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Note that the following lectures include animations and PowerPoint effects such as fly ins and transitions that require you to be in PowerPoint's Slide Show mode (presentation mode).


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

D. Luminosity

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

light as a wave 1
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
Distances to Stars

d in parsec (pc) p in arc seconds



d =


Trigonometric Parallax:

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.

the trigonometric parallax
The Trigonometric Parallax


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
Proper Motion

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.

intrinsic brightness absolute magnitude
Intrinsic Brightness/ Absolute Magnitude

The more distant a light source is, the fainter it appears.

brightness and distance
Brightness and Distance


intrinsic brightness absolute magnitude 2
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):



F ~


Star A

Star B


Both stars may appear equally bright, although star A is intrinsically much brighter than star B.

distance and intrinsic brightness
Distance and Intrinsic Brightness


Recall that:


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

distance and intrinsic brightness 2
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.


absolute magnitude
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
Absolute Magnitude (2)

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

the distance modulus
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 – MV

= -5 + 5 log10(d [pc])

Distance in units of parsec


d = 10(mV – MV + 5)/5 pc

the size radius of a star
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:

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

example star radii
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
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


Temperature (or spectral type)


Absolute mag.

Hertzsprung-Russell Diagram




Spectral type: O B A F G K M

the hertzsprung russell diagram
The Hertzsprung-Russell Diagram

Most stars are found along the Main Sequence

the hertzsprung russell diagram 2
The Hertzsprung-Russell Diagram (2)

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).

Giant Stars

Same temp., but fainter → Dwarfs

the radii of stars in the hertzsprung russell diagram
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

luminosity classes
Luminosity Classes

Ia Bright Supergiants



Ib Supergiants


II Bright Giants


III Giants

IV Subgiants



V Main-Sequence Stars

example luminosity classes
Example Luminosity Classes
  • Our Sun: G2 star on the Main Sequence:G2V
  • Polaris: G2 star with Supergiant luminosity: G2Ib
spectral lines of giants
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
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 stellarmasses.

the center of mass
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
Center of Mass


estimating stellar masses
Estimating Stellar Masses

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)

examples estimating mass
Examples: Estimating Mass

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.

visual binaries
Visual Binaries

The ideal case:

Both stars can be seen directly, and their separation and relative motion can be followed directly.

spectroscopic binaries
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
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
Spectroscopic Binaries (3)

Typical sequence of spectra from a spectroscopic binary system


eclipsing binaries
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
Eclipsing Binaries (2)

Peculiar “double-dip” light curve

Example: VW Cephei

eclipsing binaries 3
Eclipsing Binaries (3)


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.

masses of stars in the hertzsprung russell diagram
Masses of Stars in the Hertzsprung-Russell Diagram

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:

High masses




tlife ~ M-2.5





Sun: ~ 10 billion yr.

Low masses

10 Msun: ~ 30 million yr.

0.1 Msun: ~ 3 trillion yr.

maximum masses of main sequence stars
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
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.

Gliese 229B

Brown Dwarfs

surveys of stars
Surveys of Stars

Ideal situation:

Determine properties of all stars within a certain volume.


Fainter stars are hard to observe; we might be biased towards the more luminous stars.

a census of the stars
A Census of the 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.

new terms
New Terms

stellar parallax (p)

parsec (pc)

proper motion


absolute visual magnitude (Mv)

magnitude–distance formula

distance modulus (mv – Mv)

luminosity (L)

absolute bolometric magnitude

H–R (Hertzsprung–Russell) diagram

main sequence



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
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
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 questions48
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 questions49
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 questions50
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 questions51
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 questions52
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 questions53
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 questions54
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 questions55
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 questions56
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 questions57
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 questions58
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 questions59
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 questions60
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 questions61
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 questions62
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 questions63
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 questions64
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 questions65
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 questions66
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 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.


1. c

2. c

3. b

4. d

5. d

6. b

7. a

8. a

9. e

10. a

11. e

12. e

13. a

14. d

15. c

16. c

17. d

18. d

19. a

20. d