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HW4_SOLUTIONS

# HW4_SOLUTIONS - ASTR302 Homework#4(due in class on...

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ASTR302 Homework #4 (due in class on Wednesday, March 12, 2008) 1) White dwarfs are held up against gravitational collapse by gas degeneracy pressure. The equation of state gas degeneracy pressure is 3 / 5 ρ P . Hydrostatic equilibrium requires that 4 2 R M P . a) Use these two relations along with the definition (equation) for density to show that the radius of a white dwarf 3 / 1 - M R . Show your work. b) Do more massive white dwarf stars have larger or smaller diameters than less massive white dwarfs? How does this relate to the fact that the maximum mass of a white dwarf is 1.4 solar masses? 1a) Because P is part of both the equation of state and the hydrostatic equilibrium equations, we have: 4 2 3 / 5 R M ρ We can then substitute the formula for density into this equation: 3 3 4 R M π ρ = to give: 4 2 3 / 5 3 3 4 R M R M π Rearranging the exponent for the first term and dropping the constants gives: 4 2 5 3 / 5 R M R M Solving this gives: 3 / 1 4 5 3 / 6 3 / 5 - = M R R R M M b) As the mass increases, the radius of the white dwarf decreases due to the -1/3 exponent shown above. At 1.4 solar masses, the high mass and small radius leads to a density and pressure that cannot be held up by gas degeneracy pressure any longer and the object collapses to a neutron star rather than a white dwarf.

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HW4_SOLUTIONS - ASTR302 Homework#4(due in class on...

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