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Roger A. Freedman • William J. Kaufmann III
Universe
Eighth Edition
CHAPTER 17
The Nature of Stars
M 39 is an Open or Galactic Cluster
Today, we will learn
17-1 How we can measure the distances to the stars
Box 17-1 Stellar Motions
17-2 How we measure a star’s brightness and
luminosity
and depending on time available
17-3 The magnitude scale for brightness and
luminosity
Careful measurements of the
parallaxes of stars reveal their
distances.

The brightness of a star is not a good
indicator of distance.
e.g., Polaris is closer than Betelgeuse but
Betelgeuse appears brighter.

Distances to nearby stars can be
measured using parallax (see Section 4-3,
Fig. 4-7, Tycho Brahe).
Parallax is the apparent change in the position
of an object due to a change in observing
position.
Stellar Parallax
As Earth moves from
one side of the Sun to
the other, a nearby star
will seem to change its
position relative to the
distant background stars.
d=1/p
d = distance to nearby star in
parsecs
p = parallax angle of that star
in arcseconds
Parsec
Is the distance corresponding to a
Parallax of one arc second
1 arc sec = 1 degree /3,600
1
1 arc sec
206,265
So, 1 pc = 206,265 AU
Some Nearby Stars
Proxima Centauri: p = 0.772 arcsec, d = 1/p = 1.3 pc
Barnard’s Star:
p = 0.545 arcsec, d = 1/p = 1.83 pc
Sirius A/B :
p = 0.379 arcsec, d = 1/p = 2.64 pc
1 pc = 206,265 AU = 3.26 LY
Stellar Motions
The Doppler Effect reveals the radial velocity
Stellar Motions
The rate of change in position on
the sky is the proper motion.
If we know the distance we can
calculate the tangential velocity.
Example: Barnard’s Star
A certain iron line of wavelength l0=516.629 nm
is observed in the spectrum at a wavelength
l = 516.438 nm. Using the Doppler formula
Dl/l0 = (l- l0)/ l0 =(516.438 – 516.629)/516.629
= - 0.00037
So the radial velocity is
vr = - 0.00037 c = - 111 km/s
The negative sign means this star is approaching us.
If a star’s distance is known, its luminosity
can be determined from its brightness.



As you get farther and
farther away from a star,
it appears to get dimmer.
Luminosity, L, doesn’t
change
Apparent brightness, b,
does change following
the inverse square law for
distance.
b = L / (4pd2)
If a star’s distance is known, its luminosity
can be determined from its brightness.

A star’s luminosity can be determined from
its apparent brightness if its distance is
known:
L = 4p d 2 b
L = 4p d2 b
L/L = (d/d)2  (b/b)
Where L = the Sun’s luminosity
Example: The Sun
d = 1 AU = 1.51011 m
b = 1370 W/m2 (Solar Constant)
L = 4p d2 b = 1.256 101 2.251022 m2 1.37103 W/m2
L = 3.871026 W
Today, we will learn
17-2 How we measure a star’s brightness and
calculate its luminosity knowing the distance.
17-3 The magnitude scale for brightness and
luminosity. Introduce the distance modulus.
Example: e Eridani
d = 3.22 pc = 3.22206,265 AU
= 6.65105 AU
b = 6.7310-13 b
L/L = (6.65105)2 6.7310-13
= 0.3
e Eri has a luminosity equal to
30% of the solar luminosity.
Luminosity
Function
As stars go, our Sun
is neither extremely
luminous nor
extremely dim.
It is somewhat more
luminous than most
nearby stars – of the
30 stars within 4 pc,
only three have a
greater luminosity.
1 L = 3.86 X 1026 W
Astronomers often use the
magnitude scale to denote
brightness.
Historically, the apparent magnitude scale
for naked eye stars, runs from 1 (brightest)
to 6 (dimmest).
 Today, the apparent magnitude scale
extends into the negative numbers for
really bright objects and into the 20s and
30s for really dim objects.
 Absolute magnitude, on the other hand is
how bright a star would look if it were 10 pc
away.

