The Family of Stars

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The Family of Stars
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• Chapter 9

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Guidepost
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• If you want to study anything scientifically, the
  first thing you have to do is find a way to
  measure it. But measurement in astronomy is very
  difficult. Astronomers must devise ingenious
  methods to find the most basic properties of
  stars. As you will see in this chapter, combining
  those basic properties reveals important
  relationships among the family of stars. Your
  study of stars will reveal answers to five basic
  questions
• How far away are the stars?
• How much energy do stars make?
• How big are stars?
• What is the typical star like?

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Guidepost (continued)
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• Making measurements is the heart of science, and
  this chapter will answer two important questions
  about how scientists go about their work
• How can scientists measure properties that cant
  be directly observed?
• How do scientists accumulate and use data?
• 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 and how it works,
  you must understand the stars.

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Outline
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I. Measuring the Distances to Stars A. The
Surveyor's Method B. The Astronomer's Method C.
Proper Motion (??) II. Intrinsic Brightness 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
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Outline
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D. Interferometric Observations of Diameter E.
Luminosity Classification F. Luminosity
Classes G. 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 Survey
of the Stars A. Mass, Luminosity, and
Density B. Surveying the Stars
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The Properties of Stars
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We already know how to determine a stars

• 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.
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Distances to Stars
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d in parsec (pc) p in arc seconds (arcsec)
__
1
d
p
Trigonometric Parallax (????)
Star appears slightly shifted from different
positions of the Earth on its orbit
1 pc 3.26 LY
The farther away the star is (larger d), the
smaller the parallax angle p.
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The Trigonometric Parallax
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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 gt d 50 pc
This method does not work for stars farther away
than 50 pc.
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Proper Motion
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In addition to the periodic back-and-forth 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.
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Intrinsic Brightness/ Absolute Magnitude
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The more distant a light source is, the fainter
it appears.
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Intrinsic Brightness / Absolute Magnitude (2)
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More quantitatively The flux received from the
light is proportional to its intrinsic brightness
or luminosity (L, unit erg/s or J/s) and
inversely proportional to the square of the
distance (d)
L
__
F
d2
Star A
Star B
Earth
Both stars may appear equally bright, although
star A is intrinsically much brighter than star B.
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Distance and Intrinsic Brightness
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Example
Recall that
Betelgeuse
Magn. Diff. Intensity Ratio
1 2.512
2 2.5122.512 (2.512)2 6.31

5 (2.512)5 100
App. Magn. mV 0.41
Rigel
For a magnitude difference of 0.41 0.14 0.27,
we find an intensity ratio of (2.512)0.27 1.28
App. Magn. mV 0.14
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Distance and Intrinsic Brightness (2)
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Rigel is appears 1.28 times brighter than
Betelgeuse,
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.
Rigel
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Absolute Magnitude
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To characterize a stars intrinsic brightness,
define Absolute Magnitude (MV)
Absolute Magnitude Magnitude that a star would
have if it were at a distance of 10 pc.
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Absolute Magnitude (2)
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Back to our example of Betelgeuse and Rigel
Betelgeuse
Betelgeuse Rigel
mV 0.41 0.14
MV -5.5 -6.8
d 152 pc 244 pc
Rigel
Difference in absolute magnitudes 6.8 5.5
1.3 gt Luminosity ratio (2.512)1.3 3.3
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The Distance Modulus (????)
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If we know a stars absolute magnitude, we can
infer its distance by comparing absolute and
apparent magnitudes
Distance Modulus mV MV -5 5 log10(d pc)
Distance in units of parsec
Equivalent d 10(mV MV 5)/5 pc
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The Size (Radius) of a Star
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We already know flux increases with surface
temperature ( T4) hotter stars are brighter.
But brightness also increases with size
Star B will be brighter than star A.
A
B
Absolute brightness is proportional to radius
squared, L R2
Quantitatively L 4 p R2 s T4
Surface flux due to a blackbody spectrum
Surface area of the star
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Example Star Radii
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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 suns.
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Organizing the Family of Stars The
Hertzsprung-Russell Diagram
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We know Stars have different temperatures,
different luminosities, and different sizes.
To bring some order into that zoo of different
types of stars organize them in a diagram of
Luminosity
Temperature (or spectral type)
versus
Hertzsprung-Russell Diagram (???)
Absolute mag.
Luminosity
or
Temperature
Spectral type O B A F G K M
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The Hertzsprung-Russell Diagram
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Most stars are found along the Main Sequence
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The Hertzsprung-Russell Diagram (2)
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Same temperature, but much brighter than MS stars
Stars spend most of their active life time on the
Main Sequence (MS).
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The Brightest Stars
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The open star cluster M39
The brightest stars are either blue (gt unusually
hot) or red (gt unusually cold).
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The Radii of Stars in the Hertzsprung-Russell
Diagram
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Betelgeuse
Rigel
10,000 times the suns radius
Polaris
100 times the suns radius
Sun
As large as the sun
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The Relative Sizes of Stars in the HR Diagram
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Eyeball Balloon Sun Supergiants
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Luminosity Classes
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Ia Bright Supergiants
Ia
Ib
Ib Supergiants
II
II Bright Giants
III
III Giants
IV Subgiants
IV
V
V Main-Sequence Stars
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Example Luminosity Classes
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• Our Sun G2 star on the Main Sequence
• G2V
• Polaris G2 star with Supergiant luminosity G2Ib

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Spectral Lines of Giants
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Pressure and density in the atmospheres of giants
are lower than in main sequence stars.
gt Absorption lines in spectra of giants and
supergiants are narrower than in main sequence
stars
gt From the line widths, we can estimate the size
and luminosity of a star.
? Distance estimate (spectroscopic parallax)
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Binary Stars
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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.
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The Center of Mass
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center of mass balance point of the system
Both masses equal gt center of mass is in the
middle, rA rB
The more unequal the masses are, the more it
shifts toward the more massive star.
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Estimating Stellar Masses
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Recall Keplers 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
aAU3
____
MA MB
Py2
(MA and MB in units of solar masses)
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Examples Estimating Mass
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a) Binary system with period of P 32 years and
separation of a 16 AU
163
____
MA MB 4 solar masses
322
b) Any binary system with a combination of period
P and separation a that obeys Keplers 3. Law
must have a total mass of 1 solar mass
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Visual Binaries
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The ideal case
Both stars can be seen directly, and their
separation and relative motion can be followed
directly.
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Spectroscopic Binaries
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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
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Spectroscopic Binaries (2)
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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
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Spectroscopic Binaries (3)
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Typical sequence of spectra from a spectroscopic
binary system
Time
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Eclipsing Binaries
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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!
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Eclipsing Binaries (2)
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Peculiar double-dip light curve
Example VW Cephei
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Eclipsing Binaries (3)
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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.
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The Light Curve of Algol
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Masses of Stars in the Hertzsprung-Russell Diagram
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The higher a stars mass, the brighter it is
L M3.5
High masses
High-mass stars have much shorter lives than
low-mass stars
Mass
tlife M-2.5
Low masses
Sun 10 billion yr.
10 Msun 30 million yr.
0.1 Msun 3 trillion yr.
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Surveys of Stars
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Ideal situation for creating a census of the
stars
Determine properties of all stars within a
certain volume
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Surveys of Stars
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Main Problem for creating such a survey
Fainter stars are hard to observe we might be
biased towards the more luminous stars.
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A Census of the Stars
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Faint, red dwarfs (low mass) are the most common
stars.
Bright, hot, blue main-sequence stars (high-mass)
are very rare.
Giants and supergiants are extremely rare.