768 lines
32 KiB
Plaintext
768 lines
32 KiB
Plaintext
Earth’s charge and the charges of the Van Allen belts
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Jacob Biemond*
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Vrije Universiteit, Amsterdam, Section: Nuclear magnetic resonance, 1971-1975 *Postal address: Sansovinostraat 28, 5624 JX Eindhoven, The Netherlands Website: http://www.gewis.nl/~pieterb/gravi/ Email: gravi@gewis.nl
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ABSTRACT
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In this paper three separate charges are distinguished for the Earth and its magnetosphere. First, it is assumed, that the Earth and its nearest atmosphere bear a net negative charge QE. Secondly, a positive charge Qi and a negative charge Qo are proposed for the inner and outer Van Allen belt, respectively. Thirdly, a belt with net zero charge (electron slot) will be assumed to be present between both charged belts.
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According to the three tori model, recently developed for pulsars and black holes, equilibrium may exist between the charges QE, Qi and Qo. Three expressions for the Coulomb electric field at different distances from Earth’s centre are derived from the same model. Using available data, values for the three charges are deduced for the solar minimum and maximum, respectively. An averaged charge QE of about – 1 C is extracted for the Earth. Some other features of the model are discussed, among them the flow of charge during the change from solar minimum to maximum.
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Furthermore, it is shown that the magnitude of Earth’s magnetic field cannot be explained by the motion of the charges QE, Qi and Qo. In order to obtain a better explanation, the so-called Wilson-Blackett law is discussed. In addition, a large toroidal electric current in the Earth is proposed.
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1. INTRODUCTION
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It is generally assumed, that the naked Earth bears a large negative electric charge, Qs, generating a vertical electric field at its surface. In the fair-weather area the magnitude of this electric field is about – 100 V/m, corresponding to a charge Qs = 4πε0rs2E = – 4.5×105 C (rs = 6371 km) at Earth’s surface (see, e.g., Uman [1]). However, an almost equal amount of positive charge, is distributed throughout Earth’s nearest atmosphere. In this study it is attempted to deduce Earth’s residual charge QE, up to an altitude of about 70 km.
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Apart from QE, it is assumed that the two Van Allen belts [2] also bear a net electric charge: a positive charge Qi for the inner torus and a negative charge Qo for the outer torus. The two described belts are separated by a region with zero net charge, the so-called "electron slot". The magnetic field around the Earth is important for the orientation of the Van Allen belts, but in this paper we mainly investigate the electric interactions between the charges QE, Qi and Qo.
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Starting from the three tori model, recently developed for pulsars and black holes [3, 4], equilibrium between the charges QE, Qi and Qo appears to be possible. From the same model three different expressions for the Coulomb electric field, depending on the distance from Earth’s centre, have been derived. The deduced Coulomb electric field at the plasmapause is put equal to the so-called co-rotation field. Values for the charges QE, Qi and Qo can then be calculated for the solar maximum or minimum, respectively. The results of the idealized model are discussed.
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In addition, the contributions to Earth’s magnetic field caused by the proposed charges QE, Qi and Qo are calculated. Since the latter contributions appear to be almost negligible, a previously proposed gravitational explanation of Earth’s magnetic field is considered (see, e.g., refs. [5–9] and references therein). Especially, the so-called Wilson-
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Blackett law is discussed. In order to obtain agreement between observed and predicted magnetic fields, the existence of a large toroidal current in the Earth is assumed.
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2. ELECTRIC FIELDS IN THE MAGNETOSPHERE
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The three tori model has been developed for the explanation of quasi-periodic oscillations (QPOs) of pulsars, black holes and white dwarfs [3, 4]. In this work this model will be applied to the Earth and its magnetosphere. Apart from Earth’s residual negative charge QE, it will be assumed that the Earth is surrounded by three circular tori in the equatorial plane. First, an inner torus with radius ri, containing a total positive charge Qi. Secondly, an outer torus with radius ro, containing a total negative charge Qo (Qo and QE should have the same sign). Thirdly, a third uncharged torus with radius rm, containing a total mass mm (the subscript m stems from middle) is assumed to be present between the two other tori. Thus, it is assumed that ri < rm < ro. The idealized model is displayed in figure 1.
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z
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s
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mo Qo
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mi
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mm
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Qi
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no
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nm
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ni
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ms
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rE ri
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rm
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QE
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νi
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νm
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ro y
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νo
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x
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Figure 1. Idealized model of Earth’s charge and charges in the magnetosphere. A net negative
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charge QE (blue) is adopted for the Earth and its nearest atmosphere. In addition, three circular tori around the Earth are assumed: an inner torus with charge Qi (red), radius ri and mass mi, an outer torus with charge Qo (blue), radius ro and mass mo and a torus with charge Qm = 0 (grey), radius rm and mass mm between the other tori. The unit vector of Earth’s rotation axis s ≡ Ωs/Ωs and the unit vectors along the rotation axes of the three tori (ni, nm and no) are all assumed to coincide. In addition, the frequencies of charge elements dQi and dQo in the tori are denoted by νi and νo, whereas the frequency of a mass element dmm is denoted by νm.
