Condition of predominance of electrostatic forces over gravitational forces in space plasma

Astrophysicists seek to elucidate the nature of so-called “dark matter,”^1–6 which is thought to fill cosmic space at a density five times the average density of visible matter and manifests itself only through gravity. Dark matter explains the observed high (non-Keplerian) tangential rotation rate of distant stars around the center of the Galaxy,^7,8 as well as the effect of galaxies lensing light from distant light sources.⁹ Decades of searching for this matter have not yet been successful. Therefore, alternative hypotheses are of interest.

An alternative hypothesis that promises to solve the problem of the high tangential rotation rate of distant stars in the Galaxy is based on the electrostatic interaction of stars with the center of the Galaxy.⁴ The author suggested that the Sun and other stars have a positive electric charge. At the same time, the Sun and stars are attracted to the negative electric charge concentrated in the central bulge of the Galaxy and thus provide stronger than gravitational attraction and high tangential velocity of orbital motion of distant stars (rotation of the Galaxy as a whole as a solid structure). According to numerical estimates of the solar charge made by the author,⁴ the solar charge is about 10²² Coulomb. For such estimates, the author used the observations of radial acceleration of oxygen ions O⁵⁺ in the solar atmosphere published in Ref. 10. The positive value of the solar charge was explained by the fact that heavy positively charged protons leave the surface of the Sun more slowly than light negatively charged electrons. The possible positive charge of the Sun has also been pointed out in Refs. 11–13. There are also reports in the literature that the Sun carries a negative rather than positive charge. For example, the papers by Bailey^14,15 indicated that measurements of interplanetary magnetic fields made by the Pioneer 5, Explorer 10, Mariner 2, and Explorer 12 space probes confirmed the hypothesis of the author of these papers that the Sun is negatively electrically charged. The inferred magnitude of the charge is −10¹⁸ Coulomb. Also, a recent in Ref. 16 reported that the Parker Solar Probe detected a deficit of electrons with suprathermal velocities in the distribution function of free electrons in the near-solar medium. This feature may indicate the presence of negative electric potential of the Sun and negative electric charge. Thus, the sign and magnitude of the solar charge are subject to further clarification.

Returning to the hypothesis of electrostatic interaction of the Sun and stars with the central bulge of the Galaxy expressed in Ref. 4, the following remarks should be made. First, the electrostatic field of the Sun can be strongly shielded by the electron–proton plasma even at a distance of one solar radius from the Sun's surface. Therefore, the author's estimate of the Sun's charge value of 10²² Coulomb may be far from reality. Second, due to the screening of the electric fields of the Sun and stars by the interstellar plasma, their direct electrostatic interaction with the center of the galaxy may be impossible. Nevertheless, studies of electrostatic interactions in cosmic plasma are necessary because they are far from being exhausted. In particular, interactions that take into account the spatial correlation of charge carriers in each other's electric fields¹⁷ have not been investigated. Such studies have been performed for ion–electron and dust plasmas.^18–20 It was shown that such interactions lead to spatial self-organization of plasmas. Such studies have not yet been carried out for space plasmas.

In connection with the above, the aim of this work is as follows:

• to calculate the free energy of charged particles in the interstellar plasma, taking into account their spatial correlation in the electrostatic fields of other charged particles,

• calculation of the forces of electrostatic compression/stretching acting in the plasma,

• in calculating the value of stellar electric charge, at which the electrostatic forces of plasma compression can become comparable to or exceed the gravitational forces of interstellar attraction.

Thus, the value of the electric solar/stellar charge will be determined, which could explain the observed high orbital velocity of distant stars of the Galaxy without involving the idea of gravitational interaction of stars with “dark matter.” It is suggested that electrostatic compression forces may be accompanied by appreciable shear stress forces in the plasma, contributing to the rotation of the galaxy as a single “solid” substance. As a result, the orbital velocity of rotation of distant stars from the galactic center should be high.