Modern Magnitude Scale
1st magnitude are 100 times brighter than 6th magnitude
2.5122.5122.5122.5122.512=(2.512)5=100
A star of magnitude 4.2 is 2.512 times brighter than
another star of magnitude 5.2
A star of magnitude 3 is dimmer than 2 by 2.512
b3 = b2/2.512
A star of magnitude 7 is brighter than a 9th mag star by
b7 = b9 (2.512)2 = 6.31
Astronomers
often use the
magnitude
scale to
denote
brightness.
Apparent Magnitude m
Two stars of magnitudes m1 and m2 have brightness ratio
b2/b1 = (2.512) m1-m2
Nova Cygni 1975 brightened by 13 magnitudes in two days:
it went from m1 = 15 to m2 = 2, so
b2/b1 = (2.512) m1-m2 = 185,000.
If a star of apparent magnitude m were to be moved
to a new location 10 times farther away, its brightness
would decrease by a factor of 100. Therefore its new
apparent magnitude would be m + 5.
Absolute Magnitude M
M is the apparent magnitude a star would have at a
distance of 10 pc. Example: the sun at 10 pc
b/ b = 1/ (2,062,650)2 = 2.3510-13 = (2.512) -26.7-M
M = +4.83  5
For any star actually 10 pc distant, m = M
Any star at 100 pc appears 100 times fainter than at 10 pc
Therefore m = M + 5
Any star at 1000 pc appears 100 times fainter than at 100 pc
Therefore m = M + 10
Distance Modulus m-M
This concept enables a distance determination if you know
the absolute magnitude. The apparent magnitude is always
determined from observation.
d = 10 pc 10(m-M)/5
m-M d (pc)
-5
0
1
2
3
4
5
10
15
1.0
10
16
25
40
63
100
1000
10000
If a star’s distance is known, its luminosity
can be determined from its brightness.

A star’s luminosity can be determined from
its apparent brightness if its distance is
known:
L = 4p d 2 b
L = 4p d2 b
L/L = (d/d)2  (b/b)
Where L = the Sun’s luminosity
If a star’s luminosity is known, its
distance can be determined from its
brightness.

A star’s distance can be determined from its
apparent brightness if its luminosity is known:
L/L = (d/d)2  (b/b)
Where L = the Sun’s luminosity
(d/d )2 = (L/L) / (b/b)
And finally (see also Box 17-2)
d

d
L / L
b / b
Today, we will learn
17-4 How a star’s color indicates its temperature
17-5 How a star’s spectrum reveals its chemical
composition
17-6 How we can determine the sizes of stars
The solar spectrum
at high dispersion
shows hundreds of
absorption lines
corresponding to
hydrogen and
many other
elements.
The spectra of stars reveal their chemical
compositions as well as surface
temperatures.


In the late 19th Century,
spectra were obtained for
hundreds of thousands of
stars.
These stellar spectra were
grouped into a classification
scheme of spectral types A
through O by a team at
Harvard.
• Today we recognize the spectral types O, B, A, F, G,
K, M, L and T, running from hottest to coolest.
The spectra of stars reveal their chemical
compositions as well as surface
temperatures.
OBAFGKMLT
 hottest to coolest
 bluish to reddish
 Further refined by attaching an integer, for
example: F0, F1, F2, F3 … F9 where F1 is

hotter than F3

An important sequence to remember:

Oh Be a Fine Girl (or Guy), Kiss Me Lovingly,
Tenderly
Cecilia Payne-Gaposchkin
& Meghnad Saha
Saha’s Equation (1920)
CPG at Harvard
In her doctoral dissertation (1925) CPG demonstrated that
the spectral sequence OBAFGKM is the result of the emission
and absorption by gas of “cosmic” composition with
decreasing temperature.
Stars come in a wide variety of sizes
Stefan-Boltzmann law relates a star’s energy
output, called LUMINOSITY, to its temperature
and size.
LUMINOSITY = 4pR2sT4
LUMINOSITY is measured in joules per square meter of a
surface per second and s = 5.67 X 10-8 W m-2 K-4


Small stars will have low luminosities unless
they are very hot.
Stars with low surface temperatures must be
very large in order to have large luminosities.
Hertzsprung-Russell (H-R) diagrams
reveal the different kinds of stars.

