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Astron. Astrophys. 350, 434-446 (1999)

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3. Stellar trajectories

The epicycle approximation holds for stars that follow almost circular orbits around the center of the Galaxy. Although valid for most of our stars, some of them, with high peculiar velocities with respect to the Local Standard of Rest (LSR), perform radial galactic excursions of a few kpc in length. The galactic gravitational potential changes significatively at different points of their trajectories, and so the first order approximation is no longer valid. In addition, the vertical motion is poorly described by a harmonic oscillation as stars gain certain height over the galactic plane. For a more rigorous analysis, we need to use a realistic model of the galactic gravitational potential. Stellar orbits will be determined from this model by integrating the equations of motion (Sect. 3.1). Moreover, the addition of a constant scattering in this process will allow us to account for the disc heating effect on the stellar trajectories (3.2).

3.1. The galactic potential

Expressed in a cartesian coordinate system ([FORMULA]) centered at the position of the Sun 1 and rotating at a constant angular velocity [FORMULA],


the equations of motion of a star, assuming that the gravitational potential of the Galaxy [FORMULA] (in cylindrical coordinates centered on the GC) is known, are:


A fourth order Runge-Kutta integrator allows us to numerically solve these equations when [FORMULA] is known, obtaining the trajectory of the star. To get a realistic estimation of the galactic gravitational potential we decompose it into three parts: the general axisymmetric potential [FORMULA], the spiral arm [FORMULA], and the central bar [FORMULA] perturbations to the first contribution, i.e.


We adopt the model developed by Allen & Santillán (1991) for the axisymmetric part of the potential, [FORMULA], because of both its mathematical simplicity - which allows us to determine orbits with a very low CPU consumption - and its updated parameters. The model consists of a spherical central bulge and a disk, both of the Miyamoto-Nagai (1975) form, plus a massive spherical halo. This model is symmetrical with respect to an axis and a plane. The authors adopted the recommendations of the IAU (Kerr & Lynden-Bell, 1986) for the galactocentric distance of the Sun ([FORMULA] = 8.5 kpc) and the circular velocity at the position of the Sun ([FORMULA]= 220 km s-1). The Oort constants derived from this model are well within the currently accepted values.

Spiral arm perturbation to the potential is taken from Lin and associates' theory (e.g. Lin 1971 and references therein), that is:




[FORMULA] is the amplitude of the potential, [FORMULA] is the ratio between the radial component of the force due to the spiral arms and that due to the general galactic field. [FORMULA] is the constant angular velocity of the spiral pattern, m is the number of arms, i is the pitch angle, [FORMULA] is the radial phase of the wave and [FORMULA] is a constant that fixes the position of the minimum of the spiral potential. According to Yuan (1969), we adopt [FORMULA] (this value has been confirmed in a recent study by Fernández, 1998) and [FORMULA] km s-1 kpc-1.

We consider a classical two arm pattern ([FORMULA]) (Yuan, 1969; Vallee, 1995). If we assume that the Sagittarius arm is located at a galactocentric distance RSag = 7.0 kpc, and that the interarm distance in the Sun position is [FORMULA]3.5 kpc as suggested by observations on spiral arm tracers (Becker & Fenkart, 1970; Georgelin & Georgelin, 1976; Liszt, 1985; Kurtz et al., 1994), then the pitch angle can be determined to be:


For [FORMULA], we obtain [FORMULA] (close to the Yuan's (1969) value [FORMULA]).

For [FORMULA] and [FORMULA], the adopted value [FORMULA] leads to a minimum in the potential at the observed position of the Sagittarius arm (R = RSag = 7.0 kpc, [FORMULA]). [FORMULA] can thus be expressed as:


For [FORMULA] and [FORMULA] we obtain [FORMULA] 0.131 rad.

The central bar potential we use here is, for simplicity, a triaxial ellipsoid with parameters taken from Palous et al. (1993), i.e.


where [FORMULA] and [FORMULA], with [FORMULA] 45o (Whitelock & Catchpole, 1992). [FORMULA] and [FORMULA] are the three semi-axes of the bar, with [FORMULA] its scale length, and with [FORMULA], [FORMULA], and [FORMULA] kpc. The adopted total mass of the bar, [FORMULA], is 109 [FORMULA], and its angular velocity, [FORMULA], is 70 km s-1 kpc-1 (Binney et al., 1991). Although most of the parameters that define the bar are very uncertain, the effect of the bar on the stellar trajectories becomes important only after several galactic rotations, which requires a length of time greater than the age of the stars considered in our study.

