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What technology (at minimum), devices or minimum civilization development is needed, so that an individual member of this civilization would be able to detect that some star is going supernova?
My question is inspired by "The Inner Light" episode of Star Trek Next Generation (though I don't know what civilization development level is presented there).
I just want to know what must be invented by a civilization, to be able to detect this kind of threat?
When people think of supernovae, they often only think about the visible light emitted during and after the explosion. However, if you restrict yourself to observing in the optical part of the electromagnetic spectrum, you'll lose a whole lot of information. There are a few different astronomical messengers scientists can use to study a supernova.
In the final stages of their lives, massive stars tend to exhibit violent behavior, such as chaotic episodes of mass loss. The final stages of fusion (neon, oxygen and then silicon) take years down to days to occur, compared to the billions of years during which the Sun fuses hydrogen. Several mechanisms in these late stages could lead to mass loss events. These would be detectable in the optical and infrared bands, appearing as dimming as ejected dust enshrouds the star.
This was one of the reasons it was briefly thought that Betelgeuse could be in its final months of its live when it began dimming this past autumn. While the dimming is likely due to a major mass-loss event, it isn't necessarily due to one of those uber-late-stage mechanisms. In other words, a single large mass loss event isn't evidence of a supernova, but a sustained period of activity could wag its head indicatively.
In the hours to days) before a supernova, the progenitor star releases a flood of neutrinos carrying a lot of energy. This burst is detectable, and it's observed before the light from the actual supernova reaches the observer because it's well in advance. We observed neutrino emission from SN 1987A two or three hours before its light reached Earth.
Neutrinos are fairly hard to detect, as they don't interact strongly with matter. Even from SN 1987A, we only detected a handful. To maximize your odds of detecting any of them, you'd need something like the Supernova Early Warning System, a small network of neutrino detectors, all working together. It includes the Super-Kamiokande in Japan and Ice Cube in Antarctica.
Neutrinos can be lethal in large doses, so these neutrinos could serve as an adequate warning sign only if you're far enough away from the supernova - otherwise, you'd just be killed! According to XKCD, you'd want to be at least a few astronomical units away to survive the neutrinos - but then, of course, the actual explosion would kill you a few hours later.
You might be surprised to learn that asymmetric supernovae can produce gravitational waves (Ott et al. 2003). There are two possible forms of the gravitational wave signal:
A sharp spike followed by high-frequency oscillations, caused by a slow-rotating progenitor star.
Oscillating "bounces" of the amplitude caused by expansions and contractions.
Type 1 waves are expected to be stronger, but it appears that both types should be detectable by interferometers like LIGO and Virgo, assuming the sources are within the Milky Way. We're fortunate; their peak frequencies lie in the middle of the frequency range of these instruments. With peak strains of $h\sim10^{-21}$ for sources $\sim10\text{ kpc}$ away, supernovae on the other side of the galaxy could be detectable. Given that strain scales inversely with distance, the strain of waves from a closer source would be even greater.
During the explosion - and in the weeks and months afterward - a supernova will emit visible light, coming from hot, radioactive bits of ejecta moving through space. Astronomers can study the light curve of the event to glean information about the progenitor star and the nature of the explosion. Supernova light curves generally rise rapidly and then fade over time. Scientists can figure out what kind of supernova a supernova is by observing morphological properties of the light curve.
The interesting thing, of course, is that it does take some time for a supernova to reach peak brightness - maybe a few days to a couple weeks. It's possible that the supernova could only prove dangerous at or near that peak brightness, and so the initial increase in luminosity might be enough of a warning sign without being deadly.
Image courtesy of Wikipedia user Lithopsian.
To accurately measure the light curves from a supernova, you'll need sufficiently advanced telescopes working in a range of wavelengths. UV, infrared and visual light are your best bet, and we have many telescopes that study the universe in one or more of these wavelengths.
For one thing that would be the "scientific method" as to acquire knowledge and investigate phenomena based on empirical observation and measurable evidence - so to say objective rather than subjective.
Other than that, I'd strongly vote for a deep theoretical understanding of the internal workings of a star (that is physics) and to detect the changes in your star, obviously. Optical instruments (so optics and the associated technology chains of glass making, manufacturing of optical elements, e.g. grinding and polishing of lenses, as well as fine mechanical manufacturing for making your instruments). The idea of spectroscopy would be helpful as to measure the changes in emission wavelengths of the star with respect to its lifecycles. Measurable evidence of going supernova besides emission spectra is of course the brightness of the star - its astronomical magnitude. To measure that empirically it's advisable to have had photography invented (which is easier than inventing an electronic photometer).
Most of that hardware should have been available at say ~1850. Although that time lacks the understanding of fusion processes inside the star - kind of vital to the understanding of a supernova.
It's also more than noteworthy that going supernova is not exactly something that happens in no time. We're talking about hundreds of thousands if not millions of years (the heavier the star the faster it lives) from noticeable changes to supernova. That is to say: plenty of time to discover spaceflight and get the hell out of there.
Ghanima's answer makes the fundamental point that theoretical knowledge is required before the necessary technology is developed. HDE226868's answer is an excellent summary of the process.
There is a problem with the idea of a home star going supernova. A supernova event requires a considerable amount a mass. Further, a star's lifetime is determined by its mass. Here is a table summarizing the relationship between stellar mass and stellar lifetime :
Theory suggests that the minimum mass for a small supernova is about 1.44 solar masses. A core collapsing supernova requires 8 to 9 solar masses.
So a minimum supernova would result from a star with an expected lifespan of about 3 billion years. Consider that our sun is 4.5 billion years old, and that it has taken that long for intelligent life to evolve here on earth. This would seem to imply that a supernova would occur before intelligent life could evolve.
On the other hand, a supernova need not occur on ones home star to have catastrophic effects. Any stars and their planets in the local star group would suffer catastrophe.
