Abstract
Decays of radionuclides throughout the earth’s interior produce geothermal heat, but also are a source of antineutrinos; these geoneutrinos are now becoming observable in experiments such as KamLAND. The (angle-integrated) geoneutrino flux has been shown to provide a unique probe of geothermal heating due to decays, and an integral constraint on the distribution of radionuclides in the earth. In this paper, we calculate the angular distribution of geoneutrinos, which opens a window on the differential radial distribution of terrestrial radionuclides. We develop the general formalism for the neutrino angular distribution. We also present the inverse transformation which recovers the terrestrial radioisotope distribution given a measurement of the neutrino angular distribution. Thus, geoneutrinos not only allow a means to image the earth’s interior, but offer a direct measure of the radioactive earth, both revealing the earth’s inner structure as probed by radionuclides, and allowing a complete determination of the radioactive heat generation as a function of radius. Turning to specific models, we emphasize the very useful approximation in which the earth is modeled as a series of shells of uniform density. Using this multishell approximation, we present the geoneutrino angular distribution for the favored earth model which has been used to calculate the geoneutrino flux. In this model the neutrino generation is dominated by decays of potassium, uranium, and thorium in the earth’s mantle and crust; this leads to a very “peripheral” angular distribution, in which 2/3 of the neutrinos come from angles θ ≳ 60° away from the nadir. We note that a measurement of the neutrino intensity in peripheral directions leads to a strong lower limit to the central intensity. We briefly discuss the challenges facing experiments to measure the geoneutrino angular distribution. Currently available techniques using inverse beta decay of protons require a (for now) unfeasibly large number of events to recover with confidence the forward scattering signal from the background of subsequent elastic scatterings. Nevertheless, it is our hope that future large experiments, and/or more sensitive techniques, can resolve an image of the earth’s radioactive interior.
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References
Albarede F, van der Hilst R (2002) Phil Trans R Soc Lond A 360:2569
Apollonio, M. et al. [CHOOZ Collaboration]: 2000, Phys. Rev. D 61, 012001 [arXiv:hep-ex/9906011]
Araki T et al. (2005a) Nature 436:499
Araki, T. et al. [KamLAND Collaboration]: 2005b, Phys. Rev. Lett. 94, 081801 [arXiv:hep-ex/0406035]
Barger, V., Whisnant, K., Pakvasa, S. and Phillips, R.: 1994, in J. N. Bahcall et al. (eds.), Solar Neutrinos-The First 30 years, Addison-Wesley, Reading, p. 300
Binney J, Tremaine S (1987) Galactic Dynamics. Princeton, Princeton University Press
Bracewell RN (1986) The Fourier transform and its applications. McGraw-Hill, New York
Chandrasekhar, S.: 1950, Radiative Transfer, Oxford University Press, Oxford, Chap. 1
Dahlen FA (2004) Geophys J Int 157:315
Dziewonski, A. M. and Anderson, D. L.: 1981, Phys. Earth Plan. Int. 25, 297 http://solid_Earth.ou.edu/prem.html
Eder G (1966) Nucl Phys 78:657
Eguchi, K. et al. [KamLAND Collaboration]: 2003, Phys. Rev. Lett. 90, 021802 [arXiv:hep-ex/0212021]
Fiorentini, G., Lasserre, T., Lissia, M., Ricci, B. and Schonert, S.: 2003a, Phys. Lett. B 558, 15 [arXiv:hep-ph/0301042]
Fiorentini, G., Mantovani, F. and Ricci, B.: 2003b, Phys. Lett. B 557, 139 [arXiv:nucl-ex/0212008]
Gessmann CK, Wood BJ (2002) Earth Planet Sci Lett 200:63–78
Herndon, J. M.: 2003, PNAS 100(6), 3047
Hochmuth, K. A. et al.: 2005, Astropart. Phys. [arXiv:hep-ph/0509136] (in press)
Hofmeister AM, Criss RE (2005) Tectonophysics 395(3-4):159–177
Krauss LM, Glashow SL, Schramm DN (1984) Nature 310:191
Lee, K. K. M. and Jeanloz, R.: 2003, Geophys. Res. Lett. 30, 2212
Mantovani, F., Carmignani, L., Fiorentini, G. and Lissia, M.: 2004, Phys. Rev. D 69, 013001 [arXiv:hep-ph/0309013]
McDonough, W.: 2002, http://mahi.ucsd.edu/cathy/SEDI2002/ABST/SEDI1-2.html
McDonough WF, Sun S-S (1995) Chem Geol 120:223
McKenzie D, Richter F (1981) J Geophys Res 86:11667
Nunokawa, H., Teves, W. J. C. and Zukanovich Funchal, R.: 2003, JHEP 0311, 020 [arXiv:hep-ph/0308175]
