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Lecture 14:
Studying the stars
Astronomy 111
Monday October 17, 2016
Reminders
• Homework #7 due Monday
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Studying the stars
•
•
•
•
•
How far?
How bright?
How massive?
How big?
How hot?
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Why are distances important?
Distances are necessary for estimating:
• Total energy released by an object
(Luminosity)
• Masses of objects from orbital motions
(Kepler’s third law)
• Physical sizes of objects
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The problem of measuring
distances
Q: What do you do when an object is out of
reach of your measuring instruments?
Examples:
• Surveying & mapping
• Architecture
• Any astronomical object
A: You resort to using GEOMETRY.
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Method of trigonometric
parallaxes
June
p
December
Foreground
Star
Distant Stars
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Parallax decreases with
distance
Closer stars have larger parallaxes:
Distant stars have smaller parallaxes:
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Stellar parallaxes
All stellar parallaxes are less than 1
arcsecond
Nearest star, a Centauri, has p=0.76-arcsec
Cannot measure parallaxes with naked eye.
First parallax observed in 1837 (Bessel) for
the star 61 Cygni.
Use photography or digital imaging today.
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Parallax formula
1
d
p
p = parallax angle in arcseconds
d = distance in “Parsecs”
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Parallax Second = Parsec (pc)
Fundamental unit of distance in
astronomy
“A star with a parallax of 1 arcsecond has
a distance of 1 Parsec.”
Relation to other units:
1 parsec (pc) is equivalent to
206,265 AU
3.26 Light Years
3.085x1013 km
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Light year (ly)
Alternative unit of distance
“1 Light Year is the distance traveled by light
in one year.”
Relation to other units:
1 light year (ly) is equivalent to
0.31 pc
63,270 AU
Used mostly by journalists, Star Trek, etc.
—not generally used by astronomers!
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Examples
a Centauri has a parallax of p=0.76 arcsec:
A distant star has a parallax of p=0.02
arcsec:
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Limitations
If stars are too far away, the parallax can
be too small to measure accurately.
The smallest parallax measurable from
the ground is about 0.01 arcsec
• Measure distances out to ~100 pc
• But, only a few hundred stars are this
close to the Sun
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Hipparcos satellite
European Space Agency
Launched in 1989
Designed to measure precision parallaxes
to about ±0.001 arcseconds!
• Gets distances good out to 1000 pc
• Measured parallaxes for ~100,000
stars!
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GAIA satellite
• ESA’s new GAIA mission is way better:
• Parallax precision p ~ 10-5 arcsec for 1 billion stars
– 100x better parallaxes
– 10,000x more stars
– Stars that are ~100 times fainter than Hipparcos
The position of a billion
stars will be measured
by GAIA; this image
shows the first
preliminary data release.
From sci.esa.int/gaia
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How “bright” is an object?
• We must define “brightness”
quantitatively.
• Two ways to quantify brightness:
– Intrinsic Luminosity:
Total energy output.
– Apparent Brightness:
How bright it looks from a distance.
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Appearances can be deceiving...
• Does a star look “bright” because
– it is intrinsically very luminous?
– it is intrinsically faint but located nearby?
• To know for sure you must know:
– the distance to the star, or
– some other, distance-independent property
of the star that clues you in.
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Luminosity
• Luminosity is the total energy output
from an object.
– Measured in power units:
Energy/second emitted by the object (e.g.,
Watts)
– Independent of distance
• Important for understanding the energy
production of a star.
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Apparent brightness
• Measures how bright an object appears
to be to a distant observer.
• What we measure on earth
(“observable”)
• Measured in flux units:
Energy/second/area from the source.
• Depends on the distance to the object.
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Inverse Square Law of
Brightness
The apparent brightness of a source is
inversely proportional to the square of its
distance:
1
B  2
d
2-times Closer = 4-times Brighter
2-times Farther = 4-times Fainter
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d=1
B=1
d=2
B=1/4
d=3
B=1/9
Flux-luminosity relationship
Relates apparent brightness (Flux) and
intrinsic brightness (Luminosity) through the
Inverse Square Law of Brightness:
Luminosity
Flux =
2
4 d
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Measuring apparent brightness
• The process of measuring the apparent
brightnesses of objects is called
Photometry.
