In 1851 The Astronomical Journal published the William
Chauvenet lunar distance method to calculate longitude from sextant
observations of lunar distance (the angle between the Moon and another
celestial body). An appendix in the 1855 American Ephemeris and Nautical
Almanac has a description
of the method, tables, and two worked examples. And of course Chauvenet's
magnum opus on spherical astronomy has a thorough
explanation (the tables
are in Volume 2). He calls it "the shortest and most simple of the
approximative methods *when these are rendered sufficiently accurate by the
introduction of all the necessary corrections*." (his italics)

Here I present the method in a Windows program. It's as close as practical to a literal implementation of the Chauvenet formulas. The one deviation is the refraction model, which is a hybrid of the Cassini model and the low altitude formula in the modern Astronomical Almanac, already implemented in my SofaJpl fundamental astronomy DLL. I felt it was not worth the effort to code Chauvenet's extensive formulas, which are from Bessel. Tests of my refraction model agree well with the tables of Chauvenet.

ChauvenetLunar is distributed as a 212 k .zip file. No installer is necessary for an application this simple. An installer is a potential security vulnerability since it requires system privileges on your machine. On the other hand, with a .zip file you simply extract all files into one folder. (By default, Windows unzip creates new folder "ChauvenetLunar" as a subfolder of the one that contains the .zip.) Your system (desktop, registry, etc.) is not affected. I recommend the new folder be created somewhere in your Documents folder. To create a desktop shortcut, right click on the .exe file, select Send To, and Desktop.

The main window appears when you launch the program:

Initially, all buttons are disabled except Moon. After you visit its window, the Sun button is enabled, and so on until the OK button is the last to be enabled.

The Moon button opens a window to enter observed altitude, semidiameter, and horizontal parallax.

Enter the values in the image. Don't forget the upper limb box, otherwise lower limb is assumed. These very precise angles are a simulated observation calculated with modern software. The correct final result is known, which will let us test the accuracy of the Chauvenet formulas.

The software allows angle entry in D, DM, or DMS format. For example, you can enter a decimal degrees angle in the degrees box and nothing or zero in the minutes and seconds boxes. When no great precision is required there are no boxes for minutes and seconds. (In his book Chauvenet rounds these values to whole degrees.)

After entering all values, click OK. On the main window the Sun button is now enabled. Open the Sun window. Except for title, it's a clone of the Moon window. Enter these values:

Although the window says "Sun" it can be used for any body. If the center of a planet was observed use zero or blank for semidiameter, else the program assumes a limb observation and applies SD, however small. For stars use zero or blank SD and HP.

After you enter the Sun values, OK out of the window, and click the Atmosphere button on the main window. Enter these values:

In the heyday of lunars the air pressure instrument was a mercury barometer, which responded to both pressure and temperature. Another factor was the different temperature coefficients of the mercury and the brass scale that measured its height. Chauvenet discusses these matters at length, and I implement his correction formulas to obtain the actual air pressure (station pressure) when the "mercury barometer" box is checked. That's appropriate for solving examples in old books. On the other hand, today altimeter setting is the "barometric pressure" distributed to the public, at least in the US. In that case, un-check "mercury barometer" to disable the barometer temperature input box and compute station pressure from altimeter setting and height.

In reality, normal air pressure variations have little effect on refraction. Temperature variation is far more significant. For instance, a 2% pressure increase, say from 29.9 to 30.5 inches, is uncommon. A 2% decrease in absolute temperature, say from 70 to 60 F, has the same effect on refraction. Thus, normally it's sufficient to enter temperature, height, and a nominal altimeter setting.

Height also affects the dip correction. It's assumed your altitudes are with respect to the sea horizon, so dip is automatically subtracted unless you check the "no dip correction" box.

For this example, enter the values in the image and OK out of the window.

In the main window enter dead reckoning latitude 40° and lunar distance 67°23′16.5″. (The tab key will navigate through the input boxes and buttons in sequence.)

