Article One of the United States Constitution
The Senate's current practice is to elect a full-time President pro tempore at the beginning of each Congress, as opposed to making it a temporary office only existing during the Vice President's absence.  Since World War II, the senior (longest serving) member of the majority party has filled this position.  As is true of the Speaker of the House, the Constitution does not require that the President pro tempore be a senator, but by tradition, a senator is always chosen.