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Subject: Re: The Tables for Clearing the Lunar Distance
From: Fred Hebard (Fred@XXX.XXX)
Date: Thu May 22 2003 - 12:18:06 EDT
Bruce,
Thank you so much for laying this out for us mere mortals. I am sure
it was a lot of work to do so, and I very much appreciate the effort
you have put forth to do this.
Yours Truly,
Fred Hebard
On Thursday, May 22, 2003, at 11:57 US/Eastern, Bruce Stark wrote:
> This is a belated continuation of the April 28th and May 5th postings
> "Clearing a lunar" and "Converting a Lunar to GMT." It focuses on my
> Tables.
>
> The first thing the Tables and work form do is square away the
> details. If
> altitudes were measured along with the distance, table 1 gives the
> adjustment to
> turn them into apparent altitudes of the centers. Or, if altitudes were
> calculated rather than measured, the "W.W." tables "uncorrect" them to
> turn them
> into apparent altitudes of the centers.
>
> The W.W., (wrong way) tables were developed by working backward. Take,
> for
> example, the W.W. Parallax correction for the moon at 5° altitude and
> 61'
> horizontal parallax. Multiply the cosine of 5° by 61'. That gives the
> moon's
> parallax correction for a 5° apparent altitude. Taken to three decimal
> places the
> correction is 60.'768, additive. Apply that the wrong way. That is,
> subtract it
> from 5° to get 3° 59.'232. Then multiply the cosine of 3° 59.'232 by
> 61' to get
> 60.'852. Subtract that from 5° to get 3° 59.'148. Repeat another time.
> There's no appreciable change from 60.852, so the W.W.P. correction,
> rounded to the
> nearest tenth, is 60.'9.
>
> Table 1 and the W.W. tables are ridiculously precise, considering how
> forgiving lunars are of inaccurate apparent altitudes. But they don't
> take up much
> room, and might have other uses.
>
> The apparent altitudes of the centers, whether found with the help of
> table 1
> or the W.W. tables, are usually designated by "m" and "s." But lower
> case
> letters don't always catch the eye. I decided to use the larger, more
> noticeable,
> "Ma" and "Sa" instead. Similar thinking affected the design of the
> work sheet
> and the other tables. Navigators aren't always wide awake and at their
> brightest when working observations. They make fewer blunders if they
> can quickly
> find, and unambiguously recognize, what they are looking for.
>
> Since the true altitudes themselves are not used in the equation
> there's no
> reason to differentiate them, and (H~H) serves in place of (M~S). The
> (cos M *
> cos S)/(cos m * cos s) part of the equation was named "Q."
>
> Substituting the above symbols for the usual ones makes the equation
> for
> clearing (given in the May 5th posting) look like this:
>
> hav D = sqrt{hav[d - (Ma~Sa)] * hav[d + (Ma~Sa)]} * Q + hav(H~H)
>
> Tables 2 and 3 are refraction and parallax tables: table 2 for the
> moon,
> table 3 for the other body. Refraction is for 50° Fahrenheit and 30
> inches of
> mercury. Values agree with those in the WW II era Bowditch, where they
> are given
> to the nearest tenth of a second of arc. For convenience the tables
> also give,
> next to the refraction and parallax corrections, each body's part of Q.
>
> To fit into tables 2 and 3 the (cos M * cos S)/(cos m * cos s) version
> of Q
> was rearranged to (cos M/cos m)(cos S/cos s). The first half answers
> to the
> same arguments as the moon's refraction and parallax, and the second
> half to the
> same arguments as the other body's refraction and parallax. To save
> space,
> everything before the first significant figure was dropped before the
> values were
> put in the tables. To prevent rounding error buildup a sixth decimal
> place,
> set off by a comma, was included.
>
> The part that goes in table 2 is negative. The part that goes in table
> 3,
> given directly, would be positive. But in absolute value it would
> always be less
> than anything in table 2. It never exceeded 12,7, as I recall. So I
> applied
> -12,7 (if that's what it was) to everything that went into table 3,
> and +12,7
> to everything that went into table 2. That way the two parts can be
> added to
> get the total. A clever dodge, but an old one.
