APPENDIX 2: Radiant Energy Emission

(i). Energy density of radiation
(ii). Stellar luminosities
(iii). Cepheid variables
(iv). Supernova 1987 A
(v). Chemical reactions

(i). Energy density of radiation

The energy density of emitted radiation, as distinct from the ZPE itself, is discussed first. The energy density, ρ, of electromagnetic waves is given by this equation in references [148-150]

 εE2/4π = μH2/4π = ρ                                        (76)

Now electromagnetic waves are described by a sine function. The peak amplitude A0 of sine waves is such that E2 = (1/2)A02 since the factor of 1/2 is the mean of the square of the sine over all angles. It follows from (76) that an electromagnetic wave has an energy density given by εA02/8π, or alternatively μA02/8π as in [148]. Now the field strength of both the electric field E, and the magnetic field H are constant at the time of emission with changing c. This implies that wave amplitudes are unchanged. But both ε and μ are proportional to 1/c from (6). So the energy density of electromagnetic waves, ρ, is proportional to 1/c.  We can now write

 ρ ~ 1/c ~ U                                                (77)

The increase in U, ε and μ with time means the energy density of emitted radiation will also increase with time. So higher c values in the past went hand in hand with lower energy densities for all electromagnetic radiation because the properties of the vacuum have altered. In the case of a monochromatic wave in transit through the vacuum, E, H and A0  still remain constant, but ε and μ increase with time during travel, so the wave’s energy density will be higher at reception than at emission. But, since the speed of light has slowed in transit, the number of waves being received by the observer per unit time is less than the number that was emitted per unit time, so the radiation intensity will remain unchanged. Let us examine this.

Radiation intensity, I, given by the term ρc in [148-150] is constant from (77). By way of explanation, assume that an electromagnetic wave takes one second to pass an observer, and its measured energy density is one erg per cubic centimeter with the speed of light being one centimeter per second. The intensity of the wave will be 1 erg per square centimeter per second. Now if the speed of light at emission was 10 times faster, then from (77) this means the energy density of each wave is 1/10th erg per cubic centimetre. But the speed of light is 10 times greater, so we now have 10 waves, each of 1/10th erg per cubic centimeter, passing a given point in one second. This is true since wavelengths remain constant from the moment of emission to the moment of reception even though c drops with time. Thus the total energy at emission that passes through one square centimetre per second is the same as at reception, so radiation intensities remain unchanged. We can now discuss energy emission processes.

(ii). Stellar luminosities

Three factors determine the luminosity of a star with increasing ZPE strength, and decreasing c. First is the photon production rate, which depends on the stellar nuclear reaction rate. Second is the mass-density of the particles making up the star seen from the atomic frame of reference. Third is the star’s opacity, which inhibits the energy transport from the reaction centre out to where that energy radiates into space. Four key processes may contribute to opacity: free-free transitions, bound-free transitions, bound-bound transitions, and scattering by electrons. We note that bound-bound transitions play a negligible part in stellar interiors [151], but “typical stellar interior densities, electron scattering will predominate over bound-free and free-free absorption for temperatures in excess of, say, some 107°K” [152].

For the central area of stars, this leaves electron scattering of photons as the prime source of opacity. If ne is the number of electrons in unit volume and ρm is the mass-density [153, 154], then the opacity from scattering by free electrons Κe is given by [151, 155]

Κe = σne / ρm ~ c2 ~ 1/U2                                             (78)

Here, σ is the Thomson scattering cross-section where σ = (8π/3)[e2/(εmec2)]2 which is constant, as is ne. But ρm is proportional to 1/c2 from (16), so electron scattering opacity is proportional to c2 as in (78). Some stars do not have the interior conditions for high energy Compton or inverse Compton effects to be relevant [156]. Still a formula for the Rosseland mean opacity for Compton scattering by free electrons in the non-degenerative limit exists [157]. It differs from (78) only by a dimensionless number.

 Let us consider bound-free transitions. Harwit states “hydrogen and helium do not contribute significantly to the bound-free transitions” [158]. Instead, a factor Z is put in the equation where Z is the metal abundance expressed as a fraction of total mass, and so is constant for all c. Harwit detailed terms giving the numerical coefficient in the equation describing Kramer’s Law of Opacity for bound-free absorption [159]. Including these give the following equation

Κbf = (2/3) √(2π/3) [e6h2ρm/(ε3cmH2me3/2) [Z(1+X)/(kT)7/2][kT/(hf)]3 <gbf>/F ~ c2     (79)

Here, k is Boltzmann’s constant, which is invariant for all c. The terms mH  and  meare the masses of the hydrogen atom and electron respectively [160] and are proportional to 1/c2 and to U2 from (16). F contains correction and other factors independent of c, while <gbf> is the dimensionless mean Gaunt factor of the order of unity. X is a dimensionless number, and T is temperature. When the c-dependent terms are analysed, the result is that bound-free opacities are proportional to c2, the same as electron scattering. In [159], figure 8.3 reveals the stellar conditions allowing free-free transitions are restricted. Nevertheless Harwit repeats the above process for these free-free transitions. The resulting equation gives a result similar to (79), but an (X + Y) term is included in place of Z. Schwarzschild lists the complete equation as [161]:

Κff = (2/3) √(2π/3) [e6h2ρm/(ε3cmH2me3/2 k7/2)](1/196.5) <gff> (X+Y)(1+X)(1/T7/2) ~ c2  (80)

Comparison with (79) shows the same c2 and hence 1/U2 proportionality for free-free opacity results. Even if bound-free and free-free absorptions are less vital than electron scattering in considerations here, it still appears that all stellar opacities are proportional to c2 and hence to 1/U2. It can therefore be written that the average value for the opacity of the whole star is

ΚA ~ c2 ~ 1/U2                                                         (81)

The other factor affecting a star’s luminosity is its rate of burning nuclear fuel. Since nuclear reactions are temperature sensitive, the proton-proton reactions dominate at lower temperatures with the carbon cycle prominent later [162].  The key reaction needed to get the proton-proton sequence started is given by the equation H1 + H1 → D2 + e+ + ν . This is a beta process with a mean reaction time for any given particle of 14 billion years [162]. But this reaction rate is proportional to c, and hence inversely proportional to U, as shown by the treatment of Swihart [163]. He states that the reaction rate per unit volume is given by

Nr = {4h2DE1/6N1N2/[M3/2(kT)2/3]}e-1.89[E/{kT)]^1/3 ~ c ~ 1/U           (82)

Here the temperature is T, Boltzmann’s constant is k, and Planck’s constant is h. N1 and N2 are the numbers of interacting particles per unit volume, M is the reduced mass of the two reacting particles, D is the probability of reaction between nuclei, and E is the energy required to penetrate the Coulomb barrier. In (82), the only two factors that are c-dependent are M and h. Their ratio, h2/M3/2, is proportional to c. Stellar nuclear reactions are thereby proportional to c, or 1/U, and so will be the photon production rate. When photon production rate and opacity are then considered together, the luminosity of a star with varying ZPE should be established.  The key formula for stellar luminosity is given by Harwit as [164]:

Ls = -16πac / (3ΚAρm)r2T3(dT/dr)                            (83)

Here the term a is given by the relationship a = 8π5k4/[15(hc)3] which is constant as hc is constant and Boltzmann’s constant k is invariant. Note that ρm is the mass-density, so it is proportional to 1/c2 from (16). Therefore we can state that

 ΚAρm = constant                                               (84)

An approach by Schwarzschild is relevant [165]. He defines the fraction of energy a beam of light loses by scattering/absorption over distance dr. This gives К′ multiplied by dr so that

 Ls = -4πr2(4ac/3)(T3/К′)(dT/dr)                               (85)

In (85), the term К′ = ΚAρ agrees with Harwit, so the two formulae are basically the same. In view of (84), equations (83) and (85) both contain all the factors involved when a variable ZPE is considered, the opacity, the mass density of atomic particles, and the speed of light which reflects faster burning rate. Thus the equation for luminosity at temperature T gives us

 Lc = -4πr2τ(4ac/3)(T3Aρm)(dT/dr) ~ c                   (86)

Thus stellar photon emission rates are inversely proportional to U. But (77) shows another factor is operating since the energy density of all electromagnetic radiation is proportional to 1/c, quite apart from any redshifting process. The energy density of each photon is thus intrinsically lower, proportional to 1/c, when the ZPE strength is lower. The final result is that the radiation intensity, I, from a star is unaffected by the changes in the ZPE. This occurs since the increased production rate of electromagnetic waves (or photons) is counterbalanced by the fact that each wave (or photon) has a lower energy density. Since it can be shown that a star’s luminosity is proportional to its radiative intensity, we can write the conclusion that the star’s total emitted energy passing through unit area per second is such that

 I ~ L ~ cρ = constant                                               (86A)

An example of what these principles mean in practice is given by a consideration of Cepheid variables and their observation in distant galaxies, and a discussion about supernova 1987A.

(iii). Cepheid variables

The near surface layers of Cepheid variable stars pulsate in and out like clockwork and their brightness pulsates inversely with the same period. The more massive the Cepheid, the brighter it shines, and the slower its pulsation rate. There is a direct link between a Cepheid’s period and its intrinsic luminosity. So Cepheids are used as distance indicators for galaxies in which their typical light curve is seen. Given the observed period, the intrinsic luminosity is known, and, when compared with the observed luminosity, the distance is deduced from the inverse square law [166]. This raises two matters; the observed luminosity, and the pulsation rate.  Let us compare our observations of a distant Cepheid with one nearby us.

Equation (86) shows the number of waves, or photons emitted per unit time, from each Cepheid is proportional to c. For any planets around the distant Cepheid, (77) coupled with (86) and (86A) ensures the intensity of the radiation remains fixed with time as the ZPE varies. But when we observe the stream of waves or photons from that distant Cepheid, the reception rate is lower than at emission.  Thus, if the speed of light was 10 times its current value at the point of emission, then the reception rate for the waves or photons is only 1/10th of that at emission. But this reception rate is in exact accord with our local Cepheid, since its wave or photon emission rate has also dropped as the speed of light has dropped. Since ε and μ have also increased by a factor of 10 while the waves were in transit, the energy density of those waves at reception is now 10 times that at emission. Thus the observed luminosity of distant Cepheids is the same as for those nearby, apart from the dimming due to distance.

But what about Cepheid pulsation rates? To explain the pulsation process, Eddington suggested his ‘valve mechanism.’ If a specific layer of a star near the surface became more opaque upon compression, it would block the energy flowing towards the surface and push the surface layers upwards. Then, as this expanding layer became more transparent, trapped radiation would escape and the layer would collapse back down to begin the cycle again. He said “To apply this method we must make the star more heat-tight when compressed than when expanded; in other words, the opacity must increase with compression” [167]. The difficulty is that in most regions of a star, the opacity decreases with compression rather than increases. However, it was found that in some stars’ partial ionization zone, near the surface, this valve mechanism does operate. There, compression and rarefaction store and extract energy through ionization. As this layer compresses, its density and opacity increase and heat is absorbed. Then, as the layer expands the density and opacity decrease, so heat is released. Thus the opacity of the partial ionization zone modulates the flow of energy through that zone and is the direct cause of stellar pulsations [167]. The standard relationship has the pulsation period inversely proportional to the square root of the opacity, К, [168]. If the results of (81) are used, the stellar pulsation period at the point of emission, te, is given by

 te ~ √(1/К) ~  √(1/c2) ~ 1/c ~ U                                        (87)

This means that Cepheid periods lengthen as the strength of the ZPE increases.

Yet that is not the end of the story. At the time of reception, the wave-train carrying the information from the distant star is traveling more slowly than at the moment of emission. So the star’s period of variation appears longer at reception than at emission by a lengthening factor, c. But that period will be the same as a local Cepheid, since the local Cepheid’s period of variation has increased in inverse proportion as c has decreased. Therefore, the period of the Cepheid at reception, tr, will appear to be the period at emission, te, given in (87), multiplied by the lengthening factor, c. So compared with our local Cepheid we have

 tr ~ (1/c)(c/1) = unchanged                        (88)

Thus the period of the distant Cepheid will appear to be the same as that for our local Cepheid. So measurements of distance based on Cepheid variability will be unaffected by ZPE changes.

(iv). Supernova 1987 A

It has been claimed that two features of supernova SN1987A in the Large Magellanic Cloud (LMC) disprove changes in the ZPE and lightspeed. First was the measured exponential decay of the light intensity curve from the radioactive decay of cobalt 56. Second were the enlarging rings of light from the explosion that illuminated distant sheets of gas and dust. Since the both the distance to the LMC and the angular distance of the ring from the supernova are well-known, a simple calculation shows how long it takes light to get from the supernova to the sheets, and how long the peak intensity should take to pass.

Now, as confirmed below in Appendix 4, radioactive decay rates are proportionally faster when lightspeed is higher and the ZPE energy density is lower. This means a shorter half-life for cobalt 56 than the light intensity curve revealed. For example, if c was 10 times its current value, the half-life would only be 1/10th of what it is today, so the light intensity curve should decay in 1/10th of the time it takes today. In a similar fashion, if c was 10 times greater at the supernova, the light should have illuminated the sheets and formed the rings in only 1/10th of the time at today’s speed. Yet both the light intensity decay curve and the timing of the appearance of the rings are in accord with the speed of light equal to its current speed.

