate
corresponding to the time-average
of the flaring rate leads to effective removal of the dead-zone beyond
3 AU. When we consider the X-ray flaring model we observe similar behaviour
to that in the model without heavy metals. Interior to about 1 AU
the dead-zone structure at the time of the snapshot is essentially the
same as the one obtained when the ionisation rate has a constant value
As discussed in Ilgner & Nelson (2006a), the recombination
of free electrons in this region remains dominated by molecular ions even when
heavy metals are present, so this result is expected. Further out in the disk
the recombination becomes dominated by the heavy metal ions, M+,
and the recombination time is longer than the period of the X-ray flares.
In this region the disk again responds to the time dependent X-ray flares
as if the ionisation rate were equal to the time averaged value,
resulting in the effective removal of the dead-zone beyond
The results obtained for model3 are presented in Fig. 3.
Here, the column density of the whole disk is plotted as a function of radius
|Figure 4: Model1 - the change in the ionisation fraction is plotted against the relative change in the ionisation rate at different radial positions R along the transition zone assuming . It is a limit cycle since the orbit is closed (the arrow points forward in time). In addition, the equlibrium values are shown obtained for the unperturbed models with , and .|
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Our previous work that compared the results of different chemical networks has already established that the more complex chemical models based on the UMIST data base predict deeper and more extensive dead-zones than the simpler Oppenheimer & Dalgarno models (Ilgner & Nelson 2006a). The primary reason is the larger number of molecular ions that occur in the complex chemistry, leading to a faster recombination of free electrons. Comparing Figs. 1 and 3 confirms this. Considering the response of model3 to differing constant X-ray fluxes, we observe that very similar trends arise as in model1. For example, an increase in X-ray luminosity by a factor of 100 still leaves a substantial dead-zone beyond R> 0.5 AU.
Considering model3 with X-ray flares, we see that the dead-zone between 0.5 < R < 2 AU is of similar depth to that generated by the base value of the X-ray luminosity (with associated ionisation rate ). This arises for similar reasons to those given when describing model1 above: the local recombination time is shorter than the time period between X-ray flares, ensuring that the instantaneous ionisation fraction remains more or less in phase with the X-ray luminosity. Conversely, the dead-zone depth beyond R>2 AU approaches that predicted by the model whose ionisation rate is the time average of the X-ray flaring model. This arises because the recombination time here is longer than the time period between X-ray flares, ensuring that the local ionisation fraction responds to the average X-ray luminosity rather than the instantaneous value.
The right panel shows the effect of adding an elemental abundance of magnesium . For a constant X-ray luminosity increased above the base value by a factor of 100, the dead-zone disappears beyond R>2 AU. The dead-zone is predicted to be very shallow in this region by the X-ray flaring model. In both these cases, however, a significant dead-zone remains between 0.5 < R < 2 AU because the dominant sources of recombination here are molecular ions rather than magnesium ions.
In the previous Sect. 3.1 our analysis focused on the
chemical state of the disk at a particular point in time
midway between two X-ray flares after 10 000 yr of evolution.
Since the ionisation rate
periodically in time, so does the size of the active zone.
We now focus on this periodic time dependence.
We consider the time-dependent ionisation fraction at a few well-defined
positions within the disk, along the transition zone
defined by Figs. 2 and 3.
We refer to the ionisation fraction at the transition zone
(defined where the magnetic Reynolds number =100) as the
critical value, and denote it by
|Figure 5: Model3 - the change in the ionisation fraction is plotted against the relative change in the ionisation rate near the transition zone at for different elemental abundances . The left panel is for and the right panel is for . It is a limit cycle since the orbit is closed (the arrow points forward in time). Note that is slightly less than unity when because the position of the transition zone does not coincide precisely with a grid cell in this model.|
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Oppenheimer & Dalgarno model
We found that the change in with time t is given by a limit cycle. The limit cycle becomes apparent by plotting the ionisation fraction relative to the critical value in the transition zone against the change in the ionisation rate . The ionisation fraction repeats with a periodicity of resulting in a closed orbit. Examples of limit cycles obtained for model1 in the absence of heavy metals are shown in Fig. 4; note that the arrow points forward in time such that the cycle is traced in an anticlockwise direction. The left hand panel shows the variation of at R=1 AU for model1. The limit cycles obtained at with are quite similiar to those obtained with since the molecular ion still dominates the recombination of free electrons there. At locations where metal ions dominate when they are included, such as at 3 AU (as shown in the right panel of Fig. 4) the change in the ionisation fraction across the limit cycle is tiny and would appear as a straight line when using the same scale of Fig. 4. The results of model1 at 3 AU without heavy metals shows a modest rise in ionisation fraction during the cycle, as shown by the right panel of Fig. 4.