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By using Coulomb’s law only, it can be shown, that equilibrium between Earth’s charge QE and the charges Qo and Qi in the tori is possible [3, 4]. The following relations then apply
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QE g(x)Qi and QE x2 f (x)Qo,
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(2.1)
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where the parameter x is defined by x ≡ ri/ro. It is noticed that the expressions of (2.1) are most easily deduced for a point charge QE. It can be shown, however, that these relations remain valid for the electric interaction between a sphere with homogeneous charge density and total charge QE, charge Qi and charge Qo for any value of ri ≥ rE and ro > ri (rE is Earth’s naked radius rs, plus its nearest atmosphere). The functions g(x) and f(x) are defined by
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g(
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x)
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2
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E(x) 1 x2
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and
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f
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(
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x)
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2 x
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K
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(
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x)
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E( 1
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x) x2
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,
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(2.2)
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2
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where K(x) and E(x) are elliptic integrals of the first and second kind, respectively. Following the formalism of refs. [3, 4], the total equatorial Coulomb electric field,
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ECoul(r), due to the charges QE, Qi and Qo, at radius r in the interval rE ≤ r < ri can be calculated to be
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ECoul(r)
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k r2
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QE
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r2 ri2
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f (r/ri ) Qi x2 f (r/ro) Qo
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,
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(2.3)
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where k = 1/(4πε0) = 10−7c2 = 8.9876×109 N.m2.C−2 is the Coulomb constant (ε0 is the vacuum permittivity). It is noted that SI units are used throughout this work. The quantities f(r/ri) and f(r/ro) are analogously defined to function f(x) in (2.2). Note that function f(r/ri) displays a singularity near r = ri. Substitution of (2.1) into (2.3) yields
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ECoul(r)
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k QE r2
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1
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r2 ri2
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f (r/ri ) g(x)
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f
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(r/ro f (x)
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)
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.
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(2.4)
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When the radii r, ri and ro are known, the quantities f(r/ri) and f(r/ro), g(x) and f(x) can be calculated (see refs. [3, 4]). Substitution of the value for ECoul(r) and the other parameters into (2.4) then would yield the corresponding value for the charge QE.
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In addition, the total equatorial Coulomb electric field, ECoul(r) for a radius r in the interval ri < r < ro can be calculated to be
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ECoul(r)
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k QE r2
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r2
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1
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ri2
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g (ri / r g(x)
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)
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f
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(r/ro) f (x)
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,
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(2.5)
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where the quantity g(ri/r) is analogously defined to function g(x) in (2.2). The functions g(ri/r) and f(r/ro) can be calculated from chosen r and known ri and ro. Singularities occur for g(ri/r) near r = ri and for f(r/ro) near r = ro.
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Finally, the total equatorial Coulomb electric field, ECoul(r), for a radius r > ro can be calculated to be
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ECoul(r)
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k QE r2
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1
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g (ri / r ) g(x)
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g (ro / r ) x2 f (x)
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,
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(2.6)
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where the quantity g(ro/r) is also analogously defined to function g(x) in (2.2). It is noticed that g(ro/r) displays a singularity near r = ro.
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In presence of a magnetic field transformation properties of the electric field become important. For example, the transformation of an electric field from a rotating to a nonrotating frame of reference is given by
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E E v B,
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(2.7)
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where the field E' is measured in the rotating frame. The velocity v and the fields E and B are given in the nonrotating frame of reference. If no electric field is measured in the rotating frame (E' = 0), an electric field
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E vB
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(2.8)
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will exist in the nonrotating frame.
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3
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It is generally assumed that the plasma in the magnetosphere, subjected to Earth’s magnetic induction field B, moves nearly rigidly with the Earth. The so-called corotational electric field Ecor(r) can be calculated from (2.8), resulting into (see, e.g., Volland [10])
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Ecor
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(r
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)
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v
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B
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(Ωs
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r)
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B
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s
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(s
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r)
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B 0
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rs3 r3
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,
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(2.9)
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where v = Ωs×r is the speed of the co-rotating plasma in the equatorial plane. The field B at distance r is approximated by the field of an ideal magnetic dipole, so that B = B0 rs3/r3 (B0 is Earth’s magnetic equatorial field at distance rs). For convenience’ sake, it has been assumed that the unit vectors of the rotation axes of the Earth s ≡ Ωs/Ωs and the circular orbit of the rotating plasma coincide. When r is expressed in units rs (so r = L rs), relation (2.9) can be rewritten as
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Ecor(r)
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s rs3 B0 r2
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s rs B0 L2
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13.9 L 2 mV/m
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(2.10)
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for Ωs = 7.292×10−5 rad.s−1, B0 = 3×10−5 T and rs = 6371 km. It is noticed that the electric field of (2.10) could have been generated by a net hypothetical charge from the Earth of magnitude Qcor ≡ – k−1Ωs rs3 B0 = – 62.9 C.