• Two ways to express apparent brightness:
– as Stellar Magnitudes
– as Absolute Fluxes (energy per second per
area)
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Magnitude system
• Traditional system dating to classical times
(Hipparchus of Rhodes, c. 300 BC)
• Rank stars according to apparent brightness
– 1st magnitude are brightest stars
– 2nd magnitude are the next brightest
– and so on...
• Faintest naked-eye stars are 6th magnitude.
• Modern version quantifies this system
– 5 steps of magnitude = factor of 100 in Flux.
• Computationally convenient, but somewhat obtuse.
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Flux photometry
• Measure the flux of photons from a star
using a light-sensitive detector:
– Photographic Plate
– Photoelectric Photometer (photomultiplier
tube)
– Solid State Detector (e.g., CCD)
• Calibrate the detector by observing a set
of “Standard Stars” of known brightness.
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Measuring luminosity
• In principle you just combine
– the brightness (flux) measured via
photometry
– and the distance to the star
• using the inverse-square law.
• The biggest problem is finding the
distance.
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Measuring masses of stars:
Binary stars
• Apparent Binaries
– Chance projection of two distinct stars along
the line of sight.
– Often at very different distances.
• True Binary Stars:
– A pair of stars bound by gravity.
– Orbit each other about their center of mass.
– Between 20% and 80% of all stars are
binaries.
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Types of binaries
• Visual Binary:
Can see both stars & follow their orbits over
time.
• Spectroscopic Binary:
Cannot resolve the two stars, but can see their
orbit motions as Doppler shifts in their spectra.
• Eclipsing Binary:
Cannot resolve the two stars, but can see the
total brightness drop when they periodically
eclipse each other.
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Visual Binary
1890
1940
1990
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Center of mass
• Two stars orbit about their center of
mass:
a
a
1
2
M2
a
M1
• Measure semi-major axis, a, from projected orbit
and the distance.
• Relative positions give: M1 / M2 = a2 / a1
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Measuring masses
Newton’s Form of Kepler’s Third Law:
P
2
4 a
=
G(M1 + M 2)
2 3
• Measure period, P, by following the
orbit.
• Measure semi-major axis, a, and mass
ratio (M1/M2) from projected orbit.
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Problems
• We need to follow the orbits long
enough to trace them out in detail.
– This can take decades.
– Need to work out the projection on the sky.
• Everything depends critically on the
distance:
– semi-major axis depends on d
– derived mass depends on d3 !!
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Spectroscopic binaries
• Most binaries are too far away to see
both stars separately.
• But, you can detect their orbital motions
by the periodic Doppler shifts of their
spectral lines.
– Determine the orbital period & size from
velocities.
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Spectroscopic binaries
B
A
B
B
A
A
A
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B
Problems
• Cannot see the two stars separately:
– Semi-major axis must be guessed from
orbit
– Can’t tell how the orbit is tilted on the sky
• Everything depends critically on
knowing the distance.
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Eclipsing binaries
• Two stars orbiting nearly edge-on.
– See a periodic drop in brightness as one
star eclipses the other.
– Combine with spectra which measure
orbital speeds.
• With the best data, one can find the
masses without having to know the
distance!
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Eclipsing binaries
4
Brightness
3
1
2
1
3
2
Time
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4
Problems
• Eclipsing Binaries are very rare
– Orbital plane must line up just right
• Measurement of the eclipse light curves
complicated by details:
– Partial eclipses yield less accurate
numbers.
– Atmospheres of the stars soften edges.
– Close binaries can be tidally distorted.
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Stellar masses
• Masses are known for only ~200 stars.
– Range: ~0.1 to 50 Solar masses
• Stellar masses can only be measured
for binary stars.
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Stellar radii
• Very difficult to measure because stars are
so far away.