The time required for the system to establish a limit cycle varies depending on the local position along the transition zone, but to within an accuracy of limit cycles were achieved throughout the disk for model1, with and without heavy metals.
We differentiate between limit cycles for which the change in ionisation fraction lags substantially behind the corresponding ionisation rate , and those for which does not. The former behaviour is illustrated by the right panel of Fig. 4 where the value of continues to rise during the rise and fall of the perturbed ionisation rate during an X-ray flare. This phenomenon occurs only in those regions where the recombination time is sufficiently long that the instantaneous recombination rate is smaller than the instantaneous perturbed ionisation rate for the duration of the flare. Once the flare has died away then the ionisation fraction slowly decreases back down to its original value, just in time for the next flare to begin. This slow decrease in arises because the recombination time is longer than the time period between X-ray flares.
The left panel of Fig. 4 shows an example where the changes in ionisation fraction are more in phase with the perturbation to the ionisation rate. This occurs in regions where the recombination time becomes short compared to the perturbed ionisation rate during a flare. The onset of an X-ray flare leads to a quite dramatic rise in the ionisation fraction at the transition zone. As the flare reaches its peak and begins to subside, the recombination rate starts to exceed the ionisation rate and the ionisation fraction decreases while the flare subsides. At the point where the flare ends the perturbation to has dropped from a peak value of down to . Once the flare has subsided completely the ionisation fraction drops to the value corresponding to the steady state obtained when the X-ray luminosity has its base value (with associated ionisation rate ). This final drop in the ionisation fraction to the critical value at the transition zone occurs because the recombination time is shorter than the time period between X-ray flares.
Figure 4 also shows that the duration of the X-ray flares is too short to reach the ionisation fraction obtained for model1 with a constant ionisation rate . Indeed we found that for the unperturbed models the transition from state to occurs on time scales between (at ) and (at ), where denotes the steady fractional abundance.
|Figure 6: Model1 - column densities of the whole disk (solid line) and of the active zones (dashed and dotted lines) - refering to magnetic Reynolds numbers - for . The dashed lines refer to simulations obtained by assuming unchanging ionisation rates for different plasma temperatures. Left panel: the dotted line refers to the column density of the active zone at obtained by assuming a time-dependent ionisation rate caused by periodic variation of the plasma temperature between keV and keV. Right panel: the dotted line refers to the column density of the active zone at obtained by assuming a time-dependent ionisation rate caused by both the X-ray luminosity and the plasma temperature increasing during flares. The disk becomes entirely active when the X-ray luminosity is increased by a factor of 100 and the plasma temperature is maintained at a constant value of keV.|
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We now continue by discussing the time dependent evolution of the ionisation fraction obtained for model3 at positions in the immediate vicinity of the transition zone for times beyond 10 000 yr. Once again we found limit cycle behaviour throughout the disk, but in order to obtain periodic orbits beyond radii R> 2 AU we had to evolve the disk slightly beyond 10 000 yr (since a steady-state had not been reached by 10 000 yr). Figure 5 shows examples of the limit cycles obtained at R=1 AU in the absence of magnesium (left panel) and with an abundance of magnesium (right panel). For that region, metals do not have a dramatic effect on the ionisation fraction because molecular ions still dominate the recombination process of free electrons (see Ilgner & Nelson 2006a). It is clear that in both cases the free electron fraction remains close to being in phase with the changing X-ray luminosity, with increasing as increases, and decreasing as does. Here the recombination rate becomes larger than the ionisation rate immediately after the peak of the X-ray flare, causing the ionisation fraction to decrease as the flare intensity diminishes. The ionisation fraction returns to the value obtained for a steady X-ray flux with ionisation rate shortly after the flare finishes, and remains at this value until the onset of the next flare. As was the case with the Oppenheimer & Dalgarno model, this fall of the ionisation fraction to the base value occurs because the recombination time is shorter than the time period between X-ray flares.
