Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
286 asteroids. The essential aim is to venture into the
possibility of generating a basaltic crust on a differentiated asteroid within the initial ~6 Myr. of the
solar system as is indicated by the observational records [6], [7]. The thermal models with porous planetesimals
[7] could not generate the basaltic crust even within the initial 10 Myr.
II. Numerical Simulations
The numerical simulations associated with the thermal modeling of asteroids deal with solving the heat
conduction partial differential equation for a spherical asteroid with uniformly distributed radioactive heat
sources,
26
Al and
60
Fe equation 1. The two short-lived nuclides produce approximately 3 million electron volts
MeV of energy per β-decay. The initial energy
production rates, Q
26
Al and Q
60
Fe are ~2×10
-7
W kg
-1
and 2×10
-8
W kg
-1
, respectively, for the two short-lived nuclides with the initial abundances of ~600 ppb [2] and
~280 ppb [8], respectively, at the time of the condensation of the earliest solar system dust grains.
This time is marked as the beginning of the solar system:
26 60
2 26
60 2
Al Fe
Al t Fe t
T T
Q Q
e e
t c
c r
λ λ
κ
− ⋅
− ⋅
∂ ∂
= +
+ ∂
∂
1 The temperature dependent thermal diffusivity ‘
κ’and the specific heat ‘c’ were used to deduce the thermal
evolution of asteroids [6], [7], [9]. The thermal diffusivity and specific heat were varied over the range
of 6.4-5.4×10
-7
m
2
s
-1
and 610-830 J kg
-1
K
-1
, respectively, for the un-melted and non-porous rock. The
un-melted asteroid with an initial density of 3,560 kg m
-3
was assumed to consist of uniformly distributed 16 metallic iron, 1 nickel, 3 iron in iron-sulfide form,
with the remaining constituents as silicates, commonly referred as rock. Since we are considering the
temperature dependent
κ and c, along with the melting and differentiation of the asteroids in the simulations it is
not possible to obtain a direct analytical solution of the partial differential equation. Hence, the partial
differential equation was solved by the finite difference method, using the classical explicit approximations [10].
To avoid any numerical instability associated with the technique the temporal and spatial grid sizes of 1 year
and 0.3 km, respectively, were used in the simulations. The chosen spatial grid interval divides the asteroid into
concentric shells of thickness 0.3 km. Subsequent to the melting of the planetesimals the metallic and silicate
melts were moved across these shells by performing detailed mass balance calculations [7].
The initial temperatures of the asteroids were assumed to be 250 K. We assumed an identical temperature for
the planetary surfaces throughout their thermal evolution. The heat from the interiors of the asteroids is transferred
to its surface from where it is radiated away to interplanetary space. The accretion of the asteroid
commenced with a “seed nuclei” of size 0.3 km. The accretion was initiated after a chosen time-span, ‘T
Onset
’ Myr. from the time of the condensation of the earliest
solar system dust grains. Some of these grains have survived in the meteorites. These grains are routinely
analyzed by various analytical instruments [2]. The onset of the accretion of different asteroids can occur at
distinct times during the initial ~10 Myr. of the solar system. The accretion duration of the asteroids was
considered to be one million years in all the simulations in the present work. The accretion was numerically
executed by gradually adding a single spatial grid element of size 0.3 km to the pre-existing spatial grid
array of the asteroids at a regular interval of time till the asteroids attains a final size of either 100 km or 270 km,
depending upon the simulation. The accretion was performed linearly in time. The radiogenic heating and
the associated thermal effects were incorporated right from the onset of the accretion of the asteroids.
The two uniformly distributed radionuclides in the un- melted asteroids gradually heat the body to its melting
temperature. The melting of the metallic iron-nickel and iron-sulfide occurs over the temperature range,
∆T
metal
~1,213-1,233 K, whereas, the melting of the bulk silicate the rocky content occurs over the temperature range,
∆T
silicate
~1,450-1,850 K [7]. The latent heats L of the melting of the metals and
silicates were assumed to be 2.7×10
5
J kg
-1
and 4.0×10
5
J kg
-1
, respectively. The latent heats were incorporated in the specific heat of the equation 1 by using equation 2
over the temperature range ∆T associated with the
melting:
L c
c T
= + ∆
2 Subsequent to the melting, the metallic melt
gravitationally sink towards the center of the asteroid to form a metallic core. As mentioned earlier, the entire
asteroid was divided into concentric shells of thickness 0.3 km each in order to perform the numerical treatment
of the thermodynamical processes related with the melting and planetary differentiation. An estimate was
made after every temporal step of one year, for each spatial grid shell of 0.3 km regarding the fraction of the
silicate and metal in melt and un-melt forms. The specific heat for a specific shell was estimated using the
weighted-average mean of the specific heats of the melt and un-melt fractions.
