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I realise this may be a difficult question to answer because, AFAIK, we don't have an accurate estimate of the size of the protostellar cloud, or whether our sun formed from a subsection of a much bigger cloud.

The reason I think this question "might" have an answer, at least in terms of an order of magnitude estimate, is by comparing it to the collapse timescale of (hopefully a large sample at various stages of collapse) of other stellar systems that we have observed to date.

My second question is: does the timescale involved in the collapse of a gas cloud to a star allow for the helium present in the cloud to sink, at least partially, towards the center of the cloud?

In other words, would there be time for helium, in however small an amount, to be involved in the fusion initiation process?

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This timescale is not well defined. If you mean the time from initiation of cloud collapse to the beginning of hydrogen fusion, it is around 10 million years and determined by the Kelvin-Helmholtz timescale of the contracting protosun.

If however you mean the timescale of cloud collapse to a revealed protostar with perhaps a disk around it, this is an order of magnitude shorter. The previous phase of collapse to a core with a roughly spherically symmetric envelope is about an order of magnitude shorter again.

These timescales are estimated with respect to the first one by counting relative numbers of objects in each phase. So the absolute number depends on how well we understand the ages of young stars on their approach to the main sequence, and these could be wrong by a factor two (for various reasons to do with atmospheres, rotation and the presence of magnetic fields).

The timescales above are for stars of a solar mass. It is probably a factor of a few longer for lower mass stars and is several times quicker for higher mass stars.

As to your second question, collapsing clouds and protostars are governed by convective heat transport. This efficiently mixes the gas, such that protostars are expected to be initially chemically homogeneous. However, there are various mechanisms that can chemically fractionate the material in a circumstellar disk; gravitational settling to the midplane is one of them.

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Not knowing the age of the sun has not stopped man calculating earth's age using zircon crystals, moon rock from the Apollo mission and possible martian implants.

But earth formed by bas becoming rock and then coalesced around 4.5 Bn years ago. So we could say that the earth and the sun took $\pm$9.3 Billion years to form following the so called "big bang", The "Infinite Needle point from "which some believe this universe formed. "Hawkins attempts to show a universe with no with No Beginning nor end in time? (Carl Sagan's foreword to Stephen Hawkins A Brief History in Time 1988)

"Scientific progress is the discovery of a more and more comprehensive simplicity...The previous successes give us confidence in the future of science: we become more and more conscious of the fact that the universe is Cognizable" Georges Lemaitre (Father of the Big Bang).

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