Comparing Telescopes

In this problem, we were told about two telescopes that observe at different wavelengths and asked to compare their angular resolutions.  The CCAT is a 25 m telescope that observes at a wavelength of 850 microns.  The MMT is only a 6.5 m telescope that observes in the J-band.  A quick Google search taught me that the J-band is a range of infrared frequencies centered around wavelengths of 1.25 microns.

In class, we learned that angular resolution, $\theta = \frac{\lambda }{D}$.

$\theta_{CCAT}=\frac{850 \mu m}{25 m}\left ( \frac{1 m}{1\times 10^{-6}\mu m} \right )= 3.4\times 10^{-5} rad$

$\theta_{MMT}=\frac{1.25 \mu m}{6.5 m}\left ( \frac{1 m}{1\times 10^{-6}\mu m} \right )= 1.9\times 10^{-7} rad$

The angular resolution of the MMT is smaller, and therefore better than that of the CCAT.  This might be slightly surprising because the CCAT has a larger diameter, which, as a very loose rule of thumb, is usually better than smaller telescopes because they can collect more light.  It's still necessary to have CCAT, though, because the MMT isn't big enough to capture wavelengths as big as those captured by CCAT.