Eppure si muove        
                                         ...despite tidal braking

Léo DUBAL

dubal  @  archaeometry.org

Abstract: 

The pearled solar eclipse of 1912.04.17  
occurred 60 hours after the TITANIC disaster 
had cast its shadow upon this exciting event. 
The data collected during this most elusive eclipse,
as well as those of dozen of other solar eclipses spread over
the last 4'000 years, are compared to those generated by Xavier JUBIER's 5MCSE,
the most up-to date ergonomical solar eclipse simulation freeware,
which allows the choice of the DeltaT parameter, as well as the exact GPS coordinates of the observation site. 
For the last 4'000 years, the Earth's spin slowdown key-parameter D T(t) will be assumed to behave as
                             D T(t)  =  30.65 * t²  [s],
where t is expressed in centuries. 
An appropriate ephemeris SI day, as origin of the time axis, has been chosen as Julian Day 2'398'000.5.
Excellent agreement has been found between our Geo-Chronological Model and
the records described in World Annals of all reliable candidates of ancient solar eclipses. 
One might speculate if the seconds in excess of this Model compared to actual DeltaT values
is the consequence of  recent global warming.
 

 The accuracy in retrodiction of  past Solar Eclipses is bound to 
an adequate Modelling of the slowing-down of the Earth's spin.
The central parameter is the Length of the Day.
The secular length of the day λ is defined as:
  λ = (36524.24)^-1 [cy]
while the circadian length of the day L(D) :
  L(D) =  24[h] x 60[m/h] x 60[s/m] = 86'400 [s]

Since the introduction of Atomic clocks and of the SI seconds,
one has to introduce an ideal  D_o Day  which length is 
L_o( D_o) = 86'400 [s SI]
In obvious divergence with commonly accepted timing for D_o 
we suggest to choose for the ephemeris SI day
D_o  = JD 2'398'000.5 
i.e. 1853.05.27 / 00:00 UT
JD meaning Julian Day.
In 1583, the year Galileo GALILEI
observed that a pendulum could be used for keeping time reliably,
Joseph-Juste  SCALIGER, the chronologist born in Agen, published in Lutecia
 "De emendatione temporum"  [On the Correction of Chronology],
the genial dating system he conceived after his lucky escape to Geneva
from the St Bartholomew'  massacre.
 SCALIGER named this first linear time scale Julian Day 
 in honour to his father Julius Caesar de L'ESCALLE.
The introduction of such a Reform for Chronology
 represents a deep paradigmatic change.
Let us underline that our empirical choice for the
origin of the time-axis differs by half a century from,
 e.g.:  1800.01.01, the origin picked-up by
Kevin D. PANG, K.K. YAU & H.H. CHOU
 in Geo.Res.For. 3-4, 1998, pp. 459-488, 
for their analysis of Ancient Chinese Solar Eclipses

The time of occurrence of a solar eclipse will hereafter be referred as :
t  =  λ ^. (JD - D_o)  [cy]

From the cumulative Geophysical difference D T
DT(t)  = UT - AT  [s] ,
between the Atomic Time defining L_{o ,}and
 Universal Time defining L = L(t)
follows the difference in " LOD",  the length of the day:
D L =  L(t) - L_o  =  λ ^. dDT / dt   [s]
The rate of change of L(t) is therefore :
dL/dt =  λ d²DT/dt²  [s . cy^-1]
The average, fortnightly rate of change of secular lod  is, as expected, minute (ppm) compared to
 the rate of change of the fortnightly lod up and downs due to lunar tidal

2. Slow-down of Earth' Spin

The secular Earth's rotation rate
ω = 47.34 ^. 10⁹  [" / cy]
might be expressed in terms of the L(t)
ω = 2^.π  / L   [ rad . s^-1]
and its time variation, as: 
dω/dt = - 2π ^. L^{-2 .} dL/dt  [rad . s^-1 .cy^-1]
whence:
  = - L^{-1 .} λ ^.  d²DT/dt²   [cy^-1]

Our Geo-Chronological Model assumes a parabolic time dependence for DT(t):
D T(t)  =   c ^. t²  [s] 
whence
   =   - L^{-1 .} λ ^.  2 ^. c  = - 0.63377 ^. 10^{-9 .} c   [ cy ^-1]  

According to  J.G. WILLIAMS
("Diurnal and semidiurnal tidal contribution to lunar secular acceleration" ,
 EOS Trans Am. Geoph.Un. 73,43,126, 1992 )
the LLR (lunar laser ranging) data indicates a secular acceleration of the Moon