Pollack HN, Hurter SJ, Johnson JR (1993) Rev Geophys 31:267
Rama Murthy, V., van Westrenen, W. and Fei, Y.: 2003, Nature 423, 163
Rothschild, C. G., Chen, M. C. and Calaprice, F. P.: 1998, Geophys. Res. Lett. 25, 1083 [arXiv:nucl-ex/9710001]
Stein, C.: 1995, in T. J. Ahrens (ed.), Global Earth Physics: A Handbook of Physical Constants, AGU Reference Shelf 1, American Geophysical Union, Washington, p. 144
Van Schmus, W. R.: 1995, in T. J. Ahrens (ed.), Global Earth Physics: A Handbook of Physical Constants, AGU Reference Shelf 1, American Geophysical Union, Washington, p. 283
Vogel, P. and Beacom, J. F.: 1999, Phys. Rev. D 60, 053003 [arXiv:hep-ph/9903554]
Wasserburg G.J., MacDonald G.J.F., Hoyle F., Fowler W.A. (1964) Science 143:465
Acknowledgements
We are grateful to Stuart Freedman for very helpful discussion regarding experimental issues and prospects regarding antineutrino directional sensitivity. We thank Charles Gammie for encouragement and for alerting us to the Abel transform, and V. Rama Murthy for enlightening discussions regarding core potassium. We are particularly thankful to Georg Raffelt for guidance and insight. The work of BDF is supported by the National Science Foundation grant AST-0092939.
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Appendix A: Terrestrial Tomography: Inverting the Angular Distribution
Appendix A: Terrestrial Tomography: Inverting the Angular Distribution
We may write the intensity distribution (Equation 6) in dimensionless units as
where \( {\sigma = \sin \theta \in [0,1]}\) and \( {x = r/R_\oplus \in [0,1]}\). Thus both I and q are defined on the interval [0, 1]. Furthermore, for an experiment on the surface of the earth, we expect that I(1) = 0 = q(1) because the earth’s density goes to zero at the surface (by definition!). However, a real experiment located slightly under the surface of the earth might have a nonzero horizontal flux I(1).
Clearly, I(σ) is an integral transformation, with \( {K(\sigma,x) = x/\sqrt{x^2-\sigma^2}}\) the kernal. Specifically, Equation (A1) is a version of the Abel transform (Bracewell, 1986). In fact, the usual Abel transform is applied to a function defined over an infinite domain, but fortunately one can show that the key results carry over to our case of a finite domain.
The inverse Abel transform appropriate for our case is
where μ = cos θ, and I′(y) = dI(y)/dy is the usual derivative.
Equation (A2) thus demonstrates by construction that, given a complete knowledge of the intensity distribution, one can fully recover the radioisotope source distribution. Thus, measurement of the geoneutrino angular distribution truly does carry the promise of tomographic imaging of the earth’s interior. In addition, with q(x) in hand, one can completely determine the radiogenic heat production of the earth, both globally and as a function of depth.
Furthermore, Equation (A2) has the properties one would expect on physical grounds. The density at \( {r = R_{\oplus} x}\) depends only on the intensity derivative for the region \( {\sin \theta \ge x}\) , i.e., angles along or exterior to the tangent angle. Thus, inferring the outer density structure requires only knowledge of the peripheral intensity. On the other hand, to recover the inner density structure requires both peripheral and central intensities. This is indeed sensible if one thinks of the angular distribution roughly as a linear combination of intensities along the line of sight: outer angles have only a few “terms” in the sum, while inner angles contain all “terms.”
One consequence of this result is that the peripheral intensity constrains the central intensity, by setting a lower limit on it. If we consider I(σ) only for σ > σ0, we can infer a lower limit q min to the density distribution at x < σ0, namely
This example illustrates that even with an incomplete or low-resolution determination of the intensity pattern, one can draw powerful physical conclusions.
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Fields, B.D., Hochmuth, K.A. Imaging the Earth’s Interior: the Angular Distribution of Terrestrial Neutrinos. Earth Moon Planet 99, 155–181 (2006). https://doi.org/10.1007/s11038-006-9132-4
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DOI: https://doi.org/10.1007/s11038-006-9132-4