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Apart from the proposed equatorial Coulomb electric fields of (2.4), (2.5) and (2.6),
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other contributions to the total electric field are usually assumed. For example, an
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additional electric field in the magnetosphere arises from the solar wind, the so-called
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convection electric field, Econv(r). According to the model of Volland [10], the radial electric convection field in the equatorial plane, Econv(r), can be written as
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Econv(r
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)
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2
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A
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(kp rs
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)
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L
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sin
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,
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(2.11)
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where φ = 0º points to the midnight, φ = 90º towards dawn, and so on. The multiplier A(kp) depends on the kp index, an empirical quantity that quantifies the level of geomagnetic activity. Maynard and Chen [11] deduced the following expression for A(kp)
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A(kp )
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45 1 0.159kp 0.0093kp2
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3 V.
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(2.12)
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Note that for a constant value of A(kp) integration of Econv(r) of (2.11) from φ = 0º to φ = 360º yields a zero result.
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In this work ECoul(r) of (2.6) will be put equal to Ecor(r) of (2.10) for the radius r = rpp of the plasmapause. Substitution of all necessary parameters into (2.6) then yields a value for the charge QE in the case of the solar maximum and minimum. In table 1 values for the solar minimum and maximum of radii ri, rm and ro, extracted from observations of Vette [12], are summarized. The values for ri, rm and ro are estimated from the AE-8 max/min equatorial radial profiles of figures 45 and 40 (equatorial omnidirectional electron fluxes at electron energies of 3 MeV versus L have been considered). The values for rpp = Lpp rs are related to kp by an expression given by O’Brien and Moldwin [13]
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Lpp 0.43kp 5.9.
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(2.13)
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We estimate that Lpp = 4.5 and 5.5 for the solar maximum and minimum, respectively,
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4
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corresponding to kp values of 3.3 and 0.9. From the data the ratios ri/rpp, ro/rpp and x ≡ ri/ro can be calculated, whereas the quantities g(ri/rpp), g(ro/rpp), g(x) and f(x) in (2.6) can be obtained from their respective series expansions (compare with tables 1 in refs. [3, 4]).
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Likewise, the other functions f and g can be obtained. The values of Ecor(rpp) can be calculated from (2.10) from the chosen values of Lpp. Subsequently, the value of QE can be calculated from (2.6) and those of Qi and Qo from (2.1), respectively. All results have been given in table 1.
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Table 1. Radii ri, rm and ro are extracted from data from Vette [12], whereas rpp is estimated from (2.13). From these data the ratios ri/rpp, ro /rpp and x ≡ ri/ro, and quantities g(ri/rpp), g(ro/rpp), g(x) and f (x) in (2.6) are calculated. Other functions f and g can be obtained in the same way. The quantity Ecor(rpp) is calculated from (2.10). QE is then obtained from (2.6) and Qi and Qo from (2.1).
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radius ( rs)
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ri 1.55 rm 2.4
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ro 3.7 rpp 4.5
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ratio
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ri/rpp 0.344
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f
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g
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solar maximum
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f (rs/ri) 0.587
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g(ri/rpp) 1.100
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ro/rpp 0.822
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f (rs/ro) 0.147
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g(ro/rpp) 2.468
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x ≡ ri/ro 0.419
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f (x) 0.260
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g(x) 1.158
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E (r) (mV/m)
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ECoul(rE) a – 0.251
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Ecor(rpp) b – 0.688
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Q (C)
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Qi + 1.00
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Qm 0 Qo – 25.5
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QE – 1.16
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solar minimum
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ri
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ri/rpp
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f (rs/ri)
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1.55
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0.282
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0.587
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rm 2.4
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ro
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ro/rpp
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f (rs/ro)
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4.2
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0.764
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0.127
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rpp 5.5
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x ≡ ri/ro 0.369
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f (x) 0.217
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a Calculated from (2.4). b Calculated from (2.10).