Open the Almanac window and enter the values below:

These are geocentric lunar distances at 1800, 1900, and 2000. In the old days they were tabulated in the almanac, normally every three hours. However, my program does not restrict you to that interval. Also, if the distances are unknown it will calculate them from the coordinates of the bodies. But for this example we use the distances I have provided.

For optimum accuracy the observation should occur between the first two entries, in this case between 1800 and 1900. But the math still works if you're outside that interval. The third distance (labeled "distance at t2") is not required. If present, it helps correct for the non-constant lunar distance rate of change. In this case, if you compare lunar distances at t0 and t1 vs. t1 and t2, the hourly rates are 2.7″ different. A correction for this "second difference" is applied with the formula of Chauvenet.

If you only need the "cleared" lunar distance (i.e., the observed distance reduced to Earth center), the almanac lunar distances are irrelevant and you don't need to enter values. However, you must still enter the window and exit with the OK button, else the OK button in the main window is not enabled.

For this demonstration, enter the values above and exit. In the main window click OK:

The correct angle d ("cleared" lunar distance, i.e., the equivalent geocentric angle) is 67°36′49.6″ and the correct UT is 17:50:00, so the result above is highly satisfactory. (Date is 2020 October 21, position N40° W050°.)

Although the time of 1750 was not within the data in the Almanac window, the result was still good. However, for best accuracy the observation should be in the interval between the first two almanac distances, though in this case it makes no difference. In fact, the third distance (which provides the optional second difference correction) can be omitted with no effect.

However, my demonstration was not too sensitive to these factors since the observation was close to one of the tabulated distances, and the tabulation interval was only one hour. It was usually three hours in the old days.

In this program angles may be entered in D, DM, or DMS format. For example, you can enter decimal degrees and leave the minutes and seconds boxes empty. (An empty box is the same as zero). For some non-critical angles, such dead reckoning latitude, it's sufficient to use the nearest degree or tenth degree, so there are no boxes for minutes and seconds.

If at least one of degrees, minutes, or seconds is negative, the program
considers the entire angle negative. For example, you can enter -22°13′45″
with a negative sign on 22° only. (But it's not an error to put a negative
sign on all three number.) Be careful with negative angles less than one
degree. The negative sign must be in a *nonzero* box. For example, to
input -0°0′0.5″, put the negative sign on the seconds.

It's not necessary that the values fit the boxes. The program reads all digits even if they overflow the box. (That can happen if you copy and paste from another document.)

The angle format selector in the main window affects only the output of d, the "cleared" (geocentric) lunar distance. Output precision is fixed at 0.1″, or the equivalent in other formats.

In this test, all inputs are decimal degrees. And since this is an accuracy test, precision is far beyond what is attainable in practice. Values are from my Lunar 4.4 software at 2019-01-01 19:00:00 UT1, +1m08.627s delta T, JPL DE431 ephemeris, IAU 2006 precession model and 2000B nutation. Polar motion is ignored.

Moon 15.44833 alt UL 0.25517 SD 0.93678 HP -13.8 dec Sun 26.66954 alt LL 0.27109 SD 0.00248 HP -23.0 dec 30.3 inches Hg, 32 F, height 0 40 lat, 48.44996 distance near to near almanac distances 49.14341, 47.67513, 46.21022 first distance at 18h, interval 3h 48°39′13.4″, 18h59m59s computed d and UT 48°39'13.0", 19h00m00s correct values

The result should be exactly midnight, but that makes the time solution too easy since it would normally coincide with a tabulated almanac distance. So to make the problem more difficult I provide distances every two hours, with the solution falling midway between the first and second distances.

