Lambda over D

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 in the infrared J-band? What's "J-band"?

The MMT (Multiple Mirror Telescope). It only has one primary mirror.

The key to this problem is a very simple equation, one that draws on the Airy disk (which I discussed the origins of in my previous post) to determine angular resolution. Essentially, in order to resolve two nearby objects as being truly separate, the center of one needs to be far enough from the center of the other that it is located at the first "null," or minimum, of the other object's Airy disk. This gives us a crucially important formula, known as the Rayleigh Criterion, or the diffraction limit:\[\theta = 1.22\lambda/D\] Where $\theta$ is the smallest possible angle a telescope can resolve, $D$ is the telescope's diameter, and $\lambda$ is, as always, the wavelength of incoming light (the 1.22 coefficient comes from calculating the first zero of a Bessel function).

So, having looked up the infrared J-band and knowing that it's centered at $\lambda = 1250$ nanometers (confusingly, there is also a J-band in the radio, ranging rom $\lambda = 1.5$ to $3$ centimeters), here is all the information we need:

$\lambda_{MMT} = 1.25 \times 10^{-6}$ meters

$\lambda_{CCAT} = 8.5 \times 10^{-4}$ meters

$D_{MMT} = 6.5$ meters

$D_{CCAT} = 2.5 \times 10$ meters

Thus, $\theta_{MMT} = \frac{1.25 \times 10^{-6}}{6.5} = 2 \times 10^{-7}$ radians, and $\theta_{CCAT} = \frac{8.5 \times 10^{-4}}{2.5 \times 10} = 3 \times 10^{-5}$ radians. So, despite being considerably smaller, the MMT is actually giving better angular resolution than the CCAT when observing in much shorter wavelengths. There are caveats to this, of course--it might have better angular resolution, but the MMT's light-collecting area is always going to be much smaller than the CCAT's. But this relationship between wavelength and diameter is still very important, and is actually the reason why the folks at the CfA are able to do meaningful radio astronomy research using a 1.2-meter telescope right here in Cambridge.

The CfA 1.2-meter