A Landau-type instability mechanism for generating spiral waves is suggested.

Two populations of stars, Populations I and II, are considered, the second one with mean rotational velocity zero. Then a dispersion relation is derived which is reduced to the Lin-Shu dispersion relation in the case of vanishing Population II. The amplification of the wave is of the same type as the two-stream instability. It occurs if the angular velocity of the spiral pattern Ωs is smaller than the angular velocity of the Population I stars. A value of Ωs = 22–25 km s−1 kpc−1 was found, as well as the growth parameter. Spiral arms are formed in 108–109 yr, while trailing and leading waves grow at the same rate.

A quasi-linear theory is developed to account for the limited growth of the spiral waves.

Detailed accounts of the theory and of its implications are contained in recent publications (Marochnik, 1969; Marochnik and Suchkov, 1969a; 1969b; Marochnik and Ptitzina, 1969; Marochnik et al., 1969).

By numerical methods we have found an unstable two-armed density wave on a flat galactic model. We present the results in a form of four plots, and briefly discuss the observational implications as well as the uncertainties involved in the models and the calculations.

An ‘anti-spiral theorem’ holds with limited validity for the neutral modes of oscillation in a stellar disk - namely, whenever the effects of stellar resonances can be ignored. In the regions between Lindblad resonances, a group of spiral waves will propagate in the radial direction with the group velocity found by Toomre. This propagation occurs with the conservation of ‘wave action’, wave energy, and wave angular momentum.

The observational evidence that may indicate the presence of large scale modes of oscillation in galaxies is reviewed, and some results of theoretical modal calculations are described.

To maintain the spiral structure over long periods in spite of the chaotic disturbances so often seen in galaxies it is probably necessary to have a large energy store which is gradually tapped to generate and maintain the spiral wave. Now in a sense stable axially symmetrical model galaxies are configurations of minimum energy for a given angular momentum structure. By a given angular momentum structure I mean that the function μ(h)dh giving the total mass with specific angular momentum between h and h + dh is given and in making the minimum energy statement I have assumed that there is sufficient ‘random’ motion to satisfy Toomre's local stability criterion. To tap a store of energy one must either tap the energy of the whole rotational structure by finding disturbances that may change μ(h), or one may tap the smaller energy store of the ‘random’ motions. Clearly the larger energy store is the more attractive. Now μ(h) is conserved for any axially symmetrical time dependent motions so the only disturbances that can change μ(h) are non-axially symmetrical. If one considers a single star moving in nearly circular motion and interacting with a weak disturbance in the gravitational potential which may be Fourier analysed into components of the form S(R) exp[i(mΦ+ωt)] then one finds that the energy and angular momentum of the star oscillate but do not change in the mean. This statement is untrue when the force due to the wave does not average to zero or when it resonates with the natural oscillation of the star about the circular motion. The angular frequency of the force seen by the star is ω + mΩ, where Ω(R) is the rotational angular velocity of the galaxy, so the exceptions occur at the resonances Here κ(R) = (4B(B-A))1/2, the epicyclic angular frequency, and A(R) and B(R) are Oort's constants. At the radius where ω + mΩ(R) = 0 the pattern of the disturbance moves around at the same velocity as the circular motion of the stars epicentre. The other resonances that were discovered by B. Lindblad lie one within and one outside that circle.

This progress report on the deduction of an almost grazing orbit of the Large Magellanic Cloud from the warped shape of our Galaxy focused on two issues of special relevance to this Symposium:

1. (i) Distance R0. - An implausibly large mass (> 3 × 1010M⊙) of the LMC is needed to account for the observed warp even with the most optimal orbit, using Schmidt's (1965) disk model of our Galaxy, or any close variant thereof. Although as yet only tentative, the inference seems to be that our distance R0 from the galactic center has been overestimated; instead of the now ‘standard’ R0 = 10 kpc, these disk-bending calculations suggest a value more like the older R0 ≅ 8 kpc.

2. (ii) Forced spiral waves. - Even with the revised R0, the sense of the LMC orbit remains ambiguous (though not the perigalactic distance of roughly 20 kpc nor the low absolute inclination of that orbit relative to the galactic plane). Neither the detailed shapes of the vertically bent model disks, nor any computed tidal effects of the Galaxy upon the LMC and SMC, either separately or as a system, rule out an LMC passage that is retrograde with respect to the galactic rotation, but all favor a direct passage as being the more plausible. The latter kind of passage, however, presents an embarrassment of riches.

A method is described for the construction of a density wave of finite amplitude and in the form of a tightly wound spiral in a pressureless, self-gravitating disk of infinitesimal thickness in an external gravitational field. Waves of one kind are found for systems in which the law of rotation departs only slightly from that of a solid body. Waves of a second kind are found in systems possessing appreciable differential rotation; however, such waves can occur only if the self-gravitation of the disk is small compared with the external gravitation. The latter waves are to be identified with the waves described by linear theories.

