The Family of StarsChapter 9

If you want to study anything scientifically, the first thing you have to do is find a way to measure it. But measurement in astronomy is very difficult. Astronomers must devise ingenious methods to find the most basic properties of stars. As you will see in this chapter, combining those basic properties reveals important relationships among the family of stars. Your study of stars will reveal answers to five basic questions:

• How far away are the stars?

• How much energy do stars make?

• How big are stars?

• What is the typical star like?

Guidepost

Making measurements is the heart of science, and this chapter will answer two important questions about how scientists go about their work:

• How can scientists measure properties that can’t be directly observed?

• How do scientists accumulate and use data?

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 and how it works, you must understand the stars.

Guidepost (continued)

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

II. Intrinsic BrightnessA. Brightness and DistanceB. Absolute Visual Magnitude (絕對星等 )C. Calculating Absolute Visual MagnitudeD. Luminosity

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

Outline

D. Interferometric Observations of DiameterE. Luminosity ClassificationF. Luminosity ClassesG. Spectroscopic Parallax (視差 )

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 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 (三角視差 ):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 (arcsec)

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 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.

Intrinsic Brightness / Absolute Magnitude (2)

More quantitatively:

The flux received from the light is proportional to its intrinsic brightness or luminosity (L, unit: erg/s or J/s) 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 Modulus (距離模數 )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 [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:

A BStar 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 starSurface flux due to a blackbody spectrum

Example: Star Radii

Polaris (北極星 ) 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)

Lum

inos

ity

Temperature

Spectral type: O B A F G K M

Hertzsprung-Russell Diagram (赫羅圖 )or

Abs

olut

e m

ag.

The Hertzsprung-Russell Diagram

Most stars are found along the

Main Sequence

The Hertzsprung-Russell Diagram (2)

Stars spend most of their

active life time on the M

ain

Sequence (MS).

Same temperature,

but much brighter than

MS stars

The Brightest StarsThe 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

10,000 times the

sun’s radius

100 times the

sun’s radius

As large as the sun

Rigel Betelgeuse

Sun

Polaris

Eyeball:Balloon = Sun:Supergiants

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

IIIII

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

systemBoth 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 ____ 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 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 Diagram

The higher a star’s mass, 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.

Low

masses

High masses

Mass

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.