|
|
|
|
|||||
|
||||||
Subject: Re: Approximate methods for clearing the Lunar distances - some details; corrected version
From: Arthur Pearson (arthurpearson@XXX.XXX)
Date: Sat Apr 19 2003 - 22:53:20 EDT
Jan,
I have re-read this posting and continue to find it fascinating. It
paints a vivid picture the extraordinary efforts expended to find an
accurate method of clearing the distance that was also convenient for
use by the average sea captain. The derivation of these formulae is
brilliant in itself. The ingenuity required to transform them in ways
amenable to tabular solution is stunning. The determination to grind
through 30,000 LD solutions and another 50,000 interpolations to
actually produce a table is beyond imagination. Sending a man to the
moon offers nothing more impressive than the achievement of making
lunars a practical tool.
Because I need to condense things to understand them, I have amended my
diagram to illustrate all the variables you mention and copied over the
polynom, the substitutions, and the approximate formula just below it.
It is still at
http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG. Please
let me know if you see any mistakes in my transcription of formulae or
placement of variables. Needless to say, links to both branches of this
thread are now on the Nav-L section of www.LunarDistance.com.
My only remaining questions: What exactly is a polynom and where in the
derivation of formulae are those perpendiculars used?
Thanks again,
Arthur
-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@XXX.XXX] On Behalf Of Jan Kalivoda
Sent: Sunday, April 13, 2003 12:16 PM
To: NAVIGATION-L@XXX.XXX
Subject: Approximate methods for clearing the Lunar distances - some
details; corrected version
I am sorry, but when posting this article for the first time, symbols of
halving and of squaring were distorted in formulas on the web. Therefore
I send it for the second time. Erase the first version, please and take
the symbols "P2", "sin2" as "P squared", "sin squared" onwards. The
original article follows:
=====================
In my previous article on classification of methods of clearing the
lunar distances, I didn't want to burden its text by greater details.
But some of you replied that they would like to read more about the
approximate methods for clearing the lunar distances ("lunars", LD's)
and about the deduction of their basic formula. Therefore I can perhaps
try to give more details now - only for the interested guys.
Meanwhile Arthur Pearson prepared a nice picture for this purpose on his
delightful website devoted to "lunars" (http://www.lunardistance.com/).
I thank him very much, it was beyond my graphic abilities. Use this
picture as the reference, as I will do onward:
http://members.verizon.net/~vze3nfrm/Nav_L_Graphs/ApproxLD.JPG
(In this picture and in the following text you can always take a star or
a planet for the Sun ! )
Consider the spherical triangle: Zenith - true Moon - true Sun; and the
second one: Zenith - apparent Moon - apparent Sun. They have the common
angle Z'. Therefore we can deduce various strict fomulas for the true LD
between the true Sun and the true Moon, using the apparent LD and four
apparent and true zenith distances in these two triangles. This is the
way of rigorous methods.
But as the distances apparent Sun - true Sun and apparent Moon - true
Moon are very small in comparison to other elements of both triangles,
we can also search only for small corrections that can be algebraically
added to the apparent=observed LD to obtain the true LD. This is the way
of approximate methods.
These corrections can be found by two procedures: by the calculus or by
using the spherical trigonometry of perpendiculars dropped between the
vertices and sides of two triangles mentioned (see the lower part of the
picture). The end formula CAN be the same in both cases (as always in
the nautical astronomy, several equivalent solutions are possible,
according to the aim of their inventors).
The trigonometrical deduction would require some hundred lines; so I
confine myself to the calculus procedure.
