Chronology Discussion

The chronology model leads to an accretion rate of comets on the Earth that is steadily diminishing with time. This rate is displayed on Fig. 2.4 for the first two and a half billion years after "age zero" (the epoch of dust sedimentation from the nebular gas). The contribution of the asteroid belt and of Jupiter's comets comes early; it is half finished during the first 200 million years. Since the different zones decay at different rates, the total accretion rate represented by the dotted line subsides drastically after 600 million years. The contribution of the zone of Jupiter prevails for the first 400 million years,

Moon formation

Fig. 2.4. Accretion rate of comets on the Earth, in grams per million years, as a function of time since dust sedimentation in the midplane of the accretion disk 4.56 billion years ago. The contributions of the zones of the giant planets start at different times and decay at different rates. The mass of each contribution is deduced from Table 2.2 and the exponential time scales are from Table 2.3. The exponential time scales for orbital diffusion have been normalized by using Everhart's (1977) results of random walk numerical experiments for the orbital diffusion of comets by the giant planets, and the exponential lifetimes of decay were chosen in proportion to the orbital periods of the giant planets. Each of the contributions of the different planet zones is a straight (solid) line on this logarithmic diagram. The sum of all contributions is the dashed line with an exponential rate varying with time, to be compared with the cratering rate on the Moon (Carr et al., 1984), whose data have been normalized to our units. The agreement is surprisingly good, not only for the mass rates, but also for the subsiding rate with time (reproduced, with permission, from Delsemme (2000)).

Age zero

Age of the solar system (bri years)

Fig. 2.4. Accretion rate of comets on the Earth, in grams per million years, as a function of time since dust sedimentation in the midplane of the accretion disk 4.56 billion years ago. The contributions of the zones of the giant planets start at different times and decay at different rates. The mass of each contribution is deduced from Table 2.2 and the exponential time scales are from Table 2.3. The exponential time scales for orbital diffusion have been normalized by using Everhart's (1977) results of random walk numerical experiments for the orbital diffusion of comets by the giant planets, and the exponential lifetimes of decay were chosen in proportion to the orbital periods of the giant planets. Each of the contributions of the different planet zones is a straight (solid) line on this logarithmic diagram. The sum of all contributions is the dashed line with an exponential rate varying with time, to be compared with the cratering rate on the Moon (Carr et al., 1984), whose data have been normalized to our units. The agreement is surprisingly good, not only for the mass rates, but also for the subsiding rate with time (reproduced, with permission, from Delsemme (2000)).

then Saturn takes over up to 1 billion years, then Uranus up to 1.4 billion years, and finally Neptune. Neptune's contribution contains the 2:1 resonance with Neptune in the Kuiper Belt; after 4.6 billion years (beyond the limit of Fig. 2.4) it still contributes 5 x 1018 g/Myr, or 50 comet collisions per million years, coming from the short-period comets that we still observe nowadays. This is a fair estimate for such a simplistic model. The difference between the times needed by the (short) runaway growth of Jupiter and the (longer) orderly growth of the Earth is the decisive factor that marshals the fraction of volatiles imprisoned in the Earth mantle. If our chronology model is correct, when the chondritic and cometary bombardment induced by Jupiter started, the Earth embryo had probably reached some 90% of the present mass of the Earth, and numerous collisions with 1022-1023 g bodies were still to come.

We presume that the accumulation of the Earth was essentially finished in 40-50 million years, assuming that it was somewhat enhanced at the beginning by the presence of nebular gas. It is not implausible either that it ended by a small period of runaway growth due to a large value of the gravitational focusing factor for the largest two or three bodies involved. However, if the nebular gas had dissipated much earlier and if there was no runaway growth, the standard orderly growth would yield 100 million years for the accumulation of the Earth, in which case one half of the chondritic contribution of the asteroid belt and one fourth of the comets coming from Jupiter's zone would be buried deep into the Earth's mantle. This represents a total of somewhat less than 1024 g of volatile material buried in the mantle, as opposed to a veneer of 6 x 1025 g brought after the accumulation of the Earth is essentially finished. Of course, this 1% of the total mass of exogenous bodies brought early onto the Earth does not seem very significant, until it is realized that its volatile material may explain a nonnegligible amount that would be recycled again and again in the mantle.

The chronology model of Table 2.3 implies the more likely view of a 40 million year accumulation for the Earth. In this case, the contribution to the lower mantle is minimal, and the early contribution to the upper mantle becomes undistinguishable from the late major contribution of the next 600 million years. This comes from the "gardening" of the crust and upper mantle by large impacts and because of the subsequent convection of the upper mantle.

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