- Text
- Plasma,
- Density,
- Electron,
- Equilibrium,
- Binary,
- Radius,
- Stellar,
- Eclipsing,
- Parameters,
- Equation,
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Book Boris V. Vasiliev Astrophysics

Chapter 12 The Theory of Earth 12.1 The introduction Seismic measurements show that as the depth grows, the density of the Earth’s matter increases at first more or less monotonically. However, on the core-mantle interface it rises abruptly. There exist a series of Earth models [23],[24] for the description of the density growth, for the explanation of the reason of the separation of the Earth matter into the core and the mantle and the determination of the density jump on their interface. Quite separately there exist models for the explanation of the origin of the terrestrial magnetism. Among these models, there is one based on the hydrodynamic hypothesis. It was developed in the 1940’s-1950’s. At present it is generally adopted. All these models try to decide the basic problem - to obtain the answer: why the main magnetic field of the Earth near the poles is of the order of 1 Oe? Such statement of the basic problem of terrestrial magnetism models nowadays is unacceptable. Space flights, started in 1960’s, and the further development of astronomy have allowed scientists to obtain data on magnetic fields of all planets of Solar system, as well as some their satellites and a number of stars. As a result, a remarkable and earlier unknown fact has been discovered. It appears that the magnetic moments of all space bodies (those which have been measured) are proportional to their angular momenta. The proportionality coefficient is approximately equal to G 1/2 /c, where G is the gravitational constant, c is the velocity of light (fig.7.1). Amazing is that this dependence remains linear within 20 orders of magnitude! This fact makes it necessary to reformulate the main task of the model of terrestrial magnetism. It should explain, first, why the magnetic moment of the Earth, as well as of other space bodies, is proportional to its angular momentum and, second, why the proportionality coefficient is close to the above given ratio of world constants. The aim of this paper is to show that it is possible to construct the theory of the Earth using the full energy minimization method. This theory allows us to apply a new 95

approach to the separation of the Earth onto the core and the mantle and avoid the shortcomings of previous models connected with their independent existence. 12.2 Equation of state First, to create a theory of the Earth, we need to find the radial dependence of the terrestrial matter density γ(r). 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 γ(r) on the pressure p is described by Hook’s law: γ = γ0 1 − p (12.1) B i.e. at small pressures the equation of state is ( ) p = B 1 − γ0 , (12.2) γ where γ 0 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 B = αγ 1+1/k . (12.3) where α is a constant, k is a polytropic index describing the elastic property of matter (k = 0 describes incompressible matter). Thus, the equation of state can be written as ( ) p = αγ 1+1/k 1 − γ0 . (12.4) γ At small pressures it transforms into Hook’s law and at higher pressures it transforms into the standard polytropic equation p = αγ 1+1/k . (12.5) 12.3 The core and the mantle 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 k = 1. 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 k = 3/2. Plasma is electrically polarized matter and we can 96

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Boris V.Vasiliev ASTROPHYSICS and a

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5 The main parameters of star 29 5.

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theories of stars that are not purs

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But plasma is an electrically polar

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can be described by definite ration

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But even at so high temperatures, a

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2.2 The energy-preferable state of

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20

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3.2 The equilibrium of a nucleus in

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As a result, the electric force app

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and in accordance with Eq.(4.11), t

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The total kinetic energy of plasma

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we obtain η = 20 = 6.67, (5.24) 3

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One can show it easily, that in thi

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2.10 log (TR/R o T o ) 1.55 1.00 me

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averaged temperature and averaged p

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log R/R o 1.5 1.3 1.1 0.9 measured

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