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Re: Method of Lunars
From: Peter Smith
Date: 1997 Sep 03, 12:18 PM

Daniel K. Allen (Visual C++)  asked:
>
> Since the method of lunars has been considered obsolete for about a
> century now, I have found it hard to get many details on the subject.
> Can anyone familiar with the method of lunars give any information about
> how it works, and what kinds of calculations must be made in order to
> figure longitude without time?

Since the Moon moves with respect to the Sun and stars, by measuring
the distance between it and a body near its path, one can interpolate
the time at which that distance was current from a table of lunar
distances -- although such tables haven't been published since the
beginning of this century (US Nautical Almanac dropped them in 1912).

Bowditch editions printed before before 1914 give several methods of
solving this.  Each is based on three simultaneous observations:
lunar altitude, a second body's altitude, and the angular distance
from the moon to the second body.  While the key to the solution is
the moon-body distance, the altitudes are required to compute the
refraction and parallax corrections necessary to correct the observed
distance from the moon's limb to the second body's limb to an actual
distance from the moon's center to the second body's center.

When only a single observer or instrument is available, the navigator
must approximate the altitudes at the time of the distance sight by
taking timed altitudes of both bodies before and after the distance
sight.  The altitudes at the time of the distance sight are then
interpolated by proportional logs.  Clearly, this will only be accuate
when both the moon and the second body are well off the meridian and
their altitudes are changing fairly linearly.  However, since the
corrections for refraction and parrallax do not change too rapidly at
moderate altitudes, the errors introduced by imprecition here shouldn't
be large.

Once the true lunar distance is calculated, the navigator enters the
almanac.  In Bowditch's day, the Nautical Almanac tabulated the true
distance from the moon to the sun, four major planets, and nine stars
for every three hours.  One would use the actual distance observed
to interpolate between tabulations and approximate the Greenwich time
of the observation.  Given the non-linearities in the motions of the
two bodies, this was not precise. Ol' Nat has this to say for the
expected accuracy:

As the moon moves in her orbit about 1' in 2m of
time, it follows that if her angular distance can be
ascertained from the sun or star within 1', the time
at Greenwich will be known within 2 minutes, and the
longitude within 30 miles.

As for the details of the calculation behind the now-extinct tables,
I attach a posting by once (and perhaps, still) listmember Jeff Gottfred,
who's grasp of theory and wealth of experience far exceeds mine.

[begin quote]

From- gottfred@agt.net (Jeff Gottfred)
Subject: Lunar Distances
Date: Mon, 18 Dec 1995 17:09:23 -0700

Why do I do lunars?

The short answer is for fun. A slightly longer answer is that I am very
interested in the navigational techniques of Alexander Mackenzie, David
Thompson, Peter Fidler, Lewis & Clark, &c. For a hobby I do workshops
and demonstrations of the techniques that these men used at various
historical sites.

[As a small aside, this summer, while giving a demonstration at the
Lewis and Clark festival in Great Falls, I took a noon sight for the
latitude of the Giant Springs of the Missouri River. To some
embarrassment, I discovered that Lewis was in error by some 17 nautical
miles. (Oops!). Even if you ignore refraction, it's hard to explain that
kind of error. I wonder if he was just having a bad day, or if all their
data was that bad? Perhaps some massive frontal system had just gone
through or something...]

In this message, I talk about my first successful attempts at lunars,
and then describe an historical method by Young (1848).

If anyone has fiddled around with lunars before, please let me know, I'd
be keen to know how it worked out for you and what methods you used.

If any of you are really keen to try this stuff using the methods,
tables, &c used from about 1790-1820 let me know, I can provide you with
them.

What is a Lunar?

The idea behind lunar distance observations is conceptually easy.
First, you must set your watch to local time-- I use solar time because
its easier, and my watch is then useful for other purposes, but you
could use any convenient body. Once you have set your watch to local
apparent time, you apply the equation of time correction (as found at
the bottom of each page in the Nautical Almanac) to set your watch to
local mean time (NB, this is NOT the same as mean zone time!!) so you
can relate it to the time data in the almanac, which uses the mean sun.

Next, you then measure the angular distance of the moon from some other
object on the ecliptic (the sun is my fave), and then look up in a lunar
distance table when the moon would be seen to be that distance by an
observer in Greenwich. Voila!, you now know the relationship between
Greenwich and local time, and therefore (at four minutes per degree)

Actually doing one:

O.K., problem #1, just how do you correct for refraction & parallax &c
when you make this observation?

This is called the "clearing the distance" problem.

For my first attempt (in the absence of better information-- but more on
that later), I simplified the problem of parallax by making my
observation at the moment of lunar transit, and ignoring refraction by
making sure to keep my observation above 20 degrees apparent altitude
for both bodies. To make it easy with a reflecting horizon, I used the
sun as the second body.

In this case, to clear the distance I had to apply corrections for the
semi-diameter of both bodies (I used near limb to near limb, therefore
added both corrections), and also the small correction for lunar
augmentation. At the moment of lunar transit, there is no parallax in
the plane perpendicular to the meridian, only the normal parallax in
altitude (PA) which is derived from the horizontal parallax value in the
almanac (H.P.). The PA is computed simply as:

PA = HP cos A

where A is the apparent altitude of the moon.

I have therefore constructed a spherical triangle that looks (something)
like this:

Z
.
/|
/ |
/  |
/   |
S/----+ Ma
\.  |
`\|
Mo

Where Z is the zenith, Ma is the actual position of the moon. Mo is the
observed position of the moon, and S is the sun;
and where Z-Mo is the apparent zenith distance of the moon, and Z-S is
the apparent zenith distance to the sun, and Ma-Mo the parallax in
altitude of the moon (PA).

