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Stellar Properties 1) Distance trig parallax d(pc) = 1/p (arcsec) 2) Velocity (Vspace)2 = (Vrad)2 + (Vtan)2 3) Brightness mag = -2.5 log (flux) + constant; L 4) Temperature B-V; spectral class 5) Mass spectroscopic binary; K, P, i 6) Radius eclipsing spectroscopic binary tan p = 1AU / d (AU) 1. Distance for small angles p =1 AU/ d(AU) d (AU) = 1/p where p is in radians 1 radian = 206265 arcsec d (AU) = 206265 / p (arcsec) (define 1pc = 206265 AU) d (pc) = 1 / p (arcsec) 1 AU p d QuickTime™ and a Video decompressor are needed to see this picture. Parallax measurements nearest star ~ 0.8” (d~ 1.3 pc) ground limit ~ 0.01” (d~ 100 pc) HST limit ~ 0.001” (d~ 1000pc) Hipparcos (1989-1993) [120,000 stars to 0.001”; 1 million stars to 0.02”] GAIA (2013-2018) [1 billion stars] to 0.000020” 2. Velocity (Space V)2 = (Radial V)2 + (Tangential V)2 Radial V from Doppler: / = v/c Tangential V from proper motion arcsec/yr : Vt = 4.74 /p km/s proper motion Proper Motion Vr sin = = Vt/d Vt Vt = d = /p pc/yr rad d arcsec, pc Vt = 4.74 /p km/s km, yr sec depends on d, speed and direction Barnard’s star (d=1.85pc) has largest =10”/yr http://youtu.be/yxPPDDP5kyQ QuickTime™ and a Video decompressor are needed to see this picture. 3. Brightness - T, size, d Magnitude scale: backwards, logarithmic Energy scale: luminosity, flux L=4R2T4 ergs/s Magnitudes: each mag is factor of 2.5 fainter 1 mag = 2.5 5 mag = 100 10 mag = 10,000 mag = -2.5 log flux + constant m2 - m1 = -2.5 log f2 / f1 Apparent mags (m) as seen from Earth Absolute mags (M) if object at 10 pc m Sun moon Venus Sirius Vega eye 30in 5m faintest -26.5 -12.5 -4 -1.4 0 6 15 20 28 M +5 +19 +27.5 +1.4 0.5 4.2 6.0 2.0 5.0 3.3 10 pc 1.3 5 pc -26.5 2.0 5 pc 10 pc apparent mags 15 pc absolute mags 2.0 0.0 Absolute mag M: if star were viewed at 10pc Apparent mag: star as viewed from earth m-M = -2.5 log (E/d2) - (-2.5 log (E/102)) = -2.5 log E + 5 log d + 2.5 log E - 5 m-M = -5 + 5 log d distance modulus Color Index mb - mv = Mb - Mv = B-V mv - mr = V-R B-V gives temperature Common filters: U,B,V,R,I,J,H,K Johnson ugriz Sloan visual filter Hot star looks blue B-V ~ - 0. 5 Cool star looks red B-V ~ 1. 5 T B-V Bolometric Magnitude: Brightness over all ~ L Mbol = Mv + BC Mbol* - Mbol = -2.5 log L*/L Mbol ~ 4.74, L ~ 4x1033 ergs/s QuickTime™ and a decompressor are needed to see this picture. Brightness - depends on T, R, d Magnitudes (backwards, logarithmic)= -2.5 log(flux) + C or m2 - m1 = -2.5 log (f2/f1) • m (apparent mag - as seen from earth - includes d) • M (absolute mag - object at 10 pc - eliminates d) • m-M = -5 + 5 log d (distance modulus) • MBOL (bolometric mag - over all ) = MV + BC • B-V (color index) - gives T Energy (luminosity, flux) • L= total energy from star/sec = 4R2T4 ergs/s • Flux = energy received at earth at = L/4d2 ergs/cm2/s/Å • MBOL* - MBOL(sun) = -2.5 log (L*/Lsun) 4. Temperature (B-V, Spectral Class) Mv -6 +1 +5 +15 Class Lines O(5-9) B(0-9) A F G K M R N S L T Temp >30,000K HeII 11-30,000 He H strong 7500-11000 6000-7500 CaII metals 5000-6000 metals,bands 3500-5000 TiO 2000-3500 B-V -0.3 -0.1 0.0 0.3 0.6 1.0 1.5 B-V=-0.865 + 8540/T T~ 9000/[(B-V)+0.93] sun = G2V K with C M with C ZrO hydrides <2000 methane <1300 Luminosity Class: I, II=SG, III, IV=Giant, V=dwarf (main sequence) Spectral Class Mnemonics Oh, Be A Fine Girl(Guy), Kiss Me Right Now Smack Oh Brother, Astronomy Finally Gruesomely Killed Me Right Now *Slump* Oven Baked