# CCAT vs. Keck: Angular Resolution and Telescope Size

by: Eric S. Mukherjee, Nathan Baskin

This is not the actual write up I was going to do for this week. I just thought this problem was a cool look at the many factors that go into designing a telescope and decided to write it up anyway. The other one will follow.

Abstract
It's very easy to assume that a bigger telescope is always the more precise one. The question is, is this assumption always true? In this post, we examine an order of magnitude solution of problem 3 of the problem set on optics to determine whether this is the case.

Introduction
The angular resolution of a telescope is the smallest angular separation at which the telescope can resolve two distinct objects. Any angle smaller than this registers as one larger object when viewed by the telescope in question. Caltech is building CCAT, a 25m telescope which observes in the 350-850 micron wavelength. We would like to know how the angular resolution compares to that of Keck, a 10m telescope observing in the near-infrared J-band at 1.25 microns.

Methods
We want to find angular resolution, which is given by the small angle approximation:

We then convert all of our known quantities into the same units. In this case, meters. As we can see, the angular resolution has a direct dependence on the wavelength, so we need only convert the minimum wavelength for CCAT:

Immediately we can see that these differ by two orders of magnitude while the diameters, 25 m and 10 m, respectively, are on the same order of magnitude. We can ignore the factor of 1.22 since it is shared by the two equations and is close to 1 and find that

Conclusions
We find that despite the greater diameter of CCAT, there is a difference of roughly a factor of 100 between the angular resolutions of the two telescopes. This was immediately obvious from the differences between the two wavelengths, since the diameters of the two telescopes have the same order of magnitude. Solving the equation for angular resolution we get precisely:

We find in the end that in order to determine a telescope's power we must refer to the wavelength at which it is observing as well as the diameter and that in order to have an equivalent angular resolution at the desired wavelength, CCAT would have to be a 2500 m telescope.

## 2 Responses to CCAT vs. Keck: Angular Resolution and Telescope Size

1. OK, I see your point. But I'm wondering about a related issue: Why would an institution building a new telescope not build the "most powerful" one possible? Could be cost or something else similarly mundane. however, is it possible that given what is known about th e science of whatever one wishes to observe with a given telescope under development, the scientist simply say "Having a more powerful telescope won't give us better/deeper understanding?" is such thinking an element of the design?

2. It's helpful to astronomers to express the angles in arcseconds since we generally know the size of the object we want to observe in arcseconds or arcminutes, rather than in radians.

Are there any 2500 m telescope dishes out there? we do make radio observations at the same angular resolution as Keck gets in optical. How do we get that resolution without a dish that's more than a mile wide?