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Slide 1
The Family of StarsChapter 9
Slide 2
Part 1: measuring and classifying the stars
• What we can measure directly: – Surface temperature and color– Spectrum– Apparent magnitude or intensity– Diameter of a few nearby stars– Distance to nearby stars
• What we usually cannot: – Distance to most stars– Luminosity (energy radiated per
second)– Diameter and mass
Slide 3
Surface temperature and color indices
Color indices:B-V, U-B
Color filters
Differences in apparent magnitudes observedthrough different filters
K)(
103)nm(
6
T
Slide 4
Spectral Classification of Stars
Mnemonics to remember the spectral sequence:
Oh Oh Only
Be Boy, Bad
A An Astronomers
Fine F Forget
Girl/Guy Grade Generally
Kiss Kills Known
Me Me Mnemonics
Slide 5
How to find distances to the stars?
• Parallax (only for stars within ~1500 ly)
• From stellar motions
• For moving clusters
• Using “standard candles” (model-dependent)
• Using mass-luminosity relation (for main-sequence stars) or period-luminosity relations (for binaries and variable stars; model-dependent)
Slide 6
When d is too large, angles A and B become too close to 900
The larger the baseline, the longer distances we can measure
Slide 7
Apparent shift in the position of the star: parallax effect
The longest baseline on Earth is our orbit!
Angular shift; we can measure it directly
Slide 8
Effect is very small: shift is less than 1 arcsec even for closest stars
Aristotle used the absence of observable parallax to discard heliocentric system
Larger shift
Smaller shift
Slide 9
Half of the angular shift is called parallax angle p and used to define new unit of distance
Slide 10
The parallax angle p
arcsec) in(
AU1206265
pd
Define 1 parsec as a distance to a star whose parallax is 1 arcsec
d (in parsecs) = 1/p1 pc = 206265 AU = 3.26 ly
dp
AU1206265arcsec) in(
Small-angle formula:
Slide 11
The Trigonometric Parallax
Example:
Nearest star, 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
With Hipparcos satellite: parallaxes up to 0.002 arcsec, i.e. d up to 500 pc.
118218 stars measured!
Slide 12
Proper MotionIn 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.
Slide 13
Barnard’s star: highest proper motion10 arcsec per year, or one lunar diameter per 173 yrApproaches us at 160 km/secFourth closest star
Slide 14
Brightness and distance
• Apparent magnitude: tells us how bright a star looks to our eyes
Intensity, or radiation flux received by the telescope: Energy of radiation coming through unit area of the mirror per second (J/m2/s)
Slide 15
Brightness and Distance
The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):
24 d
LI
R
d
L
Slide 16
d1
d2
2221
21 44 IdIdL 2
11 4 d
LI
2
22 4 d
LI
24 d
LI
Slide 17
Intrinsic Brightness, or luminosity
The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):
I = L__
4d2
Star AStar B Earth
Both stars may appear equally bright, although star A is intrinsically much brighter than star B.
Slide 18
Brightness and Distance
(SLIDESHOW MODE ONLY)
Slide 19
Define the magnitude scale so that two objects that differ by 5 magnitudes have an intensity ratio of 100.
100;5 B
AAB I
Imm 512.2100;1 5
B
AAB I
Imm
AB mm
B
A
I
I )512.2(
B
AAB I
ILogmm 5.2)(
Order of terms matters!
Recall the definition of apparent magnitude:
Slide 20
However, the apparent magnitude mixes up the intrinsic brightness of the star (or luminosity) and the effect of distance (which has nothing to do with the luminosity of the star).
Inverse square law: 2
1
22
2
1
2
12
;4 d
d
L
L
I
I
d
LI
Slide 21
Slide 22
Distance and Intrinsic Brightness
Betelgeuse
Rigel
Example:
App. Magn. mV = 0.41
Recall that:
Magn. Diff.
Intensity Ratio
1 2.512
2 2.512*2.512 = (2.512)2 = 6.31
… …
5 (2.512)5 = 100
App. Magn. mV = 0.14For a magnitude difference of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28
Slide 23
Distance and Intrinsic Brightness (2)
Betelgeuse
Rigel
Rigel is appears 1.28 times brighter than Betelgeuse,
Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times more luminous than Betelgeuse.
