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

## 3. Kinematics

In the epicyclic approximation used to describe the orbits of stars moving in a separable potential, the equations of motion are usually written in a rotating cartesian reference frame whose axes are respectively directed towards the galactic center, the direction of galactic rotation, and the North galactic pole. The equations of motion can then be written as

where I use a notation similar to that of Comerón et al. 1997 1, adding Eq. (4c) to describe the motion perpendicular to the galactic plane. The angular velocity of an object in a circular orbit around the galactic center at the position momentarily occupied by the Sun is which, by definition, is also the angular velocity of the chosen reference frame with respect to an inertial one. is the usual Oort constant for the case of pure galactic differential rotation, is the epicyclic frequency in the galactic plane, and is the oscillation frequency perpendicular to the galactic plane for orbits whose amplitude is well below the scale height of gravitating matter in the galactic disk. and describe the position of the guiding center of the epicyclic orbit, is the amplitude of the epicycle in the galactocentric direction, is the amplitude of the vertical oscillation, and and define the position of a star in its orbit at an instant t.

Developing the arguments of the trigonometric functions in Eqs. (4), and using the values of the coordinates and velocities at the initial instant , it is easy to show that Eqs. (4) can be written in the following compact form:

where , , . The elements of the matrices , are:

If the stars had a common origin so that the present age of the system is t, then the initial positions are related to the present ones and to the initial velocity pattern by

On the other hand, if the initial positions of stars were distributed on a plane tilted with respect to the galactic plane, then their initial positions fulfilled the relationship

where is the vector perpendicular to the plane and defines the location of the plane with respect to the origin of coordinates. Replacing Eq. (6) in Eq. (7),

It is then possible to show that, if the pattern of initial velocities can be expressed as a linear function of the initial coordinates of the stars, then the tilted plane remains as a tilted plane as the positions of its stars evolve with time under the influence of the galactic potential. Let the linear combination be expressed in a general form as

Using Eq. (6), one obtains after some algebra:

where is the identity matrix. Replacing this in Eq. (8),

This can be expressed simply as

which is again the equation of a plane, whose perpendicular vector is now

and the present location of the plane is defined by

Given the geometry of stellar initial positions and velocities, it may be more convenient to write them in a reference frame whose axes are aligned parallel or perpendicular to the plane, and whose origin is chosen so as to simplify the expression of the initial law of motion. In such a reference frame, in which the position vector is denoted by , the initial law of motion can be written as

where the relation between and is

The rotation of axes is explicitly decomposed as the product of two rotations whose matrices are

The initial inclination of the plane with respect to the galactic plane is , and the longitude of the nodal line defined by the intersection of both planes is . An expression analogous to Eq. (16) can be written for the velocity:

The initial law of motion expressed in the usual epicyclic motion base is thus

which gives the expression of

The time evolution of the nodal line and the inclination are thus given by that of the vector , whose components are

This derivation allows a rather straightforward connection between the matrix and the local values of the Oort constants, which can be measured from the observations. Developing Eq. (5) with the use of Eq. (9), one obtains:

Taking the time derivative of Eq. (21),

Using Eq. (21) to isolate , and replacing it in Eq. (22), one obtains

Taking spatial derivatives now, one obtains

where the elements of , , are

It should be noted that Eq. (24) is valid for any system of stars for which the epicyclic approximation applies, and whose initial positions and velocities are related by Eq. (9). The additional condition (7), implying that the stars are distributed in a plane, has not been used in deriving Eq. (24). This condition must be implicitly included in the expression of the spatial derivatives that will be actually used here in the calculation of the Oort constants, namely:

where the p subindex denotes the derivatives measured on the plane, and is either x or y. Using these definitions, the Oort constants become

to which one can add the derivatives involving the velocity component perpendicular to the galactic plane, giving the components of the axis of vertical oscillation (see Eq. (3)):

© European Southern Observatory (ESO) 1999

Online publication: November 3, 1999
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