Article
Astronomy
December 2010 Vol.55 No.35: 4010–4017 doi: 10.1007/s11434-010-4197-x

SPECIAL TOPICS:

Experimental measurement of growth patterns on fossil corals: Secular variation in ancient Earth-Sun distances
ZHANG WeiJia1,3,4*, LI ZhengBin2,3 & LEI Yang1
1 2

Department of Physics, Peking University, Beijing 100871, China; Department of Electrical Engineering and Computer Science, Peking University, Beijing 100871, China; 3 State Key Laboratory of Advanced Optical Communication Systems & Networks, Peking University, Beijing 100871, China; 4 Committee of Yuanpei Honors Program, Peking University, Beijing 100871, China Received June 3, 2010; accepted July 22, 2010

In recent years, much attention has been given to the increase in the Earth-Sun distance, with the modern rate reported as 5–15 m/cy on the basis of astronomical measurements. However, traditional methods cannot measure the ancient leaving rates, so a myriad of research attempting to provide explanations were met with unmatched magnitudes. In this paper we consider that the growth patterns on fossils could reflect the ancient Earth-Sun relationships. Through mechanical analysis of both the Earth-Sun and Earth-Moon systems, these patterns confirmed an increase in the Earth-Sun distance. With a large number of well-preserved specimens and new technology available, both the modern and ancient leaving rates could be measured with high precision, and it was found that the Earth has been leaving the Sun over the past 0.53 billion years. The Earth’s semi-major axis was 146 million kilometers at the beginning of the Phanerozoic Eon, equating to 97.6% of its current value. Measured modern leaving rates are 5–14 m/cy, whereas the ancient rates were much higher. Experimental results indicate a special expansion with an average expansion coefficient of 0.57H0 and deceleration in the form of Hubble drag. On the basis of experimental results, the Earth’s semi-major axis could be represented by a simple formula that matches fossil measurements. growth pattern, Earth’s semi-major axis, planetary Hubble expansion
Citation: Zhang W J, Li Z B, Lei Y. Experimental measurement of growth patterns on fossil corals: Secular variation in ancient Earth-Sun distances. Chinese Sci Bull, 2010, 55: 4010−4017, doi: 10.1007/s11434-010-4197-x

The coral rhythm, a cyclic pattern of physiological changes or changes in activity in both living and fossil coral epitheca, is the result of physiological activity in response to a cyclic change in physical conditions, such as a diurnal light intensity change, monthly tidal fluctuation and seasonal temperature variation. The original publication of Wells [1] on coral growth stimulated considerable interest in “biological clocks.” Wells submitted the idea that small ridges and annulations on the epitheca of corals might represent, respectively, daily and annual variations in skeletal deposition. Because this idea is very appealing and could provide direct evidence of the relation between the geological time scale and the number of days per year [2,3], it stimulated
*Corresponding author (email: itaisa@pku.edu.cn) © Science China Press and Springer-Verlag Berlin Heidelberg 2010

numerous works on the implications of the few preliminary figures [4–8]. It is now widely accepted that coral fossils have three kinds of clear growth patterns: daily growth lines resulting from diurnal increments in calcium carbonate deposition, lunar month bands controlled by reproductive cycles and annual annulations that reflect alternations in the density of internal tissues [1,6,9–12]. Furthermore, recent research proved such mechanisms via biochemical and biophysical methods [13–15]. In addition, research on modern biological clocks has increasingly used modern facilities and new materials and laboratory techniques over the past decade [16–21]. On the other hand, corals are also now widely used in ancient climate and environment studies, as indicators of variations in the sea surface temperature or CO2 concentration for example [22–24]. csb.scichina.com www.springerlink.com

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However, former biological clock studies seldom considered the possibility that the ancient Earth’s revolution could be studied by fossil measurements. These studies simply suggested a constant year length in the discussion of the variation in the day length. In fact, presumption of a constant Earth orbit is unreasonable. Lammerzah et al. [25] concluded there are observations which at least until now and after many years of studies have not yet found any convincing explanation, including the recently realized increase in the mean Earth-Sun distance (the astronomical unit). From analysis of radiometric measurements, the modern increase rate was measured by Krasinsky and Brumberg [26] and Standish [27] as 5–15 m/century. Iorio [28] reported that the semi-major axis of the Earth’s orbit, approximately equal to the astronomical unit, should be increasing at 5 m/century now, which is in agreement with radiometric measurements. If the increase can be verified as having a dynamic nature, instead of comprising systematic errors in observations or incompleteness of the model used for light propagation, our understanding of cosmology and geology can be greatly improved. Furthermore, there is a faint young Sun paradox: early in the Earth’s history, the Sun’s output would have only been 70% of its modern value; thus, if the Earth-Sun distance has been stable, this solar output would have been insufficient to maintain a liquid ocean [29]. In this article, we take an interdisciplinary approach to studying the ancient Earth’s orbit. First, fossils with bioclock data covering the entire Phanerozoic eon were collected worldwide and new technologies were used for analysis. Biological clock measurements confirmed the lunar recession rate and provided the ancient Earth’s average rotation slowdown rate through celestial mechanical calculations, without suggesting a constant year length. Second, using the ancient lunar recession rate and lunar rhythm data, ancient year lengths were derived. The first fitting curve of the relationship between the Earth’s semi-major axis and geological time was obtained using Kepler’s third law. Third, by combining the ancient rotation slowdown rate and daily rhythm data, a second fitting curve was derived. As a result, ancient Earth-Sun distances were obtained. The obtained modern variation rates were consistent with the astronomical observations. Furthermore, a new evolution model of the Earth’s orbit, according to the relationship between the ancient Earth-Sun distance and geological time, the average leaving constant (~0.57H0), and deceleration in the form of Hubble drag, is provided. On the basis of experimental results and mathematical techniques, the semi-major axis of the Earth’s revolution can be written as a simple formula that provides ancient values matching fossil measurements.

1 Materials and methods
There are three hierarchies of growth patterns on fossil corals, as illustrated in Figure 1. Each corresponds to an astro-

nomical period. (1) Daily patterns. Fine ‘ridges’ on the surface of the coral epitheca are parallel to the growing edge of the corallum (Figure 1(c)). The vertical thickness of the ridges also varies. It has been long accepted that these fine ridges, or growth lines, are an expression of the daily fluctuation of the periodicity of constrictions [1]. Among diurnal animals, activity increases in daylight. Kawaguti and Sakumoto [30] initially inferred that calcification occurs in the light but not in the dark by measuring variations in the levels of Ca2+ in seawater in which they incubated corals. Goreau [12] proved that the rate of calcium deposition is directly proportional to the light intensity. Moya et al. [15] confirmed this point with experiments on modern coral tissues. (2) Lunar patterns. The coral epitheca also shows constrictions along its length, demarcating successive groups of ridges for which the term ‘bands’ is proposed. Ditch-like constrictions divide the epitheca into several bands, each containing 27–34 diurnal ridges, indicating that the bands are lunar. Usually the constrictions are simply deep grooves around the circumference of the epitheca, but occasionally they are emphasized by a change in the thickness of the corallite, most often being of smaller diameter above than below the groove, as shown in Figure 1(a) and (b) (enlargement). Each strip in Figure 1(a) and (b) is actually a lunar band. Such bands have been interpreted as being controlled by lunar-month breeding periodicity [6]. Here the motivating factor appears to be moonlight [14], with breeding usually falling between the full moon and last quarter. (3) Annual patterns. On many fossil corals, several major ‘annulations’ in the form of swells can be distinguished, as presented in Figure 1(a) and (b) (enlargement). There are 12–14 lunar bands between neighboring annulations on all specimens. A typical lunar band on fossil coral contains 27–34 daily ridges. Annual periodicity in coral growth has long been accepted [1,6,9–12]. This is related to seasonal variation in the water temperature [11]. Knutson et al. [10] took autoradiographs and X-radiographs through the centers of reef corals from Eniwetok and found that the cyclic variations in radial density revealed by X-radiography are annual. Therefore, by taking the statistics of quantitative relationships, both the ancient number of lunar months per year and the ancient number of days per year can be obtained. Following basic principles developed by Wells [1], modern technologies of pattern recovery including the fast Fourier transform and ImagePro Phase matching are used here to overcome the abrasion of the fossils. Maximum counting on each individual [4] and consecutive counting [8] are employed to yield a higher, more accurate value. Fossils with more than two consecutive annulations are chosen and analyzed. In addition, a few bivalve and brachiopod patterns are used to provide data for early ages when coral specimens are not available. On certain species, bivalve and brachiopod patterns also showed annual patterns, fortnightly patterns and daily patterns, similar to corals. Their mechanisms

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Figure 1 Diagram of coral fossil patterns. The specimen is a typical Devonian coral specimen G-1-9 (~391.8 Ma) from Ahbach Fm, Mullertchen, Germany. (a) is an overview of the specimen, showing hundreds of clear lunar bands. (b) is an enlargement of three annual patterns from (a), containing 39 lunar bands. Three lunar bands from (b) are amplified and presented in (c), from which 94 ridges are counted. On this specimen, ridge patterns were examined under 400× magnification (c).

were already illustrated by Pannella [8] and many later works [16,18,20]. Because shell patterns are merely supplements, here we simply adopt their methods [8,16,18,20].

where TEarth is the ancient Earth’s revolution period, PMoon is the ancient Moon’s revolution period, and TEarth/PMoon = Ls. According to eqs. (1), (2) and (3) and Kepler’s third law, the ancient Earth’s revolution period is

2 Results
The best 879 specimens with clear growth patterns are taken from a total of 7062 specimens collected in China, the former USSR, Germany, Spain and the United States. Specimens constitute myriads of concrete species, including Tachylasma Grabau, Caninia Michelin, Heliophyllum Hall, Waagenophyllum Hayasaka, etc. Data are summarized in Table 1 for these well-preserved specimens, each having a number of well-developed ridges and bands like those shown in Figure 1. Data presented here are typical and representative because of the wide temporospatial distribution of the samples. The decreasing number of lunar months per year in Table 1 (plotted in Figure 2 along with the results of former studies [5–8]) confirms lengthening of the lunar period. Irrespective of whether the Earth’s revolution period is a constant or is increasing as reported [26,27], the number of lunar months per year should be constant or increasing unless the Moon is leaving. From Table 1, the ancient lunar leaving rate can be derived without the suggestion of a constant year length; setting the average ancient leaving rate as vl, the ancient Moon’s semi-major axis al can be written as al = R0 − vlt,
7

⎛ 4π 2 ⎞ 2 3 TEarth (t ) = ( Ll + 1) ⋅ ⎜ ( 38.44 − vl t ) ⋅ ⎟ . GM Earth ⎠ ⎝

1

(4)

If the Earth’s rotation is slowing at an average rate of R (ms/cy), then
PDay = 86400 − Rt ,

(5)

where PDay is the ancient day length and t is the geological age. TEarth can then be obtained as

N Day ⋅ (86400 − Rt ) ≡ N Day ⋅ PDay ≡ TEarth ,

(6)

where NDay is the number of ancient days per year, as provided

(1)

where R0 = 38.44 (×10 m) is the current value and t is the geological age. According to Lambeck [2], Ls = Ll + 1, (2) where Ls is the number of sidereal months per year and Ll is the number of lunar months per year. Therefore, we have Ls ⋅ PMoon ≡ TEarth ⋅ PMoon ≡ TEarth , PMoon (3)

Figure 2 Number of lunar months in an ancient year. Black points are data from this research while color points are data from former studies [5–8]. However, random environmental disturbances or private individual traumatic events can interrupt the continuity of the record. The influence of this factor is nearly –1% according to Pannella [8] and Wells [1], and could be counterbalanced by a correction. Such correction will shift the result curve, but leave the slope of the curve unchanged. Therefore, the result of the average leaving rate would not be affected.

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Table 1 Statistics of specimens and the results of the number of days per year and number of lunar months per year from 535 to 1 Ma Area and formation Qiongzhusi Fm, Maotianshan, Chenjiang, China (Brachiopods) Kaili Fm, Guizhou, China (Brachiopods) Manger Fm, West Junggar Basin, China Bonar Village, Leon, Spain Ahbach Fm-Nohn Fm, Eifel Mountains, Germany Lower Hongguleleng Fm, China Middle Hongguleleng Fm, China Upper Hongguleleng Fm, China Hebuke River Fm, China Liuzhou, Guangxi, China Xibeikulasi Fm,West Junggar Basin, China Wuxuan, Guangxi, China Shiqiantan Fm, East Junggar Basin, China Former USSR, from several formations in current Russian Federation Guanling Fm, Panxian, Guizhou, China (Bivalves) Wayao Fm, Xinpu, Guanling, Guizhou, China (Bivalves) Treasured in Nanjing Institute of Geology and Paleontology, CAS Tuchengzi Fm, Daohugou Beds, China (Bivalves) Yixian Fm, Liaoning, China Caloosahatchee Fm, Florida, USA. Geological period Lower Cambrian, Terreneuvian Early Middle Cambrian Devonian, Pragian Lower Devonian, Emsian Middle Devonian, Eifelian Devonian, Frasinian Devonian, Famennian Devonian, Famennian Devonian, Famennian Lower Carboniferous, Tournaisian Lower Carboniferous, Visean Lower Carboniferous, Visean Carboniferous, Moskovian Carboniferous-Ordovician Lower Triassic, Anisian Upper Triassic, Carnian Cretaceous-Silurian Upper Jurassic, Kimmeridgian Lower Cretaceous, Aptian Quaternary, Pleistocene Absolute age used (Ma) ~535 ~525 ~410 ~405 391.8–397 ~375 ~370 ~365 ~360 ~350 ~340 ~335 ~310 260–440 ~240 216–228 66–423 ~150 ~120 ~2 Sum 17 3 57 13 51 23 246 231 26 1 3 1 20 22 21 47 90 1 4 2 Average lunar months per year (Ll) 13.3±0.1 13.2±0.1 13.10±0.04 13.1±0.1 13.07±0.07 13.05±0.05 13.0±0.1 13.0±0.1 13.0±0.1 13.0±0.1 13.03±0.05 13.0±0.1 12.98±0.05 Average days per year (Nday) 436±5 435±8 407±1 405.8±0.5 406.3±0.5 406.2±0.5 405.1±0.5 404.2±0.5 401.5±1.0 399±5 394±5 392±5 392.5±0.5

(12.77–13.17)±0.06, (374–413)±1, varies varies with age with fossil age 12.70±0.05 12.65±0.05 373.0±2 371.8±4

(12.45–13.14)±0.03, (370.0–403.6)±0.5, varies with age varies with fossil age 12.6±0.1 12.52±0.05 12.4±0.1 372.5±3 380.0±2 361.8±3

in Table 1. By fitting data in Table 1 to eqs. (4) and (6), the ancient vl corresponding to different values of TEarth is presented in Table 2. Since the modern rate of the increase in the mean Earth-Sun distance has already been reported [26,27], the lunar leaving rate under a constant Earth-Sun distance, 3.01 cm/cy, should be regarded as a minimum value. According to eq. (4), if the ancient Earth-Sun distance were less, then the ancient lunar leaving rate would be higher. In addition, geological observations suggest that the average lunar leaving rate should be no higher than 5.8 cm/yr [31], while modern measurements suggest that the average rotation slowdown rate should be no higher than 3.0 ms/cy [32]. Hence, Table 2 gives a lunar leaving rate of 3–5 cm/cy. The
Table 2 Lunar leaving rates under different situations Ancient Earth-Sun distance (Semi-major axis) Kept constant If Earth’s rotation slowdown at 2.4 ms/cy If Earth’s rotation slowdown at 2.6 ms/cy Varies with time If Earth’s rotation slowdown at 2.7 ms/cy If Earth’s rotation slowdown at 2.8 ms/cy If Earth’s rotation slowdown at 3.0 ms/cy Obey Planetary Hubble Expansion (eq. (14))

middle value of 4 cm/cy just matches the modern measurement value of 3.82 cm/cy [33]. In all situations, the fitting result of 4.1 cm/cy gives the best correlation. This value can be also obtained through celestial mechanical analysis. The relationship between the lunar leaving rate and lunar distance is depicted by James and Zahnle’s formula [34], expressed as (7) v=v0(d/d0)–11/2, where d0 is the modern Earth-Moon distance, v0 denotes the modern leaving rate, d is the ancient Earth-Moon distance, and v represents the ancient leaving rate. Because of the Moon’s recession, the ancient distance was less, and therefore, the ancient leaving rate should be higher. Iterative calculation of eq. (7) gives an average value of 4.1–4.2 cm/yr

Lunar leaving rate (cm/yr) (G is constant) 3.01 3.05 3.70 4.03 4.36 5.02 4.16

Lunar leaving rate (cm/yr) (G decreases 10−11/yr [35]) 3.16 3.81 4.14 4.47 5.15

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for the past 0.53 billion years. Coincidentally, later we will find this value is the result of a simple theoretical calculation using eq. (14). Considering all the above, the most credible range of the ancient lunar leaving rate is 3.8–4.2 cm/yr. It seems that the current measured value, 3.82 cm/yr [33], appears reasonable. According to the measurement data, the laser-measured 3.82 cm/yr corresponds to an ancient average slowdown rate of the Earth’s rotation of 2.6 ms/cy. The evidence here again seems to confirm the Earth’s slowdown rate as several milliseconds per century as extrapolated from modern observations [32]. In fact, it is acceptable even for the ancient average rate to have been slightly higher because ancient tidal friction should be stronger with a shorter Earth-Moon distance. Therefore, without suggesting a constant year length, modern observations of both the Moon’s retreat and the Earth’s slowdown are confirmed. Indeed, it is shown in Table 2 that the most reasonable leaving rate corresponds to a variable year length. As the year length reflects the Earth-Sun distance, the actual ancient Earth-Sun distances can be determined from the pattern data obtained. The precise ancient number of days and lunar months per year in each geological period, the ancient Earth’s rotation slowdown rate and the ancient lunar leaving rate are now obtained. To be prudential, in the following analysis, the lower bound of 3.8–4.2 cm/yr, as well as the laser-measured 3.82 cm/yr, is used as the average leaving rate (One can also use 4 or 4.2 cm/yr, and even 3 or 5 cm/yr, for which the results below do not change significantly).

geological time is plotted in Figure 3. Precambrian tidal rhythmite data at 650 Ma from Williams [17] are also used. The Earth was closer to the Sun over the past 0.53 billion years. In addition, the semi-major axis of the Earth was 146 million km at the beginning of the Phanerozoic Eon (535 Ma). The physical meaning and mathematical expression of the curve will be given in the discussion section. 3.2 Ancient Earth-Sun distances derived from daily pattern data An alternative approach employing a different method and different part of the dataset can also provide another set of ancient Earth-Sun distances. Using eqs. (6) and (8) and taking R = 2.6 ms/cy (26 s/Ma), another description of the ancient revolution orbit of the Earth is derived as
2 GM ⎛ Sun ⎞ r = ⎜ ( N day ⋅ (86400 − 26t ) ) ⋅ ⎟ 4π 2 ⎠ ⎝ 1/ 3

.

(10)

3 Analysis
3.1 Ancient Earth-Sun distances derived from lunar pattern data By incorporating the number of ancient lunar months per year from Table 1 into eq. (4), we can calculate the ancient year length. Using Kepler’s third law, the ancient year length can be then transformed into the Earth’s semi-major axis r:

As daily pattern data providing Nday, the relationship between the ancient Earth-Sun distance and geological time is plotted in Figure 4 (Data for 200–300 Ma are not included in the second curve because the suggestion of a linear slowdown is not correct during this period, as reported by Pannella [8], probably owing to a series of dramatic changes beginning in the Permian [36]). Supplementary data from former studies [1,6,7,9–12] are also used. The second fitting curve shown in Figure 4 is very similar to the first. Although the methods and patterns used are different (but the specimens are the same), both curves 1 and 2 describe the same nature of the Earth-Sun relationship; i.e. the ancient Earth was closer to the Sun. Even though the Earth’s semi-major axis gradually increased, the rate of increase decreased, indicating a certain deceleration. The underlying dynamics are described in the discussion section. Daily patterns are more sensitive to abrasion than lunar patterns, for daily ridges are much smaller.

GM ⎞ ⎛ r = ⎜ TEarth 2 ⋅ 2 ⎟ 4π ⎠ ⎝

1/ 3

.

(8)

Therefore, by merging eq. (4) with eq. (8) and setting the Moon’s average leaving rate as 3.8 cm/yr (0.0038×107 m/Ma), the ancient r can be obtained as

⎛ ⎞ ⎛ ( 38.44 − 0.0038t )3 ⎞ ⎜ ( Ll + 1)2 ⋅ ⎜ ⎟ ⋅ M Sun ⎟ r= ⎜ ⎟ ⎜ ⎟ M Earth ⎝ ⎠ ⎝ ⎠

1/ 3

,

(9)

where t has units of millions of years (Ma). Both the Earth and Sun masses are known: MEarth = 5.97×1024 kg and MSun = 1.99×1030 kg. As the lunar pattern data providing Ll, the relationship between the ancient Earth-Sun distance and

Figure 3 Fitting curve 1 (relationship between the ancient Earth-Sun distance and geological time) obtained from the lunar pattern data (R2 = 0.9, fitted by OriginPro 8).

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Figure 4 Fitting curve 2 (relationship between the ancient Earth-Sun distance and geological time) obtained from daily pattern data (R2 = 0.5, fitted by OriginPro 8).

The original leaving rate in the Early Cambrian (535 Ma) can be determined from Figures 3 and 4, providing a mean value of 5×10–7 m/s. Similarly, the average leaving rate over the past 0.53 billion years and the recent leaving rate are 2×10–7 m/s and 3×10–9 m/s (approximately 9 m/cy: 5 m/cy for Figure 3 and 14 m/cy for Figure 4), respectively. In short, fossil measurements provide a modern leaving rate of 5–14 m/cy, whereas ancient rates were much higher.

data indicate that the variation in the coefficient k is less than 5%. Hubble drag is an inertial force like the centrifugal force. Because proper velocity is measured in a non-inertial expanding system, an additional force, known as Hubble drag, should be added [38]. It seems that such an increase in the Earth-Sun distance may be a special Hubble expansion with certain deceleration that is in direct proportion to the Hubble drag. Note that Hubble drag is not a true force but an inertial effect similar to the centrifugal force, which should be counterbalanced by a certain force to provide a deceleration. Because the centrifugal force of the Earth’s revolution is counterbalanced by gravity, some readers might suggest that such a Hubble drag effect is also counterbalanced by gravity. Although such an explanation is reasonable, we do not wish to make any suggestion in this paper. Because we wish to present a purely experimental result, we make only a mathematical simplification by introducing a fictional velocity vF caused by such deceleration, which in fact cannot be observed solemnly. What can be observed is the net velocity v, which is the combination of Hubble expansion (Hr) and such deceleration-caused velocity (vF). However, neither the specific form of vF nor its origin is considered. Considering eq. (12), we have the mathematical expression for such a strange expansion as
2 ⎧ ( Hr − vF ) dv ⎪a = F = K , ⎪ dt r ⎨ dr ⎪ ⎪v = dt = vH − vF = Hr − vF . ⎩

4 Discussion
Krasinsky and Brumberg [27] have already excluded tidal friction and other factors from the explanation of the modern leaving rate, for their effects are insufficient. However, it is noted that the experimental data above have some interesting features. (1) Both Figures 3 and 4 indicate an expansion coefficient h that is close to the Hubble constant:

(13)

Bearing in mind the inverse relationship between the Hubble constant and the age of the universe, eq. (13) can be transformed into a special Riccati equation with an analytic solution written as r = R(n)t K +1 ⋅ [cos( K (ln t − ln C ))] K +1 ,
1 1

h=

v = (41 ± 5)(km/s)/Mpc ≈ 0.57 H 0 , r

(11)

(14)

where v is the ancient average leaving rate, r denotes the Earth-Sun distance, and H0 is the Hubble constant ((70.8±1.6) (km/s)/Mpc) [37]. (2) Both fitting curves above indicate deceleration-a drag in the reverse direction. When curves (3) and (4) are divided into 10 sections, an interesting relationship emerges: from 535 Ma to recent times, in each section, the deceleration is in direct proportion with v2/r. In addition, if we want to match the deceleration and v2, an Earth-Sun distance must be added to the denominator to compensate for the difference of two orders of magnitude. Thus, such deceleration is in direct proportion to the Hubble drag [38]. On average, Δv v2 (12) = 2.94 × 10 −23 m/s 2 ≈ k , areverse = Δt r where k is a coefficient. Among all sections, experimental

where t is the age of the universe (currently, t is 13.70 billion years [39]) and K, R(n) and lnC are constants. The curve shape is determined by K. Although eq. (12) gave K = 110, the shapes of the curves in Figures 3 and 4 correspond to K = 116 and K = 90, respectively. However, the initial rate of increase at 535 Ma of 5×10–7 m/s used in eq. (12) is not highly precise. Therefore, by taking the average of Figures 3 and 4, K = 103 is obtained as a reasonable value. Because 0.53 billion years is only a short interval compared with the ages of the universe and solar system, eq. (14) can simply use the average value of K. To determine the variation over an even longer time span, further study of K is required. Boundary conditions were used to determine R(n) and lnC. For the current Earth, r is 1.496×1011 m [28]. For 530 Ma, r is selected as 1.46×1011 m from Figures 3 and 4. Incorporating these values into eq. (14), we obtain R(n) =

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1.02×1011 and lnC = –0.142. This equation is plotted in Figure 5, and the coincidence with fossil measurement results speaks for itself. Note that the theoretical result excellently matches with fossil measurement results, but does not need any extra information. As presented in the figures, the deceleration reshaped such expansion. The Earth is gradually approaching a stable point. We now retrospectively investigate the lunar recession discussed previously. Whether such an expansion can apply to the Moon remains unknown. Even if it could, the lunar recession can still be regarded as a linear process; astrono-

mers agree that the Earth-Moon system formed about 3.9 billion years ago or later [40], and therefore, the situation of the Moon was like that of the Earth at 600 Ma. The Moon would be still far from a stable point. According to the curves presented above, the leaving rate at 600 Ma had not been significantly affected by the deceleration, corresponding to approximately 0.8H0. Therefore, we can estimate the contribution of such expansion to lunar recession, if it exists, as ~1.6 cm/yr. Van Flandern [41] reported that tidal friction contributes only half of the lunar recession. We ask if the other half is contributed by such expansion. However, this is only one of many possibilities.

Figure 5 Comparison of the ancient Earth-Sun distances derived from the theoretical equation (a) and measurement data ((b), (c)). (b) is from Figure 1 while (c) is from Figure 2.

5

Conclusions

In this paper, growth patterns on fossils are used to study ancient Earth-Sun distances. The biological clock measurement of a large number of well-preserved specimens and new technology confirmed the ancient Earth’s rotation slowdown rate (2.6 ms/cy on average) and the ancient lunar recession rate (3–5 cm/yr), without suggesting a constant year length. The most credible range was 3.8–4.2 cm/yr. Two different fitting curves of the relationship between the ancient Earth-Sun distance and geological time were derived from different kinds of pattern data. They jointly and strongly suggest that the ancient Earth was closer to the Sun. Furthermore, the variation in the Earth’s leaving rate indicates deceleration, which made such expansion totally different from the Hubble expansion of galaxies. According to fossil measurements, the Earth’s current leaving rate is 5–14 m/cy, which is much smaller than the average rate over the past 0.53 billion years. No study of the ancient Earth leaving rate has yet been reported (because of the limitations of traditional methods). The leaving rate calculated in this study fitted exactly with recent discoveries [26–28]. Therefore, geology and astronomy are again in

agreement. On the basis of the relationship between the ancient Earth-Sun distance and geological time, which is derived from experimental data, the average leaving constant (~0.57H0), and the deceleration (having the form of Hubble drag), a new planetary orbit model was established. Experimental results and mathematical calculations suggest that the semi-major axis of the Earth’s revolution can be simply written as r = R (n)t K +1 ⋅ [cos( K (ln t − ln C ))] K +1 , from which each ancient point matches fossil measurements. Substantive geological materials provide important records and boundary conditions for astronomy. A detailed evolution history of the Earth can be studied, given such a large database. This article presents a preliminary study and should be regarded as an experimental report instead of a final conclusion. Further studies may either focus on dynamics or a search for new materials, such as tidal rhythmites or stromatolites. Although the dynamics of these materials are still being debated, they could provide an even longer time span for testing.
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We wish to thank Professor Sun Yuanlin (Peking University) and Professor Liao Weihua (Chinese Academy of Sciences) for their support. We gratefully acknowledge Professor Krasinsky (Russian Academy of Sciences) for his help and advice. Professor Iorio (INFN-Sezione di Pisa, Italy) sent us important references. Materials used in this work are preserved at the Research Institute of Paleontology (Peking University) and Nanjing Institute of Geology and Palaeontology, CAS. This work was supported by the National Basic Research Program of China (2010CB328201), the National High Technology Research and Development Program of China (2006AA12Z310) and the National Natural Science Foundation of China (60772003). 1 2 3 4 5 6 7 8 9 Wells J W. Coral growth and geochronometry. Nature, 1963, 197: 948–950 Lambeck K. The Earth’s Variable Rotation: Geophysical Causes and Consequences. New York: Cambridge University Press, 1979. 449 Runcorn S K. Changes in the Earth’s moment of inertia. Nature, 1964, 204: 823–825 Mazzullo S. Length of the year during the Silurian and Devonian Periods: New values. Geol Soc Am Bull, 1971, 82: 1085–1086 Johnson G A L, Nudds J R. Growth Rhythms and the History of the Earth’s Rotation. London: John Wiley, 1975. 27–41 Scrutton C T. Periodicity in Devonian coral growth. Palaeontology, 1964, 7: 552–558 Berry W B N, Barker R M. Fossil bivalve shells indicate longer month and year in Cretaceous than present. Nature, 1968, 217: 938 Pannella G. Paleontological evidence on the Earth rotational history since Early Cambrian. Science, 1972, 16: 212–237 Scrutton C T. Periodic growth features in fossil organisms and the length of the day and month. In: Tidal Friction and the Earth’s Rotation. Berlin: Springer-Verlag, 1978. 154–196 Knutson D W, Buddemeier R W, Smith S V. Coral chronometers: Seasonal growth bands in reef corals. Science, 1972, 177: 270–272 Ma T Y H. On the seasonal change of growth in some Palaeozoic corals. Proc Imp Akad Tokyo, 1933, 9: 407–409 Goreau T F. The physiology of skeleton formation in corals. I. A method for measuring the rate of calcium deposition by corals under different conditions. Biol Bull Mar Biol Lab, 1959, 116: 59–75 Al-Horani F A, Tambutté É, Allemand D. Dark calcification and the daily rhythm of calcification in the scleractinian coral, Galaxea fascicularis. Coral Reefs, 2007, 26: 531–538 Levy L, Appelbaum W, Leggat Y, et al. Light-responsive cryptochromes from a simple multicellular animal, the coral Acropora millepora. Science, 2007, 318: 467–470 Moya A, Tambutté S, Bertucci A, et al. Study of calcification during a daily cycle of the coral Stylophora pistillata: Implications for ‘light-enhanced calcification’. J Experiment Bio, 2006, 209: 3413– 3419 Thomas G, Mehmet S. Growth dynamics of red abalone shell: A biomimetic model. Mater Sci Eng C, 2000, 11: 145–153 Williams G E. Late Precambrian tidal rhythmites in South Australia and the history of the Earth’s rotation. J Geol Soc London, 1989, 146: 97–111 Richardson C A, Peharda M, Kennedy H, et al. Age, growth rate and season of recruitment of Pinna nobillis (L) in the Croatian Adriatic determined from Mg:Ca and Sr:Ca shell profiles. J Exp Mar Biol Ecol, 2004, 299: 1–16 Qu Y G, Xie G W, Gong Y M. Relationship between Earth-SunMoon 1000 Ma ago: Evidence from the stromatolites. Chinese Sci

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Supporting information (Appendix)

The supporting information is available online at csb.scichina.com and www.springerlink.com. The responsibility for scientific accuracy and content remains entirely with the authors.

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Supporting information (Appendix)
Appendix I—Detailed methods and measurements Data collection is restricted by the fact that the growth periodicity only exists on the epithecal surface. Therefore, research data have been derived from fossil growth increment records by counting features on specimens and in photographs. In this study, specimens were captured and examined with a scanning electron microscope, EMLPKU, or stereoscopic zoom microscope, SMZ1500. Distortion was reduced and chromatic aberration corrected to a high degree, resulting in natural-looking stereoscopic images. The counting involved either the direct recording of increments between successive higher order features or the continuous counting of increments over a number of cycles with the mean number of increments per cycle calculated by division, depending on the specimen’s condition. The 879 specimens selected from a total of 7000 have clearly developed growth patterns on the epitheca and are among the most complex. Compared with former studies, we improved the manual counting and largely employed modern techniques, such as the use of an ImagePro Plus (IPP) processing facility and fast Fourier transform (FFT) techniques.

1 Principles
(1) Maximum count for each individual. It must be emphasized that the average count for each individual is subject to two factors. First, under adverse conditions during the life of the organism involved, only very limited secretion may occur, and second, the resulting growth lines may be obscure. Hence, there is inherent disagreement between the observed number of daily growth increments contained within an annual zone on a particular fossil and the actual number of days per year in the period during which the organism lived. If used, the average count method would yield a minimum value for the number of days per month or year, while the maximum count method would overcome this systematic error and yield a higher, more accurate value. (2) Mean-value calculation for all individuals. The maximum count for an individual could overcome the error caused by fossil abrasion while the mean-value calculation for all individuals could overcome the error in counting. (3) Consecutive counting. A factor that affects the precision of the data is the difficulty of determining where to start and finish counting patterns that usually merge into each other. This difficulty is partly eliminated using only consecutive counts (at least three or four). It must be emphasized that the longer the sequence of consecutive counts, the more reliable the data (derived by dividing the total number of lines of the sequence by the number of periodical patterns). In this way, the subjective decision of the limits of the patterns is reduced to the two extreme patterns of the series. Fossils with more than four consecutive annulations were chosen and analyzed.

2 General procedures and pattern information recovery technologies
We attempted to improve measurements in the following waves. (1) Each lunar band was analyzed under the scanning electron microscope EMLPKU or stereoscopic zoom microscope SMZ1500 and pictures were taken with sufficient amplification to exactly distinguish each diurnal ridge. (2) ImagePro Plus (IPP) processing facility from Germany was used to eliminate abrasion and recover the specimen information, and the FFT could eliminate abrasion frequencies and recover ridge patterns. First, a photograph of patterns was converted into a monochrome figure. Second, the FFT extracted a detailed frequency distribution from this monochrome figure. Because the frequency of growth lines was centralized, at the same magnitude, the noise frequency which frets growth lines can be determined by trial and error. The researcher simply deleted the suspected area of noise using the IPP, and then applies an inverse FFT to see whether the growth lines were recovered without abrasion. Finally, the noise that distorted the pattern most significantly was removed, leaving the growth pattern clear again, like it was millions of years ago (see Figure S1). The importance of the FFT could be easily understood in the following analysis. In most cases, a major problem in the interpretation is how to deal with some seemingly wide ridges (see Figure S2). Poor preservation results in abrades peaks of adjacent ridges, making them look like an integrated whole. However, variations in the climate and solar radiation would also produce extra-saturated ridges. In addition, seasonal changes in the ridge width are also significant (see Figure S3(c)). Therefore, the numbers of ridges in those situations are different.

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Figure S1 Bands before (a) and after (b) IPP-FFT processing. Ridges were recovered and became discernable again in the right band. The band is from specimen Hgll-8-9 (the fifth band).

Figure S2 This example of a lunar band under a stereoscopic microscope (80×) helps clarify our method of counting. While scanning from the left border to the right border, the researcher comes across ridges that seem to be too wide, which is a location of wearing. By checking the surrounding area, we can find the clearest part of the same ridge and continue counting from there. Only the number obtained from the clearest areas can be recorded. Specimen name and depositary: DZ-30-16.

To identify the actual number of ridges, biochemical measurements of recent corals in Guangzhou and a physical calculation were made to model clearly the process of daily calcium deposition. The calcification equation is
2 photons Ca 2 + + 2HCO3− ←⎯⎯⎯ CaCO3 + H 2 CO3 . →

(S1)

To determine the impact of the light intensity on the saturating rate, calcification rates were measured under different light intensities at Sun Yat-sen University, China. The effect of light intensity on the calcification rates of Stylophora pistillata micro-colonies were presented in Figure S3(a). Data suggested that the relation between the light intensity and calcification rate could be written as y=0.00667x2+0.433x+60, (S2) where y is the calcification rate and x is the light intensity. On the other hand, in the present study, the daily solar radiation variation was also measured at Sun Yat-sen University, and the results are presented in Figure S3(b). By combining the results, a relation between the time and daily calcification was derived using eq. (S2), as plotted in Figure S3(c). Note that this was not the final pattern on coral epitheca because the daily traverse growth rate also depends on the light level and temperature. According to the study of Al-Horani et al. [13] on Galaxea fascicularis, the coral’s traverse growth rate in light is almost twice that under dark conditions. Hence, the final shape should be plotted as shown in Figure S3(d). After applying the FFT, some wide ridges transform into two narrow ridges with the above theoretical shape. (3) It is possible, even for very fine fossils, to be partly worn. However, ridge information could not be totally lost. Therefore, one should search for the best preserved ridge (usually the most closely grained) in each lunar band and have the precognition of this band or specimen. Then, according to the maximum counting rule, each potential ridge should be marked

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Figure·S3 (a) Effect of light intensity on the calcification rates of Stylophora pistillata micro-colonies; (b) daily solar radiation variation measured at Sun Yat-sen University, China; (c) relation between time and daily calcification rate derived from y = 0.00667x2 + 0.433x + 60; (d) daily ridge model derived from analysis of coral light-enhanced calcification.

without overlapping. The number of ridges in each band was counted at approximately where the maximum count could be obtained. There should be no wide spaces left blank because ridges in the same band have similar width and are consecutive. Therefore, while counting the ridges, we should search for residuals of another ridge. Simply counting in a direct line may lose information. (4) ImageProPlus phase matching technique was used. All authorities agree that the skeletal density banding, whatever its cause, is an annual rhythm. However, a problem remains unsolved: the difficulty of determining where to start and end the counting of an annual pattern. Here, by means of modern laboratory techniques and computer simulations, an ideal coral shape in a tropical area was calculated and found to show clear annual annulation without uncertainty. Therefore, by phase matching technique, we identified where corresponds to the start and the end. Figures S4(a) and (b) show the sea temperature variation in tropical oceans (the monthly tropical sea temperature variation obtained from the Japanese Oceanic Information Centre, Naha (26°12′N) and Ishigaki-jima (24°20′N)) and its relationship with coral skeletal growth. In addition, by fitting data from Zhang and Yu [42], a relationship between mean surface temperature (MST) and coral skeletal growth was determined (see Figure S4(c)):

Growth rate = 2.97×MST–64.9.

(S3)

In addition, the radius of the coral (if like a cylinder) also changes according to seasonal variations. We calculated the functional relation between the coral’s growth density and MST from the ORIGIN fitting (see Figure S4(d)): Density = 4.4–0.12×MST.
2

(S4)

We simply suggested a secretion constant as 1, according to V=m/ρ=πr h, where V is the volume, m is the secreted mass, h is the width of lunar bands, and r is the coral radius. We then have (Figure S4(f))

Coral radius r =

1/ρ = hπ

1/(4.4 − 0.12MST) . π(2.97MST − 64.9)

(S5)

Our predicted annulation was somewhat protrudent. In fact, because of a lack of nutrition in winter, the secretion constant would be smaller than 1 in winter, resulting in less annulation than calculated and a better fit to the observations. However,

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Figure S4 Theoretical widths of lunar bands and coral radii based on calculations made. (a) Tropical sea temperature monthly variation from Japanese Oceanic Information Centre, Naha (26°12′N) and Ishigaki-jima (24°20′N); (b) relationship between MST and coral skeletal growth; (c) lunar band width model; (d) recorded coral density under different MST from 29 reefs of the Great Barrier Reef, data averaged across colonies from 14 Hawaiian Archipelago reefs, data averaged across colonies from a reef at Phuket, Thailand [42]; (e) calcification rates in the light, the dark and for dead colonies of Galaxea fascicularis from Al-Horani et al. [13]; (f) coral lunar radius model.

the annulation was already sufficient for deciding where to start or finish counting in the IPP phase matching method. This method compared fossils with a theoretical shape, disregarded noise, and was better than simple human estimation. (5) The processing was checked again by another researcher and corrected as necessary. (6) After the recording of data, we examined the results with the measurement of increment thickness; i.e. our counting result should be close to the approximation derived by the thickness measurement. We choose several clear ridges from the same band of the same specimen and measure their width. We then measure the band’s width. Therefore, the approximation is the width of the band divided by the mean width of the ridges. (7) Finally, the ImagePro Plus graphic processing facility was used to check whether or not the result was in the confidence interval. IPP can provide frequency information of a specimen. Each peak reflects a height change on the epitheca. Systematic error Two factors, however, can interrupt the continuity of the record: one is random environmental disturbances or private individual traumatic events and the other is the preservation of fossils. The influence of the first factor, as discussed above, is nearly 1% according to Pannella [8] and Wells [1], and could be counterbalanced with a correction. Such a correction will shift the result curve, leaving the shape of the curve unchanged. In our study, the result would be the same. Another factor that affects the precision of the data is not intrinsically related to a faulty record but to our faulty interpretation of ambiguous patterns, or to the difficulty of determining where to start and end counting patterns that usually merge into

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each other. This deviation is significantly reduced by (1) the IPP phase matching technique and modeling of coral patterns and (2) using only consecutive counts (at least three or four annual patterns).

Appendix II—Solution to the Riccati equation for the Earth-Sun distance Equations (13) can be transformed into a special Riccati equation: d⎛ dr ⎞ K ⎛ dr ⎞ ⎜ Hr − dt ⎟ = r ⎜ dt ⎟ , dt ⎝ ⎠ ⎝ ⎠
2

(S6)

which can be rewritten as
H dr dH d 2 r 1 dr +r − = K ( )2 . dt dt dt 2 r dt

(S7)

Setting z=lnr, we obtain d2 z dz dz dH =H − ( K + 1)( ) 2 + . dt dt dt dt 2

(S8)

Then setting y=dz/dt, the equation transforms into

dy dH = Hy − ( K + 1) y 2 + . dt dt
According to the inverse relation between H and t,
H= A , t

(S9)

(S10) (S11)

∴

dy A A = y − ( K + 1) y 2 − 2 . dt t t

This is a special Riccati equation that has an analytic solution. Changing the independent variable y1=(K+1)y, (S11) transforms into

dy1 A2 ( K + 1) A A = − − ( y1 − ) 2 . dt 4t 2 2t t2
Then setting y2=y1−A/2t, we obtain dy 2 A2 − (4 K + 2) A 2 + y2 = . dt 4t 2

(S12)

(S13)

If b =

1 2 2K + 1 A − A , we obtain a standard Riccati equation: 4 2 dy 1 + y2 = b 2 . dx x
(S14)

Setting y3=y2t, (S14) transforms into
2 dy3 b + y3 − y3 = , dt t

(S15)

where
1 1 2K + 1 1 1 1 2 b + y3 − y3 = −( y3 − ) 2 + A2 − A + = −( y3 − ) 2 + [ A2 − (4 K + 2) A + 1]. 2 4 2 4 2 4

(S16)

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NASA summarized existing data to obtain a constant of (70.8±1.6) (km/s)/Mpc if space is assumed to be flat, or (70.8±4.0) (km/s)/Mpc if otherwise [37]. According to a recent Wilkinson Microwave Anisotropy Probe measurement, the age of the universe is (13.70±0.05) billion years [39]. If A=1, then from A2−(4K+2)A+1 1), Ω0 2 1 | 1 − Ω0 | 2 cos s −1 ( − 1) + , cos s = ⎨ 2(Ω0 − 1) 1 − Ω0 Ω0 ⎩cosh , (Ω0 < 1).

(S19)

Two situations are considered. (1) When Ω0>1, because H0t0=1 and we expect W0≠0. Therefore
1 2 Ω0 − 1 arc cos( 2

Ω0

− 1) = 1 ⇔

2

Ω0

− 1 = cos(2 Ω0 − 1).

(S20)

There are infinite positive roots in the range Ω0>1. (2) When Ω0