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Astron. Astrophys. 356, 1089-1111 (2000)

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4. The rate of escape

4.1. Formulation of the problem

The diffusion Eq. (60) is similar, in structure, with the Kramers-Chandrasekhar equation:

[EQUATION]

introduced in the case of colloidal suspensions and in stellar dynamics (Kramers 1940, Chandrasekhar 1943a,b). In this equation, [FORMULA] governs the velocity distribution of the particles in the system. The first term in the r.h.s is a pure diffusion and the second term is a dynamical friction . These terms model the encounters between stars or the collisions between the colloidal particles and the fluid molecules. Comparing Eqs. (60) and (86), we see that the position in (60) plays the role of the velocity in (86) and the capture time [FORMULA] the role of the friction time [FORMULA]. In particular the friction force and the drift term are linear in [FORMULA] and [FORMULA] respectively. Eq. (86) was used by Chandrasekhar (1943b) to study the evaporation of stars in globular clusters (this is similar to the Kramers problem for the escape of colloidal suspensions over a potential barrier). Due to collisions, some stars may acquire very high energies and escape from the system (being ultimately captured by the gravity of nearby objects). Similarly, in our situation, turbulent fluctuations allow some dust particles to diffuse toward higher and higher radii and finally leave the vortex (being eventually transported away by the local Keplerian shear). In each case, the friction force or the drift acts against the diffusion and can reduce significantly the escape process.

We try now to evaluate the rate of particles that leave the vortex on account of turbulent fluctuations. To that purpose, we formulate the problem in terms of the density probability [FORMULA] that a particle located initially in the annulus between [FORMULA] and [FORMULA] will be found in the surface element around [FORMULA] at time t. According to Eq. (60), the time evolution of the probability [FORMULA] is given by

[EQUATION]

with initial condition

[EQUATION]

where [FORMULA] stands for Dirac's [FORMULA]-function. We assume that when the particle reaches the vortex boundary at [FORMULA], it is immediately transported away by the Keplerian shear. In other words, we adopt the boundary condition:

[EQUATION]

We call

[EQUATION]

the current of probability, i.e. [FORMULA] gives the probability that a particle crosses an element of length dl between t and [FORMULA] ([FORMULA] is a unit vector normal to the element of length under consideration).

We first introduce the probability [FORMULA] that a particle located initially in the annulus between [FORMULA] and [FORMULA] reaches for the first time the vortex boundary between t and [FORMULA]. According to what was just said concerning the interpretation of (90), we have:

[EQUATION]

The total probability [FORMULA] that the particle has reached the vortex boundary between 0 and t is

[EQUATION]

Finally, we average [FORMULA] over an appropriate range of initial positions in order to get the expectation [FORMULA] that the particle has left the vortex at time t. We have

[EQUATION]

where [FORMULA] governs the initial probability distribution of the particles in the vortex. In terms of the function Q, the rate of escape of the particles can be written

[EQUATION]

As mentioned already, this problem is similar to the diffusion of colloidal suspensions over a potential barrier or to the evaporation of stars in globular clusters. As will soon become apparent, it reduces to solving a pseudo Schrödinger equation for a quantum oscillator in a "box". In this analogy, the rate of escape appears to be related to the fundamental eigenvalue of the quantum problem. An explicit expression for the rate of escape can be obtained in two limits: when [FORMULA] or [FORMULA], the drift term can be ignored and the Fokker-Planck equation reduces to a pure diffusion equation (Sect. 4.2). On the other hand, when [FORMULA], the drift term is dominant and a perturbation approach inspired by the work of Sommerfeld and Chandrasekhar can be implemented to determine the ground state of the artificially limited quantum oscillator (Sects. 4.3 and 4.4).

4.2. The rate of escape when the drift term is ignored

When the drift term can be ignored (i.e. for very light or very heavy particles, see inequalities (75)(76)), the Fokker-Planck equation (87) reduces to a pure diffusion equation

[EQUATION]

which has to be solved in a circular domain with boundary conditions (88) and (89). The solution of this classical problem is

[EQUATION]

where [FORMULA] is the Bessel function of order m and the [FORMULA]'s denote the roots of the Bessel function [FORMULA]. The probability [FORMULA] that a particle with an initial position [FORMULA] has reached the vortex boundary between t and [FORMULA] is [see Eq. (91)]:

[EQUATION]

and the total probability that the particle has escaped during the interval [FORMULA] is [see Eq. (92)]:

[EQUATION]

Averaging the foregoing expression over all [FORMULA]'s in the range [FORMULA] with equal probability [FORMULA] (this corresponds to an initially homogeneous distribution of particles in the vortex), we obtain

[EQUATION]

To sufficient accuracy, we can keep only the first term in the series. The expectation that the particle has left the vortex at time t is therefore:

[EQUATION]

Since [FORMULA], this term represents [FORMULA] of the value of the series (99) and the approximation (100) is reasonable.

