Declination and Observations

In this post I'll be doing #4 from this week's worksheet on celestial coordinates and observations. I chose it because it's actually really simple, unless you overthink it like I did. For me, personally, the concept of celestial coordinates was very confusing, mainly because I tried too hard to visualize everything and spent too long challenging my own thought process, resulting in an obfuscation of a concept that was simple to begin with.

Here's the problem:

You are planning a space mission to observe a small area of the sky (maybe 2 sq deg) very deeply (meaning you're staring at the same spot on the sky for a long time) to make a catalog of faraway galaxies at visible wavelengths, say 550 nm. What declination range should you point your space telescope at so that ground telescopes in both the Very large Telescope (VLT) and the Keck Telescopes can follow up on the galaxies you discover? The declination range should be observable from all observatories for at least 6 months a year.

This question really just hinges on a solid understanding of what declination is. At first, I tried to visualize what the sky would look like from the VLT and the Keck Telescopes; that's probably the wrong way to go, though. It's much easier to view the problem from a third-person perspective, looking at the earth from outside, like this:

The black line on the earth is an observer - theoretically, its field of view is from horizon to horizon. In other words, it can see 90 degrees on each side of straight up. Viewed from an exterior perspective, it's much easier to see the relation between latitude of the observer and declination of your observed target. Consider an extreme point of this scenario - an observer at the equator. An observer at the equator would have a 180 degree field of view along the tangent to the surface of the Earth. So, an equatorial observer would see from declination -90 to +90 (in degrees; we are assuming the observer can see infinitely far, so the furthest he/she could see would be at +- 90 degrees):

An observer between the equator and the north pole, then, would see 180 degrees, shifted by their latitude. An a latitude of x, he/she would be able to see from (-90 + x) to (90 + x).

If the observer were in the northern hemisphere, however, the upper bound of (90 + x) is actually 90, since declination is only measured from -90 to 90 degrees. If you think about it, after some time has passed, the Earth will have rotated, so the (90 + x) declination that the observer has seen before has become (90 - x). Because of this, it makes more sense to measure declination from -90 to 90 degrees, eliminating the confusion of angles greater than 90 degrees.

Let's turn back to the problem at hand. The VLT is at a latitude of -24 degrees, and the Keck Telescopes are at a latitude of 20 degrees. The observable declinations of the VLT, then, range from (-90 - 24) to (90 - 24), or -90 degrees to 66 degrees. The Keck Telescopes could theoretically view declinations from (-90 + 20) to (90 + 20), or -70 degrees to 90 degrees. Declinations viewable from both, therefore, would lie in the intersection of these two ranges, or -70 degrees to 66 degrees. Since these bounds represent the limit of viewability, let's take a smaller intersection, say: -60 degrees to 56 degrees. The following diagram shows this scenario graphically:

So, this problem is actually fairly simple. The important part was to consider an observer located at an arbitrary latitude, diagrammed from a point outside the Earth. Trying to imagine this problem on the surface of the Earth causes headaches (at least it did for me), and is less efficient. What about the part about being observable for at least 6 months a year? It's actually a red herring; the declination range observable from a latitude remains constant throughout the year.