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Abstract We report the Earth's rate of radiogenic heat production and (anti)neutrino luminosity from geologically relevant short‐lived radionuclides (SLR) and long‐lived radionuclides (LLR) using decay constants from the geological community, updated nuclear physics parameters, and calculations of theβspectra. We track the time evolution of the radiogenic power and luminosity of the Earth over the last 4.57 billion years, assuming an absolute abundance for the refractory elements in the silicate Earth and key volatile/refractory element ratios (e.g., Fe/Al, K/U, and Rb/Sr) to set the abundance levels for the moderately volatile elements. The relevant decays for the present‐day heat production in the Earth (19.9 ± 3.0 TW) are from40K,87Rb,147Sm,232Th,235U, and238U. Given element concentrations in kg‐element/kg‐rock and densityρin kg/m3, a simplified equation to calculate the present‐day heat production in a rock isurn:x-wiley:ggge:media:ggge22244:ggge22244-math-0001 The radiogenic heating rate of Earth‐like material at solar system formation was some 103to 104times greater than present‐day values, largely due to decay of26Al in the silicate fraction, which was the dominant radiogenic heat source for the first∼10 Ma. Assuming instantaneous Earth formation, the upper bound on radiogenic energy supplied by the most powerful short‐lived radionuclide26Al (t1/2= 0.7 Ma) is 5.5×1031 J, which is comparable (within a factor of a few) to the planet's gravitational binding energy.
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A C α finite difference method for the Caputo time‐fractional diffusion equation
Abstract We begin with a treatment of the Caputo time‐fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro‐differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1‐method for the time variable and a standard fourth‐order approximation in the spatial variable. The main method derived in this article has a rate of convergence ofO(kα + h4)foru(x,t) ∈ Cα([0,T];C6(Ω)),0 < α < 1, which improves previous regularity assumptions that requireC2[0,T]regularity in the time variable. We also present a novel alternative method for a first‐order approximation in time, under a regularity assumption ofu(x,t) ∈ C1([0,T];C6(Ω)), while exhibiting order of convergence slightly more thanO(k)in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1‐method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.
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- Award ID(s):
- 2012235
- PAR ID:
- 10452591
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Numerical Methods for Partial Differential Equations
- Volume:
- 37
- Issue:
- 3
- ISSN:
- 0749-159X
- Format(s):
- Medium: X Size: p. 2261-2277
- Size(s):
- p. 2261-2277
- Sponsoring Org:
- National Science Foundation
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