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    Simulating Parallactic Retardation in Lunar Distances
    From: Arthur Pearson
    Date: 2002 Nov 1, 23:05 -0500

    Last March, the discussion on lunar distances delved into the topic of
    �parallactic retardation� as described by George Huxtable.  To refresh
    your memory, George defined parallactic retardation as a slowing of the
    apparent motion of moon with respect to the stars as a result of the
    parallax of the moon.  Specifically, it was argued that the moon�s
    apparent distance as viewed from the surface of the earth might change
    only half as much as the computed geocentric distance over a given hour.
    George argued that the magnitude of this effect would vary with the
    height of the moon as determined by the location of the observer, with a
    high moon having the greatest retardation.  This retardation makes it
    harder to get accurate time from a lunar as the lower rate of change in
    distance makes the lunar �clock� less sensitive. George�s full
    explication of this effect can be found in section 4.4 of his �About
    Lunars, part 4� at http://www.i-DEADLINK-com/lists/navigation/0203/0038.html.
    One thread of the discussion suggested this effect could be quantified
    and illustrated over an observation period by calculating geocentric
    distances at regular intervals during the observation period, and then
    calculating the corresponding apparent distances from an assumed
    position of observation. This is essentially working the lunar backwards
    from a calculated distance to an apparent distance for each interval.
    The key is to reverse the parallax adjustment we normally use to work
    from apparent to geocentric (or �observed�) distance.  This is achieved
    by first by calculating geocentric altitudes of the moon and the
    comparing body from the observation point as well as a geocentric
    distance between them. Then reverse corrections for parallax and
    refraction are calculated and applied to the geocentric altitudes to
    arrive at apparent altitudes.  The spherical law of cosines is then
    applied to calculate an apparent distance.
    With encouragement and formulas from George I have built a spreadsheet
    that makes the above calculations for every half hour over a 9 hour
    period.  I ran several 9-hour simulations with combinations of a
    mid-latitude observer, an equatorial observer, a lagging moon, a leading
    moon, after moonrise and after moonset.  Stars were used as comparing
    bodies.  After the first experiments, it was decided to ignore
    refraction in order to focus on the effects of parallax (the incremental
    effects of including refraction are discussed below). The results of
    selected simulations have been graphed and are available at
    http://members.verizon.net/~vze3nfrm/files.html.  There are six .JPG
    files that can be viewed and/or downloaded.  They will be referred to by
    name on the discussion below.
    What follows is a definition of the abbreviations I have used, an
    explanation of data sets and format of the graphs, and my own brief
    comments on each of the named graphs.  At the bottom of this posting is
    full disclosure on the procedures and formulae used for the
    calculations.  It is my hope that presenting the data to list members
    will stimulate discussion and possibly greater insight into these
    Mc � Geocentric (Calculated) Altitude of the moon
    Ma � Apparent Altitude of the moon (observer�s viewpoint after
    correcting for index error, dip and semi-diameter but before correcting
    for parallax and refraction)
    Sc � Geocentric (Calculated) Altitude of the star
    Sa � Apparent Altitude of the star (observer�s viewpoint after
    correcting for index error and dip but before correcting for refraction)
    Dc � Geocentric (Calculated) Lunar Distance
    Da � Apparent Lunar Distance (observer�s viewpoint after correcting for
    index error and semi-diameter but before correcting for parallax and
    The observation period for the simulations is March 29, 2002 from 00:00
    to 09:00 GMT. The comparing bodies used are Regulus to illustrate the
    moon lagging the star across the sky, and Zuben�ubi to illustrate the
    moon leading the star across the sky.  To illustrate different
    observation periods relative to moonrise or set, I simply changed the
    observer�s longitude rather than enter Almanac data for different hours
    of the day.  For instance, the hours of 00:00 to 09:00 from longitude
    75� W illustrate the 9 hours following moonrise.  The same hours
    observed from longitude 75� E illustrate 8 hours after sunset.  During
    the observation period, the moon�s declination was roughly 0� to 2� S
    and HP was 61.�3.
    The X axis of each graph is GMT for the observation period.  On the
    right hand Y axis is the altitude of the bodies in degrees.  Mc and Sc
    are plotted against this axis so you can see their track through the sky
    during the observation period.  On the left hand Y axis is the change in
    distance over each half hour in arc-minutes.  For example, at 05:45 we
    plot the change in distance from 05:30 to 06:00.  The changes in Dc and
    Da are plotted against this axis so you can see how they differ from
    each other and how they vary during the observation period.  For a well
    selected comparing body, the change in Dc should vary very little over
    the observation period.  Where we find change in Da less than the change
    in Dc, we are observing parallactic retardation. I did not plot the
    actual distances as they are not the focus of this analysis.  The
    distances for the moon lagging Regulus were around 45�, and those for
    the moon leading Zuben�ubi were around 30�.
    This graph is of the 9 hour period after moonrise for the moon and
    Regulus viewed from latitude 45� N. The altitude curves show Regulus
    rises first and is leading the moon across the sky. The moon crosses our
    meridian at 5:30 at about 43� altitude (right hand axis).  The Dc
    (geocentric distance) is increasing at a fairly constant rate of about
    19 arc-minutes per half hour (left hand axis) throughout the simulation.
