Abstract

From a linear Hubble nebular red-shift vs. distance law a controversy follows: the size of the distant galaxies – corresponding to the larger red-shifts – seems larger. Moreover, it appears that the farther is the galaxy the larger it is. The discovery of photons’ half-life and the following from it exponential red-shift vs. distance (modified) Hubble law resolves this controversy.

Photon’s Half-life

From Pioneer 10 excess red-shift it was determined that the Hubble discovered nebular red-shift is distance caused. A physical cause was proposed in form of photon’s half-life: the photon progressing through space uninterrupted loses the half of its’ energy – doubles its’ wavelength – in equal time intervals. [9]

It could be written as:

nu = nu0 / (2^( t / Hd)) (1)

where t – time, age of photon

nu – frequency at t time

nu0 – frequency of photon at the time t = 0, at generation

Hd – photon’s half-life, Hubble wavelength doubling constant, from Pioneer 10 measurements Hd = 8-9 Gyr.

This photon half-life satisfies the requirement that from the same source any electromagnetic radiation arrives to the observer located at R = t * c (where c – light speed in vacuum, Einstein’s constant) radial distance with the same red-shift and the observations will coincide if made on the same ray train at different time intervals. The red-shift is defined as:

z = (nu0 – nu) / nu = (2^( t / Hd) - 1) (2)

where z – red-shift. Equation (2) represents only the Hubble nebular red-shift, cosmological red-shift, the Doppler effect has to be deducted prior this treatment from the total red-shift – if any Doppler effect present in the nebular red-shift observed.

Galactic size and red-shift

The observations of galaxies are limited to the angular size or solid angle and the red-shift of known spectral line combinations. The red-shift is used to find the distance from the observer to the galactic source of rays and the solid angle indicates the size of the galaxy as a function of distance.

omega = A / R ^2 à R^2 = (A / omega) (3)

z = (2^( t / Hd) - 1) à R = c * t = c * Hd * ln(z + 1) / ln (2) (4)

using light years for the units of distance and combining the equations (3) and (4)

(A / omega) = Hd^2 * (ln(z + 1) / ln (2))^2 à omega = A * (ln (2))^2/ Hd^2 / (ln(z + 1))^2 (5)

The controversy was resolved by replacing the z^2 with a much smaller value of

(ln(z + 1) / ln (2))^2, which is represented in equation (5).

The change in the expected solid angle for an assumed similar constant is represented by equation (6)

(ln (2))^2 * z^2 / (ln(z + 1))^2 (6)

Equation (6) results for z =1 red-shift value or about 17 billion light years distance the same solid angle seen for the same galaxy as it was expected with linear Hubble red-shift vs. distance law, using the same constant for proportionality. The z = 1.5 value results a difference of 1.29 times larger solid angle is expected with exponential Hubble law, the z = 2 corresponds to 1.6 times, the z = 3 – 2.25 times larger expected solid angle than it would be with linear Hubble law. The less then 1 z values result in smaller expected view angles then it would be with linear Hubble law, down to about the half for z <= 0.25 and 0.73 for z = 0.5. It also could be considered with a shift, starting from the about equal representation for the small red-shift galaxies results say twice as big for z =1, 2.6 times at z = 1.5, 3.2 times at z = 2 and 4.5 times at z = 3. (Due to the inability to define a “linear” Hubble distance constant this second consideration is the valid. Also the limit for small R/Hd values – where the relationship was employed – z = R / Hd / ln (2) resulting a ratio of expected sizes 1 / ln(2) –1 = 0.4427 – or 2.26 times the linear expected solid angle at z = 1 for exponential law.)

This evaluation enables one to revise the actual galactic sizes, defined from the viewed solid angles and the thought linear Hubble red-shift vs. distance law. The found on the linear Hubble law basis controversy [3 - 4] will be cleared by this latest finding.

Conclusion

Considering the photon half-life – half of energy is lost in equal time intervals – resolves the galactic size vs. red-shift controversy. At z = 1 already more then twice as large solid angle expected for the galaxy, as it was thought based on the linear red-shift vs. distance Hubble law. This finding by itself represents a testable prediction offering additional evidence in support of photon half-life cause of the Hubble discovered nebular red-shift vs. distance law.

Electronic address: alistolmar@3dresearch.com

References

1. Two methods of investigating the nature of nebular red-shift – Edwin Hubble and Richard C. Tolman, Bibcode: 1935ApJ....82..302H

2. The 2dF QSO Redshift Survey - I. The Optical QSO Luminosity Function – Boyle B.J at al. astro-ph/0005368

3. The Space Density of low redshift AGN – Londish D. at al. astro-ph/0006022

4. Effects of Red Shifts on the Distribution of Nebulae - Hubble Edwin Bibcode: 1936ApJ....84..517H

5. The 200-inch telescope and some problems it may solve - Edwin Hubble Bibcode: 1947PASP...59..153H

6. The Apparent Anomalous, Weak, Long-Range Acceleration of Pioneer 10 and 11 - Slava G. Turyshev at al. gr-qc/9903024

7. Indication, from Pioneer 10/11, Galileo, and Ulysses Data, of an Apparent Anomalous, Weak, Long-Range Acceleration - John D. Anderson at al. gr-qc/9808081

8. The Other Possible Cause of Red-shift – Aladar Stolmar (unpublished)

9. Pioneer 10 Anomalous Acceleration Is Distance Caused Hubble Red-shift - Aladar Stolmar (unpublished)

10. Universe 1993 – Aladar Stolmar – ISBN 9630295121