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When reading some astrophysical papers I saw things like this:

The sources are variable on very short timescales, implying a compact emission region.

How can we conclude or measure the source size, or emission region size (or its compactness) from the variability of the spectrum?

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A source cannot coherently vary on timescales shorter than the source size divided by the speed of light.

That is because there is no way that different points on the source could "communicate" with each other and arrange for a coordinated increase or decrease in brightness.

Therefore the shortest timescale of variability gives an upper limit to the size of the emission region.

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