A letter about a shadow

Dear friend, I stood a one-metre stick upright in flat ground and marked the end of its shadow every hour. In the morning the shadow was long, towards midday it got shorter and shorter, and then it grew long again.

Dear friend, I stood a $1$ m stick upright and marked its shadow every hour: long, shorter, shortest at midday, then long again.

Stick $h=1$, shadow $s(t)$; noon means $s(t)=s_{\min }$.

The shortest shadow is the true noon. In winter it was two metres long; in summer the same stick had a shadow of only thirty centimetres.

Shortest shadow means noon. Winter: $s=2$ m. Summer: $s=0.3$ m.

Winter $s_w=2$, summer $s_s=0.3$.

The shadow and the height of the sun are linked: shadow equals the stick's height divided by the tangent of the sun's height angle. A two-metre shadow means the sun was about twenty-seven degrees high; a thirty-centimetre shadow means about seventy-three degrees.

$s=h/\tan \alpha$ with $h=1$ m. Winter: $\tan \alpha =1/2\Rightarrow \alpha \approx 26.6^{\circ }$. Summer: $\tan \alpha =1/0.3\approx 3.33\Rightarrow \alpha \approx 73.3^{\circ }$.

$s=h/\tan \alpha$; ${\alpha }_w=\arctan (1/2)\approx 26.6^{\circ }$, ${\alpha }_s=\arctan (1/0.3)\approx 73.3^{\circ }$.

The difference between the seasons is about forty-seven degrees: twice the tilt of the earth's axis, which is twenty-three and a half degrees. A stick and a ruler can measure the tilt of a planet. Measure your own shadow and write back!

$73.3-26.6=46.7\approx 2\times 23.5$: twice the tilt of the earth's axis. A stick measures a planet. Write back with your shadow!

${\alpha }_s-{\alpha }_w=46.7\approx 2\times 23.5$.