Question B1 [This question will required roughly two single-sided A4 pages to answer.] Consider a hypothetical, spherically symmetric star of mass M and radius R, for which the den- sity profile is represented by p(r) = Pc [1 − (r/R)²], where pc is the central density and r is the radial distance from the center of the star. a) If the star is in hydrostatic equilibrium, by considering the forces acting on a volume ele- ment, show that the following expression must apply: dP dr Gm(r) p 12 where P is the pressure as a function of radial position and m(r) is the mass interior to the radius r. b) For the hypothetical star, with the given density as a function of radius p(r) = pc [1-(r/R)²], show that m(r) can be written as m(r) = 1½πr³ p 31.2 5R² c) Show that the average density p of the star (i.e., total mass divided by total volume) is given by p = 0.4pc. d) e) Given that the pressure is zero when r = R, show that the central pressure is given by Pe= 15 G M² 16π R4 ΜΗ The star is composed of an ideal gas with an equation of state P = pT. Show that the central temperature is GM μmн Te = 2R k where MH is mass of hydrogen atom and μ is the mean molecular weight.

Question B1 [This question will required roughly two single-sided A4 pages to answer.] Consider a hypothetical, spherically symmetric star of mass M and radius R, for which the den- sity profile is represented by p(r) = Pc [1 − (r/R)²], where pc is the central density and r is the radial distance from the center of the star. a) If the star is in hydrostatic equilibrium, by considering the forces acting on a volume ele- ment, show that the following expression must apply: dP dr Gm(r) p 12 where P is the pressure as a function of radial position and m(r) is the mass interior to the radius r. b) For the hypothetical star, with the given density as a function of radius p(r) = pc [1-(r/R)²], show that m(r) can be written as m(r) = 1½πr³ p 31.2 5R² c) Show that the average density p of the star (i.e., total mass divided by total volume) is given by p = 0.4pc. d) e) Given that the pressure is zero when r = R, show that the central pressure is given by Pe= 15 G M² 16π R4 ΜΗ The star is composed of an ideal gas with an equation of state P = pT. Show that the central temperature is GM μmн Te = 2R k where MH is mass of hydrogen atom and μ is the mean molecular weight.