According to recent news, Kepler-444, a 11.2 billion year old star, has a system of five planets. It has a mass of 3/4 of solar mass. So I believe both Kepler 444 and the Sun will share the fate of 'orange main sequence stars'. If so, while scientists fear for life on Earth in the coming billion years, which is only one sixth for our solar system lifetime, how is it that the Kepler-444 planets have existed for a longer period of time?
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3$\begingroup$ Scientists don't fear for the Earth. They fear for the life on it, specifically for the human variety. High surface temperatures will render the Earth uninhabitable in a billion years or so, but the planet itself will be just fine for a while after that. $\endgroup$– David HCommented Feb 7, 2015 at 8:14
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The rate of evolution of main sequence stars is highly dependent on their mass. Roughly speaking, the time on the main sequence is proportional to $M^{-5/2}$, where $M$ is the mass of the star. Thus if Kepler 444 has $M=0.75\ M_{\odot}$, it can live for $0.75^{-5/2} = 2.05$ times as long as the Sun on the main sequence.
Another way of saying the same thing is that the Sun evolves 2.05 times as fast as Kepler 444. This is important because even whilst on the "stable" main sequence, burning hydrogen in their cores, stars are still changing because the average mass of a particle in their cores is increasing due to the build up of a Helium ash. This slow change means that the temperature of the core must gradually increase, the nuclear burning rate increases and the star becomes more luminous.
In terms of the Sun this means, depending on the atmospheric composition at the time, that the Earth could become much hotter in a billion years or so, and probably too hot for life (as it currently exists on Earth). Thus it is not the existence of Earth that is in jeopardy, it is the surface temperature that may destroy life.
None of the discovered planets in the Kepler 444 system are (or ever were) far enough from their parent star that they are in the "habitable zone" where liquid water could exist. The significance of Kepler 444 is that it shows that rocky planetary systems can have formed as long ago as 11 billion years, giving such systems more time, and therefore perhaps more chance, to have formed life.