We will simplistically assume that the space plasma is an electrically neutral plasma not experiencing hydrodynamic currents, consisting of particles of three varieties: (1) positively charged protons with radius, concentration, and temperature, respectively, R₊, N₊, and T₊; (2) negatively charged electrons with corresponding parameters R₋, N₋, and T₋; and (3) negatively charged stars with charge Z in units of elementary charge and with radius, concentration, and temperature, respectively, R_−z, N_−z, and T_−z. The negative charge of stars is chosen for certainty. (It will be shown below that the sign of the charge of the stars is not of fundamental importance and does not affect the conclusions of the paper.) We will also consider the plasma to be classical (non-quantum-mechanical), in spite of the submicrometer scales of distances at which the potentials of protons and electrons will have to be considered. It will be shown below that the classical approximation is good enough to obtain results not contradicting the experimental data.

We will assume that collisions of plasma particles are rare and that interparticle interactions are weak so that the plasma can be considered as ideal. The temperatures of protons and electrons can, generally speaking, differ because of the presence of electric fields in the plasma. In an electric field, light electrons are accelerated, and their temperature can “break away” significantly from the proton temperature because of the large difference between the electron and proton masses and because of the low efficiency of their mutual energy exchange.

We will calculate the free energy of each individual particle and the total free energy of all particles.

The free energy (2) includes two summands: the free energy of the proton as a representative of an ideal gas (the first summand) and the electrostatic correction to it (the second summand). The expression for the second summand is obtained by integrating over temperature the thermodynamic Gibbs–Helmholtz relation,²²  $UT2=−∂∂TFTV$⁠, where U is the internal energy, equal in our case to $eψ+$⁠, and F is the free energy. The subscript “V” means that the operation is performed at a fixed volume.

In the Poisson–Boltzmann equation (3), the first and second summands contain exponents. They describe the Boltzmann statistics to which electrons and protons in a multitemperature plasma obey. In the third summand of Eq. (3), the exponent is absent, since stars, due to their large mass, do not keep up with the motion of light protons in space and are unable to ensure the fulfillment of the above statistics. Quantitatively, the settling time of the proton life in a fixed point of space, approximately equal to the period of proton-sonic oscillations of the plasma, is less than the analogous time for a star by a factor of $MZM+=1028$⁠.¹⁷ (The ratio of star and proton masses is given under the root sign.)

Let us make the following remark concerning the limits of applicability of the Poisson–Boltzmann equation (3). The Boltzmann distribution for protons and electrons used in Eq. (3) is valid for an ideal plasma.¹⁷ In an ideal plasma, particle collisions are rare, and electrostatic interactions at the average interparticle distance are weak compared to the energy of thermal motion of the particles. At weak interparticle interactions, the problem of determining the energy of the whole plasma reduces to the problem of determining the energy of individual particles, and the Gibbs distribution over the energies of the whole system reduces to the Boltzmann distribution over the energies of individual particles. Let us focus attention on the fact that the energy of interparticle interaction in an ideal plasma should be small exactly at mean interparticle distances. In space plasma, at average distances between protons and electrons $r=N−1/3=0.03 m$ (N₋ ≈ 4 × 10⁴ m⁻³ is the average concentration of electrons²⁶), the electrostatic interaction energy of particles $e24πε0r$ is 10⁹ times smaller than the energy of thermal motion kT at T = 10⁶ K. Thus, the cosmic electron–proton plasma is ideal, and the Poisson–Boltzmann equation can be applied for it. At small interparticle distances, $≪10−10 m$⁠, the exponents in the Boltzmann distribution can exceed unity. However, the probability of realization of such cases is small. They can be considered as rare “Coulomb” collisions, which do not lead to violation of the plasma ideality.

