Terrestrial Neutrino Spectra and Energies

Abstract

Neutrino spectrum literature is investigated, both for solar and atmospheric (cosmic ray decay) sources. Comparison to solar radiation (electromagnetic flux) gives interesting results, even with moderate levels of uncertainty in the flux predictions and measurements.

Since neutrino availability affects commercial prospects of energy extraction (though neutrino availability appears to not affect investment extraction), the author hopes the exercise of extracting energy totals from existing spectrum information is done properly.

Solar Neutrino Spectrum and Intensity

No convenient tabulation of energy availability for neutrinos was found. Review of Particle Physics 2020 ¹ shows neutrino flux for solar sources in Figure 14.1 page 292 and references original sources for solar neutrino and atmospheric neutrino spectra. The following table is developed from the original sources:

Table of Solar Neutrino Energy, Mev/(cm²s)

Source Type	Flux Exponent	Flux Mantissa	Energy/ν MeV	Energy MeV	Percent of Total
pp	10	5.98	0.259	1.549E+10	75.9067
pep	8	1.44	1.400	2.016E+08	0.9880
hep	3	8.04	10.000	8.040E+04	0.0004
7Be	9	5.00	0.850	4.250E+09	20.8290
8B	6	5.58	5.000	2.790E+07	0.1367
13N	8	2.96	0.700	2.072E+08	1.0155
15O	8	2.23	1.000	2.230E+08	1.0929
17F	6	5.52	1.000	5.520E+06	0.0271
eN	5	2.34	2.220	5.195E+05	0.0025
eO	5	0.88	2.754	2.424E+05	0.0012
eF	3	3.24	2.761	8.946E+03	0.0000
————				————-	————-
Total				2.040E+10	100.0000

Serenelli(2016)² Figure 3 is the source of Figure 14.1 mentioned above, which shows a vertical log scale per 100keV. The 2018 version of the same diagram had both per 100keV and per MeV noted as the vertical scale. Table 3 lists the solar flux under various models, the SFII GS98 chosen by Review. The data has not changed since Serenelli(2011)³. Tabulation of fluxes is in Table 3. The SFII-GS98 column had appeared in Review 2018 but not 2020. Energies used for the fluxes are measured from the diagram, taking the value slightly to the left of the peak of the distributions (reason below).

Due to the variation in stated vertical scale, the author expanded the figures and performed a graphical integration of flux for the pp curve, yielding a value of 6.738e+10 compared to the published 5.98e+10. This is considered confirmation of the published magnitudes of the fluxes and the 100MeV vertical scale and, when scaled to the published flux, showed an average energy of 0.259 GeV just to the left of the peak flux shown in the graph.

Villante(2014)⁴ page 3 provides the three electron capture flux numbers ecCNO for ¹³N, ¹⁵O and ¹⁷F as count per centimeter squared per second. The GS98 column agrees better with measurements. Energies are taken from the same page, just below equation (1). The ecCNO neutrinos make up about 1% of the solar neutrino budget. Or less.

Converting from the realm of particle physics to SI units or energy requires a very tiny conversion factor: 1eV = 1.602176634E-019 J, though converting from MeV adds 6 to the exponent of 10 and converting from square centimeters to square meters adds another 4.

Atmospheric Neutrinos

Honda(2015) ⁵ has calculated atmospheric neutrino spectra for various locations around the globe, with varying atmospheric densities and conditions, compared to the long used US-standard ’76 atmosphere which has no seasonal or location parameters. The year averaged data is available from the author(s) https://www.icrr.u-tokyo.ac.jp/~mhonda/nflx2014/index.html. Variations in site averaged over the year were up to 10% in the arctic regions. The tropics were seen as 5% different. Choosing the Kamiokande year round data Solar-min and Solar-max without mountain looming over for comparison with Solar neutrino intensity shows data as flux (count) per meter squared per second per steradian per GeV (since it is a continuous spectrum) on a log scale from 100MeV to 10,000GeV, for four neutrino types, ν_μ ν̅_μ ν_e and ν̅_e. Data is shown in 20 ranges of cosine Z, averaged over 360 degrees of azimuth. The tables start at 100MeV neutrinos, so some low energy neutrinos are missing, principally from muons hitting rock or ice. The data show that the spectrum energy per GeV is dropping to less than half the peak around 500MeV neutrinos, so the loss is expected to be minor.

Calculating a single number for intensity on a horizontal square meter requires calculating the incident neutrinos on a horizontal plane from above and below, averaging solar minimum and solar maximum data (which differed by 5%), and summing the energies of all four neutrino types. The year round average was 8430 GeV/(m²s) or 1.34x10^-6 J/(m²s). A tiny number, hiding very rare but VERY high energy neutrinos! Uncertainties of 10 to 20% or even 50% do not turn the atmospheric neutrino energy availability into a big number.

