How could events triggered by a supernova accelerate the death of a main sequence star?
My story revolves around a star going supernova. The 'Earth' in this story is about 100 light-years or more away (a safe distance, but not too far away). I need this supernova to somehow accelerate the death of this 'Earth''s star such that it will slowly become uninhabitable in the next 1 million years.
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They're fine.
Here, I discussed the effects of a supernova on life on Earth. Note that a supernova would have to be within about ten parsecs (~33 light-years) to have harmful effects on life on Earth. Furthermore, the effects would be indirect, destroying the ozone layer (partially or in full) rather than incinerating the planet. At 100 light-years, we're fine.
On to your actual question. The lifetime, $\tau$, of a star on the main sequence is approximately $$\tau=10^{10}\text{ years} \cdot \left[\frac{M}{M_{\odot}} \right]^{-2.5}$$ where $M$ is the mass of the star and $M_{\odot}$ is the mass of the Sun. From this, we can see that a star with more mass will spend less time on the main sequence.
So you need to add mass to the "Sun", and you need to add mass that can be used for nuclear fusion. I can get you specific data for the density of a supernova remnant, which you could use to figure out to amount of material in a given area near the "Sun", but I think it's safe to say that there will not be a significant amount of material useful for fusion anywhere near the "Sun".
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