r_j = (M_{j rev} / 2*3.14 m_j )^1/2(T_j)^1/2                                          (1)

where r_j and T_j  - an orbital radius and period, M_{j rev} - moment of revolving momentum. The equation can be easily derived from the law of conservation of moments of revolving momentum.

The law of conservation of moments of angular momentum (M_jang) creates a similar picture for the rotations of planets and the Sun around their central axes. The form of celestial bodies, including the solar and terrestrial ellipsoids, has been transformed by the gravity of other planets and that automatically changes the period (T_j) or the frequency (F_j) of rotation, the radii (R_j) of ellipsoids, or amplitudes (A_j ) of the variations:

R_j = (5M_{j ang} / 2*3.14² m_j )^1/2(T_j)^1/2                                                            (2)

Comparison of equations ( 1 and 2) has shown that we have had specific oscillations, which display the following relationship between Aj and Tj (Berry, 1991):

A_j²F_j = c_j ,                 A_j = (c_j T_j)^1/2,           or         A_j = s (T_j)^1/2,                       (3)

where s = (c_j)^1/2, c_ang=5M_{j ang}/2*3.14² m _j , c _rev =M _{j rev} / 2*3.14 m _j . Periods  T_j  and amplitudes A_j are variable, but s  is constant for homogeneous series of natural cycles.         These spectra (3) relate to basic physical laws and have detected in all processes, for example, in various natural and technical processes: fluctuation noise, flicker effect in radio systems, variations of biologic, climatic, geologic and other processes. In the most cases these vibrations are investigated as accidental processes with specific amplitude-frequency spectrum. It is important that the coefficient s be used to classify them (Berry, 1998).

            We will check the existence of equation (3), obtained for dendrochronological periods under 230y, for much longer time intervals. If we extrapolate the series 1 - 3 (Table) up to values A_j /a = 1, when the

Coefficients ( S _DR) for series of harmonic components (4 - 230y) of the dendrochronological row (DR) of larches:

amplitudes of variations in tree rings become equal to the average value of annual growth and, therefore, correspond to the death of all trees, we shall be able to approximate the critical periods providing for global transitions of forest limits. For the series 1-3, these time intervals have the duration of 2.3, 5.4, and 13 ka. The results do not contradict data from paleogeographical studies.

            Let’s calculate the amplitude of the 100 ka period for series 3. This is the main cold period of the Pleistocene (Pl). To preserve the physical sense of this extrapolation, we will use the amplitudes of air temperatures in the Northern Hemisphere, which correlate with the amplitudes of the dendrochronological row. For this case S_DR3 = 0.0061, °C/y^1/2. In accordance with formula (3), the Northern Hemisphere temperature can be changed to

                                                A_Pl = 1.93°C                                                               (4)

and that corresponds to Imbries’ (1988) estimation (2 °C).

            Using the same series 3, we will try to assess the age of the Earth’s atmosphere  (T_A ). We will find the period that is associated with the average variation of the Earth’s surface temperature about 300°C. This period is equal to:

                                                T_A = 2.3 Ga,                                                                (5)

i.e., this time almost coincides with the time of the appearance of the first known living things. There are evidences of bacteria about 3.5Ga ago.

  References:

Berry, B.L. 1991. “Variations and interrelation between helio-geophysical characteristics”. In: Glaciers-Ocean-Atmosphere Interactions. IAHS n.208, 385-394.

Berry, B.L., 1998. Regularities of natural cycles, prediction of climate and surface conditions. Hydrol. Process. 12, 2267-2278.