Review from last class:
• Properties of photons
• Flux and luminosity, apparent magnitude and absolute magnitude, colors
• Spectroscopic observations. Doppler’s effect and applications
• Distance measurements and standard candles
• The Hubble constant and redshifts as distances
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Stars in galaxies
Why do we bother to study stars?
Galaxies are systems of stars. The stellar population is one of the most importantdefining properties of galaxies.
Stars are also the hosts of planets.
Here only a very brief review
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Basic properties of stars
Except for very nearby ones, stars are not resolved. We cannot observe directlytheir internal structure. Stars are observed as bright points
• Quantities that can be observed: flux, color and spectrum
• For the sun:
– The solar constant (the energy flux we receive on Earth): f = 1360Wm−2 ,– The distance of the sun to us is r = 1AU≈ 1.5×1013 cm– Conservation of energy, the luminosity of the sun is
L� = 4π f r2 ∼ 3.9×1026W.– The luminosity is an intrinsic property of the sun
• The flux is an observable, it depends on the distance of the observer
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Color of a star
Color: the difference between the magnitudes in two broad bands.
(mX−mY) =−2.5log( fX/ fY) . (1)
Color is related to the surface temperature:
A normal star may be considered as a blackbody:
• Blackbody: it can absorb all kinds of photons; but blackbody may not be black
• Blackbody radiation has the Planck spectrum:
Bλ(T ) =2hc2
λ5
1ehc/λkT −1
, (2)
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where T is the temperature of the body; h Planck constant; k is the Boltzmannconstant.
[Bλ(T )dλ] = Wm−2
• This spectrum depends only on temperature.
• Wien displacement law: Maximum radiation occurs at a wavelength given by
dBλ(T )dλ
= 0 ,
which gives
λmax =(
5KThc
)−1
• What is Bν(T )dν?
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• Radiation energy density:
Erad =4π
c
Z∞
0Bλ(T )dλ = aT 4
where a = 7.56×10−16Jm−3K−4 is the radiation constant.
• Stefan-Boltzmann law for blackbody radiation: Surface Brightness (regardlesswavelength):
f = σT 4 ,
where σ = 5.7×10−8W/(m2K4) is the Stefan-Boltzmann constant.
• ThusL = 4πR2 f ,
where R is radius of the star.
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• A way to estimate the radius of a star:
R =(
L4πσT 4
)1/2
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Stellar spectrum
• Continuum: blackbody radiation
• Spectral lines: mainly absorptionlines due to absorption of stellaratmosphere.
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Classification of stars according to color
O blue >25000 KB blue 11,000 - 25,000 KA blue 7,500-11,000 KF Blue-white 6,000-7,500 KG White-yellow 5,000 - 6,000 KK orange red 3500-5000 KM red 3500 K
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The H-R diagram
Color-magnitude relation ortemperture-luminosity relation
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• Main sequence: hydrogen burning in the core
• subgiant: transition from core to shell H burning, core shrink, outer layersexpand, luminosities regulated by photon diffusion to be more or less constant
• red giant: inert helium core, hydrogen shell burning; cooling causesconvection so that temperature remains constant while luminosity increases
• helium flash, causing shell expansion and reducing rate of hydrogen shellburning, luminosity drops rapidly. Because here the core is supported bydegeneracy pressure, instead of thermal pressure, it is a run-away process
• horizontal branch: helium burning in the core, luminosity almost constant
• asymptotic giant branch: double shell burning, inert carbon core
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• planetary nebula
• white dwarf
The most important for us: main sequence and red-giant sequence.
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Stellar structure and evolution
Normal stars are very simple systems:
• Almost spherical
• Almost static: evolving very slowly.The sun has not changed for the past 5 Gyr, and will not change for another5 Gyr.
• Basic equations governing stellar structure: spherical symmetry and static,and so all quantities are functions of r
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The Main sequence age
Basic equations:
• Mass conservation:dM(r)
dr= 4πr2
ρ(r)
• Hydostatic equilibrium:dPdr
=−GM(< r)ρ(r)r2
• Energy flux and luminosity
F =−λdTdr
; L = 4πr2F ,
where λ is the energy transport coefficient.
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Energy transport due to photon diffusion:
L∼ 4πr3aT 4
td,
where td is the time it takes for a photon to diffuse from the star interior to theexterior:
td = r2/(cl) ,
where l is the mean-free path of a photon in a star.
• For stars with low to medium mass:
l ∝ T 3.5/ρ2 .
• For stars with high to very high masses:
l ∝ 1/ρ .
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Simple scaling relations for a star of mass M, radius R
• Mass and pressure:
ρ ∝MR3 , P ∝
GMρ
R∝
GM2
R4 ,
• For stars with low to high mass, pressure is dominated by gas:
P ∝ ρT
For stars witt very high mass, pressure is dominated by radiation:
P ∝ T 4 .
