Of course, there are many alleged limitations with such a huge planet. For example, oceans or mountains that cannot be crossed, but such restrictions will not be taken into account. We assume that the conditions of movement on the Sun-sized Earth are similar to our planet.
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According to scientists, our ancestor Homo Erectus appeared about 2 million years ago and spread throughout parts of Africa. His descendant Homo Sapiens later migrated from Africa to the rest of the world.
The last territories that were colonized by the primitive people are the islands of Polynesia, inhabited approximately from 700 to 1000 years BC. For the initial period of settlement of the planet, it takes 2 million years.
The surface area of the earth is 510.1 million km². We will not take into account the polar regions, they are practically not populated. The habitable area, including the sea, is 400 million km².
That is, a human spreaded at a speed of:
400 million km² / 2,000,000 years = 200 km²/year on average over the past 2,000,000 years.
The surface area of the Sun is 6 trillion km². This is 12,000 times more than the surface of the Earth. Thus, intuitively, the settlement will take 12,000 times more time or 24 billion years, which is almost twice the age of the universe.
But the use of the average value is not entirely fair; in reality, species coverage will grow exponentially (compound interest). To move from the population of 1% of the surface to 100%, the occupied area should increase by about a quarter every 100,000 years.
We apply this calculation, given that the surface area of the Sun is 1,200,000% of the surface area of the Earth. Our species would spread over the surface of the Sun-sized Earth between about
4,000,000 – 4,100,000 years AD.
Even if the area of our planet was 12,000 times larger, it would take only 3 times longer to populate it. Of course, this simplified model does not take into account the invention of ships, planes, cars, which significantly accelerate the process.