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Astron. Astrophys. 361, 303-320 (2000)

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3. The formation and dynamics of clouds

The linear stability analysis in the previous section has shown that there are perturbations of a minimum wavelength that can grow beyond any limit. However, restricting ourselves to small amplitude perturbations we cannot follow the instability into the nonlinear regime. Qualitatively we expect that the unstable waves will affect the whole wall, which will become heavily corrugated. Bending of the field lines will become so severe that magnetic reconnection will occur across antiparallel lines of force coming close to each other. Therefore the field cannot ultimately prevent fragmentation of the wall into blobs of typical diameter [FORMULA]. Before we can calculate the size of such a cloudlet, we have to discuss the properties of Loop I and the Local Bubble, from which the pressure difference across the wall, driving the instability, can be inferred.

3.1. Properties of the Loop I superbubble

The plasma properties of the Loop I superbubble were determined by spectral analysis of ROSAT PSPC soft X-ray data along several radial cuts across the North Polar Spur (Egger 1993, 1995). Deprojection under assumption of spherical symmetry resulted in radial n- and T-profiles of the bubble. Large variations are found in density as well as in temperature, and in some regions the uncertainties are considerable. However, close to the bright edge of the Spur, i.e. the shock heated rim of the bubble, the data are relatively clear and the [FORMULA] statistical errors only about [FORMULA] (Egger 1993). To be conservative we use a value of [FORMULA]. Values derived from spectral fitting are generally model dependent. Here we have carefully reanalyzed the data obtained by Egger (1993) and have used the standard Raymond-Smith collisional ionization equilibrium (CIE) model with solar abundances. As a result the derived parameters for n and T are not statistically independent. Typical values of the shocked interior of the bubble close to the shell are [FORMULA] K and [FORMULA]. To lowest order, variations in [FORMULA] and [FORMULA] are such that the pressure [FORMULA] is uniform on a global scale, except for some regions, as mentioned above. Since the errors of n and T are of the same order the maximum and minimum allowable pressure for Loop I are [FORMULA] and [FORMULA] K, respectively. In the same manner we obtain for the Local Bubble from the conservative assumption of CIE [FORMULA] and [FORMULA] for the density and temperature, respectively (Egger 1993). Errors are of the same order so that the maximum and minimum permissible pressure range is [FORMULA] and [FORMULA] K, respectively.

From absorption measurements of soft X-rays with ROSAT it is known that the HI column density of the wall is [FORMULA] (Egger 1998). Thus the wall thickness is

[EQUATION]

where we have written [FORMULA]. Note that the column density of the wall [FORMULA] is presumably about twice that of the Loop I shell, [FORMULA]; however since the observational uncertainties of the shell thickness can be larger than a factor of 2, we do not discriminate in the following.

The pressure difference between Loop I and the Local Bubble leading to the acceleration of the wall ranges [FORMULA] to [FORMULA]. Thus even in the worst case there is a substantial pressure difference that pushes the wall into the Local Bubble and ensures the existence of an instability. In the following estimates we use for convenience the mean value [FORMULA].

The z-component of the "effective gravitational" acceleration of the wall is then given by

[EQUATION]

with [FORMULA] being the mean interstellar atomic mass, and [FORMULA] (see Sect. 4.1).

The average radius of the Loop I bubble is [FORMULA] and therefore the density of the ambient medium is (Egger 1993)

[EQUATION]

3.2. Properties of the unstable region

The wall and the Loop I shell are illuminated by the stellar content of the Sco-Cen association. The dynamical age of the stellar cluster is about [FORMULA] years and therefore the number of SN candidate stars is about 40 of spectral type between B3 and B1 (Egger 1993). The Lyman continuum output rate of a B1 star is [FORMULA] photons s-1 (Spitzer 1978). In order to keep the wall ionized a photon rate of

[EQUATION]

is required, where [FORMULA] for [FORMULA] (see below) is the recombination coefficient for hydrogen into the atomic level [FORMULA], excluding recombinations into the ground state which produce an ionizing photon to be absorbed "on the spot". Therefore the number of early type stars falls short by 3 orders of magnitude and the shell mainly consists of moderately warm neutral HI . The same is true for the wall, because in the Local Bubble there is no known stellar cluster that could contribute to ionization.

