## 3. Stellar trajectoriesThe 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 ## 3.1. The galactic potentialExpressed in a cartesian coordinate system
() centered at the position of the
Sun the equations of motion of a star, assuming that the gravitational potential of the Galaxy (in cylindrical coordinates centered on the GC) is known, are: A fourth order Runge-Kutta integrator allows us to numerically solve these equations when 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 , the spiral arm , and the central bar perturbations to the first contribution, i.e. We adopt the model developed by Allen & Santillán (1991)
for the axisymmetric part of the potential,
, 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 (
= 8.5 kpc) and the circular velocity at the position of the Sun
(= 220 km s Spiral arm perturbation to the potential is taken from Lin and associates' theory (e.g. Lin 1971 and references therein), that is: where is the amplitude of the
potential, is the ratio between the
radial component of the force due to the spiral arms and that due to
the general galactic field. is the
constant angular velocity of the spiral pattern, We consider a classical two arm pattern
() (Yuan, 1969;
Vallee, 1995). If we
assume that the Sagittarius arm is located at a galactocentric
distance R For , we obtain (close to the Yuan's (1969) value ). For and
, the adopted value
leads to a minimum in the potential
at the observed position of the Sagittarius arm (R =
R For and we obtain 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 and
, with
45 ## 3.2. Disc heating in the stellar trajectoriesThe observational increase in the total stellar velocity dispersion
() with time ( where is the dispersion at birth
and the "apparent diffusion
coefficient" (Wielen, 1977). The constants 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 yr,
given the quality and uniformity of our ages and the size of this
sample (2 061 stars). A standard nonlinear least-squares method
(
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 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 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 ,
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 . During the
process of orbit determination, the velocity of the star is
instantaneously changed by every
yr. This
is normally distributed around zero
with a (constant) isotropic dispersion
in each component. The equations of
motion are integrated using only the axisymmetric part of the galactic
potential (). This allows us to
introduce the heating effect only through the parameter
, avoiding possible redundances,
such as the scattering of disc stars by spiral arms. The best
approximation to the observational heating law is obtained when
km s
© European Southern Observatory (ESO) 1999 Online publication: October 4, 1999 |