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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**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.