The magnetic field strengths inside Loop I and in the shell are largely unknown. Since Loop I was first observed as an annular structure in radio wavelengths (Berkhuijsen et al. 1971), and from spectral index and polarization measurements it became clear that the radio emission is due to synchrotron radiation, there must be a non-negligible magnetic field associated with the shell. Conversely, there is a lack of radio emission from the bubble itself, so that [FORMULA]. We assume that [FORMULA] and [FORMULA], with [FORMULA], [FORMULA] and [FORMULA] (see Sect. 4.1) as typical values. In Table 1 we have listed the properties of the interaction region as a function of [FORMULA]. The temperature of the magnetized wall is then roughly given by

[EQUATION]

with the average pressure of the wall being [FORMULA], which takes into account the pressure gradient across the shell.


[TABLE]

Table 1. Temperature of the wall, [FORMULA], plasma beta [FORMULA], Alfvén speed, [FORMULA], growth time of the most unstable mode, [FORMULA], diameter, [FORMULA], and mass, [FORMULA], of the clouds, respectively, as a function of magnetic field strength, [FORMULA], in the unstable region, for [FORMULA], [FORMULA] and [FORMULA] (see Sect. 4.1).


We know that [FORMULA]; hence the growth time of the fastest growing mode is given by

[EQUATION]

The instability will eat its way through the wall, steadily producing blobs of size

[EQUATION]

As it turns out, the size of the blobs is comparable to or less than the thickness of the interaction zone and thus fragmentation of the wall and creation of holes or depressions in neutral hydrogen column density by detachment of blobs seems very likely. This is achieved by significant mass motions induced by the fastest growing mode. The creation of blobs will be assisted by magnetic reconnection, which in case of Petschek's (1964) mechanism typically occurs at a rate [FORMULA], where l is the typical distance between antiparallel lines of force and [FORMULA] is the Alfvén speed in the wall. Once the waves will become nonlinear (within a few growth times 3) and turbulence will be fully developed, l will be sufficiently small for reconnection to proceed due to tangling and twisting of field lines.

The plasma beta is given by [FORMULA]; therefore the dynamics of the plasma in the shell will be largely determined by the thermal pressure as long as [FORMULA].

3.3. Cloud dynamics

Here we consider blobs to be driven by the pressure force [FORMULA] acting on the wall and hence also on individual blobs. Once the blobs have detached, they will move ballistically through the Local Bubble without any noticeable deceleration, because the mass of hot gas swept up by a blob is in general negligible in comparison to the mass of the blob itself. For simplicity we assume the blobs to be spherically symmetric (see discussion below).

The equation of motion is then given by

[EQUATION]

where [FORMULA] and [FORMULA] denote the mass and the velocity of a blob, respectively. The mass can be written as

[EQUATION]

The cloud mass is a very sensitive function of the field strength, and values of [FORMULA] are above the estimated mass of the LIC (see Table 1), which is more of the order of [FORMULA].

The acquired velocity increase [FORMULA] by the accelerating force over a time [FORMULA] is

[EQUATION]

Note, that if [FORMULA], the blobs will have a more flattened shape, and [FORMULA] [FORMULA] and therefore [FORMULA]; as expected, the geometry factor in Eq. (63) is of order unity. With the values adopted for the interaction region this occurs for [FORMULA].

The duration of the applied force is approximately the timescale of the formation of the blob, which is a few growth times (for the instability to become fully nonlinear), i.e. [FORMULA], where [FORMULA] is typically a factor of up to a few. Using Eq. (59), we obtain

[EQUATION]

Using the previous numbers, we find that [FORMULA] (for [FORMULA]) and hence [FORMULA] km s-1. For a reasonable value of [FORMULA] it is possible, for example, to recover the observed cloud velocities of [FORMULA] km s-1.

The pressure in the blob, when it is generated, is subject to a gradient, which will level off after it has detached within a few sound crossing times, [FORMULA] yrs, typically less than the growth time of the instability; in the following subscript "b" denotes blob quantities. The pressure is shared between the thermal, magnetic and the turbulent pressure (resulting from the instability). The magnitude of the latter is given by [FORMULA], where the fluctuation amplitude [FORMULA] is of the order of the Alfvén speed of the perturbed field, which is of the order of the background field or even larger in the nonlinear regime. The temperature is therefore approximately given by [FORMULA] K.

After detachment from the wall the blob expands until it is in pressure equilibrium with the surrounding Local Bubble medium. Assuming conservation of mass, [FORMULA], and [FORMULA], for isothermal expansion, we can estimate the density of the cloud, [FORMULA], from [FORMULA]. Here [FORMULA] is the average pressure of the blob, ranging from [FORMULA], for the values derived in Sect. 3.1, [FORMULA] and [FORMULA] in their respective units. As we shall see, the size of the blobs is probably less than the thickness of the wall and therefore the average pressure of the blob may vary between [FORMULA] and [FORMULA], thus [FORMULA]. Accordingly, the cloud radius after cloud expansion is estimated to be [FORMULA]. The cloud mass is given by Eq. (62).

