F. Hersant^{1,2,3} - B. Dubrulle^{1} - J.-M. Huré^{4,5}
1 - CNRS URA 2464 GIT/SPEC/DRECAM/DSM, CEA Saclay, 91191
Gif-sur-Yvette Cedex, France
2 -
LESIA CNRS UMR 8109, Observatoire de Paris-Meudon, Place Jules
Janssen, 92195 Meudon Cedex, France
3 -
Institut für Theoretische Astrophysik, Tiergartenstraße 15,
69121 Heidelberg, Germany
4 -
LUTh CNRS UMR 8102, Observatoire de Paris-Meudon, Place Jules
Janssen, 92195 Meudon Cedex, France
5 -
Université Paris 7 Denis Diderot, 2 Place Jussieu, 75251
Paris Cedex 05, France
Received 31 January 2003 / Accepted 16 August 2004
Abstract
We investigate the relationship between circumstellar disks and the
Taylor-Couette flow. Using the Reynolds similarity principle, this
results in a number of parameter-free predictions about stability of
the disks, and their turbulent transport properties, provided the
disk structure is available. We discuss how the latter can be deduced
from interferometric observations of circumstellar material. We use
the resulting disk structure to compute the molecular transport
coefficients, including the effect of ionization by the
central object. The resulting control parameter indicates that the
disk is well into the turbulent regime. The analogy is also used to
compute the effective accretion rate, as a
function of the disk
characteristic parameters (orbiting velocity, temperature and
density). These values are in very good agreement with experimental,
parameter-free predictions derived from the supposed relationship. The turbulent
viscosity is also computed and found to correspond to an
-parameter
.
Predictions regarding fluctuations are also checked: luminosity
fluctuations
in disks do obey the same universal distribution as energy fluctuations
observed in a laboratory turbulent flow. Radial velocity
dispersion in the outer part of the disk is predicted to be of the
order of 0.1 km s^{-1},
in agreement with available observations.
All these issues provide
a proof of the turbulent character of
circumstellar disks, as well as a parameter-free theoretical
estimate of effective accretion rates.
Key words: turbulence - Solar system: formation - stars: formation - accretion, accretion disks
Stars form by gravitational collapse of molecular clouds. During this process, proto-stars get surrounded and plausibly are fed by an envelope/disk made of gas and dust, which can, under certain conditions, coagulate to form planetary embryos. There is little doubt that gas motion, usually considered as turbulent, plays a major role. Turbulent motion enhances transport properties, thereby accelerating the evolution of the temperature and density in the envelope/disk. Also, turbulence may catalyze planet formation thanks to the trapping of dust particle inside large-scale vortices (Barge & Sommeria 1995; Tanga et al. 1996; Chavanis 2000). As of now, the assertion that circumstellar disks are turbulent (what we shall refer to as the "turbulent hypothesis'') has never been properly checked. It mainly relies on the fact that the luminosity produced by the disk interacting with the central star is very large (see e.g. Hartmann et al. 1998). In certain cases (FU Orionis-type systems), this luminosity is so high that it even supersedes the stellar component. The most widely accepted scenario so far to account for the abnormal luminosities of young stellar objects involves a magnetospheric accretion for classical T Tauri stars. In this case, the matter in the inner parts of the disk is coupled with the stellar magnetic field and falls onto the stars along the field lines at the free fall velocity (see e.g. Gullbring et al. 1998; Muzerolle et al. 1998). This entails an accretion shock at the stellar surface in which almost all the visible and UV excess is produced (veiling the stellar lines). This scenario has shown many successes in interpreting spectral features of T Tauri stars like sodium and hydrogen line profiles (see e.g. Muzerolle et al. 1998). In the case of Fu Ori stars, the currently advocated picture involves a wide boundary layer (see e.g. Popham et al. 1996). One of the main weaknesses of this scenario is its low predictive power since it relies on an adjustable parameter (the accretion rate) which must be postulated a posteriori by comparison with observational data.
