In order to find longitude, you need to know what time it is. Before the invention of the chronometer, finding the longitude was a difficult problem. See Wikipedia article for a discussion of the history of the problem.
The lunar method was one method devised for telling time while at sea when no accurate chronometer is available.
In a nutshell, the lunar method relies on the fact that the moon travels rapidly across the sky — one full revolution every 27.3 days, 13 degrees every day, 32 minutes (its own diameter) every hour.
If you can plot the Moon's position accurately on the sky, you can tell time with a reasonable degree of accuracy.
The basic process is to measure the angular distance between the Moon and a nearby star or planet or the Sun, apply certain corrections, and compare that distance against a table of distances and times.
In theory, computing time using the lunar distance method should be as simple as measuring the angular distance between the Moon and a nearby object, and comparing the distance with a table. Unfortunately, atmospheric refraction throws a wrench into the entire procedure.
Editions of Bowditch older than 1916 contain sections on the Lunar Method. After that date, the method was considered obsolete and no longer covered in Bowditch. Tables have not been printed in quite some time.
See Omar Reis' page on the subject for more information and diagrams.
See Centennia Software for discussion on computing Lunars with a modern calculator.
One minute of error at the sextant when measuring the Moon's position equals roughly two minutes of time error, which can equal up to 30 nautical miles of position error.
The Moon is relatively close to the earth, which can lead to parallax errors of up to one degree. In fact, the observer's position on the Earth not only affects the Moon's position on the stars, but the even Moon's apparent diameter. These errors need to be compensated.
Atmospheric refraction can introduce errors in measurements. A 40° change in temperature can result in an error of up to 2 nm simply because of temperature's effect on refraction. For best results, all objects observed should be ten degrees or higher above the horizon.
These are difficult measurements to make if the ship is rolling at sea.
There are other obvious limitations as well, such as weather which might prevent you from sighting on the Moon.
This method is best outlined in Easy Lunars.
Measure the altitudes of the moon and the other object above the horizon. Note times. Measure the distance between the moon and the other object as accurately as you can. Repeat several times and average for best results. Note time. Measure the altitudes of the moon and the other object again. Note times. Average the altitude values.
If the other object was the Sun, find its semidiameter from the almanac. Add to the measured distance.
Find the semidiameter for the Moon from the almanac. Add or subtract from the measured distance, depending on which limb you used.
Find the augmentation of the Moon's semidiameter. You can use
0.3' * sin(H_moon) as a decent approximation, or
refer to Table 18 for more accurate
results. Add or subtract from the measured distance.
First, apply the usual altitude correction tables and adjustments for semidiameter for the Moon and the other object in order to get corrected altitudes.
Easy Lunars gives this equation for correcting the distance:
corrected_LD = observed_LD + dh_Moon*A + dh_Obj*B + Q
Where:
The "Q" term is usually less than a minute of arc, and can often be ignored, especially if the distance is close to 90, the Moon's altitude is greater than 60 degrees, or when the Moon and other object are roughly aligned vertically.
Obtain a table of lunar distances. These were traditionally published in 3-hour increments. Find the distances for the hours before and after the time you made your observations. Your adjusted distance should lie between those distances. If not, a mistake has been made somewhere.
Use ordinary interpolation to find the corrected time of observation.
Based on the 1906 edition of Bowditch, pp.288-333, the procedure goes like this:
The Nautical Almanac publishes a table, in 3-hour intervals, the angular distances of the center of the moon from the center of the Sun, the brightest planets, and certain bright stars. The tables also indicate "E" or "W" to let you know on which side of the moon the object can be found. Given the approximate time, and the values from the table, you can easily find the body in question.
Bowditch recommends that the Measurements be made by a team of four people: one person to measure the distance between the Moon the the celestial body, one to measure the altitude of the Moon, one to measure the altitude of the body, and one to note times and write down measurements from the other three people on the team.
The moon and the celestial body should be at least 10° above the horizon, to reduce refraction error.
Your best sextant should be used for the measurement between Moon and body, since this is the most crucial.
Greenwich Date: Correct chronometer time for its error and compute approximate Greenwich date and time.
Nautical Almanac: Use the Almanac with the Greenwich day and hour to find the Moon's semidiameter and horizontal parallax. If the Sun was observed, get the semidiameter and horizontal parallax. If a planet was observed, get the horizontal parallax.
Apparent Altitude of the Moon: Apply index correction of the sextant, subtract the dip of the horizon (from table 14). Add or subtract semidiameter of the moon depending on which limb was observed. "☾'s App. Alt"
Apparent Altitude of the Sun, planet, or star: Apply index correct of the sextant, subtract the dip of the horizon (from the table). Add or subtract semidiameter of the Sun, if appropriate, depending on which limb was observed. "☉'s or *'s App. Alt"
Apparent Distance: Correct for index error. Add or subtract the Moon's semidiameter, depending on which limb was used. For best results, also add or subtract the augmentation of the Moon's semidiameter from table 18. If the Sun was observed, add the Sun's semidiameter. "App. Dist."
