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

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From: Arthur Pearson
Date: 2003 Feb 1, 13:09 -0500

```Bill,

My experience is that the altitudes do make a difference, and I suspect
the way I handled them may have introduced some error. So I welcome any
critique of my method which was as follows:

I took altitudes of sun and moon before and after the lunar series (what
is listed below are the raw data).  I created a spreadsheet to make the
calculations. The spreadsheet requires altitudes (Hs) of sun and moon
that are simultaneous to the selected lunar. The limb of either body can
be upper or lower, and the reduction assumes they both were taken from
the same height of eye (HoE) for the dip correction. The peculiarities
of my altitudes required some manipulations in advance:

* My moon altitudes were upper limb before but lower limb after. Using
CelestNav, I found Ho for the LOWER limb observation, then by trial and
error I found an Hs UPPER limb that produced the same Ho.  That gave me
Hs for the UPPER limb both before and after. I interpolated linearly
between them to get an Hs simultaneous to the selected Ds.
* My second sun altitude came down on a small island so I had to reduce
it with dip short. I used 15.2' for 8 foot HoE with shoreline 0.3nm
away. As above, I found Ho for the altitude, then worked back to an
equivalent Hs that when reduced with the standard dip for 8 foot HoE
gave the same Ho. That gave me comparable altitudes before and after,
and I interpolated between them to get Hs sun simultaneous to selected
Ds.
* For the record, the simultaneous altitudes I got for my selected Ds
were:
For GMT = 19:42:04
Hs Sun = 27� 23.1' LL
Hs Moon = 65� 29.3' UL
HoE = 8 feet

The problems I suspect could affect the results are:
* Linear interpolation assumes a linear change in altitude which isn't
strictly the case.  I took about 32 minutes between before and after
altitudes. The moon was approaching its meridian passage. I am not sure
how this might bias the results or by how much. I suspect (hope) that it
is not significant.
* The manipulations I went through before interpolating may have errors
in logic or execution. Again, I suspect (hope) that they are not
significant.

I am delighted to hear any comments, critiques or suggestions.

Regards,
Arthur

-----Original Message-----
[mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of William Allen
Sent: Saturday, February 01, 2003 1:37 AM

Arthur,

Thanks for sharing your data with us.

Let me ask a question, though.  I see you have taken a series of Ds
sights over a period of about 17 minutes.  You chose one as your best
using graphs.  But then you took the heights of the moon and the sun
about 10 and 13 minutes after the time of your selected Ds?

I would have thought that you would have needed to get the moon and sun
heights at a time that was as close as possible to your Ds observation?
Or take the height of the moon and sun at the beginning of your Ds run
and then again at the end and average the two?

Maybe this doesn't make that big a difference, I don't know, but I would
be curious what you and others think about that.

Regards,
Bill Allen

-----Original Message-----
[mailto:NAVIGATION-L{at}LISTSERV.WEBKAHUNA.COM] On Behalf Of Arthur Pearson
Sent: Friday, January 31, 2003 8:48 AM

In early January I had the opportunity to take two sets of lunar
distance observations while on a 40' bareboat charter in the Grenadines
in the southeastern Caribbean.  Having only taken lunars on land
previously, it was a great fun wrestling with the practical
considerations of taking lunars more or less at sea.  This posting will
share the data for those who would like to work with them and offer some

FIRST SET
My first observations composed a complete series intended to find
latitude and longitude by the "1800 method".  I took a local apparent
noon sight (LAN) for latitude.  About 3 hours later, I took sun and moon
altitudes before and after a series of 7 lunar distances.  The first sun
sight was used to determine local apparent time (LAT) which was
converted to local mean time (LMT).  I graphed the lunars and selected
the fourth for solution.  Interpolating the altitudes to the time of the
selected lunar, I cleared the lunar and solved for GMT per lunar. The
difference between GMT and LMT was converted to longitude per lunar.