Indeed, in this framework, the luminosity is directly related to the amount of energy released by the disk. The necessity for turbulence comes from the hypothesis that no laminar motions can produce the amount of energy dissipated required to explain observed luminosities (see e.g. Pringle 1981 and references therein). However, no attempt has been made to substantiate this claim in a quantitative manner. The questions we address here are: what are the luminosities produced by a laminar disk and by a turbulent disk? How do these compare with observations? These two questions are equally difficult to answer, but they hide different levels of difficulties. One is of a theoretical nature: the physical processes at work in this disk/star interaction region are complex. A correct description should simultaneously include the resolution of turbulence (with compressibility effects), radiative transfer (accounting for UV-irradiation by the star), magnetic processes, chemistry, the disk flaring, phase separation, time evolution, etc. The second difficulty is of an observational nature. At the present time, information has been gathered about the temperature, density and velocity distributions in the outer parts of disks, at 100 AU typically, thanks to high resolution interferometry and clever data analysis (Guilloteau & Dutrey 1998). Unfortunately, basic parameters (mean free path, sound velocity and viscosity) connected with the gas dynamics and dissipation are still not known in the inner regions, and especially in the region where the disk and the star interact.
Because of these difficulties, we choose to adopt a radically different approach than the classical model: instead of trying to build a fully "realistic'' circumstellar disk, we use a simplified hydrodynamic model ("zero order model'') and study in detail its physical properties. In the future, we will slowly increase its complexity (and reality!) by adding new ingredients like magnetic field, stratification, radiative transfer. Here, we show that our zero order model is analogous to an incompressible rotating shear flow. It is therefore amenable to treatment as a simple laboratory prototype, the Taylor-Couette flow. From theoretical and experimental studies of the properties of this prototype, one can then build general laws in circumstellar disks by a simple use of the Reynolds similarity principle. Taylor-Couette flow is a classical laboratory flow, and it has been the subject of many experiments. A recent review about stability properties and transport properties for use in astrophysical flows has been made by Dubrulle et al. (2004). As a result, they derive critical conditions for stability, and simple scaling laws for transport properties, including the influence of stratification, magnetic field, boundary conditions and aspect ratio. In the present paper, we apply these results to circumstellar disks and derive the expression of the characteristic parameters of the model as a function of astrophysical observables. We propose a procedure of quantitative estimation of the observables using the observational results of Guilloteau & Dutrey (1998) and derive parameter-free predictions about turbulence and turbulent transport in circumstellar disks. These predictions are tested against observational data from T Tauri and FU Ori stars.
Observation of circumstellar disks suggests that they have sizes
between 100 and 1000 astronomical units. In the following, we will focus
only on the part of the disk expected to behave like an incompressible fluid.
An estimate of the importance of compressibility can be obtained via
the Mach number, the ratio of the typical velocity to thermal
velocities. It is generally admitted that compressibility effects
start playing a role when this number reaches values of unity. In the
outer part of the disk, this ratio has been estimated by Guilloteau
& Dutrey (1998) from CO line profiles. Its value is about
0.2-0.3. In the inner part of the disk (radius ranging from 1 to 30
astronomical units), we may use the disk structure inferred from
the D/H ratio measured in the Solar System
(Drouart et al. 1999; Hersant et al. 2001), which leads
to a Mach number of the order of 0.05 to 0.1. These figures
indicate that both the inner and the outer part can be treated
as
incompressible fluids. Closer to the star, the situation is less
clear. On the one hand, temperature tends to increase strongly, leading
to an increase of the sound velocity and a decrease of the Mach
number. On the other hand, as one gets closer to the boundary, one
may expect larger typical velocities induced by larger velocity
gradients, and thus an increase of the Mach number. There are no direct
observations supporting one scenario or the other. We shall then
consider two scenarios: one in which the Mach number Ma never exceeds
unity. In this case, the whole disk is incompressible, and connects
smoothly onto the star at the star radius. The inner boundary is thus
defined as
.
