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Chapter 9
The Family of Stars
Guidepost
Science is based on measurement, but measurement
in astronomy is very difficult. To discover the properties
of stars, astronomers must use their telescopes and
spectrographs in ingenious ways to learn the secrets
hidden in starlight. The result is a family portrait of the
stars.
Here you will find answers to five essential questions
about stars:
• How far away are the stars?
• How much energy do stars make?
• How big are stars?
• How much mass do stars contain?
• What is the typical star like?
Guidepost (continued)
With this chapter you leave our sun behind and begin
your study of the billions of stars that dot the sky. In a
sense, the star is the basic building block of the
universe. If you hope to understand what the universe
is, what our sun is, what our Earth is, and what we are,
you must understand the stars.
Once you know how to find the basic properties of
stars, you will be ready to trace the history of the stars
from birth to death, a story that begins in the next
chapter.
Outline
I. Measuring the Distances to Stars
A. The Surveyor's Method
B. The Astronomer's Triangulation Method
C. Proper Motion
II. Apparent Brightness, Intrinsic Brightness, and
Luminosity
A. Brightness and Distance
B. Absolute Visual Magnitude
C. Calculating Absolute Visual Magnitude
D. Luminosity
III. The Diameters of Stars
A. Luminosity, Radius, and Temperature
B. The H-R Diagram
C. Giants, Supergiants, and Dwarfs
Outline
D. Interferometric Observations of Diameter
E. Luminosity Classification
F. Spectroscopic Parallax
IV. The Masses of Stars
A. Binary Stars in General
B. Calculating the Masses of Binary Stars
C. Visual Binary Systems
D. Spectroscopic Binary Systems
E. Eclipsing Binary Systems
V. A Census of the Stars
A. Surveying the Stars
B. Mass, Luminosity, and Density
The Properties of Stars
We already know how to determine a star’s
• surface temperature
• chemical composition
• surface density
In this chapter, we will learn how we can
determine its
• distance
• luminosity
• radius
• mass
and how all the different types of stars
make up the big family of stars.
Distances to Stars
d in parsec (pc)
p in arc seconds
1
d = __
p
Trigonometric Parallax:
Star appears slightly shifted from different
positions of the Earth on its orbit
The farther away the star is (larger d),
the smaller the parallax angle p.
1 pc = 3.26 LY
The Trigonometric Parallax
Example:
Nearest star, a Centauri, has a parallax of p = 0.76 arc seconds
d = 1/p = 1.3 pc = 4.3 LY
With ground-based telescopes, we can measure
parallaxes p ≥ 0.02 arc sec
=> d ≤ 50 pc
This method does not work for stars
farther away than 50 pc.
Proper Motion
In addition to the
periodic back-andforth motion related to
the trigonometric
parallax, nearby stars
also show continuous
motions across the
sky.
These are related to
the actual motion of
the stars throughout
the Milky Way, and
are called proper
motion.
Intrinsic Brightness/
Absolute Magnitude
The more distant a light source is,
the fainter it appears.
Intrinsic Brightness /
Absolute Magnitude (2)
More quantitatively:
The flux received from the light is proportional to its
intrinsic brightness or luminosity (L) and inversely
proportional to the square of the distance (d):
L
__
F~ 2
d
Star A
Star B
Both stars may appear equally bright, although
star A is intrinsically much brighter than star B.
Earth
Distance and Intrinsic Brightness
Example:
Recall that:
Magn.
Diff.
Intensity Ratio
1
2.512
2
2.512*2.512 = (2.512)2
= 6.31
…
…
5
(2.512)5 = 100
For a magnitude difference of 0.41
– 0.14 = 0.27, we find an intensity
ratio of (2.512)0.27 = 1.28
Betelgeuse
App. Magn. mV = 0.41
Rigel
App. Magn. mV = 0.14
Distance and Intrinsic Brightness (2)
Rigel is appears 1.28 times
brighter than Betelgeuse,
but Rigel is 1.6 times further
away than Betelgeuse.
Thus, Rigel is actually
(intrinsically) 1.28*(1.6)2 =
3.3 times brighter than
Betelgeuse.
Betelgeuse
Rigel
Absolute Magnitude
To characterize a star’s intrinsic
brightness, define Absolute
Magnitude (MV):
Absolute Magnitude
= Magnitude that a star would have if it
were at a distance of 10 pc.
Absolute Magnitude (2)
Back to our example of
Betelgeuse and Rigel:
Betelgeuse Rigel
mV
0.41
0.14
MV
-5.5
-6.8
d
152 pc
244 pc
Betelgeuse
Rigel
Difference in absolute magnitudes:
6.8 – 5.5 = 1.3
=> Luminosity ratio = (2.512)1.3 = 3.3
The Distance Modulus
If we know a star’s absolute magnitude, we can
infer its distance by comparing absolute and
apparent magnitudes:
Distance Modulus
= mV – M V
= -5 + 5 log10(d [pc])
Distance in units of parsec
Equivalent:
d = 10(mV – MV + 5)/5 pc
The Size (Radius) of a Star
We already know: flux increases with surface
temperature (~ T4); hotter stars are brighter.
