The Family of Stars

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The Family of Stars
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• Chapter 8

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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
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d
p
Trigonometric Parallax
Star appears slightly shifted from different
positions of 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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Intrinsic Brightness / Absolute Visual Magnitude
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The more distant a light source is, the fainter
it appears.
The same amount of light falls onto a smaller
area at distance 1 than at distance 2 gt smaller
apparent brightness.
Area increases as square of distance gt apparent
brightness decreases as inverse of distance
squared
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Intrinsic Brightness / Absolute Visual
Magnitude(II)
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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
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
Betelgeuse
App. Magn. mV 0.41
Rigel
App. Magn. mV 0.14
For a magnitude difference of 0.41 0.14 0.27,
we find an intensity ratio of (2.512)0.27 1.28
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Distance and Intrinsic Brightness
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Rigel 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 Visual Magnitude
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• To characterize a stars intrinsic brightness,
  define absolute visual magnitude (MV)

Apparent visual magnitude that a star would have
if it were at a distance of 10 pc.
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Absolute Visual Magnitude(II)
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Back to our example of Betelgeuse and Rigel
Betelgeuse
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
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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
versus
Temperature (or spectral type)
Absolute mag.
Hertzsprung-Russell Diagram
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 (II)
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Same temperature, but much brighter than MS stars
? Must be much larger
Stars spend most of their active life time on the
Main Sequence.
? Giant Stars
Same temp., but fainter ? Dwarfs
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Radii of Stars in the Hertzsprung-Russell Diagram
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Rigel
Betelgeuse
10,000 times the suns radius
Polaris
100 times the suns radius
Sun
As large as the sun
100 times smaller than the sun
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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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Luminosity effects on the width of spectral lines
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Same spectral type, but different luminosity

• Lower gravity near the surfaces of giants
• smaller pressure
• smaller effect of pressure broadening
• narrower lines

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Examples
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• Our Sun G2 star on the main sequence G2V
• Polaris G2 star with supergiant luminosity G2Ib

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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
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a) Binary system with period of P 32 years and
separation of a 16 AU
163
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MA MB 4 solar masses.
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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 (II)
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The approaching star produces blueshifted lines
the receding star produces redshifted lines in
the spectrum.
Doppler shift ? Measurement of radial velocities
? Estimate of separation a
? Estimate of masses
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Spectroscopic Binaries (III)
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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, 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 (II)
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Peculiar double-dip light curve
Example VW Cephei
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Eclipsing Binaries (III)
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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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Masses of Stars in the Hertzsprung-Russell Diagram
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Masses in units of solar masses
The higher a stars mass, the more luminous
(brighter) it is
High masses
L M3.5
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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The Mass-Luminosity Relation
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More massive stars are more luminous.
L M3.5
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Surveys of Stars
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Ideal situation
Determine properties of all stars within a
certain volume.
Problem
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.