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Measuring the Stars
The Sun’s Neighborhood A plot of the 30 closest stars to the Sun, projected so as to
reveal their three-dimensional relationships. Notice that many are members of
multiple-star systems. All lie within 4 pc (about 13 light-years) of Earth. The gridlines
represent distances in the Galactic plane. The next nearest neighbor to the Sun
beyond the Alpha Centauri system is called Barnard’s Star.
2 Ways to Measure
Star Distance
Stellar Parallax
Stellar Brightness
(Specroscopic Parallax)
We already discussed stellar parallax
earlier in the course. Remember, the
more the star changes its relative
position in the sky over the course of
a year, the closer the star is to us.
Inverse-square Law As light
moves away from a source
such as a star, it is steadily
diluted while spreading over
progressively larger surface
areas (depicted here as
sections of spherical shells).
Thus, the amount of radiation
received by a detector (the
source’s apparent brightness)
varies inversely as the
square of its distance from
the source.
Can you name any bright stars?
Good! How about we look at Sirius, in the constellation Canis Major
and Rigel in the Orion constellation? Sirius is only 2.7 parsecs away,
but Rigel, in Orion, is 240 parsecs away. So, as you might expect
Sirius looks brighter in the night sky than Rigel.
The magnitude scale used in astronomy ranks the brightness of a
star with a number. But the smaller the number the brighter the star!
Distance
Apparent
Magnitude
Sirius
2.7 pc
-1.46
Rigel
240 pc
+0.14
Question: What if we could
move these two stars so
that they were each the
same distance from Earth then which would be
brighter?
Distance
Apparent
Magnitude
Absolute
Magnitude
Sirius
2.7 pc
-1.46
+1.4
Rigel
240 pc
+0.14
-6.8
A star's absolute magnitude is its apparent magnitude when viewed
from a distance of 10 parsecs. This allows astronomer's to compare
stars with each other.
A magnitude difference of 5
corresponds to 100 times in
brightness.
For example, if the apparent
magnitude of a star that is 100 pc
away is +6. Because it is 10
times farther away that means
the object will appear 100 times
dimmer (inverse-square law).
The 100 times dimmer in brightness corresponds to 5 magnitudes. So, the star has an
absolute magnitude of +6 – 5 =
+1.
 dist . 
(ap. mag .  ab. mag .)  5  log 

10
pc


Apparent
Magnitude
apparent
magnitude
absolute
magnitude
Star A
1
1
Star B
1
2
Star C
5
4
Star D
4
4
1) Which object appears brighter from Earth: Star C or Star D? Explain.
Star D – smaller apparent magnitude means the star is brighter.
2) Which object is actually brighter: Star A or Star D? Explain.
Star A – it has a smaller absolute magnitude.
3) Rank the objects in order of their distances from Earth. Explain.
Farthest to nearest: C, A&D tie, B
4) How would the apparent and absolute magnitudes of Star A change if it were
now moved to a distance of 40 parsecs from Earth? Explain.
Absolute magnitude would not change, but apparent
magnitude would get larger (it would go to about 4).
Spectral Classification
Astronomers categorize stars in spectral classes. You can remember the
spectral classes in order of decreasing temperature if you remember the
mnemonic:
"Oh, Be A Fine Guy/Girl, Kiss Me".
Almost stars fit somewhere into this sequence of O, B, A, F, G, K, and M.
O stars are hottest and thus appear bluish white;
G stars are less hot and thus appear yellowish; and
M stars are coolest and thus appear reddish.
NOTE: To understand the relationship between color and temperature, think
about the toaster element in your toaster or the heating element on your electric
stove. As it heats up, it starts to glow red. As it gets even hotter, it begins to
glow orange. As it gets even hotter, the color that it glows will move up the
spectrum to yellow (1275 K) and eventually blue-white (1425 K and higher). So
hotter objects give off higher frequency light. Keep in mind the colors of the
visible spectrum – Red, Orange, Yellow, Green, Blue, Violet.
Astronomers then divide each of these letter classes into subclasses.
subclasses are labeled from 0 to 9, going from hottest to coolest.
The
 For example, a G3 star is hotter than a G8 star.
 But a G8 star is hotter than a K9 star.
Look at the following charts that summarize the stellar spectral classes, and
show some of the key stars in the winter sky.
SPECTRAL
CLASS
COLOR
SURFACE TEMPERATURE (K)
PRINCIPAL FEATURES
EXAMPLES
O
Bluishwhite
30,000
Relatively few absorption lines. Lines
of highly ionized atoms. Hydrogen
lines appear only weakly.
Naos
B
Bluishwhite
11,000 - 30,000
Lines of neutral helium. Hydrogen
lines more pronounced than in O-type
stars.
Rigel, Spica
A
Bluishwhite
7,500 - 11,000
Strong lines of hydrogen. Also lines
of singly ionized magnesium silicon,
iron, titanium, calcium, and others.
Lines of some neutral metals show
weakly.
