We use a three-component model of a spiral galaxy, given by (Huang et al., 1979):

1. 1. An outer halo with a gravitational potential of the form where R2 = x2 + y2 + z2, M1 = 1.2 × 1011M⊙ and b1 = 1.1 Kpc.

2. 2. A nucleus, with a gravitational potential given by where M0 = 1.1 × 107M⊙, and b0 = 0.61 × 10−2 Kpc.

3. 3. A self-gravitating disk containing N stars. The density distribution in the z-direction is assumed to be: where α = 2.1 Kpc−1 is the equivalent semi-thickness of the galaxy, and σ(r) is the surface density. We consider three different initial expressions for σ(r):

   1. i. A Toomre (1963) disk.

   2. ii. A uniform distribution.

   3. iii. An exponential distribution.