Download Basics about stars

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 :
F
m = −2.5log .
F0
It is related to the luminosity L of the star by
L
F =
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 :
F
m − M = −2.5log
F10
m − M = −2.5log(
10pc 2
)
d[pc]
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 − 2.5log
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L
L
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 ± 350Å
• B, the blue magnitude at 4400 ± 500Å
• V, the visual magnitude at 5500 ± 450Å
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
2hν 3 /c
Bν (T ) = hν/kT
e
−1
Figure 3: Blackbody radiation spectrum
The Planck function peaks at a wavelength λmax which
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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 = 4πr 2 σTe4
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
O
Characteristics
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 temperature.
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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
Ia-O
Type of star
Extreme, luminous supergiants
Ia
Luminous supergiants
Ib
Less luminous supergiants
II
Bright giants
III
Normal giants
IV
Subgiants
V
Main-sequence stars
VI,sd
D
Subdwarfs
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 cluster 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:
shell.
Hydrogen burning in a thick
• 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
= 1.15log10 P d + 2.47
L
MV = −2.80log10 P d − 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 position 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 PeriodLuminosity 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
Te2
r
L
4πσ
Calculating the according mean density delivers stars
with average densities between 10−6 g/cm3 and 106 g/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
4π 2
P =
a3
G(m1 + m2 )
2
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
Mr ρ
dP
= −G
dr
r
2. Mass conservation
dM
= 4πrρ
dr
3. Energy conservation
dL
= 4πrρ(εg + εn + εv )
dr
4. Radiative Transport
dT
3 κρ L
=−
dr
4ac T 4πr
5. Convective Transport
dT
1 T dP
= (1 − )
dr
γ P 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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