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g(ri/rpp) 1.064
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g(ro/rpp) 1.997
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g(x) = 1.117
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ECoul(rE) a – 0.202
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Ecor(rpp) b – 0.461
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Qi + 0.83
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Qm 0 Qo – 31.5
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QE – 0.93
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For comparison, choosing kp = 3.3, φ = 90º and r = rpp = 4.5 rs, combination of (2.11) and (2.12) leads to a value Econv(rpp) = – 0.32 mV/m. Compare this result with that for Ecor(rpp) = – 0.688 mV/m in table 1. According to (2.11), however, Econv(rpp) only distorts a circular orbit, but averages to zero over a full revolution around the Earth.
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It is stressed that the charges QE, Qi and Qo in table 1 have been obtained by equating the electric fields ECoul(r) of (2.6) and Ecor(r) of (2.10) for r = rpp. Extrapolation of this equalization up to r = rs would result into a predicted field of ECoul(rs) = Ecor(rs) = – 13.9 mV/m and a hypothetical charge Qcor(rs) = – 62.9 C (see (2.10)). As an example, the latter charge can be compared with the charge QE = – 1.16 C for the solar maximum, calculated from (2.6) and given in table 1. Alternatively, for rE ≈ rs substitution of QE = – 1.16 C and the other necessary parameters from table 1 into (2.4) results into the following value for ECoul(rE)
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ECoul(rE ) 0.258(fromQE ) 0.054(fromQi ) 0.061(fromQo ) 0.251mV/m. (2.14)
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This result shows that ECoul(rE) is dominated by the contribution from charge QE, whereas
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5
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the contributions from the charges Qi and Qo are an order of magnitude smaller and nearly compensate each other.
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In principle, predicted electric fields at Earth’s surface can be compared with observations. For example, an average value of about – 100 V/m has been obtained from fair-weather observations (the negative sign indicates that the electric field vector is directed downward). The corresponding charge Qs at Earth’s surface then amounts to Qs = 4πε0rs2E = – 4.5×105 C (see e.g., Uman [1]). However, a nearly equal amount of positive charge is distributed in Earth’s nearest atmosphere, resulting in a much lower residual electric field at higher altitudes. As an example, Volland [10, section 2.3] gave an approximate empirical expression for the fair-weather electric field
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E(z) 93.8exp( 4.527z) 44.4exp( 0.375z) 11.8exp( 0.121z), (2.15)
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where E(z) is the electric field in V/m and z is the altitude in km. This equation is valid at mid-latitudes below about 60 km altitude. For z = 60 km the first two terms on the right hand side of (2.15) can be neglected and a value E(z) = – 8.3 mV/m results. The latter value is smaller than the value of Ecor(rs) = – 13.9 mV/m obtained from (2.10). More recent observations, however, indicate, that the electric field at higher altitudes may still be smaller. From data given by Rycroft et al. [14] (see their figure 15b) an extrapolated value of about – 0.2 mV/m can be extracted for the fair-weather electric field at an altitude of 70 km above the Earth. The latter value is in fair agreement with calculated values for ECoul(rE) calculated from (2.4). For the solar maximum and minimum these calculated values for ECoul(rE) are about – 0.251 mV/m and – 0.202 mV/m, respectively (see table 1). In order to obtain a value for the charge QE, we also could equalize ECoul(rE) of (2.4) and the observed electric field at r = rE, instead of equating ECoul(r) of (2.6) and Ecor(r) of (2.10) for r = rpp. The latter choice, however, seems to be more reliable.
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Table 2. Calculated values for Ecor(r) from (2.10) and ECoul(r) from (2.4), (2.5) and (2.6) for the solar maximum as a function of r, expressed in units of L, in the interval rs < r ≤ r = 6.0 rs.