Sirius, Vega
F
Bluishwhite to
white
6,000 - 7,500
Hydrogen lines are weaker than in Atype star but still conspicuous. Lines
of singly ionized metals are present,
as are lines of other neutral metals.
Canopus,
Procyon
G
White to
yellowishwhite
5,000 – 6000
Lines of ionized calcium are the most
conspicuous spectral features. Many
lines of ionized and neutral metals are
present. Hydrogen lines are weaker
even than in F-type stars.
Sun, Capella
K
Yellowishorange
3,5000 - 5,000
M
Reddish
3,500
Lines of neutral metals predominate.
Strong lines of neutral metals and
molecules.
Arcturus,
Aldebaran
Betelgeuse,
Antares
STAR
SPECTRAL
CLASS
SURFACE
TEMPERATURE
(K)
RADIUS
(sun = 1.0)
DISTANCE
(light
years)
LUMINOSITY
(sun = 1.0)b
Epsilon Orionis
B0
24,800
37
1600?
470,000?
Rigel
B8
11,550
74
880?
90,000?
Regulus
B7
12,210
3.6
69
270
Sirius A
A1
9,970
1.7
8.7
23
Procyon A
F5
6,510
2.1
11
7
Sun
G2
5,780
1.00
1.6 x 10-5
1.00
Capellaa
G5
5,200
14
41
130
Epsilon Eridani
K2
5,000
0.7
11
0.28
Aldebaran
K5
3,780
61
60
700
Betelgeuse
M2
3,600
1200
1400?
21,000?
Sirius B
White dwarf
30,000
0.0073
8.7
0.003
a Capella is a double star. The temperature, radius, and luminosity are those of the brighter and cooler component.
b The luminosities and radii of Epsilon Orionis, Rigel, and Betelgeuse are only approximate because their distances
are estimated.
Stellar Spectra Comparison of
spectra observed for seven stars
having a range of surface
temperatures. These are not
actual spectra, which are messy
and complex, but simplified artists’
renderings illustrating a few
spectral features. The spectra of
the hottest stars, at the top, show
lines of helium and multiply
ionized heavy elements. In the
coolest stars, at the bottom, there
are no lines for helium, but lines
of neutral atoms and molecules
are plentiful. At intermediate
temperatures, hydrogen lines are
strongest. All seven stars have
about
the
same
chemical
composition.
radius increases
radius increases
The H-R Diagram
A plot of luminosity against
surface
temperature
(or
spectral class), known as a
Hertzsprung-Russell diagram,
is a useful way to compare
stars. Plotted here are the
data for some stars we've
discussed. The Sun, of
course, has a luminosity of 1
solar unit. Its temperature,
read off the bottom scale, is
5800 K—a G-type star.
Cooler stars that are very
luminous must be very large
in radius. Conversely, very
hot stars that are not very
luminous must be very small
in radius.
Stellar Sizes Star sizes
vary greatly. Shown here
are the estimated sizes of
several well-known stars.
Only part of the red-giant
star Antares can be shown
on this scale; (The symbol
R means “solar radius.”)
 Giants are stars having
radii between 10 and 100
times that of the Sun.
 Supergiants are larger
than giants having radii
as large as 1000 times
the solar radius.
 A dwarf refers to any star
of radius comparable to
or smaller than the radius
of the Sun (from 0.01
times the Sun's radius up
to about one solar radius)
H-R Diagram of Nearby
Stars Most stars have
properties within the
shaded region of the H–
R diagram known as the
main sequence. The
points plotted here are
for stars lying within
about 5 pc of the Sun.
Each dashed diagonal
line corresponds to a
constant stellar radius,
so that stellar size can be
indicated on the same
diagram as stellar luminosity and temperature.
H-R Diagram of Bright
Stars An H-R diagram for
the 100 brightest stars in
the sky is biased in favor
of the most luminous stars
— which appear toward
the upper left — because
we can see them more
easily than we can the
faintest stars.
Hipparcos H-R Diagram
This is a simplified version
of the most complete H-R
diagram ever compiled. It
represents
more
than
20,000 data points, as
measured by the European Hipparcos spacecraft
for stars within a few
hundred parsecs of the
Sun.
Few white dwarfs appear
because almost no white
dwarfs lie close enough to
Earth to have been bright
enough for the instrument.
About 90% of all stars in our
solar neighborhood, and presumably a similar percent-age
elsewhere in the universe, are
main-sequence stars. About
9% of stars are white dwarfs,
and 1% are red giants.
Stellar Distances Knowledge of a star’s luminosity and apparent brightness can
yield an estimate of its distance. Astronomers use this third rung on our distance
ladder, called spectroscopic parallax, to measure distances as far out as individual
stars can be clearly discerned—several thousand parsecs.
Measurement of the apparent brightness of a light source, combined with
some knowledge of its luminosity, can yield an estimate of its distance.
The procedure is as follows:
1) Determine the star’s spectral type.
2) Assuming the star lies on the main sequence, read the star’s luminosity
directly off the HR diagram.
3) Knowing the star’s luminosity, determine its distance by measuring its
apparent brightness and using the inverse-square law.
