Electrostatic Effects in Planetary, Stellar, and Interstellar Atmospheres

Ruminations on the consequences of the second law of thermodynamics for electrons in various circumstances

Trevor Turton, trevor@turton.co.za
Version 4, published 2007-08-29
See here for version 3.
© Trevor Turton 1990-2007.

Postscript added 2007-08-30:  I have created a blog open to all who want to comment on this publication.  It's at http://electron-gas.blogspot.com/.

Abstract

This paper explores the behaviour of charged particles, especially electrons, in a wide variety of situations.  It starts with a consideration of charged particles in the Earth's atmosphere.  These particles arise primarily from the irradiation of upper atmospheric gases by ultraviolet rays from the Sun.  It is suggested that many of the electrons freed in this process rise to form an electron-rich atmosphere at very high altitude, that some return to the Earth's surface along geomagnetic lines of force, and that their recombination with positive ions in the polar atmospheres releases light energy, contributing to the Aurorae Borealis and Australis.

It is further suggested that the positive ions created in the upper atmosphere gradually drift downwards to Earth, recombining at or near its surface with a return stream of negative charge flowing through the Earth's surface from the polar regions.  The paper describes how this large-scale circulation of positive ions and electrons gives rise to the massive electrical discharges that occur in thunderstorms.

The paper addresses the similar behaviour of electrons and positive ions in other planetary atmospheres, and also discusses their solar analogues – flares, sunspots, and coronal mass ejections.

The behaviour of electrons and positive ions within gas and dust clouds surrounding young stars and planetary nebulae is discussed, and also that in accretion disks surrounding massive stars, neutron stars, and black holes.  It is proposed that the same basic mechanisms play a role in the evolution of stellar jets (which in the extreme are quasars) and cosmic rays.

Finally, the role that free electrons in interstellar regions may play in cosmic expansion is explored.  It is suggested that free electrons may account for the "dark energy" that is accelerating the rate of expansion of the cosmos, and also for the vast voids that occur within the large-scale structure of the cosmos.

Contents

  1. Charged Particles in the Earth's Atmosphere
  2. Other Planetary Atmospheres
  3. The Sun's Atmosphere
  4. Accretion Disks around Young Stars
  5. Planetary Nebulae
  6. Accretion Disks around Neutron Stars
  7. Black Holes and Quasars
  8. Free Electrons and Shape of the Cosmos
  9. References
  10. Addenda:
    1. 2007-08-30  Why are elliptical galaxies elliptical?

1.  Charged Particles in the Earth's Atmosphere

There are a large number of charged particles in the Earth's upper atmosphere.  These arise primarily as a consequence of the ultraviolet components of the Sun's radiational energy impinging upon the gas particles in the upper atmosphere.  Ionized oxygen molecules recombine to form ozone (O3) molecules.  These are present in sufficient quantity in the stratosphere to have given rise to the term "ozone layer".

During the daylight hours when solar irradiation occurs, a very large number of molecules become ionized.  This leads to a large number of positively charged gas molecules and free electrons in the stratosphere.  There is of course a very strong attraction between the positively charged ions and the negatively charged electrons, which in the fullness of time would result in the recombination of these particles.  However this recombination does not complete immediately, since the great majority of the particles are uncharged gas molecules.  Charged particles collide with and rebound from uncharged particles in the same way as uncharged particles interact amongst themselves.

Where two charged particles do approach one another, electrostatic forces will come into play.  Particles with the same charge will repel one another.  The combined momentum and kinetic energy of the particles will be preserved in each such interaction.  Particles with opposite charge will attract one another.  Depending upon their relative velocities and trajectories, they may collide and merge, cancelling one another's charge, or they may not.  In either event, kinetic energy and momentum will be conserved.  All of these various interactions approximate well enough to the elastic collisions of particles in an ideal gas.

1.1  Implications of the Second Law of Thermodynamics

The second law of thermodynamics dictates that when gases of different temperatures are mixed, their temperatures (i.e. kinetic energies) will tend over time to a common mean.  Their kinetic energies will also be equally distributed across the degrees of freedom available to them, namely translation and rotation.  This is known as the equipartition of energy principle.  Free electrons in the stratosphere have very small masses compared to those of the other particles with which they interact.  The mass of an electron is about 9.1*10-31 kg, whereas that of a nitrogen molecule is about 4.7*10-26 kg.  As a result of the collisions between electrons and the uncharged particles in the stratosphere, the electrons will tend to acquire large velocities relative to the other more massive particles with which they interact until on average the particles of different sizes have the same kinetic energy.  Once the free electrons have acquired greater velocities, they are less likely to combine with positively charged ions in the atmosphere, and hence more likely to persist in the free state.