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L
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Ecor(r)
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ECoul(r)
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L
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Ecor(r)
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ECoul(r)
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L
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Ecor(r)
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ECoul(r)
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(mV/m) (mV/m)
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(mV/m) (mV/m)
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(mV/m) (mV/m)
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1
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– 13.9
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– 0.251 2.8 – 1.78
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0.404 4.5 – 0.688 – 0.688
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1.2 – 9.68 – 0.202 3.0 – 1.55
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0.546 4.6 – 0.659 – 0.613
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1.4 – 7.11 – 0.303 3.2 – 1.36
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0.808 4.8 – 0.605 – 0.501
|
||
|
||
1.55 – 5.80 1.6 – 5.44
|
||
|
||
0
|
||
|
||
3.4 – 1.21
|
||
|
||
0.971 3.6 – 1.08
|
||
|
||
1.43 5.0 – 0.557 – 0.422 4.60 5.2 – 0.515 – 0.363
|
||
|
||
1.8 – 4.30
|
||
|
||
0.262 3.7 – 1.02
|
||
|
||
0
|
||
|
||
5.4 – 0.478 – 0.318
|
||
|
||
2.0 – 3.48
|
||
|
||
0.215 3.8 – 0.965 – 5.09 5.6 – 0.444 – 0.282
|
||
|
||
2.2 – 2.88
|
||
|
||
0.225 4.0 – 0.871 – 1.77 5.8 – 0.414 – 0.252
|
||
|
||
2.4 – 2.42
|
||
|
||
0.259 4.2 – 0.790 – 1.09 6.0 – 0.387 – 0.227
|
||
|
||
2.6 – 2.06
|
||
|
||
0.308 4.4 – 0.720 – 0.784
|
||
|
||
Using the result QE = – 1.16 C for the solar maximum and other data from table 1, it is possible to calculate values for ECoul(r) for all values of r ≥ rE from (2.4), (2.5) and (2.6) for the intervals rE ≤ r < ri, ri < r < ro and r > ro, respectively. The results are given in table 2 and are displayed in figure 2. In deducing the three expressions for ECoul(r), it has been assumed that the charges Qi, Qo and QE are in equilibrium for the values L = 1.55 (ri = 1.55 rs) and L = 3.7 (ro = 3.7 rs). In these cases the conditions ECoul(ri) = 0 and ECoul(ro) = 0 follow directly from the applied formalism [3, 4].
|
||
It is noticed that the adopted equilibrium situations at L = 1.55 and L = 3.7 in our
|
||
|
||
6
|
||
|
||
E Coul (green), E cor (red) in mV/m .
|
||
|
||
model are unstable. As an illustration, an electron located at a radius r slightly different from ro = 3.7 rs will be subjected to strong repulsion from the nearby toroidal negative charge Qo. Thus, the relative large values of ECoul(r) for values slightly different from L = 1.55 and L = 3.7 may explain the large values of the observed equatorial omnidirectional electron fluxes close to these L-values (see figure 2). Moreover, between these two Lvalues the equatorial omnidirectional electron flux reaches a minimum, at L = 2.4 in our examples (see Vette [12]). Thus, the occurrence of an electron slot may also be explained by our model.
|
||
4
|
||
2
|
||
0
|
||
-2
|
||
-4
|
||
-6 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 L
|
||
Figure 2. The co-rotational field Ecor(r) from (2.10) is shown as a function of L (curve in red). In addition, the Coulomb field ECoul(r), calculated from equations (2.4), (2.5) and (2.6), respectively, is given (curve in green). It has been assumed for the solar maximum that both fields are equal at L = 4.5. See text for further comment.
|
||
Comparison of the curves for the field Ecor(r) from (2.10) and the fields ECoul(r) from equations (2.4), (2,5) or (2.6) shows, that the field predicted by (2.6) nearly coincides for values of r ≥ rpp. For values of r < rpp our model predicts electric fields that substantially deviate from those predicted by the co-rotational electric field. For r = rE the value of ECoul(rE) may, however, be in better agreement with observations (see, e.g., refs. [10, 14]. It is noticed that recent observations of the distribution of He+ by Sandel et al. [15] show that plasma in the range 2 < L < 4 most frequently rotates at a rate that is roughly 85–90 % of the co-rotation rate Ωs. As a result, the field Ecor(r) from (2.10) has to be multiplied by a factor 0.85–0.90. In that case, the calculation of QE then should start from combination of ECoul(rpp) of (2.6) and a somewhat lower value of Ecor(rpp).
|
||
It is stressed that the adopted three tori model with a net charge QE for the Earth and charges Qi and Qo for the inner and outer Van Allen belts is only a first approximation. For example, the directions of the unit vectors ni and no along the rotation axes of the inner and outer torus do not coincide with the direction of Earth’s rotation unit vector s ≡ Ωs/Ωs, but differ by an angle of about 11.5º. Moreover, the charges Qi and Qo are not confined to toroidal circles in the equatorial plane, but the belts extend to a latitude of about 60º from the magnetic equator. The presented model, however, predicts, that the orbit of the inner torus with charge Qi is stable for angles smaller than ∆ = 90º – θ0 between the unit vectors ni and no (see refs. [3, 4] for a detailed calculation of ∆ or θ0
|
||
7
|
||
|
||
for an arbitrarily chosen value of x = x0). As an example, for x = 0.419 in the solar maximum ∆ = 31.1º, whereas ∆ approaches to 35.26º in the limiting case x → 0.
