NavList:

Hello to all,

PRELIMINARY NOTE : Due to lack of available time, I could not adequately reply until now to a number of queries / comments about my post "1901 May, 22 Lunar example by French Navy Captain Arago" initially published as [Navlist 11482] on Jan 12, 2010. My apologies to all concerned.

In the various replies here-under, I am referring to the 11 page "111597.arago-lunar---occn-examples.pdf" filed attached to [Navlist 11597] on Jan 24, 2010 and which is taken from Captain Arago's Treaty, with this excerpt just detailing both a Lunar example and an Occultation example.

You will find here-under my reply to all the concerned subsequent contributions, in REVERSE CHRONOLOGICAL ORDER.

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In [Navlist 11760] on Feb 03, 2010 by waldendand---com, you wrote :

" should the time for the Arago lunar distance be UT 22:20 instead of 22:30? "

This is my reply :

In [Navlist 11597] on Jan 24, 2010 I earlier indicated:

" Longitudes are all reckoned in Time from Paris, with Paris Longitude being East of Greenwich by an amount of E -002°20'13.82", or 9m20.921s as published in the French "Introduction aux Ephémérides Astronomiques" (I own the 1997 edition) which is an absolutely remarkable document of the very same "upper-class" than our most celebrated "Explanatory Supplement to the Astronomical Almanac" (I own the 1992 edition)."

If you are interested in the Lunar Distance example, which happened on May 22, 1901, then you are right. On page 6 of the Referenced Document, Arago's result is: Tmp = 10h29m47.1's with "Tmp" meaning: Mean Paris Time (Temps moyen de Paris). To translate this same instant into Greenwich Mean Time, we need to subtract 9m20.921s, and accordingly get 10h20m26.179s in GMT time scale. With today's time reckoning conventions, we need to add 12h to get UT1 = 20h20m26.179s. So, and again, you are right in the sense that "the time for the Arago lunar Distance is UT 22:20 instead of 22:30".

On the other hand, if you are interested in the Occultation example which happened on Mar 02, 1901, then on Page 11 of the Referenced Document, you will find that Captain Arago's final result is "Tmp = 10h53m17.3s" which eventually translates into UT1 = 22h43m56.379s

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In [Navlist 11755] Feb 03, 2010 by Paul Hirose, you wrote :

The code at my site has been corrected. In addition, the observation data for the Arago lunar distance and occultation have been added to the existing examples in the source code.

My Comment:

Great! And thank you very much Paul. Your "reworked" results are probably the most accurate ones achievable given the Ephemeris sources you are using. The only remaining significant improvement would be taking in account the Limb irregular shape (see F.E.R.'s comments just hereafter). See also additional comment at the end of my reply to Frank just hereunder.

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In [Navlist 11701] Jan 31, 2010 by Frank E. Reed, you wrote :

I mentioned previously somewhere that the "real" next step required to get this order of magnitude improvement in lunar distances would require an accurate model of the lunar limb. The mountains and maria along the limb lead to undulations in the nearly circular outline of the Moon of roughly one arcsecond. The details vary over time in accordance with the "lunar libration" cycles. You cannot accurately analyze the timing of lunar occultations unless corrections are applied for the profile of the lunar limb. And in fact people who observe lunar occultations, like the members of IOTA, use lunar limb models on a regular basis to get accurate predictions. The limb models have been improved by the occultation observations, but fundamentally these current limb models are based on one which was developed c.1960-1965 in preparation for the Apollo program. An interesting footnote here is that models of the lunar surface are rapidly improving right now thanks to some recent spacecraft missions. So it should be possible to permanently remove this last uncertainty in lunar observations in the very near future. I've been playing around with this myself for the past two years.

My comments:

Great to hear more about your results taking in account the shape of the Moon Limb as affected by Libration.

All this is absolutely out of reach of a small hand held computer by lack of both space and computing power.

BTW, do you know how accurate - i.e. when compared to observations - such predictions can be? I understand that observed occultations were once used to determine delta-T, but have since and lately been replaced by observations of far remote quasars, much more accurate to get UT1 while TT is now given by the atomic clocks.

.. and by former recent experience, I subscribe to all your notes/remarks about non linear behavior of some crucial data when dealing with occultations. BTW, this is why Paul's software seems quite impressive in this respect: he did not seem to have it specifically tailored or adjusted to occultations, and it seems to work so well in all cases !

