Problem B.1: Temperature of the Sun (6 Points) Assume a constant density p of 1.4 x 103 kg-m³ for the entire Sun. The ideal gas law states that pV = NkT with the pressure p, the volume V, the number of particles N, the Boltzmann constant k (1.38 x 10-23 m²kg s 2K-¹) and the temperature T. (a) Show that the temperature T at a certain pressure p is given by T (p) pm pk with the average particle mass m within the Sun (1.02 x 10-27 kg). (b) Explain why the Sun must be in a state of hydrostatic equilibrium: dp dr = -g(r)p (c) Find the gravitational acceleration g(r) at a radius r from the Sun's center. (d) Use the condition of hydrostatic equilibrium to show that the pressure p inside the Sun at a radius of R/4 from the centre is about 1.26 x 10¹4 Pa, where R is the Sun's radius of 0.7 x 10⁹ m. (e) Determine the Sun's temperature at a radius of R/4. Why is this result only a broad estimate?