Symbols:
Z - zenith
Z' - angle at zenith (difference of azimuths of both bodies)
M, S - true Moon and Sun
m, s - apparent Moon and Sun
m', s' - angles at m, s
D - true lunar distance (distance MS)
d - apparent (observed) lunar distance (distance ms)
A, a - apparent altitudes of the Moon and the Sun over the horizon
x - the distance Mm, i.e. the difference of the Moon' parallax in A and
the refraction in A
y - the distance Ss, i.e. the difference of the Sun's parallax in a and
the refraction in a
P, p - HORIZONTAL parallaxes of the Moon and Sun
(parallaxes in altitudes A, a can be obtained by formulas: P
cos A , p cos a)
R, r - refractions in altitudes A, a
The starting point is the Taylor's polynom for the small changes x,y of
two sides Zm, Zs of the spherical triangle Zms, forming the constant
angle Z':
D = d + y cos s' - x cos m' + 1/2 (y2 sin2 s' + x2 sin2 m') cotg d + x y
sin s' sin m' cosec d ..... (an infinite number of smaller terms
follow that can be and were fully neglected)
Of course, this form is unusable at sea. Now three substitutions should
be made:
y = r nearly, as the maximal parallax of the Sun is 9 arc-seconds, of
Venus 33 arc-seconds and and of Mars 23 arc-seconds; therefore the
parallax of the second body can be reduced by a special term at the end
of procedure and neglecting it at the beginning of arrangements will not
induce a significant error during the further treatment of the formula;
Jupiter, Saturn and stars have the daily parallax zero for nautical
purposes
x = P cos A - R ; difference of Moon's parallax in its observed
altitude and of Moon's refraction in its observed altitude!
cos m' = (sin a - sin A cos d) / (cos A sin d) ; this is the clever
substitution of Israel Lyons (1766) into the term (x cos m') of polynom,
resulting from the spherical cosine theorem for the triangle Zms; this
substitution is essential for the use of the whole formula in this
branch of approximate methods
So after these substitutions and some following bothersome, but
elementary trigonometric arrangements (let me jump them over now, I can
send them to you individually, if you like), we obtain the basic formula
for the most important approximate procedures and their tables for
clearing the lunar distances from 1810 to the death of lunars:
D = d -
(1) - P sin a cosec d +
(2) + P sin A cotg d +
(3) + r cos s' + R cos m' +
(4) + 1/2 (P2 cos2 A sin2 m' cotg d) -
(5) - P R cos A sin2 m' cotg d + P r cos
A sin s' sin m' cosec d +
(6) + 1/2 (r2 sin2 s' + R2 sin2 m') cotg d -
R r sin s' sin m' cosec d -
(7) - p sin A cosec d + p sin a cotg d
Remarks:
1,2 - two greatest terms for the Moon's parallax; these two corrections
were computed by the sailor himself by logarithms to 4 places and
proportional logarithms; thanks to Lyons' substitution, sailor hadn't to
be tortured by computing the angle at the apparent Moon and worked only
with the horizontal parallax of the Moon, with both observed altitudes
and with the observed distance; Thomson has provided four auxiliary
tables to speed up these two calculations made by seaman
3 - two greatest terms for both refractions; here the "THIRD CORRECTION"
begins; ALL THE FOLLOWING (except line 7) was tabulated in ONE table
according to observed altitudes of both bodies and their observed
distance; Elford as the inventor of such sort of tables (1810), then
Norie (1815) and many others tabulated only the value of this line (3),
leaving all other terms aside
4 - another term for the Moon' paralax; the value of this line was
tabulated for the first time by Maskelyne in the table No.13 of his
"Tables Requisite" = table No.35 of Norie's "Epitome of practical
Navigation" = table No.20 of Bowditch' tables (??? I am not quite sure
of this last assertion); these tables were used in older approximate
methods (Lyons, Witchell, Maskelyne, Bowditch) preceding the Elford's
deed in 1810; they could be used together with Elford's (and similar)
tables and Norie's nomograms, too
5 - two terms for combined effects of the moon's parallax and both
refractions
6 - two smaller terms for both refractions
7 - two greatest terms for the parallax of the second body, if any; it
was given by a special table by Thomson; all smaller terms of this
parallax remained fully neglected as quite unimportant
It was my beloved Thomson, who brought this procedure to the state of
perfection. He never stated his method for computing his pithy table of
the "third correction" (firstly 1824, the 67th edition in 1880,
according to the kind information of Bruce Stark), which didn't require
any interpolation, as its steps were very small. (Compare the table No.
48 in the Bowditch' old editions (to 1851 at least), which is the
reprint of Thomson's main table, although it is called "Third correction
in Lyon's improved method" in the table of contents; but
Thompson=Thomson is cited in the main text at the "second method" for
clearing LD's.)
