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An exponential model for the growth of the Earth, proportional to the new sea floor crust, can be constructed CONFIDENTLY for the last ~200 Million years, because the Earth's growth is now robust and geometrically regular. BUT before the rifting of the modern ocean basins we are only dealing with intra-continenal crust. I believe using this modelling is only accurate for maximum 200 MY and can be used to forecast a few million years ahead in time. This is because the exponent will be relatively constant now as the Earth's growth has accelerated to a steady rate.
So, Earth growth is NOW a robust and geometrically regular process which applies to the recent (~200 MY) growth period (roughly Triassic - now) and correspond to the rifting/opening of the modern oceans. But what about going further back? Has the rate of the Earth growth always been constant exponential growth? ie. a constant exponent or power that dictates the exponential nature of the growth.
Thought it can be agreed upon, that the Earth, now, manufactures matter internally, I think that the primitive earth didn't, and grew purely from external accretion of individual atoms. When the Earth began to manufacture matter internally, will correspond to a significant geological boundary. When this geological boundary is identified, modelling the growth function of the earth will be more accurate, with a continous gradient, and allow for the model to have a paleo-radius (r0) value corresponding to an initial time when the earth first became spherical and it has been approximately spherical for a few billions of years.
This is the very reason why we cannot make an exponential curve from the "beginning" to "now" using a constant exponent. Yes, an equation will eventually be derived, BUT it may not be as simple as we'd expect because I believe that the exponent (coefficient) will not be constant, though it WILL be continous - because we are talking about something that is occuring in nature. Yes the curve will be smooth, because the "constant" will have a continous rate of change. As we go further into the future, the constant will have less deviation. Visually, the curve will become more uniform.
If we are able to identify this geological boundary (possibly by studying remanent magnetism patterns) then we could make a model that would look something like this:
R(t) = R0+Re^(g(t)t)
Where g is a function of time, as well.
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