The boundary conditions for Eq. (3) are the classical value of the electric field strength at the proton surface and the equality to zero of the field strength at the d-volume boundary,

The problem (3) and (4) is nonlinear. Debye and Hückel²¹ reduced a similar problem formulated for ionic solutions to a linear one by decomposing the exponents into a series at small values of the exponent (much less than unity) and taking into account the first two terms of the expansion. In the present work, the nonlinear problem (3) and (4) was not reduced to a linear one and was solved numerically. For this purpose, the Poisson–Boltzmann equation was reduced to an equivalent system of two ordinary differential equations of the first order, and the system was solved by the Runge–Kutta method of the fourth order with a fixed step of integration, implemented in the function rkfixed of Mathcad 11 program.²³ The solution required boundary conditions for the potential and its derivative on one of the boundaries, but not the values of the derivative on two different boundaries, as in our case (4). Therefore, the shooting technique was applied. On the boundary $r=R+$⁠, the derivative of the potential was set in accordance with the boundary condition (4a), and the value of the potential itself on this boundary was selected so that on the other boundary $r=d$ the electric field took a value equal to zero, determined by the boundary condition (4b). Note that the system of equations contains regions of rapidly and slowly varying solutions. Therefore, it was convenient to switch to a variable step of integration over r. For this purpose, we substituted the variable $r=exp(y)$ in Eq. (3) and then solved the equation by Runge–Kutta method with a fixed step of integration over the variable y. The function rkfixed returned a matrix of values of solutions of the system of differential equations given on the interval $lnR+, lnd$⁠. The number of rows of the result matrix was set equal to 10⁵.

Having determined the self-consistent proton potential $u+$⁠, we subtracted the purely Coulomb proton potential $e4πε0R+$ from it. Thereby, we determined the potential produced by the plasma on the proton surface $ψ+$ using the formula $ψ+=u+R+−u+d−e4πε0R+$⁠. In the last formula, the subtraction of the potential at the boundary of the d-volume is added. It is necessary because the boundary conditions (4) determine the potential to the nearest constant summand.

For the unit electron and star, the free energies $F−$ and $F−Z$⁠, as well as the potentials created by the plasma on their surfaces, were calculated in a similar way as for the proton.

Let us determine the numerical values of the parameters of the problem. We will consider the radius of the star to be equal to the radius of the Sun R_−Z = 7 × 10⁸ m. The proton radius, R₊ ≈ 0.9 × 10⁻¹⁵ m, is equal to the sum of the true proton radius of 0.841 × 10⁻¹⁵ m²⁴ and the electron radius. The electron radius R₋ = 4.5 × 10⁻¹⁷ m.²⁵ The star is considered to be stationary relative to the plasma. The temperature of its thermal motion in space is equal to the temperature of the solar corona T_−Z = 10⁶ K due to the exchange of kinetic energy between the star and the corona particles during collisions. The proton temperature is also equal to the solar corona temperature T₊ = 10⁶ K. The average temperature of the electron should also coincide with the temperature of the corona, but locally, in the immediate vicinity, for example, of the proton or at the surface of the star, it can increase significantly due to heating in the electric field. Presumably, the electron temperature can vary in the interval T₋ = [10⁶,10⁸] K. The interval of values of the star's charge was assumed to be Z = [1,10⁵⁰]. The charge of the Sun, according to Refs. 14 and 15, is Z = 10³⁷. The parameter χ was determined as follows. This parameter should have such a value at which the average concentration of electrons and protons would be equal to the experimentally established value of about N₋ ≈ 4 × 10⁴ m⁻³ outside the solar system,²⁶ at a distance from the Sun to the nearest Alpha Centauri stars equal to δ = 3 × 10¹⁶ m.²⁷ In accordance with the above formula $N−=QZχ1−χ−1V+$⁠, we obtain (1 − χ) = 10⁻¹⁷.

Figures 1(a) and 1(b) show the results of calculations of the spatial distributions of electric field strengths in the plasma near the star surface at Z = 10³⁰, δ = 3 × 10¹⁶ m and near the proton surface. Round markers represent similar distributions of field strengths in the absence of plasma, i.e., in vacuum. The distance between the surfaces of stars δ is related to the radius of the d-sphere by the relation $d=Q8α32R−Z+δ$⁠, where α is the coefficient of filling the three-dimensional space with spheres of the same radius. For hexagonal packing, α = 0.74. The expression linking δ and d follows from the statement that a spherical volume of radius d can be filled with Q balls of radius r with filling factor a, $4π3d3=4π3r3Qa−1$⁠. At the center of each balloon is a star. The radius of the ball is related to the radius of the star by the expression $r=R−Z+δ2$⁠.

