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Historical Lunars : take in account 'delta-T' or ignore it ?



                            PRELIMINARY NOTE :

I earlier recently started addressing the items here-under with Frank E. Reed 
(private communications). Frank indicated that these topics might better be 
covered on NavList. Therefore and after extensive and in-depth reprocessing I 
am submitting everything to all of us in NavList.

                                 *******

FOREWORD

When re-processing "historical" Lunars (i.e. "real world" Lunars observed 
during the period @1700-@1900 A.D.) with to-day computational power, should 
we take in account "delta-T" or ignore it ?

Although the effects of delta-T are (almost) negligible during this past 2 
century period, - and accordingly the topic raised here might seem almost 
void or even useless - it seems interesting to agree on what should be the 
"best way" to-day to tackle historical Lunars for the sake of respecting good 
navigation principles.

SECTION I

Note : This introduction may be skipped by knowledgeable Readers who may 
directly proceed to SECTION II here-after.

SECTION I only brings back to mind a somewhat "comprehensive big picture" 
regarding Lunars through recalling the following important points which need 
to be remembered. 

-   1 - For extended periods of time before accurate 'Chronometers / 
Timekeepers' could be manufactured - i.e. before the spectacular achievements 
by Harrison in the UK followed many years later by Le Roy and then Berthoud 
in France - Lunars were almost the only practicable method to 
reckon/reconstruct "astronomical time" by comparison between "Observed" and 
"Tabular" Lunar Distances, which in turn enabled to derive an Observer's 
Longitude, and

-   2 - While the sun RA varies by some 2.5 arc minutes per hour of elapsed 
time, the Moon RA - and to a great extent the distance between the Moon and 
every 'usable' Body - varies about half a degree during the same elapsed 
hour. As a consequence, any angular error in measuring a lunar-body distance 
will translate into an Observer's Longitude error magnified some 30 times. In 
order to minimize such errors in Longitude, it has long been recognized that 
Lunar distances must be observed and recorded with the utmost achievable 
precision and care, and

-   3 - Since Lunar distances themselves are almost non subject to the too 
often quite unpredictable effects of horizon dip, but only and to some 
limited extent to atmospheric refraction (including body shape distortion due 
to differential refraction), in the best cases Lunar Distances can be 
measured to a quite high "meaningful" accuracy. A Lunar Distance 
observational accuracy of 0.1 arcminute (6 arcseconds) can be assessed as the 
very best one a well trained observer can achieve with an excellent sextant. 
For the reason stated in �2 hereabove, it has been customary to attempt 
clearing Lunars to this level of accuracy. Given both the complexity of the 
relevant computations and the historical limitations of ephemeris accuracies 
- see �6 hereafter - such a 6 arc second intended computation accuracy level 
could not be consistently reached in the past. However - and in other words - 
there is no computational "overkill" when one attempts clearing Lunars to an 
accuracy of 0.1 arcminute, i.e. to +/- 12 seconds of elapsed time, which also 
translates into +/- 3 minutes of uncertainty in Longitude. Nowadays accurate 
lunar distances clearing computations have become fully and routinely 
achievable : see for example the GREAT on-line Lunar Computer by Frank E. 
Reed (http://www.historicalatlas.com/lunars/lunars_v4.html) Congratulations 
again Frank ! , and

-   4 - Historically, the fact that UT is not a strictly "uniform" time-scale 
has been fully ascertained only as late as of the first quarter of the 20th 
century, although UT irregularities had been suspected for at least half a 
century before. This led to establishing the distinction between an almost 
'perfectly smooth' Ephemeris Time (ET) time scale, while the UT time scale 
directly linked to the Earth irregular rotation was and has been recognized 
as being a quite 'irregular' time-scale. The difference between both Times 
scales is widely known as 'delta-T'. Given the rate of change of Lunar 
distances, certainly 'delta-T' values smaller than 12 seconds of time - 
equating to a change in the Moon RA inferior to 6 arc seconds, and a change 
in the Moon GHA inferior to 3 arc minutes - could be regarded as 'almost 
negligible', and

