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

**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 matters. DEFINITION OF ABBREVIATIONS 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 refraction) DESCRIPTION OF THE DATA SETS AND GRAPHS 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�. COMMENTS ON THE GRAPHED RESULTS Mid_Lat_Lagging_Moon_1.JPG 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. Mid_Lat_Lagging_Moon_2.JPG 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). Mid_Lat_Lagging_Moon_3.JPG 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. Equ_Lagging_Moon_1.JPG 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. Mid_Lat_Leading_Moon_1.JPG 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). Equ_Leading_Moon_1.JPG 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. PRACTICAL IMPLICATIONS 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. INPUTS TO THE SIMULATION: 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 page PROCEDURES FOR CALCULATIONS: 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)))) where 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 where 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)) ) where 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))) )