The average velocity v of a gas particle of mass m in a gas at temperature T is given by:

{1}    v = (kT/m)0.5

where k is Boltzmann's constant.  At an altitude of 30km in the Earth's atmosphere the temperature is about 220K.  We can therefore predict that free electrons at this altitude will have an average velocity of about 40km/sec as a consequence of their thermal energy.  This velocity is way in excess of the escape velocity of the Earth's gravitational field, some 11km/sec at the Earth's surface.  If gravitational forces were the only ones that needed to be considered then we could predict that free electrons in the stratosphere would escape from the Earth's gravitational field into the surrounding vacuum of space in considerable quantities.  Gravitational forces are not however the only ones acting on the electron gas. We also need to take into account the effects of electrostatic and electromagnetic forces upon them.

1.2  Electromagnetic forces

The Earth has a considerable magnetic field of its own.  This corresponds quite well to the field that would be produced by a magnetic dipole at the centre of the Earth, inclined about 11o from the Earth's axis of rotation.  This field's strength is about 5 x 10-5 Tesla at the Earth's surface in the vicinity of the equator.  Free electrons in the stratosphere would tend to follow helical paths along geomagnetic lines of force.  As they do so, they will collide with uncharged particles and interact with other charged particles as described before.  After each such interaction, the velocity of the free electron involved will be changed in both magnitude and direction.  The geomagnetic field will once again constrain it to follow a new helical path about a magnetic line of force.  This line may be at a higher or lower altitude than its prior one.  Those electrons that diffuse downwards will tend to recombine with positively charged ions, but those that diffuse upwards will eventually reach an altitude where there are insufficient positively charged ions for them to recombine with, since the positively charged ions are constrained by the Earth's gravitational field to lower altitudes.  The electrons that attain higher altitudes will persist, interact with one another, and most will continue to diffuse upwards.

1.3  Electrostatic Forces

If a steady stream of electrons escaped from the Earth's gravitational field into outer space then the Earth as a whole would gradually accumulate a net positive charge.  As this charge built up, it would exercise an increasing attractive force upon the free electrons within the stratosphere.  Eventually this force it would reach such a magnitude that it would prevent all but the most energetic electrons from escaping from the Earth.  This net outflow would be balanced by the net inflow of electrons garnered from the Solar wind.  The inflow would be constrained by the Earth's magnetic field to enter the atmosphere at the magnetic poles.

1.4  Layers of Charge within the Ionosphere

During each day a new assembly of electrons is freed from the upper atmosphere by the Sun's radiation.  A pulse of electron gas will rise into the ionosphere.  During the night that follows, relatively few electrons will be freed.  The distribution of negative charge within the ionosphere should reflect its genesis.  Alternate layers of greater and lesser negative charge should be observable.

1.5  Recombination Luminescence

Atmospheric ionization takes place most abundantly at the region of the Earth currently facing the Sun most directly.  Once freed electrons have risen to the ionosphere, their high velocities and mutual repulsion will cause them to disperse.  Electrostatic and electromagnetic forces hinder dispersion upwards into space, and hence many will disperse laterally along the Earth's magnetic field lines.  As electrons disperse in this way, they will reach the Polar regions where the lines of force converge through the Earth's upper atmosphere into its crust.  Here many electrons will be brought back into contact with atmospheric gas molecules.  The convergence of the geomagnetic lines of force will result in a high concentration of returning electrons in the polar region.

The earth's electrons may be augmented at times by electrons from other sources such as our Sun; we will return to these in later sections of this paper.

The presence of a substantial negative charge in the atmosphere above the Earth's magnetic poles will induce a corresponding positive charge on the Earth's surface in these areas, since the Earth is, approximately speaking, a large spherical conductor.  The normal passage of airflow over the Earth's surface will pick up positive charge.  Electrostatic attraction will cause these positively charged particles and the negatively charged electrons to approach and combine in the atmosphere.  Some of energy gained by this recombination will be released in the form of visible light.  This effect may be a major contributor to the Aurora Borealis during the northern winters, and the Aurora Australis during the southern winters.  The same effects will hold at the opposite (summer) pole each season, but the high level of illumination due to the sun's continuous presence under these conditions would mask the visual effects.

1.6  Atmospheric Potential Difference

The escape of electrons from the equatorial and temperate regions of the Earth's stratosphere would result in a net residue of positively charged ions in the stratosphere above these regions.  The presence of a substantial positive charge above portions of the Earth's surface would induce a corresponding negative charge to form on its surface in these areas.

This large potential difference will have interesting consequences for those seeking to build "space elevators".  Carbon nanotubes offer the most promising material for the construction of such an elevator's hoist cables, and carbon is a good conductor.

1.7  Lightning

The induced negative charge on the earth's surface would attract the positive ions in the ionosphere, causing them to diffuse down through the atmosphere to the Earth's surface, where they can be neutralized by the negative charge.  The dense packing and vigorous thermal agitation of air molecules in the Earth's lower atmosphere would retard the downward flow of positively charged ions.  If the air as a whole were relatively still, one would expect an even gradient of charge distribution to be set up, with a strong positive charge at higher altitudes, and a strong induced negative charge at the Earth's surface.  However the Earth's atmosphere is never completely still, because of the solar energy that it gains each day.  Warmed air rises, cooler air moves in behind it.  This large-scale movement of air masses will distort the even gradient of charge distribution in the atmosphere.

