hydrodynamic study of harmonic motion, they found that when the surface gravity
was strong, radial pulsation was limited to the fundamental mode, whereas when the
surface gravity was weak, multiharmonic pulsations, such as chaotic motions, tended
to appear. Actually, they observed chaotic motions in F-type supergiants with low
surface gravity (Aikawa 1991).
7.3.2 Nonradial Oscillations of Stars
The existence of nonradial oscillations was first proposed by W. T. Kelvin in 1890,
and its theoretical possibility was shown by F. R. Moulton in 1909. O. Struve carried
out spectroscopic observations of β Cephei-type stars in the 1950s and found
variations of radial velocities in some spectral lines. These stars later became the
prototype of nonradial oscillations. Theoretical studies of nonradial oscillations in
the postwar period were conducted by P. Ledoux (1974) and J. P. Cox (1976).
In Japan, Unno Wasaburo of the University of Tokyo and his coworkers Osaki
Yoji, Ando Hiroyasu, and Shibahashi Hiromoto markedly contributed to the theoretical studies of nonradial oscillations in the 1970s. In their book Nonradial
Oscillations of Stars (1979), they wrote that stars are generally analogous to musical
instruments, which play various modes of oscillation and tones. In fact, nonradial
oscillations are observable in stars of a wide range of spectral types, with various
modes of oscillations (Unno et al. 1979).
Modes of oscillations are expressed by spherical harmonics l and m, where l ¼ 0,
1, 2, . . . ., and m is associated with l as m ¼ 0, Æ1, Æ2, . . . Æ l. On the restoring force
of oscillations there are two modes: pressure-mode (p-mode) and gravity-mode
(g-mode). Analogically speaking, p-mode corresponds to sound waves in the atmosphere and g-mode to the surface waves of a water pond. A sample of a wave pattern
in the case of l ¼ 4 is shown in Fig. 7.5.
When Osaki began to study β Canis Majoris in β Cephei-type stars in 1971, there
were two models, a and b, on beat phenomena and the characteristic variation in line
widths. (a) Ledoux (1951) proposed that these stars undergo nonradial oscillations in
the presence of rotation. (b) Chandrasekhar and Lebovitz (1962) presented a model
for beat phenomena as a coupling of radial and nonradial oscillations. Osaki (1971),
based on Ledoux’s model, showed that the combinations of stellar rotation and
nonradial oscillations were able to explain most of the fundamental characteristics of
β Canis Majoris-type stars, as follows (Osaki 1971).
Osaki considered the variations of line profiles and radial velocities of nonradial
oscillations corresponding to second harmonics (l ¼ 2, m ¼ 0. Æ1, Æ2) for β Cephei
stars. Different modes yield different forms of nonradial oscillations. For example,
oscillation waves in the mode m ¼ À2 propagate in the same direction as the rotation
and take forms symmetric with respect to the stellar equator. He carried out numerical calculations in this case for radial velocity curves and variations in the halfwidths. The results of those calculations are illustrated in Fig. 7.6, where the upper
panel shows radial velocity curves and the lower panel the variation in line half184
7 Postwar Development of Astrophysics, 1946–2000 (Part II: Astrophysics)
was strong, radial pulsation was limited to the fundamental mode, whereas when the
surface gravity was weak, multiharmonic pulsations, such as chaotic motions, tended
to appear. Actually, they observed chaotic motions in F-type supergiants with low
surface gravity (Aikawa 1991).
7.3.2 Nonradial Oscillations of Stars
The existence of nonradial oscillations was first proposed by W. T. Kelvin in 1890,
and its theoretical possibility was shown by F. R. Moulton in 1909. O. Struve carried
out spectroscopic observations of β Cephei-type stars in the 1950s and found
variations of radial velocities in some spectral lines. These stars later became the
prototype of nonradial oscillations. Theoretical studies of nonradial oscillations in
the postwar period were conducted by P. Ledoux (1974) and J. P. Cox (1976).
In Japan, Unno Wasaburo of the University of Tokyo and his coworkers Osaki
Yoji, Ando Hiroyasu, and Shibahashi Hiromoto markedly contributed to the theoretical studies of nonradial oscillations in the 1970s. In their book Nonradial
Oscillations of Stars (1979), they wrote that stars are generally analogous to musical
instruments, which play various modes of oscillation and tones. In fact, nonradial
oscillations are observable in stars of a wide range of spectral types, with various
modes of oscillations (Unno et al. 1979).
Modes of oscillations are expressed by spherical harmonics l and m, where l ¼ 0,
1, 2, . . . ., and m is associated with l as m ¼ 0, Æ1, Æ2, . . . Æ l. On the restoring force
of oscillations there are two modes: pressure-mode (p-mode) and gravity-mode
(g-mode). Analogically speaking, p-mode corresponds to sound waves in the atmosphere and g-mode to the surface waves of a water pond. A sample of a wave pattern
in the case of l ¼ 4 is shown in Fig. 7.5.
When Osaki began to study β Canis Majoris in β Cephei-type stars in 1971, there
were two models, a and b, on beat phenomena and the characteristic variation in line
widths. (a) Ledoux (1951) proposed that these stars undergo nonradial oscillations in
the presence of rotation. (b) Chandrasekhar and Lebovitz (1962) presented a model
for beat phenomena as a coupling of radial and nonradial oscillations. Osaki (1971),
based on Ledoux’s model, showed that the combinations of stellar rotation and
nonradial oscillations were able to explain most of the fundamental characteristics of
β Canis Majoris-type stars, as follows (Osaki 1971).
Osaki considered the variations of line profiles and radial velocities of nonradial
oscillations corresponding to second harmonics (l ¼ 2, m ¼ 0. Æ1, Æ2) for β Cephei
stars. Different modes yield different forms of nonradial oscillations. For example,
oscillation waves in the mode m ¼ À2 propagate in the same direction as the rotation
and take forms symmetric with respect to the stellar equator. He carried out numerical calculations in this case for radial velocity curves and variations in the halfwidths. The results of those calculations are illustrated in Fig. 7.6, where the upper
panel shows radial velocity curves and the lower panel the variation in line half184
7 Postwar Development of Astrophysics, 1946–2000 (Part II: Astrophysics)