Formation of gravitationally stable systems (from stars to galaxies and galaxy clusters) Jeans criterium for condensation and collapse Condition for hydrostatic equilibrium Giant molecular cloud (star formation) Energy budget balance (virial theorem) Nuclear reactions in stellar interiors Relevant time scales of the process Limiting luminosity which limits stellar mass (Eddington limit) Galaxy cluster! (structure formation) Instabilities, oscillations (not covered in this course)
mass Color-magnitude (or H-R) diagram 22000 stars observed with Hipparcos (astrometry abs mag) R.Powell
Color-magnitude (or H-R) diagram Main sequence Lifetime ~ M 3 Mass (M ) Lifetime (yr) 1 10 billion 5 100 million 10 10 million If all H mass is converted into He, a star will shine with luminosity L for.. t = 0.7% Mc 2 /L ~ 10 11 (M/M )(L/L ) yr but only 10% is burnt 10 10 yr
Spectral classification of stars Observables: luminosities, colors, spectra (colors temperatures from BB shape) Morgan-Keenan (MK) classification: based continuum shape and relative strength of absorption lines The chemical composition of most of stars is found to be similar with fractions in mass of 72% of H, 25% of He and 3% metals in mass Abs.line strengths at excitation and ionization equilibrium depends on number of atoms at a given ionization stage q, in a given level i, N q,i, which depends on the element abundance N, the temperature and the density of electrons (Ne) available for recombination: N q,i =f (Ne,T, N), for a given T, Ne will be lower for larger stars (tenuous extended envelopes), i.e. more luminous, since: L = 4πR 2 σ Teff 4. Therefore... two params, T and L, should be enough to classify spectra of most stars: T spectral class, L Luminosity sub-class (also related to surface gravity g=gm/r 2 ) early types late types Sun: G2V In hotter stars metals are mostly ionized and abs lines are not in the visible For cooler stars neutral metals appear and molecular transitions in coolest (reddest) stars
Spectral classification of stars early types late types [Jakobi spectral library] H+K break in K stars (prominent in elliptical galaxies) due to absorption of metal lines below 4000 A, including the CaII double absorption (at ~3950 A) TiO molecular bands in cool (M) stars due to high probability of these transitions
Color-color diagram for main sequence stars The shape of the continuum, and therefore Teff, can be estimated using color indexes (B-V and U-B) The zero of the color index (B V = U B =0) is set for A0 stars Approximately: B-V ~ 7000 K/Teff 0.56 Bluer hotter Deviation from BB are due to Balmer continuum abs (in hot stars) and abs lines in cooler stars Redder cooler L = 4πR 2 σ T 4 eff# Sun: Teff~5800 K, Tphotosph~6500 K
Color-magnitude (or H-R) diagram: evolution Evolution of the Sun Structure of high mass star in late stages of evolution
Globular clusters Globular cluster: coeval population He flash Star clusters Isochrones (constant age) 47 Tuc AGB HB Turn-off MS WD Open clusters 105-6 stars tightly bound Mostly old stars, and coeval No or little gas left for SF (used or dissipated) Modelled isochrones give GC age As old as the hosting galaxy On the Galaxy halo ~103 stars, young, not (or loosely) gravitationally bound Formed from fragmentation of a giant molecular cloud Still mostly coeval with ages in the range 106-109 yr Gas and star formation on-going On the Galaxy plane
Stellar evolution cycle Stellar remnants MF>3M, R~Rs MF=1.5-3 M, R~10 km MF<1.4M, R~5000 km MF = final mass after loss Type-II SN explosion PN phase binaries Type-Ia SN? 1-6 M : C+O 6-8 M : O+Ne+Mg PN phase TMS>TU metal enrichment The very early stages (collapse of molecular cloud) and late stages (mass loss and SN) of stellar evolution are poorly understood Mass loss is larger and larger for massive stars and very hard to model, as it depends not only on mass but also on stellar rotation, and metallicity (opacity). As a result, it s hard to know the mass of SN progenitors. Binary systems (found often in massive stars) have an important role in the final stages of stellar evolution (compact objects + massive star with mass transfer) There is one core-collapsed (Type-II) supernova going off in the Universe every second!