>
> Dr. Inman gets the credit for the particular arrangement of table 2. I
> junked
> my own design after examining table 34 in the 1894 print of Inman's
> Nautical
> Tables: "Corrections of the Moon's Altitude, and the Aux. Angle A."
>
> Table 3 starts out with a 2' interval for the altitude and picks up
> speed
> until the interval reaches 1°. Ordinarily this change of interval
> would be a poor
> feature. But it doesn't matter here because there's no interpolation.
> Space
> is saved, as are page turnings.
>
> Other than the sextant, and the observer's ability to use it, nothing
> is more
> critical to the accuracy of a lunar distance than the semidiameter of
> the
> moon. The sun's semidiameter is equally important when the distance is
> from him.
> The Almanac gives these values only to 0.'1 and in the case of the
> moon, not
> often enough. Fifteen years ago, for my own use, I penned manuscripts
> of what
> are now tables 4 and 5. These tables are based on the 1987 American
> Ephemeris
> and Nautical Almanac, the astronomers' version of the Almanac. It
> gives values
> to the nearest tenth of a second of arc.
>
> The physical diameters of moon and earth are constant, so the ratio of
> the
> moon's horizontal semidiameter to her horizontal parallax is constant.
> From
> Ephemeris data I took the ratio to be 0.2725. But that's for horizontal
> semidiameter. When the moon's above the horizon she's closer, and
> appears larger.
> Directly overhead she's closer by the full radius of the earth. This
> effect is
> usually taken care of by a special "augmentation" table. To avoid the
> need of a
> separate table I concocted the formula:
>
> Augmented S.D. = (0.2725 H.P.)/(1 - sin H.P. * sin H)
>
> This is the basis of table 4. Entered with the approximate altitude
> and the
> nearest 0.'1 of H.P. from the Almanac, the table gives augmented
> semidiameter
> correct to the nearest 0.'03. Altitude is not at all critical for this
> table,
> so the apparent altitude can be used in place of H.
>
> Table 5, of the sun's semidiameter, will be two days off by the year
> 2093,
> according to my figures. At that time a revision might be worthwhile.
>
> Values in these and some of the other tables were taken to an extra
> decimal
> place so that, when added together, rounding errors don't build. It
> would be
> nonsense to interpolate between the values.
>
> Table 6 can be calculated well enough by taking the difference in
> refraction
> caused by a change of 16' in the altitude and multiplying it by the
> square of
> the cosine of the "Angle from the Vertical."
>
> The "K" table is made up of negative log haversines. I tried to
> arrange it so
> values could be quickly and easily found. "K" and "Q" are not
> initials. They
> are symbols, intended to be instantly picked out on the work sheet.
>
> A problem was presented by the + sign separating "hav(H~H)" from the
> rest of
> the equation for clearing. Not easy to get around without a cumbersome
> table,
> extra trouble in calculation, or loss of accuracy. After coming up
> with the
> "inside-out critical table" which gives every value in a (fairly)
> reasonable
> number of pages, I settled on the use of Gaussian addition logs.
> Fortunately,
> subtraction logs aren't needed.
>
> Unless you are curious about addition logs, skip the next paragraph.
>
> Suppose you have log A and log B, and need log (A+B). Subtract log B
> from log
> A. That gives you log (A/B). Enter the table with log (A/B) and it
> gives you
> log (A/B + 1). Add log (A/B + 1) to log B and you have log (A + B).
> Since, in
> my method, "log A" and "log B" are negative and the Gaussian positive,
> its
> absolute value is subtracted, rather than added.
>
> The "log Dec." table uses the same "inside-out" design as table 2. It
> gives
> negative log cosines for calculating comparing distances.
>
> Table 7 is based on ten times the number of minutes of arc in a
> degree, that
> is, 2400. Entered with 31.'8 it gives log (2400/318), which is 0.8778.
>
> Table 8 is based on the number of seconds in an hour. In the first
> edition
> this table presented the minutes across the top and the seconds down
> the side.
> This made the flow of change within the table slightly different from
> that in
> the K table and table 7. The difference led me astray one time too
> many. I
> substituted the present awkward-looking arrangement in the second
> edition. Once
> you're used to it, it works fine.
>
> Bruce
>
>
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