-   5 - There still remains to-day some uncertainty on delta-T values in 
ancient times. Intensive research has been performed here since the 1980's 
and in particular by MM. Stephenson and Morrison. Their best current results 
have been summarized in http://eclipse.gsfc.nasa.gov/SEhelp/deltaT.html and 
in http://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html . Although there 
remain significant delta-T uncertainties before the first half of the 17 th 
century, starting from the 1620's and thereafter (i.e. during all the Lunars 
epoch) , our current knowledge of the delta-T is quite reliably accurate, and

-   6 - For over 2 centuries, 'fairly good' lunar positions predictions were 
available to Navigators and in particular as soon as the 1730's. One and half 
century later greater accuracies were achieved with Hansen's Theory. However 
sooner or later - i.e. 10 to 20 years at the most after their first 'flashing 
and impressive' results - all these 'historical' theories started going 
off-track. They all needed additional 'observed corrective terms' required to 
bring predictions back into conformity with observations. Even the first 
version (completion date 1908) of Ernest William Brown's Lunar Theory  - 
albeit extraordinary for its time - and subsequently published in 1919 under 
the form of its much celebrated Tables of the Motion of the Moon included in 
particular an 'empirical term' exceeding 10 arcseconds in Longitude to best 
reconcile predictions with observations. Only since the middle of the 20th 
century, have Lunar Movement theories been established which could and can 
reliably achieve accuracies consistently better than 0.1 arc minute. The 
second version of E.W. Brown's Lunar Theory, i.e. the Improved Lunar 
Ephemeris 'I.L.E.' by Brown and Eckert published shortly after 1950 was the 
first such Moon motion theory to consistently achieve accuracies better than 
0.1 arc minute, including for computations related up to a couple of 
centuries in the past. This improvement was mainly the result of properly 
taking in account the time differences between the ET and UT Time scales (see 
� 4 hereabove). Subsequently there was no longer need for the 'empirical 
term' which was removed. In the 1980's the I.L.E. theory was then followed by 
the now 'deadly accurate' DE20x - DE40x Numerical Integrations from JPL 
altogether with the ELP 2000-xx Analytical Theories and the INPOP06 Numerical 
Integration from Bureau des Longitudes. When dealing with Lunars, we should 
keep in mind that we can consistently and accurately predict/compute the Moon 
Position with an accuracy equal or better than 6 arcseconds only since the 
middle of the 20th Century, and

-   7 - When working with an accuracy of 6 arcseconds on celestial 
coordinates, we will assume that all the previously used time 
scales/variables - whether the 'Temps Uniforme' used by Urbain Le Verrier or 
the 'Ephemeris Time' used by Simon Newcomb are the same as the TDT time scale 
and the current TT time scale and that there has been no appreciable 
discontinuity between them.



SECTION II

With the perspective of the various points summarized hereabove, how should we 
best proceed if we are to reprocess / re-compute historical Lunars with 
to-day's computation tools?

I would like to offer the following comments and will certainly be very glad 
to hear feed-back from the NavList community, especially if view points are 
different from the one I am detailing here-after.

-*-*-   First, we can state that, when they cleared their Lunars, the 18th, 
19th and (if any) the early 20th centuries Navigators were not aware of the 
"necessity" of using 2 time scales. In other words - and unless they used 
data amended with 'observed corrective terms'(see �6 here-above) - they 
(unknowingly) cleared all their Lunars with the Value delta-T=0.0 seconds of 
time. This point has 2 different consequences if we are to compute again 
to-day their historical Lunars with our now much improved theories and much 
improved computational power :

     If we want with our current "computation tools" check the quality 
of their Lunars, we should put ourselves in their exact same environment, 
i.e. reprocess their Lunar computations with delta-T = 0, and

     If on the other hand we want to best derive their observed 
longitudes - and/or positions - we should certainly then take in account 
values of delta-T.


-*-*-   Second, I am submitting here-after a fictitious example of an upcoming 
Lunar which shows that - if we were to ignore to-day the effects of delta-T 
on Lunars - we would significantly increase our error in UT determination, 
and therefore our Longitude uncertainty.

