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SOLAR ENERGY

By Charles Rhodes, P. Eng., Ph.D.

INTRODUCTION:
This web page shows that fusion of solar hydrogen will provide the Earth a reliable source of energy for about another 10 billion years.
 

ENERGY IN THE UNIVERSE:
From an energy persective the universe can be viewed as an assembly of electrons, anti-electrons (positrons), protons, anti-protons, neutrinos and photons. The energy contribution of neutrinos as compared to charged particles is thought to be relatively small.

The charged particles are almost all bound into neutral atoms, neutral molecules or neutral plasma. If a significant fraction of the charged particles were free electromagnetic radiation (photons) could not propagate long distances through the universe.

The total energy of an isolated particle can be thought of as consisting of core field energy plus extended field energy plus kinetic energy.
 

ENERGY THAT CAN DO WORK IN THE SOLAR SYSTEM:
Obtaining work from particle core field energy requires a matter-antimatter reaction that does not normally occur in the solar system for lack of antimatter. Hence, in the solar system all energy that can do work comes from pre-existing kinetic energy or from a change in overlap of extended particle fields.
 

NEUTRINOS:
Neutrinos account for the energy and spin difference between isolated neutrons and the combination of equal numbers of isolated electrons plus isolated protons. In terms of energy this difference is less than 0.1%. Neutrinos propagate at about the speed of light. The rest energy of an isolated neutrino is believed to be zero.
 

PHOTONS:
A photon is a quantum of electromagnetic energy that propagates at the speed of light. The rest energy of a photon is zero. A fundamental assumption in the formulation of quantum mechanics is that the amount of energy Ep conveyed by a photon is given by:
Ep = h Fp
where:
h = the Planck constant
= 6.626069 X 10^-34 J-s
and Fp is the electromagnetic wave frequency as seen by an inertial observer.

It can be shown that the Planck constant is really a composite of other natural physical constants and that quantization of electromagnetic radiant energy is a result of quantization of charge. However, the Planck constant frequently occurs in practical energy transition calculations, so it is convenient to treat it as a separate physical constant.
 

STABLE PARTICLE DESCRIPTIONS:
An electron is a stable particle with a charge of - 1.60217646 X 10^-19 coulombs and a rest mass of 0.00054857990946 atomic weight units. An electron has associated with it extended external electric, magnetic and gravitational fields. These extended fields contain a small portion (~ 1%) of the total electron rest mass energy.

An antielectron is a stable particle with a charge of + 1.60217646 X 10^-19 coulombs and a rest mass (rest energy) of 0.00054857990946 atomic weight units. An antielectron has associated with it extended external electric, magnetic and gravitational fields. These extended fields contain a small portion (~ 1%) of the total anti-electron rest mass energy. As compared to an electron an antielectron is believed to have a sign reversal in its gravitational field unit vector.

A proton is a stable particle with a charge of + 1.60217646 X 10^-19 coulombs and a rest mass of 1.007276466812 atomic weight units. A proton has associated with it extended external electric, magnetic field and gravitational fields. These fields contain a small portion (~.34%) of the total proton rest mass energy.

An antiproton is a stable particle with a charge of - 1.60217646 X 10^-19 coulombs and a rest mass of 1.007276466812 atomic weight units. An antiproton has associated with it extended electric, magnetic field and gravitational fields. The extended fields contain a small portion (~.34%) of the total anti-proton rest mass energy. As compared to a proton an antiproton is believed to have a sign reversal in its gravitational field unit vector.
 

CHARGED PARTICLE FORMATION AND ANNIHILATION:
In a very high radiation density environment a sufficiently energetic photon will form a proton-anti-proton pair. In a high radiation density environment a sufficiently energetic photon will form an electron-anti-electron pair.

In a low radiation density environment, on contact a proton and an anti-proton will spontaneously convert to a very high energy gamma photon. In a low radiation density environment, on contact an electron and an anti-electron will spontaneously convert to a gamma photon.

A proton and an anti-proton have opposite charges. An electron and an anti-electron have opposite charges. Hence for free normal-anti-particle pairs to exist, at formation these particles must have sufficient kinetic energy to escape from their mutual electrostatic potential energy well.

The exact origin of free electrons and free protons in interstellar space is the subject of speculative cosmological theorys that are beyond the scope of this web site. It is believed that free electrons and free protons formed in the early universe at a time when the radiation density in the universe was very high. Subsequently expansion of the universe reduced the radiation density leaving free electrons and free protons as stable particles. The free electrons and free protons then aggregated to form atomic hydrogen. In so doing they emitted characteristic radiation that is readily observed with a radio telescope. At the same time antihydrogen likely formed, but the anti-matter was gravitationally repelled away from our local universe by the normal matter.

On Earth we do not observe anti-electrons or anti-protons except as products of high energy nuclear collisions. In the our solar system the numbers of normal electrons and normal protons vastly outweigh the numbers of anti-electrons and anti-protons. Any anti-matter that is formed on Earth is soon destroyed by contact with normal matter.
 

PHOTON GENERATION AND EMISSION:
The weakly bound electrons and protons forming hydrogen are the primary source of energy in the local universe. These electrons and protons are in a relatively high rest energy state. When the vector fields of particles with opposite charges overlap the total field energy decreases. Hence, the law of conservation of energy forces the particles' kinetic energy to increase. This mechanism manifests itself as electric or magnetic attraction.

When the vector fields of particles with like charges overlap the total field energy increases. Hence, the law of conservation of energy forces the particles' kinetic energy to decrease. This mechanism manifests itself as electric or magnetic repulsion.

Randomly moving charged particles in matter gradually lose their acquired kinetic energy by emitting photons of electromagnetic radiation. Charged particles can gain energy by capturing photons of electromagnetic radiation. At steady state conditions the rate of photon energy capture equals the rate of photon energy emission. The photon emission is known as black body radiation.
 

INTER-STELLAR MATTER:
In interstellar space electrons and protons aggregate to form atomic hydrogen. Neutral atomic hydrogen atoms aggregate in pairs to form molecular hydrogen. This particle aggregation causes the hydrogen to emit photons with the spectral distribution of black body radiation. The experimentally measured cosmic background radiation has a peak at 160.2 GHz, and has slight variations with respect to direction, indicating that inter-stellar space contains a low concentration of matter at a temperature of about 2.725 degrees K. It is believed that the stars formed by gravitational aggregation of this inter-stellar matter.
 

RADIATION CONVERSION:
The Earth absorbs solar radiation and emits thermal infrared radiation with characteristic temperatures in the range 220 degrees K to 285 degrees K. The low temperature of inter-stellar space acts as a sink for the thermal infrared radiation emitted by the Earth. Thus the Earth converts absorbed solar photons into a larger number of emitted infrared photons. This photon conversion action enables work and life processes on Earth.
 

STELLAR MATTER:
Spectral analysis of star light indicates that stellar matter is mostly hydrogen (or anti-hydrogen).

In stars the gravitational potential energy well causes molecular hydrogen to gradually convert into deuterium. The deuterium then fuses into light elements, then into common elements and finally via a super-nova into heavy elements. The Earth, which contains an abundance of common and heavy elements, is believed to be a remnant from another star, not a product of our sun, because our sun has not advanced far enough along its fusion cycle to form the heavier elements present on Earth.
 

STELLAR IGNITION:
Matter in interstellar space is gravitationally attracted to star formation sites. Spherical symetry and sufficient matter causes a very large pressure at the center of stars. The kinetic energy that the matter acquires while falling into a star's gravitational potential well provides the random kinetic energy (heat) necessary to initiate deuterium formation near the center of the star. Note that some of the acquired thermal energy is radiated away before deuterium formation reactions commence. Once the concentration of deuterium is sufficient to maintain D-D fusion reactions the star will ignite.
 

PRE-FUSION HEATING:
From Newton's law of gravitation, the kinetic energy Ek acquired by a hydrogen molecule with mass Mh falling into the sun's gravitational potential well and impacting the sun at radius Rs from the center of the sun is given by:
Ek = (G Ms Mh) / Rs
where:
G = 6.674 X 10^-11 m^3 kg^-1 s^-2 = gravitational constant
Ms = 1.98892 X 10^30 kg = solar mass
Mh = 1.008 X 10^-3 kg / 6.023 X 10^23 = mass of one hydrogen molecule
Rs = 6.955 X 10^8 m = solar radius

Numerical substitution gives:
Ek = (6.674 X 10^-11 m^3 kg^-1 s^-2 X 1.98892 X 10^30 kg X 1.008 X 10^-26 kg) / (6.023 X 6.955 X 10^8 m)
= [(6.674 X 1.98892 X 1.008) / (6.023 X 6.995)] X 10^-15 m^2 kg s^-2
= 0.3176 X 10^-15 J
= 0.3176 X 10^-15 J X 1 eV / 1.602 X 10^-19 J
= 0.1982 X 10^4 eV
= 1982 eV

This is the depth of the solar gravitational potential well at the surface of the sun.
 

STELLAR EVOLUTION:
The solar gravitational potential well gets deeper toward the center of the sun. Within the sun, as the gravitational potential well gets deeper the molecular and hence atomic kinetic energy increases. At the center of the sun the gravitational potential well is so deep that atomic hydrogen gradually converts into deuterium (H-2). Once there is a sufficient concentration of H-2 at a high enough temperature, (H-2)-(H-2) fusion reactions occur. These reactions form H-3 and He-3 plus free neutrons and free protons. The H-3 and He-3 then quickly react with more H-2 to form He-4 plus more free neutrons and free protons. The free protons are simply energetic atomic hydrogen. The free neutrons are absorbed by other nuclei and via subsequent electron emission form elements with higher atomic numbers. Toward the end of a star's life the low atomic weight elements fuse with each other and with the free neutrons to form higher atomic weight elements.
 

SOLAR DENSITY:
The present solar volume Vs is given by:
Vs = (4 / 3) Pi Rs^3
= 1.333 X 3.14159 X (6.955 X 10^8 m)^3
= 1409 X 10^24 m^3

Hence the present average density of the sun is given by:
Ms / Vs = 1.98892 X 10^30 kg / 1409 X 10^24 m^3
= 1412 kg / m^3

By comparison the density of liquid hydrogen on Earth is:
67.8 kg / m^3

These calculations indicate that the hydrogen in the sun, in addition to being very hot is very highly compressed. Once deuterium formation commenced the temperature within the sun rose. Meanwhile mass accumulation from gravitational aggregation of inter-stellar matter, primarily molecular hydrogen, continued.
 

STELLAR NUCLEAR PROCESSES:
Conversion of molecular hydrogen into deuterium and fusion of deuterium into higher atomic weight atoms reduces the total aggregated electron and proton field energies. There is a corresponding increase in particle kinetic energy (temperature) that causes release of energy into space by emission of thermal photons.

In a star hydrogen molecules under extremely high pressure slowly combine to form deuterium. Direct free neuton formation is unlikely because a free neutron contains more energy than the sum of the energies of a free electron and a free proton.

The deuterium formation process requires almost direct proton-proton impacts with sufficient kinetic energy (1.02 MeV) to form an electron-positron pair. The positron is ejected leaving deuterium behind. Statistically such high energy direct impacts occur infrequently and hence this reaction is very slow. In each such interaction both energy and momentum must be simultaneously conserved. The capacity of the emitted positron to take up surplus energy and momentum is very limited. This slow rate of deuterium formation is what gives stars their long lives.

During a star's early life atomic hydrogen gradually converts to deuterium. Later, after the deuterium concentration in the star becomes sufficient, deuterium atoms fuse with each other to form tritium and helium-3. Formation of tritium releases energetic free neutrons. Formation of helium-3 releases energetic free protons. These energetic particles heat the core of the star. The tritium and helium-3 then readily combine with more deuterium to form helium-4. These processes release more energetic free neutrons and energetic free protons which further heat the star and hence increase the deuterium formation and fusion rates.

Tritium and helium-3 can also combine with helium-4 to form lithium-6. The lithium-6 will then combine with tritium or helium-3 to form more helium-4. These processes also release energetic free protons and free neutrons which further heat the star and hence further increase the deuterium formation and fusion rates.

Thus deuterium, tritium, helium-3 and lithium-6 are transition isotopes in the conversion of atomic hydrogen into helium-4. However, formation of tritium, helium-3 and lithium-6 does not occur until there is a significant deuterium concentration. Hence in the early life of a star the dominant energy production reaction is conversion of atomic hydrogen into deuterium whereas in the mid life of the star energy production reactions involving deuterium-deuterium reactions leading to production of helium-4 become more important.

The heavier elements are only produced late in the life of a star.

At each reaction stage energy is released and the average atomic energy per nucleon decreases. The surplus kinetic energy converts to photons that radiate into outer space. Once the deuterium-deuterium reactions start the sun's thermal output increases.

When there is a sufficient density of He-4 then two He-4 atoms can fuse to form Li-7 plus a free proton or Be-7 plus a free neutron. The Be-7 is unstable and decays to form more Li-7. However, the Li-7 is naturally stable and is a stepping stone to forming the heavier elements.
 

TABLE OF RELATIVE ENERGIES PER NUCLEON AS COMPARED TO CARBON-12 = 1.000000000000
Free Proton = 1.007276466812

Free Electron = 0.00054857990946

Free Proton plus Free Electron: (1.007276466812 + 0.00054857990946)
= 1.007825046721

Hydrogen
= 1.00782503207

Thus there is a drop in energy in forming hydrogen from a free proton and a free electron.

 

Deuterium 2.0141017778 / 2
= 1.0070508889

Thus there is a drop in energy when two hydrogen atoms combine to form deuterium.

 

Neutron = 1.00866491600

Tritium 3.0160492777 / 3
= 1.0053497592

Tritium + Neutron (3.0160492777 + 1.00866491600) / 4
= 1.006178548

Thus there is a drop in energy when two deuterium atoms combine to form tritium plus a neutron. A primary source of neutrons is D-D reactions in stars. Another mechanism for neutron formation is electron-positron pair formation in atomic nuclei where the positron is emitted and the electron then combines with a proton to form a neutron.
 

Helium-3 3.0160293191 / 3
= 1.0053431063

Helium-3 Plus Proton (3.0160293191 + 1.007276466812) / 4
= 1.005826446

Thus there is a drop in energy when two deuterium atoms combine to form helium-3.

 

Helium-4 4.00260325415 / 4
= 1.00065081354

Deuterium plus tritium (2.0141017778 + 3.0160492777) / 5
= 1.006030211

Helium-4 plus Neutron (4.00260325415 + 1.00866491600) / 5
= 1.002253634

Thus there is a drop in energy when deuterium and tritium combine to form helium-4.

 

Deuterium plus helium-3 (2.0141017778 + 3.0160293191) / 5
= 1.006026219

Helium-4 plus Proton (4.00260325415 + 1.007276466812) / 5
= 1.001975944

Thus there is a drop in energy when deuterium and helium-3 combine to form helium-4.

 

Lithium-6 6.015122795 / 6
= 1.002520466

 

Carbon-12 12.0000000 / 12
= 1.0000000
 

Note that as heavier stable atoms are formed the nuclear binding energy becomes more negative and hence the atomic weight per nucleon gradually decreases. At the atomic number of lead this trend reverses. Hence elements with atomic weights greater than lead are unstable.
 

SOLAR RADIATION:
The solar irradiance at the Earth's orbit measured via satelite borne instruments is:
1.361 kW / m^2.

The radius of the Earth's nearly circular orbit around the sun is about 150 X 10^6 km. Hence the surface area of a sphere with the radius of the Earth's orbit is:
4 Pi R^2 = 4 X 3.14 X (150)^2 X 10^12 km^2 X 10^6 m^2 / km^2
= 28.26 X 10^22 m^2

The power of the present total photon radiation emitted by the sun is:
1.361 kw / m^2 X 28.26 X 10^22 m^2
= 38.462 X 10^22 kW
= 3.8462 X 10^26 J / s

From Einstein's relationship of:
E = M C^2,
the corresponding rate of solar mass loss due to nuclear reactions is:
(3.8462 X 10^26 J / s) / C^2
= (3.8462 X 10^26 J / s) / (9 X 10^16 m^2 / s^2)
= .4274 X 10^10 kg / s

If the sun is in the early part of its life, when the dominant source of solar heat is conversion of hydrogen into deuterium, the corresponding rate of solar hydrogen mass loss is:
(.4274 X 10^10 kg / s) / [(1.00782503207 - 1.0070508889) / 1.00782503207]
= (.430744 X 10^10 kg / s) / (.00077414317)
= 556.414 X 10^10 kg / s

If the sun is further on in its life, when the dominant source of solar heat is fusion of deuterium into helium-4, the corresponding rate of solar deuterium mass loss is:
(.4274 X 10^10 kg / s) / [(1.00782503207 - 1.00065081354) / 1.00782503207]
= (.4307 X 10^10 kg / s) / .00717422
= 60.034 X 10^10 kg / s
 

SOLAR PARTICLE EMISSION:
Measurements of the solar wind indicate that the solar particle emission rate is about:
1.3 X 10^36 particles / second. The solar wind contains both electrons and protons. The particle velocity is about 750 km / s. For hydrogen the approximate mass per particle is given by:
(1.00782 gms) / (6.023 X 10^23 atoms)
= 0.167 X 10^-23 gm / atom.

Thus the rate of solar hydrogen mass loss via the solar wind is about:
1.3 X 10^36 particles / s X 1.67 X 10^-27 kg / particle
= 2.171 X 10^9 kg / s
which is small compared to both the rate of hydrogen mass conversion into deuterium and deuterium mass conversion into heavier elements.
 

PRESENT SOLAR HYDROGEN MASS:
The present solar mass is: 1.98892 X 10^30 kg.
As indicated by solar spectra, 73.4% of the solar mass is hydrogen or deuterium.
Hence the present solar hydrogen plus deuterium mass is:
.734 x 1.98892 X 10^30 kg
= 1.4598 X 10^30 kg
 

SOLAR LIFETIME:
Assume that both the solar hydrogen fusion rate and the solar hydrogen emission rate continue unchanged. Then the solar lifetime is given by:
Lifetime = (present solar hydrogen mass) / (present hydrogen mass loss rate)

If the sun is in its early life when the energy output is primarily due to conversion of molecular hydrogen into deuterium the projected solar lifetime is given by:
= (1.4598 X 10^30 kg) / (556.414 X 10^10 kg / s)
= .00262 X 10^20 s
= .00262 X 10^20 s X 1 h / 3600s X 1 day / 24 h X 1 year / 365.25 day
= 2.62 / (3.6 X 2.4 X 3.6525) X 10^11 years
= .0831 X 10^11 years
= 8.31 billion years

If the sun is in its later life when the energy output is primarily due to fusion of deuterium into helium-4 the projected solar lifetime is given by:
= (1.4598 X 10^30 kg) / (60.114 X 10^10 kg / s)
= .02427 X 10^20 s
= .02427 X 10^20 s X 1 h / 3600s X 1 day / 24 h X 1 year / 365.25 day
= 2.427 / (3.6 X 2.4 X 3.6525) X 10^12 years
= .0769 X 10^12 years
= 76.9 billion years
 

COMMENT ABOUT THE SOLAR LIFETIME CALCULATION:
The calculated solar lifetime strongly depends on which fusion reactions are presently dominant in the sun. However, the solar wind indicates that the sun is primarily composed of hydrogen rather than deutrerium, which sugests a solar lifetime of about 10 billion years. Clearly the solar lifetime is at least four orders of magnitude longer than humans have existed on Earth. We are not going to run out of solar energy any time soon.

Over billions of years the solar radiation output will gradually increase due to the increasing solar deuterium concentration until the Earth becomes too hot for human habitation.
 

This web page last updated March 18, 2016.

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