Having GMT and exact position from a handheld GPS, we have a gauge to
judge my accuracy.  The data and my solutions are as follows:

Date: Jan. 7, 2003
Index Correction: -1.1'

LAN Sight:
Height of Eye: 10 feet
Hs LAN: 54� 37.3'
Latitude per LAN: 12� 49.0'N

Altitudes Before Lunar:
Height of Eye: 8 feet
GMT        Hs
Sun   19:18.00   32� 15.0' Lower limb
Moon  19:21:21   63� 24.0' Upper limb

GMT        Ds
19:33:27   59� 33.2'
19:35:08   59� 34.0'
19:40:22   59� 34.8'
19:42:04   59� 35.4' (after graphing all 7, I used this one)
19:45:17   59� 36.6'
19:48:07   59� 36.0'
19:50:24   59� 37.5'

Altitudes After Lunar:
GMT        Hs
Sun   19:52.39   25� 27.2' Lower limb, dip short at 0.3 nautical miles
Moon  19:55:15   66� 18.6' Lower limb

"DR" Latitude of Lunar: 12� 37.2'N (see explanation below)
LMT of Lunar per 1st sun sight: 15:36:22 (=LMT sun sight + time elapsed
to
lunar)
GMT per Lunar:  19:44:55
Longitude per lunar: 62� 8.2'W

Time Error of Lunar:     2m 51sec fast
Distance Error of Lunar: 1.3' too long
Longitude Error:         47' too far west
Total Error in position: 46 nm including 1 mile error in latitude

Position per GPS:
12� 38.2'N
61� 21.2'W

This LAN sight was taken on a broad reach in 20-25 knots in six foot
seas, common conditions between the islands of the Grenadines at this
time of year.  The error was 1 nm compared to GPS. In the course of the
afternoon we took a varied course around and between islands to our
anchorage behind the reef of the Tobago Cays. A strict DR was not
practical, so for the "DR" latitude for the lunar, I simply applied the
1nm error in my LAN sight to the latitude per GPS of my lunar
observations.

The lunar was taken in the very rolly anchorage behind the reef. Not
open ocean, but definitely a lot of motion.  The moon happened to be
nearing its highest point, providing several challenges.  It was very
difficult to get the sun over to the moon, and the high altitude made
for a very awkward posture when I did. This sight was a prolonged
struggle; I tried several postures but found nothing comfortable.  The
first moon altitude used the upper limb, the second one used the lower
limb because that looked best at the time (a mistake; I latter
calculated that the moon had not yet reached the meridian by the second
lunar). This made interpolating between the moon altitudes cumbersome
and may have introduced error by measuring an incomplete limb.  Finally,
the second sun sight came down against one of the Cays, so I had to
correct for dip short for HOE 8ft, distance of 0.3 nm, which I calculate
to be 15.2'.

The high moon also brought George's parallactic retardation into play; I
calculated the hourly change in apparent distance as only 14.5' which is
quite slow.  All in all, I am pleased with how well it came out given
the difficulties.  I also solved this lunar using the "Lewis & Clark"
method by which I calculated the altitudes based on latitude per LAN, an
assumed longitude, and the LHA of the bodies based on the afternoon sun
sight.  This method came in only 31' too far west, an improvement of
16'. As I used the same distance observation, the difference can only be
attributed to using different altitudes to clear the distance
(calculated vs. observed). Perhaps I made errors in the unusual
corrections that were needed for the observed altitudes. In any case, it
is another demonstration of the robustness and efficiency of the "Lewis
& Clark" methodology.

SECOND SET
The second set of observations was a series of lunar distances alone. I
cleared and reduced to GMT based on calculated altitudes that were in
turn based on known GMT and position.  There is no point to this process
other than improving one's lunar technique.  This set was taken hard on
the wind in 25 knots and 6 foot seas just south of Bequia. My data and
results are as follows:

Date: Jan. 10, 2003
Index correction: -1.1'

GMT        Ds
18:29:25   92� 28.4'
18:32:31   92� 29.0'
18:36:06   92� 30.8'
18:38:47   92� 31.1' (after graphing, I used this one)
18:43:44   92� 33.8'
18:46:13   92� 33.4'

Position per GPS at time of selected distance:
12� 56.9' N
61� 16.7' W

GMT per Lunar: 18:42:59
Time error of lunar: 5m 12sec fast
Distance error of lunar: 2.3' too long

With a much lower moon (Hs~17�) and thus a much faster rate of change in
apparent distance (21' per hour), this was a better time to take a
distance than the high moon of the first set.  Although a longer
distance, the posture was much more comfortable.  I was braced solidly
in the companion way and could pivot comfortably from the waste up.
However, the motion of the boat had the sun and moon dancing
boisterously around in the scope.  I felt I got a good look at a couple
of the contacts but struggled with the others.  The graph of the
observations is more consistent than I would have guessed although I
ended up pretty far off.  I would guess that had I worked the longitude
as I did with the first set, I'd be off by over 80 miles. Practice,
practice, practice... I suppose I'll just have to spend more time in the
Caribbean!

Had I been depending on these lunars at the end of a transatlantic
passage to the Grenadines, the practical consequence would be a landfall
7 to 15 hours later than anticipated. Late landfall might mean after
dark when one was planning for daylight.  More unsettling would be
arriving early in the dark had the error gone the other way.  Clearly a
broad margin of error and extreme caution are required when depending on
these techniques.  On the other hand, I would love to know the
satisfaction of leaving the Canaries, crossing an ocean, and arriving
within a few hours of ETA using only sun, moon and a sextant.

Regards,

Arthur
arthurpearson---.com

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