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    Re: Latitude by Talcott-Horrebow Method
    From: Brad Morris
    Date: 2018 Nov 5, 14:53 -0500
    To continue the discussion on latitude lines

    A physical object which has mass possesses a 'mass moment of inertia' which describes the resistance to changes in velocity (acceleration and deceleration).  Typical shapes in every mechanical engineering textbooks are spheres, cylinders and plates.  These simplified shapes are substituted when the more complex shape is difficult to model.  The distribution of mass about the axis of rotation will define its resistance to spin about that axis of rotation. The axis of rotation will change if we change the distribution of mass.  Change the distribution of mass, change the mass moment of inertia.  

    Consider now the distribution of mass of our planet.  We are far from a perfect sphere, in fact the shape is lumpy.  Now consider the Greenland ice cap.  If that were to melt into the ocean, the distribution of mass about our axis of rotation will change.  This will change the axis of rotation and therefore the definition of the pole and further, where the infinitely narrow lines of latitude are drawn on the surface.  Is that at the 0.1 arcsecond level?  I do not know, but there will be a change.

    If I walk across the road, we will change the distribution of mass about the Earth's axis of rotation.  That is clearly not a significant enough change to affect the poles by 0.1 arcsecond.  However, we will change the definition of the pole, ever so slightly.  We just have to keep adding decimal digits until we notice it.  Latitude definition changed by walking across the street

    In the Talcott-Horrebow method, they claim significance to 0.1 arcseconds, when the non deterministic Chandler Wobble has values as large as 0.3 arcseconds.  Latitude is a fuzzy line.  You cannot define it below your threshold of uncertainty.


    On Mon, Nov 5, 2018, 11:22 AM Brad M <bradley.r.morris@gmail.com wrote:
    Hello Paul

    You wrote, quoting a publication
    polar motion. Corrections for that were applied:
    "About the middle of each year the Latitude Service of the International
    Geodetic Association publishes in the Astronomische Nachrichten
    provisional values of the coordinates of the instantaneous pole for the
    preceding calendar year, together with tables to reduce observed
    latitudes, longitudes and azimuths to the mean position of the pole."
    I probably didn't communicate my point properly.
    While the pole may be described as a mathematical point, and lines of latitude as mathematical lines, in reality they are a bit more fuzzy than some pure description with infinte decimal digits of accuracy.
    Ed described a latitude measurement to 0.1 arc seconds, or about 10 feet.  The Chandler Wobble has a magnitude of 0.3 arcseconds but has non predictable variation within.  Yes, the large elements of precession and nutation can be readily corrected for.  However, Chandler Wobble can only be corrected in retrospect, for some past instant in time and not reliably for any future instant in time due to the variation in magnitude.
    Which implies that measuring the latitude and claiming significance to 0.1 arcseconds is an empty claim, unless you can know the Chandler Wobble for that instant.  In simple point of fact, the wobble was a known problem in the 1800's, the exact latitude of pre-eminent observatories was 'changing' and much study was devoted to wringing out this error.  Was it instrumentation? Observational error? Chandler devoted a mountain of effort, pouring through astronomical records, to tease out the definition of the wobble.
    The point is you can only KNOW your latitude finer than the wobble, when you know the precise value of the wobble, which has a degree of unpredictability.
    For those interested, there is an okay book available on the Chandler Wobble.  It describes how Chandler did it in every other chapter.  Its kind of nutty, the odd numbered chapters are lightweight story telling.  The even numbered chapters are scientific history.  The book editor should be shot (figuratively, of course!)

    On Nov 5, 2018 4:40 AM, "Paul Hirose" <NoReply_Hirose@fer3.com> wrote:

    "In the zenith telescope, or Horrebow-Talcott, method of determining the
    latitude, there is substituted for the measurement of the absolute
    zenith distance of a star the measurement of the small difference of
    meridional zenith distances of two stars culminating at about the same
    time, and on opposite sides of the zenith. The effect of this
    substitution is the attainment of a much higher degree of precision,
    arising from the increased accuracy of a differential measurement, in
    general, over the corresponding absolute measurement; from the
    elimination of the use of a graduated circle for the essential part of
    the measurement; and from the fact that the computed result is affected,
    not by the error in estimating the absolute value of the astronomic
    refraction, but simply by the error in estimating the very small
    difference of refraction of two stars at nearly the same altitude."
    "A sufficient number of pairs should be observed at a station to make it
    reasonably certain that the probable error of the mean result is not
    greater than ±0″.10." (Typically, about 15 pairs were enough.)
    "The stars observed upon should be taken from "The Preliminary General
    Catalogue of 6188 Stars for the Epoch 1900" by Lewis Boss, which was
    published by the Carnegie Institution at Washington in 1910."
    "Among the requisites for a pair of stars for an observing list, are,
    that their right ascensions shall not differ by more than 20m to avoid
    too great errors arising from the instability in the relative positions
    of different parts of the instrument; nor by less than about 1m, that
    interval being required to take the readings upon the first star and
    prepare for the second star of the pair; that their difference of zenith
    distances shall not exceed the half length of the micrometer comb, 20′
    for most instruments; that each star shall be bright enough to be seen
    distinctly, not fainter than the seventh magnitude for the larger
    instruments; and that no zenith distance shall exceed 45°, to guard
    against too great an uncertainty in the refraction."
    William Bowie, "Determination of Time, Longitude, Latitude, and
    Azimuth," U.S. Coast and Geodetic Survey Special Publication No. 14, 1917.
    (In later years that was replaced by Special Publication No. 237,
    "Manual of Geodetic Astronomy.")
    Since a transit instrument is free to move only in altitude in the plane
    of the meridian, the reversal to observe the second star of each pair
    was accomplished with a mechanism which lifted the telescope trunnions
    from their V-blocks, rotated 180° about a vertical axis, and lowered
    them in the opposite blocks. In the bottom photo on this page the
    telescope assembly has been lifted and partly rotated.
    The only graduated circles on that instrument are the Talcott-Horrebow
    levels attached to the sides of the telescope tube. Those are set to the
    expected altitude of the first star. No great accuracy is necessary, it
    being sufficient that the star is well positioned in the field of view.
    What is necessary is that the telescope is accurately returned to the
    same inclination for the second star.
    Exact centering of the bubble is not assumed. Its position is recorded
    immediately after each observation, and the corresponding dislevelment
    factored into the computation.
    Both stars are measured with a movable horizontal wire of a micrometer
    as they pass through the meridian. From the difference in micrometer
    readings, the difference in zenith distance is computed, from which
    latitude is computed. For example, if the readings are identical,
    latitude equals the mean declination of the pair. Note that zenith
    distance itself is not measured in this technique.
    The illustrated instrument is quite old, but I chose it because the
    reversing mechanism is shown so well. By the time of the 1917
    publication, one of the favored instruments of the Survey had a 3-inch
    f/15 telescope and was normally operated at 100x.
    Somebody mentioned the polar motion. Corrections for that were applied:
    "About the middle of each year the Latitude Service of the International
    Geodetic Association publishes in the Astronomische Nachrichten
    provisional values of the coordinates of the instantaneous pole for the
    preceding calendar year, together with tables to reduce observed
    latitudes, longitudes and azimuths to the mean position of the pole."

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