136
SOLUTION OF THE EQUATIONS
effective temperature, and therefore roughly of the same spectral type, T c
and p c decrease as M increases. Type for type, the more massive stars are
the cooler and more rarefied.
We add here for reference the formulae for calculating central tempera
ture, density and pressure when M and R are given. These are obtained
by substituting the numerical values for n — 3 in (58-4) and (57-6).
T c = 0-856
Pc = 12-98
= 11-11 G
G pfiM
!R R
M
R 3
¡3M*
P 4
.(99-3).
Table 13 a gives the mean density and central temperature of stars
of masses 3, 10, 40 respectively and of various effective temperatures
(with roughly equivalent spectral types). The values are calculated from
(99-2), the constant being chosen to fit the mass and luminosity of Capella.
The mass 3 may be taken as typical of ordinary giant stars, and the
progression of mean density with spectral type shown in the table is of
considerable interest. It agrees very well with the densities generally
attributed to these types on observational grounds.
Table 13 a.
Mean Density and Central Temperature (p = 2-11).
Effective
Tempera
ture
Type
Maas = 3 x ®
Mass = 10 x ®
Mass = 40 x ®
Pm
T c
pm
T c
Pm
T c
deg.
3,000
M
•000446
million
deg.
4-65
•000020
million
deg.
2-45
•000002
million
deg.
1-52
4,000
K
•00177
7-36
•000078
3-88
•000007
2-40
5,500
o
•0082
12-3
•000360
6-48
•000032
4-01
7,500
F
•0363
20-1
•00159
10-6
•000140
6-57
10,500
A
•182
34-5
•00800
18-2
•00070
11-3
18,000
B
(2-42)
(81-7)
•106
43-0
•0093
26-6
27,000
O
(17-4)
(156)
(•763)
(82-1)
•067
50-8
It is probable that actual stars (other than white dwarfs) do not have
temperatures above 40 million degrees, so that the last two entries for a
star of mass 3 are fictitious. This was expected on the giant and dwarf
theory, since the density on reaching type A was considered appropriate
for the turning-point into the dwarf series. According to present views
the failure of stars of mass 3 to reach types B and O is more mysterious,
and the limit of 40 million degrees will be discussed at a later stage.