• Methods:
– Eclipsing binaries (need distance)
– Interferometry (single stars)
– Lunar occultation (single stars)
• Radii are only measured for about 500
stars
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Colors of stars
• Stars are made of hot, dense gas
– Continuous spectrum from the lowest
visible layers (“photosphere”).
– Approximates a blackbody spectrum.
• From Wien’s Law, we expect:
– hotter stars appear BLUE (T=10,00050,000 K)
– middle stars appear YELLOW (T~6000K)
– cool stars appear RED (T~3000K)
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Spectra of stars
• Hot, dense lower photosphere of a star
is surrounded by thinner (but still fairly
hot) atmosphere.
– Produces an Absorption Line spectrum.
– Lines come from the elements in the stellar
atmosphere.
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Spectral classification of stars
• Astronomers noticed that stellar spectra
showed many similarities.
• Can stars be classified or grouped
according to similarities in their spectra?
• Draper Survey at Harvard (1886-1897):
– Objective prism photography
– Obtained spectra of >100,000 stars
– Hired women as “computers” to analyze
spectra
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Harvard “Computers” (c. 1900)
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Objective prism spectra
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Harvard classification
Edward
Pickering
• Edward Pickering’s first attempt
at a systematic spectral classification:
– Sort by Hydrogen absorption-line strength
– Spectral Type “A” = strongest Hydrogen
lines
– followed by types B, C, D, etc. (weaker)
• Problem:
Other lines followed no discernible patterns.
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Annie Jump Cannon
• Leader of Pickering’s “computers”, she
noticed subtle patterns among metal
lines.
• Re-arranged Pickering’s ABC spectral
types, throwing out most as redundant.
• Left 7 primary and 3 secondary classes:
• O B A F G K M (R N S)
• Unifying factor: Temperature
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Annie Jump Cannon
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The spectral sequence
O
B
A
F
G
K
MLT
Hotter
50,000K
Cooler
2000K
Bluer
Redder
Spectral sequence is a Temperature sequence
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Spectral types
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Stellar spectra in
order from the hottest
(top) to coolest
(bottom).
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The spectral sequence is a
Temperature sequence
• Gross differences among the spectral
types are due to differences in
Temperature.
• Composition differences are minor at best.
– Demonstrated by Cecilia Payne-Gaposhkin in
1920’s
• Why?
What lines you see depends on the state of
excitation and ionization of the gas.
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Example: Hydrogen Lines
• Visible Hydrogen absorption lines come
from the second excited state.
• B Stars (15-30,000 K):
Most of H is ionized, so only very weak H lines.
• A Stars (10,000 K):
Ideal excitation conditions, strongest H lines.
• G Stars (6000 K):
Too cool, little excited H, so only weak H lines.
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O Stars
• Hottest Stars: T>30,000 K
• Strong lines of He+
• No lines of H
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B Stars
• T=15,000 - 30,000 K
• Strong lines of He
• Very weak lines of H
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A Stars
• T = 10,000 - 7500 K
• Strong lines of H
• Weak lines of Ca+
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F Stars
•
•
•
•
T = 7500 - 6000 K
weaker lines of H
Ca+ lines growing stronger
first weak metal lines appear
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G Stars
• T = 6000 - 5000 K
• Strong lines of Ca+, Fe+, & other
metals
• much weaker H lines
• The Sun is a G-type Star
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K Stars
•
•
•
•
Cool Stars: T = 5000 - 3500 K
Strongest metal lines
H lines practically gone
first weak CH & CN molecular bands
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M Stars
• Very cool stars: T  2000-3500 K
• Strong molecular bands (especially
TiO)
• No lines of H
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L & T Stars
• Coolest stars: T < 2000 K
• Discovered in 1999
• Strong molecular bands
• Metal-hydride (CrH & FeH)
• Methane (CH4) in T stars
• Probably not stars at all
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Modern synthesis:
The M-K System
• An understanding of atomic physics and
better techniques permit finer distinctions.