In conclusion, when the drift term is ignored, we find that the escape time is

[EQUATION]

It corresponds, typically, to the time needed by the particles to diffuse over a distance [FORMULA], the vortex size. For light particles, using (71), we obtain explicitly

[EQUATION]

and for heavy particles

[EQUATION]

We shall come back to these expressions in Sect. 4.5.

4.3. The effective Schrödinger equation

We now return to the general problem for the rate of escape when proper allowance is made for the drift. We find it convenient to introduce the notations

[EQUATION]

or, in words, the time is measured in terms of the capture time and the distances are normalized by the concentration length (64). We let also:

[EQUATION]

and

[EQUATION]

In terms of these new variables, the problem (87)(88) and (89) takes the form:

[EQUATION]

[EQUATION]

[EQUATION]

With the change of variables

[EQUATION]

we can transform the Fokker-Planck Eq. (107) in a Schrödinger equation (with imaginary time) for a quantum oscillator:

[EQUATION]

However, contrary to the standard quantum problem, Eq. (111) has to be solved in a bounded domain of size [FORMULA] with the boundary condition (109). In other words, our problem consists in determining the characteristic functions of a quantum oscillator in a "box".

First, we notice that a separation of the variables can be effected by the substitution

[EQUATION]

where [FORMULA] is, for the moment, an unspecified constant. This transformation reduces the Schrödinger equation in a second order ordinary differential equation:

[EQUATION]

Let [FORMULA] be the solutions of this differential equation satisfying the boundary condition [FORMULA] and [FORMULA] the corresponding eigenvalues. The eigenfunctions form a complete set of orthogonal functions for the scalar product

[EQUATION]

This system can be further normalized, i.e. [FORMULA]. Any function [FORMULA] satisfying the boundary condition (109) can be expanded on this basis, and we have the formula:

[EQUATION]

In particular:

[EQUATION]

The general solution of the problem (107) (109) can be expressed in the form

[EQUATION]

where the coefficients [FORMULA] are determined by the initial condition (108), using the expansion (116) for the [FORMULA]-function. We obtain:

[EQUATION]

Using the foregoing solution for w, the probability that a particle initially located at [FORMULA] leaves the vortex between [FORMULA] and [FORMULA] is [see Eq. (91)]:

[EQUATION]

The probability [FORMULA] that the particle has left the vortex at time [FORMULA] is therefore [see Eq. (92)]:

[EQUATION]

Finally, to obtain [FORMULA], we have to take the average of the foregoing expression over the relevant range of [FORMULA]. To make the solution explicit, it remains to determine the eigenfunctions [FORMULA] and eigenvalues [FORMULA] of the quantum oscillator.

4.4. The ground state of the quantum oscillator

The dependence of the eigenvalues [FORMULA] on the size of the "box" can be obtained from a procedure developed by Sommerfeld in his studies of the Kepler problem and the problem of the rotator in the quantum theory with "artificial" boundary conditions (Sommerfeld & Welker 1938, Sommerfeld & Hartmann 1940). This was used later on by Chandrasekhar (1943b) to determine the rate of escape of stars from globular clusters, from which our study is inspired.

With the change of variables

[EQUATION]

Eq. (113) becomes:

[EQUATION]

Another change of variables:

[EQUATION]

transforms Eq. (122) into Kummer's equation

[EQUATION]

In the case of a "free" oscillator ([FORMULA]), it is well-known that the eigenvalues are [FORMULA] (with [FORMULA]) and the eigenfunctions are proportional to the Laguerre polynomials [FORMULA]. In a bounded domain ([FORMULA]), the eigenvalues will be different but if [FORMULA] is sufficiently large, we expect that the difference be small. Therefore, we expect [FORMULA]. Accordingly, for values of [FORMULA] of the order of unity, or greater, the first term in the series (120) will provide ample accuracy. Thus

[EQUATION]

We have therefore to determine the ground state of our artificially limited quantum oscillator. To that purpose, we expand f in a series:

[EQUATION]

and substitute this expansion in Eq. (124). Identifying term by term, we obtain the recursion formula:

[EQUATION]

which is easily reduced to

[EQUATION]

We know already that the eigenvalue [FORMULA] of our artificial quantum oscillator is very small. Keeping only terms of order [FORMULA] in the products (128), we obtain:

[EQUATION]

The ground state function can therefore be written

[EQUATION]

with

[EQUATION]

where

[EQUATION]

is the Exponential integral and [FORMULA] the Euler constant. The function [FORMULA] is plotted on Fig. 11. The fundamental eigenvalue [FORMULA] is determined by the condition [FORMULA] yielding

[EQUATION]

[FIGURE] Fig. 11. The function [FORMULA].

