Summary: Galaxies are considered dead when they no longer are producing new stars. It is still unclear what causes this process but what astronomers do now know is the most common time scale of them dying -- approximately four billion years. Astronomers are calling this process strangulation.
Astronomers are able to tell how quickly the galaxy died based on the metal content remaining in the galaxy. The more stars that are being formed in a galaxy, the more metal content is seen. The way it works is simple. If a galaxy dies quickly and quite violently (for instance if the cool gas that fuels the star formation is suddenly ripped away), then the stars immediately stop producing and the metal content remains the same. But if the galaxy if cut off from the supply of cool gas but still continues to produce stars, using up the last of the supply, the metal content in the galaxy grows and grows until it "suffocates".
The difference was seen when comparing the spectra of light emitted by red, passive galaxies and blue, star-forming galaxies. With the data from Sloan they were able to see that the dead galaxies were, on average, four billion years older than the active galaxies. This is consistent with the amount of time the astronomers had calculated it would take to burn of the remaining amount of gas supply whilst strangling to death.
While the astronomers are now confident that death by strangulation is the most common way a galaxy meets it's ultimate end, it is still not clear what causes this. The most likely suspect at this point in time seems to be overcrowding which would lead to a greater probability of disruption to the gas supply, but this would need further investigation.
BBC News
Nature
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