For the melting temperature of iron and silicates we assume the values listed in Table 5. Such restriction would be true only for extreme sulfur rich compositions. At low concentrations of sulphur higher liquidus temperatures are required to melt Fe,Ni-FeS systems.
Appropriate values of the liquidus temperature for the iron melt are K for LL chondrites and K for H chondrites Taylor In additional calculation, we also test the influence of a low liquidus temperature for iron on the differentiation of planetesimals for a better comparison to previous models. In the initial state the material of the planetesimal is in a powder-like form, suggesting a much lower thermal conductivity by lattice vibration phonon conduction than that of the compacted material because in a high porous body the dust grains have only a few contact points or small contact areas to the neighbouring particles.
However, if a planetesimal experiences sintering, the material compacts and the thermal conductivity rises. We use the approximation of Krause et al. This equation is a smooth approximation of the fits to the data obtained by Krause et al. In this model several components of known conductivity are randomly orientated and distributed within a mixture; a distribution that is appropriate for planetesimals. The first four values form the bulk conductivity of the silicate component and the latter two that of the iron component, both according to Eq.
From the resulting two values and from the volume fractions of iron and silicates during and after the differentiation, the conductivity in a given volume is calculated using Once a protocore is formed, the relatively high conductivity of the Fe,Ni-FeS system results in a flat temperature profile throughout the core, whereas in the mantle the temperature equalisation proceeds rather slowly.
We neglect the energy transport by radiation between grains of the porous material. This kind of heat transport occurs at the starting stage of the thermal evolution. Yet, in Henke et al. The specific heat capacity of the iron and silicate mixture depends on the temperature and on the composition according to 32 with c p,Si T and c p,Fe T the temperature dependent heat capacity of silicate and iron, respectively.
Having c p,Fe T and c p,Si T , the specific heat capacity can be computed for any mass fractions of iron and silicates from Eq. At the temperatures of e. Table 6 Parameter values used to model the radioactive heat production. Table 7 List of models presented in the figures including the varied input parameters. The pre-factor v i is given by the equation 36 with the volume fractions and of liquid resp.
At the beginning of a simulation the heat producing nuclei are distributed homogeneously, although later this may change due to the melting and subsequent chemical differentiation. Note that we neglect that some of the 60 Fe is contained in the silicates and would migrate with them.
Compared to the overall heat production heating by this portion of nuclides is considered as negligible. The used parameter values are listed in Table 6. We first study bodies in which heat is transported by conduction only, i.
We study the differences between instantaneous and continuous accretion and vary the onset time of accretion, the duration t accr of the accretion as well as the accretion law cases PI-PA. Next we consider bodies that undergo melting and study the effects of heat transport due to melt migration, material transport and redistribution of radiogenic heat sources cf. Table 7 , cases A1—A4. To determine the evolution of the interior structure, e.
Table 4 is chosen, representative for H chondrites cases P1—P4 and a sulfur-rich composition with We further examine the influence of the initial grain size b 0 and the Ostwald ripening cases G1—G3 and the effects of accretion and different accretion laws cases L1—L3 and R1—R3 on the differentiation of the planetesimals. Table 7 provides an overview on the simulations that were performed for the current study and the parameters corresponding to the respective simulation.
The considered planetesimals accrete as porous bodies and undergo sintering due to increasing temperature and pressure. We show that the thermal evolution depends on the accretion onset time t 0 , the accretion duration t accr , the accretion law and the terminal radius D. The latter corresponds to a body with the porosity equal to zero and is prescribed at the beginning of a simulation. In the case of instantaneous formation the initial radius is larger than the terminal radius and is given by 37 the radius changes with time due to sintering but the terminal radius D will not be reached as discussed below.
If the accretion is not instantaneous the initial radius of the planetesimal without porosity is chosen to be 1 km; thus using Eq. This value corresponds to the hexagonal prismatic packing of dust particles, consistent to the computation of the effective stress Eq.
Left panel : linear accretion with varying accretion duration t accr. During the early evolution the interior of the planetesimals is heated by the decay of the radioactive elements. Sintering starts at a threshold temperature that depends on the activation energy Eq. For a pressure of 0. As in the early stages of the thermal evolution the peak temperature is attained in centre of the planetesimal, it compacts from the inside out. For planetesimals with larger radii a porosity of zero can be reached almost in the entire body.
Only in the uppermost layers the temperature conditions necessary for sintering are not reached due to the cooling through the surface and low pressure.
