where means the circular rotation speed. The equation of motion in the -direction is

and that in the direction is

and the continuity equation is

As seen in Figure 3.8, we introduce the spiral coordinate in which -axis is parallel to the spiral pattern which has a pitch angle and -axis is perpendicular to the -axis.

(3.45) | |||

(3.46) | |||

(3.47) | |||

(3.48) |

Assuming that (tightly wound spiral), equations (3.44) (3.42) and (3.43) become

Similar to 2.8 we look for a steady state solution. Equation (3.51) becomes

Using this, equation (3.49) reduces to

Equation (3.50) becomes

In these equations we used following quantities:

(3.55) | |||

(3.56) | |||

(3.57) | |||

(3.58) |

is given in equation (3.41). Equations (3.52), (3.53), and (3.54) are solved under the periodic boundary condition: . Since equation (3.53) is similar to the equations in 2.8, you may think the solution seems like Figure 2.6. However, it contains a term which expresses the effect of Coliois force , the flow becomes much complicated. Taking care of the point that a shock front exists for a range of parameters, the solution of the above equations are shown in Figure 3.9. Numerical hydrodynamical calculations which solve equations (3.49), (3.50), and (3.51) was done and steady state solutions are obtained (Woodward 1975). It is shown that %, the velocity (: velocity perpendicular to the wave) does not show any discontinuity. In contrast, for %, a shock wave appears. Steady state solution is obtained from the ordinary differential equations (3.52) (3.53), and (3.54) by Shu, Milione, & Roberts (1973). Inside the CR, (Gas has a faster rotation speed than the spiral pattern). As long as there is no shock. Increasing the amplitude of the spiral force , an amplitude of the variation in increases and finally becomes subsonic partially. When the flow changes its nature from supersonic to subsonic, it is accompanied with a shock. (An inverse process, that is, changing from subsonic to supersonic is not accompanied with a shock.) In the outer galaxy (still inside CR) since decreases with the the distance from the galactic center, decreases. In this region, and the flow is subsonic if there is no spiral gravitational force, . In such a circumstance, increasing amplifies the variation in and finally reaches the sound speed. Transonic flow shows again a shock. Summary of this section is:

- There exists a spiral density pattern of the stellar component if the Toomre's parameter is .
- This is driven by the self-gravity of the rotating thin disk.
- If the amplitude of the non-axisymmetric force is as large as % of the axisymmetric one, interstellar gas whose sound speed is as large as forms the galactic shock. This is observed when the flow is transonic.
- The amplitude of the
**gas**density fluctuation is much larger than that in the stellar density. - If stars are formed preferentially in the postshock region of the spiral arm, we expect clear spiral arms made by early-type stars, which are massive and short-lived, as seen in the -band images of spiral galaxies.