curve of growth
Common sense says that if you double the number of atoms of an element, the absorption line they make should grow twice as strong. For a while it does — but only for a while. The curve of growth is the graph that charts how a line's strength (its equivalent width) really changes as you add more and more absorbing atoms, and it reveals that the relationship bends in surprising ways.
It has three stages. When atoms are few, the line is weak and its strength rises in simple proportion to the number of atoms — the linear part. As atoms pile up, the centre of the line goes as dark as it can get, saturated, and adding more atoms barely deepens it; the line stops growing and the curve flattens — the saturated part. Pile on still more, and the line can only grow by spreading its dark wings out to the sides through pressure broadening, so its strength creeps up slowly again — the damping part, the third regime.
This curve is the essential translator from what we see to what is there. Because the line-to-abundance relationship is not a straight line, you cannot just read off how much of an element a star contains from a line's strength alone. The curve of growth supplies the correct conversion at every stage, and lets astronomers turn a tableful of measured equivalent widths into real, trustworthy chemical abundances.
Two iron lines may have nearly the same equivalent width even though one is made by far more atoms — because the stronger one has saturated and sits on the flat part of the curve of growth.
Line strength versus number of atoms: rising, then flat, then slowly rising again.
A saturated line is treacherous: because its strength barely responds to extra atoms, a small measurement error there translates into a large error in the inferred abundance. Astronomers prefer to weigh elements using weak, unsaturated lines when they can.