For a direct comparison, we use the Local Cloud (LIC). The density of the LIC is not very well determined; values for the electron density range from [FORMULA] (Lallement & Ferlet 1997) to 0.3 cm-3 (Gry et al. 1995), depending on the method used and the direction of the line of sight. To infer the gas density one has to know the degree of ionization. It has been reported by various authors (e.g. Slavin & Frisch 1998) that helium is overionized relative to hydrogen and therefore the LIC is probably not in ionization equilibrium. From the HI and HeI backscattering radiation as well as the direct measurement of HeI with Ulysses the most likely value for the total LIC density is 0.32 cm-3 (Frisch et al. 1999).

We obtain good agreement between observations and the model for a set of parameters: [FORMULA], [FORMULA], [FORMULA] for the angle of the propagating waves with respect to the B-field (cf. Table 2). Note that the temperature of the LIC in the model is not well determined due to the variation of [FORMULA]; the mean value is [FORMULA] K, for the most probable value of [FORMULA]. The velocity is reproduced for a plausible range of [FORMULA] (see above). The growth time of the instability for our chosen parameters is [FORMULA] yrs. In Table 2 we have listed results from both the classical CIE model and a non-equilibrium ionization model of the Local Bubble (Breitschwerdt & Schmutzler 1994); the latter results in a much lower Local Bubble pressure (by a factor of 5) and hence in a somewhat larger pressure difference [FORMULA] across the wall. One of the advantages of a non-CIE model is that the plasma in the Local Bubble is in pressure equilibrium with the Local Cloud and does therefore not put any restrictions on the cloud survival. During the travel of the cloud from its site of formation to the solar system it may be subject to processes like ablation, irradiation and heat conduction (although reduced by the magnetic field), which may modify the values derived above. To include these complications is beyond the scope of the present paper.


[TABLE]

Table 2. The model parameters are given for a Local Bubble model assuming collisional ionization equilibrium (CIE) and for a non-equilibrium model taken from Breitschwerdt & Schmutzler (1994; B&S), respectively (for details see Breitschwerdt 1996). A value of [FORMULA] was adopted for the column density of the wall throughout.
Notes:
1) data taken from Linsky & Redfield (1999)
2) data taken from Frisch et al. (1999)
3) data taken from Dring et al. (1997)


3.4. Deceleration of the wall

The data analysis of the Leiden-Dwingeloo Survey shows that the wall itself appears to move towards the Sun with a velocity of about 5 km s-1 (Freyberg et al., 2000). This is by a factor of five lower than the velocity of the local clouds. The discrepancy can be explained in the following way. Initially, when the two bubbles came into contact first, the magnetic field lines were locally straight lines (see Fig. 2). As the field was subsequently bulging into the Local Bubble due to the overpressure [FORMULA], magnetic tension forces began to arise. Combining Eqs. (3) and (6) we obtain for the modified Euler equation

[EQUATION]

Whereas the magnetic pressure force [FORMULA] adds to the acceleration of the wall, this effect is more than compensated by the magnetic tension forces.

For illustration, let us consider a simple case of Eq. (65). We assume axisymmetry, and for simplicity a purely azimuthal field, [FORMULA]. At [FORMULA] (when the two bubbles came into contact), we require that the radial velocity [FORMULA], which is the deviation from the mean bubble expansion, is zero. In this initial phase, the nonlinear terms [FORMULA], which are of second order, will be small compared to [FORMULA]. We then have

[EQUATION]

In this idealized setup, the longitudinal flow velocity [FORMULA] is small. We are primarily interested in the time dependence of the radial flow. Therefore integration of Eq. (67) yields

[EQUATION]

where [FORMULA] is an arbitrary function of r; the "-" sign accounts for the fact that the negative gradient of the left-hand side gives a positive acceleration. The acceleration of the radial component then becomes

[EQUATION]

and upon integration with respect to time gives

[EQUATION]

which satisfies the boundary condition. Eq. (70) shows, that as the field lines bulge into the Local Bubble, and the radius of curvature of the lines of force becomes smaller, the magnetic tension decelerates the wall.

Once the blobs have detached from the wall after [FORMULA], they will therefore travel ahead. Whether the wall will eventually stall, depends on the interplay between pressure and tension forces. However, on general grounds it is unlikely that an equilibrium configuration of such a kind does exist in three dimensions.

