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March 22 Lunar Observations
From: Arthur Pearson
Date: 2002 Mar 23, 21:10 -0500

```Gentlemen,

Here are the numbers from a set of lunar distances off the sun on March
22, 2002.  I would welcome any comments on my procedures and the
problems I encountered, and if anyone is inclined to work the
calculations on their own, I would be curious how they come out.

All times reflect GMT from my GPS unit.
Latitude and longitude are also from GPS.
Angles are expressed XX* YY.Z for degrees and minutes or YY.Z' for
minutes alone.
Index Error = 2.63' ON (determined by Bruce's averaging method)
Latitude:       N 42* 21.7
Longitude:      W 71* 15.5

I took eleven distances as follows:
GMT             Ds
18:40:46        98* 29.9
18:43:41        98* 32.2
18:44:47        98* 32.4
18:46:10        98* 31.0
18:47:16        98* 31.2
18:48:16        98* 34.6
18:49:28        98* 34.8
18:50:45        98* 39.8
18:52:25        98* 41.4
18:53:37        98* 34.8
18:54:51        98* 36.2

As you can see, I had considerable trouble getting consistent distances.
I may have misread one or more of these, or I may have bumped the knob
inadvertently. Bruce's book emphasizes the sensitivity of long angles to
parallelism of the telescope, so I tried to make these contacts in the
center of the scope. Needless to say, I have lots of room for
improvement.

My next step was to graph these distances against time and then plot a
reference line that shows the slope of the change in comparing distances
from the hour before to the hour after the observations. Here are the
Nautical Almanac data I used:

Sun GHA 18:00   88* 17.4
Sun GHA 19:00   102* 17.6
Sun Dec 18:00   0* 46.1 North
Sun Dec 19:00   0* 47.1 North
Sun SD          16.1'
Moon GHA 18:00  349* 11.9
Moon GHA 19:00  3* 37.4
Moon Dec 18:00  24* 40.0 North
Moon Dec 19:00  24* 40.2 North
Moon HP         57.8'

From this I calculated:
D1 =    97* 55.9 (Comparing distance at 18:00)
D2 =    98* 26.9 (Comparing distance at 19:00)
Slope = +5.16' every 10 minutes during this hour (for ease of plotting
the reference line)

I plotted this reference line alongside the graphed distances. I ignored
the highest two distances based on my belief that they were spurious.  I
visually established a "best fit" line parallel to the reference line
through the 9 remaining valid distances. A lot of judgment here, this is
a pseudo-scientific process, but it allowed me to select a point on the
best fit line as follows:

Selected GMT of observation:            18:50:00
Ds corresponding to selected time:      98* 34.8

As I am landlocked, my altitudes were first calculated based on my
selected time and my latitude and longitude. From the calculated
altitudes, I work backwards to get apparent altitudes (altitudes correct
for all but parallax and refraction). Bruce has tables for this, I have
to fiddle around with a sight reduction spreadsheet to come up with the
corresponding values.

Sc =    40* 41.7
Sa =    40* 42.8
Mc =    70* 27.7
Ma =    70* 8.2

With all the above in hand, I can clear the distance and determine GMT
of the observation. Here are the intermediate values and final results:

Moon's refraction and parallax =        19.53'
Sun's refraction and parallax   =       1.06' (I use Sun HP = 0.0024*)
Moon's augmented semi diameter =        16.0'
Sun's semi diameter =                   16.1'
Ds =    98* 34.8
Da =    99* 4.3
D  =    98* 19.2 (the cleared distance)
D1 = 97* 55.9
D2 = 98* 26.9
GMT per observation =   18:45:07, March 22, 2002

This would suggest that my "watch" is fast by 4 minutes and 43 seconds.
However, as I assume time by GPS is quite accurate, the error is in my
observations and calculations. Presuming the calculations are accurate,
I measured the angle as smaller than it actually was. Bruce has tables
to calculate the error in the distance, I have to fiddle with my
spreadsheet to determine that I should have measured Ds = 98* 37.3, an
error of about 2.5'. When I go back to how I placed my "best fit" line
on the graph of my sights, I might have gotten closer to the mark if I
had included the two distances I discarded as spurious, but they were
clear outliers in the series, so I would not have used them in practice
at sea.

So it's back to the sextant to shoot some more. If anyone finds errors
in my calculations or problems with my procedure, please let me know.

Regards,
Arthur

```
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