Statistics – Computation
Scientific paper
Sep 1995
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1995metic..30q.509g&link_type=abstract
Meteoritics, vol. 30, no. 5, page 509
Statistics
Computation
Chondrites, Ordinary, Cooling Rates, Extinct Radionuclides, Heat Flow, Heat Sources, Thermal Conductivity
Scientific paper
Nineteen years after the discovery of excess 26Mg in the Allende meteorite proved the presence of 26Al in the early solar system [1] and after furious debates spanning the theoretical and observational aspects of the science, the hypothesis of 26Al as a heat source, remains unresolved [2]. Several attempts at modeling the thermal history of asteroids [3,4,5,6] seem to indicate that 26Al is a plausible heat source. Previous work: Each of these approaches introduced several simplification in the physics of the system. [3] and [6] use the classical analytical series solution approach, whereas [4,5] use a finite difference method on the governing equation. In this work, the finite element method, which is believed to be numerically more robust [7], is used. [3] and [6] use a Neumann convection boundary condition at the surface of the asteroid according to which the heat flux is proportional to the surface temperature of the asteroid. [4,5] use a Dirichlet boundary condition assuming the surface temperature of the asteroid to be equal to the surrounding medium. In a real scenario, radiation is believed to be the dominant process for heat loss from an asteroid, and this is what is taken into account in this work. A Neumann radiation boundary condition is used, where the heat flux at the surface of the asteroid is proportional to the fourth power of absolute temperature. [3,4,5] assume both thermal diffusivity and specific heat capacity to be independent of temperature, whereas [6] assumes specific heat capacity to be independent of temperature. In reality these parameters are temperature dependent, which is accounted for in this model. Basic equation and methodology: Heating is assumed to be caused primarily by 26Al and 60Fe, and the heat is conducted through the body and dissipated at the surface by radiation, according to the following equation: where R= radius, T= Temperature, t= time, kappa= thermal diffusivity, rho= density, cp= specific heat capacity, Q= heat production per unit mass= Ao Al Qo A1 exp(- lA1t) + Ao Fe Qo Fe exp (- lFet), Ao= initial abundance, Qo= initial heat production per unit mass, l = decay constant. The surface temperature is determined by the following energy balance equation: where, R'= radius of the asteroid, So= solar constant, A = albedo, e = emissivity, sigma = Stephen Boltzmann constant, k = thermal conductivity = rho(cp)kappa, d= orbital semi major axis. Numerical Method: A Galerkin Weak Statement, where each weight function is identical to the corresponding trial function, is written on the heat transfer equation system (1) - (2). Thermal diffusivity and specific heat capacity are temperature dependent properties. Using the relations of [8], the values of thermal diffusivity and specific heat capacity are updated, based on finite element implementation for temperature in the previous timestep. Results: Our preliminary results indicate that there are significant differences in the calculated time-temperature profiles if: a) the temperature dependence of specific heat capacity and thermal diffusivity are taken into account, in contrast to previous models where these parameters were assumed to be constant, and b) a radiation boundary condition is used. On applying this model to ordinary chondrite parent bodies, it is observed that the peak temperatures are attained earlier, the cooling curve after this time is gentler, and the closure temperature of the Rb-Sr system is reached much later. In other words, this model indicates a larger time interval between the time peak temperatures are attained in the asteroid and the time the asteroid attains closure with respect to Rb-Sr. References: [1] Lee T. et al. (1976) Geophys. Lett., 3, 41-44. [2] Wood J. A. and Pellas P. (1991) The Sun in Time, 740-760. [3] Miyamoto M. et al. (1981) Proc. LPS 12B, 1145-1152. [4] Grimm R. E. and McSween H. Y. (1989) LPS XIX, 427-428. [5] Grimm R. E. and McSween H. Y. (1989) Icarus, 82, 244-279. [6] Bennett M. and McSween H. Y. (1995) Meteoritics, submitted. [7] Baker A. J. (1983) Finite Element Computational Fluid Mechanics, 150. [8] Fujii N. and Osaka M. (1973) EPSL, 18, 65-71.
Baker Andrew Jordan
Ghosh Arpita
McSween Harry Y. Jr.
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