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CAPSTONE Lecture 6
HR Diagram
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Two stars at distances d1 and d2 with intrinsic total
luminosities L1 and L2. (L=4pr2sT4).
Flux, f, is ergs per cm squared per second over some
frequency interval is received at Earth.
f1=L1/(4pd12), f2=L2 /(4pd22)
(1)
If L1=L2, then f1/f2=4pd22/4pd12
(2)
Take the log of both sides
Multiply both sides by -2.5
-2.5 log f1 +2.5 log f2=-2.5 (2 log d2) + 2.5 (2 log d1)
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-2.5 log f1 +2.5 log f2=-2.5 (2 log d2) + 2.5 (2 log d1)
Define m= - 2.5 log f
m1-m2=5 log d1- 5 log d2.
Finally, define a reference distance, 10 pc, which is to be
the absolute magnitude (M), an intrinsic property of the
star.
Take star 2 to be at 10 pc. (Hence, log d2=1.) And, let
m1=m, m2=M. Then,
m-M=5 log d1 –5
(3)
• This equation is called the magnitude equation
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• Stars of given type of spectrum and the same colors
have the same absolute magnitude (99.9%)
• Stars have different apparent magnitudes depending
on their distance.
• Stars behind dust clouds look redder than they
are intrinsically, so…
m-M=5 log d1 –5+ A(l)
(i.e., the star looks fainter)
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The equation only makes sense if we agree to
measure all stars in a certain way, for instance, using
a filter to define exactly the wavelengths of light
included in the quoted magnitude. For instance, a filter
often used by astronomers, known as the V filter,
covers the region from 5100 A to 5900 A. Since the
filter is called the V filter, the magnitudes of stars so
measured are denoted with the subscript V on both m
and M. Other filters are the B filter, the U filter, the R
filter, etc. The main filter system today is denoted as u,
g, r, i, z, reflecting slight differences in the wavelengths
passed.
It is handy to memorize the differences in brightness
for magnitude differences of 1,2, 2.5, 3, 4 and 5. The
relative luminosities are 2.5, 6.25, 10, 15, 40 and 100.
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Hertzsprung Russell Diagram
• Stars have color
• U, B, V; u, g, r, i, z
• Color is determined by measuring an apparent
magnitude in a blue filter, say, then a red filter and
taking the difference. Bluer stars have negative B-R
values (called “color”).
• Since mB-MB=5log r -5 for a given star at given
distance
• and mV-MV=5log r -5
• mB-MB=mV-MV
• That is, we can know the color of a star without
knowing the distance.
• Colors are easy to measure, distances are hard.
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Order from chaos
• Around 1890, spectra of thousands of stars
were obtained and it was recognized that
there were only a few types.
• Around 1923, it was found that, while a
diagram of apparent mag. Vs. spectral type
(temperature) was a scatter diagram
• If the absolute magnitude was used, the
diagram was very ordered. Something
physical had been discovered about stars,
that was understood over the next 40 years.
We will do spectral types in the next lecture.
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The diagram
• Hot, massive stars end up in the upper left; cool, low mass stars
end up in the lower right.
• In addition, there are poorly populated areas. One, in the lower
left, is populated by hot, very tiny stars (white dwarfs). Another,
in the upper right, is populated by giant stars on the giant branch
(perpendicular to the main sequence, rising to the upper right).
A band runs across the top of the diagram, called the horizontal
branch. Other, higher, horizontal bands, populated by
supergiants, exist at the top of the HR diagram.
• Using color or temperature, one axis is completely independent
of distance. The other (vertical) axis can be made independent
of distance by using absolute magnitudes. This left the job of
explaining why stars resided in different places, but at least the
pattern that all agreed on made it clear that the quest would be
worth while.
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….more
• O stars have B-V=-0.3 (that is, they are 0.3
magnitudes brighter in the B filter than the V
filter. (Remember, brighter objects, in
astronomical notation, are labelled with more
and more negative absolute magnitudes.
Stars known as A stars have B-V=0.0. G
stars have B-V=0.6. M stars have B-V>1.5
(brighter in the red than in the blue.)
• The dearth of stars in parts of the diagram
meant that stars had different ages; some
evolved faster than others.
• How old are the stars?
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Age of Earth
(I gave these points earlier)
• The Biblical age of the Earth (Bishop Usher,
circa 1825) is about 6000 years, the span of
recorded history. Darwin suggested that the
age had to be 600,000,000 years, to allow
time for species to evolve under the laws he
called natural selection. (He figured this out,
among other ways, by thinking about how
farmer’s raised cattle to have different
characteristics.)
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The rapid change
• There was a raging debate between Darwin and his
followers, on the one hand, and the physical
scientists, on the other, about the age of the Earth.
The latter, using the Kelvin Helmholtz time scale
(1860)(see next lecture), argued that the Earth was
30 million years old, 5% of the age Darwin gave.
However, by 1897, radioactivity was discovered
(Madame Curie and J. Bequerel). It was quickly
realized that the Earth could be kept warm much
longer than Kelvin assumed by assuming the Earth
was just cooling off from its formation. By 1926, the
accepted radioactive age of the Earth was 4.5 billion
years, much longer than either Kelvin or Darwin has
argued for.
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Implications
• This realization meant that the Sun had to be
very old, since the Sun must be older than
Earth. Otherwise, the oceans would have
frozen over and not melted. (We know today
that stars such as the Sun do not change
brightness significantly over their lifetimes,
otherwise the HR diagram would have a
broader locus of points on the main
sequence.)
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