It follows from Fig. 1(a) that the electric field of the star is strongly shielded by the plasma and decreases to close to zero values at a distance from the surface much smaller than the radius of the star, whereas the field in vacuum decreases much more slowly in accordance with Coulomb's law. Thus, it is obvious that direct electrostatic interaction of stars with each other in the cosmic plasma is impossible.

The electric field of the proton, Fig. 1(b), is weakly shielded and decreases with increasing distance according to the law close to Coulomb's law. The electric field of the electron behaves similarly.

Figure 2 shows the dependences of the free energies of a star, a proton, an electron, and the total free energy on the distance between the nearest stars. Recall that the total free energy is the sum of the free energy of one star and the free energies of protons and electrons per star according to the electroneutrality equation.

The free energy of the F_−Z star passes through a local maximum and further decreases, passing through a local minimum, as the distance δ decreases, Fig. 2. (The points of extrema lie well outside the boundaries of the plot.) The plot with a positive slope of the F_−Z(δ) curve indicates the presence of an electrostatic force of magnitude $dF−Zdδ$ tending to reduce the interstellar distance. This force is unrelated to the force of direct Coulomb interaction between stars. It results from the spatial correlation of protons and electrons near the surface of the star and the corresponding electrostatic energy gain during plasma compression.

The proton free energy F₊ also passes through a maximum and then decreases as δ decreases, Fig. 2. The section of the curve with a positive slope also indicates the presence of a force of magnitude $dF+dδ$ per proton. This force tends to reduce the interstellar distance. In the case of the proton, the electrostatic energy gain in plasma compression is due to the correlation of the surrounding electrons.

The electron free energy curve F₋ does not have areas with a positive slope, Fig. 2. This is because protons do not have time to spatially correlate near electrons due to the large mass of the former. The negative slope means that the electrons tend to gas-kinetic expansion of the plasma (increasing the interstellar distance).

A very important characteristic of the system is the total free energy of the particles. Figure 2 shows that the total free energy F can pass through a local maximum as the interstellar distance δ decreases. The plot with a positive slope of the $dFdδ$ curve indicates that the plasma, as a whole, can also tend to contract. It will be shown below that this tendency is mainly due to such tendency of protons.

Figure 3 shows the phase diagram of interstellar distances at the points of maxima and minima of the free energies of the star $δF−Z, max$⁠, $δF−Z, min$⁠, proton $δF+, max$⁠, and total free energy $δF, max$ as a function of the charge Z of the star. The region of phase space lying to the right of the curves $δF−Z, max$⁠, $δF−Z, min$ is the region of electrostatic compression of the plasma due to the correlation of protons and electrons near the stellar surfaces. The region lying to the right of the curve $δF+, max$ is the region of compression due to the correlation of electrons near proton surfaces. The region to the right of the curve $δF, max$ is the region of compression due to the correlations of all charged particles, i.e., the plasma as a whole.

We note that in Fig. 3 the compression region of the plasma as a whole coincides with the compression region due to protons. This means that it is protons (more precisely, the correlation processes of electrons around protons) that are mainly responsible for the electrostatic confinement of the cosmic plasma as a whole. At detailed consideration of the curves $δF+, max$ and $δF, max$⁠, it was found that the second curve is located on the graph slightly below the first one because the total free energy includes, as one of the summands, the free energy of electrons, which do not tend to compress the plasma, but only to rarefy it.

