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0. The Family of Stars. Chapter 8:. 0. 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.

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The family of stars

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The Family of Stars

Chapter 8:


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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

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Distances to Stars

d in parsec (pc) p in arc seconds

__

1

d =

p

Trigonometric Parallax:

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

1 pc = 3.26 LY

The farther away the star is (larger d), the smaller the parallax angle p.


The trigonometric parallax

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The Trigonometric Parallax

Example:

Nearest star, a 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.


Intrinsic brightness absolute visual magnitude

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Intrinsic Brightness / Absolute Visual Magnitude

The more distant a light source is, the fainter it appears.

The same amount of light falls onto a smaller area at distance 1 than at distance 2 => smaller apparent brightness.

Area increases as square of distance => apparent brightness decreases as inverse of distance squared


Intrinsic brightness absolute visual magnitude ii

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Intrinsic Brightness / Absolute Visual Magnitude(II)

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

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Distance and Intrinsic Brightness

Example:

Recall:

Betelgeuse

App. Magn. mV = 0.41

Rigel

App. Magn. mV = 0.14

For 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 brightness1

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Distance and Intrinsic Brightness

Rigel appears 1.28 times brighter than Betelgeuse,

Betelgeuse

But Rigel is 1.6 times further away than Betelgeuse

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

Rigel


Absolute visual magnitude

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Absolute Visual Magnitude

To characterize a star’s intrinsic brightness, define absolute visual magnitude (MV):

Apparent visual magnitude that a star would have if it were at a distance of 10 pc.


Absolute visual magnitude ii

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Absolute Visual Magnitude(II)

Back to our example of Betelgeuse and Rigel:

Betelgeuse

Rigel

Difference in absolute magnitudes: 6.8 – 5.5 = 1.3

=> Luminosity ratio = (2.512)1.3 = 3.3


The distance modulus

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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(mV – MV + 5)/5 pc


The size radius of a star

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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:

Star B will be brighter than star A.

A

B

Absolute brightness is proportional to radius squared, L ~ R2.

Quantitatively: L = 4 p R2s T4

Surface flux due to a blackbody spectrum

Surface area of the star


Example

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Example:

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.


Organizing the family of stars the hertzsprung russell diagram

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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)

Absolute mag.

Hertzsprung-Russell Diagram

Luminosity

or

Temperature

Spectral type: O B A F G K M


The Hertzsprung Russell Diagram

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Most stars are found along the mainsequence


The Hertzsprung-Russell Diagram (II)

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Same temperature, but much brighter than MS stars

Must be much larger

Stars spend most of their active life time on the Main Sequence.

Giant Stars

Same temp., but fainter → Dwarfs


Radii of stars in the hertzsprung russell diagram

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Radii of Stars in the Hertzsprung-Russell Diagram

Rigel

Betelgeuse

10,000 times the sun’s radius

Polaris

100 times the sun’s radius

Sun

As large as the sun

100 times smaller than the sun


Luminosity classes

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Luminosity Classes

Ia Bright Supergiants

Ia

Ib

Ib Supergiants

II

II Bright Giants

III

III Giants

IV Subgiants

IV

V

V Main-Sequence Stars


Luminosity effects on the width of spectral lines

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Luminosity effects on the width of spectral lines

Same spectral type, but different luminosity

  • Lower gravity near the surfaces of giants

  • smaller pressure

  • smaller effect of pressure broadening

  • narrower lines


Examples

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Examples:

  • Our Sun: G2 star on the main sequence: G2V

  • Polaris: G2 star with supergiant luminosity: G2Ib


Binary stars

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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

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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.


Estimating stellar masses

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Estimating Stellar Masses

Recall Kepler’s 3rd Law:

Py2 = aAU3

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:

aAU3

____

MA + MB =

Py2

(MA and MB in units of solar masses)


Examples1

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Examples:

a) Binary system with period of P = 32 years and separation of a = 16 AU:

163

____

MA + MB = = 4 solar masses.

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

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Visual Binaries

The ideal case:

Both stars can be seen directly, and their separation and relative motion can be followed directly.


Spectroscopic binaries

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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 ii

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Spectroscopic Binaries (II)

The approaching star produces blueshifted lines; the receding star produces redshifted lines in the spectrum.

Doppler shift Measurement of radial velocities

Estimate of separation a

Estimate of masses


Spectroscopic Binaries (III)

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Typical sequence of spectra from a spectroscopic binary system

Time


Eclipsing binaries

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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 (II)

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Peculiar “double-dip” light curve

Example: VW Cephei


Eclipsing binaries iii

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Eclipsing Binaries (III)

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.


Masses of Stars in the Hertzsprung-Russell Diagram

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Masses in units of solar masses

The higher a star’s mass, the more luminous (brighter) it is:

High masses

L ~ M3.5

High-mass stars have much shorter lives than low-mass stars:

Mass

tlife ~ M-2.5

Low masses

Sun: ~ 10 billion yr.

10 Msun: ~ 30 million yr.

0.1 Msun: ~ 3 trillion yr.


The Mass-Luminosity Relation

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More massive stars are more luminous.

L ~ M3.5


Surveys of stars

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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.


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A census of the stars

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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.


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