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    Lewis and Clark lunars: a request for help.
    From: George Huxtable
    Date: 2004 Apr 8, 00:40 +0100

    Lewis and Clark Lunars: a request for help.
    
    For the lunartics on this list, here's something of a challenge (and others
    should switch off now). What will be required is a computer-program that
    can calculate positions of bodies in the sky back to 200 years ago.
    
    From now on "Lewis and Clark" will be shortened to L&C.
    
    My information about L&C's recorded observations has come from vol 2 of
    Gary Moulton's 13-volume set "The Journals of the Lewis and Clark
    Expedition" (Univ. of Nebraska Press, 1986)
    
    I've been following L&C's historic journey of 200 years ago: so far, just
    that section up the Mississippi from (modern) Cairo to just above St.
    Louis, where they camped for winter, before the expedition-proper set off
    up the Missouri in May 1804. At present, I am trying to understand the
    lunar observations made, near Kaskaskia, about 50 miles below St Louis.
    Being the first lunars that L&C ever recorded, they have some historical
    value.
    
    Although the pair made many navigational mistakes, it's usually been
    possible to discover and correct them. Having done so, the underlying
    accuracy of their sextant measurements has been remarkably good.
    
    At their Kaskaskia camp, L&C (predominantly Lewis, I surmise) made a number
    of lunar-distance observations during the night of 2-3 Dec 1803, which were
    then followed up by equal-altitude measurements of the Sun around noon of 3
    Dec 03, to establish local time and the error of their chronometer.
    
    They made no attempts to determine the resulting longitude. Presumably,
    (according to the advice they received before setting off) the intention
    was to bring their observational data back East afterwards, to be computed
    by others who understood these matters. So there were no calculations made
    of Local Apparent Time or reduction of lunar distance, to compare with what
    I will offer below.
    
    In trying to obtain longitudes from their observations, to be honest, I
    just cannot make head nor tail of the matter. I can not work out what they
    are doing wrong in their recorded observations. Alternately, it may be that
    I am making errors in my own analysis (or perhaps both!).
    
    So here is the data, offered up to the combined wisdom of the Nav-L mailing
    list, for any analysis and observations they may have to offer.
    
    ==============
    
    Where were they?
    
    Nowadays, we have a distinct advantage over L&C, in that we know, within a
    mile or two, where their camp had been set up at Kaskaskia since 28 Nov.,
    using modern mapping. From the website associated with Harlan and Denny's
    "Atlas of Lewis and Clark in Missouri", this is determined as N37.9849,
    W89.9528. Thanks to Paul Hirose, for collecting lat/long positions of every
    campsite, on the rivers within Missouri State, from the interactive website
    accessible via .
    Harlan and Denny have made efforts to allow for intervening changes in the
    course of the river since the early 18th century.
    
    First, putting things in backwards order, consider L&C's Sun equal-altitude
    observations, made around noon on 3 Dec 1803.
    
    ===================
    
    Equal-altitudes.
    
    Here's the simple and clever way they did that job. Having no sea-horizon,
    Sun altitudes were observed by reflection in a water-trough (which doubled
    the sextant angle). In the morning, a few hours before noon, the sextant
    was set to correspond to an altitude the Sun hadn't quite reached yet. From
    then on, throughout the day, that sextant setting remained strictly
    unaltered. Then, chronometer times were recorded: first, at the instant
    when the Sun's upper-limb just touched its inverted reflection in the water
    surface; second, when the Sun and its reflected image coincided; third,
    when its lower-limb touched its image. These would be expected to differ by
    equal intervals; by the time it took the Sun, at that altitude on that day,
    to rise by a semidiameter.
    
    Noon would then come and go, and later in the afternoon the Sun's
    reflection would again be timed by chronometer when it coincided with its
    reflected image. First, lower-limbs; then centres; finally, upper limbs.
    Again, these should be equally-spaced in time.
    
    The centre of symmetry, between these morning and afternoon times, gives
    the chronometer time of the moment of local apparent noon, after making a
    small correction to allow for changing Sun declination through the day.
    
