The Family of Stars

Chapter 9

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?

Guidepost

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.

Guidepost (continued)

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

Outline

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

V. A Census of the Stars

A. Surveying the Stars

B. Mass, Luminosity, and Density

Outline

The Properties of StarsWe 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

Trigonometric Parallax:

Star appears slightly shifted from different

positions of the Earth on its orbit

The farther away the star is (larger d),

the smaller the parallax angle p.

d = __ p 1

d in parsec (pc)

p in arc seconds

1 pc = 3.26 LY

The Trigonometric Parallax

Example:

Nearest star, 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

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

The more distant a light source is,

the fainter it appears.

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 ~L__

d2

Star AStar B Earth

Both stars may appear equally bright, although

star A is intrinsically much brighter than star B.

Distance and Intrinsic Brightness

Betelgeuse

Rigel

Example:

App. Magn. mV = 0.41

Recall that:

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 = 0.14For a magnitude difference of 0.41

– 0.14 = 0.27, we find an intensity

ratio of (2.512)0.27 = 1.28

Distance and Intrinsic Brightness (2)

Betelgeuse

Rigel

Rigel is appears 1.28 times

brighter than Betelgeuse,

Thus, Rigel is actually

(intrinsically) 1.28*(1.6)2 =

3.3 times brighter than

Betelgeuse.

but Rigel is 1.6 times further

away than Betelgeuse.

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)

Betelgeuse

Rigel

Betelgeuse Rigel

mV 0.41 0.14

MV -5.5 -6.8

d 152 pc 244 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

The Distance ModulusIf 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

Equivalent:

d = 10(mV – MV + 5)/5 pc

The Size (Radius) of a StarWe already know: flux increases with surface

temperature (~ T4); hotter stars are brighter.

But brightness also increases with size:

AB

Star B will be

brighter than

star A.

Absolute brightness is proportional to radius squared, L ~ R2

Quantitatively: L = 4 R2 T4

Surface area of the star

Surface flux due to a

blackbody spectrum

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

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)

Lu

min

osity

Temperature

Spectral type: O B A F G K M

Hertzsprung-Russell Diagram

or

Abso

lute

mag.

The Hertzsprung-Russell Diagram

The Hertzsprung-Russell Diagram (2)

Same

temperature,

but much

brighter than

Main

Sequence

stars

The Brightest Stars

The open star cluster M39

The brightest stars are either blue (=> unusually hot)

or red (=> unusually cold).

The Radii of Stars in the

Hertzsprung-Russell Diagram

Rigel Betelgeuse

Sun

Polaris

The Relative Sizes of Stars in

the HR Diagram

Luminosity Classes

Ia Bright Supergiants

Ib Supergiants

II Bright Giants

III Giants

IV Subgiants

V Main-Sequence

Stars

IaIb

II

III

IV

V

Example: Luminosity Classes

• Our Sun: G2 star on the

Main Sequence:

G2V

• Polaris: G2 star with

Supergiant luminosity:

G2Ib

Spectral Lines of Giants

=> Absorption lines in spectra of giants and

supergiants are narrower than in main sequence stars

Pressure and density in the atmospheres of giants

are lower 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

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.

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.

Estimating Stellar Masses

Recall Kepler’s 3rd Law:

Py2 = aAU

3

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 = aAU

3____

Py2

(MA and MB in units of solar masses)

Examples: Estimating Mass

a) Binary system with period of P = 32 years

and separation of a = 16 AU:

MA + MB = = 4 solar masses163____

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

Visual Binaries

The ideal case:

Both stars can be

seen directly, and

their separation and

relative motion can

be followed directly.

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)

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)T

ime

Typical sequence of spectra from a

spectroscopic binary system

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!

Eclipsing Binaries (2)

Peculiar “double-dip” light curve

Example: VW Cephei

Eclipsing Binaries (3)

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.

Example:

Algol in the

constellation

of Perseus

The Light Curve of Algol

Masses of Stars in the Hertzsprung-

Russell DiagramThe more massive a star is,

the brighter it is:

High-mass stars have

much shorter lives than

low-mass stars:

Sun: ~ 10 billion yr.

10 Msun: ~ 30 million yr.

0.1 Msun: ~ 3 trillion yr.

L ~ M3.5

tlife ~ M-2.5

Surveys of Stars

Ideal situation

for creating a

census of the

stars:

Determine

properties of all

stars within a

certain volume

Surveys of StarsMain Problem for creating such a survey:

Fainter stars are hard to observe; we might be biased

towards the more luminous stars.

A Census of the Stars

Faint, red dwarfs

(low mass) are

the most

common stars.

Giants and

supergiants

are extremely

rare.

Bright, hot, blue

main-sequence

stars (high-

mass) are very

rare.