A specific heat of ~2,000 J kg
-1
K
-1
was assumed for the silicate and metallic iron melt fractions [6].
Due to their chemical nature,
60
Fe sinks along with the metal, whereas,
26
Al remains in silicate. Hence, the process of planetary differentiation chemically
segregates the two short-lived nuclides. This alters the further course of thermal evolution of the planetesimals.
The chemical segregation is appropriately incorporated in the simulations along with the major
Copyright © 2011 Praise Worthy Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, N. 2 Special Issue on Heat Transfer
287 segregation of the immiscible metal and silicate melts.
The segregation was performed in the presence of the gravitational field of the asteroid that controls the rate of
inward and outward flows of the immiscible fluids.
The segregation of the metallic iron-nickel melt from the silicate melt occurs on account of the density
differences, where the former has an assumed density of ~7,832 kg m
-3
compared to the density of 3,560 kg m
-3
for silicates [6]. The segregation involves the separation of two immiscible fluids. Subsequent to 0.4 fraction of
the bulk silicate melting within a shell, the metallic iron- nickel melt was segregated from the shell and moved to
the consecutive shell towards the asteroids center. The metallic melt pockets were moved towards the center to
form a metallic core, thereby, pushing the silicate melt and un-melt fractions towards the upper shells on
account of buoyancy. The gradual growth of the metallic core commenced from the center in an outward manner,
whereas, a mantle of silicate magma-ocean was formed above the metallic core. Detailed mass balance
calculations were performed to monitor the rates of fluid flows of the metallic melts towards the center, and the
outward flows of melted and un-melted silicate pockets. The fluid flow rates achieved by us are identical to the
flow rates predicted by the detailed theoretical studies based on Darcy’s law of fluid dynamics, see e.g., [7].
Subsequent to 0.5 fraction of the bulk silicate melting, the convection was set-up in the molten silicate magma-
ocean. The molten metallic core was assumed to be convective right from the onset of its formation [7]. In
order to numerically simulate thermal convection, the thermal diffusivity of the associated spatial shells
involved in the convection were enhanced by three orders of magnitude compared to the value of 5.4×10
-7
m
2
s
-1
for the un-melted rocky asteroid. Even though this is not an exact approach to incorporate convection, the
alternative approach adopted here will imitate convection that is a very efficient mode of transferring heat
compared to conduction. If not impossible, it is practically extremely difficult to incorporate convection
directly in our numerical approach. The onset of convection results in almost immediate thermal
homogenization of the interiors of the body. Further, it also results in rapid cooling of the molten magma ocean
in the mantle region of the asteroid and the metallic core as the asteroid efficiently radiates away the heat
transferred through its surface to the interplanetary space. The eventual cooling of the silicate magma-ocean
below 1,500 K results in ~80 crystallization solidification of the silicate magma-ocean. As one of
the most favored hypothesis of planetary differentiation, see e.g., [5], [7], the remaining 20 of the residual
silicate melt in the mantle is considered as the source of volcanic basalt that forms the crust of the differentiated
asteroids.
The residual silicate melts, being lighter than the crystallized silicate, move towards the surface of the
asteroids as lava, and erupt as volcanoes. These are some of the earliest volcanoes in our solar system that erupted
~4.5 billion years ago. The volcanic lava eventually solidifies as basalts on
the planetesimals surface. In the previous works on the numerical simulations of
planetary differentiation [6], [7] it was assumed that the asteroids were porous un-compacted at the time of their
accretion, with a porosity of ~50. The asteroids subsequently acquired compaction sintering upon
radiogenic heating beyond ~700 K. As the thermal diffusivity of the porous asteroids is around three orders
of magnitude low compared to the non-porous bodies, the thermal evolution of the porous body is distinct from
the non-porous body. As mentioned in the introductory section, the isotopic records of meteorites indicate that
the volcanic generated basaltic crust appeared on asteroids within the initial 6 Myr. on asteroids of radii
100-270 km [4]. The previous thermal models with the porous asteroids [7] could not explain the formation of
the basaltic crust on planetesimals even within the initial 10 Myr. The main objective of the present work is to
develop the thermal models of planetary differentiation of asteroids that were accreted as non-porous
compacted bodies. As mentioned above, the thermal evolution of these non-porous bodies would be different
from the porous bodies. Further, the main emphasis of the present work is to understand whether it would be
possible to generate the basaltic crust on the asteroids within the initial 6 Myr. that could not be achieved by
the previous work [7].
III. Results and Discussion
