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LUNARS ARE EASY

First, you have to get out your sextant and shoot a lunar. I shot a "set" some years ago in Chicago, standing next to Lake Michigan. Since my sextant had been banged around a bit recently, I started with an index correction sight. This has to be done very carefully for a lunar sight since you're trying for fraction of a minute of arc accuracy. My IC was -1.0 minutes.

Chicago is Flat
Next, I shot an altitude of the Moon's Upper Limb above the horizon of our inland sea and noted the time:
   Moon UL: 45°47' at 20:09:51 UT.

Then I shot a very rough altitude of the Sun (no horizon but easy to estimate from converging perspective lines —since Chicago is flat):
   Sun LL: 47°49' at 20:11:14 UT.

And finally one actual lunar distance sight (it's normally better to take four or five and average them, or draw a line through them but I didn't have time to spare). The Moon was just shy of half full so I estimated that its elongation from the Sun would be about 75 degrees. I preset the sextant to that angle and aimed at the Moon. The line perpendicular to a line drawn through the Moon's "horns" always points to the Sun, so I rotated the sextant until the Moon appeared "horizontal" in the horizon glass. After a minute or so of sweeping around, I found the Sun. Then I brought the Sun into rough contact with the Moon so that they were touching limb-to-limb.

Fine Adjustment
When you're shooting Sun-Moon lunars, you're always bringing the "near" limbs together. From here, it's a slow process of adjustment. I usually lower the sextant from my eye, give the micrometer a slight turn, and then look through it again holding the sides of the frame with both hands. This makes for a stable platform. It also helps to sit while doing all of this. Note that you don't have to adjust the micrometer "live" as you're used to doing with altitude sights since the lunar changes very slowly. Finally, when the limbs were just touching, I noted the distance:
   Moon-Sun Near: 80°09.3' at 20:16:38 UT.
We should immediately subtract 1.0' for index error leaving 80°08.3'.

After taking the lunar, I again measured the altitude of the Moon:
   Moon UL: 47°14' at 20:17:57 UT.
I didn't bother getting a second altitude for the Sun, but it was probably about a degree and a half lower by this time. Finally, note that my height of eye was close to ten feet, and temperature was 59° F and pressure was 29.93 inches Hg [15° C and 1014mb].

Calculating the Geocentric Lunar Distances
With the observations in hand, it's time to get out the Nautical Almanac. First, there is one long calculation to do so that we have proper geocentric lunar distances to compare against the observations. This is exactly like the case of star-star distances except the Moon and the Sun move around much more rapidly, so we have to calculate two distances bracketing the time of the observations. In the 19th century, these distances were pre-calculated in the various nautical almanacs for every three hours of Greenwich Time. Today, it may make more sense to do the calculation for every hour (you can follow the steps below or you can also get pre-computed distances here on my website: ReedNavigation.com/lunars/predict).

We go to the almanac for April 26, 2004 at 20:00 and 21:00 UT and dig out the GHA and Dec for the Sun and the Moon. Then the LD follows from the usual cosine formula:
   LD = acos[sin(Dec₁) · sin(Dec₂) + cos(Dec₁) · cos(Dec₂) · cos(GHA₂ - GHA₁)].

I find:
   LD = 79°59.0' at 20:00:00 UT,
   LD = 80°26.7' at 21:00:00 UT.

We'll also need the HP or horizontal parallax for the Moon for those hours and the semi-diameter for the Sun on that date. They are:
   HP Moon = 54.7', SD Sun = 15.9'.

Pre-Clearing the Lunar
Unike a star-star sight, there is a "pre-clearing" step in every lunar. With stars, you can measure the distance from center-to-center since a star's image is basically a point. But with lunars, you're always measuring to a limb. So the first step is to add in the semi-diameters of the Sun and the Moon (you add for a "Near Limb" lunar and subtract for a "Far Limb"). We've got the Sun's SD from the almanac. You can take the Moon's SD from there, too, by interpolating. The listed SDs are correct for noon UT. Or skip the tabulated value; it's easy to calculate directly, since it's proportional to the HP:
   Moon SD = 0.2724 · HP = 14.9'.