The excess negative charge induced on the surface of the earth will tend to congregate in small bodies on the surface of the earth, especially those with small radii of curvature.  Over land, these will mainly be soil dust and small debris from degraded vegetation.  Over the seas, charge will tend to concentrate in wave crests, the droplets of water that are drawn from them by the passing wind, and salt crystals evaporated from such droplets if the water is salt.  The wind will thus lift many small bodies that carry excess negative charge and carry them.  In areas where the atmosphere has been heated and is rising, it will carry the small bodies with negative charge upwards.  New air will be drawn in from the sides to replace the rising air, bringing with is a fresh stock of negatively charged dust or droplets.

As the air rises, reduced atmospheric pressure will cause it to expand and to cool adiabatically.  If the air contains a fair amount of water vapour then the cooling will in due course bring the air's temperature down below its critical temperature, where it is super-saturated.  The water vapour will start to condense on the burden of dust and/or droplets carried by the air, including their excess negative charge.  Clouds will form at and above the altitude corresponding to the critical temperature.  The droplets will continue to rise in the column up upwelling air.  Many of them will collide with one another, often merging to form larger droplets.

As condensation and droplet merging proceeds, some of the water droplets will become so large that their weight will overcome the tendency of the rising air mass to lift them up, and they will fall downwards, with the dust and excess charge that they initially formed on still trapped within them.  As they fall, they will reach warmer zones and start to evaporate, but there is lag in this cycle, and the larger droplets may fall right out of the cloud formation, possibly reaching the land or water surface below in the form of rain.  The smaller droplets will lose sufficient mass through evaporation to be lifted again by the updraught of air before they get below the base of the cloud.  They will still contain the dust and excess negative charge on which they initially formed.

If the updraught of air continues for an appreciable length of time then more and more excess negative charge will become trapped in the cycle of rising small droplets being augmented by moist air condensing onto them, becoming heavier, falling back down to the region of the base of the cloud, losing mass through evaporation, and rising up once again.  A net excess of negative charge will accumulate within the lower regions of the cloud.  This will induce a positive charge to form on the earth's surface below the cloud, but it will have little effect on surrounding regions, from which new air will constantly flow in to replace the net updraught.  Winds blowing in from the surrounding regions will bring with them fresh negative charge.

If this accumulation of negative charge persist for long enough, a very substantial negative charge may build up within the lower regions of cloud.  If the potential difference rises to several million volts, it will be able to overcome the natural resistance of the air to the passage of electricity, and lightning will be the result.

In the tropics the warming effects of the sun on the Earth's surface are at their maximum, and the upward motion of warm air most pronounced.  Towering cumulus clouds with their heads in the stratosphere are a common occurrence.  Given the charge gradient already existing between the stratosphere and the Earth, these formations act as natural van der Graaf generators of gigantic size.  Negative charge is taken up from the Earth's surface into the cloud very rapidly, and a great deal of electric discharge activity results.

Similar but less well-known phenomenon called elves, sprites, and blue jets accompany the large-scale movement of charge between the tops of thunderstorms and the ionosphere [6].

2.  Other Planetary Atmospheres

The patterns of ionization and net the migration of electrons from tropical to polar regions should take place on any planet that has an atmosphere and that receives substantial amounts of solar (or stellar) radiation.

3.  The Sun's Atmosphere

The conditions that prevail in the Sun's atmosphere are very different from those of any planet.  The Sun radiates huge quantities of energy from its surface, and this radiation is fairly evenly distributed across its surface.  The Sun's surface temperature is about 5,785K, while the temperature within its chromosphere is about 5*106K.  At these temperatures, all molecules are dissociated into individual atoms, and all atoms are ionized.  Using equation {1} we find that free electrons near the Sun's surface would have a mean velocity of about 296km/s due to their temperature, if they could move freely, while those in the chromosphere would have a mean velocity of about 8,700km/s, which is way in excess of the escape velocity of the Sun's gravitational field, some 618km/s.  If gravitational forces were the only ones that needed to be considered then we could predict that large quantities of free electrons in the chromosphere would escape from the Sun's gravitational field into the surrounding vacuum of space.  But the loss of electrons in this way would cause the Sun to acquire a net positive charge that would, over time, grow to the point where it prevented the further escape of electrons, or at least ensured that the inflow of cooler free electrons from space balanced the outflow of high temperature electrons.  We will return to the question of the origin of the cooler electrons from space at a later point, but the Sun and other stars will clearly produce some hot electrons, which will after emission interact with other matter and magnetic fields, and through these interactions be cooled to the average temperature of whatever region of space they are in.