|
||
The results of table 1 show, that the values of Qo are more than an order of magnitude larger than QE or Qi. This result is directly predicted by relation (2.1): the smaller the ratio x, the greater Qo compared with QE and Qi. In addition, going from solar maximum to solar minimum the different charges change as follows: ΔQo ≡ Qo(max) – Qo(min) = + 6.0 C, ΔQi ≡ Qi(max) – Qi(min) = + 0.17 C and ΔQE ≡ QE(max) – QE(min) = – 0.23 C. These figures may suggest an outward flow of some negative charge from Earth’s equator to the inner and outer torus during the transition from solar maximum to minimum. Alternatively, some positive charge may flow inward from the outer torus to Earth’s equator. By the Lorentz force both the outward and the inward flow of charge at Earth’s equator will generate toroidal currents of the same direction inside the Earth.
|
||
|
||
3. EARTH’S MAGNETIC FIELD
|
||
|
||
In this section the consequences of the proposed Earth’s net charge QE and the charges Qi and Qo of the inner and outer Van Allen belts on Earth’s magnetic field are investigated. See figure 1 as an illustration of our model. If the charge QE would be homogeneously be distributed over Earth’s volume and its nearest atmosphere, the
|
||
following relation exists between Earth’s magnetic dipole moment M(QE) and its angular momentum S (see, e.g., ref. [3])
|
||
|
||
Μ(QE
|
||
|
||
)
|
||
|
||
|
||
|
||
QE 2 ms
|
||
|
||
S .
|
||
|
||
(3.1)
|
||
|
||
The magnetic induction field at the poles at distance rE from the centre, Bp(QE), is then given by (see, e.g., refs. [3, 8])
|
||
|
||
Bp (QE )
|
||
|
||
0 4
|
||
|
||
2 Μ(QE ) rE3
|
||
|
||
|
||
|
||
0 4
|
||
|
||
QE S ms rE3
|
||
|
||
|
||
|
||
0 4
|
||
|
||
2 fs QE Ωs , 5 rs
|
||
|
||
(3.2)
|
||
|
||
where μ0 = 4π×10−7 A−1.V.m−1.s is the vacuum permeability. The angular momentum is given by S = 2/5 fs ms rs2Ωs, where fs is a dimensionless factor depending on the mass density of the spherically assumed Earth. For a homogeneous mass density fs = 1, whereas fs = 0.827 for the Earth. The radius rE on the right hand side of (3.2) has been approximated by rs. Note that the directions of Bp(QE) and Ωs are predicted to be parallel.
|
||
As an example, the value QE = – 1.16 C for the solar maximum yields a value of Bp(QE) = – 4.4×10−19 s T from (3.2), extremely small compared with the observed value of Bp(tot) = – 6.1×10−5 s T (s ≡ Ωs/Ωs). For convenience’ sake, the directions of Bp(tot) and s are assumed to be parallel throughout this paper.
|
||
In order to explain the large discrepancy between Bp(tot) and Bp(QE) from (3.2), it has previously been proposed that the basic magnetic induction field for rotating celestial
|
||
bodies like the Earth, Bp(gm), is from gravitational origin (see, e.g., refs. [5–9] and references therein). Starting from the theory of general relativity, the so-called WilsonBlackett law can be deduced [6, 7] from this approach
|
||
|
||
|
||
|
||
Μ(gm)
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
G k
|
||
|
||
|
||
|
||
|
||
|
||
S ,
|
||
|
||
(3.3)
|
||
|
||
where M(gm) is the gravitomagnetic dipole moment. The dimensionless constant β of
|
||
unity does not follow from theory; it is taken equal to β = + 1 (for a discussion see ref. [7]). From the Coulomb constant k = 1/(4πε0) = 8.9876×109 N.m2.C−2 and the gravitational
|
||
|
||
8
|
||
|
||
constant G = 6.674×10−11 m3. kg−1. s−2 the remarkable gyromagnetic constant 1/2(G/k)½ = 4.309×10−11 C. kg−1 can be calculated. Combination of (3.1) and (3.3) shows that for an electron with elementary charge e = –1.602×10−19 C and mass me = 9.109×10−31 kg the
|
||
dimensionless ratio (M(gm)/S)/(M(em)/S) for β = +1 is equal to
|
||
|
||
1
|
||
|
||
M (gm) M (em)
|
||
|
||
S S
|
||
|
||
|
||
|
||
|
||
|
||
G k
|
||
|
||
|
||
|
||
2 me e
|
||
|
||
|
||
|
||
|
||
|
||
4.900
|
||
|
||
1022.