= - 25.88 ± 0.5 [" / cy²]

According to K. LAMBECK
(The Earth's Variable Rotation", Oxford Univ. Press, 1980) 
the secular acceleration of the Moon induces a tidal braking of the Earth:

   =  + 1.07.10^-9 .   =  - 27.7 . 10^-9 [cy^-1]
Over the last four millennia, the sea-level around the Earth seems to have
not exceeded 1% in fluctuation; c_t , the tidal braking might therefore
be considered as constant, at least in 1^st approximation.
   =  - 0.63377 ^. 10^{-9 .} c_t    [ cy ^-1]  , with
c_t = - 27.7 . 10^-9  /  - 0.63377 ^. 10^-9 = 43.7 [ s . cy ^-2]  

Tidal Braking is NOT the only force in action. 
As established by PANG et al. (op.cit.) for Ancient Chinese Solar Eclipses,
c lies within two boundaries
 27.5 <  c  < 32.5  [ s . cy ^-2]

As the results presented farther suggest,
for the time laps of the last 4'000 years, 
an unique c-value, well inside Pang's criterion,
suffices to retrodict satisfactorily all available data.
 This value is :  c = 30.65 [s . cy  ^-2] .

The observed secular rate of change of the Earth's spin is therefore
   =     - 19.4 ^. 10^-9 [cy^-1]
leading to an overall secular non-tidal acceleration
    ^{=     -     =}    + 8.3 ^. 10^-9 [cy^-1]
which could be accounted for 2/3 by the secular decrease
of the Earth's oblateness, -2J₂ dot  =   + 5.5 . 10^-9 [cy^-1]
 measured by Satellite Laser Ranging.
The combined effect c_tr of Tidal Braking and post-glacial Rebound 
(or change of oblatness) is  c_tr = 35.0 [s . cy  ^-2] .
Those results are shown in the "de REYFF-Chart" for Delta T(t = λ^.[JD-D_o])
for tidal braking alone, with post-glacial rebound and for Solar eclipses data 




Post-glacial Rebound is not the only Non-Tidal forces at work.
 For other smaller components, J.O. DICKEY listed, from core outwards:
 viscous torque,  topographic coupling of the mantle to the core,
electromagnetic coupling, continental drift, earth-quakes, sea level loading,
 tsunami, ocean currents, ground water, winds, and atmospheric loading.
For the last 4'000 years, one might suggest that,
  as 1^st approximation, there has been, no change
 in the overall secular Non-Tidal acceleration of the Earth.
The limit of validity of this hypothesis is provided by the occurrence,
over the last decade of a decrease of the actual DeltaT, 
(possibly an effect of global warming ?).

 
The experimental values of  LOD measured since 1962 against extragalactic sources are
compared with the continuous slow-down Model.

The software used in the present analysis is 5MCSE of Xavier JUBIER.
To introduce into 5MCSE the DeltaT of our Model, one should add into the url "&DdT=1"
e.g, for the Hipparchus eclipse (see SAROS), of -128.11.20, the url is:
http://xjubier.free.fr/en/site_pages/solar_eclipses/xSE_GoogleMapFull.php?Ecl=-01281120&Acc=2&Umb=1&Lmt=1&Mag=0&DdT=1

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viii)     In Ovar, observatory stations were spread along the road between Carregal to Cadaval
While 5MCSE retrodicts only 1.4 km of the road to have been in the totality zone,
 all the observers who were spread over 5 km to the West of station #10 
have believed to be in the central line !!! 

At station  #10, GPS W8.59757 / N40.85432
COSTA LOBO obtained 158 images in 14 [s] showing Baily grains,
whose 40 with the grains only in the areas opposed to the Sun.
 The eclipse was therefore total for a time laps of 4.4 [s],
only along the direction of the Moon,
what confirms the film of  Léon GAUMONT.
It has been interpreted, as the relative speed of the Moon
to the Earth is about 0.38 ["/s], i.e. 693 [m/s], as due to the shape of the Moon:
   a slight flattening along the axis of rotation,
The irregular profile of the Moon shows indeed,
 between valleys and peaks typically a 1["] difference..



Let us note though that Serge KOUTCHMY has shown evidence for an "ovalisation"
 (or polar lengthening) of the Solar Chromosphere, which, in our opinion,
might well have also contributed to the direction dependant totality effect
recorded in 1912
The timing of the contacts recorded in observatory #10 
are compared to the retrodicted values with  5MCSE : 

With the description of this eclipse @ E9.37 /N47.42, in the
 Annales Sangallenses, one has the very first Chronicle description of an eclipse.
Previous annals’ reports were themselves dated on the basis of eclipses!
Year 840  is, in this sense, the beginning of modern History of solar eclipses.



The year, here, is explicitly given: ω.xl.  with the abbreviation ω standing for 800
This Chronicle is most important because it is corroborated by  the "Chronicon" of ANDREAS Bergomatis  
Indictione tertia sic fuit sol obscuratus in hoc mundo, et stellas in celo apparebant,
3.Nonas Magias, ora nona, in laetanias Domini,
quasi media ora. Facta est tribulatio magna. Cumque hoc populus intenderent,
multi extimabant, quod iam amplius hoc seculum non staret
sed dum haec angustia contemplarent, refulsit sol et
quasi tremidus in antea umbraculam fugire cepit.
Ipsa vero nocte sequenti prope matutino facta est lux quasi in die.
Haec signa in celo conperta, doctores in suorum monitiones dixerunt
Estote, fratres, parati; quia adimpletum est quod in evangelio Dominus dixit:
cum haec signa videritis, scitote, quia prope est die Domini magnus et manifestus?
Sequenti autem mense Iunio Hludowicus imperator defunctus est, suosque dies finivit in pace. 
The death of LOUIS I, just mentioned in CSG, is usually assumed to occur on 840.06.20.
It is described in ANDREAS B. Chronicon in relation to
this long eclipse over Bergamo, and the 3rd year of indiction:
May 840 matches the 3rd year of the 35th indiction,
 the 15 years long ecclesiastical fiscal period  introduced by CONSTANTIN on 312.09.01. 
For the precise day in May, both Chronicles indicate 3 days before the None's of May,
which, according to http://www.louisg.net/C_julien.htm is indeed May 5th, and for the hour, 
St-Gallen mentions between the 8th and 9th hour (of the day).
  For St-Gallen, GPS gives 03:17 UT for sunrise, 8h30 added to it gives 12:47 UT
 while 5MCSE retrodicts 12:38 UT....
click for a more in-depth discussion (in French)
 
16)    SAROS 79/29 on 71.03.20

This eclipse was total for half a minute over the city of Athens,
while it is believed that
 the poet PLUTARCH spent most of his life in his city of Chaeronea.
In De facie in orbe lunae, PLUTARCH  lets an Athenian
 (friend of his ?) called Lucius, describe a solar eclipse:
...beginning  just after noonday 
 Later, Lucius described  
"...a kind of light visible about the rim,
 keeping the shadow from being profound.."
what corresponds well to the partial corona, as coined by FLAMMARION

17)    SAROS 62/54 on 65.12.16
For the eclipse over the Kingdom of CHU,  
with magnitude of 0.997 over the capital Xuzhou @ E117.15/ N34.25
one might ask if it has not been instrumental in enforcing Buddhism.

The totality path crossed over the Hellespont,  e.g. 17 s duration @ E26.398/N40.146 
while 1/5 of the sun disk was still visible in Alexandria @E29.9 /N31.2: 
this matches exactly with the description
 given by PAPPUS and CLEOMEDRE of the characteristics
 of the eclipse used by the Greek astronomer HIPPARCHUS in his
  evaluation of DE-M, the Distance Earth-Moon, in terms of RE, the Earth Radius. 
We do not know how HIPPARCHUS actually proceeded.  
A qualified guess has been  90 Earth Radii. Another approach is:
as the height of the sun was only 12° over the horizon,
 it must have been easy to remember its
 position over the landscape and then to estimate the azimuths of the 
two observation sites, i.e. 233° and 239° as retrodicted by 5MCSE.
The ratio RE / DE-M equals the ratio of a 1/5 of the sun apparent angle 0.5°/5 
and of the 6° difference in azimuth, i.e. 0.1° / 6° = 1/60, 
The distance from Alexandria to the Moon is then 60 Earth Radii.
As expected, there exist other opinions on HIPPARCHUS...
 e.g.: On the web-site of the School of Mathematics, Uni. St Andrews,
 forwarding the claims of G J TOOMER, HIPPARCHUS on the distances
of the Sun and Moon, Arch. History Exact Sci. 14 (1974), 126-142
it is boldly stated that HIPPARCHUS used the eclipse on -189.03.14,
on his birthday so to speak...  

Out of the eclipse  @ E10.32 /N36.83 emerged, in 1995,
the hypothesis that this event might have been instrumental in the institution of the cult of the 
Goddess TANIT  in Carthage. 5MCSE retrodicts  M=0.985 at 12:39 UT

21)    SAROS 42/62 on -477.02.17