Slide 1

The Family of StarsChapter 9

Slide 2

Part 1: measuring and classifying the stars

• What we can measure directly: – Surface temperature and color– Spectrum– Apparent magnitude or intensity– Diameter of a few nearby stars– Distance to nearby stars

• What we usually cannot: – Distance to most stars– Luminosity (energy radiated per

second)– Diameter and mass

Slide 3

Surface temperature and color indices

Color indices:B-V, U-B

Color filters

Differences in apparent magnitudes observedthrough different filters

K)(

103)nm(

6

T

Slide 4

Spectral Classification of Stars

Mnemonics to remember the spectral sequence:

Oh Oh Only

Be Boy, Bad

A An Astronomers

Fine F Forget

Girl/Guy Grade Generally

Kiss Kills Known

Me Me Mnemonics

Slide 5

How to find distances to the stars?

• Parallax (only for stars within ~1500 ly)

• From stellar motions

• For moving clusters

• Using “standard candles” (model-dependent)

• Using mass-luminosity relation (for main-sequence stars) or period-luminosity relations (for binaries and variable stars; model-dependent)

Slide 6

When d is too large, angles A and B become too close to 900

The larger the baseline, the longer distances we can measure

Slide 7

Apparent shift in the position of the star: parallax effect

The longest baseline on Earth is our orbit!

Angular shift; we can measure it directly

Slide 8

Effect is very small: shift is less than 1 arcsec even for closest stars

Aristotle used the absence of observable parallax to discard heliocentric system

Larger shift

Smaller shift

Slide 9

Half of the angular shift is called parallax angle p and used to define new unit of distance

Slide 10

The parallax angle p

arcsec) in(

AU1206265

pd

Define 1 parsec as a distance to a star whose parallax is 1 arcsec

d (in parsecs) = 1/p1 pc = 206265 AU = 3.26 ly

dp

AU1206265arcsec) in(

Small-angle formula:

Slide 11

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

With Hipparcos satellite: parallaxes up to 0.002 arcsec, i.e. d up to 500 pc.

118218 stars measured!

Slide 12

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.

Slide 13

Barnard’s star: highest proper motion10 arcsec per year, or one lunar diameter per 173 yrApproaches us at 160 km/secFourth closest star

Slide 14

Brightness and distance

• Apparent magnitude: tells us how bright a star looks to our eyes

Intensity, or radiation flux received by the telescope: Energy of radiation coming through unit area of the mirror per second (J/m2/s)

Slide 15

Brightness and Distance

The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):

24 d

LI

R

d

L

Slide 16

d1

d2

2221

21 44 IdIdL 2

11 4 d

LI

2

22 4 d

LI

24 d

LI

Slide 17

Intrinsic Brightness, or luminosity

The flux received from the star is proportional to its intrinsic brightness or luminosity (L) and inversely proportional to the square of the distance (d):

I = L__

4d2

Star AStar B Earth

Both stars may appear equally bright, although star A is intrinsically much brighter than star B.

Slide 18

Brightness and Distance

(SLIDESHOW MODE ONLY)

Slide 19

Define the magnitude scale so that two objects that differ by 5 magnitudes have an intensity ratio of 100.

100;5 B

AAB I

Imm 512.2100;1 5

B

AAB I

Imm

AB mm

B

A

I

I )512.2(

B

AAB I

ILogmm 5.2)(

Order of terms matters!

Recall the definition of apparent magnitude:

Slide 20

However, the apparent magnitude mixes up the intrinsic brightness of the star (or luminosity) and the effect of distance (which has nothing to do with the luminosity of the star).

Inverse square law: 2

1

22

2

1

2

12

;4 d

d

L

L

I

I

d

LI

Slide 21

Slide 22

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

Slide 23

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 more luminous than Betelgeuse.

But Rigel is 1.6 times further away than Betelgeuse

Slide 24

A star that is very bright in our sky could be bright primarily because it is very close to us (the Sun, for example), or because it is rather distant but is intrinsically very bright (Rigel, for example). It is the "true" (intrinsic) brightness, with the distance dependence factored out, that is of most interest to us as astronomers.

Therefore, it is useful to establish a convention whereby we can compare two stars on the same footing, without variations in brightness due to differing distances complicating the issue.

Astronomers define the absolute magnitude M to be the apparent magnitude that a star would have if it were (in our imagination) placed at a distance of 10 parsecs (which is 32.6 light years) from the Earth.

To determine the absolute magnitude M the distance to the star must also be known!

Absolute Magnitude

Slide 25

Absolute magnitude

1

212 log5.2

I

Imm Recall that for two stars 1 and 2

Let star 1 be at a distance d pc and star 2 be the same star brought to the distance 10 pc.

Then 2

2

1

2

10

d

I

I 2log210logloglog 22

1

2 ddI

I

1

212 log5.2

I

Imm

5log5 dmM

5/)5(10pc)( MmdInverse:

m2 = M

Slide 26

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

Slide 27

Absolute magnitudes of two different stars 1 and 2:

1

2

1

2

L

L

I

I

1

2

1

212 log5.2log5.2

L

L

I

IMM

If two stars are at the same distance of 10 pc from the earth:

Slide 28

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

Slide 29

Is there any correlation between stellar luminosities, radii, temperature, and masses???

We learned how to characterize stars with many different parameters

Organizing the Family of Stars

Slide 30

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

But luminosity also increases with size:

A BStar B will be brighter than

star A.

Luminosity is proportional to radius squared, L ~ R2.

Quantitatively: L = 4 R2 T4

Surface area of the starSurface flux due to a blackbody spectrum

Slide 31

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.

Slide 32

However, star radius is not a convenient parameter to use for classification, because it is not directly measured.

Surface temperature, or spectral class is more convenient!

Slide 33

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

orA

bsol

ute

mag

.

Slide 34

Hertzsprung-Russell Diagram1911 1913

Abs

olut

e m

agni

tude

Color index, or spectral class

Betelgeuse

Rigel

Sirius B

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Stars in the vicinity of the Sun

Slide 38

Stars in the vicinity of the Sun

Slide 39

90% of the stars are on the Main Sequence!

Slide 40

Total radiated power (luminosity) L = T4 4R2 J/s

Check whether all stars are of the same radius:

Slide 41

No, they are not of the same radius

Slide 42

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

Slide 43

Specific segments of the main sequence are occupiedby stars of a specific mass

Majority of stars are here

Slide 44

The mass-luminosity relation for 192 stars in double-lined spectroscopic binary systems.

L ~ M3.5 much stronger than inferred from L ~ R2 ~ M2/3

Slide 45

However, this M3.5 dependence does not go forever:Cutoff at masses > 100 M and < 0.08 M

Slide 46

All stars visible to the naked eye + all stars within 25 pc

H-R diagram for nearby+bright stars:

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