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Unformatted text preview: ion pressure: P = Pgas + Prad = kT
ρ + aT 4
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
(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
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.
- Winter '08