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Chapter 9 The Family of Stars

Outline

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

1 Outline

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

Light as a Wave (1) 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

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.

2 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 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.

3 Brightness and Distance

(SLIDESHOW MODE ONLY)

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.

Distance and Intrinsic Brightness

Example: Recall that:

M Intensityagn. Ratio Betelgeuse Diff. App. Magn. m = 0.41 1 2.512 V 2 2.512*2.512 = (2.512)2 = 6.31 … … 5 (2.512)5 = 100 Rigel For a magnitude difference of 0.41 App. Magn. mV = 0.14 – 0.14 = 0.27, we find an intensity ratio of (2.512)0.27 = 1.28

4 Distance and Intrinsic Brightness (2)

Rigel is appears 1.28 times brighter than Betelgeuse,

But Rigel is 1.6 times further Betelgeuse away than Betelgeuse

Thus, Rigel is actually (intrinsically) 1.28*(1.6)2 = 3.3 times brighter than Betelgeuse. Rigel

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)

Back to our example of Betelgeuse and Rigel: Betelgeuse B etelgeuseRigel

V 0.41m 0.14

V -5.5M -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

5 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 [pc])

Distance in units of parsec

Equivalent:

d = 10(mM V – V + 5)/5 pc

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:

A Star B will be B 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 Surface area of the star 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.

6 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)

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

The Hertzsprung-Russell Diagram

M os t s ta rs M a a re in f S ou eq n u d e al nc o e ng th e

The Hertzsprung-Russell Diagram (2)

Same temperature, but much brighter than S MS stars ta rs a s ct p Ma iv en → Must be e d in lif m much larger S e o e ti st qu m o e e f n on th ce t e → Giant ( h ir Sa M e Stars me S) fa tem . int p. er → , bu Dw t arfs

7 The Radii of Stars in the Hertzsprung-Russell Diagram

Rigel Betelgeuse 1 0,0 00 s ti un me ’s s rad the ius Polaris 1 00 s tim un es ’s th rad e ius Sun

A s lar ge as th e s un

100 times smaller than the sun

Luminosity Classes

Ia Bright Supergiants Ia Ib Ib Supergiants II III II Bright Giants III Giants

IV IV Subgiants V

V Main-Sequence Stars

Example Luminosity Classes

• Our Sun: G2 star on the Main Sequence: G2V

• Polaris: G2 star with Supergiant luminosity: G2Ib

8 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)

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.

9 Center of Mass

(SLIDESHOW MODE ONLY)

Estimating Stellar Masses

Recall Kepler’s 3rd Law:

2 3 Py = aAU

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:

3 ____aAU M A + MB = 2 Py

(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:

____163 M + M = = 4 solar masses. A B 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.

10 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

11 Spectroscopic Binaries (3)

Typical sequence of spectra from a spectroscopic binary system Time

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

12 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.

The Light Curve of Algol

Masses of Stars in the Hertzsprung- Russell Diagram

The higher a star’s mass, Masses in units of the more luminous solar masses (brighter) it is: 40 3.5 L ~ M H 18 ig h High-mass stars have m a s 6 much shorter lives than se s low-mass stars: 3 1.7 t ~ M-2.5 M life 1.0 as s 0.8 0.5

L m o w a s Sun: ~ 10 billion yr. s e s 10 Msun: ~ 30 million yr.

0.1 Msun: ~ 3 trillion yr.

13 Maximum Masses of Main-Sequence Stars

Mmax ~ 50 - 100 solar masses a) More massive clouds fragment into b) Very massive stars lose smaller pieces during star formation. mass in strong stellar winds

η Carinae

Example: η 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 Gliese 229B progenitors do not get hot enough to ignite thermonuclear fusion.

→ Brown Dwarfs

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.

14 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

stellar parallax (p) red dwarf parsec (pc) white dwarf proper motion luminosity class flux spectroscopic parallax absolute visual binary stars

magnitude (Mv) visual binary system magnitude–distance spectroscopic binary formula system distance modulus eclipsing binary system

(mv – Mv) light curve luminosity (L) mass–luminosity relation absolute bolometric magnitude H–R (Hertzsprung– Russell) diagram main sequence giants supergiants

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