Week 11: WS19 #2

2. Resolving organic chemistry in disks: It is debated whether the Earth's water and organics originated from material forming around 4 AU (asteroid-like) or 30 AU (comet-like). We would therefore like to know what the differences in the organic composition is between these two locations in protoplanetary disks. One of the interesting organics CH₃CN emits strongly at 239 GHz. How big of a telescope would you need to resolve a difference in chemical composition between 4 AU and 30 AU in the nearest star-forming region? How does the required telescope size compare with the biggest steerable radio telescope in operation, the 100 m Green Bank Telescope (GBT)? To the longest baseline in the ALMA interferometer of 10 km? 

First, we should figure out what the wavelength the acetonitrile is emitting at.
\(239GHz=239\times10^9s^{-1}\)
\(\lambda=c/\nu\)
\(\lambda=\frac{3.0\times10^{10}cm/s}{239\times10^9s^{-1}}=0.126cm\)
Let's assume that the nearest star-forming region is 150 parsecs away.

Since we're comparing the signal at 4 AU and 30 AU, the top will be the difference: 26 AU. In order to calculate the angle, we can use the small angle approximation.
\(tan\theta=\frac{26AU}{150pc}=\frac{26AU}{3.09\times10^8AU}\)
\(\theta=\frac{26AU}{3.09\times10^8AU}=8.41\times10^{-8}\)
Finally we can calculate the necessary diameter of the telescope.
\(D=1.22\frac{\lambda}{\theta}\)
\(D=1.22\frac{0.126cm}{8.41\times10^{-8}}=1.83\times10^6cm=18.3km\)
To get the necessary resolution for this, the GBT would be way too small. ALMA is also a bit small, but is the right order of magnitude.