But Rigel is 1.6 times further away than Betelgeuse
Slide 24
A star that is very bright in our sky could be bright primarily because it is very close to us (the Sun, for example), or because it is rather distant but is intrinsically very bright (Rigel, for example). It is the "true" (intrinsic) brightness, with the distance dependence factored out, that is of most interest to us as astronomers.
Therefore, it is useful to establish a convention whereby we can compare two stars on the same footing, without variations in brightness due to differing distances complicating the issue.
Astronomers define the absolute magnitude M to be the apparent magnitude that a star would have if it were (in our imagination) placed at a distance of 10 parsecs (which is 32.6 light years) from the Earth.
To determine the absolute magnitude M the distance to the star must also be known!
Absolute Magnitude
Slide 25
Absolute magnitude
1
212 log5.2
I
Imm Recall that for two stars 1 and 2
Let star 1 be at a distance d pc and star 2 be the same star brought to the distance 10 pc.
Then 2
2
1
2
10
d
I
I 2log210logloglog 22
1
2 ddI
I
1
212 log5.2
I
Imm
5log5 dmM
5/)5(10pc)( MmdInverse:
m2 = M
Slide 26
The Distance ModulusIf we know a star’s 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
Slide 27
Absolute magnitudes of two different stars 1 and 2:
1
2
1
2
L
L
I
I
1
2
1
212 log5.2log5.2
L
L
I
IMM
If two stars are at the same distance of 10 pc from the earth:
Slide 28
Absolute Magnitude (2)
Betelgeuse
Rigel
Betelgeuse Rigel
mV 0.41 0.14
MV -5.5 -6.8
d 152 pc 244 pc
Back to our example of Betelgeuse and Rigel:
Difference in absolute magnitudes: 6.8 – 5.5 = 1.3
=> Luminosity ratio = (2.512)1.3 = 3.3
Slide 29
Is there any correlation between stellar luminosities, radii, temperature, and masses???
We learned how to characterize stars with many different parameters
Organizing the Family of Stars
Slide 30
The Size (Radius) of a StarWe already know: flux increases with surface temperature (~ T4); hotter stars are brighter.
But luminosity also increases with size:
A BStar B will be brighter than
star A.
Luminosity is proportional to radius squared, L ~ R2.
Quantitatively: L = 4 R2 T4
Surface area of the starSurface flux due to a blackbody spectrum
Slide 31
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.
Slide 32
However, star radius is not a convenient parameter to use for classification, because it is not directly measured.
Surface temperature, or spectral class is more convenient!
Slide 33
Organizing the Family of Stars: The Hertzsprung-Russell Diagram
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)
Lum
inos
ity
Temperature
Spectral type: O B A F G K M
Hertzsprung-Russell Diagram
orA
bsol
ute
mag
.
Slide 34
Hertzsprung-Russell Diagram1911 1913
Abs
olut
e m
agni
tude
Color index, or spectral class
Betelgeuse
Rigel
Sirius B
Slide 35
Slide 36
Slide 37
Stars in the vicinity of the Sun
Slide 38
Stars in the vicinity of the Sun
Slide 39
90% of the stars are on the Main Sequence!
Slide 40
Total radiated power (luminosity) L = T4 4R2 J/s
Check whether all stars are of the same radius:
Slide 41
No, they are not of the same radius
Slide 42
The Radii of Stars in the Hertzsprung-Russell Diagram
10,000 times the
sun’s radius
100 times the
sun’s radius
As large as the sun
Rigel Betelgeuse
Sun
Polaris
Slide 43
Specific segments of the main sequence are occupiedby stars of a specific mass
Majority of stars are here
Slide 44
The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.
L ~ M3.5 much stronger than inferred from L ~ R2 ~ M2/3
Slide 45
However, this M3.5 dependence does not go forever:Cutoff at masses > 100 M and < 0.08 M
Slide 46
All stars visible to the naked eye + all stars within 25 pc
H-R diagram for nearby+bright stars:
Slide 47
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Slide 49
Slide 50
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