## 4. Initial patterns of motionUndoubtedly, 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 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 years which are consistent with the observations. For illustrative purposes, I extend the calculations to 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
(, ;
see Sect. 2) and the small value of ,
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
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 in
, rather than a reversal in the sign
of The values adopted for the galactic constants
and
appearing in the matrices
and
are those corresponding to a flat
rotation curve at the position of the Sun with a circular angular
speed of rotation km s ## 4.1. Circular motionsThe 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 , and one obtains for : 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 (, ). Fig. 2 shows the time evolution of the position of the nodal line
, the direction of the axis of
vertical oscillation , the
inclination
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 . The projection
effect may be visualized as follows: let us assume, when
, a star lying at the coordinates
, so that the distance to the nodal
line is 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 ,
,
characteristic of such motion. The evolution of
is similar to that of
, 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 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.
## 4.2. Radial expansionLet 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 such that the pattern of motions can be described, using the system defined above, as follows: If the initial volume is negligible, then , 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, , to still obtain a maximum tilt of at the age of years, after the first galactic plane crossing.
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 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 A similar, but less marked behaviour, is found when
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
## 4.3. Expansion from a lineIn 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 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 where is the angle between the direction of expansion and that of the nodal line, and 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 and years. The angle is taken to be 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 (Figs. 6 and 7) and this is also the approximate orientation of the systemic component of the galactic magnetic field.
The case has in common with the
corresponding one in the radial expansion scenario the existence of a
null constant The early evolution when 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 term, appearing only in the first years of evolution. As to the evolution of and
## 4.4. RotationThe last model considered here concerns a tilted plane formed by stars which initially rotate with an angular velocity around a perpendicular axis. The velocity of a star on the plane is thus given by , with 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 with
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 as it would correspond to
a rigid body rotation. The initial value of © European Southern Observatory (ESO) 1999 Online publication: November 3, 1999 |