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

Isotopic Closure Ages

A spread in a specific isotopic closure age as measured by the thermal model depends on the depth to the surface, the specific heat capacity, and the peak temperature attained. The depth determines the rate of heat loss, which is almost uniform in the interior of the asteroid but rapidly increases in near-surface layers. The specific heat capacity, besides producing a change in the peak temperature, has another effect: If the same amount of heat is lost from a particular depth, a greater fall in temperature will result if the specific heat capacity is low compared to a case where cv is high. So this should produce comparatively gradual cooling in cases where the specific heat capacity is high. However, this effect is offset to an extent by the following negative feedback mechanism: Gradual cooling builds up a steeper gradient from the surface to the interior which, in turn, causes greater heat loss by thermal diffusion resulting in a faster rate of cooling.

TABLE 2. Peak temperatures and Pb-Pb, Rb-Sr, and Ar-Ar isotopic closure ages for various models of specific heat capacity.
Volumetric Proportion of Petrologic Types

We show that a change of specific heat capacity by

Specific heat capacity Peak Pb-Pb age Rb-Sr age (Jlkg/K) temperature ^{(727 K)} (570 K)

600	I J 23 K	>100 Ma	>100 Ma	>IOO Ma
700	1006 K	> IOO Ma	>J OO Ma	>100 Ma
800	9J 6 K	>J OO Ma	>J OO Ma	>J OO Ma
900	874 K	> J OO Ma	>J OO Ma	>J OO Ma
1000	79J K	>J OO Ma	>J OO Ma	>100 Ma
llOO	746 K	82.7 Ma	>J OO Ma	>J OO Ma
J 200	707 K	Not Attained	>J OO Ma	>J OO Ma
Hchondrite	955 K	>J OO Ma	>J OO Ma	>IOO Ma
L chondrite	929 K	>l OO Ma	>lOO Ma	>JOO Ma
LL chondrite	908 K	>J OO Ma	> J OO Ma	>J OO Ma
Depth = 6 km
600	989 K	11.8 Ma	28.9 Ma	51.2 Ma
700	888 K	8.67 Ma	22.3 Ma	42. J Ma
800	814 K	6.45 Ma	17.1 Ma	32.3 Ma
900	757 K	4.66 Ma	13.4 Ma	25.5 Ma
1000	71 J K	Not attained	10.8 Ma	20.7 Ma
llOO	674 K	Not attained	8.9 Ma	J 7. I Ma
1200	645 K	Not attained	7.6 Ma	J 4.5 Ma
H chondrite	849 K	7.15 Ma	19.0 Ma	35.5 Ma
L chondrite	826 K	6.7 Ma	17.6 Ma	32.7 Ma
LL chondrite	808 K	6.15 Ma	16.4 Ma	30.5 Ma

Depth = 100 km (center of the asteroid)

A major reason for the spread in closure ages in the various runs

Ar-Ar age (500 K)

200--400 J/kg/K can cause a difference of -200 K in peak temperature. This difference in temperature is consider­ able compared to the difference in peak temperature for various petrologic types (e.g., type 3 = 873 K, type 4 = 973 K, and type 5 = 1023 K; Mcsween et al., 1988). Thermal models of ordinary chondrite parent bodies, after constraining the radius of the parent body using peak temperatures and closure ages of type 6 ordinary chondrites, try to determine the depths for peak temperatures of various petrologic types in order to constrain their volumetric proportions. Because specific heat capacity has a prominent effect on peak temperature, it follows that there will be an effect on the volumetric proportions of petrologic types.

Cooling Rates

An examination of Table 3 shows that there exists no clear relation between cooling rates and change of specific heat capacity. Cooling rates are controlled primarily by loss of heat from the asteroidal surface. An equal amount of heat loss in a model which uses higher cv should cause a relatively gradual fall in temperature (and therefore higher cooling rates). However, as explained in the previous section, gradual cooling causes a steeper gradient from the surface to the interior, which

is the difference in peak temperatures. For example, the peak temperature attained in the center of the asteroid is I 005 K for cv = 700 J/kg/K, compared to 916 K for Cv = 1 JOO J/kg/K (Table 2). Thus, in the first case, the temperature must decrease by 278 K to attain closure of Pb-Pb ages (727 K); whereas in the second case, the temperature has to decrease by only 19 K. Therefore, it is no surprise that the closure age is higher in the first case.

