1. Chapter 9
The Family of Stars

2. 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.

3. 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

4. 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

5. The Amazing Power of Starlight
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.

6. Distances to Stars __ 1 d = p Trigonometric Parallax: 1 pc = 3.26 LY
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
1 pc = 3.26 LY
The farther away the star is (larger d), the smaller the parallax angle p.

7. The Trigonometric Parallax
Example:
Nearest star, α 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 parallax p ≥ 0.02 arc sec, which is d ≤ 50 pc
This method does not work for stars farther away than 50 pc.

8. 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.

9. Intrinsic Brightness/ Absolute Magnitude
The more distant a light source is, the fainter it appears.

10. Brightness and Distance
(SLIDESHOW MODE ONLY)

11. Intrinsic Brightness / Absolute Magnitude (2)
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)
Star A
Star B
Earth
Both stars may appear equally bright, although star A is intrinsically much brighter than star B.

12. Distance and Intrinsic Brightness
Example:
Recall that:
Betelgeuse
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 = 0.41
Rigel
For a magnitude difference of – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28
App. Magn. mV = 0.14

13. Distance and Intrinsic Brightness (2)
Rigel is appears 1.28 times brighter than Betelgeuse,
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.
Rigel

14. 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 parsecs (pc).

15. Absolute Magnitude (2) Back to our example of Betelgeuse and Rigel:mV
0.41
0.14 MV
-5.5
-6.8 d
152 pc
244 pc
Rigel
Difference in absolute magnitudes: 6.8 – 5.5 = 1.3
Luminosity ratio = (2.512)1.3 = 3.3

16. 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 = log10(d)
distance in units of parsec

17. 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. A B
Absolute brightness is proportional to radius squared (L ~ R2).
Quantitatively: L = 4 π R2 σ T4
Surface flux due to a blackbody spectrum
Surface area of the star

18. Example: Star Radii
Polaris (F7 star) has just about the same spectral type (and thus surface temperature) as our sun (G2 star), but it is 10,000 times intrinsically brighter than our sun.
Thus, Polaris is 100 times larger than the sun.
This means its luminosity is 1002 = 10,000 times more than the sun.

19. Organizing the Family of Stars: The Hertzsprung-Russell Diagram
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
“Hertzsprung-Russell (HR) Diagram”
Absolute mag.
or Luminosity
Temperature
O B A F G K M Spectral type

20. The Hertzsprung-Russell Diagram Analogy
It’s useful to compare an HR Diagram to a similar graph of cars with different weights and horsepower.

21. The Hertzsprung-Russell Diagram
Most stars are found along the Main Sequence

22. The Hertzsprung-Russell Diagram (2)
“Giants” (and supergiants) are same temperature, but much brighter than main sequence stars.
Giants must be much larger than m.s. stars
Stars spend most of their active life time on the main sequence (m.s.)
Dwarfs are same temperature, but fainter and smaller than m.s. stars

23. The Radii of Stars in the Hertzsprung-Russell Diagram
Rigel
Betelgeuse
10,000 times the sun’s radius
Polaris
100 times the sun’s radius
Sun
As large as the sun
100 times smaller than the sun

24. Luminosity Classes Ia Bright Supergiants Ib SupergiantsII
II Bright Giants
III
III Giants
IV Subgiants IV V
V Main-Sequence Stars

25. Spectral Lines of Giants
Pressure and density in the atmospheres of giants are lower than in main sequence stars, so:
• 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”) is found using spectral type, luminosity class and apparent magnitude

26. 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.

27. 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.

28. Center of Mass
(SLIDESHOW MODE ONLY)

29. 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:
aAU3
____
MA + MB =
Py2
(MA and MB in units of solar masses)

30. Examples: Estimating Mass
a) Binary system with period of P = 32 years and separation of a = 16 AU:
163
____
MA + MB = = 4 solar masses.
322
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.

31. Visual Binaries
The ideal case:
Both stars can be seen directly, and their separation and relative motion can be followed directly.

32. 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

33. 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

34. Spectroscopic Binaries (3)
Typical sequence of spectra from a spectroscopic binary system
Time

35. Here, we know that we are looking at the system edge-on!
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!

36. Eclipsing Binaries (2) Peculiar “double-dip” light curve
Example: VW Cephei

37. 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.

38. The Light Curve of Algol

39. 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 40
L ~ M3.5 18
High-mass stars have much shorter lives than low-mass stars:
High masses 6 3
1.7
tlife ~ M-2.5
1.0
Mass
0.8
0.5
Sun: ~ 10 billion yr.
Low masses
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.

40. Maximum Masses of Main-Sequence Stars
Mmax ~ 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.

41. 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

42. Surveys of Stars Ideal situation: Problem:
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.

43. 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.

44. 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