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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Dan Allen
Date: 2004 Apr 28, 17:11 -0700
Sent: Wednesday, April 28, 2004 4:22 PM
To: NAVIGATION-L@LISTSERV.WEBKAHUNA.COM
Subject: Easy Lunars
So you've measured some star-to-star distances and
cleared them, and now you want to do a lunar. Lunars are not difficult despite
their reputation. It's perhaps not widely known that you can clear a lunar using
nothing more than the tables in the Nautical Almanac and a basic handheld
calculator. Moon UL: 45d 46.8' at 20:09:51 GMT.
Then I shot a very rough altitude of the Sun (no horizon but easy to estimate
from converging lines (Chicago is flat):
Sun LL: 47d 49' at 20:11:14 GMT.
And finally one lunar sight (it's best 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. 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: 80d 09.3' at 20:16:37 GMT.
After taking the lunar, I again measured the altitude of the Moon:
Moon UL: 47d 13.8' at 20:17:57 GMT.
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 59F and pressure was 29.93 inches
Hg.
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
Nautical Almanac for every three hours of GMT. Today, it may make more sense to
do the calculation for every hour. So we go to the almanac for April 26, 2004 at
20:00 and 21:00 GMT and dig out the GHA and Dec for the Sun and the Moon. Then
the LD follows from the usual cosine formula:
LD = arccos[sin(Dec1)*sin(Dec2) +
cos(Dec1)*cos(Dec2)*cos(GHA2-GHA1)].
I find:
geocentric_LD = 79d 59.1' at 20:00:00 GMT, We'll also need the horizontal parallax for the Moon for those hours and the
semidiameter for the Sun on that date. They are:
HP Moon = 54.7, Unlike 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 the 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-diameter 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 GMT. It's also easy to calculate it since it's
proportional to the HP:
Moon SD = 0.2724*HP = 14.9'.
For the Moon, we also need the "augmentation" of the semidiameter. 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
0.3*sin(H_moon), or you can use this short table:
<10d 0.0' 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 semidiameters is 31.0 minutes. That
means our center-to-center lunar measurement is
observed_LD = 80d 39.3'.
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 was
corrected_dist = observed_dist + dh1*A + dh2*B
where dh1 and dh2 are the altitude corrections 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. 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
corrected_LD = observed_LD + dh_Moon*A + dh_Sun*B + Q
where Q is given by
Q = (1/2) * (dh_Moon)^2 * cot(dist) * (1-A^2) / 3438
[note: in this equation "^2" means squared and also if dh_Moon 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 degrees,
so the factor cot(dist) is a small number which guarantess that Q is less than
0.1 minutes of arc. Q is also very small when the Moon's altitude is above 60
degrees and when the Moon and the other object are roughly aligned vertically.
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). These factors are exactly the same
as in the case of star-star distances:
A = (sin(H_moon) -
cos(dist)*sin(H_sun))/(cos(H_moon)*sin(dist)), We've already got the distance, 80d 39.3'. We also need the corrected
altitudes. 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 semidiameter 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. Applying the various corrections, I get:
H_sun corrected = 47d 12' Crunching the numbers on a handheld calculator, the results are:
A = 0.909, It's only a coincidence that the two values are almost the same in this case.
There's no need to bother writing down more than 3 digits of A and 2 digits of
B. These numbers are telling us that 90% 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.
To complete the clearing process, we need accurate values for dh_Moon and
dh_Sun. These are the combined effects of parallax and refraction. And here we
can use the standard tables in the Nautical Almanac. Opening the front cover of
the Nautical Almanac, the altitude correction for the Sun at 47 degrees altitude
is -0.8 minutes. 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 minutes (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_Moon and dh_Sun. Finally putting all the pieces together, we get the
corrected distance:
corrected_LD = observed_LD + dh_Moon*A + dh_Sun*B +
Q At last, we can compare this cleared lunar distance with the predicted
geocentric lunars for 20:00 and 21:00 GMT. They were 79d 59.1' and 80d 26.7'
respectively. Just eyeballing the numbers, our cleared measurement is about 25%
of the way in 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 20:16:44. This is the GMT implied by the
measurement. Since the sight was actually taken at 20:16:37, this means we've
determined GMT to within just 7 seconds. Not bad! In fact, that's probably just
a very lucky sight. If you can determine GMT to within 30 seconds of time,
you're doing well. To convert the difference in time to a difference in
longitude, divide the difference in seconds of time by 4. That is, an 8 second
error in GMT is equivalent to a 2 minute error in longitude.
The method for clearing a lunar that I've described above is very similar to
the old methods of Witchell and Thompson and others which were popular in the
19th century (compare Witchell's method in the online copy of Norie --URL
below). 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, so you'll see tangents and cotangents,e.g.,
instead of sines and cosines, 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 --URL below). The method I've described above is also very
similar to the approach that Letcher used in his 1978 "Self-Contained Celestial
Navigation with H.O. 208". He chose to calculate refraction instead of using the
tables in the Nautical Almanac. This has some advantages, but also a few
disadvantages. Specifically, that approach is unable to handle the parallax of
the Sun, Venus, and Mars (all of which are included in the tables in the front
of the almanac) and it also cannot handle non-standard temperature and pressure
which can easiy amount to a few tenths of a minute of arc difference in the
cleared lunar. I prefer including the complete altitude corrections from the
almanac since they can accomodate these various small factors.
I have the sense from chatting with people who have studied celestial
navigation that they prefer to use the tables they know. The altitude correction
tables in the almanac are well-suited to lunars. Using the Nautical Almanac and
a handheld calculator to get A, B, and Q, anyone can easily clear a lunar
distance sight in fifteen minutes or less.
Frank E. Reed PS:
First, you have to get out your sextant and shoot a lunar. I
shot a set Monday afternoon here on Lake Michigan. Since my Plath 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. Next, I shot an
altitude of the Moon's Upper Limb above the horizon of our inland sea and noted
the time:
geocentric_LD = 80d 26.7' at 21:00:00 GMT.
SD Sun = 15.9.
<30
0.1
<56 0.2
<90
0.3
B = (sin(H_sun) -
cos(dist)*sin(H_moon))/(cos(H_sun)*sin(dist)).
H_moon corrected = 46d
39.5'
B = 0.90.
= 80d 39.3' - 36.5*0.909 + 0.8*0.90 + 0 (Q is
negligible)
= 80d 39.3' - 33.2' +
0.7'
= 80d 06.8'.
[ ] Mystic, Connecticut
[X] Chicago, Illinois
URL for method III (Witchell's) in the 1828 edition of Norie:
http://www.mysticseaport.org/library/initiative/ImPage.cfm?PageNum=3&BibId=13617&ChapterId=31
URL
for Table XXXIV in Norie:
http://www.mysticseaport.org/library/initiative/Impage.cfm?PageNum=224&bibid=13617&ChapterId=39
PPS:
The method I've described here does not include a correction for the oblateness
of the Earth. This can be added easily but it's a fairly minor issue. This
method is also prone to error when the lunar distance itself is below 20 degrees
or the altitudes of the objects are below 20 degrees, and it should not be used
under those circumstances. This is an intrinsic weakness of any "series
expansion" solution of the lunar distance problem. This method also does not
account for the flattening of the Moon and Sun by refraction when they are close
to the horizon, but this effect is only significant at altitudes below 20
degrees which are already excluded on other grounds.