Theory - Theoretical Question 1 Gravitational Red Shift and...

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Theoretical Question 1 Gravitational Red Shift and the Measurement of Stellar Mass (a) (3 marks) A photon of frequency f possesses an eFective inertial mass m determined by its energy. Assume that it has a gravitational mass equal to this inertial mass. Accordingly, a photon emitted at the surface of a star will lose energy when it escapes from the star’s gravitational ±eld. Show that the frequency shift Δ f of the photon when it escapes from the surface of the star to in±nity is given by Δ f f ’ - GM Rc 2 for Δ f ¿ f where: G = gravitational constant R = radius of the star c = velocity of light M = mass of the star. Thus, the red-shift of a known spectral line measured a long way from the star can be used to measure the ratio M/R . Knowledge of R will allow the mass of the star to be determined. (b) (12 marks) An unmanned spacecraft is launched in an experiment to measure both the mass M and radius R of a star in our galaxy. Photons are emitted from He + ions on the surface of the star. These photons can be monitored through resonant absorption by He + ions contained in a test chamber in the spacecraft. Resonant absorption accors only if the He + ions are given a velocity towards the star to allow exactly for the red shifts. As the spacecraft approaches the star radially, the velocity relative to the star ( v = βc ) of the He + ions in the test chamber at absorption resonance is measured as a function of the distance d from the (nearest) surface of the star. The experimental data are displayed in the accompanying table. ²ully utilize the data to determine graphically the mass M and radius R of the star. There is no need to estimate the uncertainties in your answer. Data for Resonance Condition Velocity parameter β = v/c ( × 10 - 5 ) 3.352 3.279 3.195 3.077 2.955 Distance from surface of star d ( × 10 8 m) 38.90 19.98 13.32 8.99 6.67 (c) (5 marks) In order to determine R and M in such an experiment, it is usual to consider the frequency correction due to the recoil of the emitting atom. [Thermal motion causes emission lines to be broadened without displacing emission maxima, and we may therefore assume that all thermal eFects have been taken into account.] (i) (4 marks) Assume that the atom decays at rest, producing a photon and a recoiling atom. Obtain the relativistic expression for the energy hf of a photon emitted in terms of Δ E (the diFerence in rest energy between the two atomic levels) and the initial rest mass m 0 of the atom. (ii) (1 mark)
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This note was uploaded on 11/08/2011 for the course PHYS 0000 taught by Professor Na during the Spring '11 term at Rensselaer Polytechnic Institute.

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Theory - Theoretical Question 1 Gravitational Red Shift and...

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