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Why do we cross match astronomical catalogues at different wavelengths? What information can we determine from combining the information from two catalogues? One of the applications I read was creating a catalogue of Active Galactic Nuclei (AGNs) by searching for optical counterparts to objects appearing in radio catalogues. However I'm not sure if this information is correct. Are there any other such similar applications of cross matching?

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I did work in the 90s on cross-matching optical catalogues with catalogues of X-ray emitting sources. The main reason for doing this was to figure out which object was responsible for the X-ray emission, since there could be one, many or no optical counterparts within the "error circle" of the X-ray source position.

Armed with a list of counterparts one could then go to a telescope and see whether the counterparts betrayed other signs that they might be responsible for the X-ray emission. In my case, I was looking for examples of rapidly rotating young stars, with high levels of coronal X-ray activity. Thus the cross-matching was used to locate these stars in the first place. The paired data could then be used to explore the dependence of coronal activity with other stellar parameters like temperature, gravity and rotation.

This is an example. Your question is very (too) broad. If I were to summarise, it is that astrophysical objects emit at multiple wavelengths and a full understanding of what is going on requires the full picture.

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If I’m understanding what you’re asking, then it boils down to two things: resource management and different physics.

Let’s say you’re looking at a given variable star. Maybe you just started researching it and you want some observations done on it to study it’s internal structure or something. Getting access to a telescope capable of doing that is not trivial; maybe you’re lucky and you have one at your institution you can use, but maybe ground observations are not very good for the data you want to collect, and so you want to get access to some space telescope time. Imagine the feelings of embarrassment if you submitted a proposal to observe this target, just to find out it’s already been observed indirectly a dozen times, and that the data you needed was already out there, a product of collateral benefit from other surveys/ missions.

Its so important for individuals to know what kind of data exists from different observations, so that we don’t end up looking at the same object a dozen times because we didn’t know we had already looked at it. That’s why websites like MAST collect data from a bunch of different catalogues to give you reliable information of what has already been done for a given object.

The next part has to do with physics; you mentioned AGN specifically. Different kinds of electromagnetic radiation are created in different processes, and these processes can give different insights as to what kind of processes are happening. Maybe high energy photons would be useful for studying the jets while you want to look at lower energy photons to study the accretion disk.

tl;dr: We got to know what we got so we don’t accidentally waste time getting it again, and we want different wave lengths to study different kinds of features

Let me know if this isn’t what you meant and I can adjust.

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