while the circular velocity
v of particles in the galactic plane, the epicyclic frequency k, and the vertical frequency v of small oscillations about the equilibrium circular orbit, can be obtained from the following expressions evaluated at z = 0 (Binney & Tremaine, 2011)
Given a gravitational potential [PHI](R, z) that satisfies the above considerations, the circular velocity [v.sub.c](R) can be obtained from the gravitational potential through the relation
Finally the [C.sub.2n] are arbitrary constants, which are chosen properly in order to adjust the circular velocity of the model and the rotation curve of some particular galaxy.
With this gravitational potential, and using the equations (4) and (5) we have that the circular velocity and the surface mass density can be written respectively as
This design not only provides a circular velocity
pattern, but, more importantly, ensures good top-to-bottom movement that is critical to optimum mixing.
The circular velocity
estimates are based on Naab's simulation .
One of the oldest and most important problems in galactic dynamics is the determination of the mass distribution based on the observations of the circular velocity or rotation curve (Pierens and Hure, 2004), defined as the speed of the stars moving in the galactic plane in circular orbits around the center.
So, once an expression for the gravitational potential has been derived, corresponding expressions for the surface mass density of the disc and for the circular velocity of the disc particles can be obtained.
By adjusting the corresponding expression for the circular velocity to the observed data of the rotation curve of some specific galaxies, some particular models will be analysed.
The physical quantities in the plane of the disk, are defined and calculated such as, the epicyclic frequency, k, the vertical frequency, v, and the circular velocity, [v.sub.c] of particles.
Then, in the section 3, we find the physical quantities in the plane the disk such as circular velocity vc of particles, the epicyclic frequency, k, the vertical frequency, v.
The physical quantities of interest are evaluated in the plane the disk, these are the circular velocity, [[upsilon].sub.c], the epicyclic frequency, [kappa], and the vertical frequency, v, of small oscillations about the equilibrium circular orbit.