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In [Navlist 11699] on 30 Jan 2010 by waldendand---com, you wrote :

Note, there is an apparent typographical error in the "calcul de l'heure de paris par une distance lunaire" example of Arago. the "formules." for both RA and Dec have the second term shown as sin squared. it seem is should be sin of twice the angle. (see for example Chauvenet)

My comment : Where exactly were you able to find such "formules" in Arago's Treaty? I cannot locate/identify them.

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In [Navlist 11684] on Jan 30, 2010 by Paul Hirose, you wrote :

"The French gives me some trouble, but I believe "les hauteurs vraies" are the unrefracted altitudes. On that basis, UT1 = 22:20:34 (10:29:55 mean astronomical time at Paris), lat = +47°49'33", lon = -79°37'23". "

My reply :

NOTE : For interested readers, the numbers just here-above deal not with the Occultation, but with the 1901 May, 05 Lunar example.

You are right, "les hauteurs varies" mean "Unrefracted altitudes", i.e. as they would be observed from Earth Center and without any Atmosphere (no refraction).

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In the same [Navlist 11684], Paul you wrote :

Arago's time nomenclature is unclear to me. Time scale A is a complete mystery. I suspect M is chronometer time, and Tmp the mean time at Paris. The PDF document says Tmp = 10:29:47, which is only 8 seconds from my estimate.

My reply :

I had to dig out in my Navigation Courses to get confirmation that M was the Clock value. So you are right here.

I can also guess from my Courses that the "A" Time scale was the "Main Chronometer" time scale.

At that time - in 1901 - they did still carry one main "Chronometer" and up to 3 "Clocks". The "Main Chronometer" never left the same spot on board while the (supposedly/expectedly less accurate) "Clocks" were carried outside to time the observations. For this reason, the advance or the delay of the Main Chronometer was the main (if not the only) parameter to be closely watched. This is why, all on-board attentions were devoted to the drift rate of the "Main Chronometer", with the observation clocks were only "copies of the A time".

If we make the reasonable assumption that both Capt. Arago's examples were "actual true world" examples and that both the same "Chronometer" and same "Clock" were used, we get interesting information about the kind of drift rates such "Chronometers" and "Clocks" could achieve :

On Mar 02, 1901 (Occultation example) with A=8h15m26s and M=6h57m28s, Arago gets Tmp=10h53m17.3s. The best current determination - not taking in account the Limb irregularities - is given by you, Paul, as being "1901-03-02T22:43:38.40 UT1".

Therefore, on Mar 02, 1901, UT1-A= 14h28m12.40s and UT1-M=15h46m10.40S

On May 22, 1901 (Lunar Distance example) with A=7h51m56s and M=6h34m42s, Arago gets Tmp= 10h29m47.1s. The best current determination - again not taking in account the Limb irregularities - is given by you, Paul, as being "1901-05-22T22:20:33.83 UT1".

Therefore, on May 22, 1901, UT1-A= 14h28m37.83s and UT1-M=15h45m51.83S

With the same (reasonable) assumption, we can see that during this 81 day period the "Clock" drifted by some 10.2 seconds, which certainly is a quite surprisingly low drift rate, especially for a "Clock", while the "Chronometer" drifted by 25.43 seconds during these 81 days. This on-board "Chronometer" was (wrongly then) credited a drift rate of only … 0.2 second of time, since on Mar 02 (Tmp-A) had been updated into 2h57m31.3s while on May 22 (Tmp-A) had been updated into 2h37m51.1s.

It would be interesting to hear whether such (French Navy) practice of keeping one "Main Timekeeper/Chronometer" while using "Clocks" to time observations was also(/still ?) in use within the Royal Navy or the US Navy, or other Navies around the world.

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In the same [Navlist 11684], Paul you also wrote :

"I compute an occultation time 18 seconds different: 22:43:38.4 UT vs. 22:43:56.38 (Arago). At his time, my program says the star is -6.4" inside the Moon's limb."

My comment :

Given the Ephemeris and data source you are using to reduce your occultation, your result is certainly the best one which can be achieved when re-working this occultation with to-day computation power. Still, the only significant remaining improvement would be taking in account the Moon Limb shape as Frank has been suggesting (see hereabove).

And a last comment: while earlier running this example on my own, I was a bit surprised at the "inaccuracy" of Arago's results, mainly due to his Lunar coordinates surprisingly and significantly inaccurate by to-day standards. On my side, through running this example with full IAU 1984 Nutation, Pierre Bretagnon's precession from VSOP 87, NGC for kCan and the ELP 2000-85 theory (alas truncated) for the Moon I get a value which differs from yours by an amount of 3.8 seconds of time, which remains well within my own 6" accuracy window.

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That's all Folks …

Antoine M. "Kermit" Couëtte
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