Electrons that diffuse above the chromosphere would be constrained to follow helical paths along the lines of the Sun's magnetic field, and many would return to the Sun's magnetic poles.  They would of course interact with one another, and the more energetic would disperse to ever higher altitudes.  There should therefore be a steady migration of electrons above all parts of the Sun's surface to its magnetic poles.  This would have to be compensated for by a return flow of electrons from the magnetic poles to all parts of the Sun's surface, and/or a contra-flow of positive ions to the poles.  But the very high temperature and violent thermal agitation of the Sun's substance would impede the flow of charge.  Over time, very large potential differences could build up across the Sun's surface in a north-south direction.  These may be relieved periodically by massive flows of positively charged material from one place on the Sun's surface to another, generally flowing from regions nearer the equator to those further from it.  Initially such flows would be constrained to follow the Sun's magnetic field, and hence flow close to the Sun's surface.  The flow of charge would create its own local magnetic field, coiled around the flow, redefining the magnetic field in the vicinity of the flow.  Flows of charge through the Sun's atmosphere would cause local heating, and the heated gases would tend to rise.  If the flow of charge persists for some time, it would arch above the Sun's surface, forming the familiar solar flare.  For most of the solar cycle, a large number of solar flares are active across its face.  The large quantities of heat that they generate as charge passes through the Sun's upper atmosphere may be a significant contributor to the extremely high temperatures found in the chromosphere.

If there is a large reservoir of charge associated with a solar flare, its flow will continue even when its path rises very high above the Sun's surface.  If the flare rises high enough, it will surmount most of the Sun's atmosphere, and the regions where its magnetic field is strongest.  At this point, the flare may act as a "puncture", sheathed by its own magnetic field, allowing material with a net positive charge to flow largely unhindered from both ends of the flare into space.  This mechanism may explain the massive coronal mass ejections (CMEs) that explode from the Sun periodically.  These would have a net positive charge, but would contain a mix of the Sun's surface substance, primarily ionised hydrogen and helium with a small admixture of heavier elements.

4.  Accretion Disks around Young Stars

A lot of star formation takes place within regions of space that contain large concentrations of gas and dust.  The stars that form in such regions attract neighbouring gas and dust that typically forms an accretion disk swirling around the star.  Matter is drawn from the inner edge of such a disk onto the surface of the star.

New stars tend to emit large amounts of ultraviolet radiation.  This would ionize the gas (mostly hydrogen) within the accretion disk.  As in planetary atmospheres, electrons freed in this way would tend to behave as gas particles, and individual electrons would acquire much higher velocities than the gas and dust particles.  The electron gas would tend to "boil off" from the accretion disk, leaving it with a net positive charge.  As material from the disk accretes onto the star's surface, so the star itself will accumulate a positive charge on its spinning surface.  This may contribute to the establishment of the star's magnetic field.  Eventually the excess positive charge on the star coupled with the excess positive charge in the inner regions of the accretion disk may become so large that it repels protons (ionized hydrogen atoms) in the accretion disk more strongly than the star's gravitational force attracts them.  Protons in the inner edge of the accretion disk would flee from the system. They would be constrained to flee along the lines of the star's magnetic field, eventually to fan out from the star's magnetic poles.

5.  Planetary Nebulae

When the cores of stars of moderate size run out of hydrogen to fuel their nuclear reactions, they expand to become red giants.  Our Sun will one day expand to include the Earth's orbit.  The surface temperature of a red giant is "only" about 3,500K, but this is still more than enough to ionize the gas in its atmosphere.  The gravitational field at the red giant's surface is greatly reduced on account of its distance from the centre; so free electrons are able to escape.  This leaves the star with a net positive charge.  The rotation of the star and hence this charge will give rise to a magnetic field with its poles aligned with the star's axis of rotation.  This field will retard the escape of electrons, as will the star's growing net positive charge.  The star will receruit free, cooler electrons from surrounding space to replace those that it has lost.

When the red giant no longer has enough hydrogen to fuel its burning, its core collapses.  The greatly increased temperature and pressure at the centre causes the fusion of helium and heavier elements into yet heavier elements, which releases bursts of energy that blow off almost all of the red giant's layers, leaving an extremely dense core that becomes an intensely hot white dwarf or neutron star, with a temperature of about 25,000K.  The white dwarf may have the mass of our Sun but only the diameter of the Earth, and its rotational period is typically measured in milliseconds.  The excess positive charge that it still carries is thus spun round at very high speed, leading to a very intense magnetic field.

The outer layers of the red giant expand to form a planetary nebula.  If gravitational forces were the only ones that shaped this expansion then one would expect a spherically symmetric bubble to emerge.  In practice, planetary nebulae are considerably more complex [4].  They are usually elongated, and often comprise two overlapping bubbles of more or less equal size with widely separated centres, sometimes with a third, smaller bubble centred on the white dwarf.  The high temperature of the white dwarf results in intense ultraviolet radiation that ionizes the expanding gas bubbles.

