Latitude from the sun or the North Star

How it is calculated

Solar declination: the calculator uses Spencer's series (1971), accurate to within 0.05°. For working by hand, Cooper's formula (1969) will do, though it is up to 1.3° off in autumn: δ = 23.44° × sin(360/365 × (284 + n)), with n the day number of the year (1 to 365, ignoring leap years). Around 21 June, n ≈ 172 and δ ≈ +23.44°; at the equinoxes (March and September) δ ≈ 0°.

With the sun to the south of the observer (typical case in the temperate latitudes of the northern hemisphere): latitude = 90° − height + δ. Example: in Madrid (latitude 40.4° N) on 21 June the sun culminates at 73.04°, so latitude = 90 − 73.04 + 23.44 = 40.4°.

With the sun to the north of the observer (typical case near the equator or in the southern hemisphere): latitude = height − 90° + δ.

With a gnomon (a vertical stick stuck in the ground): sun's height = arctangent (height of the stick ÷ length of the day's shortest shadow). Example: a 1 m stick with a shortest shadow of 1 m gives a height of arctangent(1) = 45°.

With the North Star (northern hemisphere only): its height above the horizon is approximately equal to the latitude, with an error under 1° because the North Star is not exactly over the axis of rotation.

Keep in mind

• Measure the sun's height without looking at it directly: use the shadow of a gnomon, an improvised sextant, or project its image through a small hole (pinhole camera). Looking directly at the sun damages your eyes.
• For the gnomon, note the shadow's length every few minutes around noon and use the shortest one: that instant is solar noon for the location, not 12:00 on the clock.
• The day and month are counted as a year without leap years (28 days in February); the error this introduces is under a day, insignificant for this calculation.
• Atmospheric refraction near the horizon and this simplified declination formula give a typical error of 0.5-1°: enough to orient yourself, not for precision navigation.