Describe what is meant by the Doppler effect.

One of the lines in the spectrum of atomic hydrogen has a frequency of 4.6×10¹⁶Hz as measured in the laboratory. The same line in the spectrum of star X is observed on Earth to be shifted by 1.3×10¹²Hz.

(i) State the direction of the observed frequency shift.

(ii) Determine the speed at which X is moving towards Earth stating any assumption that you have made.

The star X has a companion star Y. The distance from Earth to the stars is 1.0×10¹⁸m. The images of X and Y are just resolved according to the Rayleigh criterion by a telescope on Earth with a circular eyepiece lens of diameter 5.0×10^–2m.

(i) State what is meant by the statement “just resolved according to the Rayleigh criterion”.

(ii) The average wavelength of the light emitted by the stars is 4.8×10^–7m. Determine the separation of X and Y.

the observed change in frequency/wavelength of a wave;
emitted by a source moving away from or towards/relative to the observer;

(i) a blue-shift / towards the blue end of the spectrum / to a higher frequency / OWTTE;

(ii) \(v = \left( {\frac{{c\Delta f}}{f} = } \right)\frac{{3 \times {{10}^8} \times 1.3 \times {{10}^{12}}}}{{4.6 \times {{10}^{16}}}}\);
8.5×10³ms^-1;
assume that the speed is very much less than speed of light;

(i) the two stars are (just) seen as separate images;
if the central maximum of the diffraction image of one star coincides with the first minimum of the diffraction image of the other star / OWTTE;
Accept an appropriate diagram for second marking point.

(ii) \(\theta  = \left( {\frac{{1.22\lambda }}{b} = } \right)\frac{{1.22 \times 4.8 \times {{10}^{ - 7}}}}{{5.0 \times {{10}^{ - 2}}}}\) or 1.17×10^-5rad;
\(\theta  = \frac{d}{{1.0 \times {{10}^{18}}}}\);
(d=)1.2×10¹³m;
Award [2 max] if 1.22 is missing, giving an answer of 0.98×10¹³.