Temperature dependence of specific heat capacity and its effect on asteroid thermal models

INTRODUCTION

The original purpose of asteroid thermal models was to perform feasibility tests for postulated heat sources such as ²⁶Al decay (e.g., Herndon and Herndon, 1977). However, newer thermal models have been used to make inferences about the geologic evolution of asteroids (e.g., Akridge et al., 1998; Ghosh and McSween, 1998) and meteorite parent bodies (e.g., Miyamoto et al., 1981; Grimm and Mcsween, 1989; Bennett and McSween, 1996; Benoit and Sears, 1997). As with all computer simulations, certain simplific­ ations are commonly made to render the problem computable by ignoring nonlinearities and interdependence of variable parameters. Although such simplifications may be acceptable, it is important to test whether incorporation of the complexity causes significant variation in the output of the thermal model.

An important parameter in thermal models is the specific heat capacity, defined as the heat required to change the temperature of unit mass of a substance by one unit of temperature. Specific heat capacity can be measured at constant temperature (cv) or pressure (cp)· Experimental determinations of specific heat capacity measure cP and the most comprehensive compilation of laboratory measurements of the specific heat capacities (cp) of stoichiometric minerals was given by Robie et al. ( 1978). The heat transfer equation, however, requires the use of cv. The conversion between cp and cv is as follows (Navrotsky, 1994):

(1)

where T = temperature, V = volume, a = thermal expansivity, and

p = compressibility or reciprocal of bulk modulus. Many workers

have used cp instead of cv ( e.g., Grimm and McSween, 1989). Although the use of cp is theoretically incorrect, the difference between cp and cv is no more than a few percent except at high temperature (Navrotsky, 1994), resulting in only small variations in the output of a thermal model.

Specific heat capacity has been typically assumed to be constant in most thermal models. A summary of specific heat capacity values used by various workers is given in Table 1. Wood's (1964) model, which utilized a temperature-dependent heat capacity measured for bulk ordinary chondrites (Alexeeva, 1958), was a notable exception. Grimm and Mcsween ( 1989) used the formulation that specific heat capacity was the weighted mean of individual components but used it to discriminate between the specific heat capacity of rock and ice.

In this paper, we utilize the temperature dependence of cp for the minerals that compose a rock to compute a weighted mean value for the bulk rock and compare the result with computations using con­ stant values of specific heat capacity.

Specific heat capacity is certainly not the only important vari­ able in asteroid thermal models, but we have isolated it to illustrate the effect of its temperature dependence. Ordinary chondrite parent bodies are chosen to illustrate the point because of ease of modeling (because, unlike achondrites, they did not undergo melting and differentiation).

THERMAL MODEL - METHODOLOGY

For illustration, our model uses a generic chondritic asteroid of radius 100 km. We assume accretion to be instantaneous at 3.4 Ma after calcium-aluminum-rich inclusion (CAI) formation. The accre­ tion time is chosen such that peak temperatures realized in the thermal model are roughly consistent with the range of temperatures inferred from ordinary chondrites (e.g., Mcsween et al., 1988 ). No effort is made to match exactly the peak temperature observed in any of the ordinary chondrites. The decay of26Al is assumed to be the heat source, and the initial ratio of 26Al/27Al at the time of CAI

formation is taken to be the canonical value of 5 x 1o-s

(MacPherson et al., 1995). Aluminum-26 is assumed to be homogeneously distributed in the accreting materials. The heat transfer equation is solved by employing the finite-element method, using a radiation boundary condition and an initial temperature of 292 K (for detailed methodology, see Ghosh and Mcsween, 1998). As this is a study of the effect of varying specific heat capacity, we have kept all other parameters the same in all runs. The thermal diffussivity is assumed to be constant at I 0-7 m2/s. We have used constant values of specific heat capacity varying between 600 and 1200 J/kg/K, in 100 J/kg/K increments. We have also computed specific heat capacities for H-chondrite, L-chondrite, and LL­ chondrite compositions from their normative mineralogy calculated from bulk chemical compositions (McSween et al., 1991). The specific heat capacity of the rock was assumed to be the weighted average of the specific heat capacities of the constituent minerals. The temperature dependence of specific heat capacity on minerals was incorporated from Robie et al. ( 1978). Significantly, there are no determinations of certain endmember phases of solid solution series (e.g., ferrosilite). In such cases, we have assumed the same temperature dependence as measured in another similar mineral (e.g., the temperature dependence of ferrosilite is assumed to be the

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TABLE 1. Specific heat capacity values.*

Work Value of specific heat capacity used

Comments

Akridge el al. (1998)	Used heat capacity relation with temperature for olivine (Robie el al., 1978)
Bennett and McSween (1996)	Cannot be determined	Model did not use Cy; used temperature-dependent K and K instead.
Ghosh and Mcsween (1998)	Used temperature- and composition­
	dependent Cy (Robie et al., 1978)
Grimm and McSween (1993)	not specified
Grimm and Mcsween ( 1989)	700 J/kg/K
Haack el al. (1990)	1200 J/kg/K	Long-lived radionuclides as heat source; high initial temperature
Herbert ( 1980)	770 J/kg/K	Model for mass distribution on the Moon as a function of time
Herbert ( 1989)	1200 J/kg/K	Model for electromagnetic induction heating
Herndon and Herndon (1977)	1200 J/kg/K
Herndon and Rowe (1973)	not specified
Hort (1997)	1100 J/kg/K	Cooling and crystallization in sheet-like magma bodies (on Earth)
Lazarewicz and Gaffey ( 1980)	not specified	Heating by long-lived radionuclides
Melosh ( 1990)	1000 J/kg/K	Thermal effects of impact of Earth- and Mars-size body
Minear (1980)	1200 J/kg/K	Lunar ocean modeling
Minster and Allegre ( 1978)	200 cal/kg/K
Miyamoto et al. (1981)	625 J/kg/K
Shimazu and Terasawa (1995)	100 J/kg/K	Model for electromagnetic induction heating
Toksoz et al. (1978)	1200 J/kg/K	Thermal models of the Moon, Mercury, Mars, Venus, and minor planets
Williams and Wetherill (1993) Wood (1964)	not specified Used temperature dependent heat capacity	Thermal model of collision of large asteroids
	of chondrites (from Alekseeva, 1958)