If we take the magnetic field of the star into consideration, and also the fact that its atmosphere is ionized, then the characteristic shape of planetary nebulae may be explained by assuming that when the central core explodes, the motion of the ionized material that is expelled in any direction other than along the star's magnetic poles will be retarded and redirected by the magnetic field.  This will create some backpressure, and much of the material will be ejected in the directions of the two magnetic poles where it does not have to cross the magnetic field.  As this material clears the magnetic field, it will be free to expand, and its net positive charge will ensure that it does.  Hence two bubbles evolve.  Whatever hydrogen remains in the Red Giant's atmosphere will be most rapidly accelerated by the electrostatic repulsion, and will form the largest bubble.  Heavier elements which may not be fully ionized and which carry neutron ballast will be accelerated less, and hence form smaller bubbles.

The heaviest elements in the red giant's atmosphere may well have a high enough mass to charge ratio to pass right through the star's magnetic field, albeit with reduced velocity.  They, together with the lighter elements that they drag with them, could account for the third, smaller bubble that is centred on the remnant white dwarf.  Some elongation of this bubble in the polar direction may be expected.

Depending on the strength of the star's magnetic field and the stage that has been achieved in this process, the bubbles will be more or less apparent.  Interstellar gas and dust may distort the evolving pattern, as may stellar winds from neighbouring stars.

6.  Accretion Disks around Neutron Stars

Where an accretion disk forms around a neutron star (sometimes material drawn from a close binary partner that is undergoing an expansionary phase) the same sort of phenomena as described for Harbig-Haro objects above should occur – material should be ejected along the poles of the star's magnetic field.  An extra development should arise from the fact that as material collides with the surface of a neutron star, it does so with such high velocity that gamma rays are emitted.  When these strike material in the surrounding accretion disk, some electron and positron pairs will be created.  Some of these will recombine to liberate energy again, but a portion of the positrons will escape.  Given their extremely high positive charge to mass ratio, they will be impelled to flee from the positively charged neutron star with very high velocities.  As with fleeing protons, they will be constrained to flee along the lines of the neutron star's magnetic fields.

7.  Black Holes and Quasars

Observations suggest that there are probably extremely massive gravitationally collapsed stars (black holes) at the centres of some, perhaps most galaxies, and probably all large galaxies such as our own Milky Way.  There is usually a high concentration of gas and dust at the hubs of galaxies, so a black hole in this location would be accompanied by a large accretion disk.  The regime that obtains in this situation would be similar in many ways to one in which a neutron star is surrounded by an accretion disk, but on a much larger scale.  Matter on the inner edge of the accretion disk is dragged into the black hole's event horizon at speeds almost equal to that of light.  The extremely high velocity gives rise to frictional heating.  Up to 10% of the mass that falls into the event horizon is converted to energy and radiated in a wide spectrum, including ultraviolet and gamma rays.  As before, the ultraviolet radiation will ionise gas in the accretion disk, and electrons will tend to boil off from it, leaving it with a net positive charge.  The regions closest to the black hole's event horizon would have the strongest positive charge.  Some of the matter falling into the black hole will have a net positive charge.  The net positive charge of the accretion disk will grow over time to the point where it repels infalling protons electrostatically more than the black hole's gravitational field attracts them.  This may seem an extreme condition, but we need to bear in mind that the electrostatic repulsion between two protons exceeds their mutual gravitational attraction by a factor of about 1036 (see equation {4}).  Let us consider a black hole with one million times the mass of our Sun, i.e. a mass of about 3x1036 kg.  If it and its accretion disk contain as little as 2 kg of ionized protons without their associated electrons, then infalling protons (ionized hydrogen) will experience an electrostatic repulsion of approximately the same magnitude but of opposite sense to the gravitational attraction of the black hole.

7.1  Cosmic Rays

If the inflow of ionized gas persists, with the passage of time the black hole will acquire sufficient positive charge to repel ionized hydrogen.  But helium is relatively abundant, and it will continue to sink towards the inner regions of the accretion disk and the black hole, since it carries a ballast of neutrons, and hence experiences double the gravitational attraction per proton.  As the helium moves inwards it will become completely ionized, adding to the net positive charge until it doubles.  As soon as hydrogen in the accretion disk is ionized, its proton will experience an electrostatic repulsion twice as great as the attractive force of the black hole's gravitational field.  This force will eject protons from the vicinity of the black hole at relativistic velocities.

If the inflow of gas persists, the net positive charge will increase to the point where it repels fully ionized helium.  Although much less abundant, there are significant amounts of heavier elements in most interstellar gas and dust clouds.  They have a higher neutron to proton ratio than does helium, and so will continue to be drawn towards the black hole even when fully ionized, increasing the net positive charge yet further.  At this stage, ionized helium will be ejected from the black hole at relativistic velocities.  This scenario will repeat over time for successively more massive nuclei, although their abundance will be much lower than that of hydrogen and helium.

7.2  Jets Emanating from Black Holes

The rapidly rotating positively charged inner edge of the accretion disk will create a magnetic field, roughly toroidal in shape.  The moment hydrogen is ionized in the vicinity of a black hole, its proton nucleus will be repelled by the positively charged accretion disk and rapidly accelerate away from it to relativistic velocities.  Positrons will be formed by collisions between emitted gamma rays and material in the accretion disk.  These, together with the protons, will be accelerated at relativistic speeds from the poles of the magnetic field. 