For regions beyond we evolved the chemistry for times beyond until limit cycle behaviour was obtained. These limit cycles were very similar to those shown in the right panel of Fig. 4, having similar amplitudes for the case without magnesium, and very small amplitudes when magnesium was included with .
The requirement that we temporally resolve the X-ray flares while solving the kinetic equations for the chemistry means that a maximum time-step of 1 h was adopted. This causes the calculations to become very expensive computationally, such that for the UMIST model model3 a calculation lasting for 10 000 yr requires a run time of approximately three weeks for each grid-point of the disk model. As a consequence we have been forced to only consider the Oppenheimer & Dalgarno model model1 in the following sections. We simply note at this point that model1 tends to generate thinner dead-zones than model3, such that the results described below represent an optimistic view of which fraction of the disk can sustain MHD turbulence. We note further, however, that similar trends in the results are obtained with model1 and model3 when varying physical parameters, such that model1 gives a reasonable picture of how the ionisation fraction obtained using the more complex model3 responds to changes in ionisation rate etc.
We begin by presenting the results of a model in which we
consider the effects of increasing the plasma temperature during an
X-ray flare, keeping the X-ray luminosity
This allows us to isolate the effects of the hardening of the X-ray spectrum
due to the rising plasma temperature.
We then consider a model for which both the plasma temperature
and the X-ray luminosity increase during flaring activity.
Snapshot at 10 000 yr with constant and
The results for this calculation are shown in the left panel of Fig. 6 which shows the column density of the whole disk plotted as a function of radius using the solid line. The dashed lines show the column density of the active zone for two calculations that assumed constant plasma temperatures of keV and keV, respectively, and the dotted line corresponds to the model whose plasma temperature rose to keV during flares only without an accompanying increase in above 1030 erg s-1. Note that these figures refer to model1 with no heavy metal included.
Figure 7 shows contours of the relative ionisation rate when the plasma temperature is raised from to 7 keV. The change in is dramatic in the deeper, more shielded regions simply because of increased penetration induced by the hardening of the X-ray spectrum. Near the disk surface the change is only slight, but in the interior the local ionisation rate can increase by more than a factor of 1000.
The upper dashed line in the left panel of Fig. 6 shows that
having a constantly increased value of
reduces the depth of the dead-zone throughout the disk. The dotted line,
however, shows that at a point in time midway between two peaks in the
plasma temperature, the dead-zone is unaffected interior to R <1.2 AU
because of the fast recombination time there. Beyond
R>2 AU the dead-zone is a little thinner where the recombination time
is longer than the period between increases in plasma temperature,
corresponding closely to that which would be obtained by exposure
to the time averaged value of the time dependent ionisation rate.
Results with and
A snapshot of the results obtained after 10 000 yr when the X-ray luminosity and the plasma temperature increase during a flare is shown in the right panel of Fig. 6. The solid line gives the column density of the whole disk, and the dashed line gives the column density of the active zone obtained when the X-ray luminosity is constant and takes its base value . The disk becomes active everywhere if we take a constant ionisation rate with and keV. The dotted line corresponds to the model for which the plasma temperature and X-ray luminosity increase during a flare. Note that these models are model1 in the absence of heavy metals, and the snapshot is taken at
|Figure 7: The ratio of the effective X-ray ionisation rates . and refer to local ionisation rates for plasma temperatures and , respectively. The disk parameter are and yr-1. The contour lines refer to values of : 10, 102, and 103.|
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We now consider the time dependent evolution after 10 000 yr has elapsed. The evolution of the ionisation fraction at the disk midplane (and not at the transition zone as in previous similar figures) as a function of the perturbed ionisation rate is shown in Fig. 8 at radii R=1 AU and R=5 AU for model1 in the absence of heavy metals. The first thing to note is the large perturbation to the ionisation rate experienced at the disk midplane, as illustrated by the range of values for . Second, the ionisation fraction, relative to the critical value , at the midplane located at R=1 rises from 0.1 (implying the region is dead) to a value of 2 (implying that the region is active). Thus, the inner regions of the disk show the limit cycle behaviour, and oscillate from having a dead-zone near the midplane to having an active zone there which lasts for a duration slightly longer than half the flare (i.e. about 12 h). This region remains formally "dead'' for the other six and a half days during the X-ray flaring cycle.
