Basics about stars

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Stars - some facts

Definition : A celestial body of hot gases that

radiates energy derived from thermonuclear reactions in

the interior.

• ≈ 1022 stars in the universe

• Masses range from 0.08 M� to 60 M�

• Radii range from 1/100 R� to 1000 R�

• Luminosity: (10−4− 106)L�

• Density: (10−6− 106)ρ�

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Observation of stars

Hipparchus first classified stars into six magnitudes.

Magnitude 1 meant brightest stars about 2 times brighter

than magnitude 2 etc.

Modern definition of the apparent brightness in terms

of the radiant flux F is the following :

m = −2.5logF

F0

.

It is related to the luminosity L of the star by

F =L

4πd2

To compare stars one defines absolute magnitude M :

The absolute magnitude is the apparent magnitude of a

star located at a distance of 10 parsec.

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From the equations above and this definition directly

follows :

m − M = −2.5logF

F10

m − M = −2.5log(10pc

d[pc])2

M = m − 5log(d) + 5

m-M is therefore a measure of the distance of a star and

is called distance modulus .

For our sun one finds

m� = −26.81, Msun = 4.72

Another relation following directly from the definition of

the absolute Magnitude is :

M = Msun − 2.5logL

L�

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The color index

The apparent magnitude m and absolute magnitude

M introduced earlier are bolometric magnitudes,

measured over all wavelenghts of light emitted by a star.

In practice a UBV system is used where a star’s apparent

magnitude is measured through 3 filters.

• U(mU ), the ultraviolet magnitude at 3650 ± 350A

• B, the blue magnitude at 4400 ± 500A

• V, the visual magnitude at 5500 ± 450A

Figure 1: Sensitivity function of the UBV filters.

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If the distance is known one can also determine the

absolute color magnitudes MU , MB and MV . The

difference between B-V and U-B is called color index

C of a star. (The color index does not depend on the

distance.)

The difference of mbol − V = Mbol − MV is called the

Bolometric Correction B.C.

Figure 2: Color-Color diagram for main sequence stars.

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Blackbody radiation

Stars (and planets) are approximately blackbodies.

Planck found that the emission of light from blackbodies

follows the function

Bν(T ) =2hν3/c

ehν/kT − 1

Figure 3: Blackbody radiation spectrum

The Planck function peaks at a wavelength λmax which

– 7 –

is given by Wien’s displacement law

λmax ∗ T = 0.29cm K

The Temperature of a blackbody is related to its

luminosity by the Stefan-Boltzmann equation

L = AσT 4

Making the assumption that stars are spherical one

arrives at

L = 4πr2σT 4

e

Te is then defined as the effective temperature of a

star with luminosity L and radius r.

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Classification of stellar spectra

Spectral Type Characteristics

O Hottest blue-white stars with few lines.

Strong He II absorption lines

Stong UV continuum

B Hot blue-white stars

He I absorption line strongest at B2

H I (Balmer) absorption lines becoming stronger

A White stars

Balmer absorption lines strongest at A0

Ca II absorption lines strength increasing

F Yellow white stars

Ca II lines continue to strengthen

Neutral metal absorption lines ( Fe I,Cr I)

G Yellow stars

Solar-type spectra

Fe I and other neutral metal lines becoming stronger

K Cool orange stars

Ca II H and K lines strongest at K0,weakening afterwards

Spectra dominated by metal absorption lines

M Coolest red stars

Spectrum dominated by molecular absorption bands

Neutral metal absorption lines remain strong

Historically the ordering was done alphabetically according to

intensity ratios of Balmer lines to other lines. Few years later

the modern ordering which is based upon the effective surface

temperature was established.

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Figure 4: Spectral line strength dependence on tempera-

ture.

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The MKK System &

Stellar populations

• Baade (1944): Different statistical properties for

stars in the nucleus and spiral arm of a galaxy.

• Introduction of Population I and Population II stars.

Population I Population II

Young Old

Spiral arm Nucleus and halo

High metal content low metal content

mainly O,B type stars K,M type stars

• Stars of same spectral type (temperature) had

different line strengths.

⇒ MKK-system (Morgan, Keenan and Kellman) of

luminosity classes.

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Class Type of star

Ia-O Extreme, luminous supergiants

Ia Luminous supergiants

Ib Less luminous supergiants

II Bright giants

III Normal giants

IV Subgiants

V Main-sequence stars

VI,sd Subdwarfs

D White dwarfs

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The Hertzsprung-Russell diagram

Figure 5: Luminosity classes in the HRD.

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Figure 6: Color magnitude diagram for the globular clus-

ter M3 .

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Figure 7: Hertzsprung-Russell diagram of L vs T.

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Figure 8: Schematic HR diagram.

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Types of Stars

• Main sequence stars: Stars burning hydrogen to

helium in the core.

• Subgiant branch: Hydrogen burning in a thick

shell.

• Red-giant branch: Hydrogen burning in a thin

shell with a growing helium core till helium ignites.

• horizontal branch: ejection of parts of the

envelope, helium burning in the core and hydrogen

burning in a shell.

• Post-asymptotic branch: Final evolution to the

white dwarf.

• White dwarf: Electron degeneracy pressure is

responsible for maintaining hydrostatic equilibrium.

Cooling happens via electron conduction.

• Wolf-Rayet stars: Hot and massive stars with a

high mass loss. Strong emission lines can be seen.

Probably stars in the late stages of their evolution.

• T Tauri stars: Very young solar like stars. ”Pre

main-sequence” stars.