Moon 26.77179 alt LL 0.24528 SD 0.90047 HP -13.6 dec Aldebaran 33.38650 alt 0.0 SD 0.0 HP +16.5 dec 30.3 inches Hg, 32 F, 0 height 40 lat, 98.37973 distance, far to center almanac distances 98.10525, 97.12144, 96.13749 first distance 23h, interval 2h 97°36′48.4″, 23h59m59s computed d and UT 97°36′48.1″, 00h00m00s correct values

So far I've provided geocentric lunar distances for input in the Almanac page. However, the program can calculate the distances from spherical coordinates. At the top of the Almanac page, select either RA/dec or HA/dec. The only difference between the two modes is hours or degrees for the angle measured about the pole. The coordinate system need not be equatorial, and the "longitude" angle can increase east or west. The only requirement is that the coordinates are geocentric apparent place. To demonstrate, in this example the Almanac data are ecliptic sexagesimal latitude and longitude.

Moon 71.35924 alt LL 0.24576 SD 0.90223 HP +0.8 dec Sun 35.97069 alt LL 0.27062 SD 0.00248 HP -17.7 dec 1010 mb, 10 C, height 0 10° DR lat 63.74182 observed distance, near to near

Enter this almanac data:

OK out of the Almanac window, and OK in the main window. Results:

64°06′16.0″, 16h40m01s computed values 64°06′15.9″, 16h40m00s correct values

Chauvenet gives two examples in the 1855 Almanac and one example in his spherical astronomy book. I don't think these are the best basis for testing lunar distance algorithms since he always has one body at low altitude and atmospheric conditions are far from standard in all three examples. (I believe that's intentional, to flatter his method when compared to simpler ones.) But it means the results are sensitive to the refraction model of the algorithm under test. Nevertheless, here I present all three, re-computed with my Lunar program. Both programs share the same refraction model (except refracted semidiameter), so any discrepancy is due to the slight departure from a rigorous solution in the Chauvenet lunar method.

This modern re-computation has the same angles as the 1855 Almanac example (within a few minutes). Input parameters: 1855-09-07 08:07 UT1, +0m10.270s delta T, N 35°30′ W 030°00′, JPL DE 422 ephemeris, IAU 2006 precession, 2000B nutation.

49°30.051' moon LL 14.781' SD 54.265' HP +25.3 dec 5°23.262' Sun LL 15.879' SD 0.146' HP +6.2 dec 29.1 inches Hg altimeter setting, 75 F, 20 ft height 35.5 DR lat 43°52.709' observed distance, near to near 46°02.961', 44°41.310', 43°19.763' almanac distances 6h first distance, 3h table interval 45°05.295′, 8h07m06s computed d and UT1 45°05.341′, 8h07m00s correct values

Parameters of the modern re-computation: 1855-08-30 05:42 UT1, +0m10.276s delta T, S 55°20′ W 120°25′. Fomalhaut catalog data from the Hipparcos new reduction (van Leeuven 2007).

6°34.798' Moon LL 16.399' SD 1°00.203' HP +3.8 dec 52°42.760' Fomalhaut altitude -30.4 dec 31 inches altimeter setting, 20 F, 18 ft height -55.3 DR lat 46°30.469' Moon far to Fomalhaut 43°51.962', 45°32.554' almanac distances 3 h first distance, 3 h interval 45°22.462′, 5h41m56s computed d and UT1 45°22.484′, 5h42m00s correct values

Modern re-computation at 1856-03-10 03:14 UT1, +0m10.147s delta T, N 35° W 150°, JPL DE422 ephemeris.

52°35'03.6" Moon LL 16'20.0" SD 59'58.0" HP +14.3° dec 8°57'46.8" Sun LL 16'05.6" SD 08.8" HP -4.1° dec 29.5 inch Hg altimeter setting, 58° F, 0 height +35° DR lat 44°37'40.5" distance, near to near 45°40'52.8", 47°21'51.9" almanac distances 3h first distance, 3h interval 45°48′44.3″, 3h14m00s computed d and UT1 45°48′44.9″, 3h14m00s correct values

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(last modified 2021-02-21)