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### problem08

Course: AST 320, Fall 2009
School: University of Toronto
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Word Count: 328

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Set Mini-Problem VIII: Dating stars using Lithium: contraction due 6 Feb 2009 As pre-main-sequence stars contract, at some point their internal temperature becomes suciently hot that Lithium is destroyed. This will occur earlier for more massive stars, and hence one can estimate an age for a group of stars by nding the highest-mass star that still has Lithium. In this problem set, we will determine how stars...

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Set Mini-Problem VIII: Dating stars using Lithium: contraction due 6 Feb 2009 As pre-main-sequence stars contract, at some point their internal temperature becomes suciently hot that Lithium is destroyed. This will occur earlier for more massive stars, and hence one can estimate an age for a group of stars by nding the highest-mass star that still has Lithium. In this problem set, we will determine how stars contract and in the next two we will use this to derive an analytical relation between this critical mass and age, following Bildsten et al. (1997, ApJ 482, 442 [you do not need the reference, but feel free to look it up if you want to]). We will make two assumptions: (i) that the stars are completely convective and can be described by the appropriate, n = 1.5 polytrope; and (ii) that the surface temperature remains constant during contraction. 1. For a n = 1.5 polytope, what is the total (potential plus kinetic) energy of the star? an Given eective temperature Te and radius R, you know the luminosity; assuming this is powered by contraction, what is the contraction rate dR/dt? (Write in terms of M , R, and Te , keeping all constants.) 2. Solve the dierential equation, using that R as t 0 for the integration constant. Inserting physical constants, and scaling to useful numbers, you should nd R = 0.85 R (M/0.1 M)2/3 (Te /3000 K)4/3 (t/1 Myr)1/3 . 3. Again use the fact that the star can be described by a n = 1.5 polytrope, as well as the ideal gas law, to nd the central temperature in terms of M and R. Insert the relation for the radius above, to show that Tc = 0.88 106 K (M/0.1 M )1/3 (/0.6)(Te /3000 K)4/3 (t/1 Myr)1/3 . 4. We will turn to Lithium burning in the next mini problem set. Here, end with calculating the time it takes for a 0.1 M star to reach the main sequence (Tc = 4.3 106 K; Fig. 3.3). How does this compare with the lifetime of massive stars?
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