Main sequence stars




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Red giant stars


Stars in hydrostatic equilibrium
found on a line from the upper left
to the lower right.
Hotter is brighter
Cooler is dimmer
Red Dwarfs (on MS) & Brown
Dwarfs (not on MS): lower right
corner (small, dim, and cool)
Upper right hand corner (big,
bright, and cool)
White dwarf stars

Lower left hand corner (small,
dim, and hot)
Determining the Sizes of Stars from an HR
Diagram



Main sequence stars are
found in a band from the
upper left to the lower
right.
Giant and supergiant
stars are found in the
upper right corner.
Tiny white dwarf stars are
found in the lower left
corner of the HR diagram.
Details of a star’s spectrum reveal whether it is a
giant, a white dwarf, or a main-sequence star.
Both of these stars are spectral class B8. However, star a is a
luminous super giant and star b is a typical main-sequence star.
Notice how the hydrogen absorption lines for the more luminous
stars are narrower.
LUMINOSITY CLASS
Based on the width of spectral
lines, it is possible to tell
whether the star is a
supergiant, a giant, a main
sequence star or a white dwarf.
These define the luminosity
classes shown on the left
occupying distinct regions on
the HR diagram.
The complete spectral type of
the Sun is G2 V. The “G2” part
tells us Teff, the “V” part tells us
to which sequence or luminosity
class the star belongs.
Example: M5 III is a red giant
with Teff ~ 3500K, M=0 (or
L=100 Lsun).
HR Diagram
This template
will be used in
the upcoming
test. Please
become familiar
with it. We will
do a few
examples in
class of how to
read off the
temperature,
luminosity and
size of a star
given a full
spectral type.
HR Diagram
I expect you to
know which of the
gray sequences is
which luminosity
class. From top to
bottom:
Ia, luminous
supergiants
Ib, supergiants
III, giants
V, main sequence
Examples:
G2V The Sun
M5III
B4Ib
M5Ia
Binary star systems provide crucial
information about stellar masses.

Double star – a pair of stars located at nearly the
same position in the night sky.





Optical double stars – stars that lie along the same
line of sight, but are not close to one another.
Binary stars, or binaries – stars that are gravitationally
bound and orbit one another.
Visual binary – binaries that can be resolved
Spectroscopic binary – binaries that can only be
detected by seeing two sets of lines in their
spectra
Eclipsing binary – binaries that cross one in front
of the other.
Binary Star Krüger 60
(upper left hand corner)
About half of the stars visible in the night sky are part of
multiple-star systems.
Mizar A, z1 UMa or Zeta-one Ursae Majoris, a~0.01”, P = 20.5 d
Courtesy: Navy Prototype Optical Interferometer
http://leo.astronomy.cz/mizar/article.htm
Spectroscopy makes it possible to
study binary systems in which the two
stars are close together.
k Ari
Light curves of eclipsing binaries provide
detailed information about the two stars:
Sizes, effective temperatures, shapes, etc
Light curves of eclipsing binaries provide
detailed information about the two stars.
Light curves of eclipsing binaries provide
detailed information about the two stars.
Light curves of eclipsing binaries provide
detailed information about the two stars.
NN Ser
ESO
Discussion of
Problem 66
Key Ideas



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Measuring Distances to Nearby Stars: Distances to
the nearer stars can be determined by parallax, the
apparent shift of a star against the background stars
observed as the Earth moves along its orbit.
Parallax measurements made from orbit, above the
blurring effects of the atmosphere, are much more
accurate than those made with Earth-based telescopes.
Stellar parallaxes can only be measured for stars within
a few hundred parsecs.
The Inverse-Square Law: A star’s luminosity (total light
output), apparent brightness, and distance from the
Earth are related by the inverse-square law. If any two of
these quantities are known, the third can be calculated.
Key Ideas