|
||
|
||
|
||
|
||
(3.4)
|
||
|
||
Therefore, predicted magnetic fields from gravitomagnetic origin are usually extremely small and difficult to isolate from fields due to electric charges. From (3.3) the following gravitomagnetic field Bp(gm) can be deduced for β = +1
|
||
|
||
|
||
|
||
|
||
|
||
Bp (gm)
|
||
|
||
0 4
|
||
|
||
2 Μ(gm) rE3
|
||
|
||
0 4
|
||
|
||
G k
|
||
|
||
|
||
|
||
S rE3
|
||
|
||
0 4
|
||
|
||
G k
|
||
|
||
|
||
|
||
2 fs ms Ωs . 5 rs
|
||
|
||
(3.5)
|
||
|
||
The gravitomagnetic field Bp(gm) is identified as a magnetic induction field, resembling magnetic induction fields from electromagnetic origin, like Bp(QE) from (3.2). Substitution of Earth’s mass ms = 5.977×1024 kg and the other parameters into (3.5) yields a value of Bp(gm) = – 1.95×10−4 s T.
|
||
It is assumed that the total magnetic induction field, B(tot), consisting of the sum of the fields from electromagnetic origin, Bp(em), and Bp(gm), differs from Bp(gm) by a factor β*
|
||
|
||
Bp (tot) Bp (em) Bp (gm) Bp (gm).
|
||
|
||
(3.6)
|
||
|
||
When the total field B(tot) is only due to gravitomagnetic origin, the dimensionless factor β* reduces to β* = + 1. From the calculated value for Bp(gm) and the observed value for Bp(tot) a value β* = + 0.31 for the Earth is obtained.
|
||
Since charges may move in different ways in rotating bodies, one can hardly expect that β* is a constant. Indeed, different results for β* have been found for about fourteen rotating bodies: metallic cylinders in the laboratory, moons, planets, stars and the Galaxy [6, ch. 1]. On the other hand, from a linear regression analysis of this series an almost linear relationship between the observed magnetic moment |M| and the angular momentum |S| was obtained. This result is in fair agreement with the prediction of (3.3) (|M| and |S| vary over an interval of sixty decades!). From this analysis an average value of |β*| = 0.076 was obtained. Although this result is distinctly different from the gravitomagnetic prediction β* = 1, the correct order of magnitude of β* for so many, strongly different, rotating bodies is amazing. For pulsars a separate analysis was given [9]. Since magnetic fields from electric origin may affect the total field, the reported results may reflect the validity of the gravitomagnetic hypothesis.
|
||
Analogously to pulsars and black holes, an expression of the total electromagnetic field Bp(em) from the charges QE, Qi and Qo has previously been calculated [3, 4]. First, the contribution Bp(QE) generated by the charge QE at Earth’s pole at radius rE can be calculated (see (3.2)). For convenience’ sake, radius rE will be put equal to Earth’s radius rs. In addition, the rotational frequency at radius rE will be approximated by Earth’s rotational frequency νs. The second contribution Bp(Qi) arises from the torus with charge Qi moving with a rotational frequency νi in the circular torus of radius ri (see figure 1). The third contribution Bp(Qo) is generated by the total charge Qo moving with a rotational frequency νo in the circular torus of radius ro. Combination of the gravitomagnetic contribution Bp(gm) of (3.5) with the three contributions to Bp(em) leads to the following expression for the parameter β* (see refs. [3, 4])
|
||
|
||
9
|
||
|
||
|
||
|
||
|
||
|
||
1
|
||
|
||
|
||
|
||
|
||
current
|
||
|
||
|
||
|
||
QE
|
||
|
||
|
||
|
||
5 2
|
||
|
||
Qi
|
||
|
||
i s
|
||
|
||
ri 2 / rs2 1 ri2/rs2
|
||
|
||
3 2
|
||
|
||
|
||
|
||
5
|
||
|
||
2
|
||
|
||
Qo
|
||
|
||
|
||
|
||
o s
|
||
|
||
ro2 / rs2 1 ro2/rs2
|
||
|
||
3, 2
|
||
|
||
(3.7)
|
||
|
||
where QE' is defined by the dimensionless quantity QE' ≡ (G½ ms)–1 k½QE, Qi' by Qi' ≡ (G½ ms)–1 k½Qi and so on. In the case of co-rotation of both tori with the Earth, all rotational frequencies are equal, so that νs = νi = νo. The term β*current in (3.7) has been added to account for a possible contribution from toroidal currents in the Earth. Calculation of the last three terms on the right hand side of (3.7) from data, e.g., for the solar maximum shows, that they all are negligible compared with unity value. Calculation then shows, that β*current ≈ β* – 1 = – 0.69. Thus, the proposed magnetic field from gravitational origin may be reduced by a toroidal electric current inside the Earth.
