Resolution

Much like how cameras have resolution, telescopes have a minimum angular resolution,.where if an object's angular diameter is smaller than the angular resolution, it would not be possible to resolve the object through the telescope.

The formula for calculating the angular resolution \(\theta_{min}\) is:

\[\theta_{min} \approx \frac{1.22\lambda}{d}\]

where \(\lambda\) is the wavelength of light, and \(d\) is the diameter of the aperture.

Note

\(\theta_{min}\) here is in radians. We make the approximation that \(\sin\theta = \theta\) for very small angles, and that 1.22 is more precisely 1.21966989...

Question

QuestionAnswer

Can the Hubble Space Telescope (\(\lambda\) = 191 nm, D = 2.4 m) resolve the Aniak Crater (D = 51 km) on Mars at it's closest approach? (Assume circular orbits, \(r_{earth} = 1.496\times 10^{11}\)m, \(r_{mars} = 2.279\times10^{11}\)m)

The angular size of the crater is \(\theta = \frac{51\times10^3}{2.279\times10^{11} - 1.496\times 10^{11}} = 6.5\times10^{-7} \text{rad}\)

\[\theta_{min} = \frac{1.22\times 191\times 10^{-9}}{2.4} = 9.7\times10^{-8} \text{rad}\]

Since \(\theta > \theta_{min}\), yes it can