A formula for the number of days between the winter solstice and the   latest sunrise
  A simple closed-form expression for the number of days between the winter solstice and the latest sunrise is derived. Formulas for the summer solstice and the sunset are derived as well. These approximate formulas make it easy to see the functional dependence on the latitude of the observer. An approximate expression for the difference in time of day between the latest sunrise and the sunrise on the solstice is also derived. The formulas are not valid in the Arctic, Antarctic or tropics. 
Matthias Reinsch