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Astron. Astrophys. 351, 506-518 (1999)

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4. Initial patterns of motion

Undoubtedly, an important reason why no clear interpretation on the kinematical structure of the Gould Belt has emerged yet is that, whatever the initial pattern of velocities may have been at the time of its formation, it must have been severely distorted under the effects of galactic differential rotation during its lifetime. In principle, it should be possible to use the presently observed space velocities of the stars and their ages, together with a model of the galactic potential, to trace their orbits back in time and find both their initial positions and velocities. In practice, this would require a knowledge of velocities, distances, and ages much more accurate than is available at present in order to reach any reliable conclusions.

A different way to approach this problem consists of proposing different initial kinematical patterns and space distributions for the stars of the Gould Belt, and following their evolution with time until the best agreement with the present observations is reached. In case that a satisfactory solution cannot be found for any possible age of the Belt, the model is then discarded. This is the approach followed by Lindblad 1980, and extended by Westin 1985, to evaluate the suitability of models based on a radial expansion from a small volume or on the gravitational perturbation produced by a spiral arm to reproduce the observed values of the Oort constants. They concluded that none of those models provided an acceptable fit to the available observational material. Comerón et al. 1994 suggested an expansion from a line, rather than a point, as a possible way to achieve a better fit to the Oort constants derived from their data. Recently, Palous 1998 has presented the result of N-body numerical simulations suggesting that the dissolution of an unbound rotating system of stars with an age of [FORMULA] years is the most favoured model in explaining the observed values of the Oort constants. Such an age is consistent with that derived from the individual ages of its component stars.

The development presented in Sect. 3 is well suited to the exploration of different kinematical models for several reasons. First, its predictions include the orientation of the Belt and the characteristics of the vertical motion, in addition to the Oort constants, as a function of time. Secondly, the development being fully analytical, it is easy to explore a wide range of parameters when trying to obtain a best fit, and to ensure that some models can be really discarded for any set of initial parameters. A possible drawback is the basic assumption that the initial patterns of motion have the generic form given by Eq. (9), what in principle is a very restrictive condition. However, the fact that such patterns are able to maintain the arrangement of stars on a tilted plane with time, and that this is indeed an observed feature of the Gould Belt, suggests that it should be possible (at least as a first approximation) to describe the initial pattern of motions of the Gould Belt in such a form.

The time evolution of the orientation of the Belt and of the Oort constants are described in the next subsections for different possible initial patterns of motion. The main aspect of interest here is in the early phases of this evolution, this is, in ages of a few times [FORMULA] years which are consistent with the observations. For illustrative purposes, I extend the calculations to [FORMULA] years, which covers the first resonances with the epicyclic and galactic rotation periods and their consequences on the quantities whose evolution is studied here. Such an extension may be in principle relevant to the study of other, older Gould Belt-like structures which may be eventually discovered. However, the study of such structures would be hampered with increasing age as their more massive and brightest members end up their lives, and the less massive ones dilute in phase space due to differential rotation and to the dynamical heating mechanisms operating in the galactic disk.

The evolution of the plane orientation and the Oort constants are in general sensitive to the chosen initial parameters. For this reason, I restrict the discussion to the values that reproduce the present orientation of the Gould Belt ([FORMULA], [FORMULA]; see Sect. 2) and the small value of [FORMULA], which places the Gould Belt near its peak inclination at present. In general, it is always possible to find a set of initial parameters fulfilling these constraints for an age of the Belt compatible with the observational estimates. The consistency with these constraints thus sets rather narrow limits on the possible age of the Belt, which depend on the model adopted. As a convention, the inclination is defined here as a positive quantity, and [FORMULA] as the galactic longitude of the ascending node of the Gould Belt with respect to the galactic plane, so that the crossing of the galactic plane by the Gould Belt results in a change of [FORMULA] in [FORMULA], rather than a reversal in the sign of i. In this way, [FORMULA] means that the Gould Belt reaches its largest distance to the galactic plane in the direction to the galactic anticenter, while [FORMULA] implies that it does so in the direction to the galactic center, and [FORMULA] places the direction of maximum inclination in the direction of the galactic rotation. For the sake of simplicity, and in consistency with the observations, I will consider [FORMULA], [FORMULA]. As can be seen from Eq. (24), the Oort constants are not affected by this choice.

