The Family of StarsChapter 9

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.

Guidepost

I. Measuring the Distances to StarsA. The Surveyor's MethodB. The Astronomer's MethodC. Proper Motion

II. Intrinsic BrightnessA. Brightness and DistanceB. Absolute Visual MagnitudeC. Calculating Absolute Visual MagnitudeD. Luminosity

III. The Diameters of StarsA. Luminosity, Radius, and TemperatureB. The H-R DiagramC. Giants, Supergiants, and Dwarfs

Outline

D. Luminosity ClassificationE. Spectroscopic Parallax

IV. The Masses of StarsA. Binary Stars in GeneralB. Calculating the Masses of Binary StarsC. Visual Binary SystemsD. Spectroscopic Binary SystemsE. Eclipsing Binary Systems

V. A Survey of the StarsA. Mass, Luminosity, and DensityB. Surveying the Stars

Outline

The Amazing Power of StarlightWe 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, 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 MotionIn 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.

Brightness and Distance

(SLIDESHOW MODE ONLY)

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 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 parsecs (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)

distance in units of parsec

The Size (Radius) of a StarWe already know: flux increases with surface temperature (~ T4); hotter stars are brighter.

But brightness also increases with size.

A BStar B will be brighter than

star A.

Absolute brightness is proportional to radius squared (L ~ R2).

Quantitatively: L = 4 π R2 σ T4

Surface area of the starSurface flux due to a blackbody spectrum

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

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

Lum

inos

ity

Temperature

O B A F G K M Spectral type

“Hertzsprung-Russell (HR) Diagram”

Abs

olut

e m

ag.

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

The Hertzsprung-Russell Diagram

Most stars are found along the

Main Sequence

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

Dwarfs are same temperature, but fainter and smaller than m.s. stars

Stars spend most of their

active life time on the

main sequence (m.s.)

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

Rigel Betelgeuse

Sun

Polaris

Luminosity Classes

Ia Bright Supergiants

Ib Supergiants

II Bright Giants III Giants

IV Subgiants

V Main-Sequence Stars

IaIb

IIIII

IVV

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, so:

• 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

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 Masscenter 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

(SLIDESHOW MODE ONLY)

Estimating Stellar MassesRecall 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 ____ Py

2

(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 masses.163____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)Tim

e

Typical sequence of spectra from a spectroscopic binary system

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)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 higher a star’s mass,

the more luminous (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.

0.5

18

6

31.7

1.00.8

40

Masses in units of solar masses

Low

masses

High masses

Mass

L ~ M3.5

tlife ~ M-2.5

Maximum Masses of Main-Sequence Stars

h Carinae

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

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

Mmin = 0.08 Msun

At masses below 0.08 Msun, stellar progenitors do not get hot enough to ignite thermonuclear fusion.

Brown Dwarfs

Gliese 229B

Surveys of Stars

Ideal situation: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.

A Census of the StarsFaint, 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

stellar parallax (p)parsec (pc)proper motionfluxabsolute visual magnitude (Mv)

magnitude–distance formula

distance modulus (mv – Mv)

luminosity (L)absolute bolometric magnitude

H–R (Hertzsprung–Russell) diagram

main sequencegiantssupergiants

red dwarfwhite dwarfluminosity classspectroscopic parallaxbinary starsvisual binary systemspectroscopic binary system

eclipsing binary systemlight curvemass–luminosity relation 

New Terms