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Subject: Simulating Parallactic Retardation in Lunar Distances
From: Arthur Pearson (arthurpearson@XXX.XXX)
Date: Fri Nov 01 2002 - 23:05:50 EST
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.irbs.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))) )
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