But brightness also increases with size:
A
Star B will be
brighter than
star A.
B
Absolute brightness is proportional to radius squared, L ~ R2
Quantitatively:
L = 4 p R2 s T4
Surface area of the star
Surface flux due to a
blackbody spectrum
Example: Star Radii
Polaris has just about the same spectral
type (and thus surface temperature) as our
sun, but it is 10,000 times brighter than our
sun.
Thus, Polaris is 100 times larger than the sun.
This causes its luminosity to be 1002 = 10,000
times more than our sun’s.
Organizing the Family of Stars:
The Hertzsprung-Russell Diagram
We know:
Stars have different temperatures,
different luminosities, and different sizes.
Absolute mag.
or
Luminosity
To bring some order into that zoo of different
types of stars: organize them in a diagram of
Luminosity
versus
Temperature (or spectral type)
Hertzsprung-Russell Diagram
Spectral type: O
Temperature
B
A
F
G
K
M
The Hertzsprung-Russell Diagram
The Hertzsprung-Russell Diagram (2)
Same
temperature,
but much
brighter than
Main
Sequence
stars
The Brightest Stars
The open star cluster M39
The brightest stars are either blue (=> unusually hot)
or red (=> unusually cold).
The Radii of Stars in the
Hertzsprung-Russell Diagram
Betelgeuse
Rigel
Polaris
Sun
The Relative Sizes of Stars in
the HR Diagram
Luminosity Classes
Ia
Ia Bright Supergiants
Ib
Ib Supergiants
II
III
IV
II Bright Giants
III Giants
IV Subgiants
V
V Main-Sequence
Stars
Example: Luminosity Classes
• Our Sun: G2 star on the
Main Sequence:
G2V
• Polaris: G2 star with
Supergiant luminosity:
G2Ib
Spectral Lines of Giants
Pressure and density in the atmospheres of giants
are lower than in main sequence stars.
=> Absorption lines in spectra of giants and
supergiants are narrower than in main sequence stars
=> From the line widths, we can estimate the size and
luminosity of a star.
 Distance
estimate (spectroscopic parallax)
Binary Stars
More than 50 % of all
stars in our Milky Way
are not single stars, but
belong to binaries:
Pairs or multiple
systems of stars which
orbit their common
center of mass.
If we can measure and
understand their orbital
motion, we can
estimate the stellar
masses.
The Center of Mass
center of mass =
balance point of the
system
Both masses equal
=> center of mass is
in the middle, rA = rB
The more unequal the
masses are, the more
it shifts toward the
more massive star.
Estimating Stellar Masses
Recall Kepler’s 3rd Law:
Py2 = aAU3
Valid for the Solar system: star with 1 solar
mass in the center
We find almost the same law for binary
stars with masses MA and MB different
from 1 solar mass:
3
a
____
AU
MA + MB =
Py2
(MA and MB in units of solar masses)
Examples: Estimating Mass
a) Binary system with period of P = 32 years
and separation of a = 16 AU:
163
____
MA + MB =
= 4 solar masses
2
32
b) Any binary system with a combination of
period P and separation a that obeys Kepler’s
3. Law must have a total mass of 1 solar mass
Visual Binaries
The ideal case:
Both stars can be
seen directly, and
their separation and
relative motion can
be followed directly.
Spectroscopic Binaries
Usually, binary separation a
can not be measured directly
because the stars are too
close to each other.
A limit on the separation
and thus the masses can
be inferred in the most
common case:
Spectroscopic
Binaries
Spectroscopic Binaries (2)
The approaching star produces
blue shifted lines; the receding
star produces red shifted lines
in the spectrum.
Doppler shift  Measurement
of radial velocities
 Estimate
of separation a
 Estimate
of masses
Spectroscopic Binaries (3)
Typical sequence of spectra from a
spectroscopic binary system
Time
Eclipsing Binaries
Usually, the inclination
angle of binary systems is
unknown  uncertainty in
mass estimates
Special case:
Eclipsing Binaries
Here, we know that
we are looking at the
system edge-on!
Eclipsing Binaries (2)
Peculiar “double-dip” light curve
Example: VW Cephei
Eclipsing Binaries (3)
Example:
Algol in the
constellation
of Perseus
From the light
curve of Algol, we
can infer that the
system contains
two stars of very
different surface
temperature,
orbiting in a
slightly inclined
plane.
The Light Curve of Algol
Masses of Stars in the HertzsprungRussell Diagram
The more massive a star is,
the brighter it is:
L ~ M3.5
High-mass stars have
much shorter lives than
low-mass stars:
tlife ~ M-2.5
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.
Surveys of Stars
Ideal situation
for creating a
census of the
stars:
Determine
properties of all
stars within a
certain volume
Surveys of Stars
Main Problem for creating such a survey:
Fainter stars are hard to observe; we might be biased
towards the more luminous stars.
A Census of the Stars
Faint, red dwarfs
(low mass) are
the most
common stars.
Bright, hot, blue
main-sequence
stars (highmass) are very
rare.
Giants and
supergiants
are extremely
rare.