# Author: Hannu K. J. Poropudas Title: Using Coral as a Clock APPROXIMATE ANCIENT TIME FORMU

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Author: Hannu K. J. Poropudas
Title: Using Coral as a Clock
===============================================================================

APPROXIMATE ANCIENT TIME FORMULA BASED ON FOSSIL DATA

Hannu K. J. Poropudas
(R&D Engineer, Nokia Telecommunications)
Vesaisentie 9E, 90900 Kiiminki, Finland

ABSTRACT

Time formula, also called fossil formula, gives an answer to the question:
how many mean solar days are in a tropical year at a certain point in time.
It is the simplest possible expression that has been worked out from
experimental fossil points (Eicher/Wells 1976, Pannella MacClintock
Thompson 1968 and Scrutton 1978).
The formula is intended for use mainly in the experimental fossil point's
area, which goes back, at maximum, as far as the approximately 3,556 (+ 0,032)
-
billion years old Warrawoona stromatolite fossils (The Cambridge Encyclopaedia
of Earth Sciences 1982 diagram page 357, Schopf 1983).
At the present, the time formula have been tested over an interval as far
back as 850 (+ 50) million years and consistency with the experimental points
-
looks good (formula's safe area of use).

TIME FORMULA

The time formula is the following expression given over the widest
possible time area. The time area can be limited on the basis of
experimental measurement data, if the deviations of the measured results
are, for example, greater than five to ten days.
2
M - A(T)* T - B(T)* T
N(T) = ---------------------
1 + C(T)* T

N(T) = Number of mean solar days in tropical year at the point T
T = time in uniform centuries
M = 365,2422

Here is taken into account lengthening of measured uniform mean
solar day as a first approximation (Allen 1973).
15        1                         -8
1 + C* T = 1 +  ----- *  ----- * T  = 1 + 1,7361 * 10   T
10000    86400

The uniform shortening formula of the tropical year (Allen 1973)
-6
365,24219878 - 6,16 * 10   T

has been corrected by one nonlinear term and also the coefficient of the
linear term has been corrected. As a first approximation the coefficients
A(T) and B(T) are also assumed to be constants.
If (N1, T1) and (N2, T2) are two fossil points and if C(T) is assumed
constant value then the coefficients of the formula have the following
values:
A = T2(M-N1)/T1/(T2-T1) - T1(M-N2)/T2/(T2-T1) + C*(N2T1-N1T2)/(T2-T1)

B = (M-N2)/T2/(T2-T1) - (M-N1)/T1/(T2-T1) + C*(N1-N2)/(T2-T1)

The coefficients of the terms are achieved from two reliable fossil points.
1. From coral fossil data approximately 400 million years ago. It has been
estimated that there were little over 400 days in a year at that time.
These fossil corals were ancient marine invertebrates that secreted
a microscopically thin layer of calsium carbonate each day; the layers laid
down in summer were thicker than those laid down in winter.
Estimations are based on careful countings of calcium carbonate layers in
fossil corals formed due to the changes of lightness of night and day
(The Cambridge Atlas of Astronomy 1985, Pages 55 and 54, Photograph:
J. C. Revy. The specimen is from the collection of the Brittish National
Museum of the Natural History).
2. From fossil data approximately 180 million years ago, when there were
approximately 381 days of that time in a year (Wells 1963).
-6                    -13
A = 2,46 * 10     and  B = 1,79 * 10        (in first time formula).

If fossil points from (Eicher 1976) are used.
1. From fossil data approximately 190 million years ago, when there were
approximately 382 days of that time in a year (Eicher/Wells 1976).
2. From fossil data approximately 395 million years ago, when there was
approximately 401 days of that time in a year (Eicher/Wells 1976).
then
-6                    -14
A = 2,28 * 10    and  B = 4,74 * 10         (in second time formula)

(see for basis growth increments in fossils: Rosenberg Runcorn 1975).

This seems to be a good form for the approximation even when C is afterwards
set to depend on T. This new setting is possible because the formula allows
quite wide range of variations of C's magnitude and it still gives a good
fit to the fossil data.
A(T) = T2(M-N1)/T1/(T2-T1) - T1(M-N2)/T2/(T2-T1) + C(T)*(N2T1-N1T2)/(T2-T1)

B(T) = (M-N2)/T2/(T2-T1) - (M-N1)/T1/(T2-T1) + C(T)*(N1-N2)/(T2-T1)

The best C(T) is achieved from Geologic Rock and Fossil Record Data, if
the distance between Earth and Moon can be estimated at different points
on the geologic time scale. Solar tides must also be accounted because
these tides speeds up Earth's rotation (see fossil data of past 500 million
years: Pannella MacClintock and Thompson 1968, and Scrutton 1978, Fig 8,
Page 185).
The coefficient of the asymptotic straight line for the month formula is
-7
about 2,60 * 10  from (Scrutton 1978) and the little uncertain period of
4
(Pannella MacClintoc and Thompson 1968).
-7                                   4
P(T) = 29,53 - 2,60 * 10  * ( 1 - e*sin ( 2 P T / (429,4 * 10  )))* T

P(T)= number of mean solar days per synodic month at a certain
point T of time
0 <= e <= 1, (e = 1 in first approximation)
(This month formula is not used because it is still under testings).
4
If dT = + 5 * 10  centuries and dN = + 0,5 mean solar days/tropical year
-                            -
are the accuracy of determination of the absolute age of the fossil samples,
and latter is roughly the accuracy of determination of the number of days in
year from fossil samples, then calculations gives the same order error
magnitudes as A and B themselves are, so at present accuracy the coefficients
cannot be estimated very reliable.
The coral and bivalve fossil data suggests that the value of derivative
of the time formula at the point T = 0 is (Schopf 1983):
-6
-(9,5 + 0,9) * 10    (coral data)
-           -6
-(10,0 + 1,2) * 10   (bivalve data)
-
Further data (Schopf 1983):

(d W     / dt) / (dW          / dt) = 42 + 11 (coral data)
Earth           Earth-Moon           -
= 46 + 8  (bivalve data)
-
= 51 + 5 (modern astronomical data)
-
= 48 (present day value predicted by Kepler's third law and the
conservation of angular momentum)

TABLE OF CALCULATED VALUES COMPARED WITH THE FOSSIL DATA (FIRST TIME
FORMULA, WELL'S CORAL FOSSIL DATA)

The time formula give the following table with different values of T:

TABLE 1
4
1.  = T / 10   (centuries)
2.  = calculated value from the first time formula  (days/year)
3.  = calculated value from the second time formula (days/year)
7.  = experimental value (days/year), (Wells 1963). 8.  = era (end time).

1.  2.      3.      7.      8.
------------------------------------------------------------------------
0  365,24  365,24  365,24  Cenozoic
------------------------------------------------------------------------
- 65  370,95  370,89  371     Cretaceous
------------------------------------------------------------------------
-135  377,07  377,07  377     Jurassic
------------------------------------------------------------------------
-180  381,00  381,10  381     Triassic
------------------------------------------------------------------------
-230  385,34  385,63  385     Permian
------------------------------------------------------------------------
-280  389,67  390,22  390     Pennsylvanian
------------------------------------------------------------------------
-310  392,26  393,01  393     Mississippian
------------------------------------------------------------------------
-345  395,27  396,28  396     Devonian
------------------------------------------------------------------------
<-405  400,42  401,96 >400
------------------------------------------------------------------------
-405  400,42  401,96  402     Silurian
------------------------------------------------------------------------
-425  402,14  403,88          Ordovician
------------------------------------------------------------------------
-500  408,53  411,15  412     Cambrian
(the fossil can be older)
------------------------------------------------------------------------
-600  416,99  421,08  424     Precambrian
(no fossil evidence)
------------------------------------------------------------------------
-390- 399,14- 400,53->400     Devonian (The Cambridge Atlas of)
-400  400,00  401,48          Astronomy 1985, p.55 and p.54)
------------------------------------------------------------------------

WELL'S CORAL FOSSIL SAMPLES:

TABLE 2

WELL'S CORAL FOSSIL SAMPLES (1. Wells 1963 and 2. Wells 1970):

Era 1.          Sample 1.               Data 1.
-------------------------------------------------------------------------
Pensylvanian    Caninia                 385 day layers
-------------------------------------------------------------------------
Pennsylvanian   Lophophyllidium         390 day layers
-------------------------------------------------------------------------
Middle-Devonian Heliophyllum halli      usually about 400 day layers,
(13 annual growth
increments)            ranging between extremes of 385
Eridophyllum archiaci   and 410
Favosites
-------------------------------------------------------------------------
Lower-Silurian  Holophragma calceoloides
-------------------------------------------------------------------------
Cambrian                                An estimation= 21-hr Cambrian day
(Munk MacDonald 1960).

Era 2.           Sample 2.               Data 2.
-------------------------------------------------------------------------
Mississippian    Lithostrontion          398 day layers.
-------------------------------------------------------------------------
Mississippian    Lophophylldium          380 and 390 day layers.
-------------------------------------------------------------------------
Middle-Devonian  Heliophyllum halli      average of 398 with a range from
Eridophyllum archiaci   385 to 405.
(Fig. 3.)
Cylindrophyllum
Favosites
-------------------------------------------------------------------------
Middle-Silurian  Holophragma calceoloides.
(Fig. 1. and 2.)
-------------------------------------------------------------------------
Middle Silurian  Ketophyllum             about 400 day layers.
-------------------------------------------------------------------------
Upper Ordovician Streptelasma            about 412 day layers.
-------------------------------------------------------------------------
Cambrian         Escherichia coli        An estimation = 21-hr Cambrian
day (Halberg Conner 1961).

TABLE OF CALCULATED VALUES COMPARED WITH THE FOSSIL DATA, (SECOND TIME
FORMULA,  WELL'S CORAL FOSSIL DATA)

The time formula the following table with different values of T:

TABLE 3

7.  = experimental value (days/year), (Eicher/Wells 1976).

1.  2.      3.      7.      8.
------------------------------------------------------------------------
0  365,24  365,24  365,24  Cenozoic
------------------------------------------------------------------------
- 65  370,95  370,89  371     Cretaceous
------------------------------------------------------------------------
-136  377,16  377,16  377     Jurassic
------------------------------------------------------------------------
-190  381,87  382,00  382     Triassic
------------------------------------------------------------------------
-225  384,91  385,18  385     Permian
------------------------------------------------------------------------
-280  389,67  390,22  390     Pennsylvanian
------------------------------------------------------------------------
-325  393,55  394,41  394     Mississippian
------------------------------------------------------------------------
-345  395,27  396,28  396     Devonian
------------------------------------------------------------------------
-395  399,57  401,00  401     Silurian
------------------------------------------------------------------------
-435  402,99  404,84  405     Ordovician
------------------------------------------------------------------------
-500  408,53  411,15  412     Cambrian
------------------------------------------------------------------------
-570  414,46  418,07  421     Precambrian
------------------------------------------------------------------------

Three coral species from Middle Devonian rocks of New York and Ontario,
the numerous counts range between 385 and 410 and they accumulate near the
median of 398.

AWRAMIK'S AND VANYO'S STROMATOLITE DATA FROM THE BITTER SPRINGS FORMATION,
(Anabaria Juvensis), CALCULATED AND THE FOSSIL DATA COMPARED
(FIRST AND SECOND TIME FORMULA)

(Vanyo Awramik 1985, error limit of the age determination 50 million years)
This research was supported in part by the NSF Grant EAR 83-03754 and is
contribution no. 145 of the Preston Cloud Research Laboratory.
All samples are in collections of the Preston Cloud Research Laboratory.

TABLE 4

7.  = experimental value (days/year), (Vanyo Awramik 1985).

1.  2.      3.      7.
------------------------------------------------------------------------
-850  437,83  447,19  435 (best) or 454 or 409 or 485
------------------------------------------------------------------------

This, approximately 850 million years old, stromatolite
investigated by Vanyo and Awramik had a wavy sinusoidal year growth
"wave length" of 87 mm in the best sample and the depth of a days growth
was 0,2 mm. This gave about 435 days in year at that time. Extreme values
were measured to be 409 and 485, and one measurement gave the value 454.
This wavy nature of the growth is due the point of view that the growth has
been directed towards the Sun and the same direction is repeated annually.

THE STROMATOLITE FROM THE AREA OF GREAT SLAVE LAKE (Hearne Formation) GSM 77325

(Jones 1981 and Schopf 1983, age determination is too inaccurate)
The specimen is stored in the collection of Institute of Geological
Sciences.

TABLE 5

7. =  experimental value, Highest count=(days/month) ?,(Jones 1981,
Schopf 1983).
8.  = experimental value, Mode=(days/(month/2))-(days/month) ?,
(Jones 1981, Schopf 1983).
9.  = experimental value, Tidal cycles per year=( months/year ) ?,
(Pannella 1972 b).

1.    2.       3.       7.   8.      9.
------------------------------------------------------------------------
-1790-  510,60-  567,10-
-2170   536,42   629,58                13
------------------------------------------------------------------------

Pannella has estimated (Pannella 1972 b) that there have been about 13
tidal cycles in a year at that time. He has always counted two groups of
laminae of the stromatolite as a unit. The very clear accumulation in the
magnesium spectrum of the stromatolite laminations at the number 25 remains
unclear.

GSM 77325 spectrum:

Mg: 21-25-(33-34)-44-51-(75-76).
?  -------
Si: 5-8-10-17-21-25-(26-28)-(35-36)-(43-45)-54-73-100.
-------
Al: 5-19-21-25-27-29-(30-36)-41-50-53.
----------
Ca: 8-11-17-21-25-27-34-43-46-(50-51)-66-81-101-121.
--
( )= between.

BULAWAYAN STROMATOLITE BUL-6/BUL-7 OR UCLA-BUL-7 OR P.P.P.G 253

(Schopf 1983 , error limit 140 million years, age determination too
inaccurate, see pages 385-413)
The specimen code is P.P.R.G (Precambrian Paleobiology Research Group)
number.

TABLE 6
4
1.  = T / 10   (centuries),(Schopf 1983) .
7.  = experimental value, Highest count=(days/month)?, (Pannella 1972 a).
8.  = experimental value,
Mode=(days/(month/4))-(days/(month/2))-(days/month)?, (Pannella 1972 a).
9.  = experimental value, Tidal cycles per year (months/year)?,(important)

1.    2.       3.       7.   8.        9.
------------------------------------------------------------------------
-2460-  556,33-  693,70-  41   10-20-40  ?
-2740   571,69   762,65
------------------------------------------------------------------------

There are very clear growth layers and time marks in the
Bulawayan stromatolite ?
(Pannella 1972 a):

Mode = 10-20-40 and Highest Count = 41 .

The latter could mean 41 mean solar days in a synodic month at that time
(from new moon to new moon).
The highest number in the stromatolite's mode could mean the same.
The mode's number 20 could be a sign of the high tide repeated twice
monthly. The mode's number 10 could mean that there has been two high tides
and two low tides in synodic month at that time. If it has been so, this
would mean that the solar tides had been larger at that time than they are
at present. The age determination is too inaccurate in this case and the
names of the samples are not all clear (Pannella 1972 a, BUL-6/BUL-7 or
UCLA-Bul-7 ).

WARRAWOONA STROMATOLITE, CALCULATED VALUES

The specimens are named in reference (Schopf 1983)
P.P.R.G 001-018,046,349,511-514,517,522,534-542
P.P.R.G = Precambrian Paleobiology Research Group.
(Schopf 1983 and error limit 32 million years, sample is partially melted).

TABLE 7

7.  = experimental value, Highest count = (days/month) ?.
8.  = experimental value,
Mode= (days/(month/4))-(days/(month/2))-(days/month) ?.
9.  = experimental value, Tidal cycles per year = (months/year) ?.

1.    2.       3.       7.   8.      9.
------------------------------------------------------------------------
-3556   591,60   1009,77
------------------------------------------------------------------------

TABLE OF THE FOSSIL DATA ABOUT SOLAR DAYS PER SYNODIC MONTH

(Pannella 1972 a, Pannella 1972 b, Pannella MacClintock Thompson 1968,

PANNELLA'S FOSSIL DATA

TABLE 8

7.  = experimental value, interpreted as (days/month), (Pannella 1972 a
and Pannella MacClintock Thompson 1968).
8.  = standard deviation in experimental value (days/month)
9.  = the code of the fossil specimen
10. = Highest count=(days/month)?, (Pannella 1972 a and Pannella 1972 b).
11. = Mode=(days/(month/4))-(days/(month/2))-(days/month) ?,
(Pannella 1972 a and Pannella 1972 b ).
12. = Tidal cycles per year = (months/year) ?,
(Pannella 1972 a and Pannella 1972 b ).
U = Upper, M = Middle, L = Lower.
YPM-IP = The collection of Division of Invertebrate Paleontology,
Peabody Museum, Yale University
Approximate absolute ages are estimated from Kulp (Kulp 1961).
(*) =  in reference (Pannella 1972 a) and absolute age determination
method different than in reference (Pannella MacClintock Thompson 1968).
(**)=  Pannella 1972 b.
(***)= The well preserved specimen from Belt Supergroup.

1.       2.      3.      7.    8.    9.
----------------------------------------------------------------------------
0        365,24  365,24  29,1  1,22  YPM-IP-26310
29,6  1,35         26307
29,0  0,87         26305
29,3  1,04         26312
29,2  1,05         26309
29,1  0,90         26304
29,4  1,45         28493 (*)
----------------------------------------------------------------------------
U- -10   366,12  366,11  29,7  1,05  YPM-IP-28494 (*)
29,4  0,97         26376 (*)
29,8  1,27         28495 (*)
----------------------------------------------------------------------------
M- -17   366,74  366,71  29,4  1,08         28483 (*)
----------------------------------------------------------------------------
-18      366,83  366,80  29,4  0,97  YPM-IP-26376
----------------------------------------------------------------------------
L- -27   367,62  367,58  29,2  0,99         28482 (*)
----------------------------------------------------------------------------
L- -35   368,32  368,27  29,6  0,97         26377 (*)
----------------------------------------------------------------------------
-40      368,76  368,71  29,6  0,97  YPM-IP-26377
----------------------------------------------------------------------------
M- -45   369,20  369,14  29,9  0,90         28496 (*)
----------------------------------------------------------------------------
-46      369,29  369,23  30,0  0,88  YPM-IP-26380
----------------------------------------------------------------------------
L- -50   369,64  369,58  29,4  1,03         28480 (*)
----------------------------------------------------------------------------
L- -54   369,99  369,93  29,6  1,55         28485 (*)
----------------------------------------------------------------------------
U- -58   370,34  370,27  30,0  0,88         26380 (*)
----------------------------------------------------------------------------
U- -70   371,39  371,33  29,7  1,03         26322 (*)
----------------------------------------------------------------------------
-72      371.57  371,50  29,7  1,03  YPM-IP-26322
29,9  1,16         26801
29,7  0,70         26802
30,2  1,03         26381
30,0  0,84         26382
----------------------------------------------------------------------------
U- -70   371,39  371,33  29,8  1,69         28497 (*)
29,6  0,80         11655 (*)
----------------------------------------------------------------------------
-205     383,17  383,36  29,8  1,24  YPM-IP-26803
29,8  1,15         26804
29,3  1,25         26805
----------------------------------------------------------------------------
M- -220  384,47  384,72  30,0  1,32         26803 (*)
29,4  1,18         26804 (*)
29,4  1,16         26805 (*)
----------------------------------------------------------------------------
-290     390,53  391,15  30,7  0,70  YPM-IP-26383
30,0  0,83         26378
29,9  0,52         26384
----------------------------------------------------------------------------
-305     391,83  392,54  30,2  1,20  YPM-IP-26323
----------------------------------------------------------------------------
L- -330  393,98  394,87  30,2  1,20         26323 (*)
----------------------------------------------------------------------------
-340     394,84  395,81  30,3  1,30  YPM-IP-26806
30,5  1,28         26807
----------------------------------------------------------------------------
M- -360  396,56  397,69  30,5  1,25         26808 (*)
30,2  1,56         28499 (*)
----------------------------------------------------------------------------
-380     398,28  399,58  30,5  1,25  YPM-IP-26808
U- -420  401,71  403,40  29,8  1,40         28505 (*)
----------------------------------------------------------------------------
L- -470  405,98  408,22  30,7  1,60          ?    (*)
----------------------------------------------------------------------------
-510     409,38  412,13  31,6  3,15  YPM-IP-13849
----------------------------------------------------------------------------

MAINLY PANNELLA'S STROMATOLITE DATA:

TABLE 9

MAINLY PANNELLA'S STROMATOLITE DATA:

1.       2.      3.      10.   11.    12.  9.
----------------------------------------------------------------------------
-510     409,38  412,13  33    16-31       YPM-IP-13849
----------------------------------------------------------------------------
-510(**) 409,38  412,13               13   YPM-IP-13849
13          28473
----------------------------------------------------------------------------
-950(***)446,02  458,19  31     16-31      YPM-IP-28459
----------------------------------------------------------------------------
-900-    441,94- 452,65-
473,94  499,64
-1300(**)                             13   YPM-IP-28459
13          28471/2
----------------------------------------------------------------------------
-1750    507,73  561,10  39     18-36      YPM-IP-28510
28511
----------------------------------------------------------------------------
-1800-   511,31- 568,62-
518,34  584,13
-1900(**)                             13   YPM-IP-28465
YPM-IP-28513
38 ?              (Biwabik, Mohr 1975)
----------------------------------------------------------------------------
-1845-   514,50- 575,52-
548,81  667,16
-2370(**)                             13   YPM-IP-28469
13          28468
?               (Great Slave,GSM 77325,Jones,1981)
----------------------------------------------------------------------------
<-2500   556,33  693,70                    YPM-IP-28467
(**)
----------------------------------------------------------------------------
<-2800   571,69  762,65   41 10-20-40      BUL-6/BUL-7 ? or UCLA-BUL-7
----------------------------------------------------------------------------

Biwabik stromatolite's spectrum (Mohr 1975):

(12-13)-18-(24-26)-32-44-48  (Mohr's counting method) and
--
(12-13)-18-(24-26)-32-38-44  (Pannella's counting method), ( ) = between.
-----

THE BIVALVE FOSSIL DATA OF BERRY AND BARKER:

TABLE 10
4
1.  = T / 10  (centuries), (Kulp 1961, ( )=Berry Baker 1975 a, Fig 2, p. 21)
7.  = experimental value, interpreted as (days/month),(Berry Barker 1975 a)
9.  = the name of the bivalve fossil specimen, era and formation.

1.       2.      3.      7.    9.
----------------------------------------------------------------------------
0-       365,24- 365,24-       PLEISTOCENE
-1       365,33  365,33  29,5  Chione californiensis (Sandstone near Punta
Cholla, Sonora, Mexico)
29,5  Chione succincta  (Palos Verdes Formation,
Newport Beach, California)
29,6  Chione undatella  (San Pedro Sandstone, San
Pedro, California)
----------------------------------------------------------------------------
-1-      365,33- 365,33-       PLIOCENE
-13      366,39  366,36
(-10)                    29,6  Chione cancellata (Caloosahatchee Formation,
Southern Florida)
29,7  Codakia orbicularis (Caloosahatchee Formation,
Southern Florida)
----------------------------------------------------------------------------
-25-     367,44- 367,40-       OLIGOCENE
-36      368,41  368,36
(-40)                    29,6  Crassatella lincolnensis (Sacate-Gaviota
Formation, Santa Inez, California)
29,5  Macrocallista hornii (Sacate-Gaviota
Formation, Santa Inez, California)
----------------------------------------------------------------------------
-36-     368,41- 368,36-       ECOCENE
-58      370,34  370,28
(-45)                    29,7  Meretrix splendida (Lutetian, Morigny, France)
----------------------------------------------------------------------------
-58-     370,34- 370,28-       PALEOCENE
-65      370,95  370,89
(-50)                    29,7  Venerid (Venerid shell fragments in thin
sections of Meganos Sandstone, Byron
----------------------------------------------------------------------------
-65-     370,95- 370,89-       CRETACEOUS
-135     377,07  377,07
(-100)                   29,7  Crassatella vadosus (Ripley Formation, Coon
Creek, Tennessee)
29,6  Idonearca vulgaris (Ripley Formation, Coon
Creek, Tennessee)
29,8  Glycymeris lacertosa (Ripley Formation, Coon
Creek, Tennessee)
29,8  Lucina subundata (Cody Shale,Greybull,Wyoming)
----------------------------------------------------------------------------
-135-    377,07- 377,07-       JURASSIC
-180     381,00  381,10
(-160)                   29,8  Lima gigantea (Leamington, England)
----------------------------------------------------------------------------
-180-    381,00- 381,10        TRIASSIC
-230     385,34  385,63
(-220)                   29,8  Septocardia sp. (Luning Formation, Pilot
----------------------------------------------------------------------------
-280-    389,67- 390,22-       CARBONIFEROUS
-345     395,27  396,28
(-310)                   30,0  Astartella concentrica (Gramham Formation,
Texas)
Brownwood, Texas)
30,2  Conocardium sp. (Carboniferous, Belgium)
----------------------------------------------------------------------------
-345-    395,27- 396,28-       DEVONIAN
-405     400,42  401,96
(-350)                   30,5  Conocardium sp. (Alpena Limestone, Michigan)
----------------------------------------------------------------------------

SCRUTTON'S MIDDLE DEVONIAN CORAL FOSSIL DATA:

TABLE 11

7.  = experimental values and average, interpreted as (days/month),
(Scrutton 1964 and Scrutton 1970)
8.  = experimental average, months/year
9.  = the code of the fossil specimen and the era.
10. = Number of consecutive bands.
11. = Ridges per band  27 28 29 30 31 32 33 34 35

1.       2.      3.                            7.       8.      9.
----------------------------------------------------------------------------
-365-    397,00- 398,16-                                     Middle Devonian
-390     399,14  400,53

11.
10.   27 28 29 30 31 32 33 34 35
----------------------------------------------------------------------------
5         1     2  1     1                    30,4
4         1        1  1        1              31,5
3               1  1  1                       31,0
----
30,9     12,91   BM R44851
----------------------------------------------------------------------------
13        1  2  3  1  4  2                    30,85    12,93   OUM DT2
----------------------------------------------------------------------------
8            1  3  2     2                    30,9
2               2                             30,0
----
30,7     12,99   OUM DT3
----------------------------------------------------------------------------
16        1  1  8     4  2                    30,7     12,99   OUM DT4
----------------------------------------------------------------------------
7      1  1     1  2  2                       30,15
5               2  2  1                       30,8
----
30,4     13,12   OUM DT5
----------------------------------------------------------------------------
7               4  1  2                       30,7
5               2  1  2                       31,0
2               1     1                       31,0
----
30,8     12,95   OUM DT6
----------------------------------------------------------------------------
12           2  4  2  1  2  1                 31,0     12,87   OUM DT7
----------------------------------------------------------------------------
7         1     1  4  1                       30,6     13,04   OUM DT8
----------------------------------------------------------------------------
9            3  4  1  1                       30,0     13,3    OUM DZ32
----------------------------------------------------------------------------
8         2  1  2  2  1                       29,9     13,34   OUM DZ33
----------------------------------------------------------------------------
Silurian
BM R21668
----------------------------------------------------------------------------
Cretaceous
BM R41812
----------------------------------------------------------------------------

In the very well preserved specimen OUM DT2, the interval betveen the top
edges of the two swells is 401 ridges (days/year). This unit comprises
thirteen bands (months/year), each on average with 30,85 ridges
(days/month). Scrutton has achieved values for months/year by dividing 399
(days/year) with corresponding value days/month. The interpretation of
ridges is my own.
OUM = Oxford University Museum. BM  = British Museum (Natural History).

THE SEDIMENT SAMPLE OF WILLIAMS:

From the time about 650 million years ago Williams has investigated
sediment layers and acieved result that there has been about 13,1 synodic
months in tropical year at that time (error limit 0,5 days/month was given,
the error limit from the age determination was not given  but it is probably
about 50 million years) (Williams 1989).

TABLE 12

7.  = experimental value (months/year), (Williams 1989)
8.  = error limit for 7. (months/year)
9.  = experimental value (days/month), (Williams 1989)
10. = error limit for 9. (days/month)
11. = experimental value (days/year), (Williams 1989)
12. = error limit for 11. (days/year)

1.       2.      3.      7.    8.   9.    10.  11. 12.
----------------------------------------------------------------------------
-650     421,20  426,15  13,1  0,5  30,5  1,5  400 20
----------------------------------------------------------------------------

IMAGINED SITUATION OF THE AREAS EXTERNAL TO THE DEFINED AREA

-4,55 billion years < T < -4,5  billion years and
-4,5  billion years < T < -4,4  billion years and
-4,4  billion years < T < -3,8  billion years.

Earth was supposedly created approximately 4.55 billion years ago as
condensating from carbonaceous chondrite meteorites (The Cambridge
Encyclopedia of Earth Sciences 1982). Before 4,55 billion years ago the
"length of a day" may have been very large. The very large value of the
denominator corresponds approximately to the non-rotating galactic "dust
cloud". The "tropical year" could have had a value prior to the
denominator's large value, if the creation of the Sun had occured before
Earth's creation. This would correspond to the rotation of the galactic
"dust cloud" around the recently created Sun (compare for example: Lada
and Shu 1990). Before the first area, the number of days in a year would
have been nearly zero. On the first area, the denominator would have
decreased from the very large value to 5 - 7 of our present hours at
approximately 4,5 billion years ago. (Schopf 1983, Pages 260-263). On the
second area the denominator would have decreased further to a minimum at a
point 4.4 billion years ago, which is due to the formation of Earth's iron
core (Wilhelms McCauley Trask 1987). This point of time is very inaccurate,
because it is roughly located in the time period 4.5 to 4 billion years ago
At that time, the length of a day on Earth would have been at least four of
our present hours, due to the holding together of the globe. The length of
a day may have been about 12 - 15 present hours approximately 4,0 billion
years ago according to the reference (Schopf 1983).
The probable compression or capture or fission of the Moon has also
apparently happened 4.55 - 4,4 billion years ago. The Moon's crust is a
little thinner on the Moon's visible face than on the hidden face, which
explains the relative rarity of volcanism on the latter. This asymmetry in
the mass distribution (an excess of crust on the hidden face) can be
related to the synchronism of the rotation and revolution of our satellite
(The Cambridge Atlas of Astronomy 1985, Pages 82-117).
The radius of the Moon towards the Earth is about four kilometers greater
than it is in directions at right angles. This is in fact much greater than
any tidal effect and the Moon's elongated shape must have been formed when
Moon was younger and hotter, and the shape of the Moon has been frozen when
it cooled and formed its extensive lithosphere. The effects of tides in the
solid body of the Moon slowed down its rotation until it locked into
syncronism with its orbital motion. Today there is no net torque on the
Moon but were it to spin up or spin down, the torque on its elongated shape
would in each case act to return the rotation to synchronism; the situation
is stable (The Cambridge Encyclopaedia of Astronomy 1977).
For a long time astronomers have noted that all the Moon's topographical
features are statistically oriented in three dominant directions:
north-south, north-east-south-west and north-west-south-east. This
arrangement which has been called the 'lunar grid' is the reflection of
former disclocations caused by the slowing down of the Moon's rotation
under the effects of the tides.
The whole surface of the Moon is affected everywhere by faults and
stress-induced rilles, certifying to a tendency of the Moon's surface to be
under stress at the very moment when the volcanism of the seas was in full
development. These rilles are in the main directed radially or tangentially
to the Imbrium basin (old crater), whose presence seems to have 'oriented'
the Moon's expansion. (The Cambridge Atlas of Astronomy 1985, Pages 82-117).
After 3,8 billion years, the coefficients of the time formula can have
slight undulation in the places where the distance from the Earth to the
Moon has a minimum or maximum, if there exists any minimums or maximums in
Earth-Moon distance in ancient times.

CRITICAL EXAMINATION OF TIME FORMULA

There is some inaccuracy in the determination of the age of fossils,
there values are typically in steps of 10 million years. In determination of
the age of the oldest fossils the relative error can be large. Time limits
of the geological time scale have changed from the geological time scale
which Kulp drafted in 1961 (Kulp 1961) and many points are still uncertain.
In the future the test points from the oldest fossils will hopefully give
answer to the question: how the coefficient of T in the denominator of the
time formula depends on T. Well preserved stromatolites whose ages can be
determined accurately could have conclusive meaning in solving this problem.
It is also possible that any sharp distinction between the numerator and
the denominator of the time formulae cannot be made. Ages of the fossil
specimens should be determined at least to an accuracy of 10 million years
and experimental fossil points should be more complete in order to make more
accurate month formula for approximating of the C(T). The poor accuracy of
the age determination of the well preserved Bulawayan stromatolite as well
as in case of other stromatolites is problematic.
At present solar tides are so small that they cannot cause significant
decrease in distance between the Earth and the Sun. Bulawayan stromatolite
could indicate that solar tides were larger in past than at present.
of the Sun on the geologic time scale. One possibility is that just when the
Sun condenceded from interstellar gas it must have rotated much faster than
the present one, so the rotation rate of the Sun has possibly slowed down on
the geologic time scale from its maximum value (about 3 present hours,
Runcorn 1967) to the present value (about 27 solar days). It remains partly
open how fast this slowing has happened and what role an expanding solar
corona had when interacting with the strong solar wind to this slowing. There
are several models of the formation of sunlike stars (see for example: Lada
and Shu 1990, Schatzman 1960).
The Sun does not rotate rigidly like Earth; this is because the Sun is
gaseous throughout. The polar regions of the photosphere take 37 days to
rotate once with respect to the distant stars, whereas the equator takes
26 days. As observed from Earth, which is moving around the Sun, the
corresponding synodic periods are 41 and 27 days. It is possible that the
non-rigid rotation of our Sun is caused by rapid rotation of the solar
interior, which would lead to a shear stress, and swifter rotation, at the
equator of the visible surface (The Encyclopaedia of Astronomy 1977).
The assumption that Sun's mass has remained constant along the whole
geologic time scale is not consistent at all time points with the point of
view that at the creation phase and at so called T-Tauri phase changes of
the Sun's mass could have been be significant (Runcorn 1967, Lada and Shu
1990).
Some chaotic oscillation possibly exists in internal prosesses of the
Earth on the geologic time scale which may have significant influence on
Earth's rotation (see for example: Powell 1991). And the geology of a Earth
size planet without a Moon seems to be quite different than Earth's geology
(see for example: Reports of Venus Geology 1991).
Lastly there remains to be investigated in future somewhat open question
of tidal density wave interactions, caused by our spiral galaxy Milky Way or
by around Milky Way rotating Large and Small Magellanic Cloud galaxies, with
the Sun, with our Planetary System and with the Earth-Moon system.
Distances of these two neigbour galaxies from the Milky Way is possible
oscillatory in nature and main trend is decreasing on the geologic time scale.
It has been estimated that these distances have decreased about 50 percent
in 10 billion years. Decreasing rotation period of these neigbour galaxies is
about 500 million years at present (see for example: Dekker 1976, Yuan and
Cheng 1989, 1991, The Cambridge Atlas of Astronomy 1985, Binney Tremaine 1987).

REFERENCES CITED:

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Astrophysical Quantities.
3 edition, University of London, The Athlone Press.
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Growth Increments in Fossil and Modern Bivalves.
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Rosenberg, G. D., Runcorn, S. K. (Editors), 1975.
Growth Rhythms and the History of the Earth's Rotation.
Willey Interscience, New York, Pages 9-24.
3.  Binney, J. and Tremaine, S, 1987.
Galactic Dynamics.
Princeton University Press, Princeton, New Jersey,
Pages 17 and 428-431.
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Paleontological Evidence on the Earth's Rotational History
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Tidal Growth Patterns in Recent and Fossil Mollusc Bivalve Shells:
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I would like to acknowledge Neil Jackson for his help in translating
the text. I took this task to myself as a calendar maker's question.
(See:  sci.astro, number= 5431, name= New Calendar)
I have sent earlier version of this work also 27.11.1991 to the Geology
magazine (Geological Society of America, 3300 Penrose Place, P.O. Box 9140,
Boulder, Co 80301, U.S.A).
I sent this work is also 10.12.1991 to sci.astro (number= 5394, name=
Ancient Time Formula) for criticism.
I would be grateful of suggestions and criticism of C(T)'s and P(T)'s
formulas and their possible connections.

Kiiminki  16.12.1991

Hannu Poropudas

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