In the second scenario, the Mach number
reaches unity at some "interaction radius''
,
leading to
an inner boundary at
.
At this location, a shock
appears, in which all velocities are suddenly decreased to very small
values. In the shock, all the kinetic energy is transfered to the
thermal energy, thereby producing a strong temperature increase (by a
factor of
). This
entails an increase of ionization and the matter gets more coupled to
the stellar magnetic field.
This second situation is considered in magnetospheric
accretion models (Hartmann et al. 1998; Hartmann et al.
2002), in which case
is the Alfven radius; see Schatzman (1962, 1989). Figure 1 summarizes these two possible
configurations. From a hydrodynamical point of view, in the first
situation the boundary is similar to a free-slip boundary (with
possible non-zero velocities in the direction tangential to the star
boundary), while in the second situation, the interaction radius acts
as a no-slip boundary (with all velocities becoming zero). This
difference may be reflected in the transport properties, see Dubrulle et al. (2004). In the following, the free-slip boundary condition will be
referred to as smooth, while the no-slip boundary will be
referred to as rough.
In the laboratory experiments reviewed in Dubrulle et al. (2004), the turbulent
transport depends on the boundary conditions.
Specifically, transport is enhanced (with respect to other
boundary conditions) for boundary
conditions of the rough or no-slip type.
In the astrophysical case, it is not quite clear whether these two
boundary conditions apply, or even whether different inner and outer
boundary conditions result in an intermediate transport enhancement.
We shall therefore devise observational tests using quantities
independent of boundary conditions via a suitable
non-dimensionalisation.
Figure 1: Two possible configurations considered in the present model: a) the whole disk is incompressible and extends onto the proto-star; and b) the disk is incompressible until an "interaction radius'' imposed for instance by a magnetic field. | |
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In either case,
the angular velocity
at the inner boundary is that of the star,
namely
.
For
,
the Mach number
of the flow is less than one by construction, i.e. pressure
fluctuations vary over a time scale short compared to the dynamical
time. In such a case, one can assume hydrostatic equilibrium in the
vertical direction, implying a decoupling of the vertical and
horizontal structure. It is then convenient to describe the disk by
its
"horizontal equations'', obtained by averaging the original equations of
motion in the vertical direction. The procedure is described e.g. in
Dubrulle (1992). It leads to:
(3) |
In the opposite case,
H will rather vary like the Jeans length in the vertical direction,
as:
Equation (1) admits simple basic state, under the
shape of stationary axi-symmetric solution. The mass conservation
then implies:
The basic state depends on three constants A, B and M_{t}, which
must be specified through some sort of boundary conditions. In the
case of astrophysical flows, the boundary conditions are not very
well known and it is less easy to constrain the parameters. The
condition that the rotation velocity of the disk matches the star
velocity at the interaction radius only provides one relation between
the three parameters:
Figure 2: Velocity profile in a circumstellar disk in the viscous regime, with Keplerian velocity in the outer part, and smooth matching onto the central object at the interaction radius. For this example, the mass of the central star has been taken as a solar mass. | |
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Astrophysical disks are (weakly) compressible and radially stratified.
It is however possible to build an incompressible analog of them,
using clever boundary conditions. This remark is at the heart of the
laboratory prototype. Consider an incompressible, unstratified
fluid, enclosed within a domain with cylindrical symmetry, bounded by
inner and outer radii
and .
Its equations of motion
are given by the Navier-Stokes equations:
The basic state depends on three constants A, B and K, which
must be specified through boundary conditions. In laboratory flows,
these conditions are usually well defined and allow for a simple
determination of the constants once the rotation velocities at the
inner and outer boundaries are known (Bahl 1970):
Comparing Eq. (10) with Eqs. (17) and (18), it is possible to see that the "laboratory'' analog
of Keplerian flow is such that:
The computation of the control parameters requires an estimate of
molecular transport coefficients. These coefficients depend on the
ionization state of the gas. There are two sources of ionization.
Thermal ionization is efficient in the inner part of the disk. The
corresponding ionization fraction can be written as (Fromang et al.