The values adopted for the galactic constants [FORMULA] and [FORMULA] appearing in the matrices [FORMULA] and [FORMULA] are those corresponding to a flat rotation curve at the position of the Sun with a circular angular speed of rotation [FORMULA] km s- 1 kpc-1 (Kerr & Lynden-Bell 1986), namely [FORMULA] km s-1 kpc- 1 and [FORMULA] km s- 1 kpc-1. The vertical oscillation frequency is taken to be [FORMULA] km s-1 kpc- 1, corresponding to a vertical oscillation period of [FORMULA] years (Binney & Tremaine 1987).

4.1. Circular motions

The simplest case that can be considered, and that will be used here as a reference, is that of a plane whose stars follow circular orbits around the galactic center when their positions are projected on the galactic plane. In the epicyclic reference frame, the equations of motion are described by Eqs. (4) with [FORMULA], and one obtains for [FORMULA]:

[EQUATION]

where the possibility of a nonzero vertical initial velocity is taken into account by the terms in the last row of Eq. (29). This includes also as a particular case one in which the star-forming matter of the Gould Belt is suddenly knocked away from the galactic plane towards opposite directions ([FORMULA], [FORMULA]).

Fig. 2 shows the time evolution of the position of the nodal line [FORMULA], the direction of the axis of vertical oscillation [FORMULA], the inclination i, the value of the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G. The values of [FORMULA] and [FORMULA] are their galactic longitudes at the corresponding time, i.e., they are expressed in the rotating reference frame. The evolution of the orientation can be described fairly simply: starting from the chosen initial position, the direction of the nodal line rotates in the sense of increasing galactic longitudes, assymptotically tending to align itself with the direction of galactic rotation, as a consequence of the "stretching" of the plane produced by galactic rotation. A point of the nodal line at the initial position [FORMULA] will have moved, following the galactic circular rotation, to [FORMULA] after a time t, and the longitude of the nodal line will thus be given (neglecting the possible [FORMULA] difference due to the actual inclination) by [FORMULA], which increases indefinitely with time. The axis of vertical oscillation is coincident with the nodal line, but its [FORMULA] reversals take place at the times when the vertical motions of stars change signs, rather than at the galactic plane crossings as is the case for the nodal line.

[FIGURE] Fig. 2. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt follow circular orbits around the galactic center while oscillating around the galactic plane. In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). Note that [FORMULA] in the present case, so both curves overlap. The stars in the plane are assumed to form simultaneously with a zero vertical motion; the present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

The evolution of the inclination is also easy to understand qualitatively, being dominated by the vertical oscillation of the stars around the galactic plane. The initial decrease of the amplitude of the inclination is due to the stretching of the plane as the nodal line approaches the galactocentric direction. Once the nodal line crosses it, the plane continues to stretch, but a projection effect makes the inclination amplitude grow again, and at sufficiently large times (several times larger than the timespan shown in Fig. 2) it actually tends to [FORMULA]. The projection effect may be visualized as follows: let us assume, when [FORMULA], a star lying at the coordinates [FORMULA], so that the distance to the nodal line is y and the inclination of the plane is simply [FORMULA]. As the nodal line rotates (i.e., as [FORMULA] approaches the [FORMULA] direction), the distance d of the star to the nodal line decreases as [FORMULA], so that the inclination amplitude grows as [FORMULA], with the vertical amplitude of the oscillation, [FORMULA], being constant.

Since the motions of the stars as projected on the galactic plane are circular around the galactic center in the present case, the Oort constants have the values [FORMULA], [FORMULA], [FORMULA] characteristic of such motion. The evolution of [FORMULA] is similar to that of [FORMULA], but anticorrelated with it, as expected from the stars reaching their peak vertical velocity when they cross the galactic plane. The projection effect discussed in the previous paragraph applies to the gradients of the vertical velocity as well, what causes the amplitude of the oscillation in G to vary in a similar way to that of i.

The results remain essentially unchanged if the stars are assumed to be born in the galactic plane and expelled from there in a coherent way, with vertical velocities proportional to the distance to the nodal line. This is shown in Fig. 3, which displays the same behaviour as Fig. 2, the differences being due to the somewhat different initial parameters that are necessary to match the constraints set by the presently observed orientation and state of vertical motion of the Belt.

[FIGURE] Fig. 3. Same as Fig. 2, but now assuming that the stars are formed in the galactic plane with a nonzero vertical motion. The value of k in Eq. (28) is set to 43.2 km s-1 kpc-1 to obtain a maximum inclination of 20o at [FORMULA] years. The initial orientation in this case is [FORMULA], [FORMULA].

