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    Re: Lunar distance parallactic retardation
    From: Paul Hirose
    Date: 2020 May 19, 22:20 -0700

    In February I investigated lunar distance time determination accuracy as
    affected by "parallactic retardation" (the reduction of angular rate due
    to the rapid parallax variation when the Moon altitude is high).
    
    http://fer3.com/arc/m2.aspx/Moon-Venus-Lunar-Interpretation-results-Hirose-feb-2020-g47066
    
    http://fer3.com/arc/m2.aspx/Lunar-distance-parallactic-retardation-Hirose-feb-2020-g47085
    
    I confirmed parallactic retardation degrades the accuracy of a "lunar
    time sight" (my terminology), which is a time determination from lunar
    distance observed at a known location.
    
    However, lunar distance is not enough if location and time are both
    uncertain. You also need to observe the altitudes of both bodies. In
    that case, I found parallactic retardation did not reduce accuracy. But
    the algorithm in Lunar 4.4 is far from the usual methods in the heyday
    of lunars. By successive approximations it seeks the time and place
    where all three observed angles are duplicated.  (Typically, four
    iterations are necessary for 0.1′ accuracy if the initial values were
    off by one hour and 10° of latitude and longitude.) The geocentric lunar
    distance is displayed for the user but is not part of the time
    determination.
    
    On the other hand, the classic solutions utilized the altitude
    observations to transform the observed lunar distance to the equivalent
    geocentric angle. That angle would be compared to tabulated values in
    the almanac to find the time.
    
    Many different solutions were published in the days of lunars. In his
    magnum opus on spherical astronomy, William Chauvenet described his own
    method. (I'll provide references in a separate thread.) I implemented it
    in a Windows application.
    
    There are a few differences. All calculations are performed directly
    with his formulas without reference to tables or logarithms, since the
    power of the machine makes them superfluous. And my SofaJpl astronomy
    DLL computes refraction. Chauvenet uses the refraction model of Bessel,
    and provides the formulas. They're complicated. I didn't want to code
    and test them. A few checks showed good agreement between SofaJpl and
    the Bessel tables.
    
    I repeated my parallactic retardation experiment with Chauvenet's
    method, and found no loss of accuracy at high Moon altitude. Details
    follow. All simulated observations were generated with Lunar 4.4, which
    has the same refraction model as my Chauvenet implementation.
    
    
    > First, generate simulated Moon and Venus "observations" at 2020-01-29
    > 1200 and 1800 UT1. The times were selected for their quite different
    > lunar distance rates at the observer. In reality the observations would
    > be impractical since the Sun is above the horizon, but that doesn't
    > matter in this experiment.
    >
    > 69.4 s delta T
    > 10°N 20°W, at sea level
    > 10 C, 1010 mb
    >
    > 14°22.15' Moon apparent upper limb altitude
    > 26°23.72' Venus apparent upper limb altitude
    > 12°26.36' apparent distance, Moon near to Venus far
    > +0.356' per minute topocentric lunar distance rate
    > +0.419' per minute geocentric lunar distance rate
    
    Moon
    14.74' geocentric semidiameter
    54.10' horizontal parallax
    -4°41.54' declination
    
    Venus
    0.13' semidiameter
    0.13' horizontal parallax
    -5°18.07' declination
    
    Moon upper, Venus upper, Moon near and Venus far are the correct limbs.
    If you prefer, apply Venus semidiameter to adjust altitude and distance
    to the center of mass.
    
    geocentric apparent center to center distance on 2020-01-29
    12h UT1 11°52.08'
    13h UT1 12°17.26'
    14h UT1 12°42.53'
    
    Chauvenet solution
    11°52.08′ observed distance reduced to geocenter
    12:00:00 observation time
    
    Both values are correct to the last digit. Now increase lunar distance
    by one minute. Altitudes are not changed. The new Chauvenet solution is
    2m32s later. I.e., the sensitivity to lunar distance observation error
    is 2m32s per arc minute.
    
    
    Six hours later, Moon altitude has increased from 14° to 71°. Due to
    "parallactic retardation" the lunar distance rate at the observer has
    decreased from +0.356′ to +0.204 per minute time. Does that make the
    solution more sensitive to observational error?
    
