[The following discussion is taken from "Astrophysics I: Stars" by R. Bowers and T. Deeming, 1984]

According to equation [I.I.1] for radiative energy transfer a temperature gradient must exist in order for heat energy to be radiated out to a star's surface. We also see that if the opacity k increases, we get a large temperature gradient. Any large temperature gradient is unstable to convective flow inside the star in the same sense as in Earth's atmosphere. Let us consider an example.

Consider a parcel of stellar material with some density r. By the *ideal gas equation* it also has some pressure P. Let the parcel rise a small amount in an adiabatic sense so it neither loses nor gains any internal energy to or from the surrounding material. We also assume that the pressure remains the same as its surroundings. As the parcel rises it will encounter material which is less dense than itself and will keep rising due to buoyancy forces which overcome the local gravity. Stated another way if the *actual* temperature gradient in the star is greater than the *adiabatic* temperature gradient which the parcel would experience, then the star is unstable to convection. Stated mathematically,

By taking natural logs and using the adiabatic relation

We get

From this relation we can see that if the gas is ideal, or g is nearly one(1) then convective transport becomes important. This condition is met where ionization occurs deep in the interiors of stars like our sun and other hot stars. If conditions for convection are met then the following equation for the temperature gradient is used.