4.2. Radial expansion

Let us assume now that the stars in the Gould Belt were born simultaneously in a volume much smaller than the one they occupy at present, with their initial motions contained in a plane and directed radially away from the center of such volume. This is one of the most widely considered kinematical models for the Gould Belt, motivated by the long-recognized expansion term in the velocities of nearby young stars. The early work of Blaauw 1952describing the expansion of an unbound group of stars moving in the galactic potential was extended by Lesh 1968, Lindblad 1980, and Westin 1985. Lindblad et al. 1973and Olano 1982developed the models to account for the observed radial velocities of the gas associated to the Belt.

Let us assume an initial velocity modulus of the stars proportional to the distance to the center. At very early times, it is possible to define an expansion age [FORMULA] such that the pattern of motions can be described, using the [FORMULA] system defined above, as follows:

[EQUATION]

If the initial volume is negligible, then [FORMULA], and the pattern described by Eq. (30) is equivalent to that of a system of stars expelled from a single point with random velocities. The evolution of the orientation of the plane is then as given in Fig. 4. The main characteristic is the fast decrease of the amplitude of the tilt with time at early ages, as a consequence of the expansion of its stars away from the center while maintaining constant the amplitude of their vertical motions. An important feature of this model is the need for a large initial tilt, [FORMULA], to still obtain a maximum tilt of [FORMULA] at the age of [FORMULA] years, after the first galactic plane crossing.

[FIGURE] Fig. 4. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt are born simultaneously and expand initially from a negligibly small volume with random velocities ([FORMULA] in Eq. (30)). In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). The present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

The initial expansion along the plane from a very small volume causes a large initial gradient in the vertical component of the velocity resulting in a large value of G, which rapidly decreases as the distance of the stars to the center of the expansion grows. Afterwards, the evolution of G and i are approximately anticorrelated like in the case of circular motions, although the expansion of the system causes a small lag between the peak in i and the minimum in G: since the distance to the center of expansion increases, the inclination can start decreasing while the stars are still moving away from the galactic plane. A common feature to the expansion models, including those to be considered in the next section, is the predicted permanent alignment between the nodal line and the vertical oscillation axis.

The evolution of the Oort constants plotted in Fig. 4 reproduces the results of Lindblad 1980, giving low values at early times for all the Oort constants (including a permanent null value of B) with the exception of K. Singularities in some Oort constants and in the orientation of the plane appear around the times of the epicyclic period and the galactic rotation period, due to the alignment of the stars of the plane along a single line, what gives infinite values for some spatial derivatives of some velocity components. This is a feature common to the evolution of all the models in which the stellar orbits projected on the galactic plane are not circular.

A similar, but less marked behaviour, is found when [FORMULA] is finite. This may be regarded as an approximation to the case in which the stars formed out of an extended molecular cloud become an unbound system shortly after their formation, as a consequence of the dispersal of the gas remaining in the system. Assuming that the stars located at the outskirts of the cloud expand initially with a velocity of 1 km s-1, and that the star forming cloud had an initial radius of 50 pc, one obtains [FORMULA] years. The evolution is given in Fig. 5, and is nearly identical to that depicted in Fig. 4 except for the existence of a negative value of B. The initial nonzero extent of the Belt allows to start with an inclination of its plane somewhat smaller than in the case [FORMULA] considered before, but still very considerable despite of the small expansion velocity at the outskirts of the cloud.

[FIGURE] Fig. 5. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt are born simultaneously over a circular region, expanding with initial velocities proportional to the distance to the center. The expansion age, which relates the size of the initial volume and the expansion rate, is taken to be [FORMULA] years, as explained in the text. In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). The present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

4.3. Expansion from a line

In this Secton I consider the case in which the initial expansion takes place along a preferential direction, rather than being isotropic as it has been assumed so far. Such an expansion pattern was proposed by Comerón et al. 1994on the basis of the distribution of residual velocities of Gould Belt stars when a purely circular rotation is subtracted from the observed velocities. It was suggested in that paper that such a pattern may have arisen from the sudden compression of a gas layer precursor to the Gould Belt, threaded by a magnetic field aligned with the direction of galactic rotation. The subsequent expansion of the gas would have taken place preferentially along the magnetic field lines, and would be reflected now in the motions of the stars formed out of that gas. Rather than a radiant point marking the center of expansion, it is possible in this case to define a radiant line so that stars move initially away from it following trajectories perpendicular to it.

