All I could come up with is it depends on the atmospheric composition of the planet as they will interact with the charged solar particles which might be true but is there is there a definite reason behind the different colors of aurorae on other worlds (Saturn's blue aurora for example)?
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Ultimately, you are correct. However, there may also be some clarification in order: what blue aurora of Saturn are you referring to? The most well-known auroral images of Saturn like this one, are not true color images. That is, what we see in the photo isn't what you'd see with the naked eye; instead, the photo has been enhanced to emphasize something interesting, in this case the aurora. In reality, that beautiful blue you see in the picture isn't even of light that's visible with our eyes. It's UV light which would be completely invisible to us. People working with data taken by a specialized camera have to give the UV light some visible color or we wouldn't see it at all. The Saturn aurora images I am aware of all have been modified in this manner.
However, aurora themselves are the result of charged particles funneling down a planet's magnetic field and crashing into the planet's atmosphere. On Earth, the aurora occurs because of the atoms in the atmosphere getting energized by these collisions, and then emitting light when they de-energize. Here, it's the oxygen and nitrogen in the atmosphere that are responsible for aurora light (primarily greens from oxygen, but also reds at higher altitudes from oxygen and occasionally blues from nitrogen). In other planets' atmospheres dominated by different atoms and molecules, the reactions will be different, and the wavelengths of light emitted will be different as well. For example, Saturn's aurora are due to energized H2 molecules. On Jupiter, it can be the infalling particles themselves (sulfur and oxygen) that are responsible for auroral emission.
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$\begingroup$ 'Different' You can look that up with 0 work. Gas giant atmospheres are dominated by $H_2$. It's the $H_2$ bands that radiate in the UV when excited by elctron precipitation. They don't radiate in the optical as the hydrogen molecule doesn't have allowed lines there. $\endgroup$ Sep 2, 2015 at 2:52
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$\begingroup$ It's unclear why there's a need for sass. The original question appeared to be primarily based off of a misconception concerning published images of aurora around other planets. If my answer is not of sufficient thoroughness, you should certainly feel free to submit you own? Regardless, you are only partially correct: while the mechanism for Saturn may be dominated by H2, that is not the case for Jupiter. My answer has been edited to reflect this. $\endgroup$ Sep 2, 2015 at 23:13
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$\begingroup$ @AtmosphericPrisonEscape Auroral colours are not (at least on Earth) caused by allowed lines. $\endgroup$– ProfRobSep 7, 2015 at 21:39