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Basics about stars –1– 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 –2– 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. –3– 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 –4– 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. –5– 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. –6– 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 –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 = 4πr 2 σTe4 Te is then defined as the effective temperature of a star with luminosity L and radius r. –8– 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. –9– Figure 4: Spectral line strength dependence on temperature. – 10 – 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. – 11 – 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 – 12 – The Hertzsprung-Russell diagram Figure 5: Luminosity classes in the HRD. – 13 – Figure 6: Color magnitude diagram for the globular cluster M3 . – 14 – Figure 7: Hertzsprung-Russell diagram of L vs T. – 15 – Figure 8: Schematic HR diagram. – 16 – 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. – 17 – 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. – 18 – Figure 9: Mass luminosity relation. log10 L = 1.15log10 P d + 2.47 L MV = −2.80log10 P d − 1.43 – 19 – 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. – 20 – 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. – 21 – 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. – 22 – 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 ) – 23 – 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. – 24 – 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. – 25 – 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. – 26 – 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. – 27 –