Sextant

What this page shows

At noon the sun is on the meridian. The zenith distance is 90° minus the observed altitude. Latitude is the declination plus or minus the zenith distance (90° minus the observed altitude). The sign depends on whether the sun bore south or north of the observer. South means the observer is north of the sun, so the latitude is the declination plus the zenith distance. North means the observer is south of the sun, so the latitude is the declination minus the zenith distance. The bearing is the control above. It is not a hidden assumption.

Longitude comes from the clock. The time the sun is highest, compared with the time it is highest at Greenwich, is the longitude. One hour from Greenwich is 15°. Later than Greenwich is west.

The equation of time is what turns the clock into longitude. It is the difference between apparent solar time and mean solar time, so it moves the Greenwich meridian passage off 12:00 UTC. The page offers a stated approximation, labelled as not the Nautical Almanac: E = 9.87 sin 2B − 7.53 cos B − 1.5 sin B minutes, with B = 360° (n − 81) / 365. On day 81, B is 0 and E is −7.53 min, which checks the arithmetic and is not an almanac value. The other choice is a field for the Greenwich meridian passage from an almanac. A real sight can then be compared with a published table.

The declination, until you tick the table, is δ = 23.44° × sin(360° × (n + 284) / 365). That is the approximation already on this site. It is not a second copy of the formula, and it is not an almanac. Tick the box to type a declination from a table. The date becomes n in that 365-day year. 29 February is counted as the following day, day 60, and 31 December stays day 365 even in a leap year.

The second altitude is Marcq St. Hilaire’s intercept, as in Bowditch. You give an assumed position, a declination, and a Greenwich hour angle. The page computes the altitude there from sin h = sin φ sin δ + cos φ cos δ cos LHA, with LHA = GHA + longitude. Longitude is east positive, and the hour angle is west from Greenwich. The intercept is the observed altitude minus that computed altitude. It is toward the body when the observed altitude is greater. The line of position is perpendicular to the azimuth. The noon sight has already given a parallel of latitude. The two lines cross at a position. If they are parallel, the fix is not determined. The page does not look up the next sight.

The altitude formula is the spherical one. The straight line on the chart is the plotting line of the intercept method, used near the assumed position. The declination approximation and the equation-of-time approximation are the formulas labelled as approximations. The page computes what you enter.

A worked example

The preset is a worked example, not a historical sight. Declination +20°, observed altitude 60°, sun bearing south. The zenith distance is 90° − 60° = 30°. The sun bore south, so the latitude is 20° + 30° = 50° N. If the sun bore north, the same numbers would give 20° − 30° = 10° S. The corrections are off, so the 60° in the box is the observed altitude.

Local apparent noon at 13:00 UTC is one hour after noon at Greenwich if the equation of time is zero, and one hour is 15°, so the longitude is 15° W. The preset leaves the equation-of-time approximation on, for 1 November, day 305 of the 365-day year. That value is not zero, and the longitude card shows how far it moves the 15° W. The latitude does not use that day’s approximate declination. It uses +20°, typed, as a table value.

The second altitude in the preset is not that noon sight. The assumed position is 50° N, 15° W, the declination is +20°, and the Greenwich hour angle is 45°, so the local hour angle is 45° + (−15°) = 30°. The computed altitude is sin h = sin 50° sin 20° + cos 50° cos 20° cos 30°. The observed altitude is 52°, which is greater, so the intercept is toward the body. The line is perpendicular to the azimuth and crosses the parallel of 50° N. That crossing is a chart fix, not a recorded voyage. The assumed longitude is the same 15° W as the zero-equation longitude, so the hour angle is easy to check. It is not taken from the clock. The clock longitude is the other card.

Assumptions

• The noon sun is on the meridian. The bearing, south or north, is the control on the page.
• Zenith distance is 90° minus the observed altitude. Latitude is declination plus that distance when the sun bore south, and declination minus that distance when the sun bore north.
• Observed altitude is the sextant altitude after only the corrections whose boxes are ticked. An unticked correction is not applied. There is no refraction of the sight on this page.
• Dip is the geometric dip from eye height, the shared horizon distance divided by the radius, and it is subtracted. It is not the 34′ astronomical refraction used on the midnight-sun page.
• Index error is a signed number of degrees. A positive number raises the altitude. The default is 0. It is added only when the box is ticked.
• The solar semidiameter is about 0.27°, the same round figure as on the midnight-sun page. It is the angular size from the nominal solar radius and the astronomical unit, not a second constant. Lower limb is added. Upper limb is subtracted. Off until ticked.
• Longitude is 15° per hour. East is positive in the boxes, as on the other pages. The cards also name west and east. One hour later than Greenwich is 15° W.
• The equation of time in use is either the stated approximation or the Greenwich passage you type. The approximation is not the Nautical Almanac.
• The declination in use is either the site’s approximation or the number you type. The page does not fetch an ephemeris.
• The computed altitude is the spherical formula above. The line perpendicular to the azimuth is the plotting line, on a local plane chart. A meeting far from the assumed position is not plotted as a fix.
• The fix is not determined if the line of position is parallel to the parallel of latitude.
• The worked example is not a historical sight.
• Hold Ctrl and scroll to zoom the map. The scroll wheel alone moves the page. A click sets the assumed position. The map requests tiles from OpenStreetMap. The numbers stay in this browser.

Sources

• Bowditch, N. The American Practical Navigator. The intercept method of Marcq St. Hilaire: an assumed position, a computed altitude, an intercept toward the body when the observed altitude is greater, and a line of position perpendicular to the azimuth. This page does not reproduce the book.
• The Nautical Almanac. Declination and the time of the sun’s meridian passage at Greenwich are tabulated there. Those numbers can be typed on this page. This page does not reproduce the almanac.
• Cooper, P. I. (1969). The absorption of radiation in solar stills. Solar Energy, 12, 333–346. The declination approximation is the one already used on this site, with 23.44° in place of Cooper’s 23.45°.
• Honsberg, C. and Bowden, S. PVEducation, “Solar time”. The equation of time used here is E = 9.87 sin 2B − 7.53 cos B − 1.5 sin B minutes, with B = 360° (n − 81) / 365. They describe it as within about half a minute. It is not the Nautical Almanac. pveducation.org.
• IAU 2015 Resolution B3, nominal solar radius 6.957 × 10⁸ m. IAU 2012 Resolution B2, the astronomical unit 149,597,870,700 m. The semidiameter is the atan of those two, about 0.27°, a round figure, the same order already used on the midnight-sun page.
• The horizon distance and the GRS80 mean radius are the shared geometry module. Dip is that distance divided by the radius.
• Map tiles © OpenStreetMap contributors.