    Here are their observations for Dec 3 1803 (slightly rearranged from
    Moulton page 120). The numbers 1 to 6 show the time-order of the
    measurements.
    
    3 Dec 1803      am times          pm times   (before any adjustment)
    upper-limb  1  9h 09m 59s     6  2h 01m 10s
    centre      2  9h 11m 18s     5  1h 58m 45.5s
    lower-limb  3  9h 14m 44s     4  1h 56m 20s
    
    The sextant reading was noted as 42deg 27' 0", with its usual value of
    index correction -8' 45" (ie 8' 45" to be subtracted), but this is not
    required for calculating chronometer errors.
    
    That is the information L&C have provided about the Sun equal-altitudes,
    measured around noon on 3 Dec 1803. You can analyse that information
    whatever way suits you best. In case it's helpful, my own comments and
    analysis of that data follow-
    
    ==================
    
    As often with L&C, nothing is simple: there's something odd about those
    timings given above. Looking at the time gaps between successive readings,
    those in the afternoon seem very reasonable. Between 4 and 5 we get 2m 25.5
    sec, between 5 and 6 it's 2m 24.5s. The gaps in the morning are clearly
    discrepant: between 1 and 2, 1m 19s: between 2 and 3, 3m 26s. Clearly
    something has gone wrong here: by the nature of this method each gap must
    really be the same as the others, the time for the Sun to rise or fall by a
    semidiameter. The discrepany results from a familiar mistake L&C have made
    on several other occasions, in which the chronometer minute-hand has been
    recorded as the value not-yet-reached, rather than the value just-passed,
    so the time is noted as just 1 minute later than it should have been. This
    appears to have happened with observations 1 and 3. With some confidence,
    we can correct those recorded times, to give-
    
    3 Dec 1803      am times          pm times   ( obs. 1 and 3 adjusted)
    upper-limb  1  9h 08m 59s     6  2h 01m 10s
    centre      2  9h 11m 18s     5  1h 58m 45.5s
    lower-limb  3  9h 13m 44s     4  1h 56m 20s
    
    Now, the morning time-gaps are- 1 to 2, 2m 19s; 2 to 3, 2m 26s. These are
    now entirely reasonable, and correspond well to the afternoon gaps. Note
    that the adjustments made here will shift the moment of maximum altitude
    (and also the half-interval) by no more than 20 seconds, so it's not a big
    deal.
    
    Now the times 1, 2, 3 can be averaged, and also, 3, 4, and 5. Then the
    mid-time, which I make to be 11h 35m 02.8 sec, should be the chronometer
    time of the moment of the Sun's maximum altitude. Because, in early
    December, the Sun's declination is still moving South, that maximum
    altitude will precede the moment of Sun's meridian passage, by an interval
    I make to be 6.4 seconds. In that case, the chronometer time of meridian
    passage (which is exactly 12 hours Local Apparent Time) will be 11h 35m 9.2
    sec, or we can say that the chronometer is then slow on L.A.T. by 24m
    50.8s.
    
    If we are going to use a modern computer program to deduce positions in the
    sky, we will need to work in terms of Mean Time, not Apparent Time. (On the
    other hand, L&C used their 1803 Nautical Almanac, which in those days used
    Greenwich Apparent Time as its base).
    
    Because, roughly speaking, Greenwich is about 6 hours ahead of that part of
    the Mississippi, at local noon the time at Greenwich will be around 6pm (18
    hrs) on 3 Dec 1803, in which case the "Equation of Time" (EOT) can be
    deduced as 10m 03.9s, which is the amount by which Mean Time precedes
    Apparent Time at that moment. In that case, Local Mean Noon, at Kaskaskia
    on that day, would occur at 11h 49m 56.1s. Now we can say that L&C's
    chronometer was therefore slow on Local Mean Time by 14m 46.9 s., around
    noon on 3 Dec 1803.
    