This process of using stellar spectra to infer distances is called spectroscopic
parallax. In practice, the width of the main sequence line on the HR diagram
translates into a small (10–20 percent) uncertainty in the distance, but the
method is still valid.
Stellar Luminosities Stellar luminosity classes in the H–R diagram. A star’s location
in the diagram could be specified by its spectral type and luminosity class instead of
its temperature and luminosity.
Over the years, astronomers have developed a system for classifying stars
according to the widths of their spectral lines. Because line width depends on
pressure in the stellar photosphere, and because this pressure in turn is well
correlated with luminosity, this stellar property has come to be known as
luminosity class.
Binary Stars
Most stars are members of multiple-star systems—groups of two or more stars
in orbit around one another. The majority are found in binary-star systems,
which consist of two stars in orbit about their common center of mass, held
together by their mutual gravitational attraction. (The Sun is not part of a
multiple-star system; if it has anything at all uncommon about it, it is this lack of
stellar companions.)
Astronomers classify binary-star systems (or simply binaries) according to their
appearance from Earth and the ease with which they can be observed.
 Visual binaries have widely separated members bright enough to be
observed and monitored separately.
 In the rarer eclipsing binaries, the orbital plane of the pair of stars is
almost edge-on to our line of sight. In this situation, we observe a periodic
decrease of starlight intensity as one member of the binary passes in front
of the other. Media Clip
 The more common spectroscopic binaries are too distant from us to
be resolved into separate stars, but they can be indirectly perceived by
monitoring the back-and-forth Doppler shifts of their spectral lines as the
stars orbit one another and their line-of-sight velocities vary periodically.
In a double-line spectroscopic binary, two distinct sets of spectral lines—one
for each component star—shift back and forth as the stars move. Because we
see particular lines alternately approaching and receding, we know that the
objects emitting the lines are in orbit. Media Clip
In the more common single-line systems, one star is too faint for its spectrum
to be distinguished, so we see only one set of lines shifting back and forth.
This shifting means that the detected star must be in orbit around another star,
even though the companion cannot be observed directly. If this idea sounds
familiar, that's probably because we have discussed it before. All the
extrasolar planetary systems discovered to date were found using this singleline method.
Note: These binary categories are not mutually exclusive. For example,
an eclipsing binary may also be (and, in fact, often is) a spectroscopic
binary system.
An Example of How Stellar Mass Can Be Determined
Consider the nearby visual binary system made up of the bright star Sirius A
and its faint companion Sirius B. Their orbital period is 50 years, and their
orbital semi-major axis is 20 A.U (i.e. 7.5" at a distance of 2.7 pc), implying that
the sum of their masses is 3.2 (= 203/502) times the mass of the Sun.
Further study of the orbit shows that Sirius A has roughly twice the mass of its
companion. It follows that the masses of Sirius A and Sirius B are roughly 2.1
and 1.1 solar masses, respectively.
Stellar Masses More than any
other stellar property, mass
determines a star’s position on
the main sequence. Low-mass
stars are cool and faint; they lie
at the bottom of the main
sequence. Very massive stars
are hot and bright; they lie at
the top of the main sequence.
Chart of Stellar Mass Distribution
The distribution of masses of mainsequence stars, as determined from
careful measurement of stars in the
solar neighborhood.
 With few exceptions, main-sequence stars range in mass from about 0.1 to
20 times the mass of the Sun.
 The hot O- and B-type stars are generally about 10 to 20 times more
massive than our Sun.
 The coolest K- and M-type stars contain only a few tenths of a solar mass.
Important Note: The mass of a star at the time of its
formation determines its location on the main sequence.
Stellar Radii and Luminosities (a) Dependence of stellar radius on mass for mainsequence stars; actual measurements are plotted here. The radius increases
roughly in proportion to the mass over much of the range. (b) Dependence of
luminosity on mass. The luminosity increases roughly as the fourth power of the
mass (indicated by the straight line).
Expected Stellar Lifetimes
Dividing the amount of fuel available (that is, the star’s mass) by the rate at which
the fuel is being consumed (the star’s luminosity), one can estimate the expected
lifetime of a star. The expected lifetime of the Sun is about 10 billion years, and is
currently about 4.6 billion years old.
Because luminosity increases so rapidly with mass,
the most massive stars are by far the shortest lived!
For example, according to the mass–luminosity relationship, the lifetime of a 10solar-mass O-type star is about 1/1000 (=10 solar mass/104 luminosity) that of
the Sun, or 10 million years. So we can be sure that all the O-type and B-type
stars we observe are quite young—less than a few tens of millions of years old.
The reason is that their nuclear reactions proceed so rapidly that their fuel is
quickly depleted despite their large masses. At the opposite end of the main
sequence, the low core density and temperature of an 0.1-solar-mass M-type
star mean that its proton–proton reactions churn away much more sluggishly
than in the Sun’s core, leading to a very low luminosity and a correspondingly
long lifetime. Many of the K-type and M-type stars now visible in the sky could
shine on for at least another trillion years.
Slow and steady wins the race!!