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II-4. Stellar Size – Radius (Main Ref.: Lecture notes; Lec. 4 FK Sec.5-4, 17-6, Box 17-4) Direct Measurement: Only Sun, planets, nebula, etc. Note: Stars are ~ point source, even with telescope. Solar radius: Rsun = 696,000 km = 109 R (where R = radius of the Earth = 6378 km.) Most useful and most often used method: Use Stefan Boltzman Law F = T4, for blackbody where F = energy flux = J/m2 –sec, = Stefan-Boltzman constant = 5.67 x 10-8 W m -2 K -4 Eqn (14) 1 Also, F = L / A where A = area of the star = 4 R2 F = L / 4 R2 Eqn(15a) Eqn(15b) Eqn(15c) From Eqns (14) and (15c), get: L = 4 R2 T4, Eqn(16) • Relates L (luminosity = total power output), R (radius), and T (temperature) of the star. Divide Eqn(16) for star by that for Sun, and get R / Rsun = (L / Lsun )1/2 ( Tsun / T )2, ( Eqn(17) )1/2 means Square root. See class notes for derivation. 2 EX 18 Betelgeus: L = 60,000Lsun, T = 3500 K What is its radius? Ans: R = 670 Rsun - Red Giant! Note: Tsun = 5800 K. See class notes, FK Box 17-4 Example 1. **************************************************************************************** EX 19 Sirius B - Fainter star of Sirius binary system: L = 0.0025Lsun, T = 10,000 K What is its radius? Ans: R = 0.017Rsun = 11832km = 1.86R - white dwarf! (R = radius of the Earth.) See class notes, FK Box 17-4 Example 2. 3 How to get R from Observation? Measure distance d and brightness b, then from L = 4 d2 b Finding Key Properties of Nearby Stars Eqn(18) get L. Then, measure Color get T. Then, from L, T, and Eqn(17), get R! See class notes for derivation of Eqn(18). SUMMARY: See Fig. II-25 () and class notes for details.. Fig. II-25: How to find R 4 II-5. H-R Diagram (Main Ref.: Lecture notes; FK Sec.5-9, 17-7, 17-8) II-5a. Introduction H-R Diagram means we plot A vs B, where A = m, M, L. B = T, CI (color index), spectral type (e.g., O2, A3, etc.) Note the general location of Main Sequence (MS), Giants(G), White Dwarfs(WD), Supergiants (SG). Note that black holes and neutron stars are outside. See class notes and Fig. II-27 for the etails. (also FK Fig. 17-15(b)) II-5b. Different Objects and Radius (Main Ref.: Lecture notes; FK Sec. 17-7) Note location of various bright stars in H-R Diagram – see Fig. II-26 (also FK Fig. 17-15(a)). Note that you can find radius of a star from the star’s location on the H-R Diagram – see Fig. II-27 (also FK Fig. 17-15(b)). See class notes for the details. 5 Subgiants subgiants Fig. II-26: H-R. Diagram - I Fig. II-27: H-R. Diagram -II 6 II-5c. Luminosity Class (Main Ref.: Lecture notes; FK Sec. 17-8) Deffinition: Ia: Luminous supergiants Ib: Less Luminous supergiants II: Bright Giants III: Giants IV: Subgiants V: Main Sequence See Fig. II-28 (also FK Fig. 17-18) for location of various Luminosity class. Fig. II-28: H-R. Diagram - III 7 • Classification of Stars: Stars are classified by Spectral class (type), e.g., A1, and Luminosity class, e.g., V. ******************************************** EX 20: Vega ~ A0 V; Aldebaran ~ K5 III; Rigel B8 Ia See class notes and Fig. II-26 and 28 for the details. Note: white dwarfs and neutron stars are outside. 8 II-5d. How to distinguish between stars of different luminosity class but in the same spectral class? (Main Ref.: Lecture notes; FK Sec. 17-8) Best explained by example. EX 21: A. Rigel: B8 ( T= 13,400K); Ia (L= 58000 Lsun ), Supergiant (SG) B. Algol: B8 ( T= 13,400K); V (L= 100 Lsun ), Main Sequence (MS) Same color, same T, but different L. How to find the difference? Ans: Width of the spectral lines tells the difference Lines such as H and H , strong for B8 stars, are narrow for supergiants (e.g. Rigel), but broad for main sequence stars (e.g., Algol). Why? Due to density difference, Doppler effect,etc. See class notes and FK Sec.17-8 for the details. 9 By carefully examining a star’s spectral lines, astronomers can determine whether that star is a main-sequence star, giant, supergiant, or white dwarf Fig. II-29: Line width 10 II-5e. How to find Radius R and Distance d from the H-R Diagram? (Main Ref.: Lecture notes; FK Sec. 17-8) ▪(1) From observation, find spectral class with line width H-R Diagram get L and T Eqn(17) get R! ▪(2) From (1) get L. Then, measure m Eqn(6’)(*) get b L and b Eqn(18) get d! (*) msun m = 2.5 log ( b / bsun) Eqn(6’) See class notes and Fig. II-30 for the details. Fig. II-30:Flow Diagram for finding R and d . 11 • EX 22 Find distance d for a A0 Ib star with m = +10. Ans: d = 10,000 pc. Summary: m measured, Spectral class A0 and luminosity class Ib (from width of line measured ) H-R Diagram get M. Then, Eqn(8) gives d. (See class notes for the details.) 12