The Family of Stars

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The Family of StarsWe already know how to determine a star’s

• surface temperature

• chemical composition

Now, see 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 of Stars

Trigonometric Parallax:

Star appears slightly shifted from different positions of the Earth on its orbit

The further 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 ParallaxExample:

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 further away than 50 pc.

Sirius, the brightest star in the sky, has a trigonometric parallax of p = 0.385 arc

seconds. What is its distance from Earth?

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1. 0.385 pc

2. 0.80 light years

3. 1.255 pc

4. 2.60 light years

5. 8.47 light years

The method of trigonometric parallaxes (from ground based telescopes) allows us to

measure distances …

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1. only to objects in our solar system.

2. only to stars in our solar neighborhood within the Milky Way.

3. to stars throughout the entire Milky Way.

4. to stars and galaxies throughout the Local Group.

5. even to other clusters of galaxies.

Intrinsic Brightness / Absolute Magnitude

The further away a light is, the fainter it appears.

Intrinsic Brightness / Absolute Magnitude (II)

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

The stars A and B have the same intrinsic luminosity, but A is 5 times

further away from Earth than B. Then:

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1. Both stars will appear equally bright.

2. A will appear 5 times brighter than B.

3. B will appear 5 times brighter than A.

4. A will appear 25 times brighter than B.

5. B will appear 25 times brighter than A.

EarthStar BStar A

Distance and Intrinsic Brightness

Betelgeuze

Rigel

Example:

App. Magn. mV = 0.41

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 = 0.14

For a magnitude difference of 0.41 – 0.14 = 0.27, we found: Rigel appears 1.28 times brighter than Betelgeuze

But Rigel and Betelgeuze may be at quite different distances from us!

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.

If we know a star’s absolute magnitude, we can infer its distance by comparing

absolute and apparent magnitudes.

Absolute Magnitude (II)

Betelgeuze

Rigel

Betelgeuze Rigel

mV 0.41 0.14

MV -5.5 -6.8

d 152 pc 244 pc

Difference in absolute magnitudes:

6.8 – 5.5 = 1.3

=> Luminosity ratio = (2.512)1.3 = 3.3

Rigel is actually 3.3 times brighter than Betelgeuze!

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 B

Star B will be brighter than star A.

Specifically: Absolute brightness is proportional to radius (R) squared, L ~ R2.

Example:

Both Spica B and Sirius B are B-type stars, but Sirius B is a white dwarf star, with a

radius ~ 560 times smaller than Spica B.

Thus, since L ~ R2, Sirius B is intrinsically

5602 ≈ 320,000

times fainter than Spica B.

Polaris has just about the same spectral type (and surface temperature) as our sun,

but it is 10,000 times brighter. Thus, Polaris’ radius is … the sun’s radius.

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1. the same as

2. 100 times larger than

3. 100 times smaller than

4. 10,000 times larger than

5. 10,000 times smaller than

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

The Hertzsprung Russell Diagram

Most stars are found along the Main Sequence

The Hertzsprung-Russell Diagram (II)

Stars spend most of their active life

time on the Main Sequence (MS).

Same temperature,

but much brighter than

MS stars

→ Must be much larger

→ Giant Stars

Same temp., but

fainter → Dwarfs

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 Betelgeuze

Sun

Polaris

Luminosity Classes

Ia Bright Supergiants

Ib Supergiants

II Bright Giants

III Giants

IV Subgiants

V Main-Sequence Stars

IaIb

II

III

IV

V

Examples:

• Our Sun: G2 star on the Main Sequence:

G2V

• Polaris: G2 star with Supergiant luminosity:

G2Ib