From Thomson's results it has been ascertained that he had included ALL
terms of the approximate formula given above. He couldn't compute them
step by step, as it would be an absolutely hopeless effort for one man,
even for many. From some his remarks (and above all from remarks of
G.Coleman, the editor of Norie's Epitome after Norie's death, who was
purportedly in contact with Thomson in his youth) it is probable, that
Thomson computed only the first two terms from the lines (1) and (2)
above and then the whole difference between true and apparent distance
by some rigorous method. And by subtracting those two terms from the
whole difference found, he obtained the value of remaining terms from
the lines (3) - (6). The effect of the parallax of the second body from
the line (7) he tabulated in a special table, as said above.
The improvement of accuracy of Thomson' table in comparison with Elford
and Norie is most evident, when the lunar distance itself and both
altitudes are small (and if both altitudes are nearly the same - this
unfavourable case is seldom in lower latitudes). So for both altitudes
and lunar distance of 20 degrees, Elford gave the "third correction" of
22 arc-seconds, Norie the value of 19 arc-seconds, but Thomson (and
Bowditch) 88 arc-seconds! And this greater value is the true one. The
importance of smaller terms in the approximate formula (lines 4 - 6) is
manifest by that.
The general accuracy of Thomson's main table of the "third difference"
was very high. It has been found out by later trial calculations that
the error is under 2 arc-seconds in the most cases, and only seldom it
attains 5-6 arc-seconds. Of course, isolated errors of a much greater
degree can hide in the tables (which were never recomputed as the
whole), as in all tables for "approximative" methods. Slocum mentioned
one such case in the record of his circumnavigation - near the island
Nukahiva in Marquesas, he ascertained "an error in the important
logarithm of the tables" during clearing the lunar distance. I suppose
that he had some table for approximative methods in his hands - he could
not verify the values of the decadic logarithms themselves aboard, I
guess. Or does anybody understand this interesting place of Slocum's
book in a different manner?
But the accuracy of Thomson's tables, as mentioned, must be understood
only for their default conditions. All tables of the "third difference"
for clearing LD's by "approximate" methods were liable to the important
drawback: they were to be computed for the MEAN EQUATORIAL horizontal
parallax of the Moon (57,5 arc-minutes) and for the MEAN refraction (for
30 inches of barometric pressure, 55 Fahrenheit degrees of temperature).
And because their values were the lump of effects of both the parallax
and the refraction, allowing for the actual Moon's parallax, for the
actual atmospheric conditions and for the effects of ellipsoidal Earth's
shape separately was very difficult. I won't enter into these details
here. But the maximal error created by using only the mean Moon's
horizontal parallax (57,5 arc-minutes) was 10 arc-seconds at short LD ,
5 arc-second at LD of 40 degrees and decreasing further with growing LD.
The combined maximal amount of all three effects of ellipsoidal !
Earth's shape was 13 arc-seconds. The effect of the anomalous refraction
could be considerable, of course, above all in the cases, when the LD
ran vertically to the horizon - the error of 60 arc-seconds is then
exceptionally possible.
But one had always to expect a MINIMAL error of 30 arc-seconds in
measuring a lunar distance aboard, even after averaging a set of
observations. And the tabulated Moon's positions in almanacs (and
therefore the tabulated lunar distances as well) could be in error of 1
arc-minute before 1820 and 20 arc-seconds before 1880. So an exaggerated
accuracy of procedures and auxiliary tables for clearing lunar distances
was considered meaningless by sailors (not by theoreticians and teachers
of navigation). In the second half of 19th century the more precise
forms and tables of approximate methods appeared (Chauvenet after 1850,
Bolte in 1894 and certainly other ones). But they became so complicated
in use that they provided no advantage in comparison to rigorous
methods. So the generic line "(Lyons) - Elford - Thomson - epigons of
both" prevailed among approximate methods up to the end of lunars.
Of course, Thomson's empirical procedure and table constructed by it
could not be taken into the official navigation handbooks and courses.
Thomson's procedure and tables remained an illegitimate darling of
sailors for 80 years, but they never obtained the status of their
reputable bride.
Jan Kalivoda
|