Moon's Reduced Parallax and Refraction: Use table 19 [Bowditch, 1906, p.524] with the latitude of observation and the Moon's horizontal parallax to get a correction factor. Add this to the Moon's horizontal parallax. "☾'s Red. P."
Use Table I [Bowditch, 1906, p.295] with the Moon's apparent altitude, and get the mean reduced refraction and then apply the corrections given in Tables 21 and 22. "☾'s Red. R."
Subtract reduced refraction from reduced parallax. "☾'s Red. P. and R."
(A brief word of explanation: parallax causes the moon to appear lower in the sky than it truly is. Refraction causes it to appear higher. The "☾'s Red. P. and R." value gives us the total correction factor. The Moon is always higher than it appears because the parallax term is greater than the refraction term. We need to know how much the correction is in order to incorporate it into the calculations for correcting the distance.)
Reduced Parallax and Refraction of Sun, Planet, or Star: With the apparent altitude of the sun, planet, or star, use Table I to get the mean reduced refraction. correct with tables 21 and 22. If the sun is used, subtract its horizontal parallax (which is always 8.5") from its reduced refraction. "☉'s Red. P. and R.". If a planet is used, subtract its horizontal parallax (if known). "*'s Red. P. and R.. A star does not require any correction for horizontal parallax.
(Note that the Sun, stars, and planets are always lower than they appear, because the refraction term is greater than the parallax term. In fact, the parallax term is so small that it's not given in modern almanacs.)
Use tables II, III, IV, and V [Bowditch 1906, pp.296-307] to get the logarithms A, B, C, and D. Put these at the head of columns labeled A, B, C, and D.
Get the log of ☾'s Red. P. and R. from table IX [ibid, p.313] and put into columns A and B.
Get the log of ☉'s Red. P. and R. from table IX and put into columns C and D.
Get log sin ☾'s App. Alt. from table 44 [ibid, p.608] and put into columns A and D.
Get log sin ☉'s or *'s App. Alt. from table 44 and put into columns B and C.
Get log cot App. Dist. from table 44 and put into columns A and C.
Get log cosec App. Dist. from table 44 and put into columns B and D.
The sum of column A is the log of the First Part of ☾'s Correction. (Table IX) This is positive if apparent distance is less than 90° and negative if more than 90°.
The sum of column B is the log of the Second Part of ☾'s Correction. (Table IX). This is always negative.
The sum of column C is the log of the First Part of ☉'s or *'s Correction. (Table IX). This is negative if apparent distance is less than 90° and positive if more than 90°.
The sum of column D is the log of the Second Part of ☉'s or *'s Correction. (Table IX). This is always positive.
Add the first and second parts of ☾'s correction. This is ☾'s whole correction.
Likewise find ☉'s or *'s whole correction.
First Correction of Distance: Add the two whole corrections to get First Correction of Distance. Add to apparent distance.
Second Correction of Distance: Use Apparent Distance and First Correction of Distance with Table VI [Bowditch 1906, p.308] to get the Second Correction of Distance. Apply to distance.
Correction for the Elliptical Figure of the Moon's disk or Contraction of the Moon's Semi-diameter: use table VII A [ibid, p.311] with the ☾'s App. Alt. and ☾'s Red. P. and R.. With this number and ☾'s whole correction, enter Table VII B and get the required contraction, which is added to the app. dist. when the farther limb was observed, or subtracted when the nearer limb was used.
Correction for the Elliptical Figure of the Sun's disk or Contraction of the Sun's Semi-diameter: use table VIII A [ibid, p.312] with the ☉'s App. Alt. and ☉'s Red. P. and R.. With this number and ☉'s whole correction, enter Table VIII B and get the required contraction, which is subtracted from the app. dist..
Correction for Compression, or for the Spheroidal Figure of the Earth: Use the Nautical Almanac and the Greenwich date to get the declinations of the bodies to the nearest degree. With the moon's declination and apparent distance, use table XI A [ibid, p.332] to get the first part of N. If the declination is south, reverse the sign of the value. With the sun or star's declination and the apparent distance, use table XI B to get the second part of N, changing sign as above. Add these two values. Use table IX to get the required log N.
To log N, add the log sine of the latitude. The sum is the log of the required correction for compression. In north latitude, add log N (or subtract if negative). In south latitude, subtract (add).
You now have true distance.
Use the Nautical Almanac to find the two distances that bracket the true distance. Use the first distance, along with the Prop. Log following it, and the hours of Greenwich time over it. Find the difference between the distance from the Almanac and the true distance, and add the log of this difference (Table IX) and the Prop. Log. The sum is the log of an interval of time to be added to the hours of Greenwich time taken from the Almanac. The result is the approximate Greenwich time.
Take the difference between the two Prop. Logs in the Almanac. With this difference and the interval of time just found, use table X and take out the seconds, which are to be added to the approximate Greenwich time when the Prop. Logs are decreasing, or subtracted when the Prop. Logs are increasing. The result is the true Greenwich time.
