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Temperature in Kelvin
FIG. I. The dependence of specific heat capacity on temperature for some common minerals in chondrites (a) and various ordinary chondrite compositions (b). It should be noted that there are no measurements on ferrosillite, albite, orthoclase, or diopside. The spikes in (a) for metallic Fe exist because of phase changes.
thermal modeling of asteroids. No attempt is made to reconstruct the thermal histories of specific ordinary chondrite parent bodies as has been done by various workers (e.g., Bennett and McSween, 1996; Benoit and Sears, 1997). However, it will be clear from this work that the results of previous thermal modeling of ordinary chondrite parent bodies (and other asteroid thermal models which use temperature-independent specific heat capacity) may need to be refined.
EFFECT ON THERMAL MODELS
Peak Temperatu res
Our runs using constant values of cv from 600 to 1200 J/k.g/K yield peak temperatures of 1123 and 707 K, a spread of 416 K (Table 2) in peak temperature, whereas accounting for the temper ature dependence of cv results in peak temperatures of 955, 929, and
100 K. As a first approximation, the average specific heat capacity
of H chondrites may be said to be -700--800 J/kg/K, and that of L and LL chondrites to lie -800 J/kg/K. Thus, it is not surprising that the peak temperatures obtained in the case of H chondrites (955 K) lie between the peak temperature obtained for a value of constant cv = 700 and 800 J/kg/K ( 1006 and 916 K, respectively), whereas for L and LL chondrites, the peak temperatures (929 and 908 K, respectively) obtained are close to that for cv = 800 .1/kg/K (916 K). Hence, the value of cv from Miyamoto et al. ( 1981) would result in overestimation of peak temperatures for L chondrites by 1090 K- 916 K = 174 K, whereas the value of cv from Herndon and Herndon (1977) would result in the underestimation of peak temperature by 916--707 K = 219 K.
Accretion Time
The last section shows that a thermal model which uses a lower value of specific heat capacity obtains higher peak temperatures. Thermal models that use 26AI decay as a heat source have typically attempted to match peak temperatures obtained in meteorites by adjusting the accretion time, thereby determining the amount of live 26Al incorporated into the asteroid. However, to realize the same peak temperatures, it is necessary to assume a shorter accretion time in a thermal model that assumes a higher value of specific heat capacity. For example, the runs which use 600 and 1200 J/kg/K produce peak temperatures of 1123 and 707 K, respectively, at the center of the asteroid. Because both runs use the same accretion time (and, therefore, the same amount of live 26Al), the heat provided is equal. This can be verified by a rough calculation using
cv = 700 J/kg/K cv = 600 J/kg/K cv = 800 J/kg/K
1000
·
c
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[ill
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Cv = 1200 J/kg/K
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Cv =
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cv = 900 Jlkg!K
1000 J/kg/K
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Cv = 1100 J/kg/K
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:;;
·BOO
(ii 900 (ii
-
·c BOO .!: 700
ro .2
::J
100
Qi
600
Q)
400
300 300
Cv = 1200 J/kg/K
0 10 20 30 40 50 60 70 BO 90 100 0 10 20 30 40 50 60 70 BO 90 100
Time after accretion in Ma Time after accretion in Ma
1200 1000
H Chondrite L Chondrite [fi]
1100
·c s:
900
- BOO
LL Chondrite
(ii 900 (ii
·- 800
ro
::J
700
LL Chondrite
.2
600
L Chondrite
Qi Q)
E
0.. 0..
600 E
500
500
300 300
0 10 20 30 40 50 60 70 BO 90 100 0 10 20 30 40 50 60 70 BO 90 100
Time after accretion in Ma Time after accretion in Ma
FIG. 2. Thermal histories of an asteroid of radius 100 km, calculated for the same thermal parameters except specific heat capacity, at the center (a, b), and at a depth of 6 km (c, d). (a) and (c) use constant values of specific heat capacity, whereas (b) and (d) incorporate temperature dependence for various ordinary chondrite compositions.
Eq. (2). The heat generated per unit mass to obtain the peak temperature (= Cv dT = 600 x ( Tpeak-Timtial) = 600 x (1123-292) = 600 x 831 = 504 600 J/kg) is equal to the heat generated in the second case (= Cv dT = I 200 x ( TpeaCTinitia/ ) = 600 x (707-292) = 600 x 415 = 498 000 J/kg). To make the peak temperature equal to I 123 K in both runs, it is necessary to supply twice as much heat to the thermal model where cv = I 200 J/kg/K. This can be done by doubling the amount of live 26AI incorporated in the asteroid, which equates to a decrease i n the accretion time by 0.72 Ma. As mentioned above, average cv (over the temperature range of the simulation) for ordinary chondrites can be approximated at
-800 J/kg/K. A value of cv = 1200 J/kg/K overestimates accretion time, whereas cv = 600 J/kg/K would underestimate accretion time. Thus, an unrealistic value of specific heat capacity causes an incorrect estimation of accretion time in a model which uses 26AI heating, although the thermal model can reproduce realistic peak temperatures.
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