2003):
When the gas is neutral, the viscosity and heat diffusivity are given
by (Lang 1980):
When the gas is weakly ionized (
), the transport
coefficient must be multiplied by a factor (Lang 1980):
The resistivity of an ionized gas can be written as the sum of the
resistivity induced by electron-neutral collisions and electron-ion collisions:
(34) |
(35) | |||
(36) |
Table 1: Observational parameters for T Tauri stars (from Bouvier 1990; Hartmann et al. 1998) and for FU Ori stars (from Popham et al. 1996) considered in this study ( left) and disk physical parameters ( right). The computation of the corotation radius requires the knowledge of the stellar rotation velocity. In case this last quantity is not available, the corotation radius has been set to , the solar value. Lower and upper bound on and Re have been computed using either the star radius or the corotation radius for . Accretion rates are in yr^{-1}.
Parameters associated with the disk are , , and . The disk inner radius depends on whether the disk/star interaction is direct or indirect. In the first case, . In the second case, may not exceed the corotation radius at which the disk velocity matches the star velocity. In the following, we will consider variations of in between these two limits. The disk outer radius must be specified through the implicit relation (11). Direct observations of the disk suggest that the disk size is of the order of AU for the disk around T Tauri and even smaller, AU, for the disk around FU ORI (Kenyon 1999). Clearly, is thus the maximum size can achieve. For practical reasons, we defer its discussion to the next section, after computation of the temperature and density profile.
Parameters relative to the proto-star have been measured in some T Tauri and FU Ori stars. Table 1 gives the sample of stars we shall use in the following. It is particularly convenient and illustrative to scale all quantities in the problem with respect to values defined at the distance of AU, which is the characteristic radius corresponding to a disk with cm and AU, the two extreme limits for and . We also use, as a reference, a rotation period of the star of 8 days (typical T Tauri star), leading to s^{-1}.
The temperature and number density in circumstellar disks are
not known due to the lack of
spatial resolution. However, their magnitude can
be deduced by short radii extrapolation of measurements made on the outer
disk. The inversion method of Guilloteau & Dutrey
(1998), based on -square fitting of CO interferometric maps, yields
temperature and density profiles at
AU.
For instance, for the disk around DM Tau (
),
their method predicts
It is interesting to study the ionization state of the disk with
temperature and density observed in DM Tau. The ionization
fraction is plotted as a function of radius in
Fig. 3 for the thermal and X-ray contribution. One
sees that the thermal contribution dominates at r<1 AU, while the
X-ray contribution becomes important at larger radii. However,
comparing with the limiting ionization state Eq. (33), one
sees
that only the outer part of the disk r>100 AU is sufficiently
ionized to influence the molecular viscosity.
Figure 3: Midplane ionization fraction due to thermal contribution (dot-dashed line) and X-ray contribution (dot-dotted line). The solid line is the limiting ionization fraction, below which the ionization does not influence molecular transport. The shaded area is the region whereionization has to be taken into account in the computation of theviscosity. | |
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The stability properties of circumstellar disks can be found by comparing the physical Reynolds number Re with critical Reynolds numbers derived in laboratory experiments, in the anti-cyclonic non-linear regime. These measurements are summarized in Dubrulle et al. (2004). Disregarding any body forces, one finds a critical Reynolds number of the order of 2300, well below the disk value. Taking into account the possible stable vertical stratification observed e.g. in DM Tau (Dartois et al. 2003)^{}, one obtains a slightly larger value of the order of 4000 (Dubrulle et al. 2005). The presence of a vertical magnetic field may increase the critical Reynolds number, due to the low magnetic Prandtl number prevailing in disks. Using the scaling of Willis & Barenghi (2002), one finds a critical Reynolds number . This is still well below the observed Reynolds number. These numbers lead us to conclude that the disk is turbulent.