Returning to the original eigenfunction [FORMULA], we have established that

[EQUATION]

where [FORMULA] is a normalizing factor. The final result can therefore be expressed in the form:

[EQUATION]

with

[EQUATION]

In the expression for the amplitude, the bar indicates an average over the relevant range of [FORMULA]. In practice, A is close to unity but its precise value determines to which accuracy the approximation consisting in keeping only the dominant term in the series (120) is justified. For sufficiently large [FORMULA]'s, this will always be the case, so we shall take

[EQUATION]

This expression is consistent with the physical condition [FORMULA] as [FORMULA].

4.5. Application to the solar nebula

Let [FORMULA] denote the total mass of particles present in the vortex at time [FORMULA]. If we assume no renewal from the outside then, on account of turbulent fluctuations, some particles will escape the vortex and, according to Eq. (137), the mass will decay like:

[EQUATION]

The "evaporation" will take place on a typical time

[EQUATION]

where [FORMULA] is given by Eq. (133). Note that both [FORMULA] and [FORMULA] depend on the friction parameter [FORMULA] of the particles. For very light or very heavy particles, [FORMULA] and [FORMULA]. With the definition (106) we find:

[EQUATION]

in good agreement with the result (101) obtained when the drift term is ignored. This shows that Eq. (139) can be used with good accuracy for all types of particles. Substituting Eqs. (73)(74) and (33)(34) in Eq. (139), we obtain explicitly:

[EQUATION]

[EQUATION]

where [FORMULA] is the function defined by Eq. (131). When [FORMULA], Eq. (141) tends to

[EQUATION]

Very light particles are not concentrated in the vortices and they diffuse away after only [FORMULA] rotation periods. By contrast, for optimal particles with [FORMULA], the concentration reduces considerably the evaporation and we find

[EQUATION]

The particles are so much concentrated in the vortices ([FORMULA]) that the turbulent fluctuations are not sufficient to allow them to reach the edge of the vortex. Therefore, their escape is completely negligible. Finally, for [FORMULA], we have

[EQUATION]

For very heavy particles, the escape time goes to infinity because the particles are not coupled to the gas and therefore not affected by diffusion. Recall, however, that our study is not well suited for very heavy particles. As discussed in Sect. 2.3, these particles are more likely to cross the vortices without being captured. The curve [FORMULA] versus [FORMULA] is potted on Fig. 12 (see also Fig. 13 for the dependence of [FORMULA] on the size of the particles). The asymptotic regimes (143) and (145) are obtained for particles with [FORMULA] and [FORMULA] respectively. For these particles, [FORMULA] (see inequalities (75) and (76)), so there is no concentration. Therefore, the asymptotic limits (143) and (145) agree with the results (102) and (103) obtained when the drift term is ignored. When the drift becomes efficient, i.e. for [FORMULA], the escape time increases extremely rapidly and the particles remain emprisoned in the vortices.

[FIGURE] Fig. 12. Escape time of the particles as a function of their friction parameter.

[FIGURE] Fig. 13. Escape time of the particles as a function of their size at [FORMULA] and [FORMULA] ([FORMULA]).

These results assume that there is no renewal of the particles in the vortices. If we now take into account a capture process like in Sect. 2.4, we are led to consider the balance equation:

[EQUATION]

The first term in the right hand side accounts for a flux of particles inside the vortex (see Eq. (37)) and the second term for an exponential decay of the particle number due to "evaporation" (see Eq. (138)).

For particles which can experience gravitational collapse (see inequalities (81)(83) and (85)), the escape time is much longer than the vortex lifetime (see Fig. 12). In that case, evaporation can be neglected and Eq. (146) reduces to Eq. (37). The maximum mass captured by the vortex is determined by its lifetime and the results of Sects. 2.5 and 3.4 are unchanged.

Consider, however, the idealistic situation when [FORMULA]. For sufficienlty large times ([FORMULA]), the vortex will achieve a stationary distribution of dust particles obtained by setting [FORMULA] in Eq. (146). This gives a maximum mass

[EQUATION]

which is now limited by the evaporation time (compare Eq. (147) with Eq. (38)). The density enhancement in the vortices (taking into account the concentration effect) is

[EQUATION]

as represented on Fig. 14. We find that even if the vortex had an infinite lifetime, particles with friction parameter [FORMULA] cannot trigger the gravitational instability. This result implies (see Fig. 5) that subcentimetric particles cannot form planetesimals even in the most optimistic case. Sticking processes are necessary to produce larger particles.

[FIGURE] Fig. 14. Enhancement of the dust surface density in the vortices as a function of the friction parameter when evaporation due to small scale turbulence is taken into account ([FORMULA]).

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© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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