Geophysics dealing with terrestrial magnetism problem, are thinking often that their main task is to construct a theory that would explain the reason why the main Earth’s magnetic field near the poles is approximately equal to 1 Oe. In the second half of the twentieth century, this formulation of the problem turned out to be unacceptable. After beginning of cosmic flying this geophysical problems turned into a special case of the more general problem of magnetism of cosmic bodies.

Flying of spacecrafts and general progress of astronomical techniques have discovered a wonderful, previously unknown fact: the magnetic moments of all the celestial bodies of the Solar system, as well as a number of stars and pulsars, are proportional to the torque of these cosmic bodies (Figure 3), as it should be in accordance with the Blackett’s hypothesis.

It is remarkable that this relationship keeps linearity in the range of about 20 orders of magnitude!

All around us terrestrial matter has the atomic structure.

This means that the density of all materials in a condensed state (not in the gaseous state) is determined by the interaction between the electron shells of neighboring atoms.

The heat capacity of atomic matter is positive. Therefore, the thermal energy of the terrestrial bodies tends to the minimum (to zero) at.

The gravitational field with the acceleration generates a force if atomic substance has density.

In atomic matter, this force is balanced by the pressure gradient, which occurs in the interaction of atomic shells.

The equilibrium state for atomic matter in gravitational field is described by Euler’s equation:

Another kind of matter (non atomic) is plasma. It was discovered in the middle of the last century.

All atomic materials transform in plasma state under action of high enough pressures or temperatures. At that atoms are ionized completely or partially. The result electron gas and naked nuclei or ions form electron-nuclear or electron-ion plasma.

The properties of plasma are radically different from properties of atomic matter. Together with the lack of electron shells, their interaction disappear. As a result the action of gravity can not create a gradient of pressure in plasma.

As shown [5] , at the energetically favourable state, the hot dense plasma has finite density and finite temperature.

The equilibrium condition (21) for plasma with energetically favourable density is reached at

where is mass density, A and Z are the mass and charge number of nuclei from which composed plasma, is proton mass.

At that plasma obtains an effective charge with density

and effective electric field acting on plasma

As both equilibrium plasma density and equilibrium temperature are known, we can estimate the stellar core mass and its radius.

According to the virial theorem, the potential energy of particles executing a finite motion must be equal to twice their kinetic energy (with the opposite sign as the potential energy of the related particle is negative):

where is full number of particles in the plasma core of star.

With using obtained definitions (16) and (20), we get radius of electrically polarized stellar core

where is the Chandrasekhar’s mass.

At that the mass of stellar core

Calculations of the total mass of star show that it exceeds the mass of the core twice [5] [8] :

A thin spherical surface with radius r carrying an electric charge q at the rotation around its axis with frequency obtains the magnetic moment

The rotation of a ball charged at density will induce the magnetic moment

Thus the positively charged core of a star induces the magnetic moment

The negative charge equal to positive volume one is distributed on the surface of the stellar core.

A polarization of stellar matter located above the core―in the stellar atmosphere―we are not taken into account.

The negative surface charge creates a magnetic moment

So the total magnetic moment of the core is

Simultaneously, the torque of a ball with mass and radius is

As a result, the giromagnetic ratio for celestial bodies where the force of their gravity induces the electric polarization will depend on world constants only:

This relation was obtained for the first time by P. M. S. Blackett [3] . He shows that giromagnetic ratios of the Earth, the Sun and the star 78 Vir are really near to.

By now the magnetic fields, masses, radii and velocities of rotation are known for all planets of the Solar system and for a some stars. These measuring data are shown in Figure 3, which is taken from [4] . It is possible to see that these data are in satisfactory agreement with Blackett’s ratio. At some assumption, the same parameters can be calculated for pulsars. All measured masses of pulsars are equal by the order of magnitude [10] . It is in satisfactory agreement with the condition of equilibrium of relativistic matter [9] . It gives a possibility to consider that masses and radii of pulsars are determined. According to generally accepted point of view, pulsar radiation is related with its rotation, and it gives their rotation velocity. These assumptions permit to calculate the giromagnetic ratios for three pulsars with known magnetic fields on their poles [11] . It is possible to see from Figure 3, the giromagnetic ratios of these pulsars are in agreement with Blackett’s ratio.

First, to create a theory of the Earth, we need to find the radial dependence of the terrestrial matter density. To do this, it is necessary to write the equation of equilibrium for forces applied to the matter and the state equation of matter, i.e. the dependence of the matter density on the pressure. It is assumed that at small pressures the dependence of the matter density on the pressure p is described by Hook’s law:

i.e. at small pressures the equation of state is

where is the matter density at zero pressure; B is the bulk module of matter. At high pressures the bulk module itself starts to depend on the density. This dependence can be described by the polytropic function

where is a constant, k is a polytropic index describing the elastic property of matter (describes incompressible matter). Thus, the equation of state can be written as

At small pressures it transforms into Hook’s law and at higher pressures it transforms into the standard polytropic equation

Let us assume that the considered spherical body (the Earth or any other planet) is divided into two regions―an inner core and an outer mantle. Thus, we shall assume that the mantle is composed of hard rock of the basalt type and is characterized, as generally accepted, by a polytropic index.

Under action of ultrahigh pressure, atoms in the core lose their outer electron shell to reduce their volume and form dense plasma. In this state, substance is characterized by the polytropic index. Plasma is electrically polarized matter and we can assume that the matter inside the core may be electrically polarized by gravity if it is energetically favorable.