Compare true Greenwich time with the chronometer reading at the time of the observations to get a new correction factor for the chronometer.
At lat. 41°59' N, long. 87°54' W (dead reckoning). Approximate time is 20:09:51 GMT, April 26, 2004.
Altitudes of ☾ and ☉ taken before and after shooting the distance between them.Observed distance of ☾'s and ☉'s nearer limbs was 80°09.3' at 20:16:37. Altitude of ☾ UL is 47°13.8'. Altitude of ☉ LL is 47°49'. Barometer is 29.93; thermometer 59°. Height of eye is 10ft.
See the nautical almanac for 26 April 2004 to obtain values for the Moon and Sun.
Note: Bowditch gives values in degrees, minutes and seconds, while modern almanacs give values in degrees and decimal minutes. In this example, I have converted all values to decimal minutes.
At lat. 35°30' N, long. 30°W (dead reckoning). Local mean time is 18h8m0s, September 6, 1855.
Observed distance of ☾'s and ☉'s nearer limbs was 43°52'10". Altitude of ☾ is 49°32'50". Altitude of ☉ is 5°27'10". Barometer is 29.1; thermometer 75°. Height of eye is 20ft.
Preparation of the Data:
Local Mean Time 18:08 Longitude 30°W + 2:00 G.M.T., approx 20:08 ☾'s S.D. 14'50.0" Aug. Table 18 + 11.2 ☾'s Aug. S.D. 15 01.2 ☾'s Par., Almanac 54'19.4" Aug. Table 19 + 3.6 ☾'s Red. P. 54 23.0 Ob. Alt. ☾ 49°32'50" Dip, Table 14 - 4 23 ☾'s Aug. S.D. + 15 01 ☾'s app. Alt 49 43 28 ☾'s Red. R., Table I 1'16" Bar. 29.1, Table 21 - 3 Term. 75°, Table 22 - 4 ☾'s Red. R. 1 09 ☾'s Red. P. 54 23 ☾'s Red. R. - 1 09 ☾'s Red. P. and R. 53 14 Obs. Alt ☉ 5°27'10" Dip, Table 14 - 4 23 ☉'s S.D. + 15 55 ☉'s App. Alt 5 38 42 ☉'s Red. R., Table I 8'57" Bar. 29.1, Table 21 - 16 Therm. 75°, Table 22 - 28 ☉'s Red. R. 8 13 ☉'s Red. R. - 8 13 ☉'s Red. P. 8 ☉'s Red. P. and R. 8 05 Obs. Dist S, ☾ 43°52'10" ☾'s Aug S.D. + 15 01 ☉'s S.D. + 15 55 App. Dist 44 23 06 ☾'s Dec., Almanac 25°N ☉'s Dec., Almanac 6°N
Bowditch 1906, p.295.
This table gives atmospheric refraction for apparent altitudes from 5° to 90°. Values range from 9'54.2" at 5° to 0'0.1" at 76°.
Table is accurate at barometer 30in, temperature 50°. Further corrections are required at different temperature and pressure.
Bowditch 1906, p.313
This is a table of the log10 values of angles in seconds, from 0°0'0" to 2°59'59". E.g. the entry for 0°0'1" is 0.00, the entry for 0°0'10" is 1.00, and the entry for 0°1'40" (100 seconds) is 2.00.
Bowditch 1906, p.522
Bowditch 1906, p.524
Background: Calculations of the Moon's semidiameter are based on the physical size of the Moon and its distance from the Earth. However, since the radius of the earth is 4000 miles, and the distance to the moon averages about 239,000 miles, the semidiameter of the Moon is affected by where on the Earth you're standing when you observe it. This table provides this correction to the nearest 0.1".
This table takes the apparent altitude of the moon (0-90°), and the moon's semidiameter to the nearest 30" and returns a correction factor to the semidiameter.
0.3' *
sin(H) or from this table:
Bowditch 1906, p.524
This is a table that takes the observer's latitude (0-90) and the Moon's horizontal parallax (53', 57', or 61') and gives a correction for horizontal parallax. Outputs range from 0.0" at the equator to up to 12.0" (0.2') at the poles.
Bowditch 1906, p.608
General-purpose table, used in various calculations. Table runs from 0° to 45° (and 135°-180°) in 1-minute increments.
Matching angles to lines in the table can be a little tricky, so pay attention: (We'll use page 648 as an example.)
Angles on the left use the left minutes column. Angles on the right use the right minutes column. Angles at the top use the top column labels. Angles at the bottom use the bottom column labels.
For example, log(sin(49°3')): The "49°" label is at the lower-right of the page. We use the minutes scale on the right. We use the column labels from the bottom. 49°3' is the fourth line from the bottom. Sine is the rightmost column. log(sin(49°3')) = 9.87811
Columns are:
Note: Log values typically have +10 added to them to avoid needing to deal with negative numbers in an era without calculators. The user of these tables needs to be aware of this and subtract the appropriate multiple of 10 before computing the antilog.