TheorySolutions - Solutions to Theoretical Question 1...

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Unformatted text preview: Solutions to Theoretical Question 1 Gravitational Red Shift and the Measurement of Stellar Mass (a) If a photon has an effective inertial mass m determined by its energy then mc 2 = hf or m = hf c 2 . Now, assume that gravitational mass = inertial mass, and consider a photon of energy hf (mass m = hf c 2 ) emitted upwards at a distance r from the centre of the star. It will lose energy on escape from the gravitational field of the star. Apply the principle of conservation of energy: Change in photon energy ( hf i- hf f ) = change in gravitational energy, where subscript i initial state and subscript f final state. hf i- hf f =- GMm f - - GMm i r hf f = hf i- GMm i r hf f = hf i- GM hf i c 2 r hf f = hf i 1- GM rc 2 f f f i = 1- GM rc 2 f f = f f- f i f i =- GM rc 2 The negative sign shows red-shift, i.e. a decrease in f , and an increase in wavelength. Thus, for a photon emitted from the surface of a star of radius R , we have f f = GM Rc 2 Since the change in photon energy is small, ( f f ), m f m i = hf i c 2 . (b) The change in photon energy in ascending from r i to r f is given by hf i- hf f =- GMm f r f + GMm i r i GMhf i c 2 1 r i- 1 r f f f f i = 1- GM c 2 1 r i- 1 r f In the experiment, R is the radius of the star, d is the distance from the surface of the star to the spacecraft and the above equation becomes: f f f i = 1- GM c 2 1 R- 1 R + d (1) The frequency of the photon must be doppler shifted back from f f to f i in order to cause resonance excitation of the He + ions in the spacecraft. Thus apply the relativistic Doppler principle to obtain: f f f = s 1 + 1- where f is the frequency as received by He + ions in the spacecraft, and = v/c . That is, the gravitationally reduced frequency f f has been increased to f because of the velocity of the ions on the spacecraft towards the star. Since 1, f f f = (1- ) 1 2 (1 + )- 1 2 1- Alternatively, since 1, use the classical Doppler effect directly. Thus f = f f 1- or f f f = 1- Since f must be equal to f i for resonance absorption, we have f f f i = 1- (2) Substitution of 2 into 1 gives = GM c 2 1 R- 1 R + d (3) Given the experimental data, we look for an effective graphical solution. That is, we require a linear equation linking the experimental data in and d . Rewrite equation 3: = GM c 2 R \ + d- R \ ( R + d ) R Inverting the equation gives: 1 = Rc 2 GM R d + 1 or 1 = R 2 c 2 GM 1 d + Rc 2 GM Graph of 1 vs. 1 d 1 1 intercept = = Rc GM GM R c slope = = R 2 2 2 d The slope is Rc 2 GM R = R (A) The 1 -intercept is Rc 2 GM = (B) and the 1 d-intercept is- 1 R (C) R and M can be conveniently determined from (A) and (B). Equation (C) is redundant. However, it may be used as an (inaccurate) check if needed....
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TheorySolutions - Solutions to Theoretical Question 1...

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