Theo3_Question_Final - THEORETICAL PROBLEM No. 3 WHY ARE...

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THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine because they are fusing hydrogen into helium in their central parts. In this problem we use concepts of both classical and quantum mechanics, as well as of electrostatics and thermodynamics, to understand why stars have to be big enough to achieve this fusion process and also derive what would be the mass and radius of the smallest star that can fuse hydrogen. Figure 1. Our Sun, as most stars, shines as a result of thermonuclear fusion of hydrogen into helium in its central parts. USEFUL CONSTANTS Gravitational constant = 11 10 7 . 6 - × = G m 3 kg -1 s 2 Boltzmann´s constant = 23 10 4 . 1 - × = k J K -1 Planck’s constant = 34 10 6 . 6 - × = h m 2 kg s -1 Mass of the proton = 27 10 7 . 1 - × = p m kg Mass of the electron = 31 10 1 . 9 - × = e m kg Unit of electric charge = 19 10 6 . 1 - × = q C Electric constant (vacuum permittivity) = 12 0 10 9 . 8 - × = ε C 2 N -1 m -2 Radius of the Sun = 8 10 0 . 7 × = S R m Mass of the Sun = 30 10 0 . 2 × = S M kg
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1. A classical estimate of the temperature at the center of the stars. Assume that the gas that forms the star is pure ionized hydrogen (electrons and protons in equal amounts), and that it behaves like an ideal gas. From the classical point of view, to fuse two protons, they need to get as close as 15 10 - m for the short range strong nuclear force, which is attractive, to become dominant. However, to bring them together they have to overcome first the repulsive action of Coulomb’s force. Assume classically that the two protons (taken to be point sources) are moving in an antiparallel way, each with velocity rms v , the root-mean-square (rms) velocity of the protons, in a one- dimensional frontal collision. 1a
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This note was uploaded on 11/08/2011 for the course PHYS 0000 taught by Professor Na during the Spring '11 term at Rensselaer Polytechnic Institute.

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Theo3_Question_Final - THEORETICAL PROBLEM No. 3 WHY ARE...

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