Is it likely that the Earth will survive or more likely that it will be engulfed?
Most likely it will be engulfed, certainly it will be incinerated.
The outer atmospheres of red giants are so diffuse it's possible the Earth will continue to orbit "within" the red giant. That's what I read anyway.
Yeah, but that could only last for a limited period of time before the orbit decays catastrophically.
There is a lot of information here: http://en.wikipedia.org/wiki/Sun#Life_phases
There won't be an Earth by that point - the entire planet will have boiled in just a few centuries.
Well when the Sun becomes a red giant in approx. 5 billion years the Sun will expand to a little past Mars' orbit. So as soon as the Sun's core starts to shrink and become denser the outermost layers will expand and will engulf the Earth. Before the outermost layers reach us the Earth will be hit with intense heat and radiation. Even before the Earth is destroyed all life will be wiped out from the heat and intense radiation. Our oceans will boil and evaporate and as soon as the outermost layers reach Earth, it will be vaporized. So the answer is no, the Earth will not survive the Suns transformation.
In my opinion if life is still thriving on Earth in 5 billion years, i still don't think our technology will be advanced enough to survive this event. We perhaps might be able to travel to another planet and/or star system to survive.
There is still considerable dissent over the fate of earth during the red giant phase of sol. Some think it will be incinerated, some that it will migrate to a larger orbit due to solar mass loss and survive, and others think it may be annihilated due either to tidal stresses or a collision with mars. Only time will tell [quite a bit of it at that]. There is, however, little dissent the earth will become utterly uninhabitable within about a billion years. So, we have a lot less time than once thought to plan and implement an exit strategy.
If we continue to progress as we have the last two thousand years, you'd think life would be advanced enough after a billion years to leave and go to another star system...
Well, that assumes that it is even possible to develop interstellar travel. It's probably true, but we may be limited to extremely slow sub-lightspeed velocities where it takes hundreds or thousands of years to reach the nearest star, in which case interstellar travel is OVERWHELMINGLY difficult and expensive.
Interstellar travel will require enormous breakthrough in physics and engineering. Is a billion years enough time? Perhaps, assuming we are not too distracted by our efforts to cleanse the planet of 'infidels'.
Assuming technology progresses forward a billion years will be enough time to do absolutely crazy things. How crazy? We could just grab some planets off this list
(I realize that many of these will probably no longer be inhabitable at that point but we're likely to have found some new ones that are)
Spend a thousand years building ships for the whole human race to fly away on, then spend 20 (or 200, or 2000, whatever) years flying there on our huge spaceship habitats. Yeah it's a huge expenditure of resources to pull it off, and I realize that it seems like we are in no way organized enough to do something like this, but in a billion years we will have had a billion years of social and political development. It would not at all surprise me if the planet could organize something to this scale once people realize 'hey we have 10,000 years left to get off this planet, we either start this now or we're responsible for destroying a billion years of human progress'.
I mean, in a billion years evolution would have rendered our descendants unrecognizable to us - for all we know at that point we would have evolved the ability to survive for hours in a vacuum environment. Who the heck knows what human civilization would be like at that point. It seems like the biggest concern would be running out of necessary natural resources at some point
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