Given the strong repulsion that positrons and protons have for one another, one would expect the jets of ejecta to fan out as they leave the immediate vicinity of the black hole and its accretion disk.  As these jets of positive charges penetrate far out into space, they will attract free electrons from surrounding space, and also impact on the gas and dust that happens to lie in their path with such energy that they will knock many of its electrons free.  Electrons will flow towards and into the jets.  They will of course be strongly attracted to the positrons and protons within it, but recombination will not be immediate because of the relativistic speeds of the positrons and protons.  Some recombination will certainly occur, and this will give rise to a charge gradient, with more positive charge near the black hole and accretion disk from which the jet emanates.  Once within the jet, electrons will tend to flow towards the region of greater positive charge.  They will thus move in the opposite direction to the positive charge.

Viewed from the frame of reference of the negative charge, the positive charge will flow away from the black hole at relativistic speeds, giving rise to a strong enclosing magnetic field coiled about the stream of positive charge.  As the electrons cross this field, they will experience a force impelling them towards the centre of the stream of positive charge.

Viewed from the frame of reference of the positive charges, the free electrons will appear to flow towards them at relativistic speeds, giving rise to a strong enclosing magnetic field coiled about the stream of negative charge.  As the positive charges cross this field, they will experience a force impelling them towards the centre of the stream of negative charge.

The fact that the positive charge is inter-penetrated by substantial amounts of negative charge will reduce the electrostatic repulsion between the positive charges, and similarly for the electrons.  Taken together, these various forces may be strong enough to complete the "pinch" effect.  The positive charges and the negative charges will follow a common linear path (although they traverse it in different directions), a so-called stellar jet.  Instead of diverging, the jets of positrons and protons emanating from both poles of the black star will act as self-collimating beams, penetrating perhaps hundreds of light years out into space.  Huge jets that conform to this general description are observed emerging from what are assumed to be super-massive black holes accreting surrounding material at the centres of galaxies.

8.  Free Electrons and the Shape of the Cosmos

We have suggested that some of the matter falling into black holes carries a net positive charge, since some of its electrons are stripped from it by the intense radiation emitted from the accretion disk that encircles the black hole.  Once within the black hole's event horizon, all matter and radiation is drawn inevitably towards the singularity at the centre of the black hole, in due course entering it.  Classically, the singularity has zero volume, so the mass density and time dilation within it are infinite.  For the material that enters the singularity, time ceases to pass.  The principle of cause and effect which underpins all of the laws of physics ceases to apply to all matter and energy within the singularity, because cause and effect would be separated by an infinite amount of time.  We can therefore predict that any excess positive charge that enters the singularity can no longer exert any electrostatic effects on any other particle outside of the singularity, and hence the excess positive charge that enters the singularity becomes sequestered.  So far as the universe outside is concerned, it has ceased to exist.

Let us consider this rather radical hypothesis, explore the consequences it would have if true, and try to determine in what ways these consequences could be discerned and the hypothesis thus tested.

If positive charge from material that falls into black holes is continually sequestered in the singularities within these black holes then there should be a net build-up of negative charge (primarily free electrons) in the space surrounding black holes.  It is believed that most black holes are created at the centres of galaxies, and that larger galaxies have more, and larger, black holes.  Larger galaxies should therefore generate and contain more free electrons that should smaller ones.  The free electrons would of course repel one another, and lacking the countervailing attraction of the positive charge now sequestered, would normally flee from one another, the vicinity of the black hole, and, over time, the galaxy that contains it.  While the free electrons are very light and could accelerate to relativistic velocities very quickly, their exodus from the galaxies in which they are set free will be hindered by the gas, dust, and magnetic fields that permeate all galaxies.  Interactions with these will repeatedly serve to reduce the free electron's velocities to a value closer to the temperature of the regions through which they pass.  Larger galaxies tend to contain more and larger black holes, and should thus generate more free electrons that would smaller galaxies.  Free electrons in large galaxies will take longer to disperse and escape into intergalactic space because of their galaxies' larger sizes (diameters of some hundreds of thousands of light years).  Larger galaxies should therefore accumulate stronger apparent negative charges than smaller galaxies.

When the cosmos first came into existence there were, it is believed, no black holes.  Some relatively small ones may have come into existence as a result of extreme density fluctuations in some regions.  But the great majority of black holes, and the largest of them, were formed from the gravitational collapse of very large stars during the first half of cosmic existence.  As they formed, grew, and sequestered positive charge, a growing apparent net negative charge accumulated within galaxies and permeated into the spaces between them.  This net negative charge would exert a repulsive force on the galaxies, countering at least to some extent the force of gravity.  This repulsive force would build up steadily over time.