|
||
|
||
4. CONCLUSIONS
|
||
|
||
It is generally assumed that the naked Earth bears a large negative charge Qs of about – 4.5×105 C (see e.g., Uman [1]). In this work it is attempted to calculate Earth’s net charge, QE, including the charge of its nearest atmosphere. An estimate for QE in the solar minimum and maximum has been deduced from the so-called three tori model [3, 4]. An averaged charge of about – 1.0 C for QE can be obtained from our table 1. To our knowledge this result is the first estimate of Earth’s net charge QE. Thus, the obtained value of QE appears to be many orders of magnitude smaller than the charge Qs at Earth’s surface.
|
||
In addition, the three tori model may explain the existence of an inner and outer Van Allen belt, with charge Qi and Qo, respectively, separated by a belt with net zero charge, the so-called electron slot. Present work illustrates, that the three tori model, recently developed for the explanation of QPOs of pulsars, black holes and white dwarfs, may also be applied to the Earth and its magnetosphere.
|
||
From the deduced average value of net charge QE = – 1.0 C an averaged poloidal magnetic field of Bp(QE) = – 3.8×10−19 s T is calculated from (3.2), extremely small compared with the observed value of Bp(tot) = – 6.1×10−5 s T (s ≡ Ωs/Ωs). A much better agreement with the observed value is obtained from a previously proposed gravitomagnetic theory [5–9]. The Wilson-Blackett law following from this approach yields a value of Bp(gm) = – 1.95×10−4 s T. The difference with the observational value may be attributed to a large toroidal current inside the Earth.
|
||
|
||
REFERENCES
|
||
|
||
[1] Uman, M. A., "The Earth and its atmosphere as a leaky spherical capacitor." Am. J. Phys. 42, 1033-1035 (1974).
|
||
[2] Van Allen, J. A. and Frank, L. A., "Radiation around the Earth to a radial distance of 107,400 km." Nature 183, 430-434 (1959).
|
||
[3] Biemond, J., "Quasi-periodic oscillations, charge and the gravitomagnetic theory." arXiv:0706.0313v2 [physics.gen-ph], 20 Mar 2009.
|
||
[4] Biemond, J., "Quasi-periodic oscillations, mass and jets of black holes: XTE J1550-564 and Sgr A*." http://vixra.org/abs/1001.0021, 18 Jan 2010.
|
||
[5] Biemond, J., Gravi-magnetism, 1st ed. (1984). Postal address: Sansovinostraat 28, 5624 JX Eindhoven, The Netherlands. E-mail: gravi@gewis.nl Website: http://www.gewis.nl/~pieterb/gravi/
|
||
[6] Biemond, J., Gravito-magnetism, 2nd ed. (1999). See also ref. [5]. [7] Biemond, J., "Which gravitomagnetic precession rate will be measured by Gravity Probe B?",
|
||
arXiv:physics/0411129v1 [physics.gen-ph], 13 Nov 2004. [8] Biemond, J., "The gravitomagnetic field of a sphere, Gravity Probe B and the LAGEOS satellites",
|
||
arXiv:0802.3346v1 [physics.gen-ph], 22 Feb 2008. [9] Biemond, J., "The origin of the magnetic field of pulsars and the gravitomagnetic theory." In: Trends in
|
||
pulsar research (Ed. Lowry, J. A.), Nova Science Publishers, New York, Chapter 2 (2007) (updated and revised version of arXiv:astro-ph/0401468v1, 22 Jan 2004). [10] Volland, H., Atmospheric electrodynamics, Springer Verlag, Berlin, 1984.
|
||
|
||
10
|
||
|
||
[11] Maynard, N. C. and Chen, A. J., "Isolated cold plasma regions: Observations and their relation to possible production mechanisms." J. Geophys. Res. 80, 1009-1013 (1975).
|
||
[12] Vette, J. I., "The AE-8 trapped electron model environment." National Space Science Data Center (NSSDC) World Data Center A for Rockets and Satellites (WDC-A-R&S), 91-14, NASA, Goddard Space Flight Center, Greenbelt, Maryland (1991).
|
||
[13] O’Brien, T. P. and Moldwin, M. B., "Empirical plasmapause models from magnetic indices." Geophys. Res. Lett. 30, 1152-1156 (2003).
|
||
[14] Rycroft, M. J., Odzimek, A., Arnold, N. F., Füllekrug, M., Kułak, A. and Neubert, T., "New model simulations of the global atmospheric electric circuit driven by thunderstorms and electrified shower clouds: The roles of lightning and sprites." J. Atmos. Sol. -Terr. Phys. 69, 2485-2509 (2007).
|
||
[15] Sandel, B. R., Goldstein, J. Gallagher, D. L. and Spasojevic M., "Extreme Ultraviolet Imager observations of the structure and dynamics of the plasmasphere." Space Sci. Rev., 109, 25-46 (2003).
|
||
11
|
||
|