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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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