As a rule, this possibility is not considered at all on the basis of the fact that the electrical polarization is connected with the appearance of some additional energy and is, therefore, energetically disadvantageous. At the same time, it escapes completely everybody’s attention, that the electrical polarization changes and even can reduce other types of energy, such as gravitational and inner energy. Assuming that the core of the planet can be electrically polarized, we shall have as a purpose of our solution the determination of its radius for the minimum of its full energy. If this configuration corresponds to the body with zero, it will mean that the separation of the planet into an electrically polarized core and an unpolarized mantle is energetically disadvantageous. In view of mechanical strains, we shall assume that the planet matter has a homogeneous chemical composition with the density and the bulk module at zero pressure. We shall assume that the electrical polarization intensity inside a core is proportional to gravity

Thus, inside the core the effect of gravitation is completely compensated by the electric force

where is the density of matter inside the core, g is the gravity acceleration, is the charge density connected with polarization, is the electric field strength connected with polarization. This becomes possible because the behavior of gravitational and electric field intensities has similar descriptions:

Due to such an electrical polarization distribution a bounded volume electric charge exists inside the core and on its surface there is a surface charge of the opposite sign so that the total electric charge of the core is equal to zero. It is easy to see that the polarization jump on the core surface is immediately accompanied with a pressure jump or, speaking in terms of bounded charges, the surface charge tends to compress the charged core. Thus, although the effect of gravity in the core is compensated, its matter experiences the pressure of the entire mass over its surface

(indicates that its value is taken on the core surface) and the pressure of the surface charge [13] [14] :

This additional compression has a significant value, which is assumed to be sufficient to transform the core matter into the plasma state. In this case, the polarization sign is evident as soon as in this process the core matter acquires a positive charge while electrons are pushed out to the core surface. Estimations show that since the force of gravitation is weak compared to the electric force, the charge related to each ion is only about of the electron charge (for the case of terrestrial gravitation).

Since the compressibility of dense plasma is determined by the bulk module of Fermi gas, the polytropic index of such a matter is 3/2. In this case, the pressures inside the core and the core density are constant,

As a result, the density inside the core is higher than the one that would exist inside the planet in the absence of polarization. The equilibrium state of the mantle matter is described by

where is the mass of the matter confined to the radius r:

Thus, from Equation (50) for the mantle matter, we have

where and is the radius of the planet in the case when it is composed of incompressible matter. Under strain the full mass of the planet is naturally conserved

Solving together Equations (49), (52), and (53), we find and the ratio as functions of.

Next, we have to answer the principal question of whether the existence of an electrically polarized core is energetically advantageous. The gravitational energy of the spherical body under the definition is

It can be found for the known density distribution inside a planet. Furthermore, accounting to the thermodynamic equation for chemical potential

(where is the ion mass) from the equation of state Equation (42), the chemical potential is

and the density of the internal energy of the core is

Doing analogous calculations for the mantle, we obtain

The electric energy exists only inside the core and its density is

Since the thermal energy is neglected, to calculate the full energy of the planet, it is necessary to integrate Equations (57), (58), and (60) over the volume of the planet and sum them and Equation (54). To do this, we need to determine the values of constants composing these equations.

The mass M and radius R of the Earth are known. Therefore, we know the average density of the Earth. On the basis of the geophysical data, we accept that the density of matter and bulk module on the surface of the mantle is and. These values are characteristic for basalts [16] . Based on the above said we determine and the parameter. We can found the value of the parameter as we know the values of and and therefore we can find the ratio

Next from all possible solutions we choose the one that actually meets the condition (60). In fact this procedure is reduced to choosing the parameter, i.e. the ion mass related to each free electron in electron-ion plasma of the core. The total energy (related to) is plotted as a function of the parameter in Figure 4. The dependence of the outer radius of the planet R (related to) over the core radius at different values of is shown in Figure 5. It can be seen that of the whole family only the curve obtained on the condition that in the core there exist approximately 22 nucleons per electron of electron-ion plasma satisfies

to the ratio (Equation (60)) for.

Finally, knowing and, we can find the distribution of the density of matter inside the planet. This is illustrated in Figure 5 for (is the proton mass) and.

Thus, the calculation shows that for the Earth it is energy advantageous to have an electrically polarized core. Radius and density of the core are approximately equal to and 10 g/cm³, respectively. On the mantle-core interface, the matter density drops sharply to 5 g/cm³ and then it decreases almost linearly as the radius increases. The measured dependence of the matter density inside the Earth is shown also in Figure 6. It is determined by measuring the propagation velocity of seismic waves. Being different, the calculated and measured dependencies coincide in the principal feature, i.e. they both indicate the existence of approximately equal jumps of the density on the core-mantle interface at about the half-radius of the planet.

Knowing the matter distribution between the core and the mantle and their sizes, it is possible to calculate the moment of inertia for our theory. For a spherical body with a radial density distribution, we have

In our case, we obtain

It is in good agreement with the measured value 0.331.

It is obvious that the most bright and important result of the developed theory is the understanding of the mechanism of the generation of the terrestrial magnetic field. It is very simple: the rotation of the electrically polarized core (together with the planet) about its axis with the frequency produces the magnetic moment

Substituting appropriate values, we obtain which is almost exactly equal to one-half of the observed value of the moment.