The effect of the finite thickness of a galaxy on the propagation of density waves of the type described by Lin and his collaborators has been calculated. The calculated effect does not differ appreciably from what has been estimated previously on the basis of heuristic arguments.

Plane galactic orbits near the particle resonance of a spiral field have been calculated numerically. Besides the ring type orbits, i.e. orbits filling a ring around the galactic center, there are orbits librating near the potential maxima at the co-rotation distance (Lagrangian points). This effect tends to disrupt the spiral arms at the co-rotation distance.

We point out some properties of barred spiral galaxies which are important for the theory of their formation and spiral structure, and describe some theoretical work on the dynamics of these systems.

The asymmetric structure and dynamics of late-type barred spirals is analyzed in terms of a model consisting of a small prolate spheroid (the bar) displaced from the center of a large oblate spheroid (the disk). The non-axisymmetric motions predicted for this model are in remarkable agreement with optical and radio observations of internal motions in several magellanic barred spirals, including NGC 4027 and the Large Cloud. In particular, the model explains the displacement of the apparent center of the rotation curve from the center of the bar.

Results of an n-body calculation, containing about 120000 particles, were shown as a motion picture. Some of the particles are treated as ‘gas’, obeying a special dissipative dynamics, the rest as ‘stars’. The system was started as pure ‘gas’, and ‘stars’ were made out of the ‘gas’ in a manner closely mimicking real galaxies. Spiral density wave patterns appear in the ‘gas’, and last for about 3 ‘galactic rotations’ without substantial change of form. Various experiments are described that have been undertaken in an attempt to learn the roles of various parts of the system in the maintenance of spiral patterns.

A computer model for isolated disks of stars is used to study the self-consistent motion of large numbers of point masses as they move in the plane of the galactic disk. The Langley Research Center's CDC 6600 computers are used to integrate the equations of motion for systems containing from 50000 to 200000 stars. The results are presented in the form of a motion picture.

The present paper consists of three parts: (1) A general explanation, from a semi-empirical point of view, of the density wave theory and its ramifications, with only a few remarks on those detailed features which have not been stressed before; (2) a statement of our conclusions about the Milky Way System; and (3) a discussion of the problem of the origin of density waves of spiral form.

In order to make a direct comparison with observations of the 21-cm line of neutral hydrogen, theoretical profiles based on the ideas of the density-wave theory are constructed for a modified Schmidt model of the Galaxy and its theoretical spiral pattern. The comparison has covered galactic longitudes lII = 30° −330° with 10° intervals in the galactic plane. Good agreement is found in most of the above directions.

Observations of the neutral hydrogen in the first quadrant of galactic longitude have been analysed. The existence of large-scale streaming motions such as the streaming associated with the Sagittarius arm makes interpretation of the observations in terms of circular galactic rotation unsatisfactory. It is shown that application of the density-wave theory formulated by Lin et al. (1969) leads to a more satisfactory interpretation. Using kinematic models based on this theory the distribution and motion of the neutral hydrogen are studied. Failures of kinematic models based on circular rotation are pointed out. A map of the distribution of neutral hydrogen is produced. The Scutum arm is composed of inner and outer arcs both of which seem to be moving outward from the galactic center with velocities of the order of 30 km s−1.

The possible existence of a stationary two-armed spiral shock pattern for a disk-shaped galaxy, such as our own Milky Way System, is demonstrated. It is therefore suggested that large-scale galactic shock phenomena may very well form the large-scale triggering mechanism for the gravitational collapse of gas clouds, leading to star formation along narrow spiral arcs within a two-armed grand design of spiral structure.

The deviation of the vertex of the velocity ellipse of stars of spectral class A is shown clearly in a diagram published by Strömberg (1946); no such clear deviation is shown by the velocity ellipses of stars of other spectral types in his diagrams. This deviation of the vertex or tilt of the velocity ellipse can have some connexion with spiral structure, and there seem to be two possibilities, (a) that the tilt is confined to young stars only, and has something to do with an initial condition of their formation; and (b) that the tilt is a manifestation of the nature of the attracting field in the neighbourhood of the sun, which is not central on account of the attractions of spiral arms.

The present note examines data taken from the velocities of nearby stars, and uses the classification of the late type main sequence stars put forward by Wilson, based on observations of the reversals in the Ca+ H and K lines.

From this material it is concluded that the vertex deviation is confined to young stars, and that hypothesis (a) rather than hypothesis (b) is correct – or at least strongly indicated by available observations. The paper goes on to show that the deviation of the vertex can be explained by supposing that the stars were formed comparatively recently in a thin strip of the galaxy more or less at right angles to the direction of the galactic centre – in fact, in something like a spiral arm.

The residual velocity distribution, observed in the solar neighbourhood, exhibits several peculiarities for the stars of low epicyclic energy: vertex deviation, two-stream distribution, etc.

The photograph of the galaxy NGC 3486 shows its spiral structure in statu nascendi. The evident explanation is that of a cataclysmic origin of the visible spiral arms.