Transcript The Milky Way - University of North Texas

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Chapter 9
The Family of Stars
Guidepost
Science is based on measurement, but measurement in
astronomy is very difficult. Even with the powerful modern
telescopes described in Chapter 6, it is impossible to
measure directly simple parameters such as the diameter of
a star. This chapter shows how we can use the simple
observations that are possible, combined with the basic laws
of physics, to discover the properties of stars.
With this chapter, we leave our sun behind and begin our
study of the billions of stars that dot the sky. In a sense, the
star is the basic building block of the universe. If we hope to
understand what the universe is, what our sun is, what our
Earth is, and what we are, we must understand the stars.
In this chapter we will find out what stars are like. In the
chapters that follow, we will trace the life stories of the stars
from their births to their deaths.
Outline
I. Measuring the Distances to Stars
A. The Surveyor's Method
B. The Astronomer's Method
C. Proper Motion
II. Intrinsic Brightness
A. Brightness and Distance
B. Absolute Visual Magnitude
C. Calculating Absolute Visual Magnitude
D. Luminosity
III. The Diameters of Stars
A. Luminosity, Radius, and Temperature
B. The H-R Diagram
C. Giants, Supergiants, and Dwarfs
Outline
D. Luminosity Classification
E. Spectroscopic Parallax
IV. The Masses of Stars
A. Binary Stars in General
B. Calculating the Masses of Binary Stars
C. Visual Binary Systems
D. Spectroscopic Binary Systems
E. Eclipsing Binary Systems
V. A Survey of the Stars
A. Mass, Luminosity, and Density
B. Surveying the Stars
Light as a Wave (1)
We already know how to determine a star’s
• 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.
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.
Brightness and Distance
(SLIDESHOW MODE ONLY)
Intrinsic Brightness / Absolute
Magnitude (2)
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 (2)
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 (2)
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 (2)
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.
Center of Mass
(SLIDESHOW MODE ONLY)
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
3. 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 (2)
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 (3)
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 (2)
Peculiar “double-dip” light curve
Example: VW Cephei
Eclipsing Binaries (3)
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.
New Terms
stellar parallax (p)
parsec (pc)
proper motion
flux
absolute visual
magnitude (Mv)
magnitude–distance
formula
distance modulus
(mv – Mv)
luminosity (L)
absolute bolometric
magnitude
H–R (Hertzsprung–
Russell) diagram
main sequence
giants
supergiants
red dwarf
white dwarf
luminosity class
spectroscopic parallax
binary stars
visual binary system
spectroscopic binary
system
eclipsing binary system
light curve
mass–luminosity relation
Discussion Questions
1. If someone asked you to compile a list of the nearest
stars to the sun based on your own observations, what
measurements would you make, and how would you
analyze them to detect nearby stars?
2. The sun is sometimes described as an average star.
What is the average star really like?
Quiz Questions
1. The parallax angle of a star and the two lines of sight to the
star from Earth form a long skinny triangle with a short side of
a. 1000 km.
b. 1 Earth diameter.
c. 1 AU.
d. 2 AU.
e. 40 AU.
Quiz Questions
2. What is the distance to a star that has a parallax angle of 0.5
arc seconds?
a. Half a parsec.
b. One parsec.
c. Two parsecs.
d. Five parsecs.
e. Ten parsecs.
Quiz Questions
3. Why can smaller parallax angles be measured by telescopes
in Earth orbit?
a. Telescopes orbiting Earth are closer to the stars.
b. Earth's atmosphere does not limit a telescope's resolving
power.
c. Earth's atmosphere does not limit a telescope's light
gathering power.
d. Earth's atmosphere does not limit a telescope's magnifying
power.
e. They can be connected to Earth-based telescopes to do
interferometry.
Quiz Questions
4. At what distance must a star be to have its apparent
magnitude equal to its absolute magnitude?
a. One AU.
b. Ten AU.
c. One parsec.
d. Ten parsecs.
e. One Megaparsec.
Quiz Questions
5. Which magnitude gives the most information about the
physical nature of a star?
a. The apparent visual magnitude.
b. The apparent bolometric magnitude.
c. The absolute visual magnitude.
d. The absolute bolometric magnitude.
e. None of the above tells us anything about the physical
nature of a star.
Quiz Questions
6. For which stars does absolute visual magnitude differ least
from absolute bolometric magnitude?
a. Low surface temperature stars.
b. Medium surface temperature stars.
c. High surface temperature stars.
d. Stars closer than 10 parsecs.
e. Stars farther away than 10 parsecs.
Quiz Questions
7. The absolute magnitude of any star is equal to its apparent
magnitude at a distance of 10 parsecs. Use this definition, how
light intensity changes with distance, and how the stellar
magnitude system is set up to determine the following. If a
star's apparent visual magnitude is less than its absolute visual
magnitude, which of the following is correct?
a. The distance to the star is less than 10 parsecs.
b. The distance to the star is 10 parsecs.
c. The distance to the star is greater than 10 parsecs.
d. Its bolometric magnitude is greater than its visual magnitude.
e. Its bolometric magnitude is less than its visual magnitude.
Quiz Questions
8. What is the distance to a star that has an apparent visual
magnitude of 3.5 and an absolute visual magnitude of -1.5?
a. 100 parsecs.
b. 50 parsecs.
c. 25 parsecs.
d. 10 parsecs.
e. 5 parsecs.
Quiz Questions
9. What is the luminosity of a star that has an absolute
bolometric magnitude that is 10 magnitudes brighter than the
Sun (-5.3 for the star and +4.7 for the Sun)?
a. 1 solar luminosity.
b. 10 solar luminosities.
c. 100 solar luminosities
d. 1000 solar luminosities.
e. 10000 solar luminosities.
Quiz Questions
10. How can a cool star be more luminous than a hot star?
a. It can be more luminous if it is larger.
b. It can be more luminous if it is more dense.
c. It can be more luminous if it is closer to Earth.
d. It can be more luminous if it is farther from Earth.
e. A cool star cannot be more luminous than a hot star.
Quiz Questions
11. A star has one-half the surface temperature of the Sun, and
is four times larger than the Sun in radius. What is the star's
luminosity?
a. 64 solar luminosities.
b. 16 solar luminosities.
c. 4 solar luminosities.
d. 2 solar luminosities.
e. 1 solar luminosity.
Quiz Questions
12. The Sun's spectral type is G2. What is the Sun's luminosity
class?
a. Bright Supergiant (Ia)
b. Supergiant (Ib)
c. Bright Giant (II)
d. Giant (III)
e. Main Sequence (V)
Quiz Questions
13. A particular star with the same spectral type as the Sun
(G2) has a luminosity of 50 solar luminosities. What does this
tell you about the star?
a. It must be larger than the Sun.
b. It must be smaller than the Sun.
c. It must be within 1000 parsecs of the Sun.
d. It must be farther away than 1000 parsecs.
e. Both a and b above.
Quiz Questions
14. In addition to the H-R diagram, what other information is
needed to find the distance to a star whose parallax angle is
not measurable?
a. The star's spectral type.
b. The star's luminosity class.
c. The star's surface activity.
d. Both a and b above.
e. All of the above.
Quiz Questions
15. What is the radius and luminosity of a star that is classified
as G2 III?
a. About 0.1 solar radii and 0.001 solar luminosities.
b. About 1 solar radii and 1 solar luminosity.
c. About 10 solar radii and 100 solar luminosity.
d. About 100 solar radii and 10,000 solar luminosities.
e. About 1000 solar radii and 1,000,000 solar luminosities.
Quiz Questions
16. For a particular binary star system star B is observed to
always be four times as far away from the center of mass as
star A. What does this tell you about the masses of these two
stars?
a. The total mass of these two stars is four solar masses.
b. The total mass of these two stars is five solar masses.
c. The ratio of star A's mass to star B's mass is four to one.
d. The ratio of star B's mass to star A's mass is four to one.
e. Both b and c above.
Quiz Questions
17. For a particular binary star system the ratio of the mass of
star A to star B is 4 to 1. The semimajor axis of the system is 10
AU and the period of the orbits is 10 years. What are the
individual masses of star A and star B?
a. Star A is 1 solar mass and star B is 4 solar masses.
b. Star A is 4 solar masses and star B is 1 solar mass.
c. Star A is 2 solar masses and star B is 8 solar masses.
d. Star A is 8 solar masses and star B is 2 solar masses.
e. None of the above.
Quiz Questions
18. To which luminosity class does the mass-luminosity
relationship apply?
a. The Supergiants.
b. The Giants.
c. The Subgiants.
d. The Main Sequence.
e. The mass-luminosity relationship applies to all luminosity
classes.
Quiz Questions
19. Which luminosity class has stars of the lowest density,
some even less dense than air at sea level?
a. The Supergiant.
b. The Bright Giant.
c. The Giant.
d. The Subgiant.
e. The Main Sequence.
Quiz Questions
20. In a given volume of space the Red Dwarf (or lower main
sequence) stars are the most abundant, however, on many
H-R diagrams very few of these stars are plotted. Why?
a. Photographic film and CCDs both have low sensitivity to lowenergy red photons.
b. They are so very distant that parallax angles cannot be
measured, thus distances and absolute magnitudes are difficult
to determine precisely.
c. They have so many molecular bands in their spectra that
they are often mistaken for higher temperature spectral types.
d. They have very low luminosity and are difficult to detect,
even when nearby.
e. Most of them have merged to form upper main sequence
stars.
Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
c
c
b
d
d
b
a
a
e
a
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
e
e
a
d
c
c
d
d
a
d