1. Chapter 9
The Family of Stars

2. Guidepost
Science is based on measurement, but measurement in astronomy is very difficult. To discover the properties of stars, astronomers must use their telescopes and spectrographs in ingenious ways to learn the secrets hidden in starlight. The result is a family portrait of the stars.
Here you will find answers to five essential questions about stars:
How far away are the stars?
How much energy do stars make?
How big are stars?
How much mass do stars contain?
What is the typical star like?

3. Guidepost (continued)
With this chapter you leave our sun behind and begin your study of the billions of stars that dot the sky. In a sense, the star is the basic building block of the universe. If you hope to understand what the universe is, what our sun is, what our Earth is, and what we are, you must understand the stars.
Once you know how to find the basic properties of stars, you will be ready to trace the history of the stars from birth to death, a story that begins in the next chapter.

4. Outline I. Measuring the Distances to Stars A. The Surveyor's Method
I. Measuring the Distances to Stars
A. The Surveyor's Method
B. The Astronomer's Triangulation Method
C. Proper Motion
II. Apparent Brightness, Intrinsic Brightness, and Luminosity
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

5. Outline D. Interferometric Observations of Diameter
D. Interferometric Observations of Diameter
E. Luminosity Classification
F. 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
A Census of the Stars
A. Surveying the Stars
B. Mass, Luminosity, and Density

6. The Properties of Stars
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.

7. 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.

8. 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.

9. 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.

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

11. 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 ~ d2
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 0.41 – 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. = Magnitude that a star would have if it were at a distance of 10 pc.
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.

15. Absolute Magnitude (2) Back to our example of Betelgeuse and Rigel:
Back to our example of Betelgeuse and Rigel:
Betelgeuse
Betelgeuse
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 [pc])
Distance in units of parsec
Equivalent:
d = 10(mV – MV + 5)/5 pc

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 p R2 s T4
Surface flux due to a blackbody spectrum
Surface area of the star

18. 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.

19. 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
Luminosity
versus
Temperature (or spectral type)
Absolute mag.
Hertzsprung-Russell Diagram
Luminosity or
Temperature
Spectral type: O B A F G K M

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

21. The Hertzsprung-Russell Diagram (2)
Same temperature, but much brighter than Main Sequence stars
Stars spend most of their active life time on the Main Sequence (MS).

22. The Brightest Stars The open star cluster M39
The open star cluster M39
The brightest stars are either blue (=> unusually hot) or red (=> unusually cold).

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

24. The Relative Sizes of Stars in the HR Diagram

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

26. Example: Luminosity Classes
Our Sun: G2 star on the Main Sequence:
G2V
Polaris: G2 star with Supergiant luminosity: G2Ib

27. 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)

28. 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.

29. The Center of Mass center of mass = balance point of the system
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.

30. 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)

31. 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

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

33. 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

34. 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

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

36. Here, we know that we are looking at the system edge-on!
Eclipsing Binaries
Usually, the 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!

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

38. Algol in the constellation of Perseus
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.

39. The Light Curve of Algol

40. Masses of Stars in the Hertzsprung-Russell Diagram
The more massive a star is, the brighter it is:
L ~ M3.5
High masses
Mass
High-mass stars have much shorter lives than low-mass stars:
Low masses
tlife ~ M-2.5
Sun: ~ 10 billion yr.
10 Msun: ~ 30 million yr.
0.1 Msun: ~ 3 trillion yr.

41. Ideal situation for creating a census of the stars:
Surveys of Stars
Ideal situation for creating a census of the stars:
Determine properties of all stars within a certain volume

42. Main Problem for creating such a survey:
Surveys of Stars
Main Problem for creating such a survey:
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.