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• For stars with low to medium mass:
L ∝R3T 4l
R2 ∝R7T 7.5
M2 ∝R7
M2
(Pρ
)7.5
∝ M5.5/R0.5
For stars with high mass:L ∝ M3 .
For stars with very high mass:L ∝ M .
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For stars with low to high mass
L ∝ M4
• The age of main sequence star:
tMS ∝ M/L ∝ M−3 ,
since the total amount of fuel is ∝ M, and the rate of consumption is ∝ L.
• Radius of a star R ∝ M. Using L = 4πR2σT 4e , we have
Te ∝ L1/4/R1/2∝ M/M1/2
∝ M1/2 .
Thus, More massive stars are brighter, hotter, and have shorter lifetimes:
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Summary
Table 1: Properties of stars.Mass L Te f f Spec.type tMS tred EMS EpMS
(M�) (L�) (K) (Myr) (Myr) Gyr×L� Gyr×L�0.8 0.24 4860 K2 25000 – 10 –1.0 0.69 5640 G5 9800 3200 10.8 241.5 4.7 7110 F3 2700 900 16.2 132.0 16 9080 A2 1100 320 22.0 183.0 81 12250 B7 350 86 38.5 195.0 550 17181 B4 94 14 75.2 239.0 4100 25150 – 26 1.7 169 4040.0 240000 43650 O5 4.3 0.47 1500 112
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The idea of spectral synthesis
• To get the spectrum a galaxy which contains stars with different masses andages.
• Simple case: a coeval population of age t, initial mass function
• More complicated case: star formation rate
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The initial mass function
Note that the properties and evolution of a star is largely determined by its initialmass: L ∝ M4, Teff ∝ M1/2, R ∝ M, tMS ∝ M−3.
The initial mass function (IMF), φ(m)dm, which is normalized as
Zmφ(m)dm = M� ,
is the number of stars with birth masses in the range m → m + dm among apopulation of stars with a total mass of M�.
Thus, if the total mass that is converted into stars is M, then (M/M�)φ(m)dm isthe number of stars with masses in m±dm/2.
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What is (M/M�)mφ(m)dm?
What are
Z m2
m1
MM�
φ(m)dm
and Z m2
m1
(M/M�)mφ(m)dm
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The observed IMF
The initial mass function (IMF) is determined observationally. The Salpeter IMFhas the form
φ(m) = Am−2.35 ,
for 0.1M� ≤ m < 100M�.
The observed IMF is still uncertain; there are other forms: e.g. Scalo IMF,Kroupa IMF, Chabrier IMF, etc.
It is also unclear if the IMF is universal.
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The spectral energy distribution for an coevel populaiton
The spectral energy distribution of a star is expected to be determined by itsinitial mass m and its age τ:
Lλ(m,τ) ,
where Lλ is the energy emitted by the star at wavelength λ. Then the spectralenergy distribution of a coeval population of age τ can be written as
L (cp)λ
(τ) =Z
∞
0Lλ(m,τ)
φ(m)M�
dm ,
which is the energy emitted per unit mass.
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Star formation rate
Now suppose the star formation rate (i.e. the mass that turns into stars in a unittime) is known as a function of time, ψ(t), then the luminosity of the galaxy atany time t can be written as
Lλ(t) =Z t
0L (cp)
λ(t− t ′)ψ(t ′)dt ′ .
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So the spectrum of a galaxy is determined by three quantities:
• Star formation rate (history): ψ(t)
• The initial mass function (IMF): φ(m)
• Spectra of individual stars: Lλ(m,τ)
Since Lλ(m,τ) can be obtained from stellar evolution, and if the IMF is universal(determined from obsrvation), the spectral energy distribution may be used toinfer its star formation history.
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Synthesized spectra for coeval populations
The predicted spectra of a singlestarburst at ages 0.001, 0.01, 0.1, 0.4,1, 4, and 13 Gyr.
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Properties
• At an age younger than ∼ 107 yr, the spectrum is completetly dominated bythe blue main sequence stars, which have strong emission in the UV due totheir high effective temperature.
• At the age of about 107 yr, the most massive stars have already evolved offthe main sequence and become red supergiants, which causes a drop in UVflux and a rise in near-infrared flux in the synthesized spectrum.
• From a few times 108 yr to about 1×109 yr, the AGB stars maintain a relativelyhigh flux in the near-infrared, while the UV flux continues to drop as more andmore low-mass stars evolve off the main sequence.
• After a few times 109 yr, stars in the red giant branch take over as the maincontributors of the near-infrared flux.
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Broard-band luminosity and colors
• The broad-band luminositiesdecrease with age, and thedecrease is more rapid for abluer band, because of thedisappearance of bright mainsequence stars and supergiant.
• Galaxies also become redder withtime, as more and more starsevolve off the main sequence.
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The observed spectra of different galaxies
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