On the existence of another source of heat production for the earth and planets, and its connection with gravitomagnetism
© Elbeze; licensee Springer. 2013
Received: 7 May 2013
Accepted: 25 September 2013
Published: 5 October 2013
Recent revised estimates of the Earth’s surface heat flux are in the order of 47 TW. Given that its internal radiogenic (mantle and crust) heat production is estimated to be around 20 TW, the Earth has a thermal deficit of around 27 TW. This article will try to show that the action of the gravitational field of the Sun on the rotating masses of the Earth is probably the source of another heat production in order of 54TW, which would satisfy the thermal balance of our celestial body and probably explain the reduced heat flow Qo. We reach this conclusion within the framework of gravitation implied by Einstein’s special and general relativity theory (SR, GR). Our results show that it might possible, in principle, to calculate the heat generated by the action of the gravitational field of celestial bodies on the Earth and planets of the Solar System (a phenomenon that is different to that of the gravitational tidal effect from the Sun and the Moon). This result should help physicists to improve and develop new models of the Earth’s heat balance, and suggests that contrary to cooling, the Earth is in a phase of thermal balance, or even reheating.
Approximately fifty per cent of the heat generated by the Earth is thought to be produced by the radioactive decay of elements such as uranium, thorium and their isotopes. Geophysicists estimate heat flow from the Earth’s interior to be in the order of 47 TW (Davies and Davies 2010), which is similar to, but slightly higher than previous estimates (e.g. Pollack et al. 1993; – 44.2 TW ± 1 TW and Jaupart et al. 2007 – 46 TW ± 3 TW ).
What still remains to be understood is the quantity of heat generated from the Earth’s primitive heat and the heat produced through the decay of radioactive elements found in the mantle. The most popular model of radioactive heating is based on the Bulk Silicate Earth (BSE) model (McDonough and Sun 1995), which assumes that radioactive materials, such as uranium and thorium, are found in the Earth’s lithosphere and mantle but not in its iron core. The BSE model also states that the amount of radioactive material can be estimated by studying igneous rocks formed on the Earth and the composition of meteorites.
From this model scientists believe that approximately 20 TW (Mareschal JC et al. (1999)) of heat is created by radioactive decay (Palme and O’Neil 2003 and Bellini et al. 2010), comprised of around 8 TW from uranium (238U), 8 TW from thorium (232Th) and 4 TW from potassium (40 K). Of this, 7 TW is believed to be created in the Earth’s crust and 13 TW in the mantle.
At the same time around 8 TW has been attributed to core dissipation in solid earth. Other heat sources have also been suggested; 39 TW of surface heat flux has been attributed to mantle convection processes, which include approximately 1 TW of latent crystallization heat at the inner core boundary (gravitational energy released by the compression of the core would be of the same order), and residual heat from planetary accretion. Although this initial heat may have rapidly dissipated through the Earth’s superficial layers, slower internal processes would still continue even today (according to some authors this energy has already dissipated).
Mantle energy budget: preferred value and range
Oceanic heat loss (300 × 106 km2)
Continental heat loss (210 × 106 km2)
Total surface heat loss (510 × 106 km2)
Radioactive sources (mantle + crust)
Continental heat production (crust + lith. mantle)
Heat flux from convecting mantle
Radioactive heat sources (convecting mantle)
Heat from core
Tidal dissipation in solid earth
Gravitational energy (differentiation of crust)
Net loss (mantle cooling)
Present cooling rate, K Gy-1
Present Urey ratio b
Our argument is based on earlier work (Elbeze 2012) and takes as a starting point results related to the gravitomagnetism framework implied by Einstein’s general relativity theory (GR).
The lense-thirring effect
Here v is the projection of the vector speed along the radial radius r.
It should be noted that if G and M are relativistic and depend on speed ν, then r does not depend on this speed and remains constant in this study.
An interesting characteristic of Eq. (8) is that acceleration is no longer independent of the sign of the velocity v of the test particles making up the mass M at the source of the gravitational field.
Complete symmetry for the sun and the planets
Speed ν is defined as the relative speed between the Sun’s hemispheres and the planet in question, in this case the Earth. This does not take into account the influence of the other planets in the Solar System. The relative speed of the Sun’s rotation seen by a test body ΔMx belonging to the Earth is almost zero because it is subject to speeds +ν and −ν of both hemispheres of the Sun (see Figure 2). Extended to the total mass of the Earth this speed is considered to have no effect on the action of the gravitomagnetic field of the Sun. By applying Eq. (8) and replacing ν by zero, acceleration is equal to Newton’s classic relation.