Hydrogen pp burning: all nuclear reaction branches and neutrino spectra [Bradt]
Degeneracy in 2D phase-space [Bradt]
Equation of state of stellar matter T~ρ 1/3 ρ T~ρ 2/3 [Bradt]
Mass vs central density for typical EoS of compact objects [Shapiro & Teukolsky book]
Cross section of interactions [Bradt] Cross section σ: measures the probability for a given interaction (absorption, scattering, decay) to occur It is calculated analyzing the quantum-mechanical probabilistic effect of a given process σ(cm 2 or m 2 ) = No. of scatters per second (s 1 ) / No. photons in 1 m 2 beam per second (m 2 s 1 ) = = rate of scattering/input flux Consider the process (b): Total area blocked by all targets in a thin slab is σ Ntargets = σ na dx, therefore the fractional area blocked is: σ n dx The fraction of photons absorbed by the slab is: dn/n = σ n dx If Ni and N(x) are the initial no. photons and at position x, the cumulative absorption is for constant σ and n : log (N(x)/Ni ) = σ n x N(x) = Ni exp( σnx) Defining the mean free path: λ = 1/(σn) N(x) = Ni exp( x/λ) Introducing the matter density of absorbing material ρ (kg/m 3 ), and the opacity κ = σ n/ρ N(x) = Ni exp( κ ρ x). Also: λ = 1/(κρ) The dimensionless optical depth is defined as τ = x/λ = κρx ; in general: τ ~ 1: optically thick to radiation τ << 1: optically thin to radiation
Black holes (basic properties, see e.g. M.Longair High Energy Astrophysics ) Final stage of stellar evolution with masses >3 M and progenitor masses >~30 M (very uncertain) Only properties they can have: mass, angular momentum, electric charge Non-rotating (Schwarzschild) BH have a surface at the radius from which radiation suffer infinite gravitational redshift from an observer at infinity: i.e. no information will escape at r rg There is a last stable orbit for a Schw. BH ar radius r last = 3rg, within which particles will spiral in to r=0 For a rotating (Kerr) BH a maximal angular momentum exists: Jmax=GM 2 /c 2 for which r = rg/2 = r last BHs are the most compact objects which can exist with mass M; rg gives the scale of the BH BHs are the most powerful energy sources in astrophysics: 5.7% and 42% of the rest-mass energy of matter can be released when falling into the BH (binding energy of particles on the last stable orbit), compared to the efficiency of 0.7% of fusion of H into He Neutron star: M=M, RNS=10 km > 3rg White dwarf: M=M, RWD=5 10 3 km > 10 3 rg The general consensus is that the singularity (r=0), which RG cannot deal with, will be described by a quantum gravity theory An heuristic derivation of the Schwarzschild radius can be obtained considering the escape velocity from the surface of a spherical object of mass M and radius R and considering the minimum radius when the escape velocity would be the speed of light
photons Frad p+ e Fgrav Accretion processes Eddington luminosity Consider ionized hydrogen: the radiation pressure is due to Thomson photon scattering by electrons: Prad = F/c (momentum flux) = (L/4πr 2 )/c The outward radiation force on a single electron is Frad = σt (L/4πr 2 )/c Thompson cross-section Gravitational force act on e +p which are kept close to each other by electrostatic forces. The condition for equilibrium then is: Frad Fgrav = GM(mp+me)/r 2 GM mp /r 2 ~LX of QSOs ~LX of X-ray binaries Maximum luminosity allowed in spherical symmetry. More generally, if κ is the opacity: LEdd first introduced in stellar evolution, radiation pressure plays a critical role for the evolution of very massive stars Note: super-eddington luminosities (L > LE) can be obtained in some cases with anisotropic accretion (disk) so that outgoing radiation is well separated from infalling material If Ṁ is the mass accretion rate and η is the efficiency of the conversion into energy, then: L = η Ṁ c 2 and the maximum BH growth, ṀEdd, will occur when accreting matter reaches LE Eddington time scale (required to radiate away all the mass at the Eddington limit)
Examples of accretion processes in astrophysics on BHs or compact sources Active Galactic Nuclei At which wavelength do we expect accretion sources to be strong emitters? We can estimate a lower limit to the temperature of the source if irradiates at the Eddington limit. Any source looses energy most efficiently if it radiates as a BB. Then: for a NS: LEdd ~ 1044 (MBH /106 M ) erg/s MBH ~105 109 M LEdd ~ 1038 erg/s ~ LX M ~ M strong X-ray emitters for a WD (RWD / RNS ~500): O supergiant orbiting at 0.2 UA T~3 105 K UV radiation (Cataclysmic variables) Cygnus X-1: BH ~15 M HMXB prototype, 1964 first X-ray rocket BH, NS, WD X-ray binaries 4 π R2NS σb T4 = LEdd T>~107 K Cataclysmic Variables
Accretion processes Mass m falling from infinity: When the matter reaches the star ar r=r, decelerates and converts its kinetic energy into heat which is radiated away with luminosity: Defining the efficiency η of conversion of rest-mass into heat/radiation: (rg = 2GM/c 2 is the gravitational radius associated to the mass M) the efficiency depends on how compact the star is: for a white dwarf: η ~3 km/(2 5000 km) = 3 10 4 for a neutron star: η ~3 km/(2 10 km) = 0.15 for a BH, an accretion disk is formed and the maximum energy which can be released is given by the energy to be dissipated (via viscosity) to reach the last stable orbit; General Relativity shows that η~0.06, 0.42 for a non-rotating and rotating BH. For comparison the efficiency of fusion of H into He is η [H He] ~ 7 10 3 the efficiency of accretion processes on BH or NS is ~>10 higher than fusion of H into He which power stars
From star systems to galaxies: Stellar Population models simulating galaxy spectra Inputs:! Star-formation history ( SFR(t) ) Initial mass function (IMF) relative abundance of stars as a fnct of their initial mass (when they reach the MS) Metallicity Z: fraction in mass of metals Method:! Evolutionary tracks for each stellar mass Populate the HR diagram Produce a theoretical stellar spectrum at each point of the HR diagram Sum all spectra and follow the evolution as a fnct of cosmic time Usage:! Obtain age and metallicity of galaxies from spectra and/or spectral energy distribution (i.e. photometry in many bands) Estimate the redshift when spectroscopy is not available or is not feasible: photometric-redshifts