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    Re: Lunars.
    From: George Huxtable
    Date: 2001 Jul 10, 7:46 AM

    To a question from Nigel Gardner-
    >Anyone out there with a good algorithm and process for longitude by lunars?
    I posted the following reply on 5 June, to which I am going to make some
    additions and qualifications below-
    >I am aware that what follows is not a full answer to Nigel's question. It
    >proposes a successive-approximation procedure for correcting the
    >chronometer error: not finding the longitude directly, but a first step
    >toward it. It's a procedure which could be implemented on a computer, but
    >mould be quite inappropriate to other situations such as hand-calculation
    >using tables.
    >The produre described below has not been written or tested by me, so it is
    >rather a tentative suggestion. Some details have been omitted, such as
    >automatic decision of which limb of the moon is the appropriate one to
    >measure to. It's quite possible that there errors or ambiguities, and I
    >would be grateful if any list member would bring them to my attention. The
    >proposal has been stood up to be shot down, if necessary.
    >I would treat the problem in its simplest form to start with, not
    >attempting to establish longitude immediately, but simply to find the
    >chronometer error. Once this error is known, longitude can then be deduced
    >by measurement of an altitude of a body that's well away from the meridian,
    >as a separate problem.
    >To find the chronometer error, the observer only needs to take the lunar
    >distance and needs to measure no altitudes at all. The necessary altitudes
    >can be computed rather than measured, so no horizon is needed. Star-moon
    >distances can then be measured in pitch-darkness, when stars are bright.
    >I would presume a rough initial position (lat and long) and an assumed time
    >of the measurement: probably the chronometer reading with its unknown
    >I know of no way of expressing time as an explicit funtion of lunar
    >distance; only how to calculate lunar distance in terms of time. So the
    >method will be one of successive approximation.
    >The accurate position in the sky of the moon and the other body must be
    >calculated, to a precision of 10 sec of arc or so, for the assumed time.
    >For this I would use the methods given by Meeus, in his Astronomical
    >Algorithms, using all the many coefficients that he provides. These sky
    >positions should be converted to altitude and azimuth, as seen from the
    >assumed lat and long. These are the positions the bodies would be seen at
    >if viewed from the centre of the earth, and if our atmosphere did not
    >Correct these two altitudes for the effect of refraction and also parallax
    >of moon and sun (augmented parallax in the case of the moon). Note that
    >these corrections are to be made in the opposite sense to what we are used
    >to, because we are aiming to calculate the direction in the sky the
    >observer would see the body if he were to measure its altitude, by
    >correcting the geocentic position. Not the other way round, as we usually
    >do it. It may be worthwhile to adapt those corrections slightly because we
    >are using a slightly different angle as the argument, but this is a minute
    >Now we calculate the angle-in-the-sky between these two apparent bodies by
    >one of the standard spherical-geometry formulae. Exactly the same as
    >calculating the great-circle distance between A and B on a spherical earth.
    >This is the calculated lunar distance. Note that in this process we have
    >not needed to "clear" the lunar distance of parallax and refraction in the
    >complex way that is usually called for; it was simple addition and
    >Now we take our measured lunar distance, corrected for index error. We
    >allow for the semidiameter of the moon (remembering which was the
    >appropriate limb) and if called for, the Sun. These semidiameters and the
    >parallaxes used previously are taken from the calculated earth distance,
    >which was a by-product of the Meeus calculations.
    >After all this, we have a measured lunar distance and a calculated lunar
    >distance to compare. The difference between these quantities, if we divide
    >by the angular velocity of the moon in its orbit, is roughly the error in
    >the chronometer. Adjust the assumed time by this quantity, and if necessary
    >repeat the whole calculation again, using the new assumed time. At each
    >iteration the time-error will shrink and you choose when to stop the
    >George Huxtable
    There have since been interesting responses from Steven Wepster and from
    Herbert Prinz.
    It's become clear to me that the above posting omits a few matters which
    might puzzle anyone trying to implement the procedure I described as a
    computer program.
    I referred the reader to Meeus' invaluable book on Astronomical Algorithms,
    for the necessary calculation of the precise position of the moon and the
    sun-or-star-or-planet in the sky at each stage of the iteration. I should
    perhaps have explained that Meeus gives these as Ecliptic coordinates
    (ecliptic lat and long) which then need to be converted to equatorial (dec
    and right-ascension-or-GHA) and then to the observer's horizon coordinates
    (altitude only is required) to work out the necessary corrections for
    refraction and parallax.
    I should also make it clear that the position of the sun in the sky isn't
    given explicitly by Meeus. Because he writes for astronomers, he abides by
    the astronomers' supposition that the earth goes round the sun, whereas we
    navigators take quite a contrary view of that matter. So to get the
    direction of the sun from the earth, take Meeus' calculation of the
    direction of earth from the sun, then take the opposite direction.
    If you are ambitious enough to calculate the positions of the planets in
    the sky from earth, it's necessary to calculate the position vector of both
    earth and planet from the sun, then subtract these vectorially to get
    planet-from-earth. Not that hard, when a computer does all the work.
    I also over-simplified when I stated-
    >After all this, we have a measured lunar distance and a calculated lunar
    >distance to compare. The difference between these quantities, if we divide
    >by the angular velocity of the moon in its orbit, is roughly the error in
    >the chronometer.
    Before doing this , we should have established whether the body in question
    is to the east or to the west of the moon, because on this depends whether
    the lunar distance is increasing or decreasing with time. Not hard to do,
    but essential, if we wish to have the iteration diverge instead of converge
    in all cases.
    It's also necessary to impose restrictions on the body that we observe, to
    fit in with an assumption that the relative motion between it and the Moon
    is roughly 12 degrees per day, increasing or decreasing. This depends on
    the direction of the body from the moon being more-or-less in the direction
    of the moon's motion. This condition is always met by the sun and planets,
    which with the moon all lie close to the plane of the ecliptic, as long as
    the angle between moon and other body is restricted to values between 30
    and 120 degrees. For stars, a restricted set of stars should be kept to,
    all lying close to the ecliptic. The following set of 9 stars is commonly
    used, conveniently spaced around the ecliptic: Hamal, Aldebaran, Pollux,
    Regulus, Spica, Antares, Altair, Fomalhaut, Markab.
    Steven Wepster pointed out-
    >That is how you circumvent the 'clearing of the
    >Still, I believe that this procedure is computationally just about as
    >expensive as the
    >'classical' way. Instead of clearing the distance you have to do a
    >transformation from
    >equatorial coordinates (Dec,GHA) of the bodies to horizon coordinates
    >(Az,Alt). In
    >terms of amount of work, that is quite comparable to the clearing process
    >(I didn't
    >make a detailed count).
    Steven is quite right. I ponted out that my method circumvented the coomon
    bugbear of "clearing the distance", but that was really to explain what was
    going on, rather than to claim it as a saving in computation. Indeed, the
    proposed method is remarkably extravagant in computing power, which is only
    allowable because that power is so readily available. Not only are the
    computation steps required that Steven describes, but much more computing
    even that that is necessary to include all the harmonic terms in the
    calculation of positions of the bodies from Meeus, at each iteration.
    George Huxtable
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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