Hubble's Constant in Terms of the Compton Effect
2852 Blue Jay Way
Lafayette, CO 80026
A new red shift distance relationship, z = HD/(1 -
HD/2), is derived for the Compton effect interpretation of the cosmological red
Headings: (Cosmology:) miscellaneous, theory, distance scale
This paper shows how the Compton effect produces a red shift of the same form as the Doppler effect for the cosmological, quasar, solar and other intrinsic red shifts.
The red shift controversy has been raging ever since Hubble's and Humason's original papers (Hubble & Humason 1931, Humanson 1931) carried the footnote: "It is not at all certain that the large red shifts in the spectra are to be interpreted as a Doppler effect, but for convenience they are expressed in terms of velocity and referred to as apparent velocities." Hubble felt that the data was in better agreement with light having a loss of energy to the intervening medium proportional to the distance it travels through space by what he called "a new principle of nature" (Hubble 1937). This was because if it were Doppler the light should appear to be less bright (due to a decrease in photon flux) than if it were a loss of energy. Such a brightness correction did not fit the direct proportionality to distance data.
The Compton effect occurs when light interacts with matter. The photon bounces off the matter and transfers some of its energy and momentum to the particle. In this process the photon wavelength increases. The increase in wavelength is inversely proportional to the mass of the particle. The Compton effect wavelength change is most often observed when the photon bounces off an electron since the electron is the least massive stable particle. It is observed best when the photon is of high energy since the wavelength change per interaction is the greatest energy change. When the particle is already a free electron, the Compton effect is the only process that occurs for the interaction except for pair production. Pair production normally only occurs for high energy photons of gamma ray wavelengths and shorter, so the Compton effect is the only process that occurs for photons of longer wavelength. The change in wavelength per interaction is very small so it usually takes multiple collisions or the photon for an appreciable measurable effect to be observable for very long wavelength photons. Yet, multiple Compton collisions for a photon traveling long distances through a highly ionized gas will produce a significant red shift.
Several authors have now suggested that the Compton effect is this new principle that Hubble was looking for, for the solar, quasar and cosmological red shifts (Compton 1923, Sistero 1966, Kierein & Sharp 1968, Maric et al 1977, Reber 1977). For the quasar case being a star surrounded by a large electron cloud, this would produce an intrinsic quasar red shift (Burbidge & Burbidge 1969), allowing quasars to be local and obviating the need to explain their brightness and apparent superluminal velocities (Marscher & Scott 1980, Pearson et al 1981).
The Compton effect explains the red shift on the sun being greater at the limb than at the center because the number of electrons along the line of sight through the solar atmosphere is greater at the limb. Kierein & Sharp (1968) showed that the red shift agreed quantitatively with the increase in this number. Compton (1923) had predicted this qualitatively earlier for the solar case, when he extrapolated the Compton effect to visible wavelengths.
This idea has met with the objection that the Compton effect interpretation should produce blurred objects and spectral line broadening. This is despite the well-known experience that light will interact with a transparent medium, being slowed according to the index of refraction, without blurring or line broadening. Reber's solution to this objection was a random walk analysis that showed that the photon remained within the observed blur circle (Reber 1968, 1977).
The objection can be explained classically by considering the electrons to act as centers of Huygens' secondary wavelets that reconstruct the wave front. The Compton effect does not depend on the electric charge of the electron, but rather is a consequence of conservation of momentum and energy. Thus, the E x H vector of the photon need not be altered by the Compton process. It is the E x H vector of the wave front that is seen and that contains the information that defines the wave front's initial velocity vector.
Neves & Assis (1995) claimed the Compton effect does not produce a red shift of the same form as the Doppler effect and Hubble’s law. This letter shows how Hubble's constant can be expressed in terms of the Compton effect.
Hubble's law observes that the red shift, z, is proportional to the distance to the object:
z = Dl / l = HD or, Dl = HDl (1)
where Dl is the red shifted change in wavelength, l is the original wavelength, D is the distance to the object, and H is "Hubble's constant" of proportionality (H is sometimes conventionally expressed as H/c for convenience for the Doppler interpretation).
If one interprets this law as being due to multiple Compton effect interactions of photons starting at the distance D and interacting with an intervening medium of free particles (such as electrons) of density r particles per cubic centimeters, then the following calculations can be made:
Dl = (Dli)(Ni) (2)
where Dli is the shift per interaction given by the familiar Compton formula:
Dli = h (1 - cos q) / mc (3)
where h is Planck's constant, m is the mass of the particle (electron), c is the velocity of light, and q is the angle of deflection of the photon velocity vector. Ni in equation 2 is the number of Compton interactions occurring, so that cos q is the "average cos q" observed over the large number of interactions involved.
Ni = (Nt)T (4)
That is, the number of interactions equals the integrated probability, Nt, that an interaction is occurring at any time, times the total time of travel, T, where
T = D / c (5)
Nt = srl c / lc (6)
where s is the Thomson cross section (in the case where the particle is the electron), and lc = the Compton wavelength of the particle = h / mc
Thus, from equations 4, 5 and 6:
Ni = srl c D / lcc (7)
and from equations 2, 3 and 7 and substituting and canceling:
Dl = sr(1 - cos q)Dl (8)
Thus, from equations 1 and 8:
H = sr(1 - cos q)
This interesting result shows that the "large cosmological constant", H can be expressed in terms of the "smaller" Thomson cross-section constant so familiar in everyday physics of subatomic particles. This connection between the very large and the very small was first suggested by Dirac.
It should be noted that the l in equations 6, 7 and 8 strictly speaking is not the original wavelength of the photon, but rather the wavelength at the time of the interaction. This wavelength varies from l at the start to l + Dl at the end of the travel, so that the average wavelength should be l + Dl/2. This is a small correction for the observed cosmological (non-quasar) shifts where z is less than 1, the correction being less than the uncertainty in H and D.
Thus, when l + Dl/2 is substituted for l, the result is:
z = HD/(1-HD/2),
which leads to correspondingly shorter distances for a given z than in the case that z = HD. These distance differences can be significant for larger z, resulting in a new form for Hubble's law. As better measurements are made of the redshift distance relationship, it should be theoretically possible to determine which relationship is the observed one. Preliminary evidence of such a deviation from z=HD is usually ascribed to deceleration or acceleration (depending on how you look at it) of the big bang, but is more logically described as being due to a Compton effect red shift.
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