3.5. Can the clouds be accelerated by outflowing gas of Loop I?

We may also speculate, whether the clouds might be accelerated due to the blow-out of gas from Loop I through the holes, which are caused by the detachment of the blobs. There is no guarantee that such a mechanism might work here, and we merely mention it as an interesting possibility.

Suppose that the holes in the wall form an idealized convergent-divergent device, known as a de Laval nozzle. In Loop I, the gas flows subsonically towards the holes, becomes sonic at the throat and supersonic when it enters the Local Bubble. At the blob surface there will exist a stagnation point, and a bow shock will decelerate the supersonically impinging gas flow, thus converting part of the kinetic into thermal energy. In the following estimates we will neglect these details, because the overall effect on the blob will be acceleration, although the post-shock gas will be deflected sideways and only part of the incoming kinetic energy can be transferred to the cloud.

The force per unit area acting on the cloud due to the jet of plasma channelled through a hole is the difference between its ram pressure and the thermal pressure in the Local Bubble; hence

[EQUATION]

where [FORMULA], [FORMULA] are the velocity and density of the jet, respectively. Essentially, thermal pressure in Loop I at the base of the jet is converted into kinetic energy. The maximum effect is reached if the efficiency is 100%. In this case we obtain from Bernoulli's equation

[EQUATION]

where [FORMULA] is the speed of sound in Loop I. Thus, assuming [FORMULA], and using [FORMULA] for the values discussed in Sect. 3.1, we obtain

[EQUATION]

where for simplicity [FORMULA] has been assumed. This acceleration could be quite substantial if compared to Eq. (63); the question is: what is [FORMULA] in here? A steady jet will expand freely for a time

[EQUATION]

where L is the distance in z-direction from the hole. Therefore [FORMULA], with [FORMULA] being the maximum distance of free expansion. We don't know how well collimated the jet will be; it is supposed to have a half opening angle [FORMULA] with respect to the z-axis, with [FORMULA], where r is the radial coordinate. Its Mach number increases along the z-axis, and appreciable deceleration will occur at a distance [FORMULA], where it has roughly swept up its own mass:

[EQUATION]

where [FORMULA] is the volume of the conical jet and [FORMULA] is the mass flux through the hole. In order to evaluate [FORMULA] we use the equation of continuity and Bernoulli's equation for a polytropic gas; since there is no significant exchange of heat between the jet and the surrounding medium we use the adiabatic law [FORMULA] where C is a constant depending on the specific entropy of the gas.

[EQUATION]

with D being constant along a streamline. At the critical point of the flow Eq. (77) becomes (Courant & Friedrichs 1948)

[EQUATION]

where [FORMULA] is the speed of sound at the critical point in the flow. With increasing distance from the jet the speed of sound drops and the fluid velocity increases, so that we obtain from Eq. (78) approximately

[EQUATION]

Using the adiabatic law and Eq. (78), the mass flux into the jet is given by

[EQUATION]

where the diameter of the surface area [FORMULA] at the throat of the nozzle has been equated to the diameter of the clouds. For a proper jet we expect [FORMULA], and therefore

[EQUATION]

Using the minimum expansion time, [FORMULA], Eq. (75) yields

[EQUATION]

here we have used the positive root for a physically meaningful result and the numerical values for the respective densities given in Sect. 3.1.

It has been implicitly assumed that the half opening angle [FORMULA] will remain constant, i.e. [FORMULA], where [FORMULA] is the radius of the jet cross section at distance [FORMULA]. This is true, because the jet is still expanding freely until it hits the cloud, and the so-called working surface has not yet fully developed. It is known that in the case the ambient medium is constant (like here), the jet will reconfine (e.g. Sanders 1983; Falle 1991), in order to drive a bow shock into the ambient medium; however this is not relevant in the initial stage of expansion under consideration.

Since [FORMULA] is the maximum velocity, [FORMULA], which is

[EQUATION]

Adopting the values derived earlier for [FORMULA], [FORMULA], [FORMULA] and [FORMULA], we can now conclude that the effect of accelerating the clouds by the plasma, which is streaming out of the holes and pushing the clouds, is of the same order as the initial acceleration given in Eq. (63), if the half opening angle satisfies

[EQUATION]

and [FORMULA].

It is unlikely that such a well collimated jet exists in the present case, and the effect of accelerating the clouds by the plasma streaming out of the holes is probably rather small.

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Online publication: January 29, 2001
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