    The rate of change in Da (apparent distance) starts at 18�/half hour and
    drops to a low of about 13.8�/half hour. Note that the lowest rate of
    change in Da occurs just as the moon reaches its greatest altitude.  As
    the moon gradually loses altitude, Da gradually increases again.  This
    confirms the parallactic retardation effect and demonstrates that it
    increases with the altitude of the moon.
    This graph shows about 9 hours prior to sunset.  It shows the continued
    increase in the rate of change of Da as the moon loses altitude on its
    way to setting.  Note that just as the moon is approaching moonset,
    change in Da is approaching equality with change in Dc.  I will note at
    this point that when refraction is included in the simulations, the
    incremental effect is insignificant except where altitude is less than
    20�. At these altitudes, refraction begins to noticeably depress
    altitude, keeping change in Da relatively higher when the bodies are
    rising, and keeping it relatively lower when they are setting (for a
    lagging moon).
    This is a hypothetical case where we are calculating the distances after
    moonset. We could never observe these distances, but it illustrates a
    point George made last March when he stated that the change in Da is
    less than change in Dc only during the time when the moon is visible.
    He argued that the apparent moon �catches up� during the period when the
    moon is �under foot� and that if the earth was transparent and we could
    observe it, change in Da would be greater than change in Do during the
    period when it is below the horizon.  This simulation validates his
    argument, with the rate of change in Da surpassing that of Dc just as
    the moon sets.
    This simulation shows the point of view of an observer on the equator
    for the 9 hours after moonrise.  The moon and Regulus both reach much
    greater altitudes, almost 90� for the moon.  The rate of change in Dc is
    unchanged (as it should be), but the rate of change in Da drops much
    lower than it did at mid-latitude, hitting a minimum of 11.5�/half hour
    as the moon reaches its greatest altitude.  This is 39% lower than the
    change in Dc vs. 27% lower for the mid-latitude observer.
    This simulation shows the moon and Zuben�ubi where the moon has risen
    first and is leading across the sky after moonrise.  In this simulation,
    the right hand axis (change in distance) is a negative scale as the
    distances are decreasing when the moon is leading.  The values at the
    bottom of the scale indicate a faster rate of decrease in the distance
    and therefore a greater rate of change.  Once you adjust to the scale,
    you can see a similar story to the lagging moon simulations.  As the
    moon increases in altitude, the rate of change in Da decreases to a low
    of -14.1�/half hour. The layout of this graph allows one to see by
    inspection that the difference between change in Da and change in Dc
    appears to be greater as the moon sets.  Note that at altitude 30� for
    the rising moon, change in Da is 15.6�/half hour. At altitude 30� for
    setting moon, change in Da is 15.3�/half hour.  I have not puzzled out
    why this is the case. It may be a function of how I graph the rate of
    change over the half hour (see explanation above).
    Here is the moon leading after moonrise as observed from the equator.
    We observe the same increase in the effect of parallactic retardation
    with greater moon altitudes, with the rate of change of Da hitting a low
    of 12.2�/half hour.  We can also observe more clearly the unexplained
    difference between change in Da at 40� altitude rising moon vs. change
    in Da at 40� altitude setting moon.
    At the end of the day, all the simulations point to a pretty simple
    conclusion.  It is best to observe the lunar distance when the moon is
    as low as practical for the conditions. The lower the moon, the greater
    the rate of change in Da, the more sensitive the lunar clock.
    Latitude and Longitude of the observer
    Hourly GHA, declination and HP for the moon for the observation period
    Hourly GHA for Aries for the observation period
    SHA and declination of the star from the appropriate three-day almanac
    For each half hour interval of the observation period, Mc and Sc are
    calculated using the procedures and formulas shown in the Nautical
    Almanac, �Procedures for Sight Reduction�, pp. 277-280.
    Ma and Sa for each half hour are calculated as:
    Ma = Mc � Parallax + Refraction
    Sa = Sc + Refraction
    Parallax for the moon is calculated as:
    P = (1-0.0032*(Sin(Lat))^2) * (ATan(Cos(Mc)/((3438/HP)-Sin(Mc))))
    Lat = Latitude of the observer (in degrees)
    HP = HP of the moon (in arc-minutes)
    Refraction for the moon is calculated as:
    R = (1.02* Tan((90-0.998797*Mc�-10.3/(Mc�+ 5.11))))/60
    Mc� = Mc � P
    Refraction for the star is calculated as:
    R = (1.02* Tan((90-0.998797*Sc-10.3/(Mc�+ 5.11))))/60
    Dc for each half hour is calculated as:
    Dc = ACos( (Cos(coDecM)*Cos(coDecS))  +
    (Sin(coDecM)*Sin(coDecS)*Cos(HA)) )
    coDecM = co-declination of the moon = 90� - declination of the moon
    CoDecS = co-declination of the star = 90� - declination of the star
    HA = hour angle between moon and star = (GHA moon) ~ (GHA star)
    Da for each half hour is calculated as:
    Da = ACos( (Sin(Sa)*Sin(Ma)) + (Cos(Sa)*Cos(Ma)*(Cos(Dc)-
             (Sin(Sc)*Sin(Mc)))/(Cos(Sc)*Cos(Mc))) )

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