Marine Sextant

What is modelled

A sextant measures the angle between the sun and the visible horizon by a double reflection: the index mirror turns with the arm, the horizon mirror is fixed. Rotating a mirror by α turns the reflected ray by 2α, so the arc is graduated in double degrees. The reading then needs corrections before it is a true altitude.

Equations

• Reading: Hs = H + R - SD + D + IE for the lower limb; corrected altitude Ho = Hs - IE - D - R + SD.
• Dip of the horizon: D = 1.76' √h (geometric acos(R/(R+h)) ≈ 1.93' √h; the almanac value includes terrestrial refraction).
• Refraction (Bennett): R = cot(h + 7.31/(h+4.4)) arcmin.
• Sun altitude: sin H = sinφ sinδ + cosφ cosδ cos(15° × t); noon sight φ = 90° - Ho + δ.

Checks run in the page

• Ray-traced two-mirror geometry gives a parallel horizon ray for six altitudes.
• Geometric dip against 1.93' √h.
• Bennett refraction against 58.2" tan z at 45°.
• The daily maximum of Ho returns the set latitude.

Honest simplifications

Standard atmosphere, parallax and semi-diameter variation ignored (SD fixed at 16'), sun declination constant during the day, no mirror non-perpendicularity or prismatic error. The 2D ray picture draws the mirrors in the same frame as the sky.

• Drag the time slider to noon and read the latitude from the sun at its highest.
• Raise the eye height: the dip correction grows with the square root of height.
• Set a large index error and see that it cancels exactly in Ho.