    From measurements of the chronometer's error, made earlier in the journey,
    we can also now estimate that it is losing about 8.6 sec, from one day to
    the next. This is the only bit of information that you will have to take my
    word for, and it has little effect on the final results. For everything
    else, L&C's own observations are provided here.
    
    =====================
    
    An interesting side-issue.
    
    There's some extra mileage to be got out of these equal-altitude
    observations, that L&C never made use of in any of their many
    equal-altitude Sun observations. Knowing the measured Sun's altitude at the
    morning and afternoon observations, and the half-interval between them in
    time, and the Sun's dec., it's possible to calculate the latitude, rather
    precisely, using the expression-
    
    lat = arctan(tan dec/cos H) ? arccos(sin(arctan(tan dec/cos H))*sin alt/sin
    dec),
    
    where H is the half-interval, converted to an angle at 15deg for each hour
    (after making a tiny adjustment because it was measured on a chronometer
    that runs a bit slow).
    
    The altitude is obtained by subtrating 8' 45" (index error) from the
    sextant reading, then halving (L&C always did those in the opposite order,
    in error), and correcting for refraction and Sun parallax. The resulting
    alt. is 21deg 06' 50". The predicted Sun dec at the moment of observation
    is -22deg 04.4'.
    
    For L&C, the "? " sign in the formula above always turns out as a "+",
    because the Sun always passes to the South of the observer.
    
    Dip and semidiameter corrections are irrelevant in this context.
    
    The resulting latitude is then 37deg 58.7' (37.9783deg),  which is less
    than half a mile from the latitude of 37.9849deg deduced from the atlas.
    That should boost confidence that our adjustment to the L&C time
    observations was well-founded.
    
    ================
    
    Lunar distances.
    
    Now we can deal with the lunar distance observations, made on the previous
    night of 2 to 3 Dec, 1803. These are in three sets, which we can label A,
    B, and C. No altitudes were measured, for Moon or star, by L&C, so in each
    case these altitudes must be calculated.
    
    Sets A and C are stated to be made to Aldebaran (West of the Moon). L&C
    doesn't state to which limb of the Moon the measurement was made.  However,
    we can deduce from the position of the Sun at the time, that the Moon's
    Eastern limb was lit by the Sun, so the angle to Aldebaran should have been
    measured ACROSS the Moon from its far limb. Set B was to Regulus, stated to
    be East of the Moon, so the lunar distance should have been measured from
    the Moon's near limb.
    
    Times and angles were recorded as follows by L&C (the averages were
    calculated by me)-
    
    Data-set A (from far limb, West to Aldebaran)
    chron time pm on 2 Dec    lunar distance
        10h 51m 8s             60deg 57' 45"
        11h 04m 52s            61deg 01' 00"
        11h 15m 05s            61deg 05' 00"
        11h 20m 50s            61deg 05' 15"
        11h 24m 04s            61deg 10' 00"
        11h 27m 07s            61deg 10' 45"
       ------------            -------------
    Av. 11h 13m 51s p.m.   Av. 61deg 04' 58"
    
    ==========
    
    Data-set B (from near limb, East to Regulus)
    chron time am on 3 Dec    lunar distance
        00h 11m 12s            17deg 47' 15"
        00h 21m 59s            17deg 45' 15"
        00h 23m 58s            17deg 45' 00"
        00h 29m 49s            17deg 44' 00"
        00h 35m 04s            17deg 37' 15"
        00h 39m 56s            17deg 37' 00"
       ------------            -------------
    Av. 00h 27m 00s        Av. 17deg 42' 38"
    
    =========
    
    Data-set C (from far limb, West to Aldebaran)
    chron time am on 3 Dec     lunar distance
        01h 01m 29s             61deg 37' 30"
        01h 07m 10s             61deg 41' 00"
        01h 10m 37s             61deg 39' 30"
        01h 14m 53s             61deg 39' 15"
        01h 17m 00s             61deg 40' 30"
        01h 19m 45s             61deg 40' 15"
       -----------              -------------
    Av. 01h 11m 49s         Av. 61deg 39' 40"
    
    ===================
    
    Now, all the data necessary to compute lunar longitudes has been provided.
    It may be computed out in any way that suits you best, to obtain a
    longitude for that Kaskaskia campsite.
    
