inter4-06-sol - ²rom the two empirical relations we can...

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Astro 346, Spring Semester 2006 Interactive problem solving 4, solutions Problem 1: Stellar orbits The orbit axes will have a random orientation. The defnition oF the angle i is such that i = π/ 2 , iF the orbit axis is perpendicular to the line-oF-sight. Use polar coordinates with the line-oF-sight as axis, i as the polar angle, and φ as azimuth. The solid angle element then is d Ω = sin i di A random orientation means that the distribution oF orbit axes in solid angle is constant. dN d Ω = Const . = 1 sin i dN dφ di = 1 2 π dN d cos i The orbit axes must thus be evenly distributed in cos i . The mean value oF sin i over the range oF possible inclination angles, i [0 , π/ 2] , is < sin i > = Z 1 0 d cos i sin i = Z 1 0 dx q 1 - x 2 = π 4 Problem 2: Open clusters The main-sequence stars in cluster A appear to be brighter than those in cluster B by F A 30 F B . The main-sequence cut-o± temperature is T A 7800 K For cluster A and T B 9150 K For cluster B. The distance ratio is the inverse oF the square-root oF the brightness ratio. D A D B = s F B F A 1 / 5 . 5 Cluster B is thus more distant than cluster A by a Factor oF 5.5.
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Unformatted text preview: ²rom the two empirical relations we can derive T ∝ M . 6 and the liFetime oF a main-sequenxce star, t life ∝ M L ∝ M-3 ∝ T-4 . 8 . The main-sequence cut-o± temperatures oF the two clusters shoudl indicate their age. t A t B ’ ± T B T A ² 4 . 8 ’ 2 . 1 Cluster A should thereFore be a bit more than twice as old as cluster B. Problem 3: Cepheids This Cepheid has an absolute brightness M B =-3 . 58 . The distance is therefore D = (10 pc) 10 . 2 ( m B-M B ) = (10 pc) 10 4 . 82 ’ 650 , 000 pc ’ 2 . 1 · 10 6 light-years The observed radiation was therefore emitted two million years ago, or in other words, the star is now two million years older than it was at the time of emission. The lifetime of a Cepheid in this phase is shorter than that, and thus we would not Fnd it....
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inter4-06-sol - ²rom the two empirical relations we can...

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