2021USAAAO决赛第2题

英文原题

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.

中文翻译

2. (5 points) 太阳的对流层是太阳内部最接近表面的主要区域。它的特点是由对流迅速将热量输送到表面。当一团气体上升时，它会膨胀并变得越来越稀薄。为了使它继续上升，太阳内部的温度梯度必须比绝热梯度更陡，绝热梯度是指气体在允许自由膨胀而不输入任何热量的情况下所具有的温度。

在阳光下，绝热梯度满足 $$T {propto} p^{0.4}$$，其中$$T$$是温度，$$p$$是任一点的压力。

对流带底部约在太阳表面以下20万公里处，温度约为 2 × 10⁶ K ，密度约为 200kg/m³。估算对流区温度的上界，其中密度为1.2kg/m3。你可以假设理想气体定律在对流区成立。