8.1  Sanity Check

The claim that the observed acceleration of cosmic expansion is due to nothing more than plain old electrons is perhaps a startling one.  Let us do a few "back of the envelope" calculations to see how plausible this claim is.  Rather than trying to deal with the entire cosmos, which is rather unimaginably large, let us take our own Milky Way galaxy [15] as a yardstick.  It has been around for about 12 billion years, which is most of the time since the Big Bang.  If it has sequestered its fair share of positive charge and liberated the corresponding number of electrons, how much would that add up to?

If we work with the assumption that dark energy accounts for about 75% of the mass/energy of the cosmos as suggested in [16] then our Milky Way should have by now sequestered about 3.7*106 kg (i.e. about 3,700 tonnes) of protons in its black holes, in the process releasing about 2 tonnes of free electrons, most of which would have percolated out of the Milky Way into intergalactic space by now.  These are not big, scary numbers.  They are in fact disturbingly small and plausible.  Do we have any grounds to doubt that such quantities of positive and negative charge could have been respectively sequestered and dispersed?

8.2  Long Term Prognoses

If the hypotheses described in this paper hold true then one can expect that an ever-increasing amount of positive charge will be sequestered in the singularities of black holes.  The galaxies that contain the black holes will also eventually be consumed by them.  This process will result in an ever-increasing amount of net negative charge permeating the cosmos, which would cause the expansion of the cosmos to accelerate further.  Eventually, all matter and energy will be absorbed into a black hole somewhere in the cosmos.  According to classical physics, the cosmos would then fall into perpetual darkness, expanding endlessly.

It was once thought that black holes are forever, but Stephen Hawking has shown that over time they gradually "evaporate" away, providing the ambient temperature is low enough (some millionths of a degree for a large black hole).  As of now, the ambient temperature does not fall below the 2.7K cosmic microwave background radiation, the after-glow of the Big Bang.  But as the cosmos expands, the cosmic microwave background radiation cools.  After a very long time (e.g. a googol years) the background radiation should be cool enough for the evaporation of black holes to ensue.  For large black holes this will be a leisurely affair, but the process accelerates as the black holes get smaller, until at last they reach a critical limit and explode back into normal existence.  It is at this time still unknown whether all of the matter and energy, and its state at the time that it fell into the black hole, will be recovered, or whether it has been lost forever.  Many scientists, including Hawking, believe that the original state will be recovered.  If so, the net positive charge that has been captured by the black hole over the course of its existence will suddenly reappear, adding considerably to the vigour of the ensuing explosion.

And so perhaps a long time from now the darkness of the cosmos will be lightened again, and the positive charge that was lost will be recovered, balancing in theory the net negative charge that caused the cosmos to expand at an ever-accelerating rate.  But the news of the re-arrival of the missing positive charge will spread at "only" the speed of light, and it may be too late to ever stop the runaway expansion that ensued during its long sequestration.

8.3  That "Infinite Density" Thing

Einstein's general theory of relativity includes amongst its many predictions the possibility of black holes.  If the mass of a star is large enough, once it has burned up its fuel and starts collapsing, it will continue to collapse until a singularity forms at its centre.  The volume occupied by the singularity is zero, and within it the curvature of space, the density of matter, and time dilation become "infinite", a disturbing concept.

We can defend outselves from this concept by invoking Heisenberg's uncertainty principle.  This states that it is impossible to exactly measure both the location and the momentum of a particle, because the instruments that we use to perform the measurement (light waves, electrons, or whatever) interfere slightly with the object that we are measuring as we measure it.  The smaller the particle, the greater the uncertainty.  More audaciously, Heisenberg asserted that the particles themselves, and the other particles with which they interact, cannot know exactly where they are and exactly how fast they are travelling.  Quantum field theory has since amply proven the accuracy of Heisenberg's claims.

The singularity within a black hole can be considered to be a particle, although of prodigious mass.  Therefore, we, and it, cannot know exactly where it is and exactly how fast it is travelling.  Given its enormous mass, the degree of uncertainty as to its location is so small as to be utterly irrelevant.  But what it does mean is that we can assign a vanishingly small but greater then zero size to the singularity, corresponding to its quantum-mechanical "fuzziness".  And therefore we can assert that within it, mass density and time dilation are not truly infinite, just ridiculously large.  This offers a small crumb of comfort to those, like me, who find the notion of infinite density and space curvature somehow obscene, even when hidden behind an event horizon.

9.  References

Apart from the references given below, the book "The Road to Reality: A Complete Guide to the Laws of the Universe" by Roger Penrose, ISBN-10: 0679454438, gives an excellent introduction to the cosmos as we find it today, its evolution, and the forces that have shaped it.