For the Moon, we also need the "augmentation" of the semi-diameter. This is the slight increase in the Moon's size when it's overhead (you're closer to the Moon on average when it's high overhead). You can calculate this from 0.25' · sin(h_m), where h_m is the altitude of the Moon, or you can use this short table:
   <10°  0.0'
   <35°  0.1'
   <75°  0.2'
   <90°  0.3'

For the sights above, the augmentation is 0.2' so the Moon's augmented SD is 15.1' minutes of arc. The sum of the two semi-diameters is 31.0 minutes. That means our center-to-center "pre-cleared" lunar measurement is:
   LD_pc = 80°39.3'.

Pre-clearing the Altitudes
We also need the pre-cleared altitudes. These are the altitudes of the centers of the Sun and Moon, corrected for dip and semi-diameter (only). These altitudes do not have to be especially accurate. For the Moon's altitude, I took one sight before the lunar and one after the lunar. We can interpolate between those two values to get an altitude that is effectively simultaneous with the lunar distance observation. You can correct the altitudes for dip and semi-diameter separately but it was common in the 19th century to use a "cheap" correction that combined these: add 12' for a Lower Limb sight, subtract 20' for an Upper Limb sight. This was part of the "pre-clearing" process. Applying the various corrections, I get:
   Sun LL corrected = 47°12'
   Moon UL corrected = 46°40'

Clearing the Distance
And now to clear the distance... We need to remove the local, observer-centric effects of refraction and parallax so that we can compare our measurement with the predicted geocentric distances above. As in the case of star-star sights, we clear the altitude by determining very carefully how much each object's altitude is modified by refraction (and parallax now, too) and then figure out what percentage of that change in altitude acts along the arc from one object to the other. For star-star sights, the equation is:
   D_c = D_o + dh₁ · A + dh₂ · B
where dh₁ and dh₂ are the refraction corrections of the two stars, and A and B are the percentages or "corner cosines" for the arc in question.

With lunars, the change in the Moon's altitude can be as much as a degree. This is large enough compared to the other angles involved that we need to go one more step in this calculation —a step beyond the linear factors, A and B. There is a "quadratic" correction, call it Q, that has to be added into the puzzle. That is, the equation to correct a lunar distance is:
   LD_c = LD_pc + dh_m · A + dh_s · B + Q
where dh_m and dh_s are the altitude corrections of the Moon and Sun.

The Quadratic Correction
It looks intimidating at first, but it's frequently negligible, and historically it was found in an easy, short lookup table. In the modern era, we can calculate it directly. Q is given by
   Q = (1/2) dh_m² · [1 - A²] / tan LD.
"A" here is the same "corner cosine" factor from above [note: if dh_m is measured in minutes of arc and you want the result in minutes of arc, then Q has to be divided by 3438 (the number of minutes in a unit angle)].

This "Q" correction is usually less than a minute of arc, and it is often small enough to be ignored. In the observation I've described above, the measured lunar is about 80°, so the dividing factor tan(LD) is a large number which guarantess that Q is less than 0.1 minutes of arc. Q is also very small (small enough to ignore) when the Moon's altitude is above 60° or when the Moon and the other object are roughly aligned vertically.

The Linear Terms
Let's calculate A and B first. These "corner cosines" tell us how much of an object's altitude correction acts along the arc that we are measuring. So, for example, when you're shooting a lunar, if the Moon is exactly above the Sun or other object (so that their azimuths are identical), then the A and B factors would both be 100% (though opposite in sign):
   A = cos α = (sin h_s - cos LD · sin h_m) / (cos h_m · sin LD),
   B = cos β = (sin h_m - cos LD · sin h_s) / (cos h_s · sin LD).