    Simulated observations at the new time:
    
    > 18:00 UT1
    > 70°47.90' Moon apparent upper limb altitude
    > 58°17.74' Venus apparent upper limb altitude
    > 13°55.18' apparent distance, Moon near to Venus far
    > +0.204' per minute topocentric
    > +0.426' per minute geocentric
    
    Moon
    14.73' geocentric semidiameter
    54.09' horizontal parallax
    -3°32.55' declination
    
    Venus
    0.13' semidiameter
    0.13' horizontal parallax
    -5°10.37' declination
    
    geocentric apparent center to center distance on 2020-01-29
    18:00 UT1 14°24.32'
    19:00 UT1 14°49.92'
    20:00 UT1 15°15.56'
    
    Chauvenet solution
    14°24.328′ observed distance reduced to geocenter
    18:00:01 observation time
    
    If lunar distance is increased one minute, the new solution is 2m18s
    later. Compare that to 2m32s error at 1200 UT when the lunar distance
    rate was greater. In this case, "parallactic retardation" didn't degrade
    accuracy. In fact, parallactic retardation was greater in the second
    case, but the solution was a little *less* sensitive to lunar distance
    error.
    
    
    Now a second example with the Sun. Note that the Moon upper limb is
    observed in the first case, lower limb in the second.
    
    > For a second test of "parallactic retardation" in a lunar distance time
    > determination, I present two simulated Sun - Moon observations on
    > 2020-01-30. First, the constant conditions for both observations:
    >
    > 2020 Jan 30 UT date
    > +1m09.4s delta T
    > 10°N 20°W position, at sea level
    > 10.0 C (50.0 F) air temperature
    > 1010.0 mb (29.83″ Hg) air pressure
    > 50.0% relative humidity
    >
    > Now the observations. Caution - the Moon altitude limb is not the same
    > in both observations.
    >
    > 2020-01-30 13:00 UT1
    >    19°55.49' Moon apparent upper limb altitude
    >    60°50.10' Sun apparent lower limb altitude
    >    62°38.06' Moon near to Sun near limb
    > +0.377' per minute (topocentric)
    > +0.449' per minute (geocentric)
    
    Moon
    14.74' geocentric semidiameter
    54.12' horizontal parallax
    +0°08.17' declination
    
    Sun
    16.24' semidiameter
    0.15' horizontal parallax
    -17°42.86' declination
    
    geocentric apparent center to center distance:
    1300 UT1 62°27.46'
    1400 UT1 62°54.41'
    1500 UT1 63°21.35'
    
    Chauvenet solution:
    62°27.465′ geocentric distance
    13:00:01 observation time
    
    2m13s time error per 1′ distance error.
    
    
    Five hours later, Moon altitude has increased from 20° to 80°. Rate of
    change of distance has decreased from +0.377' per minute (topocentric)
    to +0.235'. Again test the sensitivity to error in the observed lunar
    distance.
    
    > 2020-01-30 18:00 UT1
    >
    >    79°57.88' Moon apparent lower limb altitude
    >    18°21.44' Sun apparent lower limb altitude
    >    64°02.26' computed Moon near to Sun near limb
    > +0.235' per minute time (topocentric)
    > +0.449' per minute (geocentric)
    
    Moon
    14.75' geocentric semidiameter
    54.14' horizontal parallax
    +1°06.45' dec
    
    Sun
    16.24' semidiameter
    0.15' horizontal parallax
    -17°39.44' dec
    
    geocentric apparent center to center distance
    1800 UT1 64°42.21'
    1900 UT1 65°09.17'
    2000 UT1 65°36.13'
    
    Chauvenet solution
    64°42.213′ geocentric distance
    18:00:00 observation time
    
    Re-solving with a small change in the observed lunar distance indicates
    2m22s time error per 1′ lunar distance error (vs. 2m13s in the case with
    less parallactic retardation).
    
    Based on this limited experiment I believe "parallactic retardation" —
    though a real phenomenon — does not affect the accuracy of the classic
    "lunar". However, it does degrade accuracy when time is calculated from
    lunar distance (only) at a known location.
    
    All Chauvenet solutions were accurate to 0.01′ geocentric separation
    angle and one second of time. But the fact that all four were at 10°
    latitude and standard atmospheric conditions may have something to do
    with the high accuracy. I will try Chauvenet's examples, which are more
    diverse.
    

       
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