  1. "Kennelly-Heaviside Layer", Wikipedia, http://en.wikipedia.org/wiki/Kennelly-Heaviside_layer
  2. Wåhlin, L., "Elements of fair weather electricity". J. Geophys. Res., 99, 10767-10772, 1994.
  3. Holzworth, R. H.; Dazey, M. H.; Schnauss, E. R.; Youngbluth, O., "Direct measurement of lower atmospheric vertical potential differences", Geophysical Research Letters, Volume 8, Issue 7, p.783-786, http://adsabs.harvard.edu/abs/1981GeoRL...8..783H
  4. Holzworth, Robert H., "Hy-wire measurements of atmospheric potential", Journal of Geophysical Research, Volume 89, Issue D1, p.1395-1402, http://adsabs.harvard.edu/abs/1984JGR....89.1395H
  5. Maynard, N. C.; Croskey, C. L.; Mitchell, J. D.; Hale, L. C., "Measurement of volt/meter vertical electric fields in the middle atmosphere", Geophysical Research Letters, vol. 8, Aug. 1981, p.923-926, http://adsabs.harvard.edu/abs/1981GeoRL...8..923M
  6. Matt Heavner, "Red Sprites and Blue Jets", http://elf.gi.alaska.edu/
  7. "Telluric Current", Wikipedia, http://en.wikipedia.org/wiki/Telluric_current
  8. "Surprising Jupiter", NASA JPL, http://www.jpl.nasa.gov/news/features.cfm?feature=534
  9. "Lightning on Saturn", NASA JPL, http://www.jpl.nasa.gov/multimedia/audioclips/saturn-lightning/
  10. "Jupiter's Aurora", NASA JPL, http://pds.jpl.nasa.gov/planets/captions/jupiter/aurora.htm
  11. Saturn's North Pole Hexagon and Aurora", NASA JPL, http://saturn.jpl.nasa.gov/multimedia/images/image-details.cfm?imageID=2549
  12. "Coronal Mass Ejections", http://science.nasa.gov/ssl/pad/solar/cmes.htm
  13. Bruce Balick and Adam Frank, "The Extraordinary Deaths of Ordinary Stars", Scientific America, July 2004 http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=000F0061-745A-10CF-AD1983414B7F0000
  14. Christopher J. Conselice , "The Universe's Invisible Hand", Scientific American, February 2007 http://www.sciam.com/article.cfm?chanID=sa006&colID=1&articleID=1356B82B-E7F2-99DF-30CA562C33C4F03C
  15. "Dark energy", Wikipedia, http://en.wikipedia.org/wiki/Dark_energy
  16. "Galaxy groups and clusters", Wikipedia, http://en.wikipedia.org/wiki/Galaxy_cluster
  17. "Milky Way galaxy", Wikipedia, http://en.wikipedia.org/wiki/Milky_way

10.  Addenda

The body of the paper above will not change from the date of its publication, 2007-08-29.  Any additional runinations on the topic will be captured as dated items in this, the addenda section.

2007-08-30:  Why are elliptical galaxies elliptical?

If gravity were the prime force acting on a galaxy then one would expect most galaxies to be spherical, except those that are spinning &ndash they should adopt a disc shape.  There are indeed many of these, the spiral galaxies.  But about 70% of all galaxies are elliptical in shape (see Wikipedia: Elliptical Galaxies).

Why?

A recent paper published in Arxiv by Michael J. Longo, "The Axis of Opportunity: The Large-Scale Correlation of Elliptical Galaxies" (see http://arxiv.org/ftp/arxiv/papers/0708/0708.4013.pdf) provides a possible clue.  Almost all of the elliptical galaxies in our celestial neighbourhood (that is to say, those with red shifts <0.20) have their major axes aligned, pointing to a common point in the cosmos.

Suppose elliptical galaxies started out life as spherical galaxies, with some massive black holes at their centres, helping to hold them together.  As described in the main paper, the black holes would liberate a large population of free electrons over time.  These would take time to percolate out of the galaxy, giving it an apparent net negative charge.

Suppose too that somewhere in the galactic neighbourhood of the spherical galaxy there was a very large, very old galaxy in which the central black holes had consumed most of the material of the galaxy, spitting out vast numbers of electrons in the process, and that these electrons ballooned out into the void, repelling the smaller, younger galaxies in the vicinity and creating an apparent massive void in the cosmos.

Suppose too that the smaller galaxies that are being expelled from the vicinity of the very old, large galaxy happen to contain a large amount of dark, non-baryonic matter.  This matter, we believe, would not be affected by electrostatic charge.  It would lag behind the baryonic matter of the fleeing galaxy, eventually causing the galaxy to extend along the vector of acceleration away from the central old galaxy.  The gravitational attraction between the baryonic and non-baryonic matter within the young galaxy would keep the whole together, more or less, but over time it would gradually distort into a, well, elliptical shape, for the want of a better word.

For completeness, let us consider what would happen to small rotating galaxies that have an disk structure similar to our Milky Way and that get swept away from a large, old galaxy by a tide of free electrons.  They too would get distorted into an elliptical shape as their baryonic matter leads the charge to flee from the large, old galaxy.  This would create turmoil in the rotation of the disk.  The residual angular momentum of the galaxy would cause the major axis of the ellipse to rotate, but the electrostatic repulsion would in effect apply a torque to the galaxy contrary to its existing rotation.  With the passage of much time, the fleeing galaxy's rotation would become largely stilled, and it would become in the end difficult to distinguish it from a similar-sized galaxy that had started out with relatively little rotation.