The Population of Stars: Stars of relatively low
luminosity are more common than more luminous stars.
Our own Sun is a rather average star of intermediate
luminosity.
The Magnitude Scale: The apparent magnitude scale is
an alternative way to measure a star’s apparent
brightness.
The absolute magnitude of a star is the apparent
magnitude it would have if viewed from a distance of 10
parsecs. A version of the inverse-square law relates a
star’s absolute magnitude, apparent magnitude, and
distance.
Key Ideas



Photometry and Color Ratios: Photometry measures the
apparent brightness of a star. The color ratios of a star are
the ratios of brightness values obtained through different
standard filters, such as the U, B, and V filters. These ratios
are a measure of the star’s surface temperature.
Spectral Types: Stars are classified into spectral types
(subdivisions of the spectral classes O, B, A, F, G, K, and
M), based on the major patterns of spectral lines in their
spectra. The spectral class and type of a star is directly
related to its surface temperature: O stars are the hottest
and M stars are the coolest.
Most brown dwarfs are in even cooler spectral classes called
L and T. Unlike true stars, brown dwarfs are too small to
sustain thermonuclear fusion.
Key Ideas


Hertzsprung-Russell Diagram: The HertzsprungRussell (H-R) diagram is a graph plotting the absolute
magnitudes of stars against their spectral types—or,
equivalently, their luminosities against surface
temperatures.
The positions on the H-R diagram of most stars are
along the main sequence, a band that extends from high
luminosity and high surface temperature to low
luminosity and low surface temperature.
Key Ideas


On the H-R diagram, giant and supergiant stars lie
above the main sequence, while white dwarfs are below
the main sequence.
By carefully examining a star’s spectral lines,
astronomers can determine whether that star is a mainsequence star, giant, supergiant, or white dwarf. Using
the H-R diagram and the inverse square law, the star’s
luminosity and distance can be found without measuring
its stellar parallax.
Key Ideas
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Binary Stars: Binary stars, in which two stars are held in
orbit around each other by their mutual gravitational
attraction, are surprisingly common. Those that can be
resolved into two distinct star images by an Earth-based
telescope are called visual binaries.
Each of the two stars in a binary system moves in an
elliptical orbit about the center of mass of the system.
Binary stars are important because they allow
astronomers to determine the masses of the two stars in
a binary system. The masses can be computed from
measurements of the orbital period and orbital
dimensions of the system.
Key Ideas


Mass-Luminosity Relation for Main-Sequence Stars:
Main-sequence stars are stars like the Sun but with
different masses.
The mass-luminosity relation expresses a direct
correlation between mass and luminosity for mainsequence stars. The greater the mass of a mainsequence star, the greater its luminosity (and also the
greater its radius and surface temperature).
Key Ideas
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
Spectroscopic Observations of Binary Stars: Some
binaries can be detected and analyzed, even though the
system may be so distant or the two stars so close
together that the two star images cannot be resolved.
A spectrum binary appears to be a single star but has a
spectrum with the absorption lines for two distinctly
different spectral types.
Key Ideas


A spectroscopic binary has spectral lines that shift back
and forth in wavelength. This is caused by the Doppler
effect, as the orbits of the stars carry them first toward
then away from the Earth.
An eclipsing binary is a system whose orbits are viewed
nearly edge-on from the Earth, so that one star
periodically eclipses the other. Detailed information
about the stars in an eclipsing binary can be obtained
from a study of the binary’s radial velocity curve and its
light curve.
Today we will learn
17-7 How H-R diagrams summarize our knowledge of
the stars
17-8 How we can deduce a star’s size from its spectrum
17-9 How we can use binary stars to measure the
masses of stars
17-10 How we can learn about binary stars in very close
orbits
17-11 What eclipsing binaries are and what they tell us
about the sizes of stars