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Lecture 12
The Family of Stars
Announcements
Homework assignment 6 is due today.
Homework 7 – Due Monday, March 19
Unit 52: TY4
Unit 54: P3, TY3
Unit 56: P1
Unit 58: RQ2, TY1, TY2
Exam 2
The test is this Wednesday, March 7th.
Required
Pencil/pen
Equation sheet
Recommended
Calculator
Scratch paper
Covers Units 22-25,28-30,49-52,54,56, and 58
(more or less)
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
Test Prep
• 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 Rules
• 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.
Properties of Stars
We 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.
Distances to Stars
d in parsec (pc)
p in arc seconds
1
d = __
p
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.
1 pc = 3.26 LY
The Trigonometric Parallax
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
=> d ≤ 50 pc
This method does not work for stars
farther away than 50 pc.
Proper Motion
In addition to the
periodic back-andforth 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.
Intrinsic Brightness/
Absolute Magnitude
The more distant a light source is,
the fainter it appears.
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):
L
__
F~ 2
d
Star A
Star B
Both stars may appear equally bright, although
star A is intrinsically much brighter than star B.
Earth
Distance and Intrinsic Brightness
Example:
Recall that:
Magn.
Diff.
Intensity Ratio
1
2.512
2
2.512*2.512 = (2.512)2
= 6.31
…
…
5
(2.512)5 = 100
For a magnitude difference of 0.41
– 0.14 = 0.27, we find an intensity
ratio of (2.512)0.27 = 1.28
Betelgeuse
App. Magn. mV = 0.41
Rigel
App. Magn. mV = 0.14
Distance and Intrinsic Brightness
Rigel is appears 1.28 times
brighter than 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.
Betelgeuse
Rigel
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.
Absolute Magnitude
Back to our example of
Betelgeuse and Rigel:
Betelgeuse Rigel
mV
0.41
0.14
MV
-5.5
-6.8
d
152 pc
244 pc
Betelgeuse
Rigel
Difference in absolute magnitudes:
6.8 – 5.5 = 1.3
=> Luminosity ratio = (2.512)1.3 = 3.3
The Distance Modulus
If we know a star’s absolute magnitude, we
can infer its distance by comparing absolute
and apparent magnitudes:
Distance Modulus
= mV – M V
= -5 + 5 log10(d [pc])
Distance in units of parsec
Equivalent:
d = 10(mV – MV + 5)/5 pc
The Size (Radius) of a Star
We already know: flux increases with surface
temperature (~ T4); hotter stars are brighter.
But brightness also increases with size:
A
Star B will be
brighter than
star A.
B
Absolute brightness is proportional to radius squared, L ~ R2.
Quantitatively:
L = 4 p R2 s T4
Surface area of the star
Surface flux due to a
blackbody spectrum
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.
Organizing the Family of Stars:
The Hertzsprung-Russell Diagram
We know:
Stars have different temperatures,
different luminosities, and different sizes.
Absolute mag.
or
Luminosity
To bring some order into that zoo of different
types of stars: organize them in a diagram of
Luminosity
versus
Temperature (or spectral type)
Hertzsprung-Russell Diagram
Spectral type: O
Temperature
B
A
F
G
K
M
The Hertzsprung-Russell Diagram
The Hertzsprung-Russell Diagram
Same
temperature,
but much
brighter than
MS stars
 Must be
much larger
 Giant
Stars
The Radii of Stars in the
Hertzsprung-Russell Diagram
Rigel
Betelgeuse
Polaris
Sun
100 times smaller than the sun
Luminosity Classes
Ia Bright Supergiants
Ia
Ib
Ib Supergiants
II
III
IV
II Bright Giants
III Giants
IV Subgiants
V
V Main-Sequence
Stars
Example Luminosity Classes
• Our Sun: G2 star on the Main Sequence:
G2V
• Polaris: G2 star with Supergiant luminosity:
G2Ib
Spectral Lines of Giants
Pressure and density in the atmospheres of giants
are lower than in main sequence stars.
=> Absorption lines in spectra of giants and
supergiants are narrower than in main sequence stars
=> From the line widths, we can estimate the size and
luminosity of a star.
 Distance
estimate (spectroscopic parallax)
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.
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.
Estimating Stellar Masses
Recall Kepler’s 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:
3
a
____
AU
MA + MB =
Py2
(MA and MB in units of solar masses)
Examples: Estimating Mass
a) Binary system with period of P = 32 years
and separation of a = 16 AU:
163
____
MA + MB =
= 4 solar masses.
2
32
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.
Visual Binaries
The ideal case:
Both stars can be
seen directly, and
their separation and
relative motion can
be followed directly.
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
Spectroscopic Binaries
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
Spectroscopic Binaries
Typical sequence of spectra from a
spectroscopic binary system
Time
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!
Eclipsing Binaries
Peculiar “double-dip” light curve
Example: VW Cephei
Eclipsing Binaries
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.
The Light Curve of Algol
Masses of Stars in the HertzsprungRussell Diagram
The higher a star’s mass,
the more luminous
(brighter) it is:
40
L ~ M3.5
High-mass stars have
much shorter lives than
low-mass stars:
tlife ~
M-2.5
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.
Masses in units of
solar masses
18
6
3
1.7
1.0
0.8
0.5
Maximum Masses of Main-Sequence Stars
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
h Carinae
Example: h Carinae: Binary system of a 60 Msun and 70 Msun star.
Dramatic mass loss; major eruption in 1843 created double lobes.
Minimum Mass of Main-Sequence Stars
Mmin = 0.08 Msun
Gliese 229B
At masses below
0.08 Msun, stellar
progenitors do not
get hot enough to
ignite thermonuclear
fusion.
 Brown
Dwarfs
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.
A Census of the Stars
Faint, red dwarfs
(low mass) are
the most
common stars.
Bright, hot, blue
main-sequence
stars (highmass) are very
rare
Giants and
supergiants
are extremely
rare.
For Next Time
Work on your equation sheet
I have reversed Labs 7 and 9 so that we can
do another “pencil & paper” lab after the test.