40°N · November 15 — noon altitude 30.86°, daylight 9 h 45 min

A shadow is the tangent of the altitude, read backwards

If the noon altitude is h, a pole 1 m tall casts 1 ÷ tan h metres of shadow. Measure the shadow instead and the altitude falls out, and from the altitude you can work back to a date or a latitude — that is how Eratosthenes measured the earth in antiquity. Shadows stretch fast as the sun drops: at 45° the shadow equals the height, at 30° it is 1.7 times as long, and at 5° more than eleven times.

• 1 m ÷ tan 30.86°1.67 m

What this leaves out

Sunrise here means the centre of the sun crossing the geometric horizon, an altitude of 0°. The usual convention adds refraction and the sun’s own radius and uses −0.833°, which makes the day 8 to 10 minutes longer at mid-latitudes. The clock is apparent solar time: to reach civil time you still have to add the equation of time (±16 minutes), the gap between your longitude and your time zone, and any summer time. The declination formula carries an error of up to 1.5°, leap years are not counted, so dates after February sit one day out, and neither the observer’s elevation nor hills and buildings on the horizon are in here.

A high sun is a strong sun

The higher the sun, the less air its light passes through, and the stronger the ultraviolet. As a rule of thumb, when your shadow is shorter than you are, the ultraviolet is near its peak for the day — how long skin lasts before it reddens depends on the index and the skin type.

Nearby cells

Same latitude, other dates

Same date, other latitudes

Worth knowing

• Declination δ = 23.44° × sin(360° × (284 + N) ÷ 365), where N counts the days from 1 January.
• Noon altitude = 90° − |latitude − declination|; at the June solstice that is 90° − φ + 23.44° in the north.
• Length of day = 2H ÷ 15 hours with cos H = −tan(latitude) × tan(declination); beyond |cos H| = 1 lies polar day or night.
• Shadow = height ÷ tan(altitude). Times are apparent solar time, with refraction and the equation of time left out.