    ================
    
    My lunar-distance calculation method.
    
    It may be useful to explain a bit further how I have proceeded. Please
    don't accept anything I did as correct, but question each step before you
    follow it. Even better, don't follow it at all, but do the thing your own
    way instead.
    
    Each averaged lunar distance has to be corrected for index error by
    subtracting 8' 45".
    
    In each case, I have assumed an initial longitude for the observation site
    of 90deg W. The aim is to compute the corrected lunar distance that should
    correspond to that long., to compare with the observed sextant lunar
    distance (corrected for index error). 90deg W makes a good starting point
    because it's a nice round number that we happen to know is very close to
    the map-longitude that we hope to confirm.
    
    Start from the averaged chronometer time of the observation. From the
    equal-altitude measurements, the chronometer error on Local Mean Time at
    local noon on 3 Dec was well determined. By correcting by a few seconds for
    the chronometer's loss-rate, in the interval between the observation and
    that noon, we know the chronometer error on LMT when the observation was
    made. Applying this gives the Local Mean Time of the observation. If the
    longitude is initially assumed to be 90deg W, there is exactly 6 hours
    difference (or long./ 15) in time, between Greenwich and the point of
    observation. Adding that difference gives the GMT at the observation.
    
    Using that GMT, the assumed long. of 90deg W, and the determined lat. of
    37deg 58.7' (37.9783deg), a computer program can provide precise positions
    in the sky of the Moon and star at that instant, in terms of true altitude
    and azimuth, and also the value of Moon's Horizontal Parallax. That
    provides all the necessary information to (backwards) correct those sky
    altitudes for parallax and refraction, leaving the azimuths unaltered, to
    obtain apparent positions of Moon and star. (Note that those corrections
    are in the opposite direction to the way they are normally made when
    correcting an altitude.)
    
    Then the angle across the sky, between the apparent positions of Moon and
    star, is computed, and a correction for Moon semidiameter is added or
    subtracted, as appropriate. This gives a computed lunar distance, which is
    to be compared with the observed sextant reading (after it has been
    corrected for index error).
    
    The two should correspond closely, and any divergence gives a clue to the
    error in assumed longitude. For every arc-minute of divergence, the assumed
    longitude will need to be altered by about 30 arc-minutes, the GMT has to
    be recalculated for the new longitude (this is important), and a
    reiteration should be done. When the star is West of the Moon, then the
    assumed long should be shifted Easterly if the predicted lunar distance
    exceeds the measured value, and Westerly if it's less; and vice versa if
    the star is East of the Moon. After a reiteration, the change in lunar
    distance corresponding to a change in longitude will become clear, and a
    final figure for longitude may be arrived at by a bit
    of interpolation or extrapolation, with no need for further iteration.
    
    =============
    
    Lunar-distance data-set A.
    
    At noon on 3 Dec., the chronometer was determined to be slow on Local Mean
    Time by 14m 46.9s, and we must allow for a presumed losing of 8.6 seconds a
    day. At the time of lunar observation A, then, (11h 13m 51s p.m on 2 dec)
    the chronometer was then slow on LMT by 4.5 sec less, or 14m 42.4 sec.
    Therefore the Local Mean Time at observation A is 11h 28m 33.4 sec pm on 2
    Dec 1803. For our first iteration, we have presumed the longitude to be
    90deg West (or exactly 6 hours in time), so the Greenwich Mean Time at
    observation A will be exactly 6 hours later, or 5h 28m 33.4sec on 3 Dec.
    
    The lunar distance between Moon and Aldebaran far-limb was then calculated
    as described above, taking for the  first iteration, that value of GMT, the
    presumed long of 90deg W, and the latitude of 37deg 58.7'.
    
