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The Family of Stars. Chapter 9. Guidepost.

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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

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

the amazing power of starlight
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.

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, 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.

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)

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


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 parsecs (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)

distance in units of parsec

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 π R2 σ T4

Surface flux due to a blackbody spectrum

Surface area of the star

example star radii
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.

organizing the family of stars the hertzsprung russell diagram
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




“Hertzsprung-Russell (HR) Diagram”

Absolute mag.

or Luminosity


O B A F G K M Spectral type

the hertzsprung russell diagram analogy
The Hertzsprung-Russell Diagram Analogy

It’s useful to compare an HR Diagram to a similar graph of cars with different weights and horsepower.

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)

“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

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

spectral lines of giants
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

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