120 9. STELLAR EVOLUTION
We will discuss the details further in Section 12.2.3, but the following simple relations
are approximately true.
1. All else being equal, hotter stars have a higher luminosity.
2. All else being equal, larger stars have a higher luminosity.
3. Regarding the luminosity of a star, both size and temperature matter, but temperature
matters more. Double its radius and a star is four times brighter. But double its temperature
and it will be sixteen times brighter.
A good example is the double star Albireo, in Cygnus the Swan. rough even a small
telescope one sees a beautiful pair of stars—a brighter yellow one and a noticeably dimmer
blue one. We shall see (in Section 12.2.4) that the difference in color means the blue star is
significantly hotter than the yellow one. And so it must be the case the that the yellow star is
larger. If they were the same size, the blue (hotter) star would be brighter. And so the cooler,
brighter star must be larger to make up for its lower temperature.
e approximate relationship between L, T, and R for stars can be easily captured math-
ematically. e power emitted per surface area is proportional to the fourth power of the tem-
perature. And the surface area is proportional to the square of the radius (the surface area of a
sphere is 4R
2
). us, the luminosity, L
‹
, of a given star is simply the product of the area and
the power per area:
L
‹
L
@
D
R
‹
R
@
2
T
‹
5770
4
: (9.1)
I have expressed Equation (9.1) so as to compare a particular star’s L, T, and R to that of
the Sun, which has an effective temperature of 5770 K. is is the temperature the Sun would have
if, given its radius and luminosity, it emitted its light with a perfect thermal blackbody spectrum
(see Section 12.2.3). e Sun and other stars emit light only approximately as a blackbody, and
so Equation (9.1) can be taken as only approximate for real stars. Alternatively, it can be used as
the definition of the effective temperature of a star.
9.2 THE HERTZSPRUNG–RUSSEL DIAGRAM
One of the most important tools for understanding the relationships between different stars—
and the changes that an individual star undergoes as time passes—is called the Hertzsprung–
Russel diagram, or simply H-R diagram. In its traditional form, it is a graph with luminosity
on the vertical axis and spectral type on the horizontal axis. Stars are then placed on the graph
accordingly, each point representing the measured values for an individual star.
We discuss spectral type in detail in Sections 12.2.7 and 13.2.3; it relates quite directly to
temperature. But there is a complication, and it comes from a fluke of history; the sequence of
spectral types was discovered before its physical cause was understood. And this sequence in its
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