Let us discuss the high value of the electron temperature T₋ = 2.1 × 10⁸ K chosen in the phase diagram calculations (Fig. 3). For this purpose, let us consider the graph of the dependence of the value of $δF+, max$ on the electron temperature shown in Fig. 4. The plasma compression due to protons appears at the electron temperature below 3.5 × 10⁸ K. As the temperature decreases, the curve begins to reach saturation, as shown by the dashed curve (expected curve course), but an additional process is included that bends the calculated curve sharply upward (solid curve). This process is caused by the increase of the exponent in the Boltzmann distribution for electrons to a level exceeding some critical value (Boltzmann distribution for electrons $n=N−expeu+kT−$⁠). At high values of the exponent, the Boltzmann distribution ceases to be fulfilled in practice. To eliminate the problem, researchers apply an artificial procedure of “trimming” the potential limiting the value of the exponent.²⁸ In our opinion, one can avoid the application of such a procedure by making the natural assumption that the electron near the proton can be accelerated by the electric field, reaching high velocities. (The energy of the electron near the proton, expressed in temperature units, can theoretically reach values of 10¹⁰ K.) Increasing the electron temperature also limits the value of the exponent and solves the above problem. Therefore, the value of the temperature of “hot” electrons in the calculations (in the Boltzmann distribution for electrons) was chosen to be T₋ = 2.1 × 10⁸ K, which is slightly less than the threshold value of 3.5 × 10⁸ K. The value of the exponent at temperature T₋ = 2.1 × 10⁸ K is a few tens, i.e., much greater than unity. Of course, there may exist “hot” electrons with temperatures exceeding a threshold value. Such electrons, for example, can occur when electrons move in the direction of the proton with a small aiming distance. These electrons do not lead to a gain in the free energy of protons (Fig. 4). On average in interstellar space, the temperature of electrons should be equal to the temperature of protons T₊ = 10⁶ K due to mutual energy exchange during rare collisions. In particular, in the first summand for the electron free energy $F−=−kT+lnV−−…$⁠, the electron temperature equal to the proton temperature $T+$ was chosen in the calculations.

Note that the position of the curve $δF, max$ in the phase diagram of Fig. 3 does not depend on the radius of the star, but is determined only by the charge of the star and the charge and radius of the proton and electron. Therefore, this curve is suitable for determining the phase region of plasma compression with stars of any size.

The achievement of the phase curve $δF, max$ in the phase diagram of Fig. 3 is a satisfactory indication of the distance between Sun-like stars. Thus, at charge Z = 10³⁷ (the presumed charge of the Sun), the interstellar distance is $δF,max =15 000×Z1/3$ = 3 × 10¹⁶ m. This distance coincides in order of magnitude with the distance from the Sun to the nearest stars Alpha Centauri, about 4 × 10¹⁶ m,²⁷ and Barnard's star, about 6 × 10¹⁶ m.²⁹ 

Note that in Ref. 19 a phase diagram of a dust plasma calculated by a method similar to that described above was given. It also showed the possibility of compression of the plasma as a whole due to the correlation of electrons near positively charged ions. The calculated radius of the ions was 10⁻¹⁰ m. The distance between dust particles at the top of the confinement region lay in the sub-millimeter distance region. The distance was described by the stepping function $δF,max=10−5×Z1/3$⁠. In the space plasma considered in the present work, instead of positively charged ions, protons with radius R₊ ≈ 10⁻¹⁵ m five orders of magnitude smaller than the ion radius participate. As a consequence, the region of interparticle distances on the phase curve was shifted by almost ten orders of magnitude to the region of large interstellar distances. That is, according to the approximation $δF,max =15 000×Z1/3$⁠, at Z ∼ 10³⁰–10⁴⁰, the interstellar distances amounted to 10¹⁴–10¹⁷ m. This is not contradicted by experiential data.

The question arises. Can't there arise in cosmic plasma a tendency to rarefaction (instead of a tendency to compression) at changing the charge of stars from negative to positive? The answer is that the rarefaction tendency will not arise. As it was shown above, the plasma tendency to compression appears mainly due to the free energy gain of protons due to correlation of electrons near their surfaces. An insignificant contribution to the free energy gain is given by the charge correlation near the surfaces of stars. In this case, if the stars are negatively charged, protons correlate near their surfaces, and if they are positively charged, electrons correlate. The gain of electrostatic energy and the tendency to compression is achieved in either case. Quantitatively, the free energy gain at any sign of the stellar charge, at small relative magnitude of this charge $1−χ=10−17$⁠, is determined by the concentrations of electrons and protons. These concentrations are nearly equal to each other. At negative stellar charge, the proton concentration is a tiny fraction of a percent greater than the proton concentration, and at positive charge, the electron concentration is greater than the proton concentration by the same amount. Thus, quantitatively, the free energy gain is the same in both cases.