Definition of hidden variables
Our calculations are based on a modified value for the radius of the Earth. This modification is described in Elbeze (2012). The Earth’s real radius is defined as Rrr and the differential of the real radius as dRrr. As explained in Elbeze (op. cit.) the Earth, Sun, planets and the stars in general are complex systems and their apparent dimensions cannot be used directly in calculations.
This is a result of position-dependent hidden variables that maintain these celestial bodies in their respective planetary systems. In the case of the Earth the tilt of its rotation axis with respect to the ecliptic plane defines the real radius Rrr used here. Let us assume real radius Rrr for the Sun, and the apparent radius Rcb (cb for celestial bodies) for the planets of the Solar System, modified data Ωcb and experiential data Ωdata, which is the projection of the sum of the angles of the axis of rotation and the angle of the orbit on the ecliptic plane.
Modifications of Ωdata for the planets and the Sun (Elbeze 2012 )
Angle the ecliptic makes with the projection of the axis of rotation of the planet on the ecliptic Ωdata(the data observations in radians)
modified Ωdata as Ωcb in radians
0.45789 + π
represents the latitude of the point considered.
H planet is the distance traveled in 12 hours, half the time required for a complete revolution of the Earth, from the farthest to the point nearest to the Sun. It should be noted that the real radius Rrr is used to calculate H planet and V planet. Both H planet and V planet are hidden variables, dependent on the position of the planet in the Solar System and its inclination to the ecliptic (Elbeze 2012).
Reaction between the sun’s gravitational field and the earth’s rotation
Using the polar coordinates and considering the Earth as having a quasi-continuous density in different parts from the inner core to the upper mantle we will consider mass ΔMx whose volume is defined according to the Figure 6.
Here μ is the density of the zone on the Earth and r is the vector radius r of the testing mass ΔMx (see Figure 6).
Next we calculate the effect of the gravitational field of the Sun on the mass ΔMx along its upward or downward trajectory of height H planet. From Eq. (8) and using with dW = dWu for upward and dW = dWd for downward for D1 and D2 this gives the equations below (16a and 16b).
In order to facilitate the calculation, and not introduce large errors into the final results, the following considers H planet and V planet as acceptable average values. This is preferred to more accurate calculation that takes into account the infinitesimal displacement and variable speed .
Despite this simplification, which only incurs a slight quantitative error, we must bear in mind the fact that the speed .of the infinitesimal mass dMx has a direction which varies from 0 to 2π over 24 hours.
Equations (16a) and (16b) define the two extreme points of the Earth from the center of the Sun
In reality, the planets of the Solar System travel along ecliptic orbits and Dstar should be replaced by Dstar = a · (1 − ex · cos(λ)) or a = half major axis = 149, 6 · 109 · m For the Earth, ex is the eccentricity (0.0167) and λ varies from 0 to 2π (0 in winter and π in summer; the farthest point of the Earth from the Sun).
The distance R1–R2 (from the center of the planet) represents the depth of the layer used to calculate the energy generated by the reaction with the gravitational field of the star. In the case of the Earth R1 = 0, R2 = Rearth and which gives a value of 6.4 · 1013 · Watt for Wt.
Application of Wt to the earth
Depths, densities and heat production ( Wt ) of the Earth’s interior
Data on the Earth’s Interior fromWt
2.2 to 2.9
3.8 to 5.4
5.2 to 6.2
9.9 to 12.2
12.8 to 13.1
All the Earth
Although the data used in Table 3 is relatively exact it clearly shows that the dimensions and densities are not completely accurate. If it was the case that the value of 32 TW (18 TW from the upper mantel and 14 TW from the lower mantel) had completely accounted for the 47 TW heat contribution from the gravitational action of the Sun and the heat loss by the Earth, the heat contribution from radioactive elements would be in the order of 15 TW, which is comparable with the value generated by the Bulk Silicate Earth (BSE) model.
Where cb is the celestial body subject to the gravitational field of mass Mstar (which generates the gravitational field), Dstar is the distance between the two bodies in question, Tcb is the round trip time of the celestial body, Hcb and vcb concern the body subject to the gravitational field of the mass Mstar calculated using (Eq. 22) and (Eq. 23).
Relationship between Wt and reduced heat flow
In general we assume that the mantle and crust heat flux is proportional to the average surface heat flux. Pollack and Chapman (1977) argued that mantle heat flux represents 40% of the regional average surface heat flux. Despite the fact that their measures were based on a small dataset, we consider here that they are valid up to a minimum scale of about 300 km (Mareschal and Jaupart 2004). Average heat flow data suggest an empirical relationship of the form:
Reduced heat flows for the linear data fit of individual terrain
Reduced Heat Flow mW/m2
Vitorello et al., 1980
McLaren et al., 2001
Eastern USA Phanerozoic
Roy et al., 1968
Eastern USA Proterozoic
Roy et al., 1968
Kukkonen et al., 2001
Hydman et al., 1979
Costain et al., 1986
kutas , 1984
Decker et al., 1988
In order to compare Eq. (21) with the value of Qo (reduced heat flow), we must extend Eq. (21) which calculates heat production due to the gravitational action of the Sun on the Earth, to calculate heat flow up to depths of the order of 500 km, comprising the lithosphere and the upper mantel.
