Local Thermodynamic Equilibrium
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Prelude: Types of Temperature
Local Thermodynamic Equilibrium (LTE)
Local Thermodynamic Equilibrium means
where is the local kinetic temperature. Take a sphere at surface temp , with some gas (matter) inside it. Allow to come to thermal equilibrium. Inside the sphere, you see a Planckean spectrum (). Now suppose you take a box of gas (it is both emissive and absorbent) and shoot rays of photons through that box suffer some absorption:
But those photons also pick up some intensity:
But since the photons should not pick up energy going through the box (everything is at the same temperature),
Now suppose you let all the photons out of this sphere. Those photons will no longer be in thermal equilibrium with the gas. Suppose we magically fix the temperature of the gas at this point. The source function will remain the same (it will still be the case that ) because the source function is a property of the matter alone (the absorptive and emissive properties of it). However, .
Suppose we look along a column of this gas. At each point, the source function will be: . Therefore, along that column will be: