Neutron stars are the gravitationally collapsed cores of stars. They weigh a couple solar masses and...

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Neutron stars are the gravitationally collapsed cores of stars. They weigh a couple solar masses and have radii of about 10 km.¹ Neutron star CRUSTS (uppermost 1 km) are supported against gravity by relativistic, degenerate electrons stripped from iron nuclei (nucleon number A = 56, charge Z= 26). These free electrons constitute a liquid with zero shear modulus but non-zero bulk modulus K. (a) Estimate the bulk modulus K of the neutron star crust. Express symbolically in terms of the number density of free electrons ne and fundamental constants. You need not provide a number. Hints: What units does K have? What is the relation between energy and momentum for a relativis- tic particle (the electrons in this case)? What is the relation between momentum and interparticle spacing according to the uncertainty principle equivalently, what is the relation for the de Broglie wavelength? What is the interparticle spacing if ne is the number density? (b) The bare iron nuclei in the crust are not relativistic. They arrange themselves into a solid lattice. This lattice can support shear. Estimate the shear modulus u of the neutron star crust by considering the ion lattice (neglecting the electrons). Express in terms of ne, Z, and fundamental constants. You need not provide a number. (c) Express u/K in terms of the fine structure constant and other given quantities, and estimate its value numerically. Neutron stars are the gravitationally collapsed cores of stars. They weigh a couple solar masses and have radii of about 10 km.¹ Neutron star CRUSTS (uppermost 1 km) are supported against gravity by relativistic, degenerate electrons stripped from iron nuclei (nucleon number A = 56, charge Z= 26). These free electrons constitute a liquid with zero shear modulus but non-zero bulk modulus K. (a) Estimate the bulk modulus K of the neutron star crust. Express symbolically in terms of the number density of free electrons ne and fundamental constants. You need not provide a number. Hints: What units does K have? What is the relation between energy and momentum for a relativis- tic particle (the electrons in this case)? What is the relation between momentum and interparticle spacing according to the uncertainty principle equivalently, what is the relation for the de Broglie wavelength? What is the interparticle spacing if ne is the number density? (b) The bare iron nuclei in the crust are not relativistic. They arrange themselves into a solid lattice. This lattice can support shear. Estimate the shear modulus u of the neutron star crust by considering the ion lattice (neglecting the electrons). Express in terms of ne, Z, and fundamental constants. You need not provide a number. (c) Express u/K in terms of the fine structure constant and other given quantities, and estimate its value numerically.