Longest Day Calculation

The longest day in the northern hemisphere occurs on June 21st of each year. Given a city whose latitude angle is $\theta _L = 39^{\circ}$ North , find from geometrical considerations only the expected length of the day in hours and minutes.

Note that on June 21st, the axis of rotation of Earth which is tilted by $\theta_i = 23.44^{\circ}$ lies in the plane that contains the center of Earth, and the center of the Sun, and which is perpendicular to the orbital plane of the Earth around the Sun.

Also note that the actual day length differs by a few minutes, and the difference is due to the refraction of the Sun rays in the atmosphere.