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The Family of Stars
Lecture 12
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Homework assignment 6 is due today.
Homework 7 – Due Monday, March 19
Unit 52: TY4Unit 54: P3, TY3Unit 56: P1Unit 58: RQ2, TY1, TY2
Announcements
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Exam 2
The test is this Wednesday, March 7th.
RequiredPencil/penEquation sheet
RecommendedCalculatorScratch paper
Covers Units 22-25,28-30,49-52,54,56, and 58 (more or less)
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Exam 2 Details
•85 minutes Wednesday (5-6:25 pm)
•Approximate Test Format:–20 multiple-choice questions (2 points each)–10 True/False questions (2 points each)–6 Short answer/problem questions (5 points each)–Equation sheet (10 points)
•Mostly conceptual
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• How to prepare?– Focus on the lecture material:
• These units contain a LOT more material than what we actually went over in class.
• You ARE responsible for understanding the topics covered in class (including details in the book that I may not have mentioned).
• You are NOT responsible for other stuff in these chapters not covered at all in lecture.
Test Prep
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• Forbidden– Cell phone (not even for use as calculator)– Communication with anyone other than me– Textbook or any other reference material not
written by you.• Equation Sheet
– Single page (front and back) HAND WRITTEN notes, equations, or any information you want to bring to the test.
– Max size 8.5 x 11 inches.– Will be turned in – counts 10% of your test
grade.
Test Rules
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Properties of StarsWe already know how to determine a star’s
• surface temperature
• chemical composition
• surface density
In this lecture, 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
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.
d = __ p 1
d in parsec (pc) p in arc seconds
1 pc = 3.26 LY
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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
This method does not work for stars farther away than 50 pc.
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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.
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Intrinsic Brightness/ Absolute Magnitude
The more distant a light source is, the fainter it appears.
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Intrinsic Brightness / Absolute Magnitude
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):
F ~ L__d2
Star AStar 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
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
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Distance and Intrinsic Brightness
Betelgeuse
Rigel
Rigel is appears 1.28 times brighter than Betelgeuse,
Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times brighter than Betelgeuse.
But Rigel is 1.6 times further away than Betelgeuse
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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.
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Absolute Magnitude
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
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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
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The Size (Radius) of a StarWe already know: flux increases with surface temperature (~ T4); hotter stars are brighter.
But brightness also increases with size:
A BStar B will be brighter than
star A.
Absolute brightness is proportional to radius squared, L ~ R2.
Quantitatively: L = 4 R2 T4
Surface area of the starSurface flux due to a blackbody spectrum
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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.
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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
.
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The Hertzsprung-Russell Diagram
Most stars are found along the
Main Sequence
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The Hertzsprung-Russell Diagram
Stars spend most of their
active life time on the
Main Sequence (MS).
Same temperature,
but much brighter than
MS stars
Must be much larger
Giant StarsSame temp., but
fainter → Dwarfs
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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
100 times smaller than the sun
Rigel Betelgeuse
Sun
Polaris
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Luminosity Classes
Ia Bright Supergiants
Ib Supergiants
II Bright Giants
III Giants
IV Subgiants
V Main-Sequence Stars
IaIb
IIIII
IV
V
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Example Luminosity Classes
• Our Sun: G2 star on the Main Sequence:
G2V
• Polaris: G2 star with Supergiant luminosity:
G2Ib
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Spectral Lines of Giants
=> Absorption lines in spectra of giants and supergiants are narrower than in main sequence stars
Pressure and density in the atmospheres of giants are lower than in main sequence stars.
=> From the line widths, we can estimate the size and luminosity of a star.
Distance estimate (spectroscopic parallax)
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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.
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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.
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Estimating Stellar Masses
Recall Kepler’s 3rd Law:
Py2 = aAU
3
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:
MA + MB = aAU
3 ____ Py
2
(MA and MB in units of solar masses)
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Examples: Estimating Mass
a) Binary system with period of P = 32 years and separation of a = 16 AU:
MA + MB = = 4 solar masses.163____322
b) Any binary system with a combination of period “P” and separation “a” that obeys
Kepler’s 3rd Law must have a total mass of 1 solar mass.
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Visual Binaries
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
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 BinariesThe 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 BinariesT
ime
Typical sequence of spectra from a spectroscopic binary system
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Eclipsing Binaries
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
Peculiar “double-dip” light curve
Example: VW Cephei
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Eclipsing Binaries
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.
Example:
Algol in the constellation of Perseus
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The Light Curve of Algol
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Masses of Stars in the Hertzsprung-Russell Diagram
The higher a star’s mass, the more luminous
(brighter) it is:
High-mass stars have much shorter lives than
low-mass stars:
Sun: ~ 10 billion yr.10 Msun: ~ 30 million yr.0.1 Msun: ~ 3 trillion yr.
0.5
18
6
31.7
1.00.8
40
Masses in units of solar masses
Low
masses
High masses
Mass
L ~ M3.5
tlife ~ M-2.5
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Maximum Masses of Main-Sequence Stars
Carinae
Mmax ~ 50 - 100 solar masses
a) More massive clouds fragment into smaller pieces during star formation.
b) Very massive stars lose mass in strong stellar winds
Example: Carinae: Binary system of a 60 Msun and 70 Msun star. Dramatic mass loss; major eruption in 1843 created double lobes.
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Minimum Mass of Main-Sequence Stars
Mmin = 0.08 Msun
At masses below 0.08 Msun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.
Brown Dwarfs
Gliese 229B
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Surveys of Stars
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
Faint, red dwarfs (low mass) are the most common stars.
Giants and supergiants are extremely rare.
Bright, hot, blue main-sequence stars (high-mass) are very rare
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Work on your equation sheet
I have reversed Labs 7 and 9 so that we can do another “pencil & paper” lab after the test.
For Next Time