Now for problem #2. Lunar distance tables.

Lunar distance tables haven't been produced for over 80 years so I
generated a little visual basic program that would crank out lunar
distance solutions for every ten seconds for the hour of the
observation--i.e., just enter the sun's GHA & dec, and moon's GHA and
dec, and solve the problem...

I then just go down the list until the find the distance that
corresponds to what I measure, and Bingo! I know the time in Greenwich
at the time of my obs!

When I have done this using a mechanical pocket watch which gains about
1.8 seconds per hour (and the rate varies!) I can find my longitude to
within about 16' or 10 nm at this latitude using this technique. The
to pick the exact moment of local apparent noon!

The problem with this technique is that it is so restrictive, I mean,
how many times per month do you get the moon transiting at same time
that the sun is more than 20 degrees above the horizon, and yet far
enough from the moon so that the moon is visible, and not more than 130
degrees from the moon (so you can measure the distance with a sextant)?
Well, you can count 'em on the fingers of one elbow... e.g., a couple of
times a month...

An Historical Technique:

This summer I stumbled upon the most excellent source on this stuff:
Charles H. Cotter, "A History of Nautical Astronomy", American Elsevier
Publishing Company, Inc., 52 Vanderbilt Avenue, New York, New York,
10017. 1968

Here is a method for clearing the distance first published by Young, in
1856.

First, construct the following spherical triangle (only better than this
one!):

Z
,/\
,/   `\
,/       `\
,/           `\ M
s,/--,        ,----\
,/     `-,----'      `\
,/   ,----' `-----,      `\ m
S,/----'              `-------`\
,/                               `\
H /-----------------------------------`\ O

Z is the zenith, M is the actual position of the moon, m is the apparent
position of the moon, S is the actual position of the sun, s is the
apparent position of the sun.

Zs is the apparent zenith distance to the sun, ZS is the actual zenith
distance to the sun, ZM is the actual zenith distance to the moon, and
Zm is the apparent zenith distance to the moon. HO is the observers
horizon.

Zs is less than ZS due to refraction, but Zm is greater than ZM because
the effect of the lunar parallax in altitude (PA) is (normally) greater
then the refraction.

What we see (and measure) in the sky is the observed lunar distance ms,
what we want to accomplish by "clearing the distance" is to compute the
actual distance MS -- this distance can then be compared to the lunar
distance table/output as described above.

For triangle ZSM, using the law of cosines for spherical triangles we
can write:

cos Z = (cos SM - cos ZS * cos ZM) / (sin ZS * sin ZM)

remembering your math teacher who insisted that:
sin (90-x) = cos x
cos (90-x) = sin x

we can write:

cos Z = (cos SM - sin HS * sin OM) / (cos HS * cos OM)

Next, for triangle Zsm, we can write another equation for Z:

cos Z = (cos sm - cos Zs * cos Zm) / (sin Zs * sin Zm)

An again, remembering our trig:

cos Z = (cos sm - sin Hs * sin Om) / (cos Hs * cos Om)

This gets a bit ugly, so let's simplify the notation a bit,
let's define some new variables:
D = SM, the true lunar distance

S = HS, the true altitude of the sun's center.
M = OM, the true altitude of the moon's center.
d = sm, the observed lunar distance.
s = Hs, the sun's apparent altitude.
m = Hm, the moon's apparent altitude.

we can now re-write the above equations as:

cos D - sin S * sin M
cos Z = ---------------------
cos S * cos M

and

cos d - sin s * sin m
cos Z = ---------------------
cos s * cos m

Now, add one to both sides of each equation:

cos S * cos M   cos D - sin S * sin M
1 + cos Z = ------------- + ---------------------
cos S * cos M       cos S * cos M

or,

cos D + cos S * cos M - sin S * sin M
1 + cos Z = -------------------------------------
cos S * cos M

and likewise to the second equation:

cos d + cos s * cos m - sin s * sin m
1 + cos Z = -------------------------------------
cos s * cos m

Now, remebering that wonderful trig formula (that I just now had to go
and look up again...)

cos (x + y) = cos x * cos y - sin x * sin y

we can now write:

cos D + cos (M + S)
1 + cos Z = -------------------
cos M * cos S

and,

cos d + cos (m + s)
1 + cos Z = -------------------
cos m * cos s

so, now be equating the two equations we get:

cos D + cos (M + S)   cos d + cos (m + s)
------------------- = -------------------
cos M * cos S         cos m * cos s

Multiplying both sides by cos M * cos S we get:

[cos d + cos (m + s)] * cos M * cos S
cos D + cos (M + S) = -------------------------------------
cos m * cos s

This can be written as:

cos M * cos S
cos D + cos (M + S) = [cos d + cos (m + s)] * -------------
cos m * cos s

Subtracting cos (M + S) from both sides we get (at last):

cos M * cos S
cos D = [cos d + cos (m + s)] * ------------- - cos (M + S)
cos m * cos s

This is Young's formula for clearing the distance.

In the best case, you use three observers to measure the alitiude of the
moon, the altitude of the sun, and the lunar distance at the same
instant.

If you are doing this alone, then you must take a few obs of the lunar
and solar altitudes, then measure the distance, then a few more lunar
and solar altitudes. You must then plot the altitudes and pick the
altitudes at the instant of the lunar distance measurement. One of the
benefits of Young's method is that slight errors in the altitudes do not
have a large effect on the result.

If anyone is really keen, I have more methods form Cotter's book...

[end quote -- whew! PS]
--
Peter Smith -- psmith@wellspring.us.dg.com
Data General Corp., Westboro, Massachusetts  (for whom I do not speak)
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