Determine the wavelength of the radio wave as measured by the observer on Earth.

The radio signals from two stars on opposite sides of the galaxy are detected on Earth using a radio telescope. The telescope has a circular receiving dish.

(i) State the Rayleigh criterion for the images of two point sources to be just resolved.

(ii) The galaxy is 2.0×10²¹m from Earth and the stars are separated by 5.0×10¹⁹m. Determine the minimum size of the telescope dish required to resolve the images of the two stars at a wavelength of 5.1×10^–2m.

\(\left( {{\rm{since}}\frac{v}{c} \ll 1} \right)\)\(\Delta f\left( { = \frac{v}{c}f} \right) = \frac{{6.0 \times {{10}^6}}}{{3.0 \times {{10}^8}}}6.0 \times {10^9}\left( { = 0.12 \times {{10}^9}{\rm{Hz}}} \right)\);
\(f' = f - \Delta f\left( { = 5.9 \times {{10}^9}{\rm{Hz}}} \right)\);
\(\lambda '\left( { = \frac{c}{{f'}} = \frac{{3.00 \times {{10}^8}}}{{5.9 \times {{10}^9}}}} \right) = 5.1 \times {10^{ - 2}}\left( {\rm{m}} \right)\);

Award [2 max] if ƒ'=ƒ+Δƒ used to give λ'=4.9x10^-2(m).

(i) the two (point-like) sources generate diffraction patterns with central maxima;
the central maximum of one pattern overlaps with the first minimum of the second diffraction pattern;

(ii) \(\theta  \approx \frac{d}{D} = \frac{{5.0 \times {{10}^{19}}}}{{2.0 \times {{10}^{21}}}} = 0.025({\rm{rad}})\)
\(\left( {b > 1.22\frac{{5.1 \times {{10}^{ - 2}}}}{{0.025}} = } \right)2.5\left( {\rm{m}} \right)\);

Allow [1 max] for solution that omits 1.22.