Assuming that the initial velocity of any given star has a modulus proportional to the distance to the radiant line, one obtains an initial expansion law that can be expressed in the general form (9), with

[EQUATION]

where [FORMULA] is the angle between the direction of expansion and that of the nodal line, and [FORMULA] is again the expansion age, now defined as the ratio between the initial distance of a star to the radiant line and its initial velocity. Two cases similar to those presented in Sect. 3.2 are shown in Figs. 6 and 7, corresponding to [FORMULA] and [FORMULA] years. The angle [FORMULA] is taken to be [FORMULA] for both cases, implying that the radiant line is coincident with the apsidal line. The foundation for this choice lies in the physical motivation of this expansion law, as outlined above. The present position of the nodal line implies an initial position roughly aligned with the direction of galactic rotation, especially in the case [FORMULA] (Figs. 6 and 7) and this is also the approximate orientation of the systemic component of the galactic magnetic field.

[FIGURE] Fig. 6. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt are born simultaneously on a very narrow strip, and expand initially with random velocities directed prependicular to it. The strip is assumed to coincide with the apsidal line, so that the initial expansion takes place parallel to the direction of the nodal line. In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). The present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

[FIGURE] Fig. 7. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt are born simultaneously and expand initially on a plane, with velocities that are parallel to the nodal line and proportional to their distance to it. The expansion age of the system is set to [FORMULA] years. In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). The present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

The [FORMULA] case has in common with the corresponding one in the radial expansion scenario the existence of a null constant B and the large positive initial K term, which decreases with time and eventually becomes negative when the epicyclic orbits reverse the initial expansion. However, the present case is characterized by large absolute values of A and C, unlike in the radial expansion case. The evolution of the orientation is similar between both cases, but now, due to the fact that the expansion takes place initially perpendicular to the direction of maximum inclination, the decrease in the amplitude of the inclination is much slower; this is, the initial tilt of the plane is not very different from the presently observed one, unlike in both of the radial expansion scenarios discussed before.

The early evolution when [FORMULA] years is qualitatively very similar to that of the radial expansion case with the same expansion age, with the largest differences, such as the initial [FORMULA] term, appearing only in the first [FORMULA] years of evolution.

As to the evolution of [FORMULA] and G, a behaviour similar to the radial expansion case (alignment of the axis of vertical oscillation with the nodal line, anticorrelation between i and G) is found for both cases, with the exception of the very early evolution in the case of G that now starts from a null value.

4.4. Rotation

The last model considered here concerns a tilted plane formed by stars which initially rotate with an angular velocity [FORMULA] around a perpendicular axis. The velocity [FORMULA] of a star on the plane is thus given by [FORMULA], with [FORMULA] being the position vector of the star with respect to the center of rotation. This allows one to express the initial pattern of motion in terms of

[EQUATION]

with w being the modulus of [FORMULA]. To illustrate the resulting pattern of motions and choose a value best matching the actual observations, w is set to -6.5 km s-1 kpc-1. The positive value of [FORMULA] corresponds to a prograde rotation, i.e., in the same direction as the galactic rotation (although with lower angular speed than the latter) in a fixed reference frame. The results are shown in Fig. 8.

[FIGURE] Fig. 8. The evolution of the nodal line [FORMULA], the axis of vertical oscillation [FORMULA], the inclination i, the Oort constants A, B, C, K, and the instantaneous angular velocity of oscillation G in the case in which the stars of the Gould Belt are born simultaneously in a plane which rotates like a solid body and move independently afterwards. The initial angular velocity of the plane is set to -6.5 km s-1 kpc-1, i.e., the sense of rotation in the epicyclic reference frame has a direction opposite to that of the galactic rotation. In the upper panel, the solid curve represents the direction of the nodal line, and the dashed curve the direction of the axis of vertical oscillation. The Oort constants are identified by the solid line (A), the dotted line (B), the dashed line (C), and the dot-dashed line (K). The present orientation and peak inclination are obtained with the initial values [FORMULA], [FORMULA], at an age of [FORMULA] years.

The evolution of the orientation of the plane, the Oort constants, and the parameters defining the oscillation of the stars around the galactic plane is clearly different from that in the cases studied so far. The most remarkable difference is the introduction of an offset between the nodal line and the axis of vertical oscillation, which is initially [FORMULA] as it would correspond to a rigid body rotation. The initial value of G is small but not zero, as the stars are moving across the plane and have their largest vertical velocity component at the nodal line. The offset between [FORMULA] and [FORMULA] is maintained with time, although its value varies as the initial circular rotation pattern is distorted by the differential galactic rotation. The combination of initial rigid body rotation and subsequent independent epicyclic orbits results in an anisotropic expanding motion, characterized by the positive values of C and K. The initial rotation also yields a large negative value of B (which is the rotational of the velocity field) at the earliest stages.

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© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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