The reason is the slow-down in the speed of light mentioned in Appendix 2 part (i) dealing with radiant energy density and part (iii) with Cepheid variables. Since c is the same at any instant throughout the universe, light from distant objects always arrives at earth with the current universal value. Barnet et al. proved this for light from distant quasars [212]. So if c really was 10 times its current value at the time of the supernova in the LMC, then the photons with the information have slowed in transit to 1/10th of their original speed and so bring us information at 1/10th the rate it happened. In other words we are seeing the entire event being played back in slow motion. The shorter half-life and shorter transit times are lengthened by the slow-down in c so they appear to run at the same rate as these events do today. Thus these phenomena do not negate changes in the ZPE.

 (v). Chemical reactions

Collision theory suggests that chemical reactions could be affected by a ZPE and c that is changing. In reactions, a series of steps may occur rapidly. But if the old collision theory was correct, “all chemical reactions would be completed in a fraction of a second” [169]. It is now known that collision theory “is of only limited usefulness” [170] since reaction rates are governed by the slowest step in the reaction series, called the “rate determining step” [171]. Here, an activated complex forms and disrupts, yielding products for the rest of the reaction series. The key step is the activated complex formation, written mathematically as [172]

 k* = AK                                                           (89)

Here, k* is the reaction rate constant, and K is the equilibrium constant for the activated complex. K describes the “equilibrium” between reactants and activated complexes and is unchanged with ZPE variation since temperature and kinetic energy are unchanged. The term A is the reaction probability, or chance of forming the activated complex at any time. Now masses and velocities in (16), (17) behave such that, at constant temperature, the number of approaches per second by ions or charged molecules to a potential reactant is proportional to c. We call this quantity Y*. Now the velocity of these approaches is faster when c is higher, so the time that each ion or charged molecule is in the vicinity of the potential reactant is thus proportional to 1/c. Let this quantity be X*. It can be shown that the chance of forming the activated complex, A, at any instant is the number of ions approaching the reactant per second, Y*, multiplied by the time the ion spends in the vicinity of the reactant, X*. So that

A = Y*X* ~ c(1/c) = constant                                 (90)

Thus A is constant. Since K is also constant, the reaction rate, k*, remains fixed for all c.


[148] F.A. Jenkins & H.E. White, “Fundamentals of Optics” 3rd Edition, (McGraw-Hill, 1957). pp. 412-414.
[149] S. G. Starling & A. J. Woodall, “Physics,”  Longmans, (1958) 1129.
[150]A.P. French,op. cit., p.40. (see ref. #73)
[151] M. Harwit, op. cit., p. 322. (see ref. #27)
[152] J.P. Cox & R.T. Giuli, Principles of Stellar Structure, (Gordon & Breach, New York, (1968) 383.
[153] Ibid, p. 374.
[154] M. Harwit, op. cit., p. 315.(see ref. #27, linked above)
[155] Ibid, p. 82.
[156] T. L. Swihart, Physics of Stellar Interiors,  Pachart Publishing House, Tucson, Arizona, (1972) 76.
[157] Cox and Giuli, op. cit., p. 374
[158] M. Harwit, op. cit., p. 323. (see ref.#27, linked above)
[159] Ibid, pp. 323-324.
[160] Ibid, p. 314.
[161] M. Schwarzschild, Structure and Evolution of the Stars, Princeton University Press,Princeton, New Jersey, (1958) 88.
[162] M. Harwit, op. cit., pp. 336-337 and Fig. 8.7. (see ref. #27, linked above)
[163] T. L. Swihart, op. cit., p. 82.
[164] M. Harwit, op. cit., p. 321. (see ref. #27, linked above)
[165] M. Schwarzschild, op. cit., p. 42.
[166] F. Hoyle, Frontiers of Astronomy, William Heinemann Ltd, London, (1956) 184-188.
[167] D. Ostlie & B. Carroll, An Introduction to Modern Stellar Astrophysics, Weber State University (Addison-Wesley Publishing Co. Inc., Reading, MA, 1996).
[168]. P.S. Wesson, Cosmology and Geophysics: Monographs on Astronomical Subjects, 3, Adam Hilger Ltd., Bristol, (1978) 62-63.
[169] W. J. Moore, Physical Chemistry, 3rd edition, Longmans,(1961) 548.
[170] S. V. Filseth in Chemical Systems: Energetics, Dynamics, Structure, J. A. Campbell, (Ed.) W. H. Freeman and Co., San Francisco (1970) 555-556, 583.
[171] Ibid, pp. 555-556.
[172] Ibid, p. 588.

[212] C. Barnet, R. Davis, and W. L. Sanders, Ap. J. 295 (1985), 24.

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