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Stellar Properties Distance trig parallax d(pc) = 1/p (arcsec) Velocity (V space ) 2 = (V rad ) 2 + (V tan ) 2 Brightness mag = -2.5 log (flux) + constant; L Temperature B-V; spectral class Mass spectroscopic binary; K, P, i Radius eclipsing spectroscopic binary.
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Stellar Properties • Distance trig parallax d(pc) = 1/p (arcsec) • Velocity (Vspace)2 = (Vrad)2 + (Vtan)2 • Brightness mag = -2.5 log (flux) + constant; L • Temperature B-V; spectral class • Mass spectroscopic binary; K, P, i • Radius eclipsing spectroscopic binary
tan p = 1AU / d (AU) 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. Distance 1 AU p d
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 proper motion Tangential V from proper motion arcsec/yr : Vt = 4.74 /p km/s
Proper Motion Vr sin = = Vt/d Vt = d = /p pc/yr rad arcsec, pc km, yr sec Vt = 4.74 /p km/s Vt d
depends on d, speed and direction Barnard’s star (d=1.85pc) has largest =10”/yr
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
4.2 2.0 2.0 6.0 0.0 5.0 3.3 1.3 10 pc 5 pc -26.5 2.0 5 pc 10 pc 15 pc apparent mags absolute mags
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
Bolometric Magnitude: Brightness over all ~ L Mbol = Mv + BC Mbol* - Mbol = -2.5 log L*/L Mbol ~ 4.74, L ~ 4x1033 ergs/s
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 Class Lines Temp B-V B-V=-0.865 + 8540/T T~ 9000/[(B-V)+0.93] sun = G2V 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)
Spectroscopic parallax: Use stars < 100pc to calibrate MV for spectral classes For unknown star: use CCD to measure mV use spectrograph to find spectral class use calibration from (1) to get MV d) use distance modulus to calculate d
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
d1 d2 m1d1 =m2d2 v = 2d/P so d=vP/2 m1v1P/2 = m2v2P/2 m1v1 = m2v2 m1/m2 = v2/v1 m2 x m1 center of mass physics . Kepler’s 3rd law . r m M Fg = Fc GmM/r2 = mv2/ r v = 2r/P GM/r = 42r2/P2 M = 42r3/GP2
Mass - from spectroscopic binaries need K1, K2, P, i) m1/m2 = v2 / v1= K2 / K1 for double-lined binary m1 + m2 = 42(a1 + a2)3 /GP2 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 For single-lined binary with solar mass units (m1+ m2)P2 = (a1+ a2)3 = a13(1+ a2/a1)3 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