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Widdekind
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According to the History Channel documentary Comets -- Prophets of Doom (DVD), Earth today accumulates "30 to 40 thousand tons of space dust per year".
CRUDE MODEL:
We imagine that the accumulation rate of space debris declines exponentially:
CRUDE MODEL:
We imagine that the accumulation rate of space debris declines exponentially:
[tex]rate = T \times e^{-\frac{t}{\tau}}[/tex]
We then demand that:[tex]T \times e^{-\frac{4.5 \, billion}{\tau}} = 40,000 \, tons \, year^{-1}[/tex]
and[tex]\int_{0}^{4.5 \, billion} T \times e^{-\frac{t}{\tau}} dt = M_{Earth}[/tex]
Solving this system of equations numerically yields:[tex]T \approx 3 \times 10^{13} tons \, year^{-1}[/tex]
[tex]\tau \approx 200 \, million \, years[/tex]
This suggests, that Earth only reached ~98% of its present mass after 4[tex]\tau[/tex] = 800 million years. Since Earth is ~4.6 billion years old, this suggests, that the early Earth was accumulating considerable mass until ~3.8 billion years ago. Perhaps surprisingly, this agrees w/ Geological evidence, that Earth's Late Heavy Bombardment persisted until ~3.8 billion years ago*. Before this time, Earth was significantly (2-40%) smaller than today, explaining why no prior rocks** have (to date) been found.[tex]\tau \approx 200 \, million \, years[/tex]
* James F. Luhr. (DK) Earth -- the Definitive Visual Guide, pp. 26-29.
** ibid., Zircon crystals have been found from ~4 billion years ago.
** ibid., Zircon crystals have been found from ~4 billion years ago.