• Morgan-Keenan (M-K) Classification
System:
Start with Harvard classes:
• O B A F G K M L T
Subdivide each class into numbered
subclasses:
• A0 A1 A2 A3 ... A9
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Examples
• The Sun:
G2 star
• Other bright stars:
Betelgeuse: M2 star (Orion)
Rigel: B8 star (Orion)
Sirius: A1 star (Canis Major)
Aldebaran: K5 star (Taurus)
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Summary of stellar properties
• Large range of Stellar Luminosities:
– 10-4 to 106 Lsun
• Large range of Stellar Radii:
– 10-2 to 103 Rsun
• Modest range of Stellar Temperatures:
– 3000 to >50,000 K
• Wide Range of Stellar Masses:
– 0.1 to ~50 Msun
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Luminosity-Radius-Temperature
Relation
• Stars are approximately black bodies.
Stefan-Boltzmann Law:
energy/sec/area = sT4
The area of a spherical star:
area = 4R2
• Predicted Stellar Luminosity
(energy/sec):
L = 4R2 sT4
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Example 1:
2 stars are the same size, (RA=RB), but star A
is 2 hotter than star B (TA=2TB):
LA (2TB )
LA 4R sT



4
L B 4R sT
LB
TB
LA
4
 2  LA  16  LB
LB
2
A
2
B
4
A
4
B
4
Therefore: star A is 16 brighter than star B
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Example 2:
2 stars are the same temperature, (TA=TB), but
star A is 2 bigger than star B (RA=2RB):
LA (2R B )
LA 4R sT



2
L B 4R sT
LB
RB
LA
2
 2  LA  4  L B
LB
2
A
2
B
4
A
4
B
2
Therefore: star A is 4 brighter than star B
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Hertzsprung-Russell Diagram
• Plot of Luminosity versus Temperature:
– estimate T from Spectral Type
– estimate L from apparent brightness &
distance
• Done independently by:
– Eljnar Hertzsprung (1911) for star clusters
– Henry Norris Russell (1913) for nearby
stars
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Eljnar Hertzsprung
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Henry Norris Russell
H-R Diagram
Luminosity (Lsun)
106
Supergiants
104
102
Giants
1
10-2
10-4
40,000
White Dwarfs
20,000
10,000
5,000
Temperature (K)
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2,500
Main Sequence
• Most nearby stars (85%), including the
Sun, lie along a diagonal band called
the
Main Sequence
• Ranges of properties:
– L=10-2 to 106 Lsun
– T=3000 to >50,0000 K
– R=0.1 to 10 Rsun
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Giants & Supergiants
• Two bands of stars brighter than Main
Sequence stars of the same
Temperature.
– Means they must be larger in radius.
• Giants
R=10 -100 Rsun L=103 - 105 Lsun T<5000 K
• Supergiants
R>103 Rsun L=105 - 106 Lsun T=3000 50,000 K
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White Dwarfs
• Stars on the lower left of the H-R
Diagram are fainter than Main
Sequence stars of the same
Temperature.
– Means they must be smaller in radius.
– L-R-T Relation predicts:
R ~ 0.01 Rsun (~ size of Earth!)
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Hipparcos
H-R Diagram
4902 single stars
with distance
errors of <5%
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Luminosity classification
• Absorption lines are Pressure-sensitive:
– Lines get broader as the pressure increases.
– Larger stars are puffier, which means lower
pressure, so that
• Larger Stars have Narrower Lines.
• This gives us a way to assign a
Luminosity Class to a star based solely on
its spectrum!
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Luminosity Effects in Spectra
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Luminosity Classes:
Ia = Bright Supergiants
Ib = Supergiants
II = Bright Giants
III = Giants
IV = Subgiants
V = Dwarfs = Main-Sequence
Stars
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Spectral + Luminosity Classification
of Stars:
• Sun:
G2v (G2 Main-Sequence star)
• Winter Sky:
Betelgeuse: M2 Ib (M2 Supergiant star)
Rigel: B8 Ia (B8 Bright Supergiant star)
Sirius: A1v (A1 Main-Sequence star)
Aldebaran: K5 III (K5 Giant star)
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From Stellar Properties to Stellar
Structure
• Any theory of stellar structure must
explain the observed properties of stars.
