How is the continuous spectrum of a star formed?

The reason for the formation of a continuous spectrum is predominantly free-bound transitions, i.e., the recombination of electrons to the corresponding energy levels. Since the electron velocities can be arbitrary, the frequency of the emitted quanta is different. Optically dense gas, such as the photosphere of a star, gives off bright continuous radiation, against the background of which dark lines are visible, arising from bound-coupled transitions in colder gas.

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