However, due to the huge variation of the transport coefficients
across the disk, one may wonder how strong this conclusion is. A way
to answer this question is to see how locally in the disk, the
stability criterion are satisfied using a "local'' non-dimensionalized
parameter, built by replacing
by r the distance to the
central object. The result of this procedure is plotted in Fig. 4. One sees that at any radius, such an effective
local Reynolds number is well above any critical Reynolds number due
to body forces.
This strengthens our conclusion.
Figure 4: Physical local Reynolds number in circumstellar disks as a function of radius (dotted-line with symbols). The dashed-dotted line and the full line are the critical Reynolds number deduced from laboratory experiments, see Dubrulle et al. (2004). | |
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Figure 5: Comparison between the non-dimensional energy dissipation predicted from laboratory measurements in the wide gap limit ( dotted lines) or in the small gap limit ( plain lines) and observed in circumstellar disks ( symbols), as a function of the Reynolds number . For an easier comparison, the mean energy dissipation has been translated into the non-dimensional accretion rate (computed using Eq. (45) and observationally determined parameters reported in Table 1). The symbols report the value using , which provides an upper bound of the energy dissipation. The circles report the value using , which provides the lower bound of the energy dissipation. All the quantities have been computed using temperature and density estimated for the DM Tau system using the results of Guilloteau & Dutrey (1998), so there is no adjustable parameter in this plot. | |
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We find the following scaling for
:
At , our model (Eqs. (45) and (46) predicts that , resulting in yr^{-1} for T Tauri stars. This is in good agreement with the observed values ranging from to yr^{-1} (Hartmann et al. 1998). Disks around FU Ori are characterized by a smaller disk radius, leading to higher values of according to Eq. (47), by a factor 5. This is not quite enough to reach values of up to yr^{-1}, associated with disks around FU Ori stars (Kenyon 1994). Such values could be obtained if the typical disk density is higher in disk around FU Ori, resulting in more massive disks. This is plausible, since FU Ori stars are younger than T-Tauri stars.
A graphical representation of this discussion can be obtained by plotting the computed as a function of using the values listed in Table 1 as input parameters. To remove the problem of our ignorance of the actual value of , we have used the relation and computed the corresponding and . The actual dissipation lies in between the two corresponding estimates. These estimates are plotted in Fig. 5. For comparison, we have added the theoretical predictions (Eqs. (45) and (46)) giving the minimum and the maximum expected values in the turbulent case, as well as the laminar value. This last value is very much lower than the turbulent values, and is never even nearly approached by any stars we considered. This may be seen as a proof of the turbulent character of these disks.
We also see that energy dissipation for disks around T Tauri stars has a tendency to cluster between the minimal and maximal values allowed by the theoretical predictions. The relative position of the cluster of points is slightly better in the case where is computed with , which may be an indication that is actually closer to the corotation radius than to the star radius. However, given the error bars stressed above, this is probably not enough to conclude that disks around T Tauri stars are connected through a magneto-sphere, rather than through a boundary layer. In the case of FU Ori, the points are clearly above the maximum allowed by our choice of parameters. The discrepancy is slightly lower for the case when , a choice which should probably be favored by the possible signatures of boundary layer in these objects (Kenyon 1994). In that case, an increase of the surface density by a factor 10 with respect to our values would be enough to solve the discrepancy.
Our estimate neglects the influence of the magnetic field. Laboratory experiments using liquid metals have proved that this can potentially change the intensity of the transport with respect to the pure hydro-dynamical case. However, no experiment has been performed so far to study the magnetic influence in regimes relevant to astrophysical disks.
(50) |
Up to now, we have considered only the mean energy dissipation and its
luminous counterpart, but we can also obtain interesting information from the
luminosity fluctuations which reflect the dynamics of the underlying
turbulent flow. In laboratory experiments with smooth boundary
conditions, turbulent fluctuations are
observed to follow a universal (i.e. Reynolds
number-independent), log-normal
distribution (Lathrop et al. 1992) with variance
.
The universal distribution occurs
for variables normalized by their mean. Energy dissipation is
proportional to the wall shear stress squared. Since the functional
shape of the log-normal distribution is unchanged by squaring,
the distribution of energy dissipation should also be log-normal.
