Chapter 1.7, Problem 33ES

The number of hours of daylight on a given day at a given point on the Earth’s surface depends on the latitude $\lambda$ of the point, the angle $\gamma$ through which the Earth has moved in its orbital plane during the time period from the vernal equinox (March $21$ ), and the angle of inclination $\varphi$ of the Earth’s axis of rotation measured from ecliptic north $\left(\varphi \approx 23.45°\right)$ . The number of hours of daylight $h$ can be approximated by the formula $h=\left\{\begin{array}{ll}24,\hfill & D\ge 1\hfill \\ 12+\frac{2}{15}{\sin }^{-1}D,\hfill & \left|D\right|<1\hfill \\ 0,\hfill & D\le -1\hfill \end{array}\right.$ Where $D=\frac{\sin \varphi \sin \gamma \tan \lambda }{\sqrt{1-{\sin }^2\varphi {\sin }^2\gamma }}$ and ${\sin }^{-1}D$ is in degree measure. Given that Fairbanks, Alaska, is located at a latitude of $\lambda =65°\text{N}$ and also that $\gamma =90°$ on June $20$ and $\gamma =270°$ on December $20,$ approximate

(a) the maximum number of daylight hours at Fairbanks to one decimal place

(b) the minimum number of daylight hours at Fairbanks to one decimal place.