Crunching the numbers on a handheld calculator, the results for this lunar are:
   A = cos α = 0.909,
   B = cos β = 0.91.
It's only a coincidence that the values are almost the same in this case. There's no need to bother writing down more than three or four digits of A and two or three digits of B. These numbers are telling us that 91% of the Sun's altitude correction acts along the arc of the lunar distance and 90.9% of the Moon's altitude correction acts along the arc.

Altitude Corrections from the Almanac
To complete the clearing process, we need accurate values for dh_m and dh_s. These are the combined effects of parallax and refraction. And here we can use the standard tables in any modern Nautical Almanac. Opening the front cover of the Nautical Almanac, the altitude correction for the Sun at 47° altitude is -0.8'. This is the average of the corrections for UL and LL. Opening the back cover of the Nautical Almanac, the altitude correction for the Moon is 36.5' (again found by taking the average of the corrections for UL and LL —this is the same procedure that's used for bubble sextant sights). I also checked the temperature-pressure correction table in the front of the almanac. The conditions were close to standard, so there was no additional correction for either object. And there's one more important step: change the signs on both dh_m and dh_s. Finally putting all the pieces together, we get the corrected distance:

   LD_c = LD_pc + dh_m · A + dh_s · B + Q
  = 80°39.3' - 36.5' · 0.909 + 0.8' · 0.91 + 0 (Q is negligible)
  = 80°39.3' - 33.2' + 0.7'
  = 80°06.8'.

Interpolating to find the Longitude
At last, we can compare this cleared lunar distance with the predicted geocentric lunars for 20:00 and 21:00 UT. The predicted lunar distances were 79°59.0' and 80°26.7' respectively. Just eyeballing the numbers, our cleared measurement is about 25% of the way between the two predicted distances, so this is telling us that the time we've measured must be close to 20:15. That's good.

Interpolation yields a time for the lunar of 20:16:54. This is the UT implied by the measurement. Since the sight was actually taken at 20:16:38, this means we've determined UT to within just 16 seconds. Not bad! If you can determine UT to within 30 seconds of time, you're doing well. Key scale to remember: every tenth of a minute of accuracy (or error!) corresponds to 12 seconds in the final UT.

To convert the difference in time to a difference in longitude, divide the difference in seconds of time by 4. That is, an error of 16 seconds in UT is equivalent to a 4' error in longitude (which is 4 nautical miles at the equator or 3 nautical miles in the latitude of Chicago).

This Approach in Historical Context
The method for clearing a lunar that I've described here is very similar to the old methods of Witchell, Mendoza Rios, Bowditch, Thompson and others which were popular in the 19th century. Those methods depended on logarithms for their calculation of A and B and they used various trigonometric identities to put those calculations in a form suitable for logarithms, but that's just a re-arrangement of terms. There is no fundamental difference. The quadratic term which I have called "Q" was usually calculated in a special table that was included in Bowditch, Norie, etc. (table 35 in Norie, for example).

The method I've described above is also very similar to the approach that John Letcher used in his 1978 Self-Contained Celestial Navigation with H.O. 208 (a great book on celestial navigation generally, which I still recommend, though copies at a reasonable price are now hard to find). Letcher included a nice trick to calculate refraction instead of using the tables in the Nautical Almanac. This has some advantages, but also a few disadvantages.

Little details to worry about...
There is an additional correction for the oblateness of the Earth (the globe is a slightly flattened spheroid; not a perfect sphere). This detail can be added easily but it's small, rarely amounting to more than 0.1'. This method is also prone to error when the lunar distance itself is below 15° or the altitudes of the objects are below 15°, and it should not be used under those circumstances.

There's an App...
Sometimes you shoot a lunar, and you just want to find out how well you did. Try my web app for clearing lunars: ReedNavigation.com/lunars/clear. Here it is with all the data pre-filled for the lunar scenario above: Chicago lunar example.

You want more?
Finally, a note to the future lunarian: I wrote this essay years ago. It's a good start to get you thinking about lunars, but there's so much more to the story. Want more?? Register for my Lunars workshop. And if that's not enough... if you want to be a true lunarian... consider my Advanced Lunars workshop.

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA