O n t h e T h e r m a l H i s t o r y o f t h e E a r t h (*> B . J . L E V I N a n d S . V . M A J E V A ( * * ) R i c e v u t o il 7 dicembre 1960 The thermal history of the E a r t h has been investigated during last decade in the Otto Schmidt Institute of Physics of the E a r t h . In the analogous investigations of other authors (Jacobs and Allan, MacDonald) both the cold and the hot initial state of the E a r t h were admitted. In our Institute all calculations were based on the theory of the origin and evolution of the E a r t h developed from Prof. Schmidt's ideas. All cal- culations were made for the initially cold E a r t h heated by radioactivity. At first the thermal history of the uniform E a r t h model taking into account t h e previously neglected time-dependence of radioactive heat production was calculated by E . A. Lubimova (1952). I t was shown t h a t the mean content of radioactive elements in meteorites is sufficient for the explanation of the present incandescent state of the E a r t h ' s interior. In subsequent calculations (Lubimova, 1955-58) the non-uniform distri- bution of radioactive elements in the E a r t h a t the present time as well as the dependence of heat transport on temperature and pressure were taken into account. In the present paper the results of some new calculations are discus- sed. Their main purpose was: a) to study the thermal history of the outer parts of the E a r t h assuming a longenduring (not instantaneous) formation of the crust resulting from the differentiation of the mantle, and b) to compare the calculated heat flows for the continental and oceanic parts of the E a r t h . We wished also to clarify if it is possible to fit the observed heat flow with the smaller content of radioactive elements in the E a r t h then adopted in previous calculations. Our results were already in press when a paper by MacDonald (1959) was published. He had calculated a thermal history for a great series (*) P a p e r r e a d a t t h e Helsinky Assembly of t h e I . U . G . G . , 1960. (**) 0 . J . S c h m i d t I n s t i t u t e of P h y s i c s of t h e E a r t h , Moscow, U S S R Acad. Sci. 1 4 6 B . J . L E V I N - S . V . M A J E V A of models. B u t in the time-dependent solutions only t h e distribution of temperature in the mantle is calculated. The core is supposed to be thermally insulated and t h e existance of t h e crust enriched b y radioac- tive elements is not taken into account. Nevertheless MacDonald's p a p e r gives an independent confirmation to some conclusions obtained earlier by Lubimova. H, Cal/g.szc. Fig. 1 - H e a t o u t p u t f r o m t h e c r u s t (a) a n d f r o m t h e d e p l e t e d p a r t of t h e m a n t l e (b). - n o n - d e p l e t e d p a r t of t h e m a n t l e a n d core. depleted p a r t of t h e m a n t l e Our calculations were made for t h e following E a r t h ' s model: F o r t h e dense core of t h e E a r t h Ramsey's hypothesis is adopted, a n d therefore t h e core is supposed to exist f r o m t h e very beginning of t h e time interval for which the t h e r m a l calculations were made. I n accordance with t h e formation of t h e E a r t h by accumulation of cold particles t h e initial distri- bution of radioactive elements is adopted t o be uniform (per unit of mass) throughout t h e whole E a r t h . The E a r t h began to accumulate 5 x l 0 9 O N T I I E T H E R M A L H I S T O R Y OF T H E E A R T H 151 years ago. F r o m t h i s moment is measured the time t used in t h e calcul- ations. The real formation process of the crust was probably the gradual growth of its thickness. For the sake of simplicity of calculations this was replaced b y t h e gradual growth of the concentration of radioactive elements in t h e crust of constant thickness. For the continental parts of the E a r t h the thickness of t h e crust d = 30 km was adopted and for the oceanic crust d = 10 km The concentration of radioactive elements in the crust began to increase a t t — 2 X109 i.e. 3 billions years ago. This occured as a result of the uniform depletion of these elements in t h e upper p a r t of the m a n t l e down ot t h e depth of 1000 k m . F o r t h e crust the fol- lowing formula for the time-dependence of heat generation was adopted H(t) = ( l + k - ) S fl, (0) [1] As previously, the chemical composition of the E a r t h was supposed to be the same as t h e true mean composition of the meteoritic m a t t e r The mean content of radioactive elements in the meteoritic m a t t e r , obtained by A. Starkova in 1955 m u s t be seriously corrected because the new analyses of meteorites give as a rule much smaller content of uranium and thorium. Therefore the calculations were made not only for the mean content obtained by Starkova and used in the last paper by Lubimova (variant C, see table), b u t also for the variant Ci/2 in which the potassium content is not changed b u t the content of uranium and thorium is two times smaller. E l e m e n t variant C variant C'i/2 U 5 . 2 - 1 0 8 2 - 6 . 1 0 ' 8 Th 21-10"8 10-5.10"8 K 0.7-10" 3 0-7.10" 3 The results by J . E . Starik and M. Shatz (1956) who obtained a very great content of uranium and thorium in meteorites, inspired us to make some calculations for the variant C2 in which t h e content of U and Th is two times greater. B u t this variant leads to excessively great values of t h e temperature in the E a r t h ' s interior and of the heat flow through the surface. The present heat generation in the crust was adopted to be equal to the mean generation in 1 p a r t (by volume) of granit and 2 parts of 146 B . J . L E V I N - S . V . M A J E V A basalt. According to the d a t a b y Birch (1954) this h e a t generation is 9,7 X 1 0 1 4 cal/g sec or 30,6 times greater t h a n t h e generation in t h e non- differentiated E a r t h ' s m a t t e r according to variant C. Therefore in for- m u l a f l ] we obtain A-=29,6. The present generation of heat in the depleted p a r t of the mantle follows to be 6,3 X 1 0 1 6 cal/g sec, the same as in dunite Fig. 2 - T o t a l c o n d u c t i v i t y versus d e p t h X (T, p). C, d = 30, e = 200, c = 0,2. according to Birch. F o r t h e v a r i a n t Oi/2 the same value of k gives t h e heat generation in the crust to be 6,7 X HI"11 and in t h e depleted p a r t of the mantle 3,5 X 10 lfi. The initial radial distribution of t h e tempera- ture a t t h e time of the practical end of t h e E a r t h ' s growth was t a k e n according to Y. S. Safronov (1959) for t h e case of the duration of t h e growth equal to 0,23 X 109 years. The subsequent changes of the tempe- r a t u r e (after t = 0,23 X 10") were calculated on t h e hydraulic integrating machine constructed b y V. S. L u k y a n o v (1939). O N T I I E T H E R M A L H I S T O R Y O F T H E E A R T H 151 I n t h e core t h e metallic h e a t c o n d u c t i v i t y X — 0,5 cal/cm sec degree i n d e p e n d e n t on t e m p e r a t u r e and pressure was a d o p t e d . I n t h e m a n t l e and crust t h e lattice a n d r a d i a t i v e conductivitis were t a k e n i n t o a c c o u n t . The d e p e n d a n c e of lattice c o n d u c t i v i t y on t e m p e r a t u r e a n d pressure was t a k e n according to V. N. Z h a r k o v (1958) XM = const - y - [2] where t h e pressure f u n c t i o n /(p) increases 25 times f r o m t h e surface t o t h e m a n t l e a n d core b o u n d a r y . T h e c o n s t a n t in [2] was e v a l u a t e d a d o p t - ing X — 1,2 x 10"2 cal/cm sec degree (dunite) in t h e conditions of t h e E a r t h ' s surface. I n t h e f o r m u l a f o r t h e r a d i a t i v e c o n d u c t i v i t y 16 ctm2 m = -R T3 [3] o e t h e absorption coefficient s in t h e m o s t p a r t of calculations was t a k e n equal t o 200 c m 1 a n d in some calculations 40 cm"1. O u r results would r e m a i n valid also if it were shown t h a t n o t t h e r a d i a t i v e c o n d u c t i v i t y b u t t h e exciton c o n d u c t i v i t y is of i m p o r t a n c e in t h e E a r t h ' s interior. An e x a m p l e of t h e change of t h e t o t a l c o n d u c t i v i t y w i t h t h e d e p t h is shown on fig. 2. The h e a t c a p a c i t y was t a k e n c = 0,2 cal/g a n d in some cases c = 0,3 According to J a c o b s (1956) t h e h e a t c a p a c i t y of t h e m a n t l e is within these limits. I t m u s t b e t h e same also for t h e core if t h e l a t t e r has t h e same composition according to R a m s e y ' s hypothesis a d o p t e d b y us. T h e general c h a r a c t e r of t h e radial distribution of t e m p e r a t u r e f o r different m o m e n t s of t i m e can be seen on fig. 3 (the v a r i a n t C of t h e con- t e n t of r a d i o a c t i v e elements; h e a t capacity c = 0,2 a n d 0,3 cal/g). The knick of t h e curves a t t h e mantle-core b o u n d a r y arises because t h e t o t a l h e a t c o n d u c t i v i t y of t h e m a n t l e , even if t h e radiatice c o n d u c t i v i t y is t a k e n i n t o a c c o u n t , remains smaller t h a n t h e metallic c o n d u c t i v i t y of t h e core. T h e outflow of h e a t f r o m t h e core is small and therefore with c = 0,2 t h e central t e m p e r a t u r e is a b o u t 1,5 times greater t h a n with c = 0,3. F o r t h e v a r i a n t G t h e present central t e m p e r a t u r e is contained within 4400-6500 ° I t a n d for t h e v a r i a n t Cik — within 3000-4500 °K. F o r t h e g r e a t e r role of r a d i a t i v e c o n d u c t i v i t y (e = 40 cm"1) t h e d e p t h of outer zone f r o m which t h e h e a t flows out is also g r e a t e r a n d i t reaches t h e 146 B . J . L E V I N - S . V . M A J E V A b o u n d a r y of the core. Therefore the temperature of the core comes to be about 100° lower t h a n for s = 200. The comparison of temperatures calculated for the present m o m e n t of time with the melting temperatures is given on fig. 4. F o r the m a n t l e t h e melting curves according to Uffen (1952) and to Y.N. Zharkov are given, and for the core — t h e melting curve for iron according to Zharkov (1959). (According to d a t a by Strong the melting temperatures of iron are substantially lower). For the v a r i a n t C of the content of radioactive elements if c = 0,2 the t e m p e r a t u r e curve for t h e present time passes over the melting curves. Probably this does not correspond to t h e real C = 0.2 Fig. 3 - T e m p e r a t u r e v a r i a t i o n w i t h d e p t h . situation. F o r c = 0,3 the t e m p e r a t u r e distribution is almost the same as for variant Ci/2 and c = 0,2. Remembering t h e u n c e r t a i n t y of t h e melting curves we can say t h a t this distribution of t e m p e r a t u r e (or a similar one passing somewhat lower) agrees with t h e concept of the partial melting of the mantle which began a b o u t 3 billion years ago. I n order to obtain t h e solid inner core and the fluid outer p a r t of it we m u s t sup- pose t h a t the m a t t e r of t h e core being in metallic state has melting tem- perature intermediate between Zharkov'? ans Strong's data for the iron. MacDonald has made all his calculations for the minimal values of uranium and thorium content, obtained in recent analyses of meteorties by the neutron-activation method (U — 1,1 X 10~8; Th — 4,4 X 10 8). O N T I I E T H E R M A L H I S T O R Y O F T H E E A R T H 1 5 1 I n this case the generation of heat is so small t h a t the melting of the outer p a r t s of t h e mantle occures only for the present time and only if t h e high initial temperature of the E a r t h is supposed. B u t in t h e real E a r t h t h e partial melting of the mantle was t h e cause of the formation of the crust which hegan about 3 billion years ago. Moreover the great initial t e m p e r a t u r e of the E a r t h can be due to the shortlived isotopes only and this requires the formation of t h e E a r t h from the m a t t e r which had shoitly vefore passed t h e process of nucleogenesis. Very small uran- ium and thorium contents adopted by MacDonald can lead also to dif- Fig. 4 - T e m p e r a t u r e s calculated for t h e p r e s e n t m o m e n t of t i m e as c o m p a r e d t o t h e melting t e m p e r a t u r e s . 1 - a f t e r E . U f f e n (1952). 2,3 - V. N. Z h a r k o v (1959). Acuities in explaining the observed heat flow. MacDonald agrees t h a t t h e lattice conductivity decreases with temperature (see form [2]) b u t in his calculations he p u t s it to be constant. H e does not t a k e into account t h e low-conductivity layer a t the depth 100-200 km (see fig. 2) and therefore he overestimates the heat flow through the surface. When t h e longenduring character of the redistribution of radioac- tive elements is taken into account, falls off the substantial cooling of outer layers obtained by Lubimova for the case of instantaneous redistri- bution. I n the v a r i a n t C for the continental p a r t of the E a r t h (thickness of the crust d = 30 km) we obtained some cooling at the depth of 200-600 k m which does not exceed 200° and for the oceanic p a r t (d = 10 km) there is no cooling a t all (see fig. 5). F o r the variant Ci/2 t h e result is analogous. 146 B . J . L E V I N - S . V . M A J E V A At the same time — and t h a t is very i m p o r t a n t — we obtained for all variants t h a t the t e m p e r a t u r e a t the crust-mantle b o u n d a r y continues to increase. I t is due to the increase of the generation of heat in t h e crust because with t h e adopted linear law of concentration of radioactive ele- ments into t h e crust their income from the m a n t l e overcompensates their decay (see fig. 1). Fig. 5 - T e m p e r a t u r e d i s t r i b u t i o n in t h e u p p e r l a y e r s of t h e E a r t h . a) c o n t i n e n t a l p a r t of t h e E a r t h b) oceanic p a r t of t h e E a r t h . e = 40. e = 200. The t e m p e r a t u r e changes in the lower p a r t of t h e crust are directly connected with the changes of t h e heat flow t h r o u g h t h e E a r t h ' s surface (fig. 6). The almost constant flow was reached a t t h e end of t h e stage of uniform distribution of radioactive elements (for t h e uniform model it would begin to decrease) and a f t e r the beginning of the redistribution it began to increase again and this increase continues till now. I n t h e case of instantaneous formation of the crust there would be a sharp increase of h e a t flow followed by a decrease. O N T I I E T H E R M A L H I S T O R Y OF T H E E A R T H 151 For both variants of content of radioactive elements (C and Ci/a) all calculations give the present heat flow which is in accordance with t h e observed one within t h e u n c e r t a i n l y of the latter. For the case of a thin Fig. 6 - Superficial h e a t (low v a r i a t i o n w i t h t h e t i m e . 1 - C, d = 2 0 , e = 2 0 0 , c = 0 , 3 2 - c, 10, 2 0 0 , 0 , 2 3 - Cl/o 3 0 , 4 0 , 0 , 2 4 - c, 3 0 , 2 0 0 , 0 , 3 5 - c, 3 0 , 2 0 0 , 0 , 2 0 - c, 10, 4 0 , 0 , 2 7 - c, 3 0 , 2 0 0 , 0 , 2 (Curve 7 corresponces t o t h e i n s t a n t a n o e u s f o r m a t i o n of t h e c r u s t ) oceanic crust the h e a t flow is of course smaller t h a n for the case of a thicker continental crust. B u t the heat flow originates not only in the crust b u t also inflows from the mantle and therefore t h e difference of 146 B . J . L E V I N - S . V . M A J E V A calculated heat flows is much less t h a n the difference in thickness between t h e continental and oceanic crust. The increase of t h e role of radiative conductivity decreases the difference of heat flows B u t even if t h e role of radiative conductivity is small, this difference in h e a t flows a m o u n t s only to 30-40% and is in accordance with the conclusion b y Bullard, Maxwell and Revelle (1956) t h a t the flows observed on continents and oceans are equal within the limits of their uncertainty. Therefore this observed equality of heat flows does not contradict t h e concept of t h e gradual differentiation of the crustal m a t t e r from the mantle, which in continental p a r t s had proceeded f a r t h e r t h a n in oceanic ones. I t m u s t be stressed t h a t the inference upon the continuing increase of the t e m p e r a t u r e of the lower p a r t of t h e crust and also of t h e h e a t flow is directly connected with t h e linear law of income of radioactive elements into the crust, which was adopted arbitrarily. The change of this law and also the rejection of the supposition of uniform extraction of radioactive elements from the upper 1000 km of the m a n t l e would lead to different distributions of temperature in the outer p a r t s of t h e E a r t h and to different changes of this t e m p e r a t u r e with t h e time. F o r this calculation we m u s t receive from geologists and geophysicists reliable d a t a upon the redistribution of radioactive elements. W i t h o u t them our calculations cannot confirm or cast doubt on t h e supposition of some authors t h a t in t h e past both tectonic and volcanic activities were greater. SUMMARY The thermal history is calculated for several models of the initially cold Earth. The formation of the Earth's crust is regarded as a longenduring process, ivhich started 3 X 10s years ago. In the case of great radiative conductivity the calculated heat floiv through the surface is in agreement with the observed one for S7naller content of radioactive elements than it was admitted earlier. The difference of heat flows calculated for the continental and oceanic parts of the crust amounts only 30-40%. This can explain why the observation does not reveal systematic differences between heat flows on continents and on oceans. O N T I I E T H E R M A L H I S T O R Y O F T H E E A R T H 151 RIASSUNTO La storia termica viene calcolata per vari modelli della terra inizial- mente fredda. La formazione della crosta terrestre viene considerata come un processo che si protrae nel tempo e che e cominciato 3 x 10" anni or sono. Nel caso di grande conduttivita radiattiva, il flusso di calore calcolato at- traverso la superficie e in accordo con quello osservato per un contenuto mi- nore di elementi radioattivi di quello che era stato precedentemente ammesso. La differenza di flusso di calore calcolato per le parti continentali e oceaniche della crosta raggiunge solo il 30-40%. Questo pud spiegare perche Vosservazione non rivela differenze sistematiche tra il flusso di calore registrato nei continenti e quello negli oceani. R E F E R E N C E S BIRCH F . , Nuclear geology. ( E d . H . F a u l ) , C h . 5, 1 4 8 - 1 7 4 , ( 1 9 5 4 ) . BULLARD E . C . , MAXWELL A . E . , REVELLE R . , Advances in geophysics, 153- 1 8 1 , ( 1 9 5 6 ) . JACOBS J . A., Handb. Phys. (Hrsg. S. Fliigge), 47, 364-406, (1956). LUBIMOVA E . A., Bull. Acad. Sci. USSR, geophys. ser., 2-3, 14, (1952). — a) Publ. Geophys. Inst. Acad. Sci. USSR, 26 (153), 39-50, (1955). — b) Bull. Acad. Sci. USSR, geophys. ser., 5, 416-424, (1955). — a) OR (Doklady) Acad. Sci. USSR, 107, 1, 55-58, (1956). — b) Bull. Acad. Sci. USSR, geophys. ser., 10, 1145-1160, (1956). — Bull. Acad. Sci. USSR, geophys. ser., 5, 673-676 (1957). — Geoph. Joum., 1, 2, 1 1 5 - 1 3 4 , ( 1 9 5 8 ) . LUKYANOV V. S., Bull. Acad. Sci. URSS, Otdeleniye Technich. Nauk, 2, 5 3 - 6 7 , ( 1 9 3 9 ) . MACDONALD G . , Joum. Geophys. Res. 6 4 , 11, 1 9 6 7 - 2 0 0 0 , ( 1 9 5 9 ) . SAFRONOV V. S., Bull. Acad. Sci. USSR, geophys. ser., 1, 139-143, (1959). S T A R I K J . E . a n d S H A T Z M . M . , Geochemistry ( U S S R ) , 2 , 1 9 - 2 6 , ( 1 9 5 6 ) . S T A R K O V A A . ' G . , Meteoritics ( U S S R ) , 1 3 , 1 9 - 3 2 , ( 1 9 5 5 ) . U F F E N R . , Trans. Am. Geoph. Un., 3 3 , 6, 8 9 3 - 8 9 6 , ( 1 9 5 2 ) . ZHARKOV V. N., Bull. Acad. Sci. USSR, geophys. ser., 11, 1342-1350 (1958). — Bull. Acad. Sci. USSR, geophys. ser., 3, 465-470, (1959).