•••••••   23 Dec 2009   Height of 
Eye = 17 ' T = 59�F , P = 29.92 ' Estimated UT / Position :  16h58m53.0s / 
N3707.1W01253.7 - All observations assumed to occur at the same time - 
Distance SUN-MOON = 77�53'7 , SUNL = 5�50'1 , MOONL = 50�05'0 . All sextant 
measures are corrected for Instrument error (and only instrument error), i.e. 
all other corrections (dip, refraction, SD, Parallax ... ) need to be 
performed from the "raw data" hereabove.

*** If we compute observer's position from the observed heights :

o   With delta-T = 66.7 seconds (close to current value for Dec 23, 2009) :

    • Lunar Distance UT = 17h00m00.7s - i.e. TT = 17h01m07.4s - hence UT 
error = -1m07.7s, cleared distance = 78�34.00', and
    • We also get the following position at time of Lunar Distance :
N3659.9W01300.3 (result of 2 body fix), and
    • Crosscheck with Frank's on-line computer (you need to enter both 
heights) : Error in Lunar : -0.1', approximate Error in Longitude 03.9',      
 cleared distance = 78�33.9'

o   With delta-T = 0.0 second :

    • Lunar Distance UT  = 17h01m07.4s - i.e. TT = 17h01m07.4s -  hence 
UT error = -2m14.4s, cleared distance = 78�34.00', and
    • We also get the following position at time of Lunar Distance :
N3659.9W01317.1 (result of 2 body fix), and
    • If it were possible to use Frank's on-line Lunar Clearing computer 
with delta-T=0.0 s for this example, this would offer additional independent 
cross-check of these results.

Note : In either case (delta-T=0.0s or delta-T=66.7 s) and since we assume 
unchanged Latitude (using same both Heights), the recorded Lunar Distance 
77�53'7 defines the very same Terrestrial Time TT = 17h01m07.4s.

By comparison with the "delta-T=0.0 second" position, and since the Earth has 
"slowed down", the Observer's geographical position is shifted 66.7 seconds 
of time - or 16.6' - to the East. In either case we have here exactly the 
same Bodies configuration as seen from the Earth surface. Only the Observer's 
support - i.e. the Earth itself - has slowed down and accordingly the 
Observer's Terrestrial position has become shifted to the East.


*** If we use and consider estimated position - i.e. N3707.1W01253.7 - as 
being the "true/exact Observer's position" :

o   With delta-T = 66.7 seconds (close to current value for Dec 23, 2009) :

    • Lunar Distance UT =  17h01m44.0s - i.e. TT = 17h02m50.7s - hence 
UT error = -2m51.0s, cleared distance = 78�34.79', and
    • Crosscheck with Frank's on-line computer (do not enter any heights)  :
Error in Lunar : -0.1', approximate Error in Longitude 03.1', cleared distance = 78�34.7'

o   With delta-T = 0.0 second :

    • Lunar Distance UT = 17h06m09.4s - i.e. TT = 17h06m09.4s - hence UT 
error = -7m16.4s, cleared distance = 78�36.32', and
    • If it were possible to use Frank's on-line Lunar Clearing computer 
with delta-T=0.0 s for this example, this would offer additional independent 
cross-check of these results.

Note : It is interesting to remark here that neither bodies positions (whether 
celestial coordinates or heights) are the same because*** TT are different 
for each value of delta-T (*** BTW, why do we get different TT values ?... 
any takers here ?). Also, one can easily observe the effect of the apparent 
Sun starting "slowing down" due to the increasing refraction which becomes 
appreciable near the horizon. This explains why, for a delta-T value equal to 
66.7 seconds, we have a difference of 3m18.7s between both TT values, hence 
cleared distances different by 1.53'.


LAST NOTE : Finally I am also aware that some of the computation results given 
here-above in the previous example point our more directly towards 
"Time-keeper / Chronometer error" rather than towards "Longitude Errror" or 
"Lunar Distance Error"(these last 2 itemns being immediate results presented 
on Frank's Computer). Therefore, and although both concepts are very closely 
related, it might be interesting to cover this subject regarding "Lunars 
Philosophy" on NavList(under a separate file ?).


Comments from our NavList Community are most welcome, so that we all benefit 
from each other's contributions.

Thanks to all

Antoine M. "Kermit" Couette

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