    However, in calculating lunar A, things go wrong at the first iteration. My
    prediction of lunar distance comes out as 62deg 0.9', compared with a
    sextant observation (after index correction) of 60deg 56.2'. For a lunar,
    this is a GROSS discrepancy, of more than a degree! It would have to
    correspond to a longitude error of about 30 degrees, giving rise to a
    longitude toward 60deg West, somewhere in the Western Atlantic! Something
    is seriously wrong, and there's no point in taking the iteration further.
    
    What might the error be due to? Well, I could easily have made a simple
    mistake in calculation, so I would welcome any checking on my results, if
    possible using a different method. L&C could have made an observational
    error, as they were always prone to do. If they measured to the wrong part
    of the Moon's disc, that couldn't produce such a big error. Perhaps, they
    could have mis-identified another star as Aldebaran. I have checked for
    Rigel as a possible alternative, and that predicts 60deg 45.4', much closer
    to the observed 60deg 56.2 but this is still nearly 11 arcminutes from the
    observed value, far exceeding observational error. It's a mystery.
    
    ===============
    
    Lunar-distance data set B.
    
    This was observad at 00h 27m 00s am,on 3 Dec. Allowing for chronometer loss
    of 4.0 sec in the interval between that moment and the following noon, the
    chronometer will be slow on LMT by (14m 46.9s less 4.0) or 14m 42.9 sec at
    the moment of observation B, so that obs. was made at 0h 41m 42.9s LMT on 3
    Dec. Again, GMT is just 6h ahead, at 6h 41m 42.9
    
    The corrected lunar distance between the Moon near-limb and Regulus was
    computed for that GMT, 90deg W long. and 37deg 58.7' lat. This comes out to
    be 17deg 38.7', compared with the observed lunar distance (after index
    correction), of 17deg 33.9'. Here is a discrepancy of 4.9', which would
    make sense only if the longitude was about 92.5 deg rather than 90deg.
    Again, this is a serious discrepancy (though nothing like as bad as for
    data-set A). There seems no point in reiterating further.
    
    ===============
    
    Lunar-distance data-set C.
    
    This observation was at chronometer time 01h 11m 49s am, on 3 Dec. Loss of
    time, by chronometer, in the interval to the following noon, is 3.8 sec, so
    at the time of obs. it will be slow on Local Mean Time by 14m 43.1s. In
    that case the obs. was at 1h 26m 32s LMT on 3 Dec. For an assumed long. of
    90deg West, the GMT will again be 6h greater, at 7h 26m 32s.
    
    The corrected lunar distance between Moon far-limb and Aldebaran was
    computed for that GMT, 90deg W long, and 37deg 58.7' lat. This gives a
    value of 62deg 50.6', compared with the observed lunar distance (after
    index correction) of 61deg 30.9'.  Here we have an even greater discrepancy
    than in dataset A! Again, something is seriously wrong.
    
    ==================
    
    Here's a suspicion.
    
    If you plot out lunar distance against time, showing data-sets A and C on
    the same graph, it doesn't look right. This lunar distance is changing with
    time by only about 17' per hour, where normally a rate-of change of about
    30' per hour is expected. True, this can be reduced by the effect of
    parallactic retardation, especially if the Moon is high in the sky, but in
    this case the Moon is not particularly high. Could it be that measurements
    A and C were made, not to Aldebaran at all, but to another star which is
    well away from the direction of the Moon's path across the sky? It would
    have to be some star to the West of the Moon, because the lunar distance is
    increasing with time. I have considered, and eliminated, Rigel. Perhaps
    data-sets A and C could involve two different stars, both mis-identified as
    Aldebaran.
    
    Even if we manage to explain the A and C observation, there remains a large
    discrepancy in observation B to understand.
    
    Either that, or else, perhaps, the whole thing can be explained by a simple
    common error, either mine or L&C's. I have searched for that error without
    success, and have decided the time has come to seek help from Nav-L. Is
    anyone prepared to meet that challenge?
    
    In hope, George.
    
    
    
    
    
    
    
    
    
    
    
    ================================================================
    contact George Huxtable by email at george---.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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