3.2. Disc heating in the stellar trajectories

The observational increase in the total stellar velocity dispersion ([FORMULA]) with time (t), or disc heating , can be approximated by an equation of the form:


where [FORMULA] is the dispersion at birth and [FORMULA] the "apparent diffusion coefficient" (Wielen, 1977). The constants n, [FORMULA] and [FORMULA] give one important information on the physical mechanism responsible for the disc heating (Lacey, 1991; Fridman et al., 1994).

Our sample of B and A main sequence type stars, described in Paper I, allows us to determine an accurate and detailed disc heating law for the last [FORMULA] yr, given the quality and uniformity of our ages and the size of this sample (2 061 stars). A standard nonlinear least-squares method (Levenberg-Marquardt method) has been applied to fit the heating coefficients to our data (Fig. 1). In this way we obtain [FORMULA] km s-1, [FORMULA] and [FORMULA] (km s-1)nyr-1, similar to Lacey's (1991) results for [FORMULA], i.e. [FORMULA] km s-1 and [FORMULA] (km s- 1)5yr-1. When we fix n to 2, we obtain [FORMULA] km s-1 and [FORMULA] (km s- 1)2 yr-1, which is again in agreement with Lacey's (1991) ([FORMULA] km s-1 and [FORMULA] (km s- 1)2 yr-1) and Wielen's (1977) ([FORMULA] km s-1 and [FORMULA] (km s- 1)2 yr-1) fits with [FORMULA].

[FIGURE] Fig. 1. Fit of Eq. 10 to our data (see text), using stars closer than 300 pc from the Sun, with 100 stars per bin. Solid line : [FORMULA], [FORMULA] and n are fitted; dashed line : [FORMULA] and [FORMULA] are fitted, whereas n is fixed to 2. The error bars on points include only statistical uncertainties, calculated as [FORMULA], where N is the number of stars per bin

Our accurate observational heating law (circles in Fig. 1) is far from being smooth. Two special features can be observed on this law: first, the velocity dispersion shows a steep increase during the first [FORMULA] yr, probably due to the phase mixing of young stars. After this point, this increase becomes less pronounced. Second, an almost periodic oscillation seems to be superimposed over the "continuous" heating law. This oscillation (period [FORMULA] yr) could be the signature of an episodic event (see for instance Binney & Lacey 1988, Sellwood 1999).

Since we do not know the details about the mechanisms that perturb the stellar orbits and cause the disc heating effect, we assume for simplicity that the accelerating processes acting on a given star can be approximated by a sequence of independent and random perturbations of short duration (Wielen, 1977). In order to evaluate the magnitude of these perturbations we generated a sample of 500 stars located around the position of the Sun at [FORMULA], and uniformly distributed in a 100 pc/side cube. Initially, these stars are moving with the same velocity as the LSR, with an isotropic dispersion in velocity [FORMULA]. During the process of orbit determination, the velocity of the star is instantaneously changed by [FORMULA] every [FORMULA] yr. This [FORMULA] is normally distributed around zero with a (constant) isotropic dispersion [FORMULA] in each component. The equations of motion are integrated using only the axisymmetric part of the galactic potential ([FORMULA]). This allows us to introduce the heating effect only through the parameter [FORMULA], avoiding possible redundances, such as the scattering of disc stars by spiral arms. The best approximation to the observational heating law is obtained when [FORMULA] km s-1 for [FORMULA] yr, and [FORMULA] 15 km s-1. It can be demonstrated that [FORMULA] and [FORMULA] are related with the "true diffusion coefficient" D introduced by Wielen (1977) just as [FORMULA]. On the other hand [FORMULA], as demonstrated by Wielen (1977) using the epicycle theory. We obtain in this way an independent estimation of [FORMULA], i.e. [FORMULA] (km s- 1)2 yr-1. This value is in an excellent agreement with our previous estimation. In Fig. 2 the evolution of the total velocity dispersion for this simulated sample is compared with the fit of Eq. 10 to the observational data when [FORMULA], showing an excellent match. This procedure will be used in Sect. 5 to simulate the evolution of a stellar system under the influence of the disc heating.

[FIGURE] Fig. 2. Velocity dispersion vs. time, determined by integrating the equations of motion of a simulated sample when an isotropic diffusion is applied at each time-step. Dashed line : observational fit to disc heating with a fixed slope n=2

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999