2021USAAAO决赛第2题

2. (5 points) The convection zone of the sun is the major region of the solar interior that is closest to the surface. It is characterized by convection currents that quickly carry heat to the surface. As a pocket of gas rises, it expands and becomes less and less dense. For it to continue to rise, the temperature gradient in the sun must be steeper than the adiabatic gradient, which is the temperature that the gas would have if it were allowed to expand without any heat input.

In the sun, the adiabatic gradient satisfies $$T {\propto} p^{0.4}$$, where $$T$$ is the temperature and $$p$$ is the pressure at any given point.

The bottom of the convection zone is about 200,000 kilometers beneath the surface of the sun, and has a temperature of about 2 × 10⁶ K and a density of about 200kg/m³. Estimate an upper bound for the temperature of the convection zone where the density is 1.2kg/m³ (the density of ait). You may assume the ideal gas law holds in the convective zone.