Compared to the solar neutrinos, atmospheric neutrinos are interesting but rare and offer little energy per unit area.

Solar Electromagnetic Spectrum and Intensity

For comparison, solar spectrum has been summarized in ASTM G-173-03 ⁶. Standard sea level and extraterrestrial sprectra are available. Seventy percent of the solar electromagnetic energy is in the visible spectrum.

Table of Solar Electromagnetic Radiation Energy

Location	watts/(m2)	J/(m2s)
Outside Atmosphere	1348.2	1348.2
Global Tilt	1000.6	1000.6
Direct+Circumsolar	900.3	900.3

Lunar Electromagnetic Spectrum and Intensity

Christopher et al(2017)⁷ answers How bright is moonlight? The moon, on a clear fall night when the moon is close to the Earth, on a plane perpendicular to the moon, measures 0.3 lux (0.26 lux horizontal). A few hours a year, in the tropics, this would reach 0.32 lux. When “in the extremely unusual case of a near-perigee full Moon, near-zenith, under near-ideal atmospheric conditions, the maximum possible horizontal photopic illuminance is approximately 0.3 lux.”

For the solar spectrum, a standard value is 1 lux = 0.0079 watts/m², so the lunar intensity ranges up to .0024 w/m² or .0024 J/m²/s.

Conclusion

A table of relative energies is simple:

Neutrino and Electromagnetic Radiation: Energy Available

Source	Energy J/(m2s)	Relative to sunlight
full sunlight	1000 J/(m2s)	100%
solar neutrinos	32 J/(m2s)	3%
full moonlight	<.0024 J/(m2s)	.00002%
atmospheric neutrinos	0.00000135 J/(m2s)	.000000135 or 1.35x10-7%

In a moderately cloudy environment, integrated over 24 hours, neutrinos might represent 30% of the energy available from sunlight, requiring over 3 times the area to generate an equivalent amount of energy over a day, assuming similar efficiencies.

Even the large error bars are not expected to allow development of a vigorous energy production industry based on neutrino energy. Already the financial markets are seeing development of a vigorous investment industry based on neutrino energy. Search “neutrino energy harvesting” for investment opportunities. Good luck to that. Speculators may choose not to dump their battery stocks just yet.

Appendix - Musings on Neutrinos as Particles in the mnp Model

The mnp Model sees neutrinos as having energy, no rest mass, and as being distinguished only by energy level, unlikely by length, and even less likely by transverse pattern of mediator distribution. Neutrino change is seen as non-quantized recruiting of mediators as the neutrino passes through mass and gravitational fields. Since that recruiting is seen as proportional to neutrino energy, the change might be exponential with a very low constant in neutrino energy and traversed mass. The Model has no effective picture or response to the “tiny mass” and “anti-neutrino” descriptions. Early attempts at explaining those variations are now seen as disasters. See the mnp Manual - Appendix J - Journal of Negative Results.

Appendix - My Notes on Programs

To find the tiny numbers offered by atmospheric neutrinos, the author spent a lot of energy installing a notebook server and getting it to work locally. A frustrating but ultimately satisfying experience, not recommended for those without gurus. The web is NOT a guru.

More information on sources, page numbers, and calculation techniques than would be required in a paper is included here, as “notes to self” and as compensation for a less than eidetic memory.

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1. P.A. Zyla et al. (Particle Data Group), Prog. Theor. Exp. Phys. 2020, 083C01 (2020) https://pdg.lbl.gov/2021/web/viewer.html?file=../download/Prog.Theor.Exp.Phys.2020.083C01.pdf↩︎

2. A. Serenelli, Eur. Phys. J A52 4, 78 (2016) https://arxiv.org/pdf/1601.07179.pdf↩︎

3. A.M. Serenelli, W.C. Haxton, and C. Peña-Garay, Astrophys. J. 743, 24 (2011) https://arxiv.org/abs/1104.1639v1↩︎

4. F. L. Villante, Phys. Lett. B742, 279 (2015) https://arxiv.org/pdf/1410.2796.pdf.↩︎

5. M. Honda et al., Phys. Rev. D92, 2, 023004 (2015), https://arxiv.org/pdf/1502.03916.pdf.↩︎

6. ASTM International (2020) ASTM G173-03 - Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface (ASTM International, West Conshohocken, PA). https://www.nrel.gov/grid/solar-resource/assets/data/astmg173.xls Retrieved 2022-02-21↩︎

7. Christopher C M Kyba, Andrej Mohar, Thomas Posch, How bright is moonlight?, Astronomy & Geophysics, Volume 58, Issue 1, 1 February 2017, Pages 1.31–1.32, https://doi.org/10.1093/astrogeo/atx025↩︎