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Variables

Variable (pulsating) stars are very important for modern

astronomy. First detected in 1595 (’o Ceti’/Mira)

they serve today as standard candles to determine

extragalactic distances.

There are different types of variables appearing in the

HRD. Basically, whenever the ”instability strip” overlaps

with an actual stellar population variable stars are found.

• δ-Cepheids : Luminosities range from 300−3000L�

and periods between 1-50 days. Brightness variation

during 1 period can be up to 1 magnitude (visual)

• W Virginis stars : Metal-deficient Population II

cepheids. They are less luminous than ”classical”

cepheids with the same period (about a factor of 4).

Periods range from 2-45 days.

• RR Lyrae stars : Luminosities between 50−100L�

and periods of about 1.5-24 h. These are Population

II stars found in globular clusers.

• δ-Scuti stars : Short periods of about 1-3 hours.

• ZZ Ceti stars : Pulsating white dwarfs. Very

low luminosities and periods between 100 and 1000

seconds.

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Figure 9: Mass luminosity relation.

log10

L

L�

= 1.15log10Pd + 2.47

MV = −2.80log10Pd− 1.43

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Pulsating stars are the result of sound waves resonating

in their interior.

Eddington was the first to suggest a valve mechanism.

He proposed that in some layers of the star the opacity

shall increase with compression. Later these layers were

identified with partial ionization zones where heat was

absorbed during compression to increase the number of

ions.

The zone which is driving the pulsation for the variables

in the instability strip is the He II partial ionisation zone.

This is called the κ-mechanism.

Cepheids pulsating in the first overtone have a slightly

modified period-luminosity relation.

Figure 10: period-luminosity relation for cepheids.

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Distance of stars

1.) Trigonometric parallax :

Using the baseline of the earth’s orbit around the sun

(2 AE) distances up to 1000 parsec (spacecraft!) can be

measured.

1 parsec (parallax second) is defined as the distance of a

star with a parallax angle of 1 second.

2.) Spectroscopic parallax :

Determining the luminosity class of a star and its po-

sition in a HR diagram one can deduce its absolute

magnitude. Together with the apparent magnitude the

distance can then be calculated.

2b.) Main-sequence fitting :

Observing the color and apparent luminosities of stars in

a cluster and comparing with a ”calibrated” HR diagram,

one identifies the main-sequence and from the difference

of apparent to absolute magnitude the distance.

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3.) Variable stars :

Variable stars as RR Lyrae and Cepheids have a Period-

Luminosity relation. By measuring the distance to

nearby Cepheids (using eg method 1 or 2) one can

calibrate the PLR to absolute magnitudes.

A measurement of a cepheids period can then be used

to determine the absolute luminosity and therefore the

distance. RR Lyrae can be used for distances within our

galaxy, cepheids for distances up to 107 parsecs.

Cepheids are used as ”standard” candles and are called

primary distance indicators

4.) Secondary distance indicators :

One determines absolute luminosities for brightest blue

stars, brightest red stars, brightest globular clusters,

planetary nebulae, etc. For galaxies even further away

one uses these information to determine the distance.

5.) Tertiary distance indicators Giant elliptic

galaxies and giant spiral galaxies are brightest galaxies

in the universe. Making use of the methods described

in 4.) one determines the distance to the nearest giant

Sc or giant E galaxy and then compare those with giant

galaxies even further away to get an estimate of their

distance.

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Size and density of stars

There are several methods determining the radius of a

star:

• Interferometrie (close and big stars preferrably)

• eclipsing binaries

• the Stefan-Boltzmann law

R =1

T 2e

√

L

4πσ

Calculating the according mean density delivers stars

with average densities between 10−6g/cm3 and 106g/cm3

(For comparison : ρ� = 1.41g/cm3)

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Determination of masses

Masses of stars are determined from binary systems.

Using Keplers third law

P 2 =4π2

G(m1 + m2)a3

and measuring the radial separations from r1, r2 from

the center of mass, the masses can be determined.

Figure 11: Mass luminosity relation.

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Structure of stars

1. Hydrostatic Equilibrium

dP

dr= −G

Mrρ

r

2. Mass conservation

dM

dr= 4πrρ

3. Energy conservation

dL

dr= 4πrρ(εg + εn + εv)

4. Radiative Transport

dT

dr= −

3

4ac

κρ

T

L

4πr

5. Convective Transport

dT

dr= (1 −

1

γ)T

P

dP

dr

Vogt-Russell Theorem :

The mass and composition of a star uniquely determine

its radius, luminosity, internal structure, as well as its

subsequent evolution.

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The fate of stars

• White dwarfs : Stars with ZAMS mass below 8M�

end as white dwarfs. Consisting of a Carbon Oxygen

core electron degeneracy supports the star against

the pull of gravity.

The Chandrasekhar limit prohibits WD‘s more

massive than 1.44M�

• Neutron stars : Stars with initially up to 25M�

end their life as Neutron stars. A supernova Type II

is associated with the formation of a neutron star

• Black hole : For more massive stars even the

neutron degeneracy pressure can not stop the core

collapse. A black hole is formed.

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Summary

• Stars have a broad range of temperatures, masses,

luminosities.

• Stars (MS stars) are located on a small band in the

HRD.

• MS stars can in a first approximation be parametrized

by 1 parameter - the mass.

• The evolution of stars from the proto star - MS -

final stage can be traced by looking at a HRD.

• The MS of the HRD when compared to the MS

of a far away galaxy can be used to determine

extragalactic distances.

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