Sunday, February 15, 2015

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 in the infrared J-band? 



Telescope resolution is dependent upon the size of the primary mirror, and the wavelength of light that it observes.  

\(\theta=?\)
\(\lambda=850um,1.25um\)
\(d=25m,6.5m\)
\[\theta=1.220\frac{\lambda}{d}\]
This equation gives the resolution of a telescope (in terms of the smallest resolvable angle) in which \(\lambda\) refers to the wavelength of light and D refers to the aperture diameter.  The factor of 1.220 is a constant to assume a circular aperture.  

First let's solve for the CCAT.

\[\theta=1.220\frac{\lambda}{d}=1.220\frac{850um}{25m}=1.220\frac{8.5\times 10^{-4}m}{25m}=\boxed{4.1\times 10^{-5} radians}\]

Next lets solve for the MMT.  The MMT observes the J-Band of radiation, an infrared band centered at 1.25um.  

\[\theta=1.220\frac{\lambda}{d}=1.220\frac{1.25um}{6.5m}=1.220\frac{1.25\times 10^{-6}m}{6.5m}=\boxed{2.3\times 10^{-7} radians}\]

What does this mean? This means that the MMT can see 177x more clearly than the CCAT, despite it's smaller size.  

J-Band: http://en.wikipedia.org/wiki/J_band
Formula: http://en.wikipedia.org/wiki/Angular_resolution

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