The calculation of the gravitational action of the Sun on the Earth is shown in Eq. (21). However, for the calculation of heat flow and heat production, we will base the calculation on a 1 m2 column in a lithosphere approximately 550 km deep. In this area, heat transfer mainly occurs through thermal conduction, which enables us to assume that the heat produced as a result of the gravitational effect of the Sun in this area is equal across wide areas and therefore comparable to the reduced heat flow shown in Eq. (24). Heat propagation is lowest in the lower mantle; it is no longer completely the result of thermal conduction but various according to the geography of the area and is comparable to heat production shown in Eq. (24). We will see later that only a portion of the heat production generated by the gravitational field of the Sun in the lower mantel is taken into account in the calculation of the Earth’s heat loss.
Other explanations for the Earth’s internal heat can be found on the Internet.
ΔR is a basic unit of distance. It can be a meter or take an arbitrary value. Here, we use the meter because it is directly related to the unit area m 2 or the unit volume m 3 which leads to a definition of heat flow and heat production in (mW/m 2 or μW/m 3.).
Earlier literature on the Earth’s heat sources has also suggested the possibility of an external heat source. For example, Jaupart et al. (2007) comment that Qo could be due to an external input of heat and differences of the radiogenic heat of the Earth’s crust.
Reaction between the planets’ gravitational field and the earth’s rotation
From Eq. (21), we can calculate the gravitational effect of the planets and the Moon on the Earth. To do this we replace the data relative to the effect of the Sun on the Earth with those of the planet or satellite in question (e.g. the Moon). According to Table 2 the values to be used are i = 1 for Mercury, i = 8 for Neptune and i = 3 for the Moon.
These calculations assume that the distances between the planets and the Earth remain constant over the period of the Earth’s rotation. Although we know that this is not the case, the values of Wt shown in Figure 12 provide a relatively precise glimpse of the heat generated in the Earth. The effect of the Moon is the strongest, generating about 1 TW (particularly compared to tidal dissipation in solid earth of about 0.1 TW).
C i is heat capacity measured in Jkg − 1 · K − 1; Mcb is the mass of the planet or celestial body. The data shown in this graph is imprecise as the value of C i is not well established.
Extending earlier studies on the rotation of planets (Elbeze 2012) and particularly the relativistic effect of gravitational action (see Eq. 8 and following) this paper shows that there is another heat source, external to the Earth itself and the action of its radioactive elements. Our calculations suggest that the gravitational effect of the Sun on the Earth generates a total power equal to about 54 TW.
This external heat is due to the action of land masses moving in the gravitational field of the Sun, and depends on the relative speed ±v (velocity depends on the rotation of the Earth on its axis). This occurs because there is an asymmetry between the direction of the relative speed and its effect on the moving masses (as described by Eq. 8 and following). Infact it is an example of the gravitomagnetism phenomenon described in the study by Elbeze (2012).
This study has shown that the production of heat in the lithosphere exactly matched the reduced heat flow Qo shown in Figure 10, and the heat lost from the lower mantle forms part of the overall heat lost by the Earth. However, if the heat flow created in the lower mantel does form part of the total 47 TW of heat lost by the Earth, then heat produced by radioactive substances in the upper mantle must be less than the current estimate of 20 TW. Similarly, if the gravitational action of the Sun in the lower mantel creates a heat loss by the Earth of about 14 TW, the production of radiogenic heat would be about 15 TW (or less), which is comparable with estimates based on the Bulk Silicate Earth value (BSE) model. The 14 TW produced by the gravitational action of the Sun on the Earth would vary from one area to another depending on the distribution of sedimentary rocks. This energy would add to the part of the heat produced by radioactivity found in the crust and the lower and upper mantels to form the heat flow lost by the Earth. In this case, the balance of the Earth’s heat production would be positive, rather than negative. The overall effect of the gravitational action of the Sun on the Earth would be to increase heat generation by about 25 TW (see Table 3), which corresponds to an increase in temperature of the order of 170° K per billion years.
aPlanetary fact sheet; can be found on the Internet.
bLatitude is a geographic coordinate that specifies the north–south position of a point on the Earth’s surface. Lines of constant latitude (parallels) run east–west parallel to the Equator. Latitude is an angle which ranges from 0° at the Equator to 90° at the north and south poles.
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