Worksheet 4, Problem 2: Comparing Telescopes

CCAT is a 25-meter telescope that will detect light with wavelengths up to 850 microns. How does the angular resolution of this huge telescope compare to the angular resolution of the much smaller MMT 6.5-meter telescope observing the infrared J-band? 

The angular resolution of a telescope is the telescope's ability to see objects or details that are an angular distance \(\theta\) apart. The angular resolution of a telescope is given by the equation:

\(\theta =\dfrac{1.2\lambda}{D}\)

Where \(\theta\) is the angular resolution in radians, \(\lambda\) is the wavelength of the incident light, and \(D\) is the diameter of the telescope's lens aperture. 

The telescopes, CCAT, has diameter \(D=25\) m, and is observing light at \(\lambda =850\times 10^{-6}\) m. Putting these values into the equation for angular resolution, we get:

\(\theta _{CCAT} = \dfrac{1.2(850\times 10^{-6}\text{m})}{25\text{m}}\)

\(\theta_{CCAT} = 4.08\times 10^{-5}\text{ radians}\)

           \(= 2.3\times 10^{-3}\text{ degrees}\)

This means that the CCAT can resolve objects or details within angular distance of \(4.08\times 10^{-5}\) radians apart or \(2.3\times 10^{-3}\) degrees apart.

The smaller telescope, the MMT, has a diameter of \(D=6.5\) m, and is observing wavelengths in the J-band, which is the range of wavelengths centered on 1.25 micrometers. Plugging these values into the equation, we have:

\(\theta _{MMT}= \dfrac{1.2(1.25\times 10^{-6}\text{m})}{6.5}\)

\(\theta_{MMT} = 2.3\times 10^{-7}\text{ radians}\)

            \( =1.2\times 10^{-5}\text{ degrees}\)

The MMT can resolve much smaller objects than the CCAT telescope can at their respective wavelengths. In fact it is able to resolve objects that are 177 times smaller than those that the CCAT can.

I would like to thank Charles Law for his help in solving this problem and for proof reading this blog post.