|Figure 8: model1 - the change in the ionisation fraction is plotted against the relative change in the ionisation rate at different radial positions R with z/R = 0 (midplane). denotes the ionisation fraction at the transition layer. Each graph describes a limit cycle since the orbit is closed (the arrow points forward in time). The perturbation in the ionisation rate is based on the assumption that both the X-ray peak luminosity and the temperature of the hot plasma component at the flare peaks.|
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The right panel of Fig. 8 shows the evolution of versus the perturbation to the local ionisation rate at the disk midplane at R=5 AU. Here it is clear that limit cycle behaviour is again obtained, with the ionisation fraction showing a closed orbit. In this part of the disk, there remains a thin dead-zone throughout the cycle when heavy metals are not present. The addition of heavy metals with abundance renders the disk fully active in this region for the whole duration of the X-ray cycle when both the luminosity and plasma temperature vary.
In the ideal MHD limit the fastest growing mode of the MRI
has wave number
is the Alfvén speed
is the local Keplerian angular velocity.
The growth rate associated with this mode is
Balbus & Hawley 1991).
Ohmic resistivity strongly affects the growth of linear modes with wave number
k when the associated diffusion rate
exceeds the linear growth rate.
Comparing these rates for the fastest growing mode provides a condition
for the effects of resistivity to dominate over the growth of the mode
due to the MRI:
For the purpose of illustration, let us consider a situation where in the central regions of our disk near the location R=0.5 AU, where the inverse of the Keplerian angular velocity days. An X-ray flare arises, raising the value of for 12 h, allowing exponential growth of the fastest growing mode for this time such that an amplification of 2.5% occurs. Given our definition of it is clear that any field amplification that occurred while is diffused away within the next 12 h after drops below 100, and the growth of this mode is unable to amplify the field or drive the disk toward a turbulent state. The situation can in principle be different for longer wavelength modes. Consider the longest wavelength mode that can fit within the disk vertical extent with wavelength , and associated wavenumber . The wave number associated with the fastest growing mode has if , such that . The diffusion time associated with the longest wavelength mode is an order of magnitude longer than for the fastest growing mode (whereas the growth rate is approximately half of the maximum value). Thus any field amplification that occurs during the 12 h when is diffused away over the next 5 days. This suggests that an X-ray flaring cycle with a periodicity of less than five days could in principle lead to gradual field amplification over successive cycles. The question of whether fully developed turbulence can arise in such a scenario is unclear, and can only be addressed by means of non linear simulations that explicitly account for periodic rises in the ionisation rates due to X-ray flaring and electron recombination.
The calculations presented in this paper consider only gas-phase chemistry, and ignore the effects of dust. As such they are relevant to a stage in protoplanetary disk evolution when substantial grain growth has occurred and a dense dust layer has settled near the midplane. It is well known, however, that small dust grains are able to sweep up free electrons and substantially reduce the ionisation fraction (e.g. Sano et al. 2000; Ilgner & Nelson 2006a), and the assumption of gas-phase chemistry is only really valid when the abundance of dust grains has been depleted by a factor of between 10-4-10-8 below the canonical concentration of 10-12 (Ilgner & Nelson 2006a). An open question that we have not addressed is what happens to species that are adsorbed onto the surfaces of small grains during X-ray flares, and in particular what happens to the adsorbed electrons. Najita et al. (2001) considered the effect of nonthermal desorption of grain mantles due to X-rays, but did not include the desorption of electrons into the gas phase. We speculate that if grain mantles and electrons are desorbed into the gas phase during X-ray flares, and are adsorbed back onto the grains during the time between flares, then the behaviour of the dead-zones throughout the disk will be similar to that already observed in the central regions between 0.5 < R < 2 AU. This is because electrons adsorb onto grains very rapidly throughout the disk, such that the time scale of which electrons will be removed from the gas phase is shorter than the time between flares. An analysis of this issue will be presented in a future publication.
Questions remain, however, about the disk response in regions where the dead-zone is removed during an X-ray flare, but reappears during the low state. In particular it is not clear that MHD turbulence can be generated there as the growth time of the MRI is longer than the duration of the flares. This issue needs to be addressed using non linear simulations that take account of the time dependent ionisation rates induced by X-ray flares.
This research was supported by the European Community's Research Training Networks Programme under contract HPRN-CT-2002-00308, "PLANETS''. The calculations presented here were performed using the QMUL HPC facility purchased under the SRIF initiative. We wish to thank Eric Feigelson for information provided concerning X-ray flares during the early stages of this project.