Ants, Fried Gently, Kept Moist, Retain Natural Succulence (Largely True) He Vega H Sun metals Betelgeuse molecules Jacoby atlas 1984, ApJS, 56, 257 Info from Spectra: • abs= normal star, emission = disk or jet • composition of outer layers (if line present, element present • temperature of outer layers (from knowledge of energy levels of element) • density (narrow lines imply low density) • pressure (wide lines imply high pressure) • rotation (high rotation makes wider lines) • binarity (see spectra of two different stars) • wind (strange P Cygni line profiles with absorption + emission) • magnetic field (Zeeman splitting of lines) WD spectrum Spectra of giants P Cygni features Spectroscopic parallax: 1. Use stars < 100pc to calibrate MV for spectral classes 2. For unknown star: a) use CCD to measure mV b) use spectrograph to find spectral class c) use calibration from (1) to get MV d) use distance modulus to calculate d Different Kinds of Temperature Type From Observe Brightness Planck fctn F Color Planck fctn B-V Effective (T4) Stefan-Boltzman L & R Excitation Boltzman Ratio of lines Ionization Saha Ratio of lines Kinetic Thermal Doppler Width of lines 5. Mass (double - lined spectroscopic binaries) m1/m2 = v2/v1 m1 + m2 = 42a3/GP2 (a=vP/2) v1 sin i, v2 sin i, P come from radial velocity curve of binary Alcor and Mizar are just neighbors but Mizar itself is a visual binary and Mizar A and Mizar B are each binaries QuickTime™ and a Video decompressor are needed to see this picture. QuickTime™ and a Video decompressor are needed to see this picture. d1 m1 d2 m2 x center of mass physics m1d1 = m2d2 v = 2d/P so d=vP/2 m1v1P/2 = m2v2P/2 m1v1 = m2v2 m1/m2 = v2/v1 Kepler’s 3rd law Fg = Fc GmM/r2 = mv2/ r GM/r = 42r2/P2 M = 42r3/GP2 v = 2r/P m . r . M Mass - from spectroscopic binaries need K1, K2, P, i) m1/m2 = v2 / v1= K2 / K1 m1 + m2 = 42(a1 + a2)3 /GP2 for double-lined binary K1 = v1 sin i = 2a1sin i / P a1 = PK1 / 2 sin i a1 + a2 = P (K1 + K2) / 2 sin i m1 + m2 = (42 / GP2)P3(K1+K2)3/83sin3i = P(K1+K2)3/2G sin3i (m1+ m2)P2 = (a1+ a2)3 = a13(1+ a2/a1)3 For single-lined binary with solar mass units a2/a1 = m1/m2 (m1+ m2)P2 = a13(1+ m1/m2)3 = a13(m1+ m2)3/m23 f(m1, m2) = m23sin3i/(m1+m2)2 = a13/P2 = K13P/83 mass function gives a lower limit to m2 Mass of Sun (from planet orbits) = 2 x 1033 g Star masses range from 0. 07 M to 100 M QuickTime™ and a Video decompressor are needed to see this picture. Which is star 1 and which star 2? Which star is more massive? K1 K2 m2/m1 = K1/K2 ~ ? 6. Radius • from lunar occultation • from interferometry (for supergiants) • from T, L (R = [L/4T4]1/2) • from eclipsing, spectroscopic binaries (need eclipse times, K1, K2) D1 = (K1+ K2) ta-b where a-b is ingress or egress time D2 = (K1+ K2) ta-c where a-c is ingress/egress + eclipse time Radii of stars range from 1/100 R to 400 R QuickTime™ and a Video decompressor are needed to see this picture. a a b d c primary eclipse b d secondary eclipse c a-b or c-d moves diameter of small star Ds a-c or b-d moves diameter of large star DL Ds = V x ta-b DL = V x ta-c where V = Ks + KL P=? e=? i=? Rs/RL= ? Ms/ML=? a =? Ds=? DL=? Ms=? ML=? Ls/LL=? 1 5 10 15 20 25 Ts/TL=? P=24 hr e=0 i=90 Rs/RL= 1/3 Ms/ML=2 a= 3x106=.02AU Ds=8.1x105=0.6Dsun DL=2.4x106=1.7Dsun Ms=0.4Msun ML=0.8Msun Ls/LL=1.51 1 5 10 15 20 25 Ts/TL=1.9 100 M mass 400 R 10-6 g/cm3 density radius 0.01R 106 g/cm3 0.07M Location depend on: Mass Age Composition uses ~20,000 stars Mass - Luminosity Relation