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On 16 Dec 2004 at 10:01, Ken Muldrew wrote:

> On 16 Dec 2004 at 9:38, Alexandre Eremenko wrote:
>
> > But if Thompson's time is his local
> > apparent time (that is the time elapsed from
> > his local Sun culmination),
> > then his star altitude is close to the
> > correct value, and we still have no explanation
> > why his Lunar was so much off.
>
> I think the key problem is that the data he extracts from the almanac for
> the moon's RA and declination show an interval of only 3 minutes between
> the two lunar distance shots while his watch shows an interval of 9
> minutes. Since his DR has to be the same for both, he must have made a
> mistake in determining Greenwich apparent time for one of the shots, losing
> 6 minutes somehow.

The error in Thompson's lunars from Nov. 26 seem to stem from a single
mistake that he made when going into his almanac to work his time sights
(or at least all his data can be explained by a single error; without any
record of his rough work, this must remain conjecture).

Since the sun has already set, Thompson takes the altitudes of two stars
(Capella and Lyra (Vega)) to get his local time. He must then get the
sun's right ascension from the almanac in order to get the hour angle that
separates the sun from these stars. In order to do that, he has to know
Greenwich time. Of course the whole purpose of taking lunars is to find
out what Greenwich time is, but it is an exercise in successive
approximation, rather than getting an exact answer de novo. He knows
approximately where he is and his pocket watch gives him a reasonable
approximation to local time. By converting his dead reckoning longitude to
hours and adding it to his local time, he gets an estimate of Greenwich
time for his almanac interpolations.

Thompson's journal contains a list of courses for the journey along with
his latitude and longitude for each way point. The positions have been
updated to accord with his celestial sights, proportionally dividing up
the differences between his account and his actual position for each point
between measurements. If we just use his courses to update his starting
position, then we should have a reasonable idea of how his account looked
on Nov. 26.

At Rocky Mountain House where his journey started, he has the following
positions in November of 1800:

Latitude:
1800 9-Apr 52?21'29"

Longitude:
1800 17-Apr 115?12'00"
     18-Apr 114?57'45"

Using these values as starting points, his courses will look like:

Co. true Co. dist   N    S     E      W    52?21'29"  115? 4'52"
SE    S24E   10        9.14   4.07         52?13'32"  114?58'24"
ESE   S46E   14        9.73  10.07         52? 5' 4"  114?42'23"
SE    S24E    1        0.91   0.41         52? 4'17"  114?41'44"
SSW   S43W    1.5      1.10          1.02  52? 3'20"  114?43'21"
S48E  S19E    2        1.89   0.65         52? 1'41"  114?42'18"
SEBE  S35E    1.5      1.23   0.86         51? 0'37"  114?40'55"
ESE   S46E    4.5      3.13   3.24         51?57'54"  114?35'46"
ESE   S46E    5        3.47   3.60         51?54'52"  114?30' 3"
ESE   S47E    1.5      1.02   1.10         51?53'59"  114?28'18"
SEBE  S36E    3        2.46   1.72         51?51'51"  114?25'33"
S     S21W    8        7.47          2.87  51?45'21"  114?30' 6"
S     S21W    9        8.40          3.23  51?38' 3"  114?35'13"
S22E  S1E     1.5      1.50   0.03         51?36'45"  114?35'10"
SEBS  S13E   11       10.72   2.47         51?27'25"  114?31'14"
SSE   S1E    12       12.00   0.21         51?16'59"  114?30'54"
SEBS  S13E   13       12.67   2.92         51? 5'59"  114?26'15"
S30E  S9E     2        1.98   0.31         51? 4' 5"  114?25'45"
SEBS  S13E   12       11.69   2.70         50?54' 5"  114?21'27"
S32E  S11E   11       10.80   2.10         50?44'42"  114?18' 6"
S32E  S11E   17       16.69   3.24         50?30'11"  114?12'56"
SEBE  S35E    4        3.28   2.29         50?27'20"  114? 9'17"
S36E  S15E    5        4.83   1.29         50?23' 8"  114? 7'13"
SSE   S1E    12       12.00   0.21         50?12'42"  114? 6'53"
N30W  N9W     4  3.95                0.63  50?16' 9"  114? 7'52"
N75W  N54W   13  7.64               10.52  50?22'47"  114?24'34"
N85W  N59W    8  4.12                6.86  50?26'22"  114?35'27"

His time sight of Capella is given as:

8:32:20  98?58' 0"
8:33:10  99?13' 0"
8:34:00  99?26'45"
-----------------
8:33:10  99?12'35"
   7:54    -22'22"
-----------------
8:41:04  98?50'13"

So his watch reads 8:33:10 at the time of his altitude measurement (the
correction of 7:54 is added later once he calculates the exact local time
that corresponds to the measured altitude). His DR longitude of 114?35'27"
converts to 7h38m21s. Greenwich time should then be 8:33:10 + 7:38:21 =
16:11:31. The almanac gives the sun's RA as 16h6'14.3" on the 26th (at
noon) and 16h12'30.6" on the 27th. Thompson would have added the
proportional log of the difference between these two values, the
proportional log of Greenwich time (multiplied by 60 so that hours become
minutes, etc.) and the proportional log of 24. The inverse proportional
log would have given him the time to add to 16h6'14.3" to get the sun's RA
at the time of his Capella altitude. If I do that I get a value of
16h10'28" for the sun's RA at that time. Thompson has written down
16h11'13" for the sun's RA in his notebook and he has used this to find
his local time. Using the correct value for RA I get a local time that is
45 seconds ahead of Thompson's calculated time.

What I think he's done is to mistakenly add 8h33'10" and 10h33'10" to get
Greenwich time. I don't know why he would do this; working by candlelight
with little extra paper, I guess. But for whatever reason, that sum gives
a sun RA of 16h11'13" and puts his local time late by 45 seconds. When
taking out the RA and declination of the moon for his calculated altitudes
of Aldebaran and Altair, he correctly adds his longitude by account to his
local time, but that time is now 45s off, so those values are similarly
shifted. If I calculate the GAT from the values of RA and declination for
both of his lunars I find that the Aldebaran time is 45 seconds slow and
Altair 45 seconds fast.

If I recalculate altitudes based on the correct local time, and then re-
clear his lunars based on the proper values, and finally use the improved
lunar distances from Frank Reed's online almanac, then I get the following
values for the two lunars:

Altair: 114?14'54"
Aldebaran: 114?45'59"

The average is 114?30'27" which is a pretty reasonable value considering
that his true position is 114?19' for an error of 11' of longitude.
Thompson was merely lucky with the combination of his error and the errors
in the almanac that gave him the position 114?11'.

Bruce Stark has sent me the almanac pages covering the lunars that
Thompson took at Rocky Mountain House. Comparing errors in the original
almanacs to the correct values should tell us whether the rather large
variance in Thompson's longitudes came from poor sights or poor lunar
theory.

Ken Muldrew.

   

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