Table 2 shows that different values of specific heat capacity produce a spread of >20 Ma in Pb-Pb closure ages of ordinary chondrites at the asteroid center. For values of cv 1000 J/kg/K, Pb-Pb age closure at the center takes place beyond the timeframe of the simulation (i.e., 100 Ma; for cv = 1100 J/kg/K closure takes place at 83 Ma; and for cv > 1100 J/kg/K, the peak temperature attained is less than the closure temperature of the Pb-Pb system). The difference in isotopic ages at the center is >40 Ma at a depth of 53 km and >6 Ma at a depth of 6 km.

Radius of Parent Body

Thermal models of ordinary chondrite parent bodies have tried to constrain asteroidal radius from peak temperatures and closure ages. For example, Miyamoto et al. ( 1981) obtained radii of 85 km for both the H- and L-chondrite parent body based on a peak temperature of 1150 K for type 6 ordinary chondrites and a Rb-Sr age of I 00 Ma (Wasserburg et al., 1969; Gray et al., 1973). Bennett and McSween (1996) obtained radii of 80-95 km for both the H­ and L-chondrite parent bodies based on peak temperatures for H6 and L6 of 1273 and 1298 K, respectively (McSween et al., 1988), and a Pb-Pb age of 60 Ma (Gopel et al., 1994). We showed above

in tum causes greater heat loss by thermal diffusion, thereby resulting in faster cooling.

Summary

The schematic illustration in Fig. 3 summarizes the effect of increasing specific heat capacity on various outputs obtained from thermal models. Heating is assumed to take place by decay of short­ lived radionuclides. The accretion time is kept constant, thereby keeping the heat generation term constant. An increase in cv, with no increase in the heat generation term, results in lower peak temperature. The cooling rate is governed by the spatial gradient of temperature, the specific heat capacity (which governs the amount of heat which needs to lost per degree change in temperature), and the peak temperature attained in the asteroid. In this case, the last factor

Specific heat capacity (J/kg/K)	Cooling rate (K/Ma)	Temperature at which cooling rate was measured in K
600	21.0	750
700	26.5	750
800	29.3	750
900	J9.3	750
1000	19.6	700
llOO	18.9	650
1200	18.6	600
H chondrite	28.8	750
L chondrite	29.1	750

TABLE 3. Computed cooling rates• from various runs of the code obtained at a depth of 6 km.

^{LL chondrite} 29.4 750

that the variation of specific heat capacity causes differences in both

peak temperature and closure ages. Because the radius of the asteroid is calculated by matching these two parameters, it is no surprise that specific heat capacity will have an effect on the computed size of the parent body.

*Metallographic cooling rates and fission tracks measure cooling rates over 870 to 670 K (Lipschutz et al., 1989, and references therein). Because all runs have not spanned the entire temperature interval, we have computed instantaneous cooling rates from the slope of the time-temperature curve at a specific temperature.

Accretion time = constant

Cv t

Cooling Rate + Peak Temperature +

t
UJ /.


!
Radius of parent body
Volumetric proportion of higher petrologic type t

FIG. 3. Schematic illustration of the effect of increasing cv on other thermal parameters.

dominates to produce, on average, a slower rate of cooling. The closure age, which is governed both by peak temperature and cooling rate, will decrease. The radius of the body is determined by the peak temperature and the closure age. Thermal models have tried to match experimental peak temperature and closure age measurements to the model output. An increase in specific heat capacity causes a decrease in both peak temperature and closure ages so that experimental peak temperatures are no longer reproduced in the thermal model. In such a situation, it is possible to generate the same peak temperature (i.e., the peak temperature before cv was increased) by increasing the radius of the asteroid. When the radius of the asteroid is increased, the proportional volume of the asteroidal interior which stays hot will increase, leaving a proportionally lower amount of cool surficial material. The increase in radius, therefore, will cause the volume proportion of the highest petrologic type to increase.