• Seek clues in correlations among the
observed properties, in particular:
– Mass
– Luminosity
– Radius
– Temperature
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H-R Diagram
Luminosity (Lsun)
106
Supergiants
104
102
Giants
1
10 -2
10 -4
40,000
White Dwarfs
20,000
10,000
5,000
Temperature (K)
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2,500
• Main Sequence:
– Strong correlation between Luminosity and
Temperature.
– Holds for 85% of nearby stars including the
sun
• All other stars differ in size:
– Giants & Supergiants:
Very large radius, but same masses as M-S stars
– White Dwarfs:
Very compact stars: ~Rearth but with ~Msun!
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Mass-Luminosity Relationship
• For Main-Sequence stars:
 L   M 

  

 Lsun   M sun 
3.5
In words:
“More massive M-S stars are more luminous.”
Not true of Giants, Supergiants, or White Dwarfs.
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Luminosity (Lsun)
104
102
LM3.5
1
10-2
0.01
0.1
1
10
Mass (Msun)
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100
Stellar Density
• Density = Mass  Volume
• Main Sequence: small range of density
– Sun: ~1.6 g/cc
– O5v Star: ~0.005 g/cc
– M0v Star: ~5 g/cc
• Giants: 10 -7 g/cc
• White Dwarfs: 105 g/cc
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Interpreting the Observations:
• Main-Sequence Stars:
– Strong L-T Relationship on H-R Diagram
– Strong M-L Relationship
Implies they have similar internal structures
& governing laws.
• Giants & White Dwarfs:
– Must have very different internal structures
than Main-Sequence stars of similar mass.
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Summary
• Distance is important but hard to measure
• Trigonometric parallaxes
– direct geometric method
– only good for the nearest stars (~500pc)
• Units of distance in Astronomy:
– Parsec (Parallax second)
– Light Year
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Summary
• Luminosity of a star:
– total energy output
– independent of distance
• Apparent brightness of a star:
– depends on the distance by the inversesquare law of brightness.
– measured quantity from photometry.
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Summary
• Types of binary stars
– Visual
– Spectroscopic
– Eclipsing
• Only way to measure stellar masses:
– Only ~150 stars
• Radii are measured for very few stars.
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Summary
• Color of a star depends on its
Temperature
– Red Stars are Cooler
– Blue Stars are Hotter
• Spectral Classification
– Classify stars by their spectral lines
– Spectral differences mostly due to
Temperature
• Spectral Sequence (Temperature
Sequence)
• O B A F G K M L T
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Summary
• The Hertzsprung-Russell (H-R) Diagram
– Plot of Luminosity vs. Temperature for stars.
• Features:
– Main Sequence (most stars)
– Giant & Supergiant Branches
– White Dwarfs
• Luminosity Classification
• Mass-Luminosity Relationship
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Summary
• Observational Clues to Stellar Structure:
– H-R Diagram
– Mass-Luminosity Relationship
– The Main Sequence is a sequence of Mass
• Equation of State for Stellar Interiors
– Perfect Gas Law
– Pressure = density  temperature
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Questions
• What makes it necessary to launch
satellites into space to measure very
precise parallax?
• Would it be easier to measure parallax
from Jupiter? From Venus?
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Questions
• How much does the apparent brightness
of stars we see in the sky vary? Why?
• Stars have different colors? So is the
amount of light at different wavelengths
the same?
• Can we tell the difference between a very
luminous star that is far away and an
intrinsically low luminosity star that is
nearby?
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Questions
• What star do we know the mass of very
precisely?
• Why is it so unlikely that binaries are in
eclipsing systems?
• Most binaries are seen as
spectroscopic. Why?
• How can we know the sizes of more
stars than masses?
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Questions
• What does the temperature of a star
mean?
• Are there stars with temperatures higher
than 50000K?
• Are hotter stars brighter than cooler
stars? Are they more luminous?
• Why did it take so long to find L & T
stars?
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Questions
• Why don’t stars have just any
Luminosity and Temperature?
• Why is there a distinct Main Sequence?
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