Next, let us compare the electrostatic force of compression acting from the plasma side on the stars with the gravitational force of interstellar attraction and determine the magnitude of the stellar charge at which the electric force will be dominant. The electrostatic force due to the electrostatic energy gain of protons is $f=Z1−χ−1dF+dδ$⁠, in accordance with the electroneutrality equation (1). The gravitational force is equal to $fg=GM2δ2$⁠, where G is the gravitational constant, and M is the mass of the star equal to the mass of the Sun. Figure 5 shows the dependences of the electrostatic force (at two values of the stellar charge Z = 10³³ and 10³⁷) and the gravitational force of attraction of stars on the distance between them. The gravitational force declines with increasing distance in proportion to the inverse square of the distance, and the electrostatic force declines more slowly, in proportion to the inverse of the distance to the first degree. At the value of charge Z = 10³³, the electrostatic force of plasma compression is close in value to the gravitational force. At the value of charge Z = 10³⁷ (the presumed value of solar charge), the electrostatic force exceeds the gravitational force in hundreds and thousands of times. Thus, at the charge of the star more than 10³³, electrostatic forces in the cosmic plasma can prevail over gravitational forces. For stars similar to the Sun, such predominance, apparently, is fulfilled. This circumstance may help in explaining the high (non-Keplerian) velocities of orbital motion of stars in the Galaxy remote from its center.

Note that the plasma compression force due to the charge correlation near the stars itself $f−Z=dF−Zdδ$ is negligible. It can exceed the gravitational force only at a very large stellar charge Z ∼ 10⁵⁰.

We summarize the new results obtained in the present work.

On the basis of the Poisson–Boltzmann equation, the free energy of the cosmic plasma consisting of electrically charged particles—stars, protons, and electrons—is calculated. The value of the free energy takes into account the ideal gas energy of the particles and the energy of their electrostatic interaction with the surrounding plasma. It was found that the free energy of the plasma as a whole in a certain range of values of the interstellar distance has a positive slope. This means that there is a compressional force in the plasma tending to reduce the interstellar distance. This force owes its origin to protons, around which electrons correlate and provide a gain of electrostatic energy during plasma compression. A phase diagram of plasma states in the coordinates “interstellar distance—star charge” is constructed.

The threshold charge of the star Z = 10³³, above which the electrostatic forces of compression in the plasma become stronger than the interstellar gravitational ones, is estimated. In the case of the Sun with charge Z = 10³⁷, the electrostatic force of compression can exceed the gravitational force of attraction by hundreds and thousands of times.

Thus, the hypothesis put forward in Ref. 4 about the possibility of predominance of electrostatic forces in space over gravitational forces is confirmed in the present work under the condition that the charge of stars exceeds the value of 10³³ at the masses of stars equal to the mass of the Sun. According to the author of the paper,⁴ electrostatic interaction can explain the high linear velocity of orbital motion of stars distant from the center of the Galaxy. At the same time, the attraction of the idea of “dark matter” to explain the astronomical data is not required. Indeed, according to the conclusions of our work, the cosmic plasma compression tendency may be accompanied by significant shear stress forces in the plasma. Shear stress can support high orbital velocity of distant stars in the Galaxy.

Further experimental estimates of the stellar charge, as well as theoretical estimates of the charge based on convincing physical models that take into account the size, temperature, and age of stars, are needed to further substantiate the conclusions of this work. Further comparative analysis of the results of calculations with numerous astronomical data is necessary.

The work was carried out under the state task FWRZ-2021-0007.

The authors have no conflicts to disclose.

A. V. Shavlov: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). V. A. Dzhumandzhi: Investigation (equal); Methodology (equal); Visualization (equal). A. A. Yakovenko: Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.