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
Stars in galaxies Why do we bother to study stars? Galaxies are systems of stars. The stellar population is one of the most important defining properties of galaxies. Stars are also the hosts of planets. Here only a very brief review
Basic properties of stars Except for very nearby ones, stars are not resolved. We cannot observe directly their 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 10 13 cm Conservation of energy, the luminosity of the sun is L = 4π f r 2 3.9 10 26 W. The luminosity is an intrinsic property of the sun The flux is an observable, it depends on the distance of the observer
Color of a star Color: the difference between the magnitudes in two broad bands. (m X m Y ) = 2.5log( f X / f Y ). (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: B λ (T ) = 2hc2 λ 5 1 e hc/λkt 1, (2)
where T is the temperature of the body; h Planck constant; k is the Boltzmann constant. [B λ (T )dλ] = Wm 2 This spectrum depends only on temperature. Wien displacement law: Maximum radiation occurs at a wavelength given by db λ (T ) dλ = 0, which gives What is B ν (T )dν? λ max = ( 5KT hc ) 1
Radiation energy density: E rad = 4π c Z 0 B λ (T )dλ = at 4 where a = 7.56 10 16 Jm 3 K 4 is the radiation constant. Stefan-Boltzmann law for blackbody radiation: Surface Brightness (regardless wavelength): f = σt 4, where σ = 5.7 10 8 W/(m 2 K 4 ) is the Stefan-Boltzmann constant. Thus where R is radius of the star. L = 4πR 2 f,
A way to estimate the radius of a star: R = ( L ) 1/2 4πσT 4
Stellar spectrum Continuum: blackbody radiation Spectral lines: mainly absorption lines due to absorption of stellar atmosphere.
Classification of stars according to color O blue >25000 K B blue 11,000-25,000 K A blue 7,500-11,000 K F Blue-white 6,000-7,500 K G White-yellow 5,000-6,000 K K orange red 3500-5000 K M red 3500 K
The H-R diagram Color-magnitude relation or temperture-luminosity relation
Main sequence: hydrogen burning in the core subgiant: transition from core to shell H burning, core shrink, outer layers expand, luminosities regulated by photon diffusion to be more or less constant red giant: inert helium core, hydrogen shell burning; cooling causes convection so that temperature remains constant while luminosity increases helium flash, causing shell expansion and reducing rate of hydrogen shell burning, luminosity drops rapidly. Because here the core is supported by degeneracy 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
planetary nebula white dwarf The most important for us: main sequence and red-giant sequence.
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 another 5 Gyr. Basic equations governing stellar structure: spherical symmetry and static, and so all quantities are functions of r
The Main sequence age Basic equations: Mass conservation: dm(r) dr = 4πr 2 ρ(r) Hydostatic equilibrium: Energy flux and luminosity dp dr = GM(< r)ρ(r) r 2 F = λ dt dr ; L = 4πr2 F, where λ is the energy transport coefficient.
Energy transport due to photon diffusion: L 4πr3 at 4 t d, where t d is the time it takes for a photon to diffuse from the star interior to the exterior: t d = r 2 /(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 /ρ 2. For stars with high to very high masses: l 1/ρ.
Simple scaling relations for a star of mass M, radius R Mass and pressure: ρ M R 3, P GMρ R GM2 R 4, 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.
For stars with low to medium mass: L R3 T 4 l R7 T 7.5 R 2 M 2 R7 M 2 ( ) 7.5 P M 5.5 /R 0.5 ρ For stars with high mass: For stars with very high mass: L M 3. L M.
For stars with low to high mass L M 4 The age of main sequence star: t MS 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 = 4πR 2 σt 4 e, we have T e L 1/4 /R 1/2 M/M 1/2 M 1/2. Thus, More massive stars are brighter, hotter, and have shorter lifetimes:
Summary Table 1: Properties of stars. Mass L T e f f Spec.type t MS t red E MS E pms (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 24 1.5 4.7 7110 F3 2700 900 16.2 13 2.0 16 9080 A2 1100 320 22.0 18 3.0 81 12250 B7 350 86 38.5 19 5.0 550 17181 B4 94 14 75.2 23 9.0 4100 25150 26 1.7 169 40 40.0 240000 43650 O5 4.3 0.47 1500 112
The idea of spectral synthesis To get the spectrum a galaxy which contains stars with different masses and ages. Simple case: a coeval population of age t, initial mass function More complicated case: star formation rate
The initial mass function Note that the properties and evolution of a star is largely determined by its initial mass: L M 4, T eff M 1/2, R M, t MS M 3. The initial mass function (IMF), φ(m)dm, which is normalized as Z mφ(m)dm = M, is the number of stars with birth masses in the range m m + dm among a population 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 is the number of stars with masses in m ± dm/2.
What is (M/M )mφ(m)dm? What are and Z m2 m 1 Z m2 m 1 M M φ(m)dm (M/M )mφ(m)dm
The observed IMF The initial mass function (IMF) is determined observationally. The Salpeter IMF has the form φ(m) = Am 2.35, for 0.1M m < 100M. The observed IMF is still uncertain; there are other forms: e.g. Kroupa IMF, Chabrier IMF, etc. Scalo IMF, It is also unclear if the IMF is universal.
The spectral energy distribution for an coevel populaiton The spectral energy distribution of a star is expected to be determined by its initial mass m and its age τ: L λ (m,τ), where L λ is the energy emitted by the star at wavelength λ. Then the spectral energy distribution of a coeval population of age τ can be written as L (cp) λ (τ) = Z which is the energy emitted per unit mass. 0 L λ (m,τ) φ(m) M dm,
Star formation rate Now suppose the star formation rate (i.e. the mass that turns into stars in a unit time) is known as a function of time, ψ(t), then the luminosity of the galaxy at any time t can be written as L λ (t) = Z t 0 L (cp) λ (t t )ψ(t )dt.
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: L λ (m,τ) Since L λ (m,τ) can be obtained from stellar evolution, and if the IMF is universal (determined from obsrvation), the spectral energy distribution may be used to infer its star formation history.
Synthesized spectra for coeval populations The predicted spectra of a single starburst at ages 0.001, 0.01, 0.1, 0.4, 1, 4, and 13 Gyr.
Properties At an age younger than 10 7 yr, the spectrum is completetly dominated by the blue main sequence stars, which have strong emission in the UV due to their high effective temperature. At the age of about 10 7 yr, the most massive stars have already evolved off the main sequence and become red supergiants, which causes a drop in UV flux and a rise in near-infrared flux in the synthesized spectrum. From a few times 10 8 yr to about 1 10 9 yr, the AGB stars maintain a relatively high flux in the near-infrared, while the UV flux continues to drop as more and more low-mass stars evolve off the main sequence. After a few times 10 9 yr, stars in the red giant branch take over as the main contributors of the near-infrared flux.
Broard-band luminosity and colors The broad-band luminosities decrease with age, and the decrease is more rapid for a bluer band, because of the disappearance of bright main sequence stars and supergiant. Galaxies also become redder with time, as more and more stars evolve off the main sequence.
The observed spectra of different galaxies