Chapter 16--Properties of Stars

surface temperature to Betelgeuse, but far dimmer because
of its much smaller size. White dwarfs are usually designated
with the letters wd rather than with a Roman numeral.
The Main Sequence
The common trait of main-sequence stars is that, like our
Sun, they are fusing hydrogen into helium in their cores.
Because stars spend the majority of their lives fusing hydrogen,
most stars fall somewhere along the main sequence
of the H–R diagram.
Why do main-sequence stars span such a wide range
of luminosities and surface temperatures? By measuring
the masses of stars in binary systems, astronomers have discovered
that stellar masses decrease downward along the
main sequence (Figure 16.11). At the upper end of the main
sequence, the hot, luminous O stars can have masses as
high as 100 times that of the Sun (100M Sun ). On the lower
end, cool, dim M stars may have as little as 0.08 times the
mass of the Sun (0.08M Sun ). Many more stars fall on the
lower end of the main sequence than on the upper end,
which tells us that low-mass stars are much more common
than high-mass stars.
The orderly arrangement of stellar masses along the
main sequence tells us that mass is the most important
attribute of a hydrogen-burning star. Luminosity depends
directly on mass because the weight of a star’s outer layers
determines the nuclear burning rate in its core. More weight
means the star must sustain a higher nuclear burning rate
in order to maintain gravitational equilibrium [Section 15.3].
Almost all stars are too distant for us to measure their radii directly.
However, we can calculate a star’s radius from its luminosity with
the aid of the thermal radiation laws. As given in Mathematical
Insight 6.2, the amount of thermal radiation emitted by a star of
surface temperature T is:
emitted power per unit area sT 4
where the constant s 5.7 108 watt/(m2 Kelvin 4 ).
The luminosity L of a star is its power per unit area multiplied
by its total surface area. If the star has radius r, its surface
area is given by the formula 4pr 2 .Thus:
L 4pr 2 sT 4
With a bit of algebra, we can solve this formula for the star’s
radius r:
r L
4 4psT
Example: Betelgeuse has a luminosity of 38,000LSun and a surface
temperature of about 3,400 K. What is its radius?
Solution: First, we must make our units consistent by converting
the luminosity of Betelgeuse into watts. Remembering that
LSun 3.8 1026 watts, we find:
534 part V • Stellar Alchemy
Table 16.2 Stellar Luminosity Classes
Class Description
I Supergiants
II Bright giants
III Giants
IV Subgiants
V Main sequence
The nuclear burning rate, and hence the luminosity, is very
sensitive to mass. For example, a 10M Sun star on the main
sequence is about 10,000 times more luminous than the Sun.
The relationship between mass and surface temperature
is a little subtler. In general, a very luminous star must
either be very large or have a very high surface temperature,
or some combination of both. Stars on the upper end
of the main sequence are thousands of times more luminous
than the Sun but only about 10 times larger than the
Sun in radius. Thus, their surfaces must be significantly
hotter than the Sun’s surface to account for their high luminosities.
Main-sequence stars more massive than the
Sun therefore have higher surface temperatures than the
Sun, and those less massive than the Sun have lower surface
temperatures. That is why the main sequence slices
diagonally from the upper left to the lower right on the
H–R diagram.
Mathematical Insight 16.5 Calculating Stellar Radii
LBet 38,000 LSun 38,000 3.8 1026 watts
1.4 1031 watts
Now we can use the formula derived above to calculate the radius
of Betelgeuse:
r L
4psT
4
1.4 1031 watts
4p 5.7 108 watt
m2 (3,400 K)4
3.8 1011 1.4 10
m
31 watts
9.6 107 watts
m2
The radius of Betelgeuse is about 380 billion meters or, equivalently,
380 million kilometers. Note that this is more than twice
the Earth–Sun distance of 150 million kilometers.
K 4