Thermal equilibrium in a gravitational field

A system is in thermal equilibrium if the temperature of the system is the same everywhere. Suppose the system is in a gravitational field. Will the temperature be the same everywhere?

When we hike up a mountain, it gets cooler as we climb higher. What is the quantitative relation between temperature and elevation?

Imagine a gas molecule. If it rises higher, its potential energy increases. Its kinetic energy should decrease. Temperature is the average kinetic energy of molecules. Temperature therefore should decrease.

Suppose a kilogram of gas rises for one thousand meters. Its gain in potential energy is 9.8*1000= 9800 J. Here 9.8 is the acceleration rate of the gravitational force. We might expect the corresponding kinetic energy drops 9800 J as well. For most ideal gases at constant pressure and volume, the specific heat capacity is approximately 1000 J/(kg·K) The drop of temperature should be 9800/1000= 9.8 degrees. This is a decrease of 9.8 degrees of temperature. 

Hence, a system in thermal equilibrium may not have the same temperature if it is in a gravitational field. The definition of thermal equilibrium needs to be revised in a more general condition. But how to revise the definition?

Empirically, the decrease of temperature are less than 9.8 degrees. Typically, it is about 6.5 degrees. What factor accounts for the difference?