Interferometers
Interferometers are giant telescopes comprised of multiple smaller telescopes. Imagine two 1-meter telescopes in the CHARA array, separated by 330 meters (you should look CHARA up later). Use the double-slit experiment to think about how these two telescopes might be able to measure the angular diameter of a star with \(R = 5*R_\odot \) at a distance of 150 light years. What is the maximum wavelength \(\lambda\) at which this star can be resolved?
From the double slit experiment earlier in the worksheet, we discovered that angular resolution follows a simple relationship of \[\theta = \frac{\lambda}{D}\] From this we can extrapolate out to the CHARA example by considering the 330m separation to be the distance between our "slots", the 150 light years to be the distance L to get to the detecting plane, R to be the distance y between constructive light points. (See diagram below).
To begin we must convert all units to the same standard. The length to the star becomes \[150 \space light \space years = 1.42*10^18 m\] and \[5*R_\odot \approx 3.5*10^9 m \] Thus we can begin to solve for theta by finding the tangent. \[tan \theta = \frac{3.5*10^9}{1.42*10^18} \] which when using the small angle approximation of \(tan \theta = \theta \) we end up with \[ \theta = 2 *10^{-9} \space radians\] Now with \(\theta\) we can solve for the wavelength using the above equation, \(\theta = \frac{\lambda}{D}\). \[\theta = \frac{\lambda}{D} \Rightarrow 2*10^{-9} = \frac{\lambda}{330m} \] Which gives us the desired final quantity: \[\lambda_{max} \approx 700nm\] which means that we would be observing this star as a dark red visible light!
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