Thus, the surface retains its porosity during the entire evolution and serves as an insulating layer. The loss of porosity is accompanied by the shrinkage of the radius.
The shrinkage results in a larger surface-to-volume ratio and hence in a more effective cooling of the sintered body — in addition to the higher thermal conductivity of the compacted material in comparison to the low conductivity of the porous material. In the following, we focus on the effects of a non-instantaneous accretion of a body on its thermal evolution. For a detailed discussion of the thermal evolution of instantaneously formed porous bodies that undergo sintering by hot pressing we refer to Henke et al.
In Fig. The instantaneous accreting planetesimal starts as a porous Sintering then leads to the shrinkage of the body with a final radius of The non-instantaneous accreting body starts from a 1. During accretion the central regions reach the threshold temperature for sintering and begin to compact. Because accretion and sintering proceed simultaneously, the latter does not result in the shrinkage, but in a slightly dampened growth of the radius.
The loss of porosity continues during the growth of the body — the already accreted surface material gets covered by the new layers, heats up and gets exposed to higher temperatures and pressures. Once accretion is finished, sintering leads to the actual shrinkage of the body to the final radius of Thus, planetesimals with the same mass differ in their final size dependent on the accretion time and — as we will show later — on the accretion law. The difference in size can be explained by the difference in the thermal evolution.
In the case of non-instantaneous accretion the threshold temperature for sintering is reached at later times, if at all. After accretion and sintering, this results in a smaller sintered region and a thicker porous layer in contrast to an instantaneously accreting body.
A simple relationship can be deduced between the accretion duration and the amount of sintering. The longer it takes for a body to accrete to the same terminal radius abiding by the same accretion law , the lower is the central temperature and the average temperature at any time t , leading to a thicker regolith layer.
When a planetesimal cools down and the temperature drops below the critical sintering temperature, no further changes of the porosity due to the hot pressing are possible. Figure 3 left panel shows final porosity profiles of cooled bodies that accreted in the same manner but with different t accr. In the presented case the thickness of the porous layer increases from about 2 km for instantaneous accretion to about 5 km assuming an accretion time of 1.
Similarly, for a fixed accretion duration t accr and fixed terminal radius, linearly accreting bodies will sinter more efficient than exponentially accreting ones, and those, again, more efficient than the asymptotically accreting cases Fig. However, the differences in the final porosity profiles of those three cases are almost negligible.
In addition to differences in the final radius and the thickness of the outer porous layer, sintering in combination with accretion influences the central peak temperatures of the planetesimals. The peak temperature depends on several factors: the radius of the planetesimal determines the surface-to-volume ratio and hence controls the efficiency of the heat loss. The accretion onset time influences the radiogenic heat source density.
The accretion law controls the change of the radius during the evolution. The accretion duration either catalyses or inhibits the change of the radius and finally, the presence of porous material influences the heat conductivity. Consideration of both the heat transport by melt and the redistribution of radioactive elements solid line , consideration of the heat transport by melt and neglection of the redistribution of radioactive elements dashed line , consideration of the redistribution of radioactive elements and neglection of the heat transport by melt dot-dashed line , neglection of both the heat transport by melt and the redistribution of radioactive elements dotted line.
Figure 4 middle panel shows the dependence of the central temperature on the accretion duration for the linear accretion law and Fig. An instantaneously formed body heats up to its peak central temperature and cools down with the reduction of the concentration of 26 Al and 60 Fe. The sintering takes place in the beginning of the evolution and influences only the increase of the temperature before it reaches the maximum, see Fig.
The thermal evolution of non-instantaneously accreting bodies is associated with the interplay of heating, sintering and accretion. Under favourable conditions e. Regions around the centre lose their porosity and the heat loss by conduction becomes more efficient. As the accretion continues, the higher surface to volume ratio dampens the temperature increase. The longer the accretion the stronger is the dampening and may even result in a decrease of the temperature.
Later, after the accretion, the temperature can rise again leading to a second maximum. Note that although non-instantaneously accreting bodies have lower peak temperature, they cool down significantly slower Fig. It can be concluded that the central peak temperature in initially highly porous planetesimals is significantly higher than assuming initially compact planetesimals.
The temperature difference is in particular large for small sized planetesimals of less than a few km, consistent with the results of Henke et al. Accretion dampens the temperature increase and thus the sintering process compared to the instantaneous formation — the longer the accretion the less efficient is the sintering and thus the porosity loss. This influence of accretion on the porosity loss differs, however, between very small bodies that will not sinter at all due to the unsufficient pressure and those with larger final radii and higher pressures that will compact and reach higher thermal conductivity.
In the first case, prolonged accretion does not alter the peak temperature significantly, whereas in the second case the temperature increase depends on t accr and the accretion law. In the following we discuss the effects of melting on the chemical composition, internal structure and thermal evolution of planetesimals. We study in particular the influence of accretion duration and accretion law , the liquidus temperature of iron and the grain size, which is an important parameter constraining the flow velocities of melt.
Case A1 neglects the heat transport of melt and the redistribution of radioactive elements. Case A2 considers the heat transport of melt but neglects the redistribution of radioactive elements. Case A3 considers the redistribution of the radiogenic heat sources due to melt migration but neglects the heat transport of melt. Finally, case A4 considers both the heat transport of melt and the redistribution of radioactive elements. Figure 5 right panel shows the temperature profiles at the instant when the maximal peak temperature in the planetesimal is reached.
These temperatures and the associated times are given in Table 8. Table 8 Peak temperatures and instances of peak temperatures for the simulations A1—A4. The temperature increase in the centre is slightly buffered due to sintering after about 1.
Then, at the temperature interval between and K, i. After melting of the iron, the temperature rises rapidly again for the cases A1 and A2 neglection of the transport of radioactive elements until the silicates start to melt. For the cases A3 and A4 the iron melt sinking to the centre displaces the 26 Al-enriched compacted matrix. Thus, the heat source density is strongly reduced in the centre and the phase of a constant central temperature lasts longer, until heating from above causes a further temperature increase cf.
The consideration of both the heat transport by melt and the redistribution of radioactive elements leads to a dampening of the temperature in the region around the centre and to relative higher temperature in the upper layers e. The peak temperature is about K lower when considering the effect of the heat transport by melt and redistribution of the heat sources and occurs at about 1.
Table 7. Apart from the amount of melting, the magnitude of dampening depends also on the assumed Darcy exponent, the Darcy coefficient and on the grain size, which will be discussed below. The melt migration of iron and silicates is also associated with the differentiation and thus a change of the interior structure of the planetesimal with time. We distinguish in particular between the iron-silicate separation associated with core formation and the silicate-silicate separation.
We consider four exemplary bodies starting as porous objects with the radii of 9. Furthermore, the relative radius is given at which the first melt starts to migrate as well as the relative size and volume of the obtained reservoirs, i. The melt velocity depends in addition to the melt fraction and hence temperature also on the gravity.
Thus, the location of the first melt movement is not at the centre, where g equals to zero. At the temperature of e. Hence, no extensive melt migration is expected to occur at temperatures below the silicate solidus. Table 9. This is due to the considerably lower liquidus temperature and the associated higher degree of melting and larger partially molten zone causing higher migration velocities of the iron melt.
Table 9 Formation times and extent of the different reservoirs for planetesimals assuming the liquidus temperature of iron equal to K a and the liquidus temperature of iron equal to K b. Plotted data show the internal structures at the time when the core formation is completed, i.
Note also that after iron melt percolates downwards and a core forms, some of the iron recrystallises again with decreasing temperature. See Table 9 for further information. Depending on the prescribed liquidus temperature of the iron material in planetesimals, we can distinguish between two differentiation scenarios.
Downward movement of the iron melt starts from above the distance halfway from the centre. Molten iron sinks to the centre moving the compacted silicate matrix upwards.
Depending on the specific case cp. PL1—PL4 the iron-silicate separation ends either before silicates start to melt or at almost negligible silicate melt fractions. In this case both iron and silicate melts migrate simultaneously, assuming migration along separate veins. Independently on the liquidus temperatures except case P1, for which no iron core is formed , the first differentiated layer is located about halfway between the centre and the surface.
The following final structure develops: a high density core consisting of a mixture of Fe, Ni and FeS with a radius less than about 0. The larger the planetesimal the thinner is the regolith layer for the chosen cases. Lower iron liquidus temperature results in larger core, thicker mantle and smaller undifferentiated volume see Table 9. The thickness of the regolith that is controlled by the temperature and the pressure does not change significantly.
This layer serves as an insulating blanket that inhibits the heat loss through the surface. Figure 6 shows the volume fractions of solid black and liquid red iron and solid green and liquid yellow silicates as functions of the relative radius for the cases P1—P4 and L1—PL4. The interior structure is depicted at the time when a planetesimal cools below the iron solidus and no further differentiation is possible or at the time t 2 when iron melt migration ceases although molten iron is still present.
The migration of silicate melt from the silicate matrix exceeds that time but changes the interior structure only minor. The silicate melt is not able to penetrate the upper layers, it freezes at depth before reaching the surface — no basaltic crustal layer can form. Note that sintered and unsintered regions are not indicated in this figure.
Table 10 Influence of the grain size on the differentiation process. The duration of the core formation, i. In the cases PL1—PL4, i. Table 9 b and stays below 1 Ma. This coincides with the findings of Sahijpal et al. Tables 9 — Table 11 Influence of the accretion duration and accretion law on the differentiation process. Although the metal-silicate separation ends at the time t 2 , some melt migration still occurs in the partially molten mantle. The silicate-silicate separation is thus less efficient than the metal-silicate separation.
Silicate melt rises to the colder upper mantle and solidifies subsequently. The time of the last silicate melt migration inside a pure silicate matrix coincides roughly with the time when the temperature in the mantle falls below the silicate solidus temperature cp. Typically, the silicate-silicate separation exceeds the separation of iron by a factor of two to three and can last up to 20 Ma after CAI formation.
The differentiation of the planetesimals and thus the interior structure depends also strongly on the formation time Fig. Assuming instantaneous accretion, a formation time simultaneously with the CAIs and an H-chondritic composition, planetesimals larger than about 6 km can be partly differentiated and those larger than 8 km can even form a core. Note that in the region between the dashed and solid line, melt segregation is only minor and a pure iron core cannot be formed.
With increasing radius the melt segregation and thus differentiation is more efficient resulting in a larger final core radius. For later formation times, the occurrence of differentiated planetesimal is shifted to larger planetesimals and a formation time of about 3. The light grey area indicates the parameter range for which no melting occurs. This region is separated dashed line to the small region where melting occurs but only partial differentiation, i.
The dark grey region indicates the parameter range for which the planetesimals is differentiated into an iron core and silicate mantle. The sizes of the core and the mantle increase with increasing terminal radius and decreasing accretion time white arrow. Left panel corresponds to the case G1, middle panel to G2 and right panel to G3.
To demonstrate the dependence of the efficiency of differentiation on the grain size we compare three cases. In the third case G3 b is equal to 1 cm and does not change.
Figure 8 shows the fractions of solid iron black , liquid iron red , solid silicates green and liquid silicates yellow for cases G1, G2 and G3. A straightforward relationship can be identified between the grain size and the efficiency of the differentiation by porous flow.
In the centre the iron fraction increases and dominates, whereas the mid-depth layers consist almost entirely of the silicate material. However, no pure iron or silicate layer forms. This is the result of some small scale melt migration which takes more than 10 Ma see Table Imposing Ostwald ripening on the same initial grain size leads to larger grains during melting and consequently to larger melt migration velocities. Hence, the case G2 differentiates into an iron core, a silicate mantle and a relatively thick around one fourth of the radius undifferentiated outer layer.
Note that in this case the grains do not grow to 1 cm, as there is not enough melting. The most efficient differentiation is obtained for case G3. Due to large grains and hence high melt migration velocities the body differentiates as in the second case but has a larger core, a thicker mantle and a thinner undifferentiated layer. The onset of the melt migration and the duration of the differentiation also correlates with the grain size.
As the Earth began to melt, gases would be released in a fashion similar to that observed in contemporary volcanoes. Release of enormous quantities of volatiles during wholesale melting, resulted in the generation of the primitive atmosphere and hydrosphere. I use the term primitive, because these volcanic gases would have been very toxic, would have contained little or no free oxygen, and certainly would have been incapable of supporting any life.
It would take an additional billion years or so, before primitive organisms could develop and begin the process of photosynthesis necessary to generate an atmosphere at all like the one we enjoy and rely upon today. The diagram to the left presents a very simplified model showing how convection in the mantle might be the driving force of plate tectonics.
Contents Search. Differentiation, Planetary. Living reference work entry First Online: 02 May How to cite. Definition Planetary differentiation is the separation of different constituents of planetary materials resulting in the formation of distinct compositional layers.
This is a preview of subscription content, log in to check access. Abe Y Thermal evolution and chemical differentiation of the terrestrial magma ocean. Armstrong RL Radiogenic isotopes. The case for crustal recycling on a near-steady-state no-continental-growth Earth.
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