To check this prediction, we have computed the distribution of the
luminosity fluctuations observed from the disk around BP Tau and from
the disk around V1057 Cygni. The results are shown in Figs. 6
and 7. One sees that the
fluctuations in the disk around V1057 Cyg are very well fitted by a
log-normal distribution, with a variance similar to that of
laboratory experiments. In the case of BP Tau, however, the comparison
is not as good. This difference between the two systems may be traced
to different boundary conditions. If we accept that disks around
T Tauri stars follow the magnetospheric accretion scenario, while the disk
around FU Ori is connected to the star through a boundary layer, it may not be
surprising that only the disk around FU Ori follows the laboratory,
smooth boundary condition distribution. Since we do not have any
measurements for rough boundary conditions, we cannot say whether the
discrepancy comes from the different boundary conditions, or from the
presence of other physical effects, like accretion shock or a magnetic
field.
Figure 6: Distribution of luminosity fluctuations observed disk around BP Tau ( symbols) compared with a log-normal distribution of various variance ( plain line). The value of is indicated beside each line. | |
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Figure 7: Distribution of luminosity fluctuations observed disk around V1057 Cyg ( symbols) compared with a log-normal distribution of variance ( plain line). | |
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Laboratory measurements provide interesting clues about the intensity of velocity fluctuations. Since such fluctuations may be potentially observable in disks using non-thermal line widening, they may be used as additional constraints or observational tests of the analogy between laboratory flows and circumstellar disks. From results of Dubrulle et al. (2004), it appears that azimuthal velocity fluctuations should be proportional to the mean azimuthal velocity, with a proportionality factor depending weakly on the Reynolds number, like . With and (see Sect. 3.1), the factor is of the order of 0.01. Using Eq. (10), we can compute the azimuthal velocity dispersion for a typical circumstellar disk around a T Tauri. The azimuthal velocity dispersion decreases from about 0.6 km s^{-1} in the inner part, to 0.03 km s^{-1} in the outer part, at 100 AU The total velocity dispersion depends on the anisotropy of the turbulence. In laboratory experiments, the radial relative velocity dispersion is observed to be about twice the azimuthal velocity dispersion. There was no measure of the vertical velocity dispersion, but it can be expected to be much smaller than the horizontal dispersion due to the rotation-induced anisotropy (Dubrulle & Valdettaro 1992).
The velocity dispersion in disks has been measured by Guilloteau & Dutrey (1998) at r>100 AU They obtain a value of the order of 0.1 km s^{-1}, which would correspond to a value of about 0.05 km s^{-1} for the azimuthal component. This is close to the values found from comparison with laboratory flows.
In this paper, we have derived and studied the analogy between circumstellar disks and the Taylor-Couette flow. This analogy results in a number of parameter-free predictions about stability of the disks, and their turbulent transport properties, provided an estimate of the disk structure is available. We have proposed to get this estimate from interferometric observations of circumstellar disks, and checked that the energy dissipation, the turbulent transport, and the fluctuations in circumstellar disks all follow behavior compatible with the prediction from the analogy. This check can first be used as a clear proof of the turbulent character of circumstellar disks. A second interesting application would be to build from this analogy a parameter-free model of circumstellar disks. In this respect, the proportionality between the turbulent viscosity and the so-called "accretion rate'' (a quantity easily accessible by observation) is very interesting because it opens the possibility to infer the disk structure from the observation of its luminosity. For this, a model has to be built linking the turbulent transport and the disk structure. This is the subject of ongoing work.
We note finally that our model could also possibly apply to other types of disks (e.g. around black holes, or in close binaries) given minor adaptations.
Acknowledgements
We thank A. Dutrey for useful discussions and comments on the manuscript. We are indebted to the anonymous referee whose helpful remarks led us to clarify the paper and our thoughts. This work has received support from the Programme national de Planétologie. F.H. acknowledges support from an ESA research fellowship.