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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

Moon
  19°38.03' unrefracted center altitude
     14.82' unrefracted semidiameter
      2.64' refraction
  19°55.49' apparent upper limb altitude
  93°37.66' azimuth

Sun
  61°05.80' unrefracted center altitude
    -16.24' unrefracted semidiameter
      0.54' refraction
  60°50.10' apparent lower limb altitude
163°28.00' predicted azimuth

topocentric apparent Moon to Sun angle
  63°11.32' center to center, unrefracted
      2.23' refraction
  63°09.09' center to center, refracted
    -14.79' Moon near limb refracted SD
    -16.23' Sun near limb refracted SD
  62°38.06' Moon near to Sun near limb
+0.377' per minute (topocentric)
+0.449' per minute (geocentric)

====================

2020-01-30 18:00 UT1

Moon
  80°12.69' unrefracted center altitude
    -14.98' unrefracted semidiameter
      0.17' refraction
  79°57.88' apparent lower limb altitude
202°53.37' predicted azimuth

Sun
  18°34.79' unrefracted center altitude
    -16.24' unrefracted semidiameter
      2.88' refraction
  18°21.44' apparent lower limb altitude
247°24.27' predicted azimuth

topocentric apparent Moon to Sun angle
  64°36.13' center to center, unrefracted
      2.70' refraction
  64°33.43' center to center, refracted
    -14.98' Moon near limb refracted SD
    -16.20' Sun near limb refracted SD
  64°02.26' computed Moon near to Sun near limb
+0.235' per minute time (topocentric)
+0.449' per minute (geocentric)

====================

Solve for time with observed lunar distance from the known position. (I
call this a "lunar time sight," a not fully satisfactory term since it
could be confused with a conventional time sight with the Moon. But I
don't know what else to call it.) Lunar distances are increased 0.5' to
simulate observational error. Solutions:

13:01:13 at 1300
18:02:29 at 1800

====================

Solve for time with the lunar distances and altitudes. To simulate
observational error, lunar distances are increased 0.5' and altitudes
2'. Results:

13:01:06 at 1300
18:01:01 at 1800

====================

As in my previous lunar distance experiment with Venus, the Moon is at
low altitude in one "observation" and high altitude in the other. At
high altitude the lunar distance rate is much reduced (+0.235' per
minute vs. +0.377') due to the rapid variation of horizontal parallax
when the Moon is near the zenith. ("parallactic retardation")

Thus, it's no surprise that the lunar time sight is most sensitive to
observational error at high altitude. But that does not seem to be true
for the traditional lunar where altitudes are observed too.

I should make clear that my reduction does not follow the traditional
scheme which reduces the observed distance to the equivalent geocentric
angle. Instead, my program works entirely with the three angles observed
at the topocenter (observer's position). With an iterative algorithm it
seeks the time and place where all three angles should have occurred,
and thereby includes (to the extent they can be computed) the effects of
parallax, refraction, and even the oblateness of the Earth.

In this case, with the initial time estimate in error by one hour and
position in error by 10 degrees of latitude and longitude, the program
needed four iterations to match all angles within 0.01'.

Are the classic methods of the lunars era likewise able to overcome
parallactic retardation? I don't know. In fact, I'm not sure the
phenomenon is — IN GENERAL — harmless to the accuracy of the 3-angle lunar.

   

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