Homework 1

Define pgasp show that p 3r 4 a4 1 3 1 4 1 3 4

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Unformatted text preview: ion pressure: P = Pgas + Prad = kT 1 ρ + aT 4 µmH 3 where k is the Boltzmann constant, µ is the mean molecular weight and a is the radiation constant. Define β = Pgas/P. Show that P= 3R 4 aµ4 1/ 3 1−β β4 1/ 3 ρ4 / 3 And hence that main sequence stars could be (approximately) described by an n = 3 polytrope (assuming µ and β are constant in the interior, which they are not). (i) [10 pt] For this polytrope, use your result for the ratio of average to central density and central pressure to derive the Eddington’s quartic relation for β: 1−β = µ4 β 4 M MEdd 2 and evaluate the constant Eddington mass MEdd in solar masses. (j) [10 pt] The Sun has a mass M = 1.99x1033 g and bulk composition X = 0.73, Y = 0.25 and Z = 0.02. Assuming that most of the interior is ionized, compute the mean molecular weight µ and solve for β using the above equation. From this, and assuming the gas/radiation polytrope relation above, determine the central gas, radiation and total pressure; gas density; and temperature. How do these compare with the more carefully modeled values of Pc = 2.3x1017 dyne/cm2, ρc = 146 g/cm3 and Tc = 1.5x107 K? (k) [10 pt] For an ultrarelativistic, fully ionized, fully electron degenerate plasma, the electron pressure dominates the total pressure: 2π hc P= 3 Z 3ρ A 8π mp 4/3 (we’ll derive this later) where Z/A is the average proton/nucleon ratio for the ions in the plasma. Show that the mass of such a star is a constant Mch (the Chandrasekhar mass) and evaluate Mch in solar mass units for Z/A = 0.5. How does Mch compare to MEdd?...
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This document was uploaded on 02/28/2014 for the course PHYS 223 at UCSD.

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