Takeda regarded a comet as a gas sphere and applied Emden’s gas-sphere theory.
If a gas-sphere oscillates, the mean gas density can be derived from its oscillation
period. Takeda obtained the mean density of the Brooks comet as
ρ ¼ 3:6 Â 10
À5
ρ 0 ,
ð4:3Þ
where ρ 0 denotes the Earth’s mean density.
Takeda thought, however, that this analysis was not definitive without additional
observational data for other comets that exhibited periodic light variations. He
expressed the need for further observations (Takeda 1927).
4.3.2 Homologous Contraction of Stars
Takeda accepted the mass annihilation hypothesis in connection with the energy
source of stars, as did Araki, and he considered the evolution of stars (Takeda 1931).
He supposed that stars gradually contracted, retaining a state of equilibrium due to
the loss of mass and energy. He defined homologous contraction in the case of mass
annihilation as a contraction that maintains mechanical equilibrium throughout the
contraction process. When stellar mass is conserved, homologous contraction takes
place under Lane’s law, that is, the gas density and potential energy increase as R
À3
and R
À4 , respectively, and the temperature increases as R
À1 , where R denotes the
stellar radius. Takeda extended the homologous contraction to the case of masslosing stars and derived the relation between stellar mass, M, and radius, R, in
the form
Mβ
R
1Àn
¼ constant,
ð4:4Þ
where β is the ratio of the gas pressure to the total (gas + radiation) pressure, which
depends only on M, and n is a constant to be determined by observations. Takeda
examined the loci of eq. (4.4) on the observed (log R vs. log Mβ) diagram, and he
found good coincidence in two cases as shown in Fig. 4.16. One is the case of n ¼ 0
for the main-sequence stars. In Takeda’s theory, n ¼ 0 corresponds to isothermal
contraction. He supposed that the main-sequence stars were maintaining nearly the
same central temperature, about 40 million K, as suggested by H. N. Russell (1925),
and evolved along the main sequence losing mass and energy. Another case is
n ¼ 0.55 for the Cepheid variables. He argued that the Cepheids evolve from longer
to shorter periods, under a homologous contraction of n ¼ 0.55. Takeda’s paper was
contained in the Astronomischer Jahresbericht, but there was no international
response. This may have been a result of the loss of popularity of the massannihilation hypothesis at that time.
96
4 Astronomy in Early Showa. II. Kyoto 1926–1945
If a gas-sphere oscillates, the mean gas density can be derived from its oscillation
period. Takeda obtained the mean density of the Brooks comet as
ρ ¼ 3:6 Â 10
À5
ρ 0 ,
ð4:3Þ
where ρ 0 denotes the Earth’s mean density.
Takeda thought, however, that this analysis was not definitive without additional
observational data for other comets that exhibited periodic light variations. He
expressed the need for further observations (Takeda 1927).
4.3.2 Homologous Contraction of Stars
Takeda accepted the mass annihilation hypothesis in connection with the energy
source of stars, as did Araki, and he considered the evolution of stars (Takeda 1931).
He supposed that stars gradually contracted, retaining a state of equilibrium due to
the loss of mass and energy. He defined homologous contraction in the case of mass
annihilation as a contraction that maintains mechanical equilibrium throughout the
contraction process. When stellar mass is conserved, homologous contraction takes
place under Lane’s law, that is, the gas density and potential energy increase as R
À3
and R
À4 , respectively, and the temperature increases as R
À1 , where R denotes the
stellar radius. Takeda extended the homologous contraction to the case of masslosing stars and derived the relation between stellar mass, M, and radius, R, in
the form
Mβ
R
1Àn
¼ constant,
ð4:4Þ
where β is the ratio of the gas pressure to the total (gas + radiation) pressure, which
depends only on M, and n is a constant to be determined by observations. Takeda
examined the loci of eq. (4.4) on the observed (log R vs. log Mβ) diagram, and he
found good coincidence in two cases as shown in Fig. 4.16. One is the case of n ¼ 0
for the main-sequence stars. In Takeda’s theory, n ¼ 0 corresponds to isothermal
contraction. He supposed that the main-sequence stars were maintaining nearly the
same central temperature, about 40 million K, as suggested by H. N. Russell (1925),
and evolved along the main sequence losing mass and energy. Another case is
n ¼ 0.55 for the Cepheid variables. He argued that the Cepheids evolve from longer
to shorter periods, under a homologous contraction of n ¼ 0.55. Takeda’s paper was
contained in the Astronomischer Jahresbericht, but there was no international
response. This may have been a result of the loss of popularity of the massannihilation hypothesis at that time.
96
4 Astronomy in Early Showa. II. Kyoto 1926–1945