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Re: Letcher's Lunars Math
From: George Huxtable
Date: 2004 May 21, 16:52 +0100

```Frank Reed wrote-

>In Part 3 of George H's compendium on lunar distances, he talked a little
>about Letcher's method for clearing lunar distances as described in his book
>"Self-Contained Celestial Navigation with H.O. 208"...
>
>George H wrote:
>"The way he proposes for its use would be suitable only for lunar distances
>up to 90 degrees, whereas any lunar observer would certainly require its
>application to angles up to 120 degrees. I will therefore give a modified
>calculation method that will cover the full range of useful sextant angles."
>
>This isn't right. I can't figure out what George was thinking here, but the
>equation in Letcher's book is fine just the way it is and can be used without
>modification.

==========================

That old posting was headed "About lunars, part 3", and dated 13 Feb 2002,
if anyone wishes to look it up in their in-box or on the "irbs" archive.
I'm grateful to Frank for blowing the dust off and exposing it to the
light-of-day once again.

However, I stand by what I wrote then. Indeed, it's rather surprising that
Frank didn't become aware of the problem himself, if he tried to reduce an
example, Letcher's way, with a lunar distance exceeding 90deg.  It arises
in the correction for refraction.

Letcher says, as part of his correction process for a lunar distance-

"Next, the correction for refraction given in Table 15.2 is applied,
entering with the altitudes and distance already corrected for
semidiameters, and interpolating to the nearest tenth of a minute."

Table 15.2 is part of his chapter on "interstellar distances", using, very
ingeniously, a nomograph and a table. Unfortunately, that table handles
"distances", angles across the sky, up to 90deg, and no further. I think
there would have been no problem in extending the table to 120deg, but
Letcher didn't do that. So once a user gets to correcting for refraction,
with a lunar distance greater than 90, he is outside the limits covered by
table 15.2. What does he do, then?

Thoughtfully, in the caption to that table 15.2, Letcher has provided the
expressions that it is based on.

So, in my explanation about using Letcher's method, I have avoided any
reference to table 15.2. Instead, in "About lunars, part 3", I simplified
those expressions to--

R = .95* (sin s/sin m + sin m/sin s - 2*cos d) / sin d , in minutes of arc.
This works for all non-zero angles.

That's what I was getting at, when writing the passage that Frank quoted.

===================

However, it's not the only difficulty faced by the user of Letcher's book,
if he needs to handle lunar distances that exceed 90deg. 100 years ago,
such distances were tabulated in the Nautical Almanac, so there was no
problem. Recently, Steven Wepster has provided a similar service.

However, Letcher suggests that the user calculates his own lunar distances,
between the moon and another body, by using standard "tables of altitude
and azimuth". If, in such a table, you replace the observer's assumed lat
and long by the dec and GHA of the Moon, and input the dec and lat of the
other body, and subtract the resultant "altitude" from 90deg, this result
is the "lunar distance", the angle across the sky between the two bodies.
It's exactly the same as calculating the great-circle distance (in degrees)
between two points on the Earth's surface.

If a body is more than 90deg away from the Moon across the sky, then that
would require an "altitude" of less than zero degrees, from the
altitude-azimuth tables. Not surprisingly, compilers of these tables
haven't seen any virtue in providing such "altitudes" below (or even near
to) zero, and have saved paper and ink by omitting them. For example, my
1951 copy of the Admiralty's tables HD 486 vol IV (itself copied from the
US HO 214) omits all alt-az pairs where the altitude is less than about
5deg. How would one use those tables to  predict lunar distances that
exceeded 85deg?

For alt-az. calculations, Letcher includes the Dreisonstok tables (HO 208).
I have no familiarity with those tables at all, but it looks to me as
though they are usable down to altitudes of 0deg. Is there, I wonder, a
trick to allow them to calculate negative altitudes? I can't find any such
reference in the instructions for using HO 208.

It seems to me that Letcher neglects to draw attention to this difficulty,
and skates around it by avoiding all mention of lunar distances greater
than 90deg.

Of course, there's no great difficulty in calculating such lunar distances,
from the positions of the bodies, by other means. Bruce Stark's ingenious
lunar tables allow that to be done, with no restriction on the resulting
angle. I have quoted an expression, to use in a calculator or computer, in
"Calculating predicted lunar distances for yourself", in my earlier posting
"About lunars, part 2".

George.

>George H wrote:
>"I am not sure where this method comes from, as Letcher does not quote any
>reference, nor does he show how it is derived from first-principles. "
>
>I strongly suspect he derived it himself since it is not all that difficult
>to do so. It is similar to the equation that I posted in "Easy Lunars" except
>that he prefers a (rather neat) graphical method for calculating the refraction
>part of the linear terms and he uses specific mathematical forms for both
>refraction and the Moon's parallax instead of using the altitude correction
>tables in the Nautical Almanac.
>
>The clearing equation that I posted in "Easy Lunars" is
>
>d = d0 + dh1*A + dh2*B + Q
>
>where d0 is the measured lunar (converted to a "center-to-center" distance by
>adding/subtracting semidiameters of the two objects), dh1 is the altitude
>correction for the Moon, dh2 is the altitude correction for the Sun or other
>body, Q is the small "quadratic correction", and A and B are the "corner
>cosines"
>which tell us how much of each altitude correction acts along the arc
>connecting the two bodies. A is given by
>
>A = (sin(h2)-cos(d)*sin(h1))/(sin(d)*cos(h1))
>
>and B is the same with h1 and h2 swapped. Q is given by
>
>Q = 0.5*(dh1)^2*cot(d)*(1-A^2).
>
>Note that if you use 0.55 instead of 0.5 above, the equation covers a wider
>range of cases as I've mentioned before. I'm leaving it in the original form
>here since that's closer to Letcher's version. Also note that if dh1 is
>measured
>in degrees, then you have to divide Q by 57.3. If dh1 is measured in minutes
>of arc, then Q is divided by 3438.
>
>This equation can be derived from straight-forward calculus. You take the
>equation for the standard spherical triangle for the lunars problem and
>then do a
>two-dimensional Taylor series expansion. The resulting series has an infinite
>number of terms, but by knowing upper limits on dh1 and dh2, only the first
>few terms have to be kept to achieve the required accuracy for lunar distance
>sights. Historically, these series expansions were called "approximate" or
>"approximative" solutions of the lunar distance problem. This terminology is
>misleading. They are as accurate as they need to be when done right.
>
>Letcher, I suspect, derived the same equation as above and then he chose to
>insert specific mathematical forms for dh1 and dh2, which was also a common
>practice in 18th and 19th century clearing methods. For dh1, the altitude
>correction of the Moon, Letcher used
>
>  dh1 = HP*cos(h1)-0.95*cot(h1)
>
>for the linear term, and for the quadratic term he used
>
>  dh1 = HP*cos(h1).
>
>Dropping second part, the refraction, for the Moon in the quadratic term is
>not unreasonable since the quadratic term is very small but it does lead to an
>error of a little more than 0.1 minutes of arc in some cases where the
>distance is above 90 degrees or so and the altitude of the other body is
>low. For the
>other body's altitude correction, Letcher uses
>
>  dh2 = -0.95*cot(h1).
>
>This terms gives the altitude change due to refraction only and, for both the
>Moon and the other body, the refraction is accurate only at standard
>temperatures and pressures. It does not include an option for parallax of
>the other
>body. Also this simple expression for refraction is valid only down to 15
>degrees of altitude. It's not a good idea to shoot lunars when objects are
>at lower
>altitudes anyway, so this isn't really a problem. As Letcher notes in his
>book, his choices for the functional form of dh1 and dh2 mean that his equation
>will have errors amounting to as much as 0.3 minutes of arc. In fact for
>Mars or
>Venus (when close to the Earth) and under unusual atmospheric conditions, the
>error could be even larger than this.
>
>If you decide to crunch through the algebra to verify that you can get
>Letcher's equation by following the steps above, you'll find that his term
>"y" is
>the same as the quantity "A" in my version (A is the corner cosine for the
>Moon)
>multiplied by cos(h1). This makes little difference to the calculational
>steps involved, but it also removes the simple geometric interpretation of this
>intermediate quantity (which is very useful for teaching purposes and
>error-detection to a lesser extent) as the "percentage" of the altitude
>correction that
>acts along the arc of the lunar.
>
>Frank R
>[ ] Mystic, Connecticut
>[X] Chicago, Illinois
>
>FAMILY="SANSSERIF" FACE="Arial" LANG="0">In Part 3 of George H's
>compendium on lunar distances, he talked a little about Letcher's method
>for clearing lunar distances as described in his book "Self-Contained
>Celestial Navigation with H.O. 208"...
>
>George H wrote:
>"The way he proposes for its use would be suitable only for lunar
>distances up to 90 degrees, whereas any lunar observer would certainly
>require its application to angles up to 120 degrees. I will therefore give
>a modified calculation method that will cover the full range of useful
>sextant angles."
>
>This isn't right. I can't figure out what George was thinking here, but
>the equation in Letcher's book is fine just the way it is and can be used
>without modification.
>
>
>George H wrote:
>"I am not sure where this method comes from, as Letcher does not quote any
>reference, nor does he show how it is derived from first-principles. "
>
>I strongly suspect he derived it himself since it is not all that
>difficult to do so. It is similar to the equation that I posted in "Easy
>Lunars" except that he prefers a (rather neat) graphical method for
>calculating the refraction part of the linear terms and he uses specific
>mathematical forms for both refraction and the Moon's parallax instead of
>using the altitude correction tables in the Nautical Almanac.
>
>The clearing equation that I posted in "Easy Lunars" is
>
>d = d0 + dh1*A + dh2*B + Q
>
>where d0 is the measured lunar (converted to a "center-to-center" distance
>by adding/subtracting semidiameters of the two objects), dh1 is the
>altitude correction for the Moon, dh2 is the altitude correction for the
>Sun or other body, Q is the small "quadratic correction", and A and B are
>the "corner cosines" which tell us how much of each altitude correction
>acts along the arc connecting the two bodies. A is given by
>
>A = (sin(h2)-cos(d)*sin(h1))/(sin(d)*cos(h1))
>
>and B is the same with h1 and h2 swapped. Q is given by
>
>Q = 0.5*(dh1)^2*cot(d)*(1-A^2).
>
>Note that if you use 0.55 instead of 0.5 above, the equation covers a
>wider range of cases as I've mentioned before. I'm leaving it in the
>original form here since that's closer to Letcher's version. Also note
>that if dh1 is measured in degrees, then you have to divide Q by 57.3. If
>dh1 is measured in minutes of arc, then Q is divided by 3438.
>
>This equation can be derived from straight-forward calculus. You take the
>equation for the standard spherical triangle for the lunars problem and
>then do a two-dimensional Taylor series expansion. The resulting series
>has an infinite number of terms, but by knowing upper limits on dh1 and
>dh2, only the first few terms have to be kept to achieve the required
>accuracy for lunar distance sights. Historically, these series expansions
>were called "approximate" or "approximative" solutions of the lunar
>distance problem. This terminology is misleading. They are as accurate as
>they need to be when done right.
>
>Letcher, I suspect, derived the same equation as above and then he chose
>to insert specific mathematical forms for dh1 and dh2, which was also a
>common practice in 18th and 19th century clearing methods. For dh1, the
>altitude correction of the Moon, Letcher used
>
>  dh1 = HP*cos(h1)-0.95*cot(h1)
>
>for the linear term, and for the quadratic term he used
>
>  dh1 = HP*cos(h1).
>
>Dropping second part, the refraction, for the Moon in the quadratic term
>is not unreasonable since the quadratic term is very small but it does
>lead to an error of a little more than 0.1 minutes of arc in some cases
>where the distance is above 90 degrees or so and the altitude of the other
>body is low. For the other body's altitude correction, Letcher uses
>
>  dh2 = -0.95*cot(h1).
>
>This terms gives the altitude change due to refraction only and, for both
>the Moon and the other body, the refraction is accurate only at standard
>temperatures and pressures. It does not include an option for parallax of
>the other body. Also this simple expression for refraction is valid only
>down to 15 degrees of altitude. It's not a good idea to shoot lunars when
>objects are at lower altitudes anyway, so this isn't really a problem. As
>Letcher notes in his book, his choices for the functional form of dh1 and
>dh2 mean that his equation will have errors amounting to as much as 0.3
>minutes of arc. In fact for Mars or Venus (when close to the Earth) and
>under unusual atmospheric conditions, the error could be even larger than
>this.
>
>If you decide to crunch through the algebra to verify that you can get
>Letcher's equation by following the steps above, you'll find that his term
>"y" is the same as the quantity "A" in my version (A is the corner cosine
>for the Moon) multiplied by cos(h1). This makes little difference to the
>calculational steps involved, but it also removes the simple geometric
>interpretation of this intermediate quantity (which is very useful for
>teaching purposes and error-detection to a lesser extent) as the
>"percentage" of the altitude correction that acts along the arc of the
>lunar.
>
>Frank R
>[ ] Mystic, Connecticut
>[X] Chicago, Illinois

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
================================================================

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