What happens to the earth when the sun dies?

Answer

It is not yet entirely clear whether the Earth will survive the last phases of evolution of the Sun. Because towards the end the Sun will get bigger and bigger, and some models suggest it will get bigger than the current Earth orbit, and then the Earth will burn up.

This is not entirely certain, however. It’s hard to predict exactly how big the Sun will one day get: the outer layers of the star are very thin, and a small change in what happens inside can change a lot about the outside. But we also know that the exterior of such red giants is not a smooth sphere, but rather a turbulent mass of gas, with bubbles enriching sometimes further and sometimes less than the average radius; with that in mind, it is very unlikely that we will escape destruction. On the other hand, it is true that the Sun will already have lost a fair amount of mass before it gets to that point, which indeed implies that the Earth will be in a wider orbit, with a longer orbital period. The problem is that we cannot predict well enough how much mass the Sun will lose after its ‘first red giant phase’.

So you are right when you claim that the Earth’s orbital period will be longer, but be careful when you express that in ‘days’ or ‘months’, because these are precisely units that are connected to the Earth, and are therefore not absolute ! A ‘day’ is associated with the rotation period of the Earth on its axis, and that period tends to increase, due to the tides exerted by the Moon and Sun. In the period we are now experiencing, the days are therefore getting longer, and in fact we have fewer ‘days’ in a year every year! For the same reason, the Moon moves away from us, and so the ‘month’ also lengthens, so that there are fewer ‘months’ each year. Five hundred million years ago there were about 400 days and 13 months in a year, but the length of the year itself has not changed appreciably!

Presumably, the tidal forces will greatly diminish from 500 million years into the future. Due to the increased brightness of the Sun by then, the oceans will evaporate. And it is the friction of the seas with the continents that causes the loss of energy. So it is likely that there will indeed be more days and months in a year as the Earth moves into a further orbit, and the real year does get longer.