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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.
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
III. The Diameters of Stars
A. Luminosity, Radius, and Temperature
B. The H-R Diagram
C. Giants, Supergiants, and Dwarfs
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
We already know how to determine a star’s
In this chapter, we will learn how we can determine its
and how all the different types of stars make up the big family of stars.
d in parsec (pc) p in arc seconds
Star appears slightly shifted from different positions of the Earth on its orbit
1 pc = 3.26 LY
The farther away the star is (larger d), the smaller the parallax angle p.
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.
In 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.
The more distant a light source is, the fainter it appears.
(SLIDESHOW MODE ONLY)
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):
Both stars may appear equally bright, although star A is intrinsically much brighter than star B.
App. Magn. mV = 0.41
For a magnitude difference of 0.41 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28
App. Magn. mV = 0.14
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.
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.
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
If we know a star’s absolute magnitude, we can infer its distance by comparing absolute and apparent magnitudes:
= mV – MV
= -5 + 5 log10(d [pc])
Distance in units of parsec
d = 10(mV – MV + 5)/5 pc
We already know: flux increases with surface temperature (~ T4); hotter stars are brighter.
But brightness also increases with size:
Star B will be brighter than star A.
Absolute brightness is proportional to radius squared, L ~ R2.
Quantitatively: L = 4 p R2s T4
Surface flux due to a blackbody spectrum
Surface area of the star
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.
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
Temperature (or spectral type)
Spectral type: O B A F G K M
Most stars are found along the Main Sequence
Same temperature, but much brighter than MS stars
Must be much larger
Stars spend most of their active life time on the Main Sequence (MS).
Same temp., but fainter → Dwarfs
10,000 times the sun’s radius
100 times the sun’s radius
As large as the sun
100 times smaller than the sun
Ia Bright Supergiants
II Bright Giants
V Main-Sequence Stars
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)
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 stellarmasses.
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.
(SLIDESHOW MODE ONLY)
RecallKepler’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:
MA + MB =
(MA and MB in units of solar masses)
a) Binary system with period of P = 32 years and separation of a = 16 AU:
MA + MB = = 4 solar masses.
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.
The ideal case:
Both stars can be seen directly, and their separation and relative motion can be followed directly.
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
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
Typical sequence of spectra from a spectroscopic binary system
Usually, inclination angle of binary systems is unknown uncertainty in mass estimates.
Here, we know that we are looking at the system edge-on!
Peculiar “double-dip” light curve
Example: VW Cephei
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 higher a star’s mass, the more luminous (brighter) it is:
Masses in units of solar masses
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.
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: h Carinae: Binary system of a 60 Msun and 70 Msun star. Dramatic mass loss; major eruption in 1843 created double lobes.
Mmin = 0.08 Msun
At masses below 0.08 Msun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.
Determine properties of all stars within a certain volume.
Fainter stars are hard to observe; we might be biased towards the more luminous stars.
Faint, red dwarfs (low mass) are the most common stars.
Bright, hot, blue main-sequence stars (high-mass) are very rare
Giants and supergiants are extremely rare.
stellar parallax (p)
absolute visual magnitude (Mv)
distance modulus (mv – Mv)
absolute bolometric magnitude
H–R (Hertzsprung–Russell) diagram
visual binary